% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
%                                                                         %
% Project Gutenberg's Conic Sections Treated Geometrically, by W.H. Besant%
%                                                                         %
% This eBook is for the use of anyone anywhere at no cost and with        %
% almost no restrictions whatsoever.  You may copy it, give it away or    %
% re-use it under the terms of the Project Gutenberg License included     %
% with this eBook or online at www.gutenberg.org                          %
%                                                                         %
%                                                                         %
% Title: Conic Sections Treated Geometrically                             %
%        and, George Bell and Sons Educational Catalogue                  %
%                                                                         %
% Author: W.H. Besant                                                     %
%                                                                         %
% Release Date: September 6, 2009 [EBook #29913]                          %
%                                                                         %
% Language: English                                                       %
%                                                                         %
% Character set encoding: ISO-8859-1                                      %
%                                                                         %
% *** START OF THIS PROJECT GUTENBERG EBOOK CONIC SECTIONS ***            %
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\begin{verbatim}
Project Gutenberg's Conic Sections Treated Geometrically, by W.H. Besant

This eBook is for the use of anyone anywhere at no cost and with
almost no restrictions whatsoever.  You may copy it, give it away or
re-use it under the terms of the Project Gutenberg License included
with this eBook or online at www.gutenberg.org


Title: Conic Sections Treated Geometrically
       and, George Bell and Sons Educational Catalogue

Author: W.H. Besant

Release Date: September 6, 2009 [EBook #29913]

Language: English

Character set encoding: ISO-8859-1

*** START OF THIS PROJECT GUTENBERG EBOOK CONIC SECTIONS ***
\end{verbatim}

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Produced by K.F. Greiner, Joshua Hutchinson, Nigel Blower
and the Online Distributed Proofreading Team at
http://www.pgdp.net (This file was produced from images
generously made available by Cornell University Digital
Collections)
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%-----------------------------------%
%     Cornell University Library    %
%                                   %
%            THE GIFT OF            %
%        Professor L. A. Wait       %
% to the Evans Mathematical Library %
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% M.408     MATHEMATICS     28/V/13 %
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\begin{frontcatlist}
\fitemspace[ARITHMETIC\@.] With 8000 Examples. By Charles Pendlebury, M.A.,
F.R.A.S., Senior Mathematical Master of St.\ Paul's, late Scholar of
St.\ John's College, Cambridge.

\textsc{Complete.} With or without Answers. 7\textit{th edition}. 4\s.~6\d.

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\fitem[EXAMPLES IN ARITHMETIC,] extracted from the above, 5\textit{th.\ edition},
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\fitemspace[CHOICE AND CHANCE\@.] An Elementary Treatise on Permutations,
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\fitemspace[EUCLID\@.] Books I.--VI. and part of Book XI\@. Newly translated from the
original Text, with numerous Riders and Miscellaneous Examples in Modern
Geometry. By Horace Deighton, M.A., formerly Scholar of Queen's College,
Cambridge; Head Master of Harrison College, Barbados. 3\textit{rd
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\fitem[EXERCISES ON EUCLID and in Modern Geometry,] containing
Applications of the Principles and Processes of Modern Pure Geometry.
By J.~McDowell, M.A., F.R.A.S., Pembroke College, Cambridge, and
Trinity College, Dublin. 3\textit{rd edition, revised}. 6\s.

\fitemspace[ELEMENTARY TRIGONOMETRY\@.] By J.~M. Dyer, M.A., and the
Rev.\ R.~H. Whitcombe, M.A., Assistant Mathematical Masters, Eton College.
2\textit{nd edition, revised}. 4\s.~6\d.

\fitemspace[INTRODUCTION TO PLANE TRIGONOMETRY\@.] By the Rev.\ T.~G.
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Mathematical Master of Charterhouse. 3\textit{rd edition, revised and corrected}.
3\s.~6\d.

\fitemspace[ANALYTICAL GEOMETRY FOR BEGINNERS\@.] Part 1\@. The
Straight Line and Circle. By the Rev.\ T.~G. Vyvyan, M.A\@. 2\s.~6\d.

\fitemspace[CONIC SECTIONS, An Elementary Treatise on Geometrical.] By H.~G.
Willis, M.A., Clare College, Cambridge, Assistant Master of Manchester
Grammar School. 5\s.

\fitemspace[CONICS, The Elementary Geometry of.] By C.~Taylor, D.D., Master of
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of the Hyperbola}. 4\s.~6\d.

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\begin{center}
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\frontmatter

\thispagestyle{empty}
\pdfbookmark[0]{Conic Sections}{Conic Sections}

\vspace*{\stretch{1}}

\begin{center}
{\LARGE\textit{CAMBRIDGE MATHEMATICAL SERIES}\par}

\bigskip

\bigskip

\Huge\TitleStretch CONIC SECTIONS\\[50mm]

\end{center}

\vspace{\stretch{1}}

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\thispagestyle{empty}

\vspace*{\stretch{1}}

\begin{center}

GEORGE BELL \& SONS

\bigskip

\footnotesize
LONDON: YORK STREET, COVENT GARDEN

AND NEW YORK, 66, FIFTH AVENUE

CAMBRIDGE: DEIGHTON, BELL \& CO.\\[50mm]

\end{center}

\vspace{\stretch{1}}

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\begin{center}

{\TitleHuge\TitleStretch CONIC SECTIONS}

\vspace{1cm}

{\TitleStretch TREATED GEOMETRICALLY}

\vspace{\stretch{1}}

BY

\vspace{1cm}

{\Large W.~H. BESANT Sc.D. F.R.S.}

{\tiny FELLOW OF ST JOHN'S COLLEGE CAMBRIDGE}

\vspace{\stretch{1}}

\textit{NINTH EDITION REVISED AND ENLARGED}

\vspace{\stretch{1}}

LONDON

GEORGE BELL AND SONS

1895

\end{center}

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\thispagestyle{empty}

\vspace*{\stretch{1}}

\begin{center}

$\mathfrak{Cambridge}$:

\medskip

{\footnotesize PRINTED BY J. \& C.~F. CLAY,}

\medskip

{\footnotesize AT THE UNIVERSITY PRESS.}\\[50mm]

\end{center}

\vspace{\stretch{1}}

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\pagestyle{fancy}
\fancyhead{}
\renewcommand{\headrulewidth}{0pt}
\fancyhead[R]{\thepage}

\thispagestyle{empty}

\csection{PREFACE TO THE FIRST EDITION}
\cheader{PREFACE TO THE FIRST EDITION}
\pdfbookmark[1]{Prefaces}{Prefaces}

In the present Treatise the Conic Sections are defined
with reference to a focus and directrix, and I have endeavoured
to place before the student the most important
properties of those curves, deduced, as closely as possible,
from the definition.

The construction which is given in the first Chapter
for the determination of points in a conic section possesses
several advantages; in particular, it leads at once to the
constancy of the ratio of the square on the ordinate to
the rectangle under its distances from the vertices; and,
again, in the case of the hyperbola, the directions of the
asymptotes follow immediately from the construction. In
several cases the methods employed are the same as those
of Wallace, in the Treatise on Conic Sections, published
in the \textit{Encyclopaedia Metropolitana}.

The deduction of the properties of these curves from
their definition as the sections of a cone, seems \textit{à priori} to
be the natural method of dealing with the subject, but
experience appears to have shewn that the discussion of
conics as defined by their plane properties is the most
suitable method of commencing an elementary treatise, and
%-----File: 014.png-----
accordingly I follow the fashion of the time in taking that
order for the treatment of the subject. In Hamilton's book
on \textit{Conic Sections}, published in the middle of the last
century, the properties of the cone are first considered, and
the advantage of this method of commencing the subject,
if the use of solid figures be not objected to, is especially
shewn in the very general theorem of Art.~(\cartref{156}). I have
made much use of this treatise, and, in fact, it contains
most of the theorems and problems which are now regarded
as classical propositions in the theory of Conic
Sections.

I have considered first, in Chapter~\cchapref{I.}, a few simple
properties of conics, and have then proceeded to the particular
properties of each curve, commencing with the parabola
as, in some respects, the simplest form of a conic
section.

It is then shewn, in Chapter~\cchapref{VI.}, that the sections of
a cone by a plane produce the several curves in question,
and lead at once to their definition as loci, and to several
of their most important properties.

A chapter is devoted to the method of orthogonal projection,
and another to the harmonic properties of curves,
and to the relations of poles and polars, including the
theory of reciprocal polars for the particular case in which
the circle is employed as the auxiliary curve.

For the more general methods of projections, of reciprocation,
and of anharmonic properties, the student will
consult the treatises of Chasles, Poncelet, Salmon, Townsend,
Ferrers, Whitworth, and others, who have recently developed,
%-----File: 015.png-----
with so much fulness, the methods of modern Geometry.

I have to express my thanks to Mr R.~B. Worthington,
of St John's College, and of the Indian Civil Service, for
valuable assistance in the constructions of Chapter~\cchapref{XI.}, and
also to Mr E.~Hill, Fellow of St John's College, for his
kindness in looking over the latter half of the proof-sheets.

I venture to hope that the methods adopted in this
treatise will give a clear view of the properties of Conic
Sections, and that the numerous Examples appended to
the various Chapters will be useful as an exercise to the
student for the further extension of his conceptions of these
curves.

\bigskip

\begin{flushright}
W.~H. BESANT.\mbox{\qquad}
\end{flushright}

\bigskip

{\scriptsize
\begin{tabular}{@{}r@{}l}
\textsc{Cambridge}&,\\[.5ex]
\textit{March}&, 1869.
\end{tabular}
}
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\clearpage
\thispagestyle{empty}

\csection{PREFACE TO THE NINTH EDITION}
\cheader{PREFACE TO THE NINTH EDITION}

In the preparation of this edition I have made many
alterations and many additions. In particular, I have
placed the articles on Reciprocal Polars in a separate
chapter, with considerable expansions. I have also inserted
a new chapter, on Conical Projections, dealing however
only with real projections.

The first nine chapters, with the first set of miscellaneous
problems, now constitute the elementary portions
of the subject. The subsequent chapters may be regarded
as belonging to higher regions of thought.

I venture to hope that this re-arrangement will make
it easier for the beginner to master the elements of the
subject, and to obtain clear views of the methods of
geometry as applied to the conic sections.

A new edition, the fourth, of the book of solutions of
the examples and problems has been prepared, and is being
issued with this new edition of the treatise, with which it
is in exact accordance.

\bigskip

\begin{flushright}
W.~H. BESANT.\mbox{\qquad}
\end{flushright}

\bigskip
{\scriptsize \textit{December} 14, 1894.}
\clearpage
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\cheader{CONTENTS}
\pdfbookmark[1]{Contents}{Contents}
{\stretchyspace
\tableofcontents
}

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\mainmatter
\thispagestyle{empty}

\vspace*{40pt}
\begin{center}
\Huge \textbf{CONIC SECTIONS.}
\end{center}
\vspace{40pt}

\addtocontents{toc}{%
\protect\begin{flushright}
\vspace{48pt}% to ensure contents page break occurs at a suitable point
\textsc{page}
\protect\end{flushright}
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\csectiontoc{Introduction}
\cheader{INTRODUCTION}

\csubsection{DEFINITION}

If a straight line and a point be given in position in a
plane, and if a point move in a plane in such a manner that
its distance from the given point always bears the same
ratio to its distance from the given line, the curve traced out
by the moving point is called a Conic Section.

The fixed point is called the Focus, and the fixed line
the Directrix of the conic section.

When the ratio is one of equality, the curve is called a
Parabola.

When the ratio is one of less inequality, the curve is
called an Ellipse.

When the ratio is one of greater inequality, the curve is
called an Hyperbola.

These curves are called Conic Sections, because they
can all be obtained from the intersections of a Cone by
planes in different directions, a fact which will be proved
hereafter.

It may be mentioned that a circle is a particular case of
an ellipse, that two straight lines constitute a particular
case of an hyperbola, and that a parabola may be looked
upon as the limiting form of an ellipse or an hyperbola,
under certain conditions of variation in the lines and
magnitudes upon which those curves depend for their form.
%-----File: 020.png-----

The object of the following pages is to discuss the general
forms and characters of these curves, and to determine their
most important properties by help of the methods and
relations developed in the first six books, and in the eleventh
book of Euclid, and it will be found that, for this purpose, a
knowledge of Euclid's Geometry is all that is necessary.

The series of demonstrations will shew the characters and
properties which the curves possess in common, and also the
special characteristics wherein they differ from each other;
and the continuity with which the curves pass into each
other will appear from the definition of a conic section as a
Locus, or curve traced out by a moving point, as well as from
the fact that they are deducible from the intersections of a
cone by a succession of planes.
%-----File: 021.png-----

\cchapter{CHAPTER I}

\csection{PROPOSITION I}
\ctoc{\tocext}{The Construction of a Conic Section, and General Properties}
\cheader{CONIC SECTIONS}

\cisubsection{The Construction of a Conic Section}

\carticle{1.} Take $S$ as the focus, and from $S$ draw $SX$ at right
angles to the directrix, and intersecting it in the point $X$.

\subparagraph{Definition.} \textit{This line $SX$, produced both ways, is called
the Axis of the Conic Section.}

In $SX$ take a point $A$ such that the ratio of $SA$ to $AX$
is equal to the given ratio; then $A$ is a point in the curve.

\includegraphicsmid{001a}

\subparagraph{Def.} \textit{The point $A$ is called the Vertex of the curve.}

In the directrix $EX$ take any point $E$, join $EA$, and $ES$,
produce these lines, and through $S$ draw the straight line
$SQ$ making with $ES$ produced the same angle which $ES$
produced makes with the axis $SN$.
%-----File: 022.png-----

Let $P$ be the point of intersection of $SQ$ and $EA$ produced,
and through $P$ draw $LPK$ parallel to $NX$, and intersecting
$ES$ produced in $L$, and the directrix in $K$.

\includegraphicsmid{001a} %repeated figure

Then the angle $PLS$ is equal to the angle $LSN$ and
therefore to $PSL$;
\begin{flalign*}
&\indent\text{Hence}
 &                  SP &=  PL.      &&\phantom{\indent\text{Hence}}\\
&\indent\text{Also}
 &             PL : AS &:: EP : EA  &&\\
&&                     &:: PK : AX\,; &&\\
&& \therefore  PL : PK &:: AS : AX\,; &&\\
&\text{and}
 & \therefore  SP : PK &:: AS : AX. &&
\end{flalign*}

The point $P$ is therefore a point in the curve required,
and by taking for $E$ successive positions along the directrix
we shall, by this construction, obtain a succession of points
in the curve.

If $E$ be taken on the upper side of the axis at the same
distance from $X$, it is easy to see that a point $P$ will be
obtained below the axis, which will be similarly situated
with regard to the focus and directrix. Hence it follows
that the axis divides the curve into two similar and equal
portions.
%-----File: 023.png-----

\includegraphicsright[12]{001c}

Another point of the curve, lying in the straight line
$KP$, can be found in the following manner.

Through $S$ draw the
straight line $FS$ making the
angle $FSK$ equal to $KSP$,
and let $FS$ produced meet
$KP$ produced in $P'$.

Then, since $KS$ bisects
the angle $PSF$,
\begin{align*}
           SP' : SP  &:: P'K : PK; \\
\therefore SP' : P'K &:: SP  : PK,
\end{align*}
and $P'$ is a point in the curve.

\carticleplus{2.}{Def.} \textit{The Eccentricity. The constant ratio of the
distance from the focus of any point in a conic section to
its distance from the directrix is called the eccentricity of
the conic section.}

\includegraphicsright[12]{002a}

\textit{The Latus Rectum.} If $E$ be so taken that $EX$ is equal
to $SX$, the angle $PSN$,
which is double the angle
$LSN$, and therefore double
the angle $ESX$, is a right
angle.

For, since $EX = SX$, the
angle $ESX = SEX$, and, the
angle $SXE$ being a right
angle, the sum of the two
angles $SEX$, $ESX$, which is
equal to twice $ESX$, is also
equal to a right angle.

Calling $R$ the position of $P$ in this case, produce $RS$ to
$R'$, so that $R'S=RS$; then $R'$ is also a point in the curve.

\subparagraph{Def.} \textit{The straight line $RSR'$ drawn through the focus
at right angles to the axis, and intersecting the curve in $R$,
and $R'$, is called the Latus Rectum.}

It is hence evident that the form of a conic section is
determined by its eccentricity, and that its magnitude is
%-----File: 024.png-----
determined by the magnitude of the latus rectum, which
is given by the relation
\[
SR:SX::SA:AX.
\]

\carticleplus{3.}{Def.} \textit{The straight line $PN$ \textrm{(Fig.\ Art.~\cartref{1})}, drawn
from any point $P$ of the curve at right angles to the axis,
and intersecting the axis in $N$, is called the Ordinate of
the point $P$.}

If the line $PN$ be produced to $P'$ so that $NP'=NP$,
the line $PNP'$ is a \textit{double ordinate} of the curve.

The latus rectum is therefore the double ordinate passing
through the focus.

\subparagraph{Def.} \textit{The distance $AN$ of the foot of the ordinate from
the vertex is called the Abscissa of the point $P$.}

\subparagraph{Def.} \textit{The distance $SP$ is called the focal distance of
the point $P$.}

It is also described as the radius vector drawn from the
focus.

\carticle{4.} We have now given a general method of constructing
a conic section, and we have explained the nomenclature
which is usually employed. We proceed to demonstrate a
few of the properties which are common to all the conic
sections.

For the future the word conic will be employed as an
abbreviation for conic section.

\cheader{CONICS}

\subparagraph{Prop.~II\@.} \textit{If the straight line joining two points $P$, $P'$
of a conic meet the directrix in $F$, the straight line $FS$ will
bisect the angle between $PS$ and $P'S$ produced.}

\includegraphicsmid{004a}
%-----File: 025.png-----

Draw the perpendiculars $PK$, $P'K'$ on the directrix.
\begin{flalign*}
&\indent\text{Then} & SP : SP' &:: PK : P'K'
&&\phantom{\indent\text{Then}}
\\
&&                      &:: PF : P'F. &&
\end{flalign*}

Therefore $FS$ bisects the outer angle, at $S$, of the triangle
$PSP'$. (Euclid~\textsc{vi.}, A.)

\subparagraph{Cor.} \textit{If $SQ$ bisect the angle $PSP'$, it follows that $FSQ$
is a right angle.}

\cpagmedskip

\carticleplus{5.}{Prop.~III\@.} \textit{No straight line can meet a conic in more
than two points.}

Employing the figure of Art.~\cartref{4}, let $P$ be a point of the
curve, and draw any straight line $FP$.

Join $SF$, draw $SQ$ at right angles to $SF$, and $SP'$ making
the angle $QSP'$ equal to $QSP$; then $P'$ is a point of the curve.

For, since $SF$ bisects the outer angle at $S$,
\begin{flalign*}
&&            SP' : SP   &:: P'F  : PF, &&\\
&&                       &:: P'K' : PK  &&\\
&\text{or} &  SP' : P'K' &:: SP   : PK, &&\phantom{or}
\end{flalign*}
and therefore, $P'$ is a point of the curve, also, there is no
other point of the curve in the straight line $FPP'$.

For suppose if possible $P''$ to be another point; then, as
in Article (\cartref{4}), $SQ$ bisects the angle $PSP''$; but $SQ$ bisects
the angle $PSP'$; therefore $P''$ and $P'$ are coincident.

\includegraphicsright[14]{006a}

\cpagdummypar

\carticleplus{6.}{Prop.~IV\@.} \textit{If $QSQ'$ be a focal chord of a conic, and
$P$ any point of the conic, and if
$QP$, $Q'P$ meet the directrix in $E$
and $F$, the angle $ESF$ is a right
angle.}

For, by Prop.~II., $SE$ bisects
the angle $PSQ'$, and $SF$ bisects
the angle $PSQ$;

hence it follows that $ESF$
is a right angle.

This theorem will be subsequently
utilised in the case in
which the focal chord $Q'SQ$ is
coincident with the axis of the
conic.
%-----File: 026.png-----

\includegraphicsright[22]{007a}

\carticleplus{7.}{Prop.~V\@.} \textit{The straight lines joining the extremities
of two focal chords intersect in the directrix.}

\cheader{FOCAL CHORDS}

If $PSp$, $P'Sp'$ be the
two chords, the point in
which $PP'$ meets the
directrix is obtained by
bisecting the angle
$PSP'$ and drawing $SF$
at right angles to the
bisecting line $SQ$. But
this line also bisects the
angle $pSp'$; therefore
$pp'$ also passes through
$F$.

The line $SF$ bisects
the angle $PSp'$, and
similarly, if $QS$ produced,
bisecting the
angle $pSp'$, meet the
directrix in $F'$, the two
lines $Pp'$, $P'p$ will meet
in $F'$. It is obvious that
the angle $FSF'$ is a right
angle.

\carticleplus{8.}{Prop.~VI\@.} \textit{The semi-latus rectum is the harmonic
mean between the two segments of any focal chord of a conic.}

\includegraphicsright[8]{008a}

Let $PSP'$ be a focal chord,
and draw the ordinates $PN$,
$P'N'$.

Then, the triangles $SPN$,
$SP'N'$ being similar,
\begin{align*}
SP : SP' &:: SN : SN' \\
&:: NX - SX : SX - N'X \\
&:: SP - SR : SR - SP',
\end{align*}
since $SP$, $SR$, $SP'$ are proportional to $NX$, $SX$, and $N'X$.
%-----File: 027.png-----

\subparagraph{Cor.} Since $SP : SP - SR :: SP \ctrdot SP' : SP \ctrdot SP' - SR \ctrdot SP'$,
and $SP' : SR - SP' :: SP \ctrdot SP' : SR \ctrdot SP - SP \ctrdot SP'$,
it follows that
\begin{flalign*}
&&   SP \ctrdot SP'- SR \ctrdot SP'
  &= SR \ctrdot SP - SP \ctrdot SP';  &&\\
&\therefore & SR \ctrdot PP' &= 2SP \ctrdot SP'. &\phantom{\therefore}&
\end{flalign*}

Hence, if $PSP'$, $QSQ'$ are two focal chords,
\[
  PP' : QQ' :: SP \ctrdot SP' : SQ \ctrdot SQ'.
\]

\carticleplus{9.}{Prop.~VII\@.} \textit{A focal chord is divided harmonically at
the focus and the point where it meets the directrix.}

Let $PSP'$ produced meet the directrix in $F$, and draw
$PK$, $P'K'$ perpendicular to the directrix, fig.\ Art.~\cartref{8}.
\begin{flalign*}
\indent\text{Then }      PF : P'F &:: PK : P'K'      &&\\
                             &:: SP : SP'            &&\\
                             &::PF - SF : SF - P'F;  &&
\end{flalign*}
that is, $PF$, $SF$, and $P'F$ are in harmonic progression, and
the line $PP'$ is divided harmonically at $S$ and $F$.

\carticle{10.} \textit{Definition of the Tangent to a curve.}

\cheader{TANGENTS}

\textit{If a straight line, drawn through a point $P$ of a curve,
meet the curve again in $P'$, and if the straight line be turned
round the point $P$ until the point $P'$ approaches indefinitely
near to $P$, the ultimate position of the straight line is the
tangent to the curve at $P$.}

\includegraphicsmid{010a}
%-----File: 028.png-----

Thus, if the straight line $APP'$ turn round $P$ until the
points $P$ and $P'$ coincide, the line in its ultimate position
$PT$ is the tangent at $P$.

\subparagraph{Def.} \textit{The normal at any point of a curve is the straight
line drawn through the point at right angles to the tangent at
that point.}

Thus, in the figure, $PG$ is the normal at $P$.

\subparagraph{Prop.~VIII\@.} \textit{The straight line, drawn from the focus to
the point in which the tangent meets the directrix, is at right
angles to the straight line drawn from the focus to the point of
contact.}

\includegraphicsmid{010b}

It is proved in Art.~(\cartref{4}) that, if $FPP'$ is a chord, and if
$SQ$ bisects the angle $PSP'$, $FSQ$ is a right angle.

Let the point $P'$ move along the curve towards $P$; then, as
$P'$ approaches to coincidence with $P$, the straight line $FPP'$
approximates to, and ultimately becomes, the tangent $TP$
at $P$.

But when $P'$ coincides with $P$, the line $SQ$ coincides with
$SP$, and the angle $FSP$, which is ultimately $TSP$, becomes a
right angle.

\includegraphicsright[13]{011a}

Or, in other words, the portion of the tangent, intercepted
between the point of contact and the directrix, subtends a
right angle at the focus.
%-----File: 029.png-----

\carticleplus{11.}{Prop.~IX\@.} \textit{The tangent at the vertex is perpendicular
to the axis.}

If a chord $EAP$ be drawn through
the vertex, and the point $P$ be near
the vertex, the angle $PSA$ is small,
and $LSN$, which is half the angle
$PSN$, is nearly a right angle.

Hence it follows that when $P$
approaches to coincidence with $A$, the
point $E$ moves off to an infinite
distance and the line $EAP$, which is
ultimately the tangent at $A$, becomes
parallel to $LSE$, and is therefore perpendicular
to $AX$.

\carticleplus{12.}{Prop.~X\@.} \textit{The tangents at the ends of a focal chord
intersect on the directrix.}

For the line $SF$, perpendicular to $SP$, meets the directrix
in the same point as the tangent at $P$; and, since $SF$ is
also at right angles to $SP'$, the tangent at $P'$ meets the
directrix in the same point $F$.

\includegraphicsmid{012a}

Conversely, if from any point $F$ in the directrix tangents
be drawn, the chord of contact, that is, the straight line
joining the points of contact, will pass through the focus and
will be at right angles to $SF$.

\subparagraph{Cor.} Hence it follows that the tangents at the ends of
the latus rectum pass through the foot of the directrix.
%-----File: 030.png-----

\carticleplus{13.}{Prop.~XI\@.} \textit{If a chord $P'P$ meet the directrix in $F$,
and if the line bisecting the $PSP'$ meet the curve in $q$ and $q'$,
$Fq$ and $Fq'$ will be the tangents at $q$ and $q'$.}

Taking the figure of Art.~\cartref{7}, the line $SQ$ meets the curve
in $q$ and $q'$, and, since $SF$ is at right angles to $SQ$, it follows,
from Art.~\cartref{12}, that $Fq$ and $Fq'$ are tangents.

Hence if from a point $F$ in the directrix tangents be
drawn, and also any straight line $FPP'$ cutting the curve in
$P$ and $P'$, the chord of contact will bisect the angle $PSP'$.

\carticleplus{14.}{Prop.~XII\@.} \textit{If the tangent at any point $P$ of a conic
intersect the directrix in $F$, and the latus rectum produced
in $D$,}
\[
                      SD : SF :: SA : AX.
\]

\includegraphicsright[12]{014a}

Join $SK$; then, observing that $FSP$ and $FKP$ are right
angles, a circle can be described
about $FSPK$, and therefore the
angles $SFD$, $SKP$ are equal.

Also the angle $FSD$
\begin{align*}
  &= \text{complement of } DSP \\
  &= SPK;
\end{align*}

$\therefore$ the triangles $FSD$, $SPK$ are
similar, and
\begin{align*}
  SD : SF &::  SP : PK \\
          &:: SA : AX.
\end{align*}

\subparagraph{Cor.~(1).} If the tangent at the other end $P'$ of the focal
chord meet the directrix in $D'$,
\begin{align*}
  SD' : SF &:: SA : AX; \\
  \therefore   SD &= SD'.
\end{align*}

\subparagraph{Cor.~(2).} If $DE$ be the perpendicular from $D$ upon $SP$,
the triangles $SDE$, $SFX$ are similar, and
\begin{gather*}
\begin{aligned}
  SE : SX &:: SD : SF  \\
          &:: SA : AX  \\
          &:: SB : SX\,;
\end{aligned}  \\
\therefore SE \text{ is equal to $SR$, the semi-latus rectum.}
\end{gather*}
%-----File: 031.png-----

\carticleplus{15.}{Prop.~XIII\@.} \textit{The tangents drawn from any point to
a conic subtend equal angles at the focus.}

Let the tangents $FTP$, $F'TP'$ at $P$ and $P'$ meet the
directrix in $F$ and $F'$ and the latus rectum in $D$ and $D'$.

Join $ST$ and produce it to meet the directrix in $K$;
\begin{flalign*}
&\text{then}          & KF : SD  &:: KT : ST    &&\\
&&                               &:: KF': SD'.  &&\\
&\indent\text{Hence } & KF : KF' &:: SD : SD' &\indent\phantom{Hence}& \\
&&                               &:: SF : SF' \text{ by Prop.\ XII.} &&\\
\multispan{6}{\ \hfill$\therefore$ the angles $TSF$, $TSF'$ are equal.\hfill}
\end{flalign*}

But the angles $FSP'$, $F'SP$ are equal, for each is the
complement of $FSF'$;
\[
  \therefore \text{ the angles } TSP,\, TSP' \text{ are equal.}
\]
\includegraphicsmid{015a}

\subparagraph{Cor.} Hence it follows that if perpendiculars $TM$, $TM'$
be let fall upon $SP$ and $SP'$, they are equal in length.

For the two triangles $TSM$, $TSM'$ have the angles $TMS$,
$TSM$ respectively equal to the angles $TM'S$, $TSM'$, and the
side $TS$ common; and therefore the other sides are equal,
and
\[
  TM = TM'.
\]

\carticleplus{16.}{Prop.~XIV\@.}  \textit{If from any point $T$ in the tangent at
a point $P$ of a conic, $TM$ be drawn, perpendicular to the focal
distance $SP$, and $TN$ perpendicular to the directrix,}
\[
    SM : TN :: SA : AX.
\]
%-----File: 032.png-----

\includegraphicsright[8]{016a}

For, if $PK$ be perpendicular to the directrix and $SF$ be
joined,
\begin{align*}
             SM : SP & :: TF : FP  \\
                     & :: TN : PK; \\
  \therefore SM : TN & :: SP : PK  \\
                     & :: SA : AX.
\end{align*}

This theorem, which is due
to Professor Adams, may be
employed to prove Prop.~XIII.

For if, in the figure of Art.~(\cartref{15}), $TM$, $TM'$ be the
perpendiculars from $T$ on $SP$ and $SP'$, and if $TN$ be the
perpendicular on the directrix, $SM$ and $SM'$ have each the
same ratio to $TN$, and are therefore equal to one another.

Hence the triangles $TSM$, $TSM'$ are equal in all respects,
and the angle $PSP'$ is bisected by $ST$.

\carticleplus{17.}{Prop.~XV\@.} \textit{To draw tangents from any point to a
conic.}

Let $T$ be the point, and let a circle be described about $S$
as centre, the radius of which bears to $TN$ the ratio of
$SA : AX$; then, if tangents $TM$, $TM'$ be drawn to the circle,
the straight lines $SM$, $SM'$, produced if necessary, will
intersect the conic in the points of contact of the tangents
from $T$.

\carticleplus{18.}{Prop.~XVI\@.} \textit{If $PG$, the normal at $P$, meet the axis
of the conic in $G$,}
\[
SG : SP :: SA : AX.
\]

\includegraphicsmid{018a}

Let the tangent at $P$ meet the directrix in $F$, and the
latus rectum produced in $D$.
%-----File: 033.png-----

\cheader{NORMALS}

Then the angle $SPG =$ the complement of $SPF=PFS$,
and $PSG =$ the complement of $FSX = FSD$;

$\therefore$ the triangles $SFD$, $SPG$ are similar, and
\[
SG: SP:: SD: SF:: SA: AX\text{, by Prop.~XII.}
\]

\includegraphicsright[11]{019a}

\carticleplus{19.}{Prop.~XVII\@.} \textit{If from $G$, the point in which the
normal at $P$ meets the axis, $GL$ be drawn perpendicular
to $SP$, the length $PL$ is equal to the semi-latus rectum.}

Let the tangent at $P$
meet the directrix in $F$, and
join $SF$.

Then $PLG$, $PSF$ are
similar triangles;
\[
  \therefore PL : LG :: SF : SP.
\]

Also $SLG$ and $SFX$ are
similar triangles;
\[
 \therefore  LG : SX :: SG : SF.
\]
\begin{flalign*}
&\text{\indent Hence}&  PL : SX &:: SG : SP  &&\phantom{\indent Hence} \\
&&                              &:: SA : AX,\text{ Art.~(\cartref{18})},  &&\\
&\text{but}&            SR : SX &:: SA : AX,\text{ Art.~(\cartref{2})};  &&\\
&&               \therefore  PL &= SR.  &&
\end{flalign*}

\carticleplus{20.}{Prop.~XVIII\@.} \textit{If from any point $F$ in the directrix
tangents be drawn, and also any straight line $FPP'$ cutting
the curve in $P$ and $P'$, the chord $PP'$ is divided harmonically
at $F$ and its point of intersection with the chord of contact.}

\includegraphicsmid{020a}
%-----File: 034.png-----

For, if $QSQ'$ be the chord of contact, it bisects the angle
$PSP'$, (Prop.\ XI.), %allow linebreak to help LaTeX out
and $\therefore$, if $V$ be the point of intersection of $SQ$ and $PP'$,
\begin{align*}
FP' : FP & :: SP' : SP \\
         & :: P'V : PV \\
         & ::FP'-FV:FV-FP.
\end{align*}

\cheader{TANGENTS}

Hence $FV$ is the harmonic mean between $FP$ and $FP'$.

The theorems of this article and of Art.~\cartref{9} are particular
cases of more general theorems, which will appear hereafter.

\carticleplus{21.}{Prop.~XIX\@.} \textit{If a tangent be drawn parallel to a
chord of a conic, the portion of this tangent which is intercepted
by the tangents at the ends of the chord is bisected at
the point of contact.}

Let $PP'$ be the chord, $TP$, $TP'$ the tangents, and $EQE'$
the tangent parallel to $PP'$.

From the focus $S$ draw $SP$, $SP'$ and $SQ$, and draw $TM$,
$TM'$ perpendicular respectively to $SP$, $SP'$.

Also draw from $E$ perpendiculars $EN$, $EL$, upon $SP$,
$SQ$, and from $E'$ perpendiculars $E'N'$, $E'L'$ upon $SP'$ and
$SQ$.
%-----File: 035.png-----

\includegraphicsmid{021a}

Then, since $EE'$ is parallel to $PP'$
\begin{flalign*}
&&                       TP : EP   & :: TP' : E'P',  &&\\
&\text{but }&            TP : EP   & :: TM  : EN,    &&\phantom{and}\\
&\text{and }&            TP': E'P' & :: TM' : E'N';  &&\\
&&           \therefore  TM : EN   & :: TM' : E'N';  &&
\end{flalign*}
but $TM = TM'$, Cor.~Prop.~\textsc{xiii.};
\[
\therefore EN=E'N'.
\]
Again, by the same corollary,
\begin{gather*}
EN = EL \text{ and } E'N' = E'L'; \\
\therefore EL = E'L',
\end{gather*}
and, the triangles $ELQ$, $E'L'Q$ being similar,
\[
EQ = E'Q.
\]

\subparagraph{Cor.} If $TQ$ be produced to meet $PP'$ in $V$,
\begin{flalign*}
&&                PV : EQ  &:: TV: TQ, &&\\
&\text{and}&     P'V: E'Q &:: TV: TQ;  &&\phantom{and}\\
&&           \therefore PV &= P'V,  &&
\end{flalign*}
that is, $PP'$ is bisected in $V$.

\smallskip
\textit{Hence, if tangents be drawn at the ends of any chord
of a conic, the point of intersection of these tangents, the
middle point of the chord, and the point of contact of the
tangent parallel to the chord, all lie in one straight line.}

\csubsection{EXAMPLES}

\begin{exampleslist}

\item[1.] Describe the relative positions of the focus and directrix, first,
when the conic is a circle, and secondly, when it consists of two straight
lines.
\cheader{EXAMPLES}

\item[2.] Having given two points of a conic, the directrix, and the
eccentricity, determine the conic.

\item[3.] Having given a focus, the corresponding directrix, and a tangent,
construct the conic.
%-----File: 036.png-----

\item[4.] If a circle passes through a fixed point and cuts a given straight
line at a constant angle the locus of its centre is a conic.

\item[5.] If $PG$, $pg$, the normals at the ends of a focal chord, intersect in
$O$, the straight line through $O$ parallel to $Pp$ bisects $Gg$.

\item[6.] Find the locus of the foci of all the conics of given eccentricity
which pass through a fixed point $P$, and have the normal $PG$ given in
magnitude and position.

\item[7.] Having given a point $P$ of a conic, the tangent at $P$, and the
directrix, find the locus of the focus.

\item[8.] If $PSQ$ be a focal chord, and $X$ the foot of the directrix, $XP$
and $XQ$ are equally inclined to the axis.

\item[9.] If $PK$ be the perpendicular from a point $P$ of a conic on the
directrix, and $SK$ meet the tangent at the vertex in $E$, the angles $SPE$,
$KPE$ are equal.

\item[10.] If the tangent at $P$ meet the directrix in $F$ and the axis in $T$,
the angles $KSF$, $FTS$ are equal.

\item[11.] $PSP'$ is a focal chord, $PN$, $P'N'$ are the ordinates, and $PK$,
$P'K'$ perpendiculars on the directrix; if $KN$, $K'N'$ meet in $L$, the
triangle $LNN'$ is isosceles.

\item[12.] The focal distance of a point on a conic is equal to the length
of the ordinate produced to meet the tangent at the end of the latus
rectum.

\item[13.] The normal at any point bears to the semi-latus rectum the
ratio of the focal distance of the point to the distance of the focus from
the tangent.

\item[14.] The chord of a conic is given in length; prove that, if this
length exceed the latus rectum, the distance from the directrix of
the middle point of the chord is least when the chord passes through
the focus.

\item[15.] The portion of any tangent to a conic, intercepted between two
fixed tangents, subtends a constant angle at the focus.

\item[16.] Given two points of a conic, and the directrix, find the locus of
the focus.

\item[17.] From any fixed point in the axis a line is drawn perpendicular
to the tangent at $P$ and meeting $SP$ in $R$; the locus of $R$ is a circle.

\item[18.] If the tangent at the end of the latus rectum meet the tangent
at the vertex in $T$, $AT=AS$.

\item[19.] $TP$, $TQ$ are the tangents at the points $P$, $Q$ of a conic, and $PQ$
meets the directrix in $R$; prove that $RST$ is a right angle.

\item[20.] $SR$ being the semi-latus rectum, if $RA$ meet the directrix in $E$,
and $SE$ meet the tangent at the vertex in $T$,
\[
AT = AS.
\]
%-----File: 037.png-----

\item[21.] If from any point $T$, in the tangent at $P$, $TM$ be drawn
perpendicular to $SP$, and $TN$ perpendicular to the transverse axis,
meeting the curve in $R$, $SM=SR$.

\item[22.] If the chords $PQ$, $P'Q$ meet the directrix in $F$ and $F'$, the angle
$FSF'$ is half $PSP'$.

\item[23.] If $PN$ be the ordinate, $PG$ the normal, and $GL$ the perpendicular
from $G$ upon $SP$,
\[
GL: PN:: SA: AX.
\]

\item[24.] If normals be drawn at the ends of a focal chord, a line
through their intersection parallel to the axis will bisect the chord.

\item[25.] If a conic of given eccentricity is drawn touching the straight
line $FD$ joining two fixed points $F$ and $D$, and if the directrix always
passes through $F$, and the corresponding latus rectum always passes
through $D$, find the locus of the focus.

\item[26.] If $ST$, making a constant angle with $SP$ meet in $T$ the tangent
at $P$, prove that the locus of $T$ is a conic having the same focus and
directrix.

\item[27.] If $E$ be the foot of the perpendicular let fall upon $PSP'$ from
the point of intersection of the normals at $P$ and $P'$,
\[
PE=SP' \text{ and } P'E=SP.
\]

\item[28.] If a circle be described on the latus rectum as diameter, and if
the common tangent to the conic and circle touch the conic in $P$ and
the circle in $Q$, the angle $PSQ$ is bisected by the latus rectum. (Refer
to Cor.~2.\ Art.~\cartref{14}.)

\item[29.] Given two points, the focus, and the eccentricity, determine
the position of the axis.

\item[30.] If a chord $PQ$ subtend a constant angle at the focus, the locus
of the intersection of the tangents at $P$ and $Q$ is a conic with the same
focus and directrix.

\item[31.] The tangent at a point $P$ of a conic intersects the tangent at
the fixed point $P'$ in $Q$, and from $S$ a straight line is drawn perpendicular
to $SQ$ and meeting in $R$ the tangent at $P$; prove that the locus
of $R$ is a straight line.

\item[32.] The circle is drawn with its centre at $S$, and touching the conic
at the vertex $A$; if radii $Sp$, $Sp'$ of the circle meet the conic in $P$, $P'$,
prove that $PP'$, $pp'$ intersect on the tangent at $A$.

\item[33.] $Pp$ is any chord of a conic, $PG$, $pg$ the normals, $G$, $g$ being on
the axis; $GK$, $gk$ are perpendiculars on $Pp$; prove that $PK=pk$.

\end{exampleslist}
%-----File: 038.png-----

\cchapter{CHAPTER II}
\csectiontoc{The Parabola}
\cheader{THE PARABOLA}

\subparagraph{Def.} \textit{A parabola is the curve traced out by a point
which moves in such a manner that its distance from a given
point is always equal to its distance from a given straight
line.}

\cisubsection{Tracing the Curve}

\includegraphicsmid{022a}

\carticle{22.} Let $S$ be the focus, $EX$ the directrix, and $SX$ the
perpendicular on $EX$. Then, bisecting $SX$ in $A$, the point
$A$ is the vertex; and if, from any point $E$ in the directrix,
$EAP$, $ESL$ be drawn, and from $S$ the straight line $SP$
meeting $EA$ produced in $P$, and making the angle $PSL$
equal to $LSN$, we obtain, as in Art.~(\cartref{1}), a point $P$ in the
curve.
%-----File: 039.png-----
\begin{flalign*}
&\indent\text{For} & PL : PK &:: SA : AX, &&\phantom{\indent\text{For}}\\
&\text{and} &              \therefore PL &= PK.       &&\\
&\indent\text{But} & SP = PL, &\text{ and } \therefore SP = PK. &&
\end{flalign*}

Again, drawing $EP'$ parallel to the axis and meeting in
$P'$ the line $PS$ produced, we obtain the other extremity of
the focal chord $PSP'$.
\begin{flalign*}
&\indent\text{For the angle} &  ESP' &= PSL = PLS &&\phantom{\indent\text{For the angle}}\\
&&                                               &= SEP',     &&\\
&\text{and} &                     \therefore SP' &= P'E,
\end{flalign*}
and $P'$ is a point in the parabola.

\includegraphicsmid{022b}

The curve lies wholly on the same side of the directrix;
for, if $P'$ be a point on the other side, and $SN$ be perpendicular
to $P'K$, $SP'$ is greater than $P'N$, and therefore is
greater than $P'K$.

Again, a straight line parallel to the axis meets the curve
in one point only.

For, if possible, let $P''$ be another point of the curve in
$KP$ produced.
\begin{flalign*}
&\indent\text{Then} & SP = PK & \text{ and } SP'' = P''K  &&\phantom{\indent\text{Then}} \\
&&                        \therefore PP'' &= SP'' - SP,                && \\
&\text{or} &                         PP'' &+ SP = SP'', &&
\end{flalign*}
which is impossible.

\carticleplus{23.}{Prop.~I\@.} \textit{The distance from the focus of a point
inside a parabola is less, and of a point outside is greater
than its distance from the directrix.}
%-----File: 040.png-----

If $Q$ be the point inside,
let fall the perpendicular
$QPK$ on the directrix, meeting
the curve in $P$.

\includegraphicsmid{023a}

Then $SP + PQ > SQ$, but
\begin{gather*}
SP + PQ = PK + PQ = QK, \\
\therefore SQ < QK.
\end{gather*}

If $Q'$ be outside, and between $P$ and $K$,
\begin{flalign*}
&&SQ' + PQ' &> SP, &&\\
&&\therefore SQ' &> Q'K.&&
\end{flalign*}

If $Q'$ lie in $PK$ produced,
\begin{flalign*}
&& SQ' + SP &> PQ', &&\\
&\text{and} & \therefore SQ' &> KQ'.  &&\phantom{and}
\end{flalign*}

\carticleplus{24.}{Prop.~II\@.} $\text{\textit{The latus rectum}} = 4 \ctrdot AS$.

For if, Fig.\ Art.~\cartref{23}, $LSL'$ be the latus rectum, drawing
$LK'$ at right angles to the directrix, we have
\begin{gather*}
LS = LK' = SX = 2AS, \\
\therefore LSL' = 4\ctrdot AS.
\end{gather*}

\carticle{25.} \textit{Mechanical construction of the Parabola.}

\includegraphicsright[8]{025a}

Take a rigid bar $EKL$, of
which the portions $EK$, $KL$ are
at right angles to each other,
and fasten a string to the end
$L$, the length of which is $LK$.
Then if the other end of the
string be fastened to $S$, and the
bar be made to slide along a
fixed straight edge, $EKX$, a pencil at $P$, keeping the string
stretched against the bar, will trace out a portion of a
parabola, of which $S$ is the focus, and $EX$ the directrix.
%-----File: 041.png-----

\includegraphicsright[11]{026a}

\carticleplus{26.}{Prop.~III\@.} \textit{If $PK$ is the perpendicular upon the
directrix from a point $P$ of a parabola, and if $PA$ meet the
directrix in $E$, the angle $KSE$ is a right angle.}

Join $ES$, and let $KP$ and
$ES$ produced meet at $L$.

Since $SA = AX$, it follows
that $PL = PK = SP$;

$\therefore P$ is the centre of the
circle through $K$, $S$, and $L$,
and the angle $KSL$ is a
right angle.

Therefore $KSE$ is a right angle.

\carticleplus{27.}{Prop.~IV\@.} \textit{If $PN$ is the ordinate of a point $P$ of a
parabola,}
\[
PN^2 = 4AS \ctrdot AN.
\]

Taking the figure above,
\begin{gather*}
PN: EX:: AN: AX \\
\therefore PN^2 : EX \ctrdot KX :: 4AS \ctrdot AN : 4AS^2.
\end{gather*}

But, since $KSE$ is a right angle,
\begin{align*}
  EX \ctrdot KX & = SX^2 = 4AS^2, \\
  \therefore  PN^2 & = 4AS \ctrdot AN.
\end{align*}

\subparagraph{Cor.} If $AN$ increases, and becomes infinitely large, $PN$
increases and becomes infinitely large, and therefore the two
portions of the curve, above and below the axis, proceed to
infinity.

\carticleplus{28.}{Prop.~V\@.} \textit{If from the ends of a focal chord perpendiculars
be let fall upon the directrix, the intercepted
portion of the directrix subtends a right angle at the focus.}

For, if $PA$ meet the directrix in $E$, and if the straight
line through $E$ perpendicular to the directrix meet $PS$ in $P'$,
it is shewn, in Art.~\cartref{22}, that $P'$ is the other extremity of the
focal chord $PS$; and, as in Art.~\cartref{26}, $KSE$ is a right angle.
%-----File: 042.png-----

\carticleplus{29.}{Prop.~VI\@.} \textit{The tangent at any point $P$ bisects the
angle between the focal distance $SP$ and the perpendicular
$PK$ on the directrix.}

Let $F$ be the point in which
the tangent meets the directrix,
and join $SF$.

\includegraphicsright[16]{029a}

We have shewn, (Art.~\cartref{10}) that
$FSP$ is a right angle, and, since
$SP = PK$, and $PF$ is common to
the right-angled triangles $SPF$,
$KPF$, it follows that these triangles
are equal in all respects, and therefore
the angle
\[
SPF = FPK.
\]

In other words, \textit{the tangent at any point is equally inclined
to the focal distance and the axis.}

\begin{small}
\subparagraph{Cor.} It has been shewn, in Art.~(\cartref{12}), that the tangents at the ends
of a focal chord intersect in the directrix, and therefore, if $PS$ produced
meet the curve in $P'$, $FP'$ is the tangent at $P'$, and bisects the angle
between $SP'$ and the perpendicular from $P'$ on the directrix.

\end{small}

\carticleplus{30.}{Prop.~VII\@.} \textit{The tangents at the ends of a focal
chord intersect at right angles in the directrix.}

\includegraphicsright[11]{030a}

Let $PSP'$ be the chord, and $PF$, $P'F$
the tangents meeting the directrix in $F$.

Let fall the perpendiculars $PK$, $P'K'$,
and join $SK$, $SK'$.

The angle $P'SK' = \tfrac{1}{2} P'SX$\\
\phantom{\indent The angle $P'SK'$ }$= \tfrac{1}{2} SPK = SPF$,\\
$\therefore SK'$ is parallel to $PF$,\\
and, similarly, $SK$ is parallel to $P'F$.

But (Art.~\cartref{28}) $KSK'$ is a right angle;
\[
  \therefore PFP' \text{ is a right angle.}
\]
%-----File: 043.png-----

\carticleplus{31.}{Prop.~VIII\@.} \textit{If the tangent at any point $P$ of a
parabola meet the axis in $T$, and $PN$ be the ordinate of $P$,
then}
\[
AT = AN.
\]

\includegraphicsright[6]{031a}

Draw $PK$ perpendicular to
the directrix.
\begin{flalign*}
\indent\text{The angle } SPT&=TPK &\\
                       &= PTS, &\\
\therefore         ST  &= SP   &\\
                       &= PK   &\\
                       &= NX.  &
\end{flalign*}
\begin{flalign*}
&\indent\text{But}&          ST &= SA+AT, &&\phantom{\indent\text{But}}\\
&\text{and}&               NX&=AN+AX; &&\\
&&\therefore \text{ since } SA&=AX, &&\\
&&                         AT&=AN.
\end{flalign*}

\subparagraph{Def.} \textit{The line $NT$ is called the sub-tangent.}

The sub-tangent is therefore twice the abscissa of the
point of contact.

\carticleplus{32.}{Prop.~IX\@.} \textit{The foot of the perpendicular from the
focus on the tangent at any point $P$ of a parabola lies on the
tangent at the vertex, and the perpendicular is a mean proportional
between $SP$ and $SA$.}

Taking the figure of the previous article, join $SK$ meeting
$PT$ in $Y$.

Then $SP = PK$, and $PY$ is common to the two triangles
$SPY$, $KPY$;
\begin{align*}
\text{also the angle } SPY &= YPK; \\
\therefore \text{ the angle } SYP &= PYK,
\end{align*}
and $SY$ is perpendicular to $PT$.

Also $SY= KY$, and $SA = AX$, \;$\therefore AY$ is parallel to $KX$.
%-----File: 044.png-----

Hence, $AY$ is at right angles to $AS$, and is therefore
the tangent at the vertex.

Again, the angle $SPY = STY = SYA$, and the triangles
$SPY$, $SYA$ are therefore similar;
\begin{gather*}
\therefore SP:SY :: SY:SA, \\
\text{or } SY^2 = SP\ctrdot SA.
\end{gather*}

\carticleplus{33.}{Prop.~X\@.} \textit{In the parabola the subnormal is constant
and equal to the semi-latus rectum.}

\subparagraph{Def.} \textit{The distance between the foot of the ordinate of $P$
and the point in which the normal at $P$ meets the axis is
called the subnormal.}

\includegraphicsmid{033a}

In the figure $PG$ is the normal and $PT$ the tangent.

It has been shewn that the angle $SPK$ is bisected by
$PT$, and hence it follows that $SPL$ is bisected by $PG$,
\begin{flalign*}
&\text{and that the angle }& SPG &= GPL = PGS;
&&\phantom{and\ that\ the\ angle}\\
&\text{hence}&     SG &= SP = ST &&\\
&&             &= SA + AT = SA + AN &&\\
&&             &= 2AS + SN; &&\\
&& \llap{$\therefore$ the subnormal } NG &= 2AS. &&
\end{flalign*}

\begin{small}
\carticleplus{34.}{Cor.}
\begin{tabular}[t]{r@{\;}l}
\multicolumn{2}{l}{If $Gl$ be drawn perpendicular to $SP$,}\\
  the angle $GPl$ &= the complement of $SPT$, \\
                  &= the complement of $STP$, \\
                  &= $PGN$,\\
\end{tabular}\\
and the two right-angled triangles $GPN$, $GPl$ have their angles equal
and the side $GP$ common; hence the triangles are equal, and
\begin{align*}
Pl &= NG = 2AS \\
&= \text{the semi-latus rectum.}
\end{align*}

It has been already shewn, (Art.~\cartref{19}), that this property is a general
property of all conics.

\end{small}
%-----File: 045.png-----

\carticleplus{35.}{Prop.~XI\@.} \textit{To draw tangents to a parabola from an
external point.}

{\stretchyspace
For this purpose we may employ the general construction
given in Art.~(\cartref{17}),
or, for the special case of the parabola, the
following construction.}

Let $Q$ be the external point, join $SQ$, and upon $SQ$ as
diameter describe a circle intersecting the tangent at the
vertex in $Y$ and $Y'$. Join $YQ$, $Y'Q$; these are tangents to
the parabola.

Draw $SP$, so as to make the angle $YSP$ equal to $YSA$,
and to meet $YQ$ in $P$, and let fall the perpendicular $PN$
upon the axis.

\includegraphicsmid{035a}

Then, $SYQ$ is a right angle, since it is the angle in a
semicircle, and, $T$ being the point in which $QY$ produced
meets the axis, the two triangles $SYP$, $SYT$ are equal in all
respects;
\[
\therefore  SP = ST, \text{ and } YT = YP.
\]
But $AY$ is parallel to $PN$;
\[
\therefore  AT = AN.
\]
\begin{flalign*}
&\indent\text{Hence}& SP &= ST = SA + AT &&\phantom{\indent\text{Hence}}\\
&&                                   &= AX + AN      &&\\
&&                                   &= NX,          &&
\end{flalign*}
and $P$ is a point in the parabola.

Moreover, if $PK$ be perpendicular to the directrix, the
angle $SPY = STP = YPK$, and $PY$ is the tangent at $P$.
(Art.~\cartref{29}.)

Similarly, by making the angle $Y'SP'$ equal to $ASY'$
we obtain the point of contact of the other tangent $QY'$.
%-----File: 046.png-----

\carticleplus{36.}{Prop.~XII\@.} \textit{If from a point $Q$ tangents $QP$, $QP'$
be drawn to a parabola, the two triangles $SPQ$, $SQP'$, are
similar, and $SQ$ is a mean proportional between $SP$ and
$SP'$.}

Produce $PQ$ to meet the axis in $T$, and draw $SY$, $SY'$
perpendicularly on the tangents. Then $Y$ and $Y'$ are points
in the tangent at $A$.
\begin{flalign*}
&\indent\text{The angle}&  SPQ &= STY &&\phantom{\indent\text{The angle}}\\
&&               &= SYA &&\\
&&               &= SQP', &&
\end{flalign*}
since $S$, $Y'$, $Y$, $Q$ are points on a circle, and $SYA$, $SQP'$ are
in the same segment.

\includegraphicsmid{035a} %repeated figure

Also, by the theorem of Art.~(\cartref{15}), the angle
\[
PSQ = QSP';
\]
therefore the triangles $PSQ$, $QSP'$ are similar, and
\[
SP: SQ:: SQ: SP'.
\]

\carticle{37.} From the preceding theorem the following, which
is often useful, immediately follows.

\textit{If from any points in a given tangent of a parabola,
tangents be drawn to the curve, the angles which these tangents
make with the focal distances of the points from which they
are drawn are all equal.}

For each of them by the theorem, is equal to the angle
between the given tangent and the focal distance of the
point of contact.

\textit{Hence it follows that the locus of the intersection of a
%-----File: 047.png-----
tangent to a parabola with a straight line drawn through the
focus meeting it at a constant angle is a straight line.}

For if $QP$ be the moveable tangent, the angle $SQP=SP'Q$,
and therefore, if $SQP$ is constant, $SP'Q$ is a given angle.
The point $P'$ is therefore fixed, and the locus of $Q$ is the
tangent $P'Q$.

\begin{small}
\carticle{38.} Since the two triangles $PSQ$, $QSP'$ are similar, we have
\begin{flalign*}
&&                  PQ : P'Q  &:: SP : SQ   &&\\
&\text{and}&        PQ : P'Q  &:: SQ : SP', &&\phantom{and}\\
&&\therefore     PQ^2 : P'Q^2 &:: SP : SP'; &&
\end{flalign*}
that is, the squares of the tangents from any point are proportional to
the focal distances of the points of contact.

This will be found to be a particular case of a subsequent Theorem,
given in Art.~\cartref{51}.

\end{small}

\carticleplus{39.}{Prop.~XIII\@.} \textit{The external angle between two tangents
is half the angle subtended at the focus by the chord of
contact.}

Let the tangents at $P$ and $P'$ intersect each other in $Q$
and the axis $ASN$ in $T$ and $T'$.

Join $SP$, $SP'$; then the angles $SPT$, $STP$ are equal,
and $\therefore STP$ is half the angle $PSN$; similarly $ST'P'$ is half
$P'SN$.

\includegraphicsmid{039a}

But $TQT'$ is equal to the difference between $STP$ and
$ST'P'$, and is therefore equal to half the difference between
$PSN$ and $P'SN$, that is to half the angle $PSP'$.

Hence, joining $SQ$, $TQT'$ is equal to each of the angles
$PSQ$, $P'SQ$.
%-----File: 048.png-----

\carticleplus{40.}{Prop.~XIV\@.} \textit{The tangents drawn to a parabola from
any point make the same angles, respectively, with the axis
and the focal distance of the point.}

\includegraphicsmid{040a}

Let $QP$, $QP'$ be the tangents; join $SP$, and draw $QE$
parallel to the axis, and meeting $SP$ in $E$.

Then, if $PQ$ meet the axis in $T$, the angle
\begin{align*}
EQP &= STP = SPQ    \\
        &= SQP'. \quad \text{(Art.~\cartref{37}.)}
\end{align*}

\textit{i.e.\ }\quad$QP$ and $QP'$ respectively make the same angles with
the axis and with $QS$.

\begin{small}
\carticle{41.} Conceive a parabola to be drawn passing through $Q$, having $S$
for its focus, $SN$ for its axis, and its vertex on the same side of $S$ as the
vertex $A$ of the given parabola. Then the normal at $Q$ to this new
parabola bisects the angle $SQE$; therefore the angles which $QP$ and
$QP'$ make with the normal at $Q$ are equal.

Hence the theorem,

\textit{If from any point in a parabola, tangents be drawn to a confocal
and co-axial parabola, the normal at the point will bisect the angle
between the tangents.}

If we produce $SP$ to any point $p$, and take $St$ equal to $Sp$, $pt$ will
be the tangent at $p$ to the confocal and co-axial parabola passing
through $p$.

Hence the theorem,

\textit{If parallel tangents be drawn to a series of confocal and co-axial
parabolas, the points of contact will lie in a straight line passing through
the focus.}

In these enunciations the words co-axial and confocal are intended
to imply, not merely the coincidence of the axes, but also that the
vertices of the two parabolas are on the same side of their common
focus.

The reason for this will appear when we shall have discussed the
analogous property of the ellipse.
%-----File: 049.png-----

\carticle{42.} If two confocal parabolas have their axes in the same straight
line, and their vertices on opposite sides of the focus, they intersect at
right angles.
\begin{flalign*}
&\indent\text{For the angle}&    TPS &= \tfrac{1}{2} PST', &&\\
&\text{and}&        T'PS &= \tfrac{1}{2} PST, &&\\
&&\therefore TPT' &= \tfrac{1}{2} (PST + PST') = \text{a right angle.} &&
\end{flalign*}

It will be noticed that, in this case, the common chord $PQ$ is
equidistant from the directrices.

For the distance of $P$ from each directrix is equal to $SP$.

\end{small}

\includegraphicsmid{042a}

\carticleplus{43.}{Prop.~XV\@.} \textit{The circle passing through the points
of intersection of three tangents passes also through the focus.}

Let $Q$, $P$, $Q'$ be the three points of contact, and $F$, $T$, $F'$
the intersections of the tangents.
%-----File: 050.png-----

In Art.~(\cartref{36}) it has been shewn that, if $FP$, $FQ$ be tangents,
the angle
\[
SQF=SFP.
\]

\includegraphicsmid{043a}

Similarly $TQ$, $TQ'$ being tangents, the angle
\begin{flalign*}
&&                                         SQT &= STQ',  &&\\
&\text{hence the angle }& SFF' \text{ or } SFP &= SQT,
&\phantom{\text{hence the angle }SFF'\text{ or}}&\\
&&                                             &= STF',  &&
\end{flalign*}
and a circle can be drawn through $S$, $F$, $T$, and $F'$.

\carticleplus{44.}{Def.} \textit{A straight line drawn parallel to the axis
through any point of a parabola is called a diameter.}

\subparagraph{Prop.~XVI\@.} \textit{If from any point $T$ tangents $TQ$, $TQ'$ be
drawn to a parabola, the point $T$ is equidistant from the
diameters passing through $Q$ and $Q'$, and the diameter drawn
through the point $T$ bisects the chord of contact.}

\includegraphicsright[11]{044a}

Join $SQ$, $SQ'$, and draw $TM$, $TM'$ perpendicular respectively
to $SQ$ and $SQ'$.

Also draw $NTN'$ perpendicular
to the diameters
through $Q$ and $Q'$, and
meeting those diameters in
$N$ and $N'$.

Then, since $TS$ bisects
the angle $QSQ'$,
\[
TM=TM';
\]
and, since $TQ$ bisects the angle $SQN$,
\begin{flalign*}
&&                             TN &= TM.  &&\\
&\text{Similarly }&           TN' &= TM',  &\phantom{Similarly, }&\\
&&                  \therefore TN &= TN'.
\end{flalign*}

Again, join $QQ'$, and draw the diameter $TV$ meeting
$QQ'$ in $V$; also let $QT$ produced meet $Q'N'$ in $R$;
\begin{flalign*}
&\text{then }&       QV : VQ' &:: QT : TR  &\phantom{then }&\\
&&                            &:: TN : TN',  &&\\
\intertext{since the triangles $QTN$, $RTN'$ are similar;}
&&              \therefore QV &= VQ'.  &&
\end{flalign*}
%-----File: 051.png-----

Hence \textit{the diameter through the middle point of a chord
passes, when produced, through the point of intersection of the
tangents at the ends of the chord.}

It should be noticed that any straight line drawn
through $T$ and terminated by $QN$ and $Q'N'$ is bisected at $T$.

\carticleplus{45.}{Prop.~XVII\@.} \textit{Any diameter bisects all chords parallel
to the tangent at its extremity, and passes through the point of
intersection of the tangents at the ends of any of these chords.}

\includegraphicsmid{045a}

Let $QQ'$ be a chord parallel to the tangent at $P$, and
through the point of intersection $T$ of the tangents at $Q$ and
$Q'$ draw $FTF'$ parallel to $QQ'$ and terminated at $F$ and $F'$ by
the diameters through $Q$ and $Q'$.

Let the tangent at $P$ meet $TQ$, $TQ'$ in $E$ and $E'$, and
$QF$, $Q'F'$ in $G$ and $G'$.
\begin{flalign*}
&\text{Then }&
  EG : TF &:: EQ   : TQ   &&\phantom{Then}\\
&&        &:: E'Q' : TQ'  &&\\
&&        &:: E'G' : TF'. &&
\end{flalign*}
But $TF = TF'$, since (Art.~\cartref{44}) $T$ is equidistant from $QG$ and
$Q'G'$,
\[
\therefore EG = E'G'.
\]
Also, $EP = EG$, since $E$ is equidistant from $QG$ and $PV$, the
diameter at $P$.
\[
\therefore  EP = E'P \text{ and } GP = PG',
\]
and \hfill$ \therefore QV= VQ'. $\hfill\phantom{and}\null
%-----File: 052.png-----

Again, since $T$, $P$, $V$ are each equidistant from the
parallel straight lines $QF$, $Q'F'$, it follows that $TPV$ is a
straight line, or that the diameter $VP$ passes through $T$.

We have shewn that $GE$, $EP$, $PE'$, $E'G'$ are all equal,
and we hence infer that
\[
  EE' = \tfrac{1}{2} GG' = \tfrac{1}{2} QQ',
\]
and consequently that $TP = \frac{1}{2} TV$, or that $TP = PV$.

Hence \textit{it appears, that the diameter through the point of
intersection of a pair of tangents passes through the point of
contact of the tangent parallel to the chord of contact, and also
through the middle point of the chord of contact; and that the
portion of the diameter between the point of intersection of the
tangents and the middle point of the chord of contact is bisected
at the point of contact of the parallel tangent.}

We may observe that in proving that $EE'$ is bisected at
$P$, we have demonstrated a theorem already shewn (Art.~\cartref{21})
to be true for all conics.

\begin{small}
\carticle{46.} When the point $T$ is on the directrix, $QTQ'$ is a right angle.

If then $Qq$ is the chord which is normal at $Q$, it is parallel to the
tangent $TQ'$, and is therefore bisected by the diameter $Q'U$ through $Q'$.

\includegraphicsmid{046a}

Since $QU$ is bisected by $TV$, it follows that
\[
  Qq = 4TQ',
\]

i.e.\ \textit{the length of a normal chord is four times the portion of the
parallel tangent between the directrix and the point of contact.}

\end{small}
%-----File: 053.png-----

\carticleplus{47.}{Def.} \textit{The line $QV$, parallel to the tangent at $P$,
and terminated by the diameter $PV$, is called an ordinate
of that diameter, and $QQ'$ is the double ordinate. The point
$P$, the end of the diameter, is called the vertex of the diameter,
and the distance $PV$ is called the abscissa of the point $Q$.}

We have seen that tangents at the ends of any chord
intersect in the diameter which bisects the chord, and that
the distance of this point from the vertex is equal to the
distance of the vertex from the middle point of the chord.

\subparagraph{Def.} \textit{The chord through the focus parallel to the tangent
at any point is called the parameter of the diameter passing
through the point.}

\subparagraph{Prop.~XVIII\@.} \textit{The parameter of any diameter is four
times the focal distance of the vertex of that diameter.}

Let $P$ be the vertex, and
$QSQ'$ the parameter, $T$ the
point of intersection of the
tangents at $Q$ and $Q'$, and
$FPF'$ the tangent at $P$.
\includegraphicsmid{047a}

Then, since $FS$ and $F'S$
bisect respectively the angles
$PSQ$, $PSQ'$, $FSF'$ is a right
angle, and, $P$ being the middle
point of $FF'$, $SP = PF = PF'$.

Hence $QQ'$, which is double
$FF'$, is four times $SP$.

\carticleplus{48.}{Prop.~XIX\@.} \textit{If $QVQ'$ be a double ordinate of a
diameter $PV$, $QV$ is a mean proportional between $PV$ and
the parameter of $P$.}

Let $FPF'$ be the tangent at $P$, and draw the parameter
through $S$ meeting $PV$ in $U$.

The angle $SUT = FPU = SPF'$ (Art.~\cartref{29}), and, since the
angles $SFQ$, $SPF$ are equal (Art.~\cartref{36}), it follows that the
angles $SFT$, $SPF'$ are equal;
%-----File: 054.png-----

$\therefore SUT = SFT$, and $U$ is a point in the circle passing
through $SFTF'$.
\cpagnewpage

\includegraphicsmid{048a}

Hence, $QV$ being twice $PF$,
\begin{flalign*}
&&                         QV^2 = 4PF^2 &= 4PU\ctrdot PT;    &&\\
&\text{but}&                       PU &= SP,                    &&\\
&\text{for the angle}&      SUP = FPU &= SPF' = PSU;            &&\phantom{for\ the\ angle}\\
&\text{and}&                       PT &= PV,                    &&\\
&&                    \therefore QV^2 &= 4SP\ctrdot PV.      &&
\end{flalign*}

{\small
\includegraphicsleft[16]{049a}

\cpagdummypar

\carticle{49.} This relation may be presented
in a different form, which is
sometimes useful.

If from any point $U$ in the tangent
at $P$, $UQ$ is drawn parallel to
the axis, $UP$ and $UQ$ are respectively
equal to the ordinate and
abscissa of the point $Q$ with regard
to the diameter through $P$, and
therefore
\[
PU^2 = 4SP \ctrdot UQ.
\]

Therefore, if $VR$ is drawn parallel
to the axis from another point $V$ of
the tangent,
\[
PU^2 : PV^2  :: UQ : VR.
\]
%-----File: 055.png-----
\begin{flalign*}
&\indent\text{Hence, since}&
    UE : VR &:: PU : PV,       &&\phantom{\indent\text{Hence, since}}\\
&&  UE^2 : VR^2 :: UQ &: VR :: UQ\ctrdot VR : VR^2, &&\\
&\text{and}&      UE^2 &= UQ\ctrdot VR. &&\\
&\indent\text{Hence}& UE : UQ :: VR &: UE :: PR : PE;  &&\\
&&\therefore    UQ : QE &:: PE : ER. &&
\end{flalign*}

In a similar manner it can be shewn that $VF^2 = UQ\ctrdot VR$, and it
follows that $VF = UE$, and therefore that $EF$ is parallel to the tangent
at $P$.
} % end \small

\carticleplus{50.}{Prop.~XX\@.} \textit{If $QVQ'$ be a double ordinate of a
diameter $PV$, and $QD$ the perpendicular from $Q$ upon $PV$,
$QD$ is a mean proportional between $PV$ and the latus rectum.}

\includegraphicsmid{050a}

Let the tangent at $P$ meet the tangent at the vertex in
$Y$, and join $SY$.

The angle $QVD = SPY = SYA$, and therefore the triangles
$QVD$, $SAY$ are similar;
\begin{flalign*}
&\text{and}&     QD^2 : QV^2 &:: AS^2 : SY^2 &&\\
&&                        &:: AS^2 : AS\ctrdot SP.        &&\\
&&                        &:: AS : SP                &&\\
&&                        &:: 4AS\ctrdot PV : 4SP\ctrdot PV,       &&
\end{flalign*}
\begin{flalign*}
&\text{but}&          QV^2 &= 4SP\ctrdot PV; &&\\
&&\therefore   QD^2 &= 4AS\ctrdot PV. &&
\end{flalign*}

\carticleplus{51.}{Prop.~XXI\@.} \textit{If from any point, within or without
a parabola, two straight lines be drawn in given directions
and intersecting the curve, the ratio of the rectangles of the
segments is independent of the position of the point.}

From any point $O$ draw a straight line intersecting the
%-----File: 056.png-----
parabola in $Q$ and $Q'$, and draw the diameter $OE$, meeting
the curve in $E$.

\includegraphicsmid{051a}

If $PV$ be the diameter bisecting $QQ'$, and $EU$ the
ordinate,
\begin{align*}
  OQ \ctrdot OQ' &= OV^2 - QV^2  \\
          &= EU^2 - QV^2 = 4SP \ctrdot PU - 4SP \ctrdot PV  \\
          &= 4SP \ctrdot OE.
\end{align*}

Similarly, if $ORR'$ be any other intersecting line and $P'$
the vertex of the diameter bisecting $RR'$,
\begin{gather*}
  OR \ctrdot OR' = 4SP' \ctrdot OE.  \\
  \therefore OQ \ctrdot OQ' : OR \ctrdot OR' :: SP : SP',
\end{gather*}
that is, the ratio of the rectangles depends only on the
positions of $P$ and $P'$, and, if the lines $OQQ'$, $ORR'$ are drawn
parallel to given straight lines, these points $P$, $P'$ are fixed.

It will be easily seen that the proof is the same if the
point $O$ be within the parabola.

If the lines $OQQ'$, $ORR'$ be moved parallel to themselves
until they become the tangents at $P$ and $P'$, we shall then
obtain, if these tangents intersect in $T$,
\[
  TP^2 : TP'^2 :: SP : SP';
\]
a result previously obtained (Art.~\cartref{38}).

Again if $QSQ'$, $RSR'$ be the focal chords parallel to $TP$
and $TP'$, it follows that
\[
  TP^2 : TP'^2 :: QS \ctrdot SQ' : RS \ctrdot SR',
\]
$\therefore$ (cor.\ Art.~\cartref{8})  $TP^2 : TP'^2 :: QQ' : RR'$.
%-----File: 057.png-----

\carticleplus{52.}{Prop.~XXII\@.} \textit{If from a point $O$, outside a parabola,
a tangent $OM$, and a chord $OAB$ be drawn, and if the
diameter $ME$ meet the chord in $E$,
\[
          OE^2 = OA\ctrdot OB.
\]
}
\includegraphicsmid{052a}

Let $P$ be the point of contact of the tangent parallel to
$OAB$, and let $OM$, $ME$ meet this tangent in $T$ and $F$.

Draw $TV$ parallel to the axis and meeting $PM$ in $V$;
\begin{flalign*}
&\text{then}&     OA\ctrdot OB : OM^2 &:: TP^2 : TM^2 \text{ (Art.~\cartref{51}),} &&\\
&&                        &:: TF^2 : TM^2, &&
\end{flalign*}
since $PM$ is bisected in $V$;
\begin{flalign*}
&\text{also} &          TF : TM &:: OE : OM; &&\phantom{also}\\
&& \therefore             OE^2 &= OA\ctrdot OB. &&
\end{flalign*}

\subparagraph{Cor.~1.} If $AL$, $BN$ be the ordinates, parallel to $OM$, of
$A$ and $B$, $ML$, $ME$, and $MN$ are proportional to $OA$, $OE$
and $OB$, and therefore
\[
          ME^2 = ML\ctrdot MN.
\]

This theorem may be also stated in the following form:

\textit{If a chord $AB$ of a parabola intersect a diameter in the
point $E$, the distance of the point $E$ from the tangent at the
end of the diameter is a mean proportional between the distances
of the points $A$ and $B$ from the same tangent.}
%-----File: 058.png-----

\subparagraph{Cor.~2.} Let $KE$ be the ordinate through $E$ parallel to
$OM$.
\begin{flalign*}
&\indent\text{Then, since}&
    ML : ME &:: ME : MN,    &&\phantom{\indent\text{Then, since}}\\
&&  AL^2 : KE^2 &:: KE^2 : BN^2 &&\\
&&  \therefore    AL : KE\: &:: KE\: : BN, &&
\end{flalign*}
so that $KE$ is a mean proportional between $AL$ and $BN$, the
ordinates of $A$ and $B$.

\carticleplus{53.}{Prop.~XXIII\@.} \textit{If a circle intersect a parabola in
four points, the two straight lines constituting any one of
the three pairs of the chords of intersection are equally inclined
to the axis.}

Let $Q$, $Q'$, $R$, $R'$ be the four points of intersection;
\begin{flalign*}
&\text{then}&   OQ\ctrdot OQ' &= OR\ctrdot OR', &&\phantom{then}
\end{flalign*}
and therefore $SP$, $SP'$ are equal, (Art.~\cartref{51}).

\includegraphicsmid{053a}

But, if $SP$, $SP'$ be equal, the points $P$, $P'$ are on opposite
sides of, and are equidistant from the axis, and the tangents
at $P$ and $P'$ are therefore equally inclined to the axis.

Hence the chords $QQ'$, $RR'$, which are parallel to these
tangents, are equally inclined to the axis.

In the same manner it may be shewn that $QR$, $Q'R'$
are equally inclined to the axis, as also $QR'$, $Q'R$.
%-----File: 059.png-----

\begin{small}
\carticle{54.} Conversely, if two chords $QQ'$, $RR'$, which are not parallel,
make equal angles with the axis, a circle can be drawn through $Q$, $Q'$,
$R'$, $R$.

For, if the chords intersect in $O$, and $OE$ be drawn parallel to the
axis and meeting the curve in $E$, it may be shewn as above that
\[
OQ\ctrdot OQ' = 4SP\ctrdot OE, \text{ and } OR\ctrdot OR' = 4SP'\ctrdot OE,
\]
$P$ and $P'$ being the vertices of the diameters bisecting the chords.

But the tangents at $P$ and $P'$, which are parallel to the chords, are
equally inclined to the axis, and therefore $SP$ is equal to $SP'$.
\begin{flalign*}
&\indent\text{Hence}&
    OQ\ctrdot OQ' &= OR\ctrdot OR', &&\phantom{\indent\text{Hence}}
\end{flalign*}
and therefore a circle can be drawn through the points $Q$, $Q'$, $R$, $R'$.

If the two chords are both perpendicular to the axis, it is obvious
that a circle can be drawn through their extremities, and this is the
only case in which a circle can be drawn through the extremities of
parallel chords.

\end{small}

\csubsection{EXAMPLES}

\begin{exampleslist}

\item[1.] Find the locus of the centre of a circle which passes through a
given point and touches a given straight line.
\cheader{EXAMPLES}

\item[2.] Draw a tangent to a parabola, making a given angle with the
axis.

\item[3.] If the tangent at $P$ meet the tangent at the vertex in $Y$,
\[
AY^2 = AS\ctrdot AN.
\]

\item[4.] If the normal at $P$ meet the axis in $G$, the focus is equidistant
from the tangent at $P$ and the straight line through $G$ parallel to the
tangent.

\item[5.] Given the focus, the position of the axis, and a tangent, construct
the parabola.

\item[6.] Find the locus of the centre of a circle which touches a given
straight line and a given circle.

\item[7.] Construct a parabola which has a given focus, and two given
tangents.

\item[8.] The distance of any point on a parabola from the focus is equal
to the length of the ordinate at that point produced to meet the tangent
at the end of the latus rectum.

\item[9.] $PT$ being the tangent at $P$, meeting the axis in $T$, and $PN$ the
ordinate, prove that $TY\ctrdot TP = TS\ctrdot TN$.
%-----File: 060.png-----

\item[10.] If $SE$ be the perpendicular from the focus on the normal at $P$,
shew that
\[
SE^2 = AN \ctrdot SP.
\]

\item[11.] The locus of the vertices of all parabolas, which have a
common focus and a common tangent, is a circle.

\item[12.] Having given the focus, the length of the latus rectum, and a
tangent, construct the parabola.

\item[13.] If $PSP'$ be a focal chord, and $PN$, $P'N'$ the ordinates, shew
that
\[
AN \ctrdot AN'=AS^2.
\]
Shew also that the latus rectum is a mean proportional between
the double ordinates.

\item[14.] The locus of the middle points of the focal chords of a parabola
is another parabola.

\item[15.] Shew that in general two parabolas can be drawn having a
given straight line for directrix, and passing through two given points
on the same side of the line.

\item[16.] $Pp$ is a chord perpendicular to the axis, and the perpendicular
from $p$ on the tangent at $P$ meets the diameter through $P$ in $R$; prove
that $RP$ is equal to the latus rectum, and find the locus of $R$.

\item[17.] Having given the focus, describe a parabola passing through
two given points.

\item[18.] The circle on any focal distance as diameter touches the
tangent at the vertex.

\item[19.] The circle on any focal chord as diameter touches the directrix.

\item[20.] A point moves so that its shortest distance from a given circle
is equal to its distance from a given diameter of the circle; prove that
the locus is a parabola, the focus of which coincides with the centre of
the circle.

\item[21.] Find the locus of a point which moves so that its shortest
distance from a given circle is equal to its distance from a given
straight line.

\item[22.] The vertex of an isosceles triangle is fixed. The extremities of
its base lie on two fixed parallel straight lines. Prove that the base is
a tangent to a parabola.

\item[23.] Shew that the normal at any point of a parabola is equal to
the ordinate through the middle point of the subnormal.

\item[24.] If perpendiculars are drawn to the tangents to a parabola
where they meet the axis they will be normals to two equal parabolas.

\item[25.] $PSP'$ is a focal chord of a parabola. The diameters through
$P$, $P'$ meet the normals at $P'$, $P$ in $V$, $V'$ respectively. Prove that
$PVV'P'$ is a parallelogram.
%-----File: 061.png-----

\item[26.] If $APC$ be a sector of a circle, of which the radius $CA$ is fixed,
and a circle be described, touching the radii $CA$, $CP$, and the arc $AP$,
the locus of the centre of this circle is a parabola.

\item[27.] If from the focus $S$ of a parabola, $SY$, $SZ$ be perpendiculars
drawn to the tangent and normal at any point, $YZ$ is parallel to the
diameter.

\item[28.] Prove that the locus of the foot of the perpendicular from the
focus on the normal is a parabola.

\item[29.] If $PG$ be the normal, and $GL$ the perpendicular from $G$ upon
$SP$, prove that $GL$ is equal to the ordinate $PN$.

\item[30.] Given the focus, a point $P$ on the curve, and the length of the
perpendicular from the focus on the tangent at $P$, find the vertex.

\item[31.] A circle is described on the latus rectum as diameter, and
a common tangent $QP$ is drawn to it and the parabola: shew that $SP$,
$SQ$ make equal angles with the latus rectum.

\item[32.] $G$ is the foot of the normal at a point $P$ of the parabola,
$Q$ is the middle point of $SG$, and $X$ is the foot of the directrix: prove
that
\[
     QX^2 - QP^2 = 4 AS^2.
\]

\item[33.] If $PG$ the normal at $P$ meet the axis in $G$, and if $PF$, $PH$,
lines equally inclined to $PG$, meet the axis in $F$ and $H$, the length $SG$ is
a mean proportional between $SF$ and $SH$.

\item[34.] A triangle $ABC$ circumscribes a parabola whose focus is $S$, and
through $A$, $B$, $C$, lines are drawn respectively perpendicular to $SA$, $SB$,
$SC$; shew that these pass through one point.

\item[35.] If $PQ$ be the normal at $P$ meeting the curve in $Q$, and if the
chord $PR$ be drawn so that $PR$, $PQ$ are equally inclined to the axis,
$PRQ$ is a right angle.

\item[36.] $PN$ is a semi-ordinate of a parabola, and $AM$ is taken on the
other side of the vertex along the axis equal to $AN$; from any point $Q$
in $PN$, $QR$ is drawn parallel to the axis meeting the curve in $R$; prove
that the lines $MR$, $AQ$ will intersect in the parabola.

\item[37.] Having given two points of a parabola, the direction of the
axis, and the tangent at one of the points, construct the parabola.

\item[38.] Having given the vertex of a diameter, and a corresponding
double ordinate, construct the parabola.

\item[39.] $PM$ is an ordinate of a point $P$; a straight line parallel to the
axis bisects $PM$, and meets the curve in $Q$; $MQ$ meets the tangent at
the vertex in $T$; prove that $3 AT = 2 PM$.

\item[40.] $AB$, $CD$ are two parallel straight lines given in position, and
$AC$ is perpendicular to both, $A$ and $C$ being given points; in $CD$ any
point $Q$ is taken, and in $AQ$, produced if necessary, a point $P$ is taken,
%-----File: 062.png-----
such that the distance of $P$ from $AB$ is equal to $CQ$; prove that the
locus of $P$ is a parabola.

\item[41.] If the tangent and normal at a point $P$ of a parabola meet the
tangent at the vertex in $K$ and $L$ respectively, prove that
\[
KL^2 : SP^2 :: SP-AS : AS.
\]

\item[42.] Having given the length of a focal chord, find its position.

\item[43.] If the ordinate of a point $P$ bisects the subnormal of a point
$P'$, prove that the ordinate of $P$ is equal to the normal of $P'$.

\item[44.] A parabola being traced on a plane, find its axis and vertex.

\item[45.] If $PV$, $P'V'$ be two diameters, and $PV'$, $P'V$ ordinates to these
diameters,
\[
PV = P'V'.
\]

\item[46.] If one side of a triangle be parallel to the axis of a parabola,
the other sides will be in the ratio of the tangents parallel to them.

\item[47.] $QVQ'$ is an ordinate of a diameter $PV$, and any chord $PR$
meets $QQ'$ in $N$, and the diameter through $Q$ in $L$; prove that
\[
PL^2 = PN \ctrdot PR.
\]

\item[48.] Describe a parabola passing through three given points, and
having its axis parallel to a given line.

\item[49.] If $AP$, $AQ$ be two chords drawn from the vertex at right
angles to each other, and $PN$, $QM$ be ordinates, the latus rectum is a
mean proportional between $AN$ and $AM$.

\item[50.] $PSp$ is a focal chord of a parabola; prove that $AP$, $Ap$ meet
the latus rectum in two points whose distances from the focus are
equal to the ordinates of $p$ and $P$ respectively.

\item[51.] If the straight line $AP$ and the diameter through $P$ meet the
double ordinate $QMQ'$ in $R$ and $R'$, prove that
\[
RM\ctrdot R'M=QM^2.
\]

\item[52.] $A$ and $P$ are two fixed points. Parabolas are drawn all having
their vertices at $A$, and all passing through $P$. Prove that the points
of intersection of the tangents at $P$ with the tangent and normal at $A$
lie on two fixed circles, one of which is double the size of the other.

\item[53.] A variable tangent to a parabola intersects two fixed tangents
in the points $T$ and $T'$: shew that the ratio $ST: ST'$ is constant.

\item[54.] Through a fixed point on the axis of a parabola a chord $PQ$ is
drawn, and a circle of given radius is described through the feet of the
ordinates of $P$ and $Q$. Shew that the locus of its centre is a circle.

\item[55.] If $SY$ be the perpendicular on the tangent at $P$, and if $YS$ be
produced to $R$ so that $SR=SY$, shew that $PAR$ is a right angle.
%-----File: 063.png-----

\item[56.] If two circles be drawn touching a parabola at the ends of a
focal chord, and passing through the focus, shew that they intersect
each other orthogonally.

\item[57.] $PSQ$ is a focal chord of a parabola, whose vertex is $A$ and
focus $S$, $V$ being the middle point of the chord, shew that
\[
PV^2 = AV^2 + 3AS^2.
\]

\item[58.] $QQ'$ is a focal chord of a parabola. Describe a circle which
shall pass through $Q$, $Q'$ and touch the parabola.

If $P$ be the point of contact and the angle $QPQ'$ a right angle, find
the inclination of $QP$ to the axis.

\item[59.] Through two fixed points $E$, $F$, on the axis of a parabola are
drawn two chords $PQ$, $PR$ meeting the curve in $P$, $Q$, $R$. If $QR$ meet
the axis in $T$, shew that the ratio $TR: TQ$ is constant.

\item[60.] A chord $PQ$ is normal to the parabola at $P$, and the angle
$PSQ$ is a right angle. Prove that $SQ=2SP$, and that the ordinate of $P$
is equal to the latus rectum. Also, if $T$ is the point of intersection of
the tangents at $P$ and $Q$, and if $R$ is the middle point of $TQ$, prove
that the angle $TSR$ is a right angle, and that $ST=2SR$.

\item[61.] A straight line intersects a circle; prove that all the chords of
the circle which are bisected by the straight line are tangents to a
parabola.

\item[62.] If two tangents $TP$, $TQ$ be drawn to a parabola, the perpendicular
$SE$ from the focus on their chord of contact passes through the
middle point of their intercept on the tangent at the vertex.

\item[63.] From the vertex of a parabola a perpendicular is drawn on
the tangent at any point; prove that the locus of its intersection with
the diameter through the point is a straight line.

\item[64.] If two tangents to a parabola be drawn from any point in
its axis, and if any other tangent intersect these two in $P$ and $Q$,
prove that $SP=SQ$.

\item[65.] $T$ is a point on the tangent at $P$, such that the perpendicular
from $T$ on $SP$ is of constant length; prove that the locus of $T$ is a
parabola.

If the constant length be $2AS$, prove that the vertex of the locus
is on the directrix.

\item[66.] Given a chord of a parabola in magnitude and position, and
the point in which the axis cuts the chord, the locus of the vertex
is a circle.

\item[67.] If the normal at a point $P$ of a parabola meet the curve in $Q$,
and the tangents at $P$ and $Q$ intersect in $T$, prove that $T$ and $P$ are
equidistant from the directrix.
%-----File: 064.png-----

\item[68.] If $TP$, $TQ$ be tangents to a parabola, such that the chord
$PQ$ is normal at $P$,
\[
  PQ : PT :: PN : AN,
\]
$PN$ and $AN$ being the ordinate and abscissa.

\item[69.] If two equal tangents to a parabola be cut by a third tangent,
the alternate segments of the two tangents will be equal.

\item[70.] If $AP$ be a chord through the vertex, and if $PL$, perpendicular
to $AP$, and $PG$, the normal at $P$, meet the axis in $L$, $G$ respectively,
$GL=\text{half}$ the latus rectum.

\item[71.] If $PSQ$ be a focal chord, $A$ the vertex, and $PA$, $QA$ be
produced to meet the directrix in $P'$, $Q'$ respectively, then $P'SQ'$ will
be a right angle.

\item[72.] The tangents at $P$ and $Q$ intersect in $T$, and the tangent at
$R$ intersects $TP$ and $TQ$ in $C$ and $D$; prove that
\[
  PC : CT :: CR : RD :: TD : DQ.
\]

\item[73.] From any point $D$ in the latus rectum of a parabola, a straight
line $DP$ is drawn, parallel to the axis, to meet the curve in $P$; if
$X$ be the foot of the directrix, and $A$ the vertex, prove that $AD$,
$XP$ intersect in the parabola.

\item[74.] $PSp$ is a focal chord, and upon $PS$ and $pS$ as diameters
circles are described; prove that the length of either of their common
tangents is a mean proportional between $AS$ and $Pp$.

\item[75.] If $AQ$ be a chord of a parabola through the vertex $A$, and
$QR$ be drawn perpendicular to $AQ$ to meet the axis in $R$; prove
that $AR$ will be equal to the chord through the focus parallel to $AQ$.

\item[76.] If from any point $P$ of a circle, $PC$ be drawn to the centre
$C$, and a chord $PQ$ be drawn parallel to the diameter $AB$, and
bisected in $R$; shew that the locus of the intersection of $CP$ and $AR$
is a parabola.

\item[77.] A circle, the diameter of which is three-fourths of the latus
rectum, is described about the vertex $A$ of a parabola as centre; prove
that the common chord bisects $AS$.

\item[78.] Shew that straight lines drawn perpendicular to the tangents
of a parabola through the points where they meet a given fixed
line perpendicular to the axis are in general tangents to a confocal
parabola.

\item[79.] If $QR$ be a double ordinate, and $PD$ a straight line drawn
parallel to the axis from any point $P$ of the curve, and meeting $QR$
in $D$, prove, from Art.~\cartref{27}, that
\[
  QD \ctrdot RD = 4AS \ctrdot PD.
\]

\item[80.] Prove, by help of the preceding theorem, that, if $QQ'$ be a
chord parallel to the tangent at $P$, $QQ'$ is bisected by $PD$, and hence
determine the locus of the middle point of a series of parallel chords.
%-----File: 065.png-----

\item[81.] If a parabola touch the sides of an equilateral triangle, the
focal distance of any vertex of the triangle passes through the point
of contact of the opposite side.

\item[82.] Find the locus of the foci of the parabolas which have a
common vertex and a common tangent.

\item[83.] From the points where the normals to a parabola meet the
axis, lines are drawn perpendicular to the normals: shew that these
lines will be tangents to an equal parabola.

\item[84.] Inscribe in a given parabola a triangle having its sides
parallel to three given straight lines.

\item[85.] $PNP'$ is a double ordinate, and through a point of the
parabola $RQL$ is drawn perpendicular to $PP'$ and meeting $PA$, or
$PA$ produced in $R$; prove that
\[
PN : NL :: LR : RQ.
\]

\item[86.] $PNP'$ is a double ordinate, and through $R$, a point in the
tangent at $P$, $RQM$ is drawn perpendicular to $PP'$ and meeting the
curve in $Q$; prove that
\[
QM : QR :: P'M : PM.
\]

\item[87.] If from the point of contact of a tangent to a parabola, a
chord be drawn, and a line parallel to the axis meeting the chord,
the tangent, and the curve, shew that this line will be divided by
them in the same ratio as it divides the chord.

\item[88.] $PSp$ is a focal chord of a parabola, $RD$ is the directrix meeting
the axis in $D$, $Q$ is any point in the curve; prove that if $QP$, $Qp$
produced meet the directrix in $R$, $r$, half the latus rectum will be
a mean proportional between $DR$ and $Dr$.

\item[89.] A chord of a parabola is drawn parallel to a given straight
line, and on this chord as diameter a circle is described; prove that
the distance between the middle points of this chord, and of the chord
joining the other two points of intersection of the circle and parabola,
will be of constant length.

\item[90.] If a circle and a parabola have a common tangent at $P$, and
intersect in $Q$ and $R$; and if $QV$, $UR$ be drawn parallel to the axis
of the parabola meeting the circle in $V$ and $U$ respectively, then will
$VU$ be parallel to the tangent at $P$.

\item[91.] If $PV$ be the diameter through any point $P$, $QV$ a semi-ordinate,
$Q'$ another point in the curve, and $Q'P$ cut $QV$ in $R$, and
$Q'R'$, the diameter through $Q'$, meet $QV$ in $R'$, then
\[
VR \ctrdot VR' = QV^2.
\]

\item[92.] $PQ$, $PR$ are any two chords; $PQ$ meets the diameter through
$R$ in the point $F$, and $PR$ meets the diameter through $Q$ in $E$;
prove that $EF$ is parallel to the tangent at $P$.
%-----File: 066.png-----

\item[93.] If parallel chords be intersected by a diameter, the distances
of the points of intersection from the vertex of the diameter are in
the ratio of the rectangles contained by the segments of the chords.

\item[94.] If tangents be drawn to a parabola from any point $P$ in the
latus rectum, and if $Q$, $Q'$ be the points of contact, the semi-latus
rectum is a geometric mean between the ordinates of $Q$ and $Q'$, and
the distance of $P$ from the axis is an arithmetic mean between the
same ordinates.

\item[95.] If $A'$, $B'$, $C'$ be the middle points of the sides of a triangle
$ABC$, and a parabola drawn through $A'$, $B'$, $C'$ meet the sides again
in $A''$, $B''$, $C''$, then will the lines $AA''$, $BB''$, $CC''$ be parallel to each
other.

\item[96.] A circle passing through the focus cuts the parabola in two
points. Prove that the angle between the tangents to the circle at
those points is four times the angle between the tangents to the
parabola at the same points.

\item[97.] The locus of the points of intersection of normals at the
extremities of focal chords of a parabola is another parabola.

\item[98.] Having given the vertex, a tangent, and its point of contact,
construct the parabola.

\item[99.] $PSp$ is a focal chord of a parabola; shew that the distance
of the point of intersection of the normals at $P$ and $p$ from the
directrix varies as the rectangle contained by $PS$, $pS$.

\item[100.] $TP$, $TQ$ are tangents to a parabola at $P$ and $Q$, and $O$ is
the centre of the circle circumscribing $PTQ$; prove that $TSO$ is a
right angle.

\item[101.] $P$ is any point of a parabola whose vertex is $A$, and through
the focus S the chord $QSQ'$ is drawn parallel to $AP$; $PN$, $QM$, $Q'M'$,
being perpendicular to the axis, shew that $SM$ is a mean proportional
between $AM$, $AN$, and that
\[
MM'=AP.
\]

\item[102.] If a circle cut a parabola in four points, two on one side
of the axis, and two on the other, the sum of the ordinates of the
first two is equal to the sum of the ordinates of the other two points.

Extend this theorem to the case in which three of the points are
on one side of the axis and one on the other.

\item[103.] The tangents at $P$ and $Q$ meet in $T$, and $TL$ is the perpendicular
from $T$ on the axis; prove that if $PN$, $QM$ be the ordinates
of $P$ and $Q$,
\[
PN \ctrdot QM=4AS \ctrdot AL.
\]

\item[104.] The tangents at $P$ and $Q$ meet in $T$, and the lines $TA$, $PA$,
$QA$, meet the directrix in $t$, $p$, and $q$: prove that
\[
tp=tq.
\]

%-----File: 067.png-----
\item[105.] From a point $T$ tangents $TP$, $TQ$ are drawn to a parabola,
and through $T$ straight lines are drawn parallel to the normals at $P$
and $Q$; prove that one diagonal of the parallelogram so formed passes
through the focus.

\item[106.] Through a given point within a parabola draw a chord which
shall be divided in a given ratio at that point.

\item[107.] $ABC$ is a portion of a parabola bounded by the axis $AB$ and
the semi-ordinate $BC$; find the point $P$ in the semi-ordinate such that
if $PQ$ be drawn parallel to the axis to meet the parabola in $Q$, the sum
of $BP$ and $PQ$ shall be the greatest possible.

\item[108.] The diameter through a point $P$ of a parabola meets the
tangent at the vertex in $Z$; the normal at $P$ and the focal distance
of $Z$ will intersect in a point at the same distance from the tangent
at the vertex as $P$.

\item[109.] Given a tangent to a parabola and a point on the curve,
shew that the foot of the ordinate of the point of contact of the
tangent drawn to the diameter through the given point lies on a fixed
straight line.

\item[110.] Find a point such that the tangents from it to a parabola
and the lines from the focus to the points of contact may form a
parallelogram.

\item[111.] Two equal parabolas have a common focus; and, from any
point in the common tangent, another tangent is drawn to each; prove
that these tangents are equidistant from the common focus.

\item[112.] Two parabolas have a common axis and vertex, and their
concavities turned in opposite directions; the latus rectum of one is
eight times that of the other; prove that the portion of a tangent to
the former, intercepted between the common tangent and axis, is
bisected by the latter.

\end{exampleslist}
%-----File: 068.png-----

\cchapter{CHAPTER III}

\csectiontoc{The Ellipse}
\cheader{THE ELLIPSE}

\subparagraph{Def.} \textit{An ellipse is the curve traced out by a point which
moves in such a manner that its distance from a given point is
in a constant ratio of less inequality to its distance from a
given straight line.}

\cisubsection{Tracing the Curve}

\carticle{55.} Let $S$ be the focus, $EX$ the directrix, and $SX$ the
perpendicular on $EX$ from $S$.

\includegraphicsmid{055a}

Divide $SX$ at the point $A$ in the given ratio; the point
$A$ is the vertex.

From any point $E$ in $EX$, draw $EAP$, $ESL$, and through
$S$ draw $SP$ making the angle $PSL$ equal to $LSN$, and
meeting $EAP$ in $P$.

Through $P$ draw $LPK$ perpendicular to the directrix and
meeting $ESL$ in $L$.
%-----File: 069.png-----
\begin{flalign*}
&\indent\text{Then the angle}&
             PSL &= LSN = SLP.  &&\phantom{\indent\text{Then the angle}}\\
&& \therefore SP &= PL.         &&
\end{flalign*}
\begin{flalign*}
&\text{Also}&     PL: PK &:: SA: AX. &&\\
&\indent\text{Hence}&
   SP: PK &:: SA: AX,  &&\phantom{\indent\text{Hence}}
\end{flalign*}
and $P$ is therefore a point in the curve.

Again, in the axis $XAN$ find a point $A'$ such that
\[
SA': A'X:: SA: AX;
\]
this point is evidently on the same side of the directrix as
the point $A$, and is another vertex of the curve.

\includegraphicsmid{055a} %repeated figure

Join $EA'$ meeting $PS$ produced in $P'$, and draw $P'L'K'$
perpendicular to the directrix and meeting $ES$ in $L'$.
\begin{flalign*}
&\indent\text{Then}&
  P'L': P'K'& :: SA': A'X  &&\phantom{\indent\text{Then}}\\
&&                 &:: SA\phantom{'} : AX, &&
\end{flalign*}
\begin{flalign*}
&\text{and the angle}&   SL'P' = L'SA &= L'SP; &&\phantom{\text{and the angle}}\\
&&\therefore P'L' = & SP'. &&
\end{flalign*}

Hence $P'$ is also a point in the curve, and $PSP'$ is a focal
chord.

By giving $E$ a series of positions on the directrix we
shall obtain a series of focal chords, and we can also, as in
Art.~(\cartref{1}), find other points of the curve lying in the lines
$KP$, $K'P'$, or in these lines produced.

We can thus find any number of points in the curve.
%-----File: 070.png-----

\carticleplus{56.}{Def.} \textit{The distance $AA'$ is the major axis.}

\textit{The middle point $C$ of $AA'$ is called the centre of the
ellipse.}

\textit{If through $C$ the double ordinate $BCB'$ be drawn, $BB'$ is
called the minor axis.}

\textit{Any straight line drawn through the centre, and terminated
by the curve, is called a diameter.}

\textit{The lines $ACA'$, $BCB'$ are called the principal diameters,
or, briefly, the axes of the curve.}

\textit{The line $ACA'$ is also sometimes called the transverse
axis, and $BCB'$ the conjugate axis.}

\carticleplus{57.}{Prop.~I\@.} \textit{If $P$ be any point of an ellipse, and $AA'$
the axis major, and if $PA$, $A'P$, when produced, meet the
directrix in $E$ and $F$, the distance $EF$ subtends a right angle
at the focus.}

\includegraphicsmid{057a}

By the theorem of Art.~\cartref{4}, $ES$ bisects the angle $ASP'$,
and $FS$ bisects the angle $ASP$;
\[
\therefore ESF\text{ is a right angle.}
\]

It will be seen that, since $ASA'$ is a focal chord, this is a
particular case of the theorem of Art.~\cartref{6}.
%-----File: 071.png-----

\carticleplus{58.}{Prop.~II.} \textit{If $PN$ be the ordinate of any point
$P$ of an ellipse, $ACA'$ the axis major, and $BCB'$ the axis
minor,}
\[
PN^2 : AN \ctrdot NA' :: BC^2 : AC^2.
\]
\includegraphicsmid{058a}
Join $PA$, $A'P$, and let these lines produced meet the
directrix in $E$ and $F$.
\begin{flalign*}
&\indent\text{Then} &      PN : AN  &:: EX : AX,  &&\phantom{\indent\text{Then}}\\
&\text{and }          &      PN : A'N &:: FX : A'X;  &&\\
&& \therefore PN^2 : AN \ctrdot NA' &:: EX \ctrdot FX : AX \ctrdot A'X  &&\\
&&                                    &:: SX^2 : AX \ctrdot A'X,  &&
\end{flalign*}
since $ESF$ is a right angle (Prop.~1.); that is, $PN^2$ is to
$AN \ctrdot NA'$ in a constant ratio.

Hence, taking $PN$ coincident with $BC$, in which case
\begin{flalign*}
&&&AN = NA' = AC,&&\\
&&                        BC^2 &: AC^2 :: SX^2 : AX \ctrdot A'X, &&\\
&\rlap{and } & \therefore PN^2 &: AN \ctrdot NA' :: BC^2 : AC^2.
\end{flalign*}

This may be also written
\[
PN^2 : AC^2 - CN^2  :: BC^2 : AC^2.
\]

\subparagraph{Cor.} If $PM$ be the perpendicular from $P$ on the axis
minor,
\begin{flalign*}
&&  &  CM = PN, PM = CN,  &&\\
&\text{and } &   CM^2 &: AC^2 - PM^2  ::  BC^2 : AC^2.  &&\\
&\text{\indent Hence }&  AC^2 &: AC^2 - PM^2  ::  BC^2 : CM^2,  &&\phantom{\indent Hence }\\
&\text{and } &  \therefore  AC^2 &: PM^2  ::  BC^2 : BC^2 - CM^2, &&\\
&\text{or } &   PM^2 &: BM \ctrdot MB'  ::  AC^2 : BC^2.  &&
\end{flalign*}
%-----File: 072.png-----

\carticle{59.} If a point $N'$ be taken on the axis major, between
$C$ and $A'$, such that $CN' = CN$, the corresponding ordinate
$P'N' = PN$, and therefore it follows that the curve is symmetrical
with regard to $BCB'$, and that there is another
focus, and another directrix, corresponding to the vertex $A'$.

\carticle{60.} By help of the theorem of Art.~\cartref{57}, we can give an
independent proof of the existence of the other focus and
directrix, corresponding to the vertex~$A'$.

In $AA'$ produced take a point $X'$ such that $A'X' = AX$,
and in $AA'$ take a point $S'$ such that $A'S' = AS$.

Through $X'$ draw a straight line $eX'f$ perpendicular to
the axis, and let $EP$, $FP$ produced meet this line in $e$ and $f$.
Join $eS'$, and $fS'$.
\includegraphicsmid{060a}

\begin{flalign*}
&\text{\indent Then }
 &  eX' : EX  &:: AX' : AX    &&\phantom{\indent Then}  \\
&&            &:: A'X : A'X'  &&\\
&&            &:: FX  : fX';  &&\\
&&  \therefore eX' \ctrdot fX' = EX &\ctrdot FX = SX^2 = S'X'^2. &&
\end{flalign*}
%-----File: 073.png-----

Hence $eS'f$ is a right angle.

Through $P$ draw $KPk$ parallel to the axis, meeting $eS'$
and $fS'$ produced in $L$ and $l$.
\begin{flalign*}
&\text{\indent Then }& PL &: Pk :: S'A : AX' :: SA' : A'X, &&\phantom{\indent Then}\\
&\text{and }&          Pl &: Pk :: S'A' : A'X' :: SA : AX, &&
\end{flalign*}
\[
\therefore PL = Pl.
\]

Moreover, $LS'l$ being a right angle,
\begin{gather*}
S'P=Pl,\\
\therefore S'P : Pk :: S'A' : A'X',
\end{gather*}
and the curve can be described by means of the focus $S'$
and the directrix $eX'$.

\begin{small}
If $SA$ be equal to $AX$, the point $A'$, and therefore the points $S'$ and
$X'$, will be at an infinite distance from $S$ and $A$.

Hence a parabola is the limiting form of an ellipse, the axis major
of which is indefinitely increased in magnitude, while the distance $SA$
remains finite.

\end{small}

\carticleplus{61.}{Prop.~III\@.} \textit{If $ACA'$ be the axis major, $C$ the centre,
$S$ one of the foci, and $X$ the foot of the directrix},
\begin{flalign*}
&&             CS &: CA :: CA : CX :: SA : AX,  &&\\
&\text{and }&  CS &: CX :: CS^2 : CA^2.         &&\phantom{and}
\end{flalign*}

\includegraphicsmid{061a}

\begin{flalign*}
&\text{\indent For }&     S'A : SA &:: AX' : AX   &&\\
&&                                 &:: A'X : AX;  &&\\
&&             \therefore SS' : SA &:: AA' : AX,  &&\\
&\text{or }&              CS  : CA &:: SA\ : AX.  &&\\
&\text{\indent Again, }&  SA' : SA &:: AX' : AX;  &&\phantom{\indent Again,}\\
&&             \therefore AA' : SA &:: XX' : AX,  &&\\
&\text{or }&              CA  : CX &:: SA\ : AX;  &&\\
&&             \therefore CS  : CA &:: CA\ : CX,  &&\\
&\text{or }&         CS \ctrdot CX &= CA^2.  &&\\
&\text{\indent Also }&     CS : CX &:: CS^2 : CS \ctrdot CX  &&\phantom{\indent Also}\\
&&                                 &:: CS^2 : CA^2.  &&
\end{flalign*}
%-----File: 074.png-----

\carticleplus{62.}{Prop.~IV\@.} \textit{If $S$ be a focus, and $B$ an extremity of
the axis minor},
\[
  SB = AC \text{ and } BC^2 = AS \ctrdot SA'.
\]

For, joining $SB$ in the figure of Art.~\cartref{58},
\begin{align*}
  SB: CX &:: SA: AX  \\
         &:: CA: CX,
\end{align*}
by the previous Article,
\[
\therefore SB = CA.
\]
\begin{flalign*}
&\text{\indent Also}&   BC^2 = SB^2 - SC^2 &= AC^2 - SC^2  &\phantom{\indent Also}&\\
&&            &= AS \ctrdot SA'. &&
\end{flalign*}

\carticleplus{63.}{Prop.~V\@.} \textit{The semi-latus rectum $SR$ is a third
proportional to $AC$ and $BC$.}

For, Prop.\ II.,
\begin{flalign*}
&&  SR^2 : AS \centerdot SA' &:: BC^2 : AC^2;      &&\\
&&\therefore SR^2 : BC^2 &:: BC^2 : AC^2, &&\\
&\text{or}&        SR : BC &:: BC : AC.   &&\\
&\text{\calignedcor{Cor.} Since}&
    SR : SX &:: SA : AX &\phantom{\calignedcor{Cor.} Since}&\\
&&          &:: SC : AC, &&
\end{flalign*}
it follows that $SX \ctrdot SC = SR \ctrdot AC = BC^2$;\\
and hence also, since  $SC \ctrdot CX = AC^2$, that
\[
  SX : CX :: BC^2 : AC^2.
\]

\carticleplus{64.}{Prop.~VI\@.} \textit{The sum of the focal distances of any
point is equal to the axis major.}

Let $PN$ be the ordinate of a point $P$ (Fig.\ Art.~\cartref{60}), then
	\begin{flalign*}
&&                    S'P : SP&:: NX' : NX; &&\\
&&  \therefore  S'P + SP : SP &:: XX' : NX, &&\\
&\text{or}&    S'P + SP : XX' &:: SP : NX   &\phantom{or}&\\
&&                            &:: SA : AX   &&\\
&&                            &:: AA': XX'; &&\\
&&        \therefore S'P + SP &= AA'        &&
\end{flalign*}
%-----File: 075.png-----
\begin{flalign*}
&\text{\calignedcor{Cor.} Since}&
            SP: NX &:: SA: AX    &&\phantom{\calignedcor{Cor.} Since}\\
&&                 &:: AC: CX;   &&\\
&&\therefore AC: SP&:: CX: NX,   &&\\
&&           \llap{$AC-SP$} : SP&:: CN: NX,   &&\\
&\text{and}& \llap{$AC-SP$} : CN&:: SA: AX.  &&
\end{flalign*}

Also, \hfill $AC-SP = S'P-AC$; \hfill\phantom{\indent Also, }
\begin{flalign*}
&&   \therefore S'P-AC: CN &:: SA: AX. &&\\
&\indent\text{Hence,}&  S'P-SP: 2CN &:: SA: AX. &&\phantom{\indent\text{Hence,}}
\end{flalign*}

\cisubsection{Mechanical Construction of the Ellipse}

\carticle{65.} Fasten the ends of a piece of thread to two pins
fixed on a board, and trace a curve on the board with a
pencil pressed against the thread so as to keep it stretched;
the curve traced out will be an ellipse, having its foci at the
points where the pins are fixed, and having its major axis
equal to the length of the thread.

\carticleplus{66.}{Prop.~VII\@.}  \textit{The sum of the distances of a point
from the foci of an ellipse is greater or less than the major
axis according as the point is outside or inside the ellipse.}

If the point be without the ellipse, join $SQ$, $S'Q$, and
take a point $P$ on the intercepted arc of the curve.

\includegraphicsright[8]{066a}

Then $P$ is within the triangle $SQS'$ and therefore, joining
$SP$, $S'P$,
\begin{flalign*}
&&SQ + S'Q &> SP+S'P, \text{ Euclid \textsc{i}.\ 21,}  &&\\
&\rlap{\textit{i.e.}}&  &SQ + S'Q>AA'. &&
\end{flalign*}

If $Q'$ be within the ellipse, let
$SQ'$, $S'Q'$ produced meet the curve
and take a point $P$ on the intercepted
arc.

Then $Q'$ is within the triangle
$SPS'$, and
\begin{flalign*}
&&\therefore SP + S'P &>SQ' + S'Q', &&\\
&\textit{i.e.}&       SQ' + S'&Q'<AA'. &&\phantom{i.e.}
\end{flalign*}
%-----File: 076.png-----

\carticleplus{67.}{Def.} \textit{The circle described on the axis major as
diameter is called the auxiliary circle.}

\subparagraph{Prop.~VIII.} \textit{If the ordinate $NP$ of an ellipse be produced
to meet the auxiliary circle in $Q$,}
\[
       PN : QN :: BC : AC.
\]

\includegraphicsmid{067a}

For (Art.~\cartref{58})
\[
       PN^2 :  AN \ctrdot NA' :: BC^2 : AC^2,
\]
and, by a property of the circle,
\[
       QN^2 = AN \ctrdot NA';
\therefore  PN : QN :: BC : AC.
\]

\subparagraph{Cor.} Similarly, if $PM$, the perpendicular on $BB'$, meet
in $Q'$ the circle described on $BB'$ as diameter,
\[
PM : Q'M :: AC : BC.\\
\]
\begin{flalign*}
&\text{\indent For }&  PM^2 : BM & \ctrdot MB' :: AC^2 : BC^2, &&\phantom{\indent For}\\
&\text{and }&                 BM & \ctrdot MB' = Q'M^2.
\end{flalign*}
%-----File: 077.png-----

\cisubsection{Properties of the Tangent and Normal}

\carticleplus{68.}{Prop.~IX\@.} \textit{The normal at any point bisects the
angle between the focal distances of that point, and the
tangent is equally inclined to the focal distances.}

Let the normal at $P$ meet the axis in $G$; then (Art.~\cartref{18})
\begin{flalign*}
&&              SG  : SP  &:: SA : AX, &&\phantom{and }\\
&\text{and}&    S'G : S'P &:: SA : AX. &&
\end{flalign*}
\includegraphicsmid{068a}
\begin{flalign*}
&\text{Hence}&     SG : S'G &:: SP : S'P, &&\phantom{Hence }
\end{flalign*}
and therefore the angle $SPS'$ is bisected by $PG$.

Also $FPF'$ being the tangent, and $GPF$, $GPF'$ being
right angles, it follows that the angles $SPF$, $S'PF'$ are equal,
or that the tangent is equally inclined to the focal distances.

Hence if $S'P$ be produced to $L$, the tangent bisects the
angle $SPL$.

\subparagraph{Cor.} If a circle be described about the triangle $SPS'$,
its centre will lie in $BCB'$, which bisects $SS'$ at right angles;
and since the angles $SPG$, $S'PG$ are equal, and equal angles
stand upon equal arcs, the point $g$, in which $PG$ produced
meets the minor axis, is a point in the circle.

Also, if the tangent meet the minor axis in $t$, the point $t$
is on the same circle, since $gPt$ is a right angle.
%-----File: 078.png-----

Hence, \textit{Any point $P$ of an ellipse, the two foci, and the
points of intersection of the tangent and normal at $P$ with the
minor axis are concyclic.}

\carticleplus{69.}{Prop.~X\@.} \textit{Every diameter is bisected at the centre,
and the tangents at the ends of a diameter are parallel.}

Let $PCp$ be a diameter, $PN$, $pn$ the ordinates of $P$
and $p$.
\begin{flalign*}
&\indent\text{Then}& CN^2 : Cn^2 &:: PN^2 : pn^2  &&\phantom{\indent\text{Then}}\\
&&                                           &:: AC^2 - CN^2 : AC^2 - Cn^2 \text{ (Art.~\cartref{58})}; &&\\
&&\therefore CN^2 &: AC^2 :: Cn^2 : AC^2. &&
\end{flalign*}
\begin{flalign*}
&\indent\text{Hence}&   CN = Cn &\text{ and } \therefore CP = Cp. &&\phantom{\indent\text{Hence}}
\end{flalign*}

Draw the focal distances; then, since $Pp$ and $SS'$ bisect
each other in $C$, the figure $SPS'p$ is a parallelogram, and the
angle
\[
SPS' = SpS'.
\]

\includegraphicsmid{069a}

But the tangents $PT$, $pt$ are equally inclined to the focal
distances;
\[
\therefore \text{ the angle } SPT=S'pt,
\]
and, adding the equal angles $CPS$, $CpS'$,
\begin{gather*}
CPT=Cpt; \\
\therefore PT \text{ and } pt \text{ are parallel}.
\end{gather*}

\subparagraph{Cor.} Since $Sp$ and $S'p$ are equally inclined to the
tangent at $p$, it follows that $SP$ and $Sp$ make equal angles
with the tangents at $P$ and $p$.
%-----File: 079.png-----

\carticleplus{70.}{Prop.~XI\@.} \textit{The perpendiculars from the foci on
any tangent meet the tangent on the auxiliary circle, and
the semi-minor axis is a mean proportional between their
lengths.}

Let $SY$, $S'Y'$ be the perpendiculars; join $S'P$, and let
$SY$, $S'P$ produced meet in $L$.

The angles $SPY$, $YPL$ being equal, and $PY$ being
common, the triangles $SPY$, $YPL$ are equal in all respects;
\begin{flalign*}
&&  \multispan{1}{\hfill$\therefore PL = SP$, $SY=YL,$\hfill} &&&\\
&\text{and}&   \multispan{1}{\hfill$S'L = S'P + PL = S'P +SP = AA'$.\hfill} &&&\phantom{and}
\end{flalign*}
Join $CY$, then $C$ being the middle point of $SS'$, and $Y$
of $SL$, $CY$ is parallel to $S'L$,
\begin{flalign*}
&\text{and}&     \therefore  S'L &= 2CY. &&\phantom{and}
\end{flalign*}
Hence $CY= AC$, and $Y$ is a point on the auxiliary circle.

\includegraphicsmid{070a}

Similarly by producing $SP$, $S'Y'$ it may be shewn that
$Y'$ is also on the auxiliary circle.

Let $YS$ produced meet the circle in $Z$, and join $Y'Z$;
then $Y'YZ$ being a right angle, $Y'Z$ is a diameter and
passes through $C$.

Hence the triangles $SCZ$, $S'CY'$ are equal, and
\[
SY\ctrdot S'Y' = SY\ctrdot SZ=AS\ctrdot SA' = BC^2.
\]
%-----File: 080.png-----

\subparagraph{Cor.~(1).} If $P'$ be the other extremity of the diameter
through $P$, the tangent at $P'$ is parallel to $PY$, and therefore
$Z$ is the foot of the perpendicular from $S$ on the tangent
at $P'$.

%\includegraphicsmid{070b} %omitted since identical to 070a and on same page

\subparagraph{Cor.~(2).} If the diameter $DCD'$, drawn parallel to the
tangent at $P$, meet $SP$, $S'P$ in $E$ and $E'$, $PECY'$ is a
parallelogram, for $CY'$ is parallel to $SP$, and $CE$ to $PY'$;
\[
  \therefore PE = CY' = AC; \text{ and similarly } PE' = CY= AC.
\]

\subparagraph{Cor.~(3).} Any diameter parallel to the focal distance of
a point meets the tangent at the point on the auxiliary
circle.

\carticleplus{71.}{Prop.~XII\@.} \textit{To draw tangents from a given point to
an ellipse.}

For this purpose we may employ the general construction
of Art.~(\cartref{17}), or the following.

Let $Q$ be the given point; upon $SQ$ as diameter describe
a circle cutting the auxiliary circle in $Y$ and $Y'$; $YQ$ and
$Y'Q$ will be the required tangents.

Producing $SY$ to $L$ so that $YL = SY$, join $S'L$ cutting
the line $YQ$ in~$P$.

The triangles $SPY$, $LPY$ are equal in all respects,
%-----File: 081.png-----
since $SY = YL$ and $PY$ is common and perpendicular to
$SL$;
\[
    \therefore  SP = PL, \text{ and } S'L = S'P + PL = S'P + SP;
\]
but, joining $CY$, $S'L = 2CY = 2AC$;
\[
    \therefore  SP + S'P = 2AC,
\]
and $P$ is therefore a point on the ellipse.

\includegraphicsmid{071a}

Also the angle $SPY = YPL$,
\begin{flalign*}
&\text{and}&   \multispan{1}{\hfill $ \therefore  QP$ is the tangent at $P$.\hfill} &&&\phantom{and }
\end{flalign*}

A similar construction will give the point of contact of
the other tangent $QP'$.

\begin{small}
Referring to Art.~\cartref{35} it will be seen that the construction is the
same as that given for the parabola, the ultimate form of the circle
being, for the parabola, the tangent at the vertex.

\end{small}

\carticleplus{72.}{Prop.~XIII\@.} \textit{If two tangents be drawn to an ellipse
from an external point, they are equally inclined to the focal
distances of that point.}

Let $QP$, $QP'$ be the tangents,
$SY$, $S'Y'$, $SZ$, $S'Z'$ the perpendiculars
from the foci on the tangents;
join $YZ$, $Y'Z'$.

Then (Art.~\cartref{70})
\begin{gather*}
    SY\ctrdot S'Y' = SZ\ctrdot S'Z'; \\
    \therefore  SY : SZ :: S'Z' : S'Y'.
\end{gather*}
%-----File: 082.png-----

\includegraphicsright[9]{072a}

The points $S$, $Y$, $Q$, $Z$ being concyclic, the angles $YSZ$,
$YQZ$ are supplementary; and similarly, $Z'S'Y'$, $Z'QY'$ are
supplementary.

Therefore the angle $YSZ = Z'S'Y'$ and the triangles
$YSZ$, $Z'S'Y'$ are similar.

Therefore the angle $SQP = SZY = S'Y'Z' = S'QP'$.

\begin{small}
\carticleplus{73.}{Def.} \textit{Ellipses which have the same foci are called confocal
ellipses.}

If $Q$ be a point in a confocal ellipse the normal at $Q$ bisects the
angle $SQS'$ and therefore bisects the angle $PQP'$.

Hence, \textit{If from any point of an ellipse tangents are drawn to a
confocal ellipse, these tangents are equally inclined to the normal at the
point.}

\end{small}

By reference to the remark of Art.~\cartref{41}, it will be seen
that this theorem includes that of Art.~\cartref{41} as a particular
case.

\carticleplus{74.}{Prop.~XIV\@.} If $PT$ the tangent at $P$ meet the axis
major in $T$, and $PN$ be the ordinate,
\[
CN\ctrdot CT = AC^2.
\]

\includegraphicsmid{074a}
Draw the focal distances $SP$, $S'P$, and the perpendicular
$SY$ on the tangent, and join $NY$, $CY$.

Then, as in Art.~\cartref{70}, $CY$ is parallel to $S'P$; therefore the
angle
\begin{align*}
CYP &= S'Pt = SPY \\
    &= SNY,
\end{align*}
since $S$, $Y$, $P$, $N$ are concyclic.
\begin{flalign*}
&\indent\text{Hence}&     CYT &= CNY, &&\phantom{\indent\text{Hence}}
\end{flalign*}
and the triangles $CYT$, $CNY$ are equiangular.
%-----File: 083.png-----

\begin{flalign*}
&\indent\text{Therefore}&   CN &: CY :: CY: CT &\phantom{\indent\text{Therefore}}&\\
&\text{or}&      CN &\ctrdot CT = CY^2 = AC^2. &&
\end{flalign*}
\begin{flalign*}
\text{\calignedcor{Cor.~(1).}} CN \ctrdot NT &= CN \ctrdot CT - CN^2 = AC^2 - CN^2&\\
&= AN \ctrdot NA'.&
\end{flalign*}

\subparagraph{Cor.~(2).} Hence it follows that \textit{tangents at the extremities
of a common ordinate of an ellipse and its auxiliary
circle meet the axis in the same point.}

For, if $NP$ produced meet the auxiliary circle in $Q$, and
the tangent at $Q$ meet the axis in $T'$,
\[
CN \ctrdot NT' = CQ^2 = AC^2,
\]
therefore $T'$ coincides with $T$.

And more generally it is evident that, \textit{If any number of
ellipses be described having the same major axis, and an ordinate
be drawn cutting the ellipses, the tangents at the points of
section will all meet the common axis in the same point.}

\carticleplus{75.}{Prop.~XV\@.} \textit{If the tangent at $P$ meet the axis minor
in $t$, and $PN$ be the ordinate,}
\[
Ct \ctrdot PN = BC^2.
\]
\begin{flalign*}
&\indent\text{For,}&  Ct: PN &:: CT: NT \text{ (Fig.\ Art.~\cartref{74})}, &&\\
&&\therefore Ct \ctrdot PN : PN^2 &:: CT \ctrdot CN : CN \ctrdot NT &&\\
&& &:: AC^2 : AN \ctrdot NA' \text{ (Cor.\ 1, Art.~\cartref{74})}, &&\\
&& &::BC^2 : PN^2. &&\\
&& \multispan{2}{\hfill$\therefore Ct \ctrdot PN = BC^2$.\hfill} &&
\end{flalign*}

\carticleplus{76.}{Prop.~XVI\@.} \textit{If the tangent and normal at $P$ meet
the axis major in $T$ and $G$,}
\[
CG \ctrdot CT = SC^2.
\]

The triangles $CGg$, $CTt$, in the figure of the next article,
being similar,
\begin{gather*}
CG: Cg:: Ct: CT, \\
\therefore CG \ctrdot CT = Cg \ctrdot Ct.
\end{gather*}

But, since $t$, $S$, $g$, $S'$ are concyclic (Cor.\ Art.~\cartref{68}),
\begin{gather*}
Cg \ctrdot Ct = SC \ctrdot CS' = SC^2;  \\
\therefore CG \ctrdot CT = SC^2.
\end{gather*}
%-----File: 084.png-----

\subparagraph{Cor.} Since $CN\ctrdot CT=AC^2$, and $PN\ctrdot Ct = BC^2$,
\begin{flalign*}
&&  CG: CN &:: SC^2 : AC^2 &&\\
&\text{and}&      Cg: PN&:: SC^2 : BC^2. &&\phantom{and}
\end{flalign*}

We hence see that
\[
NG: CN:: BC^2 : AC^2.
\]

\carticleplus{77.}{Prop.~XVII\@.} \textit{If the normal at $P$ meet the axes in $G$
and $g$, and the diameter parallel to the tangent at $P$ in $F$,}
\[
PF\ctrdot PG = BC^2, \text{ \textit{and} } PF\ctrdot Pg = AC^2.
\]

Let $PN$, $PM$, perpendiculars on the axes, meet the
diameter in $K$ and $L$, and let the tangent at $P$ meet the axes
in $T$ and $t$.

\includegraphicsmid{077a}

Then, since $G$, $F$, $K$, $N$ are concyclic,
\[
PF\ctrdot PG = PN\ctrdot PK = PN\ctrdot Ct = BC^2.
\]

Similarly, since $L$, $M$, $F$, $g$ are concyclic,
\[
PF\ctrdot Pg = PM\ctrdot PL = CN\ctrdot CT=AC^2.
\]

\subparagraph{Cor.} If $SP$, $S'P$ meet the diameter $DCD'$ parallel to the
tangent at $P$ in $E$ and $E'$,
\begin{gather*}
PE = AC \text{ (Cor.\ 2, Art.~\cartref{70})};  \\
\therefore PF\ctrdot Pg = PE^2 =PE'^2,
\end{gather*}
and hence it follows that the angles $PEg$, $PE'g$ are right
angles.
%-----File: 085.png-----

\carticleplus{78.}{Prop.~XVIII\@.} \textit{If $PCp$ be a diameter, $QVQ'$ a chord
parallel to the tangent at $P$ and meeting $Pp$ in $V$, and if the
tangent at $Q$ meet $pP$ produced in $T$,}
\[
CV \ctrdot CT= CP^2.
\]

\includegraphicsmid{078a}

Let $TQ$ meet the tangents at $P$ and $p$ in $R$ and $r$, and $S$
being a focus, join $SP$, $SQ$, $Sp$.

Let fall perpendiculars $RN$, $RM$, $rn$, $rm$ upon these focal
distances;
then, since the angle $SPR=Spr$ (Cor.\ Art.~\cartref{69}),
\begin{align*}
RP : rp &:: RN : rn\\
&:: RM : rm \text{ (Cor.\ Art.~\cartref{15}),}\\
&:: RQ : rQ;\\
&:: PV : Vp.
\end{align*}
\begin{flalign*}
&\rlap{\indent Hence }&   TP : Tp &:: PV : Vp,  &&\\
&\text{or }&        CT-CP : CT+CP &:: CP-CV : CP+CV; &&\phantom{or}\\
&&             \therefore CT : CP &:: CP : CV,  &&\\
&\text{or }&     CT \ctrdot CV & = CP^2.  &&
\end{flalign*}

\subparagraph{Cor.~1.} Hence, since $CV$ and $CP$ are the same for the
point $Q'$, the tangent at $Q'$ passes through $T$.

\subparagraph{Cor.~2.}  Since $Tp : TP :: pV : VP$, it follows that
$TPVp$ is harmonically divided.

It will be seen in a subsequent chapter that this is a particular
case of a general theorem.
%-----File: 086.png-----

\cisubsection{Properties of Conjugate Diameters}

\carticleplus{79.}{Prop.~XIX\@.} \textit{A diameter bisects all chords parallel
to the tangents at its extremities.}

We have shewn in Art.~\cartref{21}, that, if $QQ'$ be a chord of a
conic, $TQ$, $TQ'$ the tangents at $Q$, $Q'$, and $EPE'$ a tangent
parallel to $QQ'$, the length $EE'$ is bisected at $P$.

Draw the diameter $PCp$; the tangent $epe'$ at $p$ is parallel
to $EPE'$ (Art.~\cartref{69}), and is therefore parallel to $QQ'$.

\includegraphicsmid{079a}

Hence $ep=pe'$, and $P$, $p$ being the middle points of the
parallels $ee'$, $EE'$ the line $Pp$ passes through $T$, and moreover
bisects $QQ'$.

Similarly, if any other chord $qq'$ be drawn parallel to $QQ'$
the tangents at $q$ and $q'$ will meet in $pP$ produced, and $qq'$
will be bisected by $pP$.

\subparagraph{Cor.} Hence, if $QQ'$, $qq'$ be two chords parallel to the
tangent at $P$, the chords $Qq$, $Q'q'$ will meet in $CP$ or $CP$ produced.

\carticleplus{80.}{Def.} \textit{The diameter $DCd$, drawn parallel to the
tangent at $P$, is said to be conjugate to $PCp$.}

A diameter therefore bisects all chords parallel to its
conjugate.

\subparagraph{Prop.~XX\@.} \textit{If the diameter $DCd$ be conjugate to $PCp$,
then will $PCp$ be conjugate to $DCd$.}

Let the chord $QVq$ be parallel to $DCd$, and therefore
bisected by $PC$, and draw the diameter $qCR$.
%-----File: 087.png-----

\includegraphicsright[8]{080a}

Join $QR$ meeting $CD$ in $U$;
then $RC = Cq$, and $QV = Vq$;

$\therefore  QR$ is parallel to $CP$.

Also  $QU : UR :: qC : CR$,
and therefore $QU = UR$.

That is, $CD$ bisects the chords parallel to $PCp$; therefore
$PCp$ is conjugate to $DCd$.

\subparagraph{Def.} \textit{Chords drawn from the extremities of a diameter to
any point of the ellipse are called supplemental chords.}

Thus $qQ$, $RQ$ are supplemental chords, and hence it
appears that supplemental chords are parallel to conjugate
diameters.

\subparagraph{Def.} \textit{A line $QV$ drawn from a point $Q$ of an ellipse,
parallel to the tangent at $P$ and terminated by the diameter
$PCp$, is called an ordinate of that diameter, and $QVq$ is the
double ordinate if $QV$ produced meet the curve in $q$.}

\carticle{81.} \textit{Any diameter is a mean proportional between the
transverse axis and the focal chord parallel to the diameter.}

\includegraphicsmid{081a}

From Art.~\cartref{70}, it appears that if $CQT$ parallel to $SP$ meet
in $T$ the tangent at $P$,
\[
      CT = AC.
\]

Draw $PV$ parallel to the tangent at $Q$;
\[
  \text{then } CQ^2 = CV \ctrdot CT = CV \ctrdot AC;
\]
%-----File: 088.png-----
but the diameter through $C$ parallel to the tangent at $Q$
bisects $Pp$ (Art.~\cartref{80}),
\begin{gather*}
\text{so that } Pp=2CV;\\
\therefore Qq^2 = Pp \ctrdot AA'.
\end{gather*}

\carticleplus{82.}{Prop.~XXI\@.} \textit{If $PCp$, $DCd$ be conjugate diameters,
and $QV$ an ordinate of $Pp$,}
\[
QV^2 : PV \ctrdot Vp :: CD^2 : CP^2.
\]

Let the tangent at $Q$ (Fig.\ Art.~\cartref{80}) meet $CP$, $CD$ produced
in $T$ and $t$, and draw $QU$ parallel to $CP$ and meeting $CD$ in
$U$.

\begin{flalign*}
&\text{\indent Then }&       CP^2 &= CV \ctrdot CT, &&\phantom{\indent Then}\\
&\text{and }&           CD^2 = CU &\ctrdot Ct = QV \ctrdot Ct; &&\\
&&         \therefore CD^2 : CP^2 &:: QV \ctrdot Ct : CV \ctrdot CT &&\\
&&                                &:: QV^2 : CV \ctrdot VT,  &&\\
&\text{and }&  CV \ctrdot VT = CV & \ctrdot CT-CV^2 = CP^2 - CV^2  &&\\
&&                           = PV & \ctrdot Vp,  &&\\
&&         \therefore CD^2 : CP^2 &:: QV^2 : PV \ctrdot Vp.
\end{flalign*}

\carticleplus{83.}{Prop.~XXII\@.} \textit{If $ACA'$, $BCB'$ be a pair of conjugate
diameters, $PCP'$, $DCD'$ another pair, and if $PN$, $DM$ be
ordinates of $ACA'$,}
\begin{flalign*}
&&  CN^2 = AM \ctrdot MA', &\quad CM^2 = AN \ctrdot NA',  &&\\
&&             CM : PN &:: AC : BC,  &&\\
&\text{and }&  DM : CN &:: BC : AC.  &&\phantom{and }
\end{flalign*}

\includegraphicsmid{083a}

Let the tangents at $P$ and $D$ meet $ACA'$ in $T$ and $t$.
%-----File: 089.png-----

\begin{flalign*}
&\indent\text{Then }&
\multispan{2}{\hfill $CN\ctrdot CT = AC^2 = CM\ctrdot Ct$\hfill}; &&\phantom{\indent\text{Then }}\\
&\text{hence }& \qquad  CM : CN &:: CT : Ct &&\\
&&                 &:: PT : CD &&\\
&&                 &:: PN : DM &&\\
&&                 &:: CN : Mt, &&
\end{flalign*}
\begin{flalign*}
&&        \therefore  CN^2 = CM\ctrdot Mt &= AC^2 - CM^2 = AM\ctrdot MA', &&\\
&\rlap{and similarly, } &    CM^2 &= AN\ctrdot NA'. &&
\end{flalign*}
\begin{flalign*}
&\indent\text{Also}&
  DM^2 : AM\ctrdot MA' &:: BC^2 : AC^2, &&\phantom{\indent\text{Also}}\\
&&  \therefore  DM : CN &:: BC : AC, &&\\
&\rlap{and similarly }&   CM : PN &:: AC : BC. &&
\end{flalign*}

\subparagraph{Cor.} We have shewn in the course of the proof that
\[
    CN^2 + CM^2 = AC^2.
\]

By similar reasoning it appears that if $Pn$, $Dm$, be ordinates
of $BCB'$,
\begin{align*}
    Cn^2 + Cm^2 &= BC^2\,; \\
\therefore PN^2 + DM^2 &= BC^2.
\end{align*}

It should be noticed that these relations are shewn to
be true when $ACA'$, $BCB'$ are any conjugate diameters,
including of course the principal axes.

\carticleplus{84.}{Prop.~XXIII\@.} If $CP$, $CD$ be conjugate semi-diameters,
and $AC$, $BC$ the principal semi-diameters,
\[
    CP^2 + CD^2 = AC^2 + BC^2.
\]

From the preceding article,
\begin{flalign*}
&&        CN^2 + CM^2 &= AC^2, &&\\
&\text{and }&    PN^2 + DM^2 &= BC^2\,; &&\\
\intertext{also $ACB$ being in this case a right angle,}
&&      PN^2 + CN^2 &= CP^2, &&\\
&\text{and }&    DM^2 + CM^2 &= CD^2, &&\\
&&    \therefore  CP^2 + CD^2 &= AC^2 + BC^2. &&
\end{flalign*}
%-----File: 090.png-----

\carticleplus{85.}{Def.} \textit{If the ordinate $NP$ of a point, when produced,
meets the auxiliary circle in $Q$, the angle $ACQ$ is called the
eccentric angle of the point~$P$.}

\subparagraph{Prop.~XXIV\@.} \textit{If $CP$, $CD$ be conjugate semi-diameters,
the difference between the eccentric angles of $P$ and $D$ is a
right angle.}

\includegraphicsmid{085a}

\begin{flalign*}
&\indent\text{From Art.~\cartref{67},}&
  RM: DM &:: AC: BC &&\\
&\text{and, from Art.~\cartref{83},}&
  CN: DM &:: AC: BC &&\phantom{\text{and, from Art.~\cartref{83},}}
\end{flalign*}
\[
\therefore RM= CN, \text{ and similarly, } QN= CM.
\]

$\therefore$ The triangles $QCN$, $CRM$ are equal, and the angles
$QCN$, $RCM$ are complementary.
\[
\therefore QCR \text{ is a right angle.}
\]

\carticleplus{86.}{Prop.~XXV\@.} \textit{If the normal at $P$ meet the principal
axes in $G$ and $g$,}
\begin{flalign*}
&& PG: CD &:: BC: AC, &&\\
&\textit{and}&      Pg: CD &:: AC: BC. &&\phantom{and }
\end{flalign*}

For, the triangles $DCM$, $PGN$ being similar,
\begin{align*}
PG: CD &:: PN: CM \\
&:: BC: AC.
\end{align*}
%-----File: 091.png-----

So also $Pgn$ and $DCM$ are similar, and
\begin{align*}
Pg: CD &:: Pn: DM \\
&:: AC: BC.
\end{align*}

\includegraphicsmid{086a}

Hence it follows that
\[
PG\ctrdot Pg =CD^2.
\]

\carticleplus{87.}{Prop.~XXVI\@.} \textit{The parallelogram formed by the
tangents at the ends of conjugate diameters is equal to the
rectangle contained by the principal axes.}

For, taking the preceding figure,
\begin{flalign*}
&& PG: BC &:: CD: AC; &&\phantom{but }\\
&\text{but }&   PG: BC &:: BC: PF \text{ (Art.~\cartref{77})}, &&\\
&& \therefore CD: AC &:: BC: PF, &&\\
&\text{and }&  CD\ctrdot PF &= AC\ctrdot BC, &&
\end{flalign*}
whence the theorem stated.

\includegraphicsright[8]{088a}

\carticleplus{88.}{Prop.~XXVII\@.} \textit{If $SP$, $S'P$ be the focal distances of
$P$, and $CD$ be conjugate to $CP$,}
\begin{flalign*}
&& \multispan{2}{\hfill$SP\ctrdot S'P = CD^2$,\hfill} &&\\
&\textit{and }& SY: SP &:: BC: CD. &&\phantom{and }
\end{flalign*}

Let $CD$ meet $SP$, $S'P$ in $E$
and $E'$, and the normal at $P$ in
$F$; then $SPY$, $PEF$, and $S'PY'$
are similar triangles;
%-----File: 092.png-----
\begin{flalign*}
&& \therefore SP: SY &:: PE: PF, &&\\
&\text{and }&    S'P: S'Y' &:: PE: PF; &&
\end{flalign*}
\begin{align*}
\therefore SP\ctrdot S'P: SY\ctrdot S'Y' &:: PE^2 : PF^2 \\
&:: AC^2 : PF^2 \\
&:: CD^2 : BC^2 \text{ (Art.~\cartref{87})}; \\
\multispan{2}{\hfill$\therefore SP\ctrdot S'P=CD^2$.\hfill}
\end{align*}
\begin{flalign*}
&\indent\text{Also }&
  SY: SP:: PF &: PE:: PF: AC, &&\\
&&\multispan{2}{\hfill$\therefore SY: SP:: BC: CD$.\hfill} &&
\end{flalign*}

\carticleplus{89.}{Prop.~XXVIII\@.} \textit{If the tangent at $P$ meet a pair of
conjugate diameters in $T$ and $T'$, and $CD$ be conjugate to $CP$,}
\[
PT\ctrdot PT'=CD^2.
\]

\includegraphicsmid{089a}

From the figure
\[
PT: PN:: CD: DM;
\]
and, if $TP$ produced meet $CB$ in $T'$,
\begin{gather*}
PT': CN:: CD: CM; \\
\therefore PT\ctrdot PT': PN\ctrdot CN:: CD^2 : DM\ctrdot CM.
\end{gather*}
\begin{flalign*}
&\indent\text{But }&
  \multispan{1}{\hfill$PN\ctrdot CN=DM\ctrdot CM$ (Art.~\cartref{83}),\hfill} &&&\\
&& \multispan{1}{\hfill$\therefore PT\ctrdot PT'=CD^2$.\hfill} &&&
\end{flalign*}

\begin{small}
\subparagraph{Cor.} Let $TQU$ be the tangent at the other end of the chord $PNQ$,
meeting $CB'$ produced in $U$; and let $CE$ be the semi-diameter parallel
to $TQ$.

\begin{flalign*}
&\indent\text{Then }&
    TP: TQ &:: PT': QU, &&\\
&& \therefore TP^2 : TQ^2 &:: PT\ctrdot PT': QT\ctrdot QU &&\\
&& &:: CD^2 : CE^2,
\end{flalign*}
that is, \textit{the two tangents drawn from any point are in the ratio of the
parallel diameters.}
%-----File: 093.png-----

In a similar manner it can be shewn that, if the tangent at $P$ meet
the tangents at the ends of a diameter $ACA'$ in $T$ and $T'$,
\begin{flalign*}
&&  \multispan{1}{\hfill$PT\ctrdot PT' = CD^2$,\hfill}  &&&\\
&&  \multispan{1}{\hfill$CD$ being conjugate to $CP$,\hfill} &&&\\
&\text{and }& \multispan{1}{\hfill$AT\ctrdot A'T' = CB^2$,\hfill} &&&\phantom{and }\\
&&  \multispan{1}{\hfill$CB$ being conjugate to $ACA'$.\hfill} &&&
\end{flalign*}

\end{small}

\carticle{90.} \textit{Equi-conjugate diameters.}

\includegraphicsright[7]{090a}

\subparagraph{Prop.~XXIX\@.} \textit{The diagonals of the rectangle formed by
the principal axes are equal and conjugate diameters.}

For, joining $AB$, $A'B$, these lines
are parallel to the diagonals $CF$,
$CE$; and, $AB$, $A'B$ being supplemental
chords, it follows that $CD$,
$CP$ are conjugate to each other.
Moreover, they are equally inclined
to the axes, and are therefore of
equal length.

\subparagraph{Cor.~1.} If $QV$, $QU$ be drawn parallel to the equi-conjugate
diameters, meeting them in $V$ and $U$,
\begin{gather*}
      QV^2 : CP^2 - CV^2 :: CD^2 : CP^2; \\
      \therefore\  QV^2 = CP^2 - CV^2 = PV\ctrdot VP',
\end{gather*}
if $P'$ be the other end of the diameter $PCP'$.
\begin{flalign*}
&\indent\text{Hence} &     QV^2 + QU^2 &= CP^2. &&\phantom{\indent\text{Hence}}
\end{flalign*}
\begin{flalign*}
&\calignedcor{Cor.~2.}&   CP^2 + CD^2 &= AC^2 + BC^2 \text{ (Art.~\cartref{84})};
  &&\phantom{\calignedcor{Cor.~2.}}\\
&&    \therefore\  2CP^2 &= AC^2 + BC^2. &&
\end{flalign*}

\carticleplus{91.}{Prop.~XXX\@.} \textit{Pairs of tangents at right angles to
each other intersect on a fixed circle.}

The two tangents being $TP$, $TP'$, let $S'P$ produced meet
$SY$ the perpendicular on $TP$ in $K$.

Then the angle $PTK = STP = S'TP'$;
\[
  \therefore\  S'TK \text{ is a right angle.}
\]
%-----File: 094.png-----
\begin{flalign*}
&\indent\text{Hence }& 4AC^2 &=S'K^2 = S'T^2 + TK^2  &&\\
&&  &=S'T^2 + ST^2\\
&&  &= 2CT^2 + 2CS^2 \text{ (Euclid, \textsc{ii.}\ 12 and 13);}\\
&&\therefore\ CT^2 &=AC^2 + BC^2,
\end{flalign*}
and $T$ lies on a fixed circle, of which $C$ is the centre.

\includegraphicsmid{091a}

This circle is called the \textit{Director Circle} of the Ellipse, and
it will be seen that when the ellipse, by the elongation of $SC$
from $S$ is transformed into a parabola, the director circle
merges into the directrix of the parabola.

\subparagraph{Cor.} If $XQ$ is the tangent to the director circle from
the foot of the directrix,
\begin{align*}
XQ^2 &=CX^2 - CQ^2 = CX^2 - CA^2 - CB^2\\
&=CX^2 - SC \ctrdot CX-SC \ctrdot SX \text{ (Arts.~\cartref{61} and \cartref{63}),}\\
&= CX \ctrdot  SX-SC \ctrdot  SX=SX^2.\\
\therefore\ XQ &= SX,
\end{align*}
and hence it follows that \textit{the directrix is the radical axis of
the director circle and of a point circle at the focus.}

\carticleplus{92.}{Prop.~XXXI\@.} \textit{The rectangles contained by the
segments of any two chords which intersect each other are in
the ratio of the squares of the parallel diameters.}

Through any point $O$ in a chord $OQQ'$ draw the diameter
$ORR'$, and let $CD$ be parallel to $QQ'$, and $CP$ conjugate to
$CD$, bisecting $QQ'$ in $V$.

Draw $RU$ parallel to $CD$.
%-----File: 095.png-----
\begin{flalign*}
&\indent\text{Then }&
  CD^2 - RU^2 : CU^2 &:: CD^2 : CP^2 \text{ (Art.~\cartref{82}),}  &&\phantom{\indent Then }\\
&&                       &:: CD^2 - QV^2 : CV^2.  &&\\
&\indent\text{But }&
         RU^2 : CU^2 &:: OV^2 : CV^2;  &&\\
&&  \therefore\ CD^2 : CU^2 :: CD^2 &+OV^2 - QV^2 : CV^2
\end{flalign*}
\includegraphicsmid{092a}
\begin{flalign*}
&\text{or }&  CD^2 : CD^2 + OV^2 - QV^2 &:: CU^2 : CV^2 &&\phantom{or } \\
&&                                      &:: CR^2 : CO^2;  &&\\
&&       \therefore\ CD^2 : OV^2 - QV^2 &:: CR^2 : CO^2 - CR^2,  &&\\
&\text{or }&      CD^2 : OQ \ctrdot OQ' &:: CR^2 : OR \ctrdot OR'.
\end{flalign*}

Similarly, if $Oqq'$ be any other chord through $O$, and $Cd$
the parallel semi-diameter,
\begin{gather*}
Cd^2 : Oq \ctrdot Oq' :: CR^2 : OR \ctrdot OR';\\
\therefore\ OQ \ctrdot OQ' : Oq \ctrdot Oq' :: CD^2 : Cd^2.
\end{gather*}

This may otherwise be expressed thus,

\textit{The ratio of the rectangles of the segments depends only on
the directions in which they are drawn.}

The proof is the same if the point $O$ be within the
ellipse.

\carticleplus{93.}{Prop.~XXXII\@.} \textit{If a circle intersect an ellipse in
four points, the several pairs of the chords of intersection are
equally inclined to the axes.}

For if $QQ'$, $qq'$ be a pair of the chords of intersection, and
if these meet in $O$, or be produced to meet in $O$, the rectangles
$OQ \ctrdot OQ'$, $Oq \ctrdot Oq'$ are proportional to the squares on
the parallel diameters.
%-----File: 096.png-----

But these rectangles are equal since $QQ'$, $qq'$ are chords
of a circle.

Therefore the parallel diameters are equal, and, since
equal diameters are equally inclined to the axes, it follows
that the chords $QQ'$, $qq'$ are equally inclined to the axes.

Conversely, if two chords, not parallel, be equally inclined
to the axes a circle can be drawn through their
extremities.

For, as in Art.~\cartref{92}, if $OQQ'$, $Oqq'$ be two chords, and $CD$,
$Cd$ the parallel semi-diameters,
\[
      OQ \ctrdot OQ' : Oq \ctrdot Oq' :: CD^2 : Cd^2;
\]
but, if $CD$ and $Cd$ be equally inclined to the axes, they are
equal, and
\[
\therefore  OQ \ctrdot OQ' = Oq \ctrdot Oq',
\]
and the points $Q$, $Q'$, $q$, $q'$ are concyclic.

\csubsection{EXAMPLES}

\begin{exampleslist}

\item[1.] If the tangent at $B$ meet the latus rectum produced in $D$, $CDX$
is a right angle.
\cheader{EXAMPLES}

\item[2.] If $PCp$ be a diameter, and the focal distance $pS$ produced meet
the tangent at $P$ in $T$, $SP = ST$.

\item[3.] If the normal at $P$ meet the axis minor in $G'$ and $G'N$ be the
perpendicular from $G'$ on $SP$, then $PN = AC$.

\item[4.] The tangent at $P$ bisects any straight line perpendicular to $AA'$
and terminated by $AP$, $A'P$, produced if necessary.

\item[5.] Draw a tangent to an ellipse parallel to a given line.

\item[6.] $SR$ being the semi-latus rectum, if $RA$ meet the directrix in $E$,
and $S'E$ meet the tangent at $A$ in $T$,
\[
      AT = AS.
\]

\item[7.] Prove that $SY : SP :: SR : PG$.

Find where the angle $SPS'$ is greatest.

\item[8.] If two points $E$ and $E'$ be taken in the normal $PG$ such that
$PE = PE' = CD$, the loci of $E$ and $E'$ are circles.
%-----File: 097.png-----

\item[9.] If from the focus $S'$ a line be drawn parallel to $SP$, it will meet
the perpendicular $SY$ in the circumference of a circle.

\item[10.] If the normal at $P$ meet the axis major in $G$, prove that $PG$ is
an harmonic mean between the perpendiculars from the foci on the
tangent at $P$.

\item[11.] The straight line $NQ$ is drawn parallel to $AP$ to meet $CP$ in
$Q$; prove that $AQ$ is parallel to the tangent at $P$.

\item[12.] The locus of the intersection with the ordinate of the perpendicular
from the centre on the tangent is an ellipse.

\item[13.] If a rectangle circumscribes an ellipse, its diagonals are the
directions of conjugate diameters.

\item[14.] If tangents $TP$, $TQ$ be drawn at the extremities, $P$, $Q$ of any
focal chord of an ellipse, prove that the angle $PTQ$ is half the supplement
of the angle which $PQ$ subtends at the other focus.

\item[15.] If $Y$, $Z$ be the feet of the perpendiculars from the foci on the
tangent at $P$; prove that $Y$, $N$, $Z$, $C$ are concyclic.

\item[16.] If $AQ$ be drawn from one of the vertices perpendicular to the
tangent at any point $P$, prove that the locus of the point of intersection
of $PS$ and $QA$ produced will be a circle.

\item[17.] The straight lines joining each focus to the foot of the perpendicular
from the other focus on the tangent at any point meet on the
normal at the point and bisect it.

\item[18.] If two circles touch each other internally, the locus of the
centres of circles touching both is an ellipse whose foci are the centres
of the given circles.

\item[19.] The subnormal at any point $P$ is a third proportional to the
intercept of the tangent at $P$ on the major axis and half the minor axis.

\item[20.] If the normal at $P$ meet the axis major in $G$ and the axis minor
in $g$, $Gg : Sg :: SA : AX$, and if the tangent meet the axis minor in $t$,
\[
      St : tg :: BC : CD.
\]

\item[21.] If the normal at a point $P$ meet the axis in $G$, and the tangent
at $P$ meet the axis in $T$, prove that
\[
      TQ : TP :: BC : PG,
\]
$Q$ being the point where the ordinate at $P$ meets the auxiliary circle.

\item[22.] If the tangent at any point $P$ meet the tangent at the extremities
of the axis $AA'$ in $F$ and $F'$, prove that the rectangle $AF$, $A'F'$
is equal to the square on the semi-axis minor.

\item[23.] $TP$, $TQ$ are tangents; prove that a circle can be described
with $T$ as centre so as to touch $SP$, $HP$, $SQ$, and $HQ$, or these lines
produced, $S$ and $H$ being the foci.
%-----File: 098.png-----

\item[24.] If two equal and similar ellipses have the same centre, their
points of intersection are at the extremities of diameters at right angles
to one another.

\item[25.] The external angle between any two tangents to an ellipse is
equal to the semi-sum of the angles which the chord joining the points
of contact subtends at the foci.

\item[26.] The tangent at any point $P$ meets the axes in $T$ and $t$; if $S$ be
a focus the angles $PSt$, $STP$ are equal.

\item[27.] A conic is drawn touching an ellipse at the extremities $A$, $B$
of the axes, and passing through the centre $C$ of the ellipse; prove that
the tangent at $C$ is parallel to $AB$.

\item[28.] The tangent at any point $P$ is cut by any two conjugate
diameters in $T$, $t$, and the points $T$, $t$ are joined with the foci $S$, $H$
respectively; prove that the triangles $SPT$, $HPt$ are similar to each
other.

\item[29.] If the diameter conjugate to $CP$ meet $SP$, and $HP$ (or these
produced) in $E$ and $E'$, prove that $SE$ is equal to $HE'$, and that the
circles which circumscribe the triangles $SCE$, $HCE'$, are equal to one
another.

\item[30.] $PG$ is a normal, terminating in the major axis; the circle, of
which $PG$ is a diameter, cuts $SP$, $HP$, in $K$, $L$, respectively: prove
that $KL$ is bisected by $PG$, and is perpendicular to it.

\item[31.] Tangents are drawn from any point in a circle through the
foci, prove that the lines bisecting the angles between the several pairs
of tangents all pass through a fixed point.

\item[32.] If a quadrilateral circumscribe an ellipse, the angles subtended
by opposite sides at one of the foci are together equal to two right
angles.

\item[33.] If the normal at $P$ meet the axis minor in $G$, and if the tangent
at $P$ meet the tangent at the vertex $A$ in $V$, shew that
\[
      SG : SC :: PV : VA.
\]

\item[34.] $P$, $Q$ are points in two confocal ellipses, at which the line
joining the common foci subtends equal angles; prove that the tangents
at $P$, $Q$ are inclined at an angle which is equal to the angle subtended
by $PQ$ at either focus.

\item[35.] The transverse axis is the greatest and the conjugate axis the
least of all the diameters.

\item[36.] Prove that the locus of the centre of the circle inscribed in the
triangle $SPS'$ is an ellipse.
%-----File: 099.png-----

\item[37.] If the tangent and ordinate at $P$ meet the transverse axis in $T$
and $N$, prove that any circle passing through $N$ and $T$ will cut the
auxiliary circle orthogonally.

\item[38.] If $SY$, $S'Y'$ be the perpendiculars from the foci on the tangent
at a point $P$, and $PN$ the ordinate, prove that
\[
      PY : PY' :: NY : NY'.
\]

\item[39.] If a circle, passing through $Y$ and $Z$, touch the major axis in
$Q$, and that diameter of the circle, which passes through $Q$, meet the
tangent in $P$, then $PQ = BC$.

\item[40.] From the centre of two concentric circles a straight line is
drawn to cut them in $P$ and $Q$; from $P$ and $Q$ straight lines are drawn
parallel to two given lines at right angles. Shew that the locus of their
point of intersection is an ellipse.

\item[41.] From any two points $P$, $Q$ on an ellipse four lines are drawn
to the foci $S$, $S'$: prove that $SP \ctrdot S'Q$ and $SQ \ctrdot S'P$ are to one another
as the squares of the perpendiculars from a focus on the tangents
at $P$ and $Q$.

\item[42.] Two conjugate diameters are cut by the tangent at any point
$P$ in $M$, $N$; prove that the area of the triangle $CPM$ varies inversely as
that of the triangle $CPN$.

\item[43.] If $P$ be any point on the curve, and $AV$ be drawn parallel to
$PC$ to meet the conjugate $CD$ in $V$, prove that the areas of the triangles
$CAV$, $CPN$ are equal, $PN$ being the ordinate.

\item[44.] Two tangents to an ellipse intersect at right angles; prove
that the sum of the squares on the chords intercepted on them by the
auxiliary circle is constant.

\item[45.] Prove that the distance between the two points on the circumference,
at which a given chord, not passing through the centre,
subtends the greatest and least angles, is equal to the diameter which
bisects that chord.

\item[46.] The tangent at $P$ intersects a fixed tangent in $T$; if $S$ is the
focus and a line be drawn through $S$ perpendicular to $ST$, meeting the
tangent at $P$ in $Q$, shew that the locus of $Q$ is a straight line touching
the ellipse.

\item[47.] Shew that, if the distance between the foci be greater than the
length of the axis minor, there will be four positions of the tangent, for
which the area of the triangle, included between it and the straight
lines drawn from the centre of the curve to the feet of the perpendiculars
from the foci on the tangent, will be the greatest possible.

\item[48.] Two ellipses whose axes are equal, each to each, are placed in
the same plane with their centres coincident, and axes inclined to each
other. Draw their common tangents.

\item[49.] An ellipse is inscribed in a triangle, having one focus at the
orthocentre; prove that the centre of the ellipse is the centre of the
nine-point circle of the triangle and that its transverse axis is equal to
the radius of that circle.
%-----File: 100.png-----

\item[50.] The tangent at any point $P$ of a circle meets the tangent at a
fixed point $A$ in $T$, and $T$ is joined with $B$ the extremity of the diameter
passing through $A$; the locus of the point of intersection of $AP$, $BT$ is
an ellipse.

\item[51.] The ordinate $NP$ at a point $P$ meets, when produced, the circle
on the major axis in $Q$. If $S$ be a focus of the ellipse, prove that
$SQ : SP :: \text{the axis major} : \text{the chord of the circle through $Q$ and $S$}$,
and that the diameter of the ellipse parallel to $SP$ is equal to the same
chord.

\item[52.] If the perpendicular from the centre $C$ on the tangent at $P$
meet the focal distance $SP$ produced in $R$, the locus of $R$ is a circle,
the diameter of which is equal to the axis major.

\item[53.] A perfectly elastic billiard ball lies on an elliptical billiard
table, and is projected in any direction along the table: shew that all
the lines in which it moves after each successive impact touch an
ellipse or an hyperbola confocal with the billiard table.

\item[54.] Shew that a circle can be drawn through the foci and the
intersections of any tangent with the tangents at the vertices.

\item[55.] If $CP$, $CD$ be conjugate semi-diameters, and a rectangle be
described so as to have $PD$ for a diagonal and its sides parallel to the
axes, the other angular points will be situated on two fixed straight
lines passing through the centre $C$.

\item[56.] If the tangent at $P$ meet the minor axis in $T$, prove that the
areas of the triangles $SPS'$, $STS'$ are in the ratio of the squares on $CD$
and $ST$.

\item[57.] Find the locus of the centre of the circle touching the transverse
axis, $SP$, and $S'P$ produced.

\item[58.] In an ellipse $SQ$ and $S'Q$, drawn perpendicularly to a pair of
conjugate diameters, intersect in $Q$; prove that the locus of $Q$ is a concentric
ellipse.

\item[59.] If the ordinate $NP$ meet the auxiliary circle in $Q$, the perpendicular
from $S$ on the tangent at $Q$ is equal to $SP$.

\item[60.] If $PT$, $QT$ be tangents at corresponding points of an ellipse
and its auxiliary circle, shew that
\[
PT : QT :: BC : PF.
\]

\item[61.] If $CQ$ be conjugate to the normal at $P$, then is $CP$ conjugate
to the normal at $Q$.

\item[62.] $PQ$ is one side of a parallelogram described about an ellipse,
having its sides parallel to conjugate diameters, and the lines joining
$P$, $Q$ to the foci intersect in $D$, $E$; prove that the points $D$, $E$ and the
foci are concyclic.
%-----File: 101.png-----

\item[63.] If the centre, a tangent, and the transverse axis be given,
prove that the directrices pass each through a fixed point.

\item[64.] The straight line joining the feet of perpendiculars from the
focus on two tangents is at right angles to the line joining the intersection
of the tangents with the other focus.

\item[65.] A circle passes through a focus, has its centre on the major
axis of the ellipse, and touches the ellipse: shew that the straight line
from the focus to the point of contact is equal to the latus rectum.

\item[66.] Prove that the perimeter of the quadrilateral formed by the
tangent, the perpendiculars from the foci, and the transverse axis, will
be the greatest possible when the focal distances of the point of contact
are at right angles to each other.

\item[67.] Given a focus, the length of the transverse axis, and that the
second focus lies on a straight line, prove that the ellipse will touch
two fixed parabolas having the given focus for focus.

\item[68.] Tangents are drawn from a point on one of the equi-conjugate
diameters; prove that the point, the centre, and the two points of contact
are concyclic.

\item[69.] If $PN$ be the ordinate of $P$, and if with centre $C$ and radius
equal to $PN$ a circle be described intersecting $PN$ in $Q$, prove that the
locus of $Q$ is an ellipse.

\item[70.] If $AQO$ be drawn parallel to $CP$, meeting the curve in $Q$ and
the minor axis in $O$, $2CP^2 = AO \ctrdot AQ$.

\item[71.] $PS$ is a focal distance; $CR$ is a radius of the auxiliary circle
parallel to $PS$, and drawn in the direction from $P$ to $S$; $SQ$ is a perpendicular
on $CR$: shew that the rectangle contained by $SP$ and $QR$ is
equal to the square on half the minor axis.

\item[72.] If a focus be joined with the point where the tangent at the
nearer vertex intersects any other tangent, and perpendiculars be
let fall from the other focus on the joining line and on the last-mentioned
tangent, prove that the distance between the feet of these
perpendiculars is equal to the distance from either focus to the remoter
vertex.

\item[73.] A parallelogram is described about an ellipse; if two of its
angular points lie on the directrices, the other two will lie on the
auxiliary circle.

\item[74.] From a point in the auxiliary circle straight lines are drawn
touching the ellipse in $P$ and $P'$; prove that $SP$ is parallel to $S'P'$.

\item[75.] Find the locus of the points of contact of tangents to a series
of confocal ellipses from a fixed point in the axis major.

\item[76.] A series of confocal ellipses intersect a given straight line;
prove that the locus of the points of intersection of the pairs of
tangents drawn at the extremities of the chords of intersection is a
straight line at right angles to the given straight line.
%-----File: 102.png-----

\item[77.] Given a focus and the length of the major axis; describe an
ellipse touching a given straight line and passing through a given
point.

\item[78.] Given a focus and the length of the major axis; describe an
ellipse touching two given straight lines.

\item[79.] Find the positions of the foci and directrices of an ellipse
which touches at two given points $P$, $Q$, two given straight lines $PO$,
$QO$, and has one focus on the line $PQ$, the angle $POQ$ being less than a
right angle.

\item[80.] Through any point $P$ of an ellipse are drawn straight lines
$APQ$, $A'PR$, meeting the auxiliary circle in $Q$, $R$, and ordinates $Qq$, $Rr$
are drawn to the transverse axis; prove that, $L$ being an extremity of
the latus rectum,
\[
Aq \ctrdot A'r : Ar \ctrdot A'q :: AC^2 : SL^2.
\]

\item[81.] If a tangent at a point $P$ meet the major axis in $T$, and the
perpendiculars from the focus and centre in $Y$ and $Z$, then
\[
TY^2 : PY^2 :: TZ : PZ.
\]

\item[82.] An ellipse slides between two lines at right angles to each
other; find the locus of its centre.

\item[83.] $TP$, $TQ$ are two tangents, and $CP'$, $CQ'$ are the radii from the
centre respectively parallel to these tangents, prove that $P'Q'$ is parallel
to $PQ$.

\item[84.] The tangent at $P$ meets the minor axis in $t$; prove that
\[
St \ctrdot PN=BC \ctrdot CD.
\]

\item[85.] If the circle, centre $t$, and radius $tS$, meet the ellipse in $Q$, and
$QM$ be the ordinate, prove that
\[
QM : PN :: BC : BC+CD.
\]

\item[86.] Perpendiculars $SY$, $S'Y'$ are let fall from the foci upon a pair
of tangents $TY$, $TY'$; prove that the angles $STY$, $S'TY'$ are equal to
the angles at the base of the triangle $YCY'$.

\item[87.] $PQ$ is the chord of an ellipse normal at $P$, $LCL'$ the diameter
bisecting it, shew that $PQ$ bisects the angle $LPL'$ and that $LP+PL'$ is
constant.

\item[88.] $ABC$ is an isosceles triangle of which the side $AB$ is equal to
the side $AC$. $BD$, $BE$ drawn on opposite sides of $BC$ and equally
inclined to it meet $AC$ in $D$ and $E$. If an ellipse is described round
$BDE$ having its axis minor parallel to $BC$, then $AB$ will be a tangent to
the ellipse.

\item[89.] If $A$ be the extremity of the major axis and $P$ any point on
the curve, the bisectors of the angles $PSA$, $PS'A$ meet on the tangent
at $P$.
%-----File: 103.png-----

\item[90.] If two ellipses intersect in four points, the diameters parallel
to a pair of the chords of intersection are in the same ratio to each
other.

\item[91.] From any point $P$ of an ellipse a straight line $PQ$ is drawn
perpendicular to the focal distance $SP$, and meeting in $Q$ the diameter
conjugate to that through $P$; shew that $PQ$ varies inversely as the
ordinate of $P$.

\item[92.] If a tangent to an ellipse intersect at right angles a tangent to
a confocal ellipse, the point of intersection lies on a fixed circle.

\item[93.] If from a point $T$ in the director circle of an ellipse tangents
$TP$, $TP'$ are drawn, the line joining $T$ with the intersection of the
normals at $P$ and $P'$ passes through $C$.

\item[94.] Through the middle point of a focal chord a straight line is
drawn at right angles to it to meet the axis in $R$; prove that $SR$ bears
to $SC$ the duplicate ratio of the chord to the diameter parallel to it, $S$
being the focus and $C$ the centre.

\item[95.] The tangent at a point $P$ meets the auxiliary circle in $Q'$ to
which corresponds $Q$ on the ellipse; prove that the tangent at $Q$ cuts
the auxiliary circle in the point corresponding to $P$.

\item[96.] If a chord be drawn to a series of concentric, similar, and
similarly situated ellipses, and meet one in $P$ and $Q$, and if on $PQ$
as diameter a circle be described meeting that ellipse again in $RS$,
shew that $RS$ is constant in position for all the ellipses.

\item[97.] An ellipse touches the sides of a triangle; prove that if one of
its foci move along the arc of a circle passing through two of the
angular points of the triangle, the other will move along the arc of a
circle through the same two angular points.

\item[98.] The normal at a point $P$ of an ellipse meets the conjugate
axis in $K$, and a circle is described with centre $K$ and passing through
the foci $S$ and $H$. The lines $SQ$, $HQ$, drawn through any point $Q$ of
this circle, meet the tangent at $P$ in $T$ and $t$; prove that $T$ and $t$ lie
on a pair of conjugate diameters.

\item[99.] If $SP$, $S'Q$ be parallel focal distances drawn towards the same
parts, the tangents at $P$ and $Q$ intersect on the auxiliary circle.

\item[100.] Having given one focus, one tangent and the eccentricity of
an ellipse, prove that the locus of the other focus is a circle.

\item[101.] $PSQ$ is a focal chord of an ellipse, and $pq$ is any parallel
chord; if $PQ$ meet in $T$ the tangent at $p$,
\[
pq : PQ :: Sp : ST.
\]

\item[102.] If an ellipse be inscribed in a quadrilateral so that one focus
is equidistant from the four vertices, the other focus must be at the
intersection of the diagonals.
%-----File: 104.png-----

\item[103.] If a pair of conjugate diameters of an ellipse be produced to
meet either directrix, prove that the orthocentre of the triangle so
formed is the corresponding focus of the curve.

\item[104.] A pair of conjugate diameters intercept, on the tangent
at either vertex, a length which subtends supplementary angles at
the foci.

\item[105.] The straight lines $TP$, $TQ$ are the tangents at the points
$P$, $Q$ of an ellipse; one circle touches $TP$ at $P$ and meets $TQ$ in $Q$
and $Q'$, and another circle touches $TQ$ at $Q$ and meets $TP$ in $P$ and
$P'$; prove that $PQ'$ and $P'Q$ are parallel, and that they are divided in
the same ratio by the ellipse.

\item[106.] If the normals at $P$ and $D$ meet in $E$, prove that $EC$ is
perpendicular to $PD$, and that the straight line joining $C$ to the
centroid of the triangle $EPD$ bisects the line joining $E$ to $T$, the point
of intersection of the tangents at $P$ and $D$.

\item[107.] A chord $PQ$, normal at $P$, meets the directrices in $K$ and $L$,
and the tangents at $P$ and $Q$ meet in $T$; prove that $PK$ and $QL$
subtend equal angles at $T$, and that $KL$ subtends at $T$ an angle which
is half the sum of the angles subtended by $SS'$ at the ends of the
chord.

\item[108.] The tangent at the point $P$ meets the directrices in $E$ and $F$;
prove that the other tangents from $E$ and $F$ intersect on the normal
at $P$.

\item[109.] If the tangent at any point meets a pair of conjugate
diameters in $T$ and $T'$, prove that $TT'$ subtends supplementary angles
at the foci.

\item[110.] $PSQ$, $PS'R$ are focal chords; prove that the tangent at $P$
and the chord $QR$ cut the major axis at equal distances from the
centre.
\end{exampleslist}
%-----File: 105.png-----

\cchapter{CHAPTER IV}

\csectiontoc{The Hyperbola}
\cheader{THE HYPERBOLA}

\csubsection{DEFINITION}

\textit{An hyperbola is the curve traced by a point which moves
in such a manner, that its distance from a given point is in a
constant ratio of greater inequality to its distance from a
given straight line.}

\cisubsection{Tracing the Curve}

\carticle{94.} Let $S$ be the focus, $EX$ the directrix, and $A$ the
vertex.

\includegraphicsmid{094a}

Then, as in Art.~\cartref{1}, any number of points on the curve
may be obtained by taking successive positions of $E$ on the
directrix.
%-----File: 106.png-----

In $SX$ produced, find a point $A'$ such that
\[
  SA' : A'X :: SA : AX,
\]
then $A'$ is the other vertex as in the ellipse, and, the
eccentricity being greater than unity, the points $A$ and $A'$
are evidently on opposite sides of the directrix.

Find the point $P$ corresponding to $E$, and let $A'E$, $PS$
produced meet in $P'$, then, if $P'K'$ perpendicular to the
directrix meet $SE$ produced in $L'$,
\[
  P'L' : P'K' :: SA' : A'X :: SA : AX,
\]
and the angle
\begin{gather*}
  P'L'S = L'SX = L'SP';  \\
  \therefore SP' = P'L'.
\end{gather*}

Hence $P'$ is a point in the curve, and $PSP'$ is a focal
chord.

Following out the construction we observe that, since
$SA$ is greater than $AX$, there are two points on the directrix,
$e$ and $e'$, such that $Ae$ and $Ae'$ are each equal to $AS$.

If $E$ coincide with $e$, the angle
\[
  QSL = LSN = ASe = AeS.
\]
\includegraphicsmid{094b}

Hence $SQ$, $AP$ are parallel, and the corresponding point
of the curve is at an infinite distance; and similarly the
curve tends to infinity in the direction $Ae'$.

Further, the angle $ASE$ is less or greater than $AES$,
according as the point $E$ is, or is not, between $e$ and $e'$.
%-----File: 107.png-----

Hence, when $E$ is below $e$, the curve lies above the axis,
to the right of the directrix; when between $e$ and $X$, below
the axis to the left; when between $X$ and $e'$, above the axis
to the left; and when above $e'$, below the axis to the right.
Hence a general idea can be obtained of the form of the
curve, tending to infinity in four directions, as in the figure
of Art.~\cartref{102}.

\csubsection{DEFINITIONS}

\indent\textit{The line $AA'$ is called the transverse axis of the hyperbola.}

\smallskip
\textit{The middle point, $C$, of $AA'$ is the centre.}

\smallskip
\textit{Any straight line, drawn through $C$ and terminated by
the curve, is called a diameter.}

\carticleplus{95.}{Prop.~I\@.} \textit{If $P$ be any point of an hyperbola, and
$AA'$ its transverse axis, and if $A'P$, and $PA$ produced, \textup{(}or
$PA$ and $PA'$ produced\textup{)} meet the directrix in $E$ and $F$, $EF$
subtends a right angle at the focus.}

\includegraphicsmid{095a}

By the theorem of Art.~\cartref{4}, $ES'$ bisects the angle $ASP'$
and $FS$ bisects $ASP$;
\[
\therefore ESF \text{ is a right angle.}
\]

$SAA'$ being a focal chord, this is a particular case of the
theorem of Art.~\cartref{6}.

\carticleplus{96.}{Prop.~II\@.} \textit{If $PN$ be the ordinate of a point $P$, and
$ACA'$ the transverse axis, $PN^2$ is to $AN \ctrdot NA'$ in a constant
ratio.}
%-----File: 108.png-----

Join $AP, A'P$, meeting the directrix in $E$ and $F$.
\begin{flalign*}
&\text{\indent Then} & PN : AN &:: EX : AX, &&\phantom{\indent Then}\\
&\text{and} & PN : A'N &:: FX : A'X;  &&\\
&& \therefore PN^2 :AN \ctrdot NA' &::EX \ctrdot FX:AX \ctrdot A'X &&\\
&&                        &::SX^2 : AX \ctrdot A'X,
\end{flalign*}
since $ESF$ is a right angle; that is, $PN^2$ is to $AN \ctrdot NA'$,
in a constant ratio.

\includegraphicsmid{095a} %repeated figure

Through $C$, the middle point of $AA'$, draw $CB$ at right
angles to the axis, and such that
\begin{flalign*}
&&BC^2 : AC^2 :: SX^2 &: AX \ctrdot A'X;&&\\
&\text{then}  &PN^2 : AN \ctrdot NA' &:: BC^2 : AC^2 &&\phantom{then}\\
&\text{or}    &PN^2 : CN^2 - AC^2 &:: BC^2 - AC^2 &&
\end{flalign*}

\subparagraph{Cor.} If $PM$ be the perpendicular from $P$ to $BC$
\begin{flalign*}
&&PM=CN, &\text{ and }  PN=CM;\\
&&\therefore  CM^2 : PM^2 &-AC^2 :: BC^2 : AC^2\\
&\text{or} &CM^2 : BC^2 &:: PM^2 - AC^2 : AC^2 &&\phantom{or}\\
&&\therefore  CM^2 + BC^2 &: BC^2 :: PM^2 : AC^2 &&\\
&\text{or} &PM^2 : CM^2 &+ BC^2 :: AC^2 : BC^2 &&
\end{flalign*}

\carticle{97.} If we describe the circle on $AA'$ as diameter, which
we may term, for convenience, \textit{the auxiliary circle}, the
rectangle $AN \ctrdot NA'$ is equal to the square on the tangent to
the circle from $N$.
%-----File: 109.png-----

Hence the preceding theorem may be thus expressed:

\textit{The ordinate of an hyperbola is to the tangent from its
foot to the auxiliary circle in the ratio of the conjugate to the
transverse axis.}

\subparagraph{Def.} \textit{If $CB'$ be taken equal to $CB$, on the other side of
the axis, the line $BCB'$ is called the conjugate axis.}

\textit{The two lines $AA'$, $BB'$ are the principal axes of the
curve.}

\textit{When these lines are equal, the hyperbola is said to be
equilateral, or rectangular.}

The lines $AA'$, $BB'$ are sometimes called major and
minor axes, but, as $AA'$ is not necessarily greater than $BB'$,
these terms cannot with propriety be generally employed.

If a point $N'$ be taken on $CA'$ produced, such that
$CN' = CN$, the corresponding ordinate $P'N' = PN$, and
therefore it follows that the curve is symmetrical with regard
to $BCB'$, and that there is another focus and directrix,
corresponding to the vertex $A'$.

\carticleplus{98.}{Prop.~III.\@} \textit{If $ACA'$ be the transverse axis, $C$ the
centre, $S$ one of the foci, and $X$ the foot of the directrix},
\begin{flalign*}
&&CS : CA :: CA &: CX :: SA : AX,\\
&\text{and}&CS : CX &:: CS^2 : CA^2.&&\phantom{and}
\end{flalign*}

Interchanging the positions of $S$ and $X$ for a new
\includegraphicsmid{098a}
figure, the proof of these relations is identical with the
proof given for the ellipse in Art.~\cartref{61}.

\carticleplus{99.}{Prop.~IV.\@} \textit{If $S$ be a focus, and $B$ an extremity
of the conjugate axis},
\[
BC^2 = AS \ctrdot SA',\text{ and } SC^2 = AC^2 + BC^2.
\]
%-----File: 110.png-----

Referring to Art.~(\cartref{98}), $SX = SA + AX$;
\begin{align*}
\therefore SX : AX &:: SA + AX : AX,\\
&:: SC + AC : AC;
\end{align*}
and similarly
\begin{gather*} %[F2: this really needs something like the flalign trick]
SX : A'X :: SC-AC : AC;\\
\therefore SX^2 : AX \ctrdot A'X :: SC^2 - AC^2 : AC^2.
\intertext{But}
BC^2 : AC^2 :: SX^2 : AX \ctrdot A'X;\\
\therefore BC^2 = SC^2 - AC^2 = AS \ctrdot SA'.
\end{gather*}

Hence
\[
SC^2 = AC^2 + BC^2 = AB^2;
\]
\textit{i.e.} $SC$ is equal to the line joining the ends of the axes.

\carticleplus{100.}{Prop.~V.\@} \textit{The difference of the focal distances of
any point is equal to the transverse axis.}

For, if $PKK'$, perpendicular to the directrices, meet
them in $K$ and $K'$,
\begin{flalign*}
&&S'P : PK' &:: SA : AX,\\
&\text{and}&SP : PK &:: SA : AX;&&\phantom{and}\\
&&\therefore S'P-SP : KK' &:: SA : AX,\\
&&&:: AA' : XX' \text{~(Art.~\cartref{98})};\\
&&\therefore S'P-SP &= AA'.
\end{flalign*}
\begin{flalign*}
&\calignedcor{Cor.~1.}&SP : NX &:: AC : CX;\\
&&\therefore SP : AC &:: NX : CX;\\
&&\therefore SP + AC : AC &:: CN : CX,\\
&\text{or}&SP + AC : CN &:: SA : AX.\\
&\indent\text{Hence also}&S'P-AC : CN &:: SA : AX.&&\phantom{\indent Hence also}
\end{flalign*}

\subparagraph{Cor.~2.} \textit{Hence also it can be easily shewn, that the
difference of the distances of any point from the foci of an
hyperbola, is greater or less than the transverse axis, according
as the point is within or without the concave side of the
curve.}
%-----File: 111.png-----

\carticle{101.} \textit{Mechanical Construction of the Hyperbola.}

\includegraphicsmid{101a}

Let a straight rod $S'L$ be moveable in the plane of the
paper about the point $S'$. Take a piece of string, the
length of which is less than that of the rod, and fasten one
end to a fixed point $S$, and the other end to $L$; then, pressing
a pencil against the string so as to keep it stretched, and a
part of it $PL$ in contact with the rod, the pencil will trace
out on the paper an hyperbola, having its foci at $S$ and $S'$,
and its transverse axis equal to the difference between
the length of the rod and that of the string.

This construction gives the right-hand branch of the
curve; to trace the other branch, take the string longer
than the rod, and such that it exceeds the length of the rod
by the transverse axis.

We may remark that by taking a longer rod $MS'L$, and
taking the string longer than $SS' + S'L$, so that the point $P$
will be always on the end $S'M$ of the rod, we shall obtain
an ellipse of which $S$ and $S'$ are the foci. Moreover, remembering
that a parabola is the limiting form of an ellipse
when one of the foci is removed to an infinite distance,
the mechanical construction given for the parabola will be
seen to be a particular case of the above.

\cisubsection{The Asymptotes}

\carticle{102.} We have shewn in Art.~\cartref{94} that if two points, $e$
and $e'$, be taken on the directrix such that
\[
Ae = Ae' = AS,
\]
the lines $eA$, $e'A$ meet the curve at an infinite distance.
%-----File: 112.png-----

These lines are parallel to the diagonals of the rectangle
formed by the axes, for
\begin{align*}
Ae': AX:: AS: AX&:: SC : AC,\\
                &:: AB: AC, \;\text{(Art.~\cartref{99}).}
\end{align*}

\subparagraph{Definition.} \textit{The diagonals of the rectangle formed by
the principal axes are called the asymptotes.}

\includegraphicsmid{102a}

We observe that the axes bisect the angles between the
asymptotes, and that if a double ordinate, $PNP'$, when
produced, meet the asymptotes in $Q$ and $Q'$,
\[
PQ=P'Q'.
\]

The figure appended will give the general form of the
curve and its connection with the asymptotes and the
auxiliary circle.

\carticleplus{103.}{Prop.~VI\@.} \textit{The asymptotes intersect the directrices
in the same points as the auxiliary circle, and the lines joining
the corresponding foci with the points of intersection are
tangents to the circle.}

If the asymptote $CL$ meet the directrix in $D$, joining
$SD$ (fig. Art.~\cartref{102}), $CL^2 = AC^2 + BC^2 = SC^2$,
\begin{flalign*}
&\text{and}& CD: CX:: CL: CA &:: SC: CA:: CA:CX;&&
\end{flalign*}

$\therefore CD= CA$, and $D$ is on the auxiliary circle.
%-----File: 113.png-----

Also
\[
CS \ctrdot CX = CA^2 = CD^2;
\]\indent
$\therefore CDS$ is a right angle, and $SD$ is the tangent at $D$.

\subparagraph{Cor.} $CD^2 + SD^2 = CS^2 = AC^2 + BC^2 \;\text{~(Art.~\cartref{99});}$
\[
\therefore SD = BC.
\]

\begin{small}
\carticle{104.} An asymptote may also be characterized as the ultimate
position of a tangent when the point of contact is removed to an infinite
distance.

It appears from Art.~\cartref{10} that in order to find the point of contact of
a tangent drawn from a point $T$ in the directrix, we must join $T$ with
the focus $S$, and draw through $S$ a straight line at right angles to $ST$;
this line will meet the curve in the point of contact.

In the figures of Arts.~\cartref{94} and \cartref{102} we know that the line through $S$,
parallel to $eA$ or $CL$, meets the curve in a point at an infinite distance,
and also that this straight line is at right angles to $SD$, since $SD$ is at
right angles to $CD$. Hence the tangent from $D$, that is the line from
$D$ to the point at an infinite distance, is perpendicular to $DS$ and therefore
coincident with $CD$.

The asymptotes therefore touch the curve at an infinite distance.

\end{small}

\carticleplus{105.}{Def.} \textit{If an hyperbola be described, having for its
transverse and conjugate axes, respectively, the conjugate and
transverse axes of a given hyperbola, it is called the conjugate
hyperbola.}

It is evident from the preceding article that the conjugate
hyperbola has the same asymptotes as the original
hyperbola, and that the distances of its foci from the centre
are also the same.

The relations of Art.~\cartref{96} and its Corollary are also true,
\textit{mutatis mutandis}, of the conjugate hyperbola; thus, if $R$ be
a point in the conjugate hyperbola,
\begin{flalign*}
&&RM^2 : CM^2 - BC^2 &:: AC^2 : BC^2,\\
&\text{and}& CM^2 : RM^2 + AC^2 &:: BC^2 : AC^2.&\phantom{and}
\end{flalign*}

\subparagraph{Def.} \textit{A straight line drawn through the centre and terminated
by the conjugate hyperbola is also called a diameter
of the original hyperbola.}
%-----File: 114.png-----

\carticleplus{106.}{Prop.~VII\@.} \textit{If from any point $Q$ in one of the
asymptotes, two straight lines $QPN$, $QRM$ be drawn at right
angles respectively to the transverse and conjugate axes, and
meeting the hyperbola in $P$, $p$, and the conjugate hyperbola
in $R$, $r$,}
\begin{flalign*}
&&QP \ctrdot Qp &= BC^2,\\
&\text{and}& QR \ctrdot Qr &= AC^2.&\phantom{and}
\end{flalign*}
\includegraphicsmid{106a}
\begin{flalign*}
&\indent\text{For}&      QN^2 : BC^2 &:: CN^2 : AC^2;&\phantom{\indent\text{For}}\\
&&\therefore QN^2 - BC^2 : BC^2 &:: CN^2 - AC^2 : AC^2\\
&&&:: PN^2 : BC^2;\\
&&\therefore QN^2 - BC^2 &= PN^2,\\
&\text{or}&\quad QN^2 - PN^2 &= BC^2;&\phantom{\text{or}}\\
&\textit{i.e.}&\quad QP \ctrdot Qp &=BC^2.&\phantom{textit{i.e.}}\\
&\indent\text{Similarly,}& QM^2 : AC^2 &:: CM^2 : BC^2;&\phantom{\indent\text{Similarly}}\\
&&\therefore QM^2 - AC^2 : AC^2 &:: CM^2 - BC^2 : BC^2,\\
&&&:: RM^2 : AC^2;\\
&&\therefore QM^2 - RM^2 &= AC^2,\\
&\text{or}& QR \ctrdot Qr &= AC^2.&\phantom{\text{or}}
\end{flalign*}

These relations may also be given in the form,
\[
QP \ctrdot Pq = BC^2,\quad     QR \ctrdot Rq' = AC^2.
\]

\subparagraph{Cor.} If the point $Q$ be taken at a greater distance
from $C$, the length $QN$ and therefore $Qp$ will be increased,
and may be increased indefinitely.
%-----File: 115.png-----

But the rectangle $QP \ctrdot Qp$ is of finite magnitude; hence
$QP$ will be indefinitely diminished, and the curve, therefore,
as it recedes from the centre, tends more and more nearly to
coincide with the asymptote.

A further illustration is thus given of the remarks in
Art.~\cartref{104}.

\carticle{107.} If in the preceding figure the line $Qq$ be produced
to meet the conjugate hyperbola in $E$ and $e$, it can be shewn,
in the same manner as in Art.~\cartref{106}, that
\[
QE \ctrdot Qe = BC^2;
\]
and this equality is still true when the line $Qq$ lies between
$C$ and $A$, in which case $Qq$ does not meet the hyperbola.

\vspace{-2ex} %reduce spacing before subsection heading, so diagram fits on page
\cisubsection{Properties of the Tangent and Normal}

\carticle{108.} In the case of the hyperbola the theorem, proofs
of which are given in Arts.~\cartref{15} and \cartref{16}, takes the following
form:

\textit{The tangents drawn from any point to an hyperbola
subtend equal or supplementary angles at either focus according
as they touch the same or opposite branches of the
curve.}

\includegraphicsmid{108a}

For, $T$ being the point of intersection of tangents to
opposite branches of the curve, let $TM$, $TM'$ be the perpendiculars
%-----File: 116.png-----
let fall from $T$ on $SP$ and $SQ$, then, as in Arts.~\cartref{15}
and \cartref{16}, $TM = TM'$;

$\therefore$ the angles $TSM$, $TSM'$ are equal, and consequently
the angles $TSP$, $TSQ$ are supplementary.

\carticleplus{109.}{Prop.~VIII\@.} \textit{The tangent at any point bisects the
angle between the focal distances of that point, and the normal
is equally inclined to the focal distances.}

Let the normal at $P$ meet the axis in $G$.

Then (Art.~\cartref{18}),
\begin{flalign*}
&&              SG : SP &:: SA : AX, \\
&\text{and}&
              S'G : S'P &:: SA : AX; &\phantom{\text{and}}\\
&&  \therefore SG : S'G &:: SP : S'P;
\end{flalign*}
and therefore the angle between $SP$ and $S'P$ produced is
bisected by $PG$.

Hence $PT$, the tangent which is perpendicular to $PG$,
bisects the angle $SPS'$.

\subparagraph{Cor.~1.} If $PT$ and $GP$ produced meet, respectively,
the conjugate axis in $t$ and $g$, it can be shewn, in exactly the
same manner as in the corresponding case of the ellipse
(Art.~\cartref{68}), that $S$, $P$, $S'$, $t$, and $g$ are concyclic.

\subparagraph{Cor.~2.} If an ellipse be described having $S$ and $S'$ for
its foci, and if this ellipse meet the hyperbola in $P$, the
normal at $P$ to the ellipse bisects the angle $SPS'$, and therefore
coincides with the tangent to the hyperbola.

\textit{Hence, if an ellipse and an hyperbola be confocal, that is,
have the same foci, they intersect at right angles.}

\carticleplus{110.}{Prop.~IX\@.} \textit{Every diameter is bisected at the centre,
and the tangents at the ends of a diameter are parallel.}

Let $PCp$ be a diameter, and $PN$, $pn$ the ordinates.
\begin{flalign*}
&\indent\text{Then}&
CN^2 : Cn^2 &:: PN^2 : pn^2,&\phantom{\indent\text{Then}} \\
&&              &:: CN^2 - AC^2 : Cn^2 - AC^2;
\end{flalign*}
hence   $CN = Cn$,  and  $\therefore CP = Cp$.
%-----File: 117.png-----

Again, if $PT$, $pt$ be the tangents,

The triangles $PCS$, $pCS'$ are equal in all respects, and
therefore $SPS'p$ is a parallelogram.

\includegraphicsmid{110a}

Hence the angles $SPS'$, $SpS'$ are equal, and therefore
$SPT = S'pt$.

But $SPC = S'pC$,

$\therefore$ the difference $TPC=\text{the difference } tpC$, and $PT$ is
parallel to $pt$.

It can be shewn in exactly the same manner, that, if
the diameter be terminated by the conjugate hyperbola, it
is bisected in $C$, and the tangents at its extremities are
parallel.

\subparagraph{Cor\@.} The distances $SP$, $Sp$ are equally inclined to the
tangents at $P$ and $p$.

\carticleplus{111.}{Prop.~X\@.} \textit{The perpendiculars from the foci on any
tangent meet the tangent on the auxiliary circle, and the
semi-conjugate axis is a mean proportional between their
lengths.}

Let $SY$, $S'Y'$ be the perpendiculars, and let $SY$ produced
meet $S'P$ in~$L$.

\includegraphicsmid{111a}

Then the triangles $SPY$, $LPY$ are equal in all respects,
\[
\text{and }SY = LY.
\]
%-----File: 118.png-----

Hence, $C$ being the middle point of $SS'$ and $Y$ of $SL$, $CY$
is parallel to $S'L$, and $S'L = 2CY$.
\begin{flalign*}
&\indent\text{But}&  S'L = S'P - PL &= S'P - SP = 2AC; &\phantom{\indent\text{But}} \\
&&\therefore  CY &= AC,
\end{flalign*}
and $Y$ is on the auxiliary circle.

So also $Y$' is a point in the circle.

Let $SY$ produced meet the circle in $Z$, and join $Y'Z$;
then, $Y'YZ$ being a right angle, $ZY'$ is a diameter and
passes through $C$. Hence, the triangles $SCZ$, $S'CY'$ being
equal,
\begin{flalign*}
&& S'Y' &= SZ, \\
&\text{and}&   SY \ctrdot S'Y' = SY \ctrdot SZ
    &= SA \ctrdot SA' = BC^2.&\phantom{\text{and}}
\end{flalign*}

\subparagraph{Cor.~1\@.} If $P'$ be the other extremity of the diameter
$PC$, the tangent at $P'$ is parallel to $PY$, and therefore $Z$
is the foot of the perpendicular from $S$ on the tangent
at $P'$.

\subparagraph{Cor.~2\@.} If the diameter $DCD'$, drawn parallel to the
tangent at $P$, meet $S'P$, $SP$ in $E$ and $E'$, $PECY$ is a
parallelogram;
\begin{flalign*}
&&\therefore PE&=CY=AC,\\
&\text{and so also}& PE'&=CY'=AC. &\phantom{\text{and so also}}
\end{flalign*}
%-----File: 119.png-----
\carticleplus{112.}{Prop.~XI\@.} \textit{To draw tangents to an hyperbola from
a given point.}

The construction of Art.~\cartref{17} may be employed, or, as in
the cases of the ellipse and parabola, the following.

\includegraphicsmid{112a}

Let $Q$ be the given point; join $SQ$, and upon $SQ$ as
diameter describe a circle intersecting the auxiliary circle
in $Y$ and $Y'$;
\[
QY \text{ and } QY' \text{ are the required tangents.}
\]

Producing $SY$ to $L$, so that $YL = SY$, draw $S'L$ cutting
$QY$ in $P$, and join $SP$.

The triangles $SPY$, $LPY$ are equal in all respects,
\begin{flalign*}
&\text{and}& &S'P - SP = S'L = 2CY = 2AC; &\phantom{and}\\
&&           &\therefore P \text{ is a point on the hyperbola.}
\end{flalign*}

Also $QP$ bisects the angle $SPS'$, and is therefore the
tangent at $P$. A similar construction will give the other
tangent $QP'$.

If the point $Q$ be within the angle formed by the asymptotes,
the tangents will both touch the same branch of the
curve; but if it lie within the external angle, they will touch
opposite branches.
%-----File: 120.png-----

\carticleplus{113.}{Prop.~XII\@.} \textit{If two tangents be drawn from any
point to an hyperbola they are equally inclined to the focal
distances of that point.}

Let $PQ$, $P'Q$ be the tangents, $SY$, $S'Y'$, $SZ$, $S'Z'$ the
perpendiculars from the foci; join $YZ$, $Y'Z'$.

\includegraphicsmid{113a}

Then the angles $YSZ$, $Y'S'Z'$ are equal, for they are the
supplements of $YQZ$, $Y'QZ'$.
\begin{flalign*}
&\indent\text{Also} & \phantom{\text{ (Art.~\cartref{111})};}
   SY \ctrdot S'Y' &= SZ \ctrdot S'Z' \text{ (Art.~\cartref{111})};  & \phantom{\indent\text{Also}}\\
&\text{or} & SY: SZ &:: S'Z': S'Y';
\end{flalign*}
\begin{center}
$\therefore$ the triangles $YZS$, $Y'S'Z'$ are similar,\\
and the angle $YZS = Z'Y'S'$.\\
But the angle $YQS = YZS$, and $Z'QS' = ZY'S'$;\\
$\therefore YQS=Z'QS'$.
\end{center}

That is, the tangent $QP$ and the tangent $P'Q$ produced
are equally inclined to $SQ$ and $S'Q$.

Or, producing $S'Q$, $QP$ and $QP'$ are equally inclined to
$QS$ and $S'Q$ produced.

In exactly the same manner it can be shewn that if
$QP$, $QP'$ touch opposite branches of the curve the angles
$PQS$, $P'QS'$ are equal.

\subparagraph{Cor.} If $Q$ be a point in a confocal hyperbola, the normal
at $Q$ bisects the angle between $SQ$ and $S'Q$ produced
and therefore bisects the angle $PQP'$.
%-----File: 121.png-----

\textit{Hence, if from any point of an hyperbola tangents be
drawn to a confocal hyperbola, these tangents are equally
inclined to the normal or the tangent at the point, according
as it lies within or without that angle formed by the asymptotes
of the confocal which contains the transverse axes.}

\carticleplus{114.}{Prop.~XIII\@.} \textit{If $PT$, the tangent at $P$, meet the
transverse axis in $T$, and $PN$ be the ordinate,}
\[
CN \ctrdot CT=AC^2
\]

Let fall the perpendicular $SY$ upon $PT$, and join $YN$,
$CY$, $SP$, and $S'P$.
\begin{flalign*}
&\indent\text{The angle}& CYT &=S'PY=SPY &
    \phantom{\indent\text{The angle}} \\
&&&=\text{the supplement of }SNY=CNY;
\end{flalign*}
\includegraphicsright[10]{114a}
\noindent also the angle $YCT$ is common
to the two triangles
$CYT$, $CYN$; these triangles
are therefore similar,
and
\[
CN:CY::CY:CT,
\]
or
\[
CN \ctrdot CT=CY^2 = AC^2
\]

\begin{flalign*}
&\calignedcor{Cor.~1\@.} \text{Hence} &
    CN \ctrdot NT &=CN^2 - CN \ctrdot CT &&
    \phantom{\calignedcor{Cor.~1.} \text{Hence}}\\
&&            &=CN^2 - AC^2      \\
&&            &=AN \ctrdot NA'
\end{flalign*}

\subparagraph{Cor.~2\@.} Hence also it follows that

If any number of hyperbolas be described having the
same transverse axis, and an ordinate be drawn cutting
the hyperbolas, the tangents at the points of section will all
meet the transverse axis in the same point.

\subparagraph{Cor.~3\@.} If $CN$ be increased indefinitely, $CT$ is diminished
indefinitely, and the tangent ultimately passes
through $C$, as we have already shewn in Art.~\cartref{104}\@.
%-----File: 122.png-----

\carticleplus{115.}{Prop.~XIV\@.} \textit{If the tangent at $P$ meet the conjugate
axis in $t$, and $PN$ be the ordinate,}
\[
Ct \ctrdot PN=BC^2.
\]
\begin{flalign*}
&\indent\text{For}& Ct : PN &:: CT : NT; \text{ (Fig.~Art.~\cartref{114})} & \phantom{\indent\text{For}} \\
&&\therefore Ct \ctrdot PN : PN^2 &:: CT \ctrdot CN : CN \ctrdot NT\\
&&&:: AC^2 : AN \ctrdot NA'.\\
&&\therefore Ct \ctrdot PN : AC^2 &:: PN^2 : AN \ctrdot NA'\\
&&&:: BC^2 : AC^2,\\
&\text{and}&Ct \ctrdot P&N = BC^2.
\end{flalign*}

In exactly the same manner as in Art.~\cartref{76}, it can be
shewn that
\begin{gather*}
CG \ctrdot CT = SC^2,\\
CG : CN :: SC^2 : AC^2,\quad Cg : PN :: SC^2 : BC^2,
\end{gather*}
\begin{flalign*}
&\text{and}&     NG : CN &:: BC^2 : AC^2. &\phantom{\text{and}}
\end{flalign*}

\includegraphicsright[15]{116a}

\carticleplus{116.}{Prop.~XV\@.} \textit{If the normal at $P$ meet the transverse
axis in $G$, the conjugate axis in $g$, and the diameter parallel
to the tangent at $P$ in $F$,}
\[
PF \ctrdot PG = BC^2, \text{ and } PF \ctrdot Pg = AC^2.
\]

Let $NP$, $PM$, perpendicular
to the axes, meet the diameter
$CF$ in $K$ and $L$;

Then $KNG$, $KFG$ being right
angles, $K$, $F$, $N$, $G$ are concyclic;
\begin{align*}
\therefore PF \ctrdot PG &= PK \ctrdot PN \\
 &=Ct \ctrdot PN=BC^2.
\end{align*}

Similarly $F$, $L$, $M$, $g$ are concyclic;
\[
\therefore PF \ctrdot Pg = PL \ctrdot PM= CT \ctrdot CN= AC^2.
\]
%-----File: 123.png-----
\carticleplus{117.}{Prop.~XVI\@.} \textit{If $PCp$ be a diameter, and $QV$ an
ordinate, and if the tangent at $Q$ meet the diameter $Pp$ in $T$,}
\[
CV \ctrdot CT=CP^2.
\]

Let the tangents at $P$ and $p$ meet the tangent at $Q$ in
$R$ and $r$;

\includegraphicsmid{117a}

Then the angle $SPR = Spr$ (Cor.~Art.~\cartref{110})\\
and therefore if $RN$, $rn$ be the perpendiculars on $SP$, $sp$, the
triangles $RPN$, $rpn$ are similar.

Draw $RM$, $rm$ perpendiculars on $SQ$.
\begin{flalign*}
&\indent\text{Then}&   TR : Tr :: RP : rp &:: RN : rn, & \phantom{\indent\text{Then}}\\
&& &:: RM : rm \text{ (Cor.~Art.~\cartref{15})}\\
&& &:: RQ : rQ.
\end{flalign*}

Hence, $QV$, $RP$, and $rp$ being parallel,
\begin{flalign*}
&&  TP : Tp &:: PV : pV;\\
&&\therefore TP + Tp : Tp-TP &:: PV+pV : pV-PV,\\
&\text{or}&     2CP : 2CT &:: 2CV : 2CP, &\phantom{\text{or}}\\
&\text{or}&         CV \ctrdot C&T=CP^2.
\end{flalign*}
%-----File: 124.png-----

\carticleplus{118.}{Prop.~XVII\@.} \textit{A diameter bisects all chords parallel
to the tangents at its extremities.}

\includegraphicsmid{118a}

Let $PCp$ be the diameter, and $QQ'$ the chord, parallel to
the tangents at $P$ and $p$. Then if the tangents $TQ$, $TQ'$
at $Q$ and $Q'$ meet the tangents at $P$ and $p$, in the points
$E$, $E'$, $e$, $e'$,
\begin{gather*}
EP=E'P \text{ and } ep=e'p, \text{ (Art.~\cartref{21})} \\
\therefore\text{the point }T\text{ is on the line }Pp; \\
\text{but }TP\text{ bisects }QQ'; \\
\text{that is, the diameter }pCP\text{ produced bisects }QQ'.
\end{gather*}

\subparagraph{Def.} \textit{The line $DCd$, drawn parallel to the tangent at
$P$ and terminated by the conjugate hyperbola, that is, the
diameter parallel to the tangent at $P$, is said to be conjugate
to $PCp$.}

A diameter therefore bisects all chords parallel to its
conjugate.

\carticleplus{119.}{Prop.~XVIII\@.} \textit{If the diameter $DCd$ be conjugate
to $PCp$, then will $PCp$ be conjugate to $DCd$.}

Let the chord $QVq$ be parallel to $CD$ and be bisected in
$V$ by $CP$ produced.

Draw the diameter $qCR$, and join $RQ$ meeting $CD$
in $U$.
%-----File: 125.png-----

Then $RC= Cq$ and $QV= Vq$; $\therefore QR$ is parallel to $CP$.
\begin{flalign*}
&\indent\text{Also}& QU : UR &:: Cq : CR, &\phantom{\indent\text{Also}} \\
&\text{and}& \therefore QU &= UR,
\end{flalign*}
that is, $CD$ bisects the chords parallel to $CP$, and $PCp$ is
therefore conjugate to $DCd$.

\includegraphicsmid{119a}

Hence, when two diameters are conjugate, each bisects
the chords parallel to the other.

\subparagraph{Def.} \textit{Chords drawn from the extremities of any diameter
to a point on the hyperbola are called supplemental chords.}

Thus, $qQ$, $QR$ are supplemental chords, and they are
parallel to $CD$ and $CP$; supplemental chords are therefore
parallel to conjugate diameters.

\subparagraph{Def.} \textit{A line $QV$, drawn from any point $Q$ of an
hyperbola, parallel to a diameter $DCd$, and terminated by the
conjugate diameter $PCp$, is called an ordinate of the diameter
$PCp$, and if $QV$ produced meet the curve in $Q'$, $QVQ'$ is the
double ordinate.}

This definition includes the two cases in which $QQ'$ may
be drawn so as to meet the same, or opposite branches of the
hyperbola.

\carticleplus{120.}{Prop.~XIX\@.} \textit{Any diameter is a mean proportional
between the transverse axis and the focal chord parallel to the
diameter.}

This can be proved as in Art.~\cartref{81}.
%-----File: 126.png-----

\cisubsection{Properties of Asymptotes}

\carticleplus{121.}{Prop.~XX\@.} \textit{If from any point $Q$ in an asymptote
$QPpq$ be drawn meeting the curve in $P$, $p$ and the other
asymptote in $q$, and if $CD$ be the semi-diameter parallel
to $Qq$,}
\[
QP \ctrdot Pq = CD^2 \textit{ and } QP =pq.
\]

Through $P$ and $D$ draw $RPr$, $DTt$ perpendicular to the
transverse axis, and meeting the asymptotes.
\includegraphicsmid{121a}
\begin{flalign*}
&\indent\text{Then}&     QP : RP &:: CD : DT, & \phantom{\indent\text{Then}}\\
&\text{and}&      Pq : Pr &:: CD : Dt;\\
&&\therefore QP \ctrdot Pq : RP \ctrdot Pr &:: CD^2 : DT \ctrdot Dt.\\
&\indent\text{But}& RP \ctrdot Pr = BC^2 = DT &\ctrdot Dt
    \text{ (Arts.~\cartref{106} and \cartref{107}),}\\
&&\therefore QP \ctrdot P&q =CD^2.
\end{flalign*}
\begin{flalign*}
&\indent\text{Similarly}& qp \ctrdot pQ &= CD^2;
    &\phantom{\indent\text{Similarly}}\\
&&\therefore QP \ctrdot Pq &= qp \ctrdot pQ;
\end{flalign*}
\begin{gather*}
\text{or, if } V \text{ be the middle point of } Qq,\\
QV^2 - PV^2 = QV^2 - pV^2.
\end{gather*}

Hence \hfill  $PV=pV$, and $\therefore PQ=pq$. \hfill\phantom{Hence\indent}

We have taken the case in which $Qq$ meets one branch
of the hyperbola. It may however be shewn in the same
manner that the same relations hold good for the case in
which $Qq$ meets opposite branches.
%-----File: 127.png-----

\subparagraph{Cor.} \textit{If a straight line $PP'p'p$ meet the hyperbola in
$P$, $p$, and the conjugate hyperbola in $P'$, $p'$, $PP'=pp'$.}

For, if the line meet the asymptotes in $Q$, $q$,
\begin{gather*}
QP'=p'q, \text{ and } PQ = qp;\\
\therefore PP'=pp'.
\end{gather*}

\carticleplus{122.}{Prop.~XXI\@.} \textit{The portion of a tangent which is
terminated by the asymptotes is bisected at the point of contact,
and is equal to the parallel diameter.}

$LEl$ being the tangent (Fig.~Art.~\cartref{121}), and $DCd$ the
parallel diameter, draw any parallel straight line $QPpq$
meeting the curve and the asymptotes.

Then $QP=pq$; and, if the line move parallel to itself
until it coincides with $Ll$, the points $P$ and $p$ coincide with
$E$, and $\therefore LE = El$.

Also\hfill $QP \ctrdot Pq = CD^2$, always;\hfill{\phantom{Also\indent}}
\[
\therefore LE \ctrdot El = CD^2, \text{ or }LE= CD.
\]

\cisubsection{Properties of Conjugate Diameters}

\carticleplus{123.}{Prop.~XXII\@.} \textit{Conjugate diameters of an hyperbola
are also conjugate diameters of the conjugate hyperbola, and
the asymptotes are diagonals of the parallelogram formed by
the tangents at their extremities.}

$PCp$ and $DCd$ being conjugate, let $QVq$, a double ordinate
of $CD$, meet the conjugate hyperbola in $Q'$ and $q'$.

Then\hfill $QV = Vq$, and $QQ' = qq'$ (Cor.~Art.~\cartref{121}),\hfill\phantom{Then\indent}
\[
\therefore Q'V = Vq'.
\]

That is, $CD$ bisects the chords of the conjugate hyperbola
parallel to $CP$.

Hence $CD$ and $CP$ are conjugate in both hyperbolas, and
therefore the tangent at $D$ is parallel to $CP$.
%-----File: 128.png-----

Let the tangent at $P$ meet the asymptote in $L$; then
\[
PL=CD \text{ (Art.~\cartref{122}).}
\]

\includegraphicsmid{123a}

Hence $LD$ is parallel and equal to $CP$;\\
but the tangent at $D$ is parallel to $CP$;
\[
\therefore LD \text{ is the tangent at } D.
\]

Completing the figure, the tangents at $p$ and $d$ are parallel
to those at $P$ and $D$, and therefore the asymptotes are
the diagonals of the parallelogram $Lll'L'$.

\subparagraph{Cor.} Hence, joining $PD$, it follows that $PD$ is parallel
to the asymptote $lCL'$, since $LP = PL'$, and $LD = Dl$,

\carticleplus{124.}{Prop.~XXIII\@.} \textit{If $QV$ be an ordinate of a diameter
$PCp$, and $DCd$ the conjugate diameter,}
\[
QV^2 : PV \ctrdot Vp :: CD^2 : CP^2.
\]

\includegraphicsmid{124a}

Let $QV$ and the tangent at $P$ meet the asymptote in
$R$ and $L$.
%-----File: 129.png-----

Then $LP$ being equal to $CD$,
\begin{align*}
RV^2 : CD^2 &:: CV^2 : CP^2;\\
\therefore RV^2 - CD^2 : CD^2 &:: CV^2 - CP^2 : CP^2.
\end{align*}

But\hfill $RV^2 - QV^2 = CD^2$. \hfill\phantom{But\indent}

Hence\hfill $QV^2 : CD^2 :: CV^2 - CP^2 : CP^2$, \hfill{\phantom{Hence}\indent}
\[
\text{or } QV^2 : PV \ctrdot Vp :: CD^2 : CP^2.
\]

\carticleplus{125.}{Prop.~XXIV\@.} \textit{If $QV$ be an ordinate of a diameter
$PCp$, and if the tangent at $Q$ meet the conjugate diameter,
$DCd$, in $t$,}
\[
Ct \ctrdot QV= CD^2.
\]

For, (Fig.~Art.~\cartref{118})
\begin{gather*}
Ct : QV :: CT : VT,\\
\text{and }\therefore Ct \ctrdot QV : QV^2 :: CV \ctrdot CT : CV \ctrdot VT.
\end{gather*}

But\hfill $CV \ctrdot CT= CP^2$, \hfill\phantom{But\indent}
\begin{align*}
\text{and } CV \ctrdot VT=CV^2 &- CV \ctrdot CT = CV^2 - CP^2;\\
\therefore Ct \ctrdot QV : QV^2 &:: CP^2 : CV^2 - CP^2,\\
&:: CD^2 : QV^2.
\end{align*}

Hence\hfill $Ct \ctrdot QV= CD^2$. \hfill\phantom{Hence\indent}

\carticleplus{126.}{Prop.~XXV\@.} \textit{If $ACa$, $BCb$ be conjugate diameters,
and $PCp$, $DCd$ another pair of conjugate diameters, and if
$PN$, $DM$ be ordinates of $ACa$,}
\begin{gather*}
CM : PN :: AC : BC,\\
\text{and } DM : CN :: BC : AC.
\end{gather*}

Let the tangents at $P$ and $D$ meet $ACa$ in $T$ and $t$;\\
then $CN \ctrdot CT = AC^2 = CM \ctrdot Ct$ (Art.~\cartref{117}),
\begin{align*}
\therefore CM : CN &:: CT: Ct,\\
&:: PT : CD,\\
&:: PN : DM,\\
&:: CN : Mt;
\end{align*}
%-----File: 130.png-----
\begin{flalign*}
&&\therefore CN^2 = CM \ctrdot Mt &= CM^2 + CM \ctrdot Ct = CM^2 + AC^2,\\
&\text{so that}&    CM^2 &= CN^2 - AC^2. &\phantom{\text{so that}}
\end{flalign*}
\includegraphicsmid{126a}
\begin{flalign*}
&\indent\text{But}&   PN^2 : CN^2 - AC^2 &:: BC^2 : AC^2;
    &\phantom{\indent\text{But}}\\
&&\therefore CM : PN &:: AC : BC;
\end{flalign*}
and, similarly,\hfill    $DM : CN :: BC : AC$. \hfill\phantom{and, similarly,}

\subparagraph{Cor.} We have shewn in the course of the proof, that
\[
CN^2 - CM^2 = AC^2.
\]

Similarly, if $Pn$, $Dm$ be ordinates of $BC$,
\begin{flalign*}
&&Cm^2 &-Cn^2 = BC^2;\\
&\text{that is,}&     DM^2 &-PN^2 = BC^2; &\phantom{\text{that is,}}
\end{flalign*}
and it must be noticed that these relations are shewn for any
pair of conjugate diameters $ACa$, $BCb$, including of course
the axes.

\carticleplus{127.}{Prop.~XXVI\@.}  \textit{If $CP$, $CD$ be conjugate semi-diameters,
and $AC$, $BC$ the semi-axes},
\[
CP^2 - CD^2 = AC^2 - BC^2.
\]

For, drawing the ordinates $PN$, $DM$, and remembering
that in this case the angles at $N$ and $M$ are right angles, we
have, from the figure of the previous article,
\begin{align*}
CP^2 &=CN^2 + PN^2,\\
CD^2 &=CM^2 + DM^2.
\end{align*}

But $CN^2 - CM^2 = AC^2$ and $DM^2 - PN^2 = BC^2$;
\[
\therefore CP^2 - CD^2 = AC^2 - BC^2.
\]
%-----File: 131.png-----

\carticleplus{128.}{Prop.~XXVII\@.} \textit{If the normal at $P$ meet the axes
in $G$ and $g$,}
\begin{flalign*}
&& PG : CD &:: BC : AC,\\
&\textit{and}&  Pg : CD &:: AC : BC. & \phantom{\textit{and}}
\end{flalign*}

For the proofs of these relations, see Art.~\cartref{86}.

Observe also that
\[
PG \ctrdot Pg = CD^2,
\]
and that
\[
Gg : CD :: SC^2 : AC \ctrdot BC.
\]

\carticleplus{129.}{Prop.~XXVIII\@.} \textit{The area of the parallelogram
formed by the tangents at the ends of conjugate diameters
is equal to the rectangle contained by the axes.}

Let $CP$, $CD$ be the semi-diameters, and $PN$, $DM$ the
ordinates of the transverse axis.

\includegraphicsright[9]{129a}

Let the normal at $P$ meet
$CD$ in $F$, and the axis in $G$.
Then $PNG$, $CDM$ are similar
triangles, and, exactly as in
Art.~\cartref{87}, it can be shewn that
\[
PF \ctrdot CD=AC \ctrdot BC.
\]

Hence it follows that, in
the figure of Art.~\cartref{123}, the
triangle $LCL'$ is of constant area.

For the triangle is equal to the parallelogram $CPLD$.

\carticleplus{130.}{Prop.~XXIX\@.} \textit{If $SP$, $S'P$ be the focal distances of
a point $P$, and $CD$ be conjugate to $CP$,}
\[
SP \ctrdot S'P = CD^2.
\]

Attending to the figure of Art.~\cartref{111}, the proof is the
same as that of Art.~\cartref{88}.
%-----File: 132.png-----

\carticleplus{131.}{Prop.~XXX\@.} \textit{If the tangent at $P$ meet a pair
of conjugate diameters in $T$ and $t$, and $CD$ be conjugate
to $CP$,}
\[
PT \ctrdot Pt=CD^2.
\]

This can be proved as in Art.~\cartref{89}.

It can also be shewn that if the tangent at $P$ meet two
parallel tangents in $T'$ and $t'$,
\[
PT' \ctrdot Pt'=CD^2.
\]

\carticleplus{132.}{Prop.~XXXI\@.} \textit{If the tangent at $P$ meet the
asymptotes in $L$ and $L'$,}
\[
CL \ctrdot CL' = SC^2.
\]

\includegraphicsmid{132a}

Let the tangent at $A$ meet the asymptotes in $K$ and
$K'$; then (Art.~\cartref{129}) the triangles $LCL'$, $KCK'$ are of equal
area, and therefore
\[
CL : CK' :: CK : CL' \text{ (Euclid, Book \textsc{vi}.),}
\]
or\hfill  $CL \ctrdot CL' = CK^2 = AC^2 + BC^2 = SC^2$. \hfill\phantom{or}

\subparagraph{Cor.} If $PH$, $PH'$ be drawn parallel to, and terminated
by the asymptotes,
\[
4 \ctrdot PH \ctrdot PH' = CS^2,
\]
for $CL = 2PH'$, and $CL' = 2PH$.
%-----File: 133.png-----

\carticleplus{133.}{Prop.~XXXII\@.} \textit{Pairs of tangents at right angles
to each other intersect on a fixed circle.}

$PT$, $QT$ being two tangents at right angles, let $SY$,
perpendicular to $PT$, meet $S'P$ in $K$.

\includegraphicsmid{133a}

Then (Art.~\cartref{113}) the angle $S'TY' = QTS$,
\begin{flalign*}
&\text{and obviously, }&&  KTP = PTS; &\phantom{and obviously, }
\end{flalign*}
therefore $S'TY'$ is complementary to $KTP$, and $S'TK$ is a
right angle.

Hence
\begin{align*}
          4AC^2 &= S'K^2 = S'T^2 + TK^2  \\
                &= S'T^2 + ST^2  \\
                &= 2 \ctrdot CT^2 + 2 \ctrdot CS^2
\text{ by Euclid \textsc{ii}.~12 and 13;}  \\
\therefore CT^2 &= AC^2 - BC^2,
\end{align*}
and the locus of $T$ is a circle.

If $AC$ be less than $BC$, this relation is impossible.

In this case, however, the angle between the asymptotes
is greater than a right angle, and the angle $PTQ$ between a
pair of tangents, being always greater than the angle between
the asymptotes, is greater than a right angle. The problem
is therefore \textit{à priori} impossible for the hyperbola, but becomes
possible for the conjugate hyperbola.

As in the case of the ellipse, the locus of $T$ is called the
director circle.
%-----File: 134.png-----

\carticleplus{134.}{Prop.~XXXIII\@.} \textit{The rectangles contained by the
segments of any two chords which intersect each other are in
the ratio of the squares on the parallel diameters.}

\includegraphicsmid{134a}

Through any point $O$ in a chord $QOQ'$ draw the diameter
$ORR'$; and let $CD$ be parallel to $QQ'$, and $CP$ conjugate to
$CD$, bisecting $QQ'$ in $V$.

Draw $RU$ an ordinate of $CP$.
\begin{flalign*}
&\indent\text{Then}&   RU^2 : CU^2 - CP^2 &:: CD^2 : CP^2;
    &\phantom{\indent\text{Then}} \\
&&\therefore CD^2 + RU^2 : CU^2 &:: CD^2 : CP^2,\\
&&&:: CD^2 + QV^2 : CV^2.\\
&\indent\text{But}&     RU^2 : CU^2 &:: OV^2 : CV^2;
\end{flalign*}
\[
\therefore CD^2 : CU^2 :: CD^2 + QV^2 - OV^2 : CV^2,
\]
\begin{flalign*}
&\text{or}& CD^2 : CD^2 + QV^2 - OV^2 &:: CU^2 : CV^2,
    &\phantom{\text{or}}\\
&&&:: CR^2 : CO^2;
\end{flalign*}
\begin{flalign*}
&&\therefore CD^2 : QV^2 - OV^2 &:: CR^2 : CO^2 - CR^2,\\
&\text{or}&  CD^2 : QO \ctrdot OQ' &:: CR^2 : OR \ctrdot OR'.
    &\phantom{\text{or}}
\end{flalign*}

Similarly, if $qOq'$ be any other chord, and $Cd$ the parallel
semi-diameter,
\begin{gather*}
Cd^2 : qO \ctrdot Oq' :: CR^2 : OR \ctrdot OR';\\
\therefore QO \ctrdot OQ' : qO \ctrdot Oq' :: CD^2 : Cd^2;
\end{gather*}
that is, the ratio of the rectangles depends only on the
directions of the chords.
%-----File: 135.png-----

\subparagraph{Prop.~XXXIV\@.} \textit{If a circle intersect an hyperbola in four
points, the several pairs of the chords of intersection are
equally inclined to the axes.}

For the proof, see Art.~\cartref{93}.

\csubsection{EXAMPLES}

\begin{exampleslist}

\item[1.] If a circle be drawn so as to touch two fixed circles externally,
the locus of its centre is an hyperbola.
\cheader{EXAMPLES}

\item[2.] If the tangent at $B$ to the conjugate meet the latus rectum in
$D$, the triangles $SCD$, $SXD$ are similar.

\item[3.] The straight line drawn from the focus to the directrix, parallel
to an asymptote, is equal to the semi-latus rectum, and is bisected by
the curve.

\item[4.] Given the asymptotes and a focus, find the directrix.

\item[5.] Given the centre, one asymptote, and a directrix, find the focus.

\item[6.] Parabolas are described passing through two fixed points, and
having their axes parallel to a fixed line; the locus of their foci is an
hyperbola.

\item[7.] The base of a triangle being given, and also the point of contact
with the base of the inscribed circle, the locus of the vertex is an
hyperbola.

\item[8.] If the normal at $P$ meet the conjugate axis in $g$, and $gN$ be the
perpendicular on $SP$, then $PN=AC$.

\item[9.] Draw a tangent to an hyperbola, or its conjugate, parallel to a
given line.

\item[10.] If $AA'$ be the axis of an ellipse, and $PNP'$ a double ordinate,
the locus of the intersection of $A'P$ and $P'A$ is an hyperbola.

\item[11.] The tangent at $P$ bisects any straight line perpendicular to
$AA'$, and terminated by $AP$, and $A'P$.

\item[12.] If $PCp$ be a diameter, and if $Sp$ meet the tangent at $P$ in $T$,
\[
SP=ST.
\]

\item[13.] Given an asymptote, the focus, and a point; construct the
hyperbola.
%-----File: 136.png-----

\item[14.] A circle can be drawn through the foci and the intersections of
any tangent with the tangents at the vertices.

\item[15.] Given an asymptote, the directrix, and a point; construct the
hyperbola.

\item[16.] If through any point of an hyperbola straight lines are drawn
parallel to the asymptotes and meeting any semi-diameter $CQ$ in $P$
and $R$,
\[
CP \ctrdot CR=CQ^2.
\]

\item[17.] $PN$ is an ordinate and $NQ$ parallel to $AB$ meets the conjugate
axis in $Q$; prove that $QB \ctrdot QB'=PN^2$.

\item[18.] $NP$ is an ordinate and $Q$ a point in the curve; $AQ$, $A'Q$ meet
$NP$ in $D$ and $E$; prove that $ND \ctrdot NE=NP^2$.

\item[19.] If a tangent cut the major axis in the point $T$, and perpendiculars
$SY$, $HZ$ be let fall on it from the foci, then
\[
AT \ctrdot A'T=YT \ctrdot ZT.
\]

\item[20.] In the tangent at $P$ a point $Q$ is taken such that $PQ$ is proportional
to $CD$; shew that the locus of $Q$ is an hyperbola.

\item[21.] Tangents are drawn to an hyperbola, and the portion of each
tangent intercepted by the asymptotes is divided in a constant ratio;
prove that the locus of the point of section is an hyperbola.

\item[22.] If the tangent and normal at $P$ meet the conjugate axis in $t$
and $K$ respectively, prove that a circle can be drawn through the foci
and the three points $P$, $t$, $K$.

Shew also that
\begin{flalign*}
&& GK : SK &:: SA : AX,\\
&\text{and}& St : tK &:: BC : CD, &\phantom{\text{and}}
\end{flalign*}
$CD$ being conjugate to $CP$.

\item[23.] Shew that the points of trisection of a series of conterminous
circular arcs lie on branches of two hyperbolas; and determine the
distance between their centres.

\item[24.] If the tangent at any point $P$ cut an asymptote in $T$, and if
$SP$ cut the same asymptote in $Q$, then $SQ=QT$.

\item[25.] A series of hyperbolas having the same asymptotes is cut by
a straight line parallel to one of the asymptotes, and through the
points of intersection lines are drawn parallel to the other, and equal
to either semi-axis of the corresponding hyperbola: prove that the
locus of their extremities is a parabola.

\item[26.] Prove that the rectangle $PY \ctrdot PY'$ in an ellipse is equal to
the square on the conjugate axis of the confocal hyperbola passing
through $P$.
%-----File: 137.png-----

\item[27.] If the tangent at $P$ meet one asymptote in $T$, and a line $TQ$
be drawn parallel to the other asymptote to meet the curve in $Q$; prove
that if $PQ$ be joined and produced both ways to meet the asymptotes
in $R$ and $R'$, $RR'$ will be trisected at the points $P$ and $Q$.

\item[28.] The tangent at a point $P$ of an ellipse meets the hyperbola
having the same axes as the ellipse in $C$ and $D$. If $Q$ be the middle
point of $CD$, prove that $OQ$ and $OP$ are equally inclined to the axes, $O$
being the centre of the ellipse.

\item[29.] Given one asymptote, the direction of the other, and the
position of one focus, determine the position of the vertices.

\item[30.] Two points are taken on the same branch of the curve, and on
the same side of the axis; prove that a circle can be drawn touching
the four focal distances.

\item[31.] Supposing the two asymptotes and one point of the curve to
be given in position, shew how to construct the curve; and find the
position of the foci.

\item[32.] Given a pair of conjugate diameters, construct the axes.

\item[33.] If $PH$, $PK$ be drawn parallel to the asymptotes from a point
$P$ on the curve, and if a line through the centre meet them in $R$, $T$,
and the parallelogram $PRQT$ be completed, $Q$ is a point on the curve.

\item[34.] The ordinate $NP$ at any point of an ellipse is produced to a
point $Q$, such that $NQ$ is equal to the sub-tangent at $P$; prove that the
locus of $Q$ is an hyperbola.

\item[35.] If a given point be the focus of any hyperbola, passing through
a given point and touching a given straight line, prove that the locus of
the other focus is an arc of a fixed hyperbola.

\item[36.] An ellipse and hyperbola are described, so that the foci of
each are at the extremities of the transverse axis of the other; prove
that the tangents at their points of intersection meet the conjugate
axis in points equidistant from the centre.

\item[37.] A circle is described about the focus as centre, with a radius
equal to one-fourth of the latus rectum; prove that the focal distances
of the points at which it intersects the hyperbola are parallel to the
asymptotes.

\item[38.] The tangent at any point forms a triangle with the asymptotes:
determine the locus of the point of intersection of the straight lines
drawn from the angles of this triangle to bisect the opposite sides.

\item[39.] If $SY$, $S'Y'$ be the perpendiculars on the tangent at $P$, a circle
can be drawn through the points $Y$, $Y'$, $N$, $C$.

\item[40.] The straight lines joining each focus to the foot of the perpendicular
from the other focus on the tangent meet on the normal and
bisect it.
%-----File: 138.png-----

\item[41.] If the tangent and normal at $P$ meet the axis in $T$ and $G$,
$NG \ctrdot CT=BC^2$.

\item[42.] If the tangent at $P$ meet the axes in $T$ and $t$, the angles $PSt$,
$STP$ are supplementary.

\item[43.] If the tangent at $P$ meet any conjugate diameters in $T$ and $t$,
the triangles $SPT$, $S'Pt$ are similar.

\item[44.] If the diameter conjugate to $CP$ meet $SP$ and $S'P$ in $E$ and
$E'$, prove that the circles about the triangles $SCE$, $S'CE'$ are equal.

\item[45.] The locus of the centre of the circle inscribed in the triangle
$SPS'$ is a straight line.

\item[46.] If $PN$ be an ordinate, and $NQ$ parallel to $AP$ meet $CP$ in $Q$,
$AQ$ is parallel to the tangent at $P$.

\item[47.] If an asymptote meet the directrix in $D$, and the tangent at
the vertex in $E$, $AD$ is parallel to $SE$.

\item[48.] The radius of the circle touching the curve and its asymptotes
is equal to the portion of the latus rectum produced, between its
extremity and the asymptote.

\item[49.] If $G$ be the foot of the normal, and if the tangent meet the
asymptotes in $L$ and $M$, $GL=GM$.

\item[50.] With two conjugate diameters of an ellipse as asymptotes, a
pair of conjugate hyperbolas is constructed: prove that if one hyperbola
touch the ellipse, the other will do so likewise; prove also that the
diameters drawn through the points of contact are conjugate to each
other.

\item[51.] If two tangents be drawn the lines joining their intersections
with the asymptotes will be parallel.

\item[52.] The locus of the centre of the circle touching $SP$, $S'P$ produced,
and the major axis, is an hyperbola.

\item[53.] If from a point $P$ in an hyperbola, $PK$ be drawn parallel to
an asymptote to meet the directrix in $K$, then $PK=SP$.

\item[54.] If $PD$ be drawn parallel to an asymptote, to meet the conjugate
hyperbola in $D$, $CP$ and $CD$ are conjugate diameters.

\item[55.] If $QR$ be a chord parallel to the tangent at $P$, and if $QL$, $PN$,
$RM$ be drawn parallel to one asymptote to meet the other,
\[
CL \ctrdot CM= CN^2.
\]

\item[56.] If a circle touch the transverse axis at a focus, and pass
through one end of the conjugate, the chord intercepted by the
conjugate is a third proportional to the conjugate and transverse
semi-axes.
%-----File: 139.png-----

\item[57.] A line through one of the vertices, terminated by two lines
drawn through the other vertex parallel to the asymptotes, is bisected
at the other point where it cuts the curve.

\item[58.] If $PSQ$ be a focal chord, and if the tangents at $P$ and $Q$ meet
in $T$, the difference between $PTQ$ and half $PS'Q$ is a right angle.

\item[59.] If a straight line passing through a fixed point $C$ meet two
fixed lines $OA$, $OB$ in $A$ and $B$, and if $P$ be taken in $AB$ such that
$CP^2 = CA \ctrdot CB$, the locus of $P$ is an hyperbola, having its asymptotes
parallel to $OA$, $OB$.

\item[60.] If from the points $P$ and $Q$ in an hyperbola there be
drawn $PL$, $QM$ parallel to each other to meet one asymptote, and
$PR$, $QN$ also parallel to each other to meet the other asymptote,
$PL \ctrdot PR=QM \ctrdot QN$.

\item[61.] Prove that the locus of the point of intersection of two
tangents to a parabola which cut at a constant angle is an hyperbola,
and that the angle between its asymptotes is double the external angle
between the tangents.

\item[62.] An ordinate $VQ$ of any diameter $CP$ is produced to meet the
asymptote in $R$, and the conjugate hyperbola in $Q'$; prove that
\[
QV^2 + Q'V^2 = 2RV^2.
\]

Prove also that the tangents at $Q$ and $Q'$ meet the diameter $CP$ in
points equidistant from $C$.

\item[63.] A chord $QPL$ meets an asymptote in $L$, and a tangent from $L$
is drawn touching at $R$; if $PM$, $RE$, $QN$, be drawn parallel to the
asymptote to meet the other,
\[
PM+QN=2 \ctrdot RE.
\]

\item[64.] Tangents are drawn from any point in a circle through the
foci; prove that the lines bisecting the angle between the tangents,
or between one tangent and the other produced, all pass through a
fixed point.

\item[65.] If a circle through the foci meet two confocal hyperbolas
in $P$ and $Q$, the angle between the tangents at $P$ and $Q$ is equal
to $PSQ$.

\item[66.] If $SY$, $S'Y'$ be perpendiculars on the tangent at $P$, and if $PN$
be the ordinate, the angles $PNY$, $PNY'$ are supplementary.

\item[67.] Find the position of $P$ when the area of the triangle $YCY'$ is
the greatest possible, and shew that, in that case,
\[
PN \ctrdot SC=BC^2.
\]

\item[68.] If the tangent at $P$ meet the conjugate axis in $t$, the areas of
the triangles $SPS'$, $StS'$ are in the ratio of $CD^2 : St^2$.
%-----File: 140.png-----

\item[69.] If $SY$, $SZ$ be perpendiculars on two tangents which meet in $T$,
$YZ$ is perpendicular to $S'T$.

\item[70.] A circle passing through a focus, and having its centre on the
transverse axis, touches the curve; shew that the focal distance of the
point of contact is equal to the latus rectum.

\item[71.] If $CQ$ be conjugate to the normal at $P$, then is $CP$ conjugate
to the normal at $Q$.

\item[72.] From a point in the auxiliary circle lines are drawn touching
the curve in $P$ and $P'$; prove that $SP$, $S'P'$ are parallel.

\item[73.] If any hyperbola is drawn confocal with a given ellipse, and
if $PN$ is the ordinate of a point of intersection of the hyperbola with
the ellipse, and $NT$ the tangent from $N$ to the auxiliary circle of the
hyperbola, prove that the angle $PNT$ is always the same.

\item[74.] Find the locus of the points of contact of tangents to a series
of confocal hyperbolas from a fixed point in the axis.

\item[75.] Tangents to an hyperbola are drawn from any point in one
of the branches of the conjugate, shew that the chord of contact will
touch the other branch of the conjugate.

\item[76.] An ordinate $NP$ meets the conjugate hyperbola in $Q$; prove
that the normals at $P$ and $Q$ meet on the transverse axis.

\item[77.] A parabola and an hyperbola have a common focus $S$ and
their axes in the same direction. If a line $SPQ$ cut the curves in $P$
and $Q$, the angle between the tangents at $P$ and $Q$ is equal to half the
angle between the axis and the other focal distance of the hyperbola.

\item[78.] If an hyperbola be described touching the four sides of a
quadrilateral which is inscribed in a circle, and one focus lie on the
circle, the other focus will also lie on the circle.

\item[79.] A conic section is drawn touching the asymptotes of an
hyperbola.  Prove that two of the chords of intersection of the
curves are parallel to the chord of contact of the conic with the
asymptotes.

\item[80.] A parabola $P$ and an hyperbola $H$ have a common focus,
and the asymptotes of $H$ are tangents to $P$; prove that the tangent
at the vertex of $P$ is a directrix of $H$, and that the tangent to $P$
at the point of intersection passes through the further vertex of $H$.

\item[81.] From a given point in an hyperbola draw a straight line
such that the segment intercepted between the other intersection
with the hyperbola and a given asymptote shall be equal to a
given line.

When does the problem become impossible?
%-----File: 141.png-----

\item[82.] If an ellipse and a confocal hyperbola intersect in $P$, an
asymptote passes through the point on the auxiliary circle of the
ellipse corresponding to $P$.

\item[83.] $P$ is a point on an hyperbola whose foci are $S$ and $H$; another
hyperbola is described whose foci are $S$ and $P$, and whose transverse
axis is equal to $SP-2PH$: shew that the hyperbolas will meet only at
one point, and that they will have the same tangent at that point.

\item[84.] $A$ point $D$ is taken on the axis of an hyperbola, of which the
eccentricity is~$2$, such that its distance from the focus $S$ is equal to
the distance of $S$ from the further vertex $A'$; $P$ being any point on the
curve, $A'P$ meets the latus rectum in $K$. Prove that $DK$ and $SP$
intersect on a certain fixed circle.

\item[85.] Shew that the locus of the point of intersection of tangents
to a parabola, making with each other a constant angle equal to half a
right angle, is an hyperbola.

\item[86.] The tangent and normal at any point intersect the asymptotes
and axes respectively in four points which lie on a circle passing
through the centre of the curve.

The radius of this circle varies inversely as the perpendicular from
the centre on the tangent.

\item[87.] The difference between the sum of the squares of the distances
of any point from the ends of any diameter and the sum of the squares
of its distances from the ends of the conjugate is constant.

\item[88.] If a tangent meet the asymptotes in $L$ and $M$, the angle
subtended by $LM$ at the farther focus is half the angle between the
asymptotes.

\item[89.] If $PN$ be the ordinate of $P$, and $PT$ the tangent, prove that
$SP : ST :: AN : AT$.

\item[90.] If an ellipse and an hyperbola are confocal, the asymptotes
pass through the points on the auxiliary circle of the ellipse which
correspond to the points of intersection of the two curves.

\item[91.] Two adjacent sides of a quadrilateral are given in magnitude
and position; if the quadrilateral be such that a circle can be inscribed
in it, the locus of the point of intersection of the other two sides is
an hyperbola.

\item[92.] The tangent at $P$ meets the conjugate axis in $t$, and $tQ$ is
perpendicular to $SP$; prove that $SQ$ is of constant length.

\item[93.] An hyperbola, having a given transverse axis, has one focus
fixed, and always touches a given straight line; the locus of the other
focus is a circle.

\item[94.] A chord $PRVQ$ meets the directrices in $R$ and $V$; shew that
$PR$ and $VQ$ subtend, each at the focus nearer to it, angles of which the
sum is equal to the angle between the tangents at $P$ and $Q$.
%-----File: 142.png-----

\item[95.] A circle is drawn touching the transverse axis of an hyperbola
at its centre, and also touching the curve; prove that the diameter
conjugate to the diameter through either point of contact is equal to
the distance between the foci.

\item[96.] A parabola is described touching the conjugate axes of an
hyperbola at their extremities; prove that one asymptote is parallel to
the axis of the parabola, and that the other asymptote is parallel to
the chords of the parabola bisected by the first.

If a straight line parallel to the second asymptote meet the hyperbola
and its conjugate in $P$, $P'$, and the parabola in $Q$, $Q'$, it may be
shewn that $PQ=P'Q'$.

\item[97.] If two points $E$ and $E'$ be taken in the normal $PG$ such that
$PE=PE'=CD$, the loci of $E$ and $E'$ are hyperbolas having their axes
equal to the sum and difference of the axes of the given hyperbola.

\item[98.] If two tangents are drawn to the same branch of an hyperbola,
the external angle between them is half the difference between the
angles which the chord of contact subtends at the foci.

If the tangents are drawn to opposite branches, the angle between
them is half the sum, or half the difference, of these angles according as
the points of contact are on the same or on opposite sides of the
transverse axis.

\item[99.] Parabolas are drawn passing through two fixed points $A$ and
$B$, and having their axes in a given direction; find the locus of the foci,
and, if a tangent be drawn at right angles to $AB$, prove that the locus
of its point of contact $P$ is an hyperbola.

\item[100.] Tangents are drawn from a point $T$ to an hyperbola whose
centre is $C$, and $CT$ produced meets the hyperbola in $P$ and the chord
of contact of the tangents in $V$. If $CD$ be the diameter conjugate to
$CP$, and $DT$, $DV$ meet the tangent at $P$ in $K$ and $U$, prove that the
triangles $PUV$, $TPK$ are equal in area.

\item[101.] One asymptote and three points $P$, $Q$, $R$ of an hyperbola are
given, construct the other asymptote.

\item[102.] If an ellipse be described having its centre on a given
hyperbola, its foci on the asymptotes, and passing through the centre
of the hyperbola, prove that the minor axis of the ellipse is equal to
the major axis of the hyperbola, and the ellipse touches the minor axis
of the hyperbola.

\item[103.] The angular point $A$ of a triangle $ABC$ is fixed, and the
angle $A$ is given, while the points $B$ and $C$ move on a fixed straight
line; prove that the locus of the centre of the circle circumscribing the
triangle is an hyperbola, and that the envelope of the circle is another
circle.

\item[104.] Given an asymptote $CQ$ and two points on an hyperbola,
$P$, $p$ on the curve, shew that the envelope of the axes is a parabola.
%-----File: 143.png-----

\item[105.] Find the locus of the middle points of a system of chords of
an hyperbola, passing through a fixed point on one of the asymptotes.

\item[106.] If a conic be described having for its axes the tangent and
normal at any point of a given ellipse, and touching at its centre the
axis-major of the given ellipse, and if another conic be described in
the same manner, but touching the minor axis at the centre, prove
that the foci of these conics lie in two circles concentric with the
given ellipse, and having their diameters equal to the sum and
difference of its axes.

\item[107.] An ellipse and an hyperbola are confocal; if a tangent to
one intersect at right angles a tangent to the other, the locus of the
point of intersection is a circle.

Shew also that the difference of the squares on the distances from
the centre of parallel tangents is constant.

\item[108.] If a circle passing through any point $P$ of the curve, and
having its centre on the normal at $P$, meets the curve again in $Q$ and
$R$, the tangents at $Q$ and $R$ intersect on a fixed straight line.

\item[109.] If the tangent at $P$ meet an asymptote in $T$, the angle
between that asymptote and $S'P$ is double the angle $STP$.

\item[110.] Four tangents to an hyperbola form a rectangle. If one side
$AB$ of the rectangle intersect a directrix in $F$, and $S$ be the corresponding
focus, the triangles $FSA$, $FBS$ are similar.

\item[111.] An ellipse and hyperbola have the same transverse axis,
and their eccentricities are the reciprocals of one another; prove that
the tangents to each through the focus of the other intersect at right
angles in two points and also meet the conjugate axis on the auxiliary
circle.

\item[112.] $ACA'$ and $BCB'$ are the transverse and conjugate axes of an
ellipse, of which $S$ and $S'$ are the foci. $P$ is one of the points of
intersection of this ellipse and a confocal hyperbola, and $aCa'$ is the
transverse axis of the hyperbola.

Prove that     $SP=Aa$, $S'P=A'a$, and $aB=CP$.

\item[113.] Prove that if $A$, $B$ and $S$ are three given points, two parabolas
can be drawn through $A$ and $B$ with $S$ as focus, and that the axes of
these parabolas are parallel to the asymptotes of the hyperbola which
can be drawn through $S$ with its foci at $A$ and $B$.

\end{exampleslist}
%-----File: 144.png-----

\cchapter{CHAPTER V}

\csectiontoc{The Rectangular Hyperbola}
\cheader{THE RECTANGULAR HYPERBOLA}

\textit{If the axes of an hyperbola be equal, the angle between the
asymptotes is a right angle, and the curve is called equilateral
or rectangular.}

\carticleplus{135.}{Prop.~I\@.} \textit{In a rectangular hyperbola}%\\
\[
CS^2 = 2AC^2, \text{ and }SA^2 = 2AX^2.
\]
\indent For \hfill $CS^2 = AC^2 + BC^2 = 2AC^2$,\hfill\phantom{For\indent}\\
and \hfill $SA : AX :: SC : AC$; \hfill\phantom{and}
\[
\therefore SA^2 = 2AX^2.
\]

Observe that, in the figure of Art.~\cartref{102}, $SDC$ is an isosceles
triangle, since
\[
SD=BC, \text{ and } CD=AC,
\]
and therefore \hfill $SD = DC$. \hfill\phantom{and therefore}

\carticleplus{136.}{Prop.~II\@.} \textit{The asymptotes of a rectangular hyperbola
bisect the angles between any pair of conjugate diameters.}

For, in a rectangular or equilateral hyperbola,
\begin{flalign*}
&& CA &= CB,\\
&\text{and therefore, since}& CP^2 - CD^2 & =CA^2 - CB^2,
    &\phantom{and therefore, since} \\
&& CP &= CD,
\end{flalign*}
$CP$, $CD$ being any conjugate semi-diameters.
%-----File: 145.png-----

Also, in the figure of Art.~\cartref{123}, the parallelogram $CPLD$
is a rhombus, and therefore $CL$ bisects the angle $PCD$.

\subparagraph{Cor.} Supplemental chords are equally inclined to the
asymptotes, for they are parallel to conjugate diameters.

\includegraphicsright[16]{137a}

\carticleplus{137.}{Prop.~III\@.} \textit{If $CY$ be the perpendicular from the
centre on the tangent at $P$, the angle $PCY$ is bisected by the
transverse axis, and half the transverse axis is a mean proportional
between $CY$ and $CP$.}

\begin{flalign*}
\indent\text{For the angle } PCL &= DCL   && \\
                                 &= YCL'; && \\
             \therefore PCA &= ACY.  &&
\end{flalign*}

Hence it follows that the
triangles $PCN$, $TCY$ are similar,
and that
\begin{gather*}
CY:CT::CN:CP; \\
\therefore CY \ctrdot CP=CT \ctrdot CN = AC^2.
\end{gather*}

Hence also, if we join $PA$
and $AY$, we observe that the
triangles $PAC$, $AYC$ are
similar.

\carticleplus{138.}{Prop.~IV\@.} \textit{Diameters at right angles to each other
are equal.}

Let $CP$, $CP'$ be semi-diameters at right angles to each
other, and $CD$ conjugate to $CP$.

Then, if $CL$, $CL'$ be the asymptotes,
\begin{gather*}
\text{the angle } P'CL' = PCL = DCL; \\
\therefore CP' = CD = CP.
\end{gather*}

Hence it follows, by help of the theorem of Art.~\cartref{120}, that
focal chords at right angles to each other are equal, and that
focal chords parallel to conjugate diameters are equal.
%-----File: 146.png-----

\carticleplus{139.}{Prop.~V\@.} \textit{If the normal at $P$ meet the axes in $G$
and $g$,}
\[
CN=NG  \text{ and } PG=Pg=CD,
\]
\textit{$CD$ being conjugate to $CP$.}

\indent{For (Art.~\cartref{115})}  $NG: CN :: BC^2 : AC^2$;
\[ \therefore NG = CN. \]
\indent{Also} \quad $PF \ctrdot PG= BC^2$  and  $PF \ctrdot Pg = AC^2$;
\[ \therefore PG = Pg. \]
\indent{Further (Art.~\cartref{128})} $PG: CD:: BC: AC$;
\[ \therefore PG = CD = CP. \]

\carticleplus{140.}{Prop.~VI\@.} \textit{If $QV$ be an ordinate of a diameter
$PCp$,}
\[
QV^2 = PV \ctrdot Vp.
\]
\begin{flalign*}
\indent & \text{For }&   QV^2 : PV \ctrdot Vp &:: CD^2 : CP^2,&&\\
 \text{and }&&                   CD &= CP;&&\\
&& QV^2 = PV \ctrdot Vp &= CV^2 - CP^2.&&
\end{flalign*}

\carticleplus{141.}{Prop.~VII\@.} \textit{The angle between a chord $PQ$, and
the tangent at $P$, is equal to the angle subtended by $PQ$ at the
other extremity of the diameter through $P$.}

\includegraphicsmid{141a}

Let $PQ$ and the tangent at $P$ meet the asymptote in $l$
and $L$. Then, if $CV$ be conjugate to $PQ$,
\begin{align*}
\text{the angle } LPQ &= PLC-VlC=LCP-VCl&&\\
&= VCP = QpP.&&
\end{align*}
%-----File: 147.png-----

\begin{small}
Or thus, let $QU$ parallel to the tangent at $P$, meet $CP$ produced
in $U$.

Then
\[
QU^2 = PU \ctrdot Up,
\]
or,
\[
QU: PU:: Up: UQ.
\]

Therefore the triangles $PQU$, $QUp$ are similar, and the angle
$QpU = PQU = LPQ$.

\end{small}

If $P$ and $Q$ are on opposite branches of the curve, the
same proof shews that
\[
\text{the angle } QpU= UQP = LPQ;
\]
\[
\therefore QPL' = QpP.
\]

\includegraphicsmid{141b}

If $QP$ is the normal at $P$, it follows that $QP$ subtends a
right angle at the other end of the diameter through $P$.

\carticleplus{142.}{Prop.~VIII\@.} \textit{Any chord subtends, at the ends of
any diameter, angles which are equal or supplementary.}

{\stretchyspace
\includegraphicsright[8]{142a}

This theorem divides itself
into four cases, which are
shewn in the appended figures.

Let $QR$ be the chord, and
$Pp$ the diameter. Then, if $LP$
be the tangent at $P$, fig.~(1),
\begin{flalign*}
\text{the angle}\quad LPQ &= QpP,\\
\text{and}\quad LPR&=RpP;\\
\therefore QPR &= QpR.
\end{flalign*}
%-----File: 148.png-----
In fig. (2), if $pl$ be the tangent at $p$, parallel to $PL$,
\[
QpR = Qpl + lpR = Qpl + pPR,
\]
\includegraphicsmid{142b}
\begin{align*}
\text{and }\qquad\qquad QPR = QPL + LPR& = QpP + LPR;\\
\therefore QpR + QPR =&lpP + LPp,
\end{align*}
that is, $QpR$ and $QPR$ are together equal to two right
angles.
} %end \stretchyspace

\includegraphicsright[8]{142c}

In fig. (3)
\begin{align*}
QPR &= QPL + LPp + pPR\\
&= QpP + Ppl + lpR\\
&= QpR.
\end{align*}

\begin{align*}
\text{In fig. (4) } QPL &= QpP, \text{ and } RPL' = RpP;\\
&\therefore QpR = QPL + RPL';
\end{align*}
therefore $QpR$ and $QPR$ are together
equal to two right angles.

\includegraphicsright[9]{142d}

Hence it will be seen that
when $QR$, or $QR$ produced, meet
the diameter $Pp$ between $P$ and
$p$, the angles subtended at $P$ and
$p$ are equal; in other cases they
are supplementary.

In the cases of the second and
third figures, if one of the angles $QPR$ is a right angle,
the other angle $QpR$ is also a right angle. The four points
$Q$, $P$, $p$, $R$ are then concyclic, and $QR$ is a diameter of the
circle.
%-----File: 149.png-----

\carticleplus{143.}{Prop.~IX\@.} \textit{If a rectangular hyperbola circumscribe
a triangle, it passes through the orthocentre.}

\subparagraph{Note.} \textit{The orthocentre is the point of intersection of the
perpendiculars from the angular points on the opposite sides.}

{\stretchyspace%
\includegraphicsright[8]{143a}

If $O$ be the orthocentre, the
triangles $LOP$, $LQR$ are similar,
and
\begin{flalign*}
LO : LP &:: LQ : LR;\\
\therefore LO \ctrdot LR&=LP \ctrdot LQ.
\end{flalign*}

But, if a rectangular hyperbola
pass through $P$, $Q$, $R$, the
diameters parallel to $LR$, $PQ$ are
equal: hence $O$ is a point on the curve.

If the angle $PRQ$ is a right angle, the line $ROL$ will be
the tangent to the curve at $R$, so that if a rectangular
hyperbola pass through the angular points of a right-angled
triangle, the hypotenuse will be parallel to the normal at the
right-angle vertex.} %end stretchyspace

\carticleplus{144.}{Prop.~X\@.} \textit{If a rectangular hyperbola circumscribe
a triangle, the locus of its centre is the nine-point circle of the
triangle.}

\includegraphicsmid{144a}

If $PQR$ be the triangle, let $L$, $L'$ be the points in which
an asymptote meets the sides $PQ$, $PR$.
%-----File: 150.png-----

Join $C$, the centre of the hyperbola, with $E$ and $F$, the
middle points of $PR$ and $PQ$.

Then $CF$ is conjugate to $PQ$, and $CE$ to $PR$; therefore
the angle
\begin{align*}
FCE &= FCL + L'CE = CLF+ EL'C\\
&= PLL' + PL'L = FPE\\
&=FDE,
\end{align*}
if $D$ be the middle point of $QR$.

$\therefore D$, $E$, $F$, $C$ are concyclic; that is, $C$ lies on the nine-point
circle.

A similar proof is applicable to the case in which the
points $P$, $Q$, $R$ lie on the same branch of the hyperbola.

\csubsection{EXAMPLES}

\begin{exampleslist}
\item[1.] $PCP$ is a transverse diameter, and $QV$ an ordinate; shew that
$QV$ is the tangent at $Q$ to the circle circumscribing the triangle $PQp$.
\cheader{EXAMPLES}

\item[2.] If the tangent at $P$ meet the asymptotes in $L$ and $M$, and the
normal meet the transverse axis in $G$, a circle can be drawn through
$C$, $L$, $M$, and $G$, and $LGM$ is a right angle.

\item[3.] If $AA'$ be any diameter of a circle, $PP'$ any ordinate to it, then
the locus of the intersections of $AP$, $A'P'$ is a rectangular hyperbola.

\item[4.] Given an asymptote and a tangent at a given point, construct
the rectangular hyperbola.

\item[5.] The points of intersection of an ellipse and a confocal rectangular
hyperbola are the extremities of the equi-conjugate diameters of the
ellipse.

\item[6.] If $CP$, $CD$ be conjugate semi-diameters, and $PN$, $DM$ ordinates
of any diameter, the triangles $PCN$, $DCM$ are equal in all respects.

\item[7.] The distance of any point from the centre is a geometric mean
between its distances from the foci.
%-----File: 151.png-----

\item[8.] If $P$ be a point on an equilateral hyperbola, and if the tangent
at $Q$ meet $CP$ in $T$, the circle circumscribing $CTQ$ touches the ordinate
$QV$ conjugate to $CP$.

\item[9.] If a circle be described on $SS'$ as diameter, the tangents at the
vertices will intersect the asymptotes in the circumference.

\item[10.] If two concentric rectangular hyperbolas be described, the
axes of one being the asymptotes of the other, they will intersect at
right angles.

\item[11.] If the tangents at two points $Q$ and $Q'$ meet in $T$, and if $CQ$,
$CQ'$ meet these tangents in $R$ and $R'$, the points $R$, $T$, $R'$, $C$ are concyclic.

\item[12.] If from a point $Q$ in the conjugate axis $QA$ be drawn to the
vertex, and $QR$ parallel to the transverse axis to meet the curve,
$QR=AQ$.

\item[13.] Straight lines, passing through a given point, are bounded by
two fixed lines at right angles to each other; find the locus of their
middle points.

\item[14.] Given a point $Q$ and a straight line $AB$, if a line $QCP$ be
drawn cutting $AB$ in $C$, and $P$ be taken in it, so that, $PD$ being a
perpendicular upon $AB$, $CD$ may be of constant magnitude, the locus
of $P$ is a rectangular hyperbola.

\item[15.] Every conic passing through the centres of the four circles
which touch the sides of a triangle, is a rectangular hyperbola.

\item[16.] Ellipses are inscribed in a given parallelogram, shew that their
foci lie on a rectangular hyperbola.

\item[17.] If two focal chords be parallel to conjugate diameters, the lines
joining their extremities intersect on the asymptotes.

\item[18.] If $P$, $Q$ be two points of a rectangular hyperbola, centre $O$,
and $QN$ the perpendicular let fall on the tangent at $P$, the circle
through $O$, $N$, and $P$ will pass through the middle point of the chord
$P$, $Q$.

\item[19.] Having given the centre, a tangent, and a point of a rectangular
hyperbola, construct the asymptotes.

\item[20.] If a right-angled triangle be inscribed in the curve, the normal
at the right angle is parallel to the hypotenuse.

\item[21.] On opposite sides of any chord of a rectangular hyperbola are
described equal segments of circles; shew that the four points, in which
the circles, to which these segments belong, again meet the hyperbola,
are the angular points of a parallelogram.

\item[22.] Two lines of given lengths coincide with and move along two
fixed lines, in such a manner that a circle can always be drawn through
their extremities; the locus of the centre is a rectangular hyperbola.
%-----File: 152.png-----

\item[23.] If a rectangular hyperbola, having its asymptotes coincident
with the axes of an ellipse, touch the ellipse, the axis of the hyperbola
is a mean proportional between the axes of the ellipse.

\item[24.] The tangent at a point $P$ of a rectangular hyperbola meets a
diameter $QCQ'$ in $T$. Shew that $CQ$ and $TQ'$ subtend equal angles at $P$.

\item[25.] If $A$ be any point in a rectangular hyperbola, of which $O$ is the
centre, $BOC$ the straight line through $O$ at right angles to $OA$, $D$ any
other point in the curve, and $DB$, $DC$ parallel to the asymptotes, prove
that $B$, $D$, $A$, $C$ are concyclic.

\item[26.] The angle subtended by any chord at the centre is the supplement
of the angle between the tangents at the ends of the chord.

\item[27.] If two rectangular hyperbolas intersect in $A$, $B$, $C$, $D$; the
circles described on $AB$, $CD$ as diameters intersect each other orthogonally.

\item[28.] Prove that the triangle, formed by the tangent at any point
and its intercepts on the axes, is similar to the triangle formed by the
straight line joining that point with the centre, and the abscissa and
ordinate of the point.

\item[29.] The angle of inclination of two tangents to a parabola is half
a right angle; prove that the locus of their point of intersection is a
rectangular hyperbola, having one focus and the corresponding directrix
coincident with the focus and directrix of the parabola.

\item[30.] $P$ is a point on the curve, and $PM$, $PN$ are straight lines
making equal angles with one of the asymptotes; if $MP$, $NP$ be produced
to meet the curve in $P'$ and $Q'$, then $P'Q'$ passes through the
centre.

\item[31.] A circle and a rectangular hyperbola intersect in four points
and one of their common chords is a diameter of the hyperbola; shew
that the other common chord is a diameter of the circle.

\item[32.] $AB$ is a chord of a circle and a diameter of a rectangular
hyperbola; $P$ any point on the circle; $AP$, $BP$, produced if necessary,
meet the hyperbola in $Q$, $Q'$, respectively; the point of intersection of
$BQ$, $AQ'$ will be on the circle.

\item[33.] $PP'$ is any diameter, $Q$ any point on the curve, $PR$, $P'R'$ are
drawn at right angles to $PQ$, $P'Q$ respectively, intersecting the normal
at $Q$ in $R$, $R'$; prove that $QR$ and $QR'$ are equal.

\item[34.] Parallel tangents are drawn to a series of confocal ellipses;
prove that the locus of the points of contact is a rectangular hyperbola
having one of its asymptotes parallel to the tangents.

\item[35.] If tangents, parallel to a given direction, are drawn to a
system of circles passing through two fixed points, the points of contact
lie on a rectangular hyperbola.
%-----File: 153.png-----

\item[36.] If from a point $P$ on the curve chords are equally inclined to
the asymptotes, the  line joining their other extremities passes through
the centre.

\item[37.] From the point of intersection of the directrix with one of
the asymptotes of a rectangular hyperbola a tangent is drawn to the
curve and meets the other asymptote in $T$; shew that $CT$ is equal to
the transverse axis.

\item[38.] The normals at the ends of two conjugate diameters intersect
on the asymptote, and are parallel to another pair of conjugate
diameters.

\item[39.] If the base $AB$ of a triangle $ABC$ be fixed, and if the difference
of the angles at the base is constant, the locus of the vertex is a
rectangular hyperbola.

\item[40.] A circle described through the angular points $A$, $B$ of a given
triangle $ABC$ meets $AC$ in $D$. If $BD$ meet the tangent at $A$ in $P$,
shew that the vertex and orthocentre of the triangle $APB$ lie on fixed
rectangular hyperbolas.

\item[41.] The locus of the point of intersection of tangents to an ellipse
which make equal angles with the transverse and conjugate axes
respectively, and are not at right angles, is a rectangular hyperbola
whose vertices are the foci of the ellipse.

\item[42.] If $OT$ is the tangent at the point $O$ of a rectangular hyperbola,
and $PQ$ a chord meeting it at right angles in $T$, the two bisectors of
the angle $OCT$ bisect $OP$ and $OQ$.

\item[43.] With two sides of a square as asymptotes, and the opposite
point as focus, a rectangular hyperbola is described; prove that it
bisects the other sides.

\item[44.] With the focus $S$ of a rectangular hyperbola as centre and
radius equal to $SC$ a circle is described, prove that it touches the
conjugate hyperbola.

\item[45.] If parallel normal chords are drawn to a rectangular hyperbola,
the diameter bisecting them is perpendicular to the join of their
feet.

\item[46.] From the foot of the ordinate $PN$ of a point $P$ of a rectangular
hyperbola, tangents $NQ$, $NR$ are drawn to the circle on $AA'$ as
diameter. Prove that $PQ$ passes through $A'$, and $PR$ through $A$, and
that, if $QR$ intersect $AA'$ in $M$, $PM$ is the tangent at $P$.

\item[47.] Shew that the angle between two tangents to a rectangular
hyperbola is equal or supplementary to the angle which their chord of
contact subtends at the centre, and that the bisectors of these angles
meet on the chord of contact.
%-----File: 154.png-----

\item[48.] Through a point $P$ on an equilateral hyperbola two lines are
drawn parallel to a pair of conjugate diameters; the one meeting the
curve in $P$, $P'$, and the other meeting the asymptotes in $Q$, $Q'$; shew
that $PP' = QQ'$.

\item[49.] If four points forming a parallelogram be taken on a rectangular
hyperbola, then the product of the perpendiculars from any
point of the curve on one pair of opposite sides equals the product of
the perpendiculars on the other pair of sides.
\end{exampleslist}
%-----File: 155.png-----

\cchapter{CHAPTER VI}

\csectiontoc{The Cylinder and the Cone}
\cheader{THE CYLINDER}

\csubsection{DEFINITION}

\carticle{145.} If a straight line move so as to pass through the
circumference of a given circle, and to be perpendicular to
the plane of the circle, it traces out a surface called a \textit{Right
Circular Cylinder}. The straight line drawn through the
centre of the circle perpendicular to its plane is the \textit{Axis} of
the Cylinder.

It is evident that a section of the surface by a plane
perpendicular to the axis is a circle, and that a section by
any plane parallel to the axis consists of two parallel lines.

\subparagraph{Prop.~I\@.} \textit{Any section of a cylinder by a plane not
parallel or perpendicular to the axis is an ellipse.}

If $APA'$ be the section, let the plane of the paper be the
plane through the axis perpendicular to $APA'$.

Inscribe in the cylinder a sphere touching the cylinder in
the circle $EF$ and the plane $APA'$ in the point $S$.

Let the planes $APA'$, $EF$ intersect in $XK$, and from any
point $P$ of the section draw $PK$ perpendicular to $XK$.

Draw through $P$ the circular section $QP$, cutting $APA'$
in $PN$, so that $PN$ is at right angles to $AA'$ and therefore
parallel to $XK$.

Let the generating line through $P$ meet the circle $EF$ in
$R$; and join $SP$.

Then $PS$ and $PR$ are tangents to the sphere;
\[
\therefore SP = PR = EQ.
\]
%-----File: 156.png-----

\begin{flalign*}
\indent & \text{But} & EQ : NX &:: AE : AX &&\\
        &&&:: SA : AX,&&\\
\text{and } &&     NX &= PK,&&\\
&&\therefore  SP : PK &:: SA : AX.&&\\
\end{flalign*}

\includegraphicsmid{145a}

Also, $AE$ being less than $AX$, $SA$ is less than $AX$, and
the curve $APA'$ is therefore an ellipse, of which $S$ is the
focus and $XK$ the directrix.

If another sphere be inscribed in the cylinder touching
$AA'$ in $S'$, $S'$ is the other focus, and the corresponding directrix
is the intersection of the plane of contact $E'F'$ with
$APA'$.

Producing the generating line $RP$ to meet the circle
$E'F'$ in $R'$ we observe that $S'P = PR'$, and therefore
\begin{flalign*}
           &     &SP + S'P &= RR'= EE'&&\\
                 &&         &= AE + AE'&&\\
                 &&         &= AS + AS';&&\\
\end{flalign*} %to allow pagebreak
\begin{flalign*}
\text{and }&     &AS'= AE' &= A'F = A'S,&&\\
             &&\therefore SP &+ S'P = AA'.&&\\
\end{flalign*}
%-----File: 157.png-----

The transverse axis of the section is $AA'$ and the conjugate,
or minor, axis is evidently a diameter of a circular
section.

\carticleplus{146.}{Def.}  If $O$ be a fixed point in a straight line $OE$
drawn through the centre $E$ of a fixed circle at right angles
to the plane of the circle, and if a straight line $QOP$ move
so as always to pass through the circumference of the circle,
the surface generated by the line $QOP$ is called a \textit{Right
Circular Cone}.

\cheader{THE CONE}

\includegraphicsmid{146a}

The line $OE$ is called the axis of the cone, the point $O$ is
the \textit{vertex}, and the constant angle $POE$ is the semi-vertical
angle of the cone.

It is evident that any section by a plane perpendicular
to the axis, or parallel to the base of the cone, is a circle;
and that any section by a plane through the vertex consists
of two straight lines, the angle between which is greatest
and equal to the vertical angle when the plane contains the
axis.

Any plane containing the axis is called a \textit{Principal
Section}.
%-----File: 158.png-----

\carticleplus{147.}{Prop.~II\@.} \textit{The section of a cone by a plane, which
is not perpendicular to the axis, and does not pass through the
vertex, is either an Ellipse, a Parabola, or an Hyperbola.}

\includegraphicsmid{147a}

Let $UAP$ be the cutting plane, and let the plane of the
paper be that principal section which is perpendicular to the
plane $UAP$; $OV$, $OAQ$ being the generating lines in the
plane of the paper.

Let $AU$ be the intersection of the principal section $VOQ$
by the plane $PAU$ perpendicular to it, and cutting the cone
in the curve $AP$.

Inscribe a sphere in the cone, touching the cone in the
circle $EF$ and the plane $AP$ in the point $S$, and let $XK$ be
the intersection of the planes $AP$, $EF$. Then $XK$ is perpendicular
to the plane of the paper.

Taking any point $P$ in the curve, join $OP$ cutting the
circle $EF$ in $R$, and join $SP$.

Draw through $P$ the circular section $QPV$ cutting the
plane $AP$ in $PN$ which is therefore perpendicular to $AN$
and parallel to $XK$.

Then, $SP$ and $PR$ being tangents to the sphere,
\[
SP = PR = EQ;
\]
%-----File: 159.png-----
\begin{flalign*}
&\text{and}&
  EQ: NX &:: AE: AX &\phantom{and}& \\
&&       &:: AS: AX. &&
\end{flalign*}
\begin{flalign*}
&\indent\text{Also}&
      NX &= PK; &\phantom{\indent\text{Also}}& \\
&&\therefore SP: PK&:: SA: AX. &&
\end{flalign*}

The curve $AP$ is therefore an Ellipse, Parabola, or Hyperbola,
according as $SA$ is less than, equal to, or greater
than $AX$. In any case the point $S$ is a focus and the corresponding
directrix is the intersection of the plane of the
curve with the plane of contact of the sphere.

\carticleplus{148.}{(1)} If $AU$ be parallel to $OV$, the angle
\begin{flalign*}
&& \multispan{1}{$AXE = OFE = OEF = AEX$,} &&& \\
&\text{so that}& \multispan{1}{\hfill$SA = AE= AX$;\hfill} &\phantom{so that}&&
\end{flalign*}
the section is therefore a parabola when the cutting plane
is parallel to a generating line, and perpendicular to the
principal section which contains the generating line.

\subparagraph{(2)} Let the line $AU$ meet the curve again in the point
$A'$ on the same side of the vertex as the point $A$.
%-----File: 160.png-----

\includegraphicsmid{148a}

Then the angle
\begin{flalign*}
&&                      AEX &= OFX  &&\\
&&                          &> FXA,  &&\\
&\text{and therefore }&  AE &< AX,  \phantom{and\ therefore} &&\\
&\text{that is }&        SA &< AX,  &&
\end{flalign*}
and the curve is an ellipse.

In this case another sphere can be inscribed in the cone,
touching the cone along the circle $E'F'$ and touching the
plane $AP$ in $S'$.

It may be shewn as before that $S'$ is a focus and that the
corresponding directrix is the intersection of the planes $E'F'$,
$APA'$.

\subparagraph{(3)} Let the line $UA$ produced meet the cone on the
other side of the vertex. The section then consists of two
separate branches.

\includegraphicsmid{148b}

\begin{flalign*}
&\text{\indent Also the angle }&  AEX &= A'FX
&\phantom{\indent Also\ the\ angle } &\\
&&                                    &< AXF,  &&\\
&\text{and therefore }&            AE &> AX,  &&\\
&\text{that is }&                  AS &> AX,  &&
\end{flalign*}
and the curve $AP$ is one branch of an hyperbola, the other
branch being the section $A'P'$.

Taking $P'$ in the other branch the proof is the same as
before that
\[
  SP': P'K':: SA: AX.
\]

In this case a sphere can be inscribed in the other
branch of the cone, touching the cone along the circle $E'F'$,
and the plane $UA'P'$ in $S'$, and it can be shewn that $S'$ is
the other focus of the hyperbola, and that the directrix is
the intersection of the cutting plane with the plane of contact
$E'F'$.

Hence the section of a cone by a plane cutting in $AU$
%-----File: 161.png-----
the principal section $VOQ$ perpendicular to it is an Ellipse,
Parabola, or Hyperbola, according as the angle $EAX$ is
greater than, equal to, or less than, the vertical angle of the
cone.

Further, it is obvious that, if any plane be drawn parallel
to the plane $AP$, the ratio of $AE$ to $AX$ is always the same;
hence it follows that all parallel sections have the same
eccentricity.

\carticle{149.} This method of determining the focus and directrix
was published by Mr Pierce Morton, of Trinity College, in
the first volume of the Cambridge \textit{Philosophical Transactions}.

The method was very nearly obtained by Hamilton, who
gave the following construction.

First finding the vertex and focus, $A$ and $S$, take $AE$
along the generating line equal to $AS$, and draw the circular
%-----File: 162.png-----
section through $E$; the directrix will be the line of intersection
of the plane of the circle with the given plane of
section.

Hamilton also demonstrated the equality of $SP$ and $PR$.

\carticleplus{150.}{Prop.~III.} \textit{To prove that, in the case of an elliptic
section,}
\[
SP+S'P= AA'.
\]

Taking the 2nd figure,
\begin{flalign*}
& &SP=PR &\text{ and } S'P=PR'; &&\\
&  &\therefore SP+S'P &= RR' = EE' &&\\
&                    &&= AE + AE' &&\\
&                    &&= AS + AS'.&&\\
&\text{\indent But }&
  A'S' = A'F' &= FF' - A'F  &\phantom{\indent But}&\\
&     &=EE'&-A'S,&&\\
&\text{also }&            A'S' + SS' &= A'S;  &&\\
&&            \therefore 2A'S' + SS' &= EE'.  &&\\
&\text{\indent Similarly }&
                           2AS + SS' &= EE';
&\phantom{\indent Similarly }&\\
&&            \therefore A'S'        &= AS,  &&\\
&\text{and }&                    AS' &= A'S.  &&\\
&\text{Hence }&             SP + S'P &= AA',  &&\\
\multispan{3}{and the transverse, or major axis} &= EE'.  &&
\end{flalign*}

In a similar manner it can be shewn that in an hyperbolic
section
\[
S'P-SP = AA'.
\]

\carticleplus{151.}{Prop.~IV.} \textit{To shew that, in a parabolic section,}
\[
  PN^2 = 4AS \ctrdot AN.
\]

\includegraphicsmid{151a}

Let $A$ be the vertex of the section, and let $ADE$ be the
diameter of the circular section through $A$.
%-----File: 163.png-----
From $D$ let fall $DS$ perpendicular to $AN$;
\begin{flalign*}
&\text{\indent then }& PN^2 &=QN \ctrdot NQ'&\phantom{\indent then }&\\
&&                          &=QN \ctrdot AE&&\\
&&                          &=4NL \ctrdot AD,&&
\end{flalign*}
if $AL$ be perpendicular to $NQ$.

But the triangles $ANL$, $ADS$ being similar,
\begin{flalign*}
&&NL : AN &:: AS : AD;&&\\
&&\therefore NL \ctrdot AD&=AN \ctrdot AS,&&\\
&\text{and}&&&\phantom{and}&\\
&&PN^2 &=4AS \ctrdot AN.&&
\end{flalign*}

\includegraphicsright[13]{152a}

\carticleplus{152.}{Prop.~V.} \textit{To shew that, in an elliptic section, $PN^2$
is to $AN \ctrdot NA'$ in a constant ratio.}

Draw through $P$ the circular
section $QPQ'$, bisect $AA'$ in $C$,
and draw through $C$ the circular
section $EBE'$.

Then
\begin{flalign*}
&&QN : AN &:: CE : AC,&&\\
&\rlap{\text{and}}& NQ' : NA' &:: CE' : A'C;&&\\
&&\therefore QN \ctrdot NQ' &: AN \ctrdot NA'&&\\
&&&:: EC \ctrdot CE':AC^2,&&\\
&\rlap{\text{or}}&&&&\\
&&PN^2 : AN \ctrdot NA' &:: EC \ctrdot CE' : AC^2;&&
\end{flalign*}
and, the transverse axis being $AA'$, the square of the semi-minor
axis $= BC^2 = EC \ctrdot CE'$.
%-----File: 164.png-----
Again, if $ADF$ be perpendicular to the axis, $AD = DF$, and,
$AC$ being equal to $CA'$, $CD$ is parallel to $A'F$,
and therefore
\[CE' = FD = AD.\]

Similarly, $CE=A'D'$, the perpendicular from $A'$ on the
axis;
\[
\therefore BC^2 = AD \ctrdot A'D',
\]
that is, \textit{the semi-minor axis is a mean proportional between
the perpendiculars from the vertices on the axis of the cone.}\bigskip

\subparagraph{Cor.} If $H$, $H'$ are the centres of the focal spheres, the
angles $HAH'$, $HA'H'$ are right angles, so that $H$, $A$, $H'$, $A'$
are concyclic.

\includegraphicsmid{152b}

It follows that the triangles $ASH$, $A'H'D'$ are similar, as
are also the triangles $A'S'H'$, $AHD$, so that
\begin{flalign*}
&&SH : A'D' :: AH& : A'H' :: AD : S'H';&\\
&\text{and }  &SH \ctrdot S'H' = A&D \ctrdot A'D' = BC^2;\phantom{and}&
\end{flalign*}

\textit{$\therefore$ the semi-minor axis is a mean proportional between the
radii of the focal spheres.}
%-----File: 165.png-----

The fact that $H$, $A$, $H'$, $A'$ are concyclic also shews that
the sphere of which $HH'$ is a diameter intersects the plane
of the ellipse in its auxiliary circle.

\carticle{153.} In exactly the same manner it can be shewn that,
for an hyperbolic section,
\begin{flalign*}
&&PN^2 &: AN \ctrdot NA'  ::  CE \ctrdot CE' : AC^2,&\\
&\text{and that}    &&CE = AD, \text{and } CE' = A'D'.&
\end{flalign*}

\includegraphicsmid{153a}

Also, as in the case of the ellipse, $BC$ is a mean proportional
between $AD$ and $A'D'$, and is also a mean proportional
between the radii of the focal spheres.

\carticleplus{154.}{Prop.~VI.} \textit{The two straight lines in which a cone
is intersected by a plane through the vertex parallel to an
hyperbolic section are parallel to the asymptotes of the
hyperbola.}

Taking the preceding figure, let the parallel plane cut
%-----File: 166.png-----
the cone in the lines $OG$, $OG'$, and the circular section
through $C$ in the line $GLG'$, which will be perpendicular to
the plane of the paper, and therefore perpendicular to $EE'$
and to $OL$.

\begin{flalign*}
&\indent\text{Hence}& GL^2 &= EL \ctrdot E'L. &\phantom{\indent\text{Hence}}& \displaybreak[1]\\
&\indent\text{But}&     EL : EC &:: OL : A'C,&&\\
&\text{and}&     E'L : E'C &:: OL : AC;&&\\
&&\therefore GL^2 : EC \ctrdot E'C &:: OL^2 : AC^2,&&\\
&\text{or}&     GL : OL &:: BC : AC;&\phantom{or}&
\end{flalign*}
therefore, (Art.~\cartref{102}), $OG$ and $OG'$ are parallel to the asymptotes
of the hyperbola.

Hence, for all parallel hyperbolic sections, the asymptotes
are parallel to each other.

If the hyperbola be rectangular, the angle $GOG'$ is a
right angle; but this is evidently not possible if the vertical
angle of the cone be less than a right angle.

When the vertical angle of the cone is not less than a
right angle, and when $GOG'$ is a right angle, $LOG$ is half
a right angle, and therefore
\begin{flalign*}
&&     OL &= LG,& \\
&\text{and}&     2 \ctrdot OL^2 &= OG^2 = OE^2,
\end{flalign*}
and the length $OL$ is easily constructed.

Hence, placing $OL$, and drawing the plane $GOG'$ perpendicular
to the principal section through $OL$, any section
by a plane parallel to $GOG'$ is a rectangular hyperbola.

It will be observed that the eccentricity of the section is
greatest when its plane is parallel to the axis of the cone.

\carticleplus{155.}{Prop.~VII\@.} \textit{The sphere which passes through the
circles of contact of the focal spheres with the surface of the
cone intersects the plane of the section in its director circle.}

Let $Q$, $Q'$ be the points in which the straight line $AA'$
is intersected by the sphere which passes through the circles
$EF$ and $E'F'$.
%-----File: 167.png-----

Then the sphere intersects the plane of the ellipse in the
circle of which $QQ'$ is the diameter.
\begin{flalign*}
&\text{\indent{Also}}   &CQ^2 - CA^2 &= AQ \ctrdot AQ' = AE \ctrdot AE' &\\
&                        &&= AS \ctrdot AS' = BC^2; &\\
&                 &\therefore   C&Q^2 = AC^2 + BC^2,&
\end{flalign*}
so that $CQ$ is the radius of the director circle.

\includegraphicsmid{155a}

Changing the figure the proof is exactly the same for the
hyperbola.

\carticleplus{156.}{Prop.~VIII.} \textit{If two straight lines be drawn
through any point, parallel to two fixed lines, and intersecting
a given cone, the ratio of the rectangles formed by the segments
of the lines will be independent of the position of the point.}

Thus, if through $E$, the lines $EPQ$, $EP'Q'$ be drawn,
parallel to two given lines, and cutting the cone in the
%-----File: 168.png-----
points $P$, $Q$ and $P'$, $Q'$, the ratio of $EP \ctrdot EQ$ to $EP' \ctrdot EQ'$ is
constant.

Through $O$ draw $OK$ parallel to the given line to which
$EPQ$ is parallel, and let the plane through $OK$, $EPQ$,
which contains the generating lines $OP$, $OQ$, meet the
circular section through $E$ in $R$ and $S$, and the plane base
in the straight line $DFK$, cutting the circular base in $D$
and $F$.

Then $DFK$ and $ERS$ being sections of parallel planes
by a plane are parallel to each other.

\includegraphicsmid{156a}

Also, $EPQ$ is parallel to $OK$;

Therefore $ERP$, $ODK$ are similar triangles, as are also
$ESQ$, $OFK$;
\begin{flalign*}
& &\therefore EP : ER  &:: OK : DK,   &&\\
\text{and }           & &EQ : ES  &::  OK : FK;   &\phantom{and}&\\
&& \therefore  EP \ctrdot EQ : ER \ctrdot ES  &::  OK^2 : DK \ctrdot FK &&\\
                               && &::  OK^2 : KT^2, &&
\end{flalign*}
if $KT$ be the tangent to the circular base from $K$.

If a similar construction be made for $EP'Q'$, we shall have
\[
EP' \ctrdot EQ' : ER' \ctrdot ES' :: OK'^2 : K'T'^2.
\]
%-----File: 169.png-----
But\hfill$ER \ctrdot ES = ER' \ctrdot ES'$;\hfill\phantom{\indent But}\\
therefore the rectangles $EP \ctrdot EQ$ and $EP' \ctrdot EQ'$ are each
in a constant ratio to the same rectangle, and are therefore
in a constant ratio to each other.

Since the plane through $EPQ$, $EP'Q'$ cuts the cone in
an ellipse, parabola, or hyperbola, this theorem includes as
particular cases those of Arts.~\cartref{51}, \cartref{58},
\cartref{82}, \cartref{92}, \cartref{96}, \cartref{124} and~\cartref{134}.

The proof is the same if the point $P$ be within the cone,
or if one or both of the lines meet opposite branches of the
cone.

If the chords be drawn through the centre of the
section $PEP'$, the rectangles become the squares of the
semi-diameters.

Hence the parallel diameters of all parallel sections of a
cone are proportional to each other.

If the lines move until they become tangents the rectangles
then become the squares of the tangents; therefore
if a series of points be so taken that the tangents from
them are parallel to given lines, these tangents are always
in the same proportion. The locus of the point $E$ will be
the line of intersection of two fixed planes touching the
cone, that is, a fixed line through the vertex.

\csubsection{EXAMPLES}

\begin{exampleslist}

\item[1.] Shew how to cut from a cylinder an ellipse of given eccentricity.
\cheader{EXAMPLES}

\item[2.] What is the locus of the foci of all sections of a cylinder of a
given eccentricity?

\item[3.] Shew how to cut from a cone an ellipse of given eccentricity.

\item[4.] Prove that all sections of a cone by parallel planes are conics
of the same eccentricity.

\item[5.] What is the locus of the foci of the sections made by planes
inclined to the axis at the same angle?
%-----File: 170.png-----

\item[6.] Find the least angle of a cone from which it is possible to cut
an hyperbola, whose eccentricity shall be the ratio of two to one.

\item[7.] The centre of a spherical ball is moveable in a vertical plane
which is equidistant from two candles of the same height on a table;
find its locus when the two shadows on the ceiling are always just in
contact.

\item[8.] Through a given point draw a plane cutting a given cone in a
section which has the given point for a focus.

\item[9.] If the vertical angle of a cone, vertex $O$, be a right angle, $P$
any point of a parabolic section, and $PN$ perpendicular to the axis of
the parabola,
\[
OP = 2AS + AN,
\]
$A$ being the vertex and $S$ the focus.

\item[10.] Prove that the directrices of all parabolic sections of a cone lie
in the tangent planes of a cone having the same axis.

\item[11.] If the curve formed by the intersection of any plane with a
cone be projected upon a plane perpendicular to the axis; prove that
the curve of projection will be a conic section having its focus at the
point in which the axis meets the plane of projection.

\item[12.] Prove that the latera recta of parabolic sections of a right
circular cone are proportional to the distances of their vertices from
the vertex of the cone.

\item[13.] The shadow of a ball is cast by a candle on an inclined plane
in contact with the ball; prove that as the candle burns down, the
locus of the centre of the shadow will be a straight line.

\item[14.] The vertex of a right cone which contains a given ellipse lies
on a certain hyperbola, and the axis of the cone will be a tangent to
the hyperbola.

\item[15.] Find the locus of the vertices of the right circular cones which
can be drawn so as to pass through a given fixed hyperbola, and prove
that the axis of the cone is always tangential to the locus.

\item[16.] An ellipse and an hyperbola are so situated that the vertices
of each curve are the foci of the other, and the curves are in planes at
right angles to each other. If $P$ be a point on the ellipse, and $O$ a
point on the hyperbola, $S$ the vertex, and $A$ the interior focus of that
branch of the hyperbola, then
\[
AS + OP = AO + SP.
\]

\item[17.] The latus rectum of any plane section of a given cone is proportional
to the perpendicular from the vertex on the plane.

\item[18.] If a sphere is described about the vertex of a right cone as
centre, the latera recta of all sections made by tangent planes to the
sphere are equal.
%-----File: 171.png-----

\item[19.] Different elliptic sections of a right cone are taken such that
their minor axes are equal; shew that the locus of their centres is the
surface formed by the revolution of an hyperbola about the axis of the
cone.

\item[20.] If two cones be described touching the same two spheres, the
eccentricities of the two sections of them made by the same plane bear
to one another a ratio constant for all positions of the plane.

\item[21.] If elliptic sections of a cone be made such that the volume
between the vertex and the section is always the same, the minor axis
will be always of the same length.

\item[22.] The vertex of a cone and the centre of a sphere inscribed within
it are given in position: a plane section of the cone, at right angles to
any generating line of the cone, touches the sphere: prove that the
locus of the point of contact is a surface generated by the revolution of
a circle, which touches the axis of the cone at the centre of the sphere.

\item[23.] Given a right cone and a point within it, there are two sections
which have this point for focus; and the planes of these sections make
equal angles with the straight line joining the given point and the
vertex of the cone.

\item[24.] Prove that the centres of all plane sections of a cone, for which
the distance between the foci is the same, lie on the surface of a right
circular cylinder.
\end{exampleslist}
%-----File: 172.png-----

\cchapter{CHAPTER VII}

\csectiontoc{The Similarity of Conics, the Areas of Conics, and the Curvatures of Conics}
\cheader{SIMILAR CONICS}

\csubsection{SIMILAR CONICS}

\carticleplus{157.}{Def.} \textit{Conics which have the same eccentricity are
said to be similar to each other.}

This definition is justified by the consideration that the
character of the conic depends on its eccentricity alone,
while the dimensions of all parts of the conic are entirely
determined by the distance of the focus from the directrix.

Hence, according to this definition, all parabolas are
similar curves.

\subparagraph{Prop.~I\@.} \textit{If radii be drawn from the vertices of two
parabolas making equal angles with the axis, these radii are
always in the same proportion.}

Let $AP$, $ap$ be the radii, $PN$ and $pn$ the ordinates, the
angles $PAN$, $pan$, being equal.

\begin{flalign*}
&\indent\text{Then} &AP^2 : ap^2  ::  PN^2 : pn^2  &
     ::  AS \ctrdot AN : as \ctrdot an.&\phantom{\indent Then}&\\
&\indent\text{But}  & AP : ap  &::  AN : an; &&\\
&                   &\therefore  AP : ap  &::  AS : as. &&
\end{flalign*}

It can also be shewn that focal radii making equal
angles with the axes are always in the same proportion.
%-----File: 173.png-----

\carticleplus{158.}{Prop.~II\@.} \textit{If two ellipses be similar their axes are
in the same proportion, and any other diameters, making
equal angles with the respective axes, are in the proportion of
the axes.}

Let $CA$, $CB$ be the semi-axes of one ellipse, $ca$, $cb$ of the
other, and $CP$, $cp$ two radii such that the angle $PCA =pca$.

Then, since the eccentricities are the same, we have, if
$S$, $s$ be foci,
\begin{flalign*}
&&                                     AC : SC  &:: ac : sc;&&\\
&&         \therefore AC^{2} : AC^{2} - SC^{2}  &::  ac^{2} : ac^{2} - sc^{2},&&\\
&\text{or}&                    AC^{2} : BC^{2}  &::  ac^{2}  : bc^{2}.&\phantom{or}&
\end{flalign*}

Hence it follows, if $PN$, $pn$ be ordinates, that
\[
PN^2 : AC^2 - CN^2  ::  pn^2 : ac^2 - cn^2;
\]
but, by similar triangles,
\begin{flalign*}
&&PN : pn  &::  CN : cn, &&\\
&\text{therefore}  &CN^2 : AC^2 - CN^2  &:: cn^2 : ac^2 - cn^2; &\phantom{therefore}&\\
&\text{and}      &CN^2 : AC^2  &::  cn^2 : ac^2. &&\\
&\text{Hence}      &CP : cp   &::   CN: cn &&\\
           &&                    &::   AC : ac.
\end{flalign*}

So also lines drawn similarly from the foci, or any other
corresponding points of the two figures, will be in the ratio
of the transverse axes.

Exactly the same demonstration is applicable to the
hyperbola, but in this case, if the ratio of $SC$ to $AC$ in
two hyperbolas be the same, it follows from Art.~(\cartref{102})
that the angle between the asymptotes is the same in both
curves.

In the case of hyperbolas we have thus a very simple test
of similarity.
%-----File: 174.png-----

\cisubsection{The Areas bounded by Conics}

\cheader{AREAS}

\carticleplus{159.}{Prop.~III\@.} \textit{If $AB, AC$ be two tangents to a parabola,
the area between the curve and the chord $BC$ is two-thirds
of the triangle $ABC$.}

Draw the tangent $DPE$ parallel to BC; then
\begin{flalign*}
&                              &AP &= PN,&&\\
&\text{and}                    &BC &= 2 \ctrdot DE;&&\\
&\text{therefore the triangle} &BPC &= 2ADE.\phantom{therefore the triangle}&&
\end{flalign*}

Again, draw the diameter $DQM$ meeting $BP$ in $M$.

By the same reasoning, $FQG$ being the tangent parallel
to $BP$, the triangle $PQB = 2FDG$.

Through $F$ draw the diameter $FRL$, meeting $PQ$ in $L$,
and let this process be continued indefinitely.

\includegraphicsmid{159a}

Then the sum of the triangles within the parabola is
double the sum of the triangles without it.
%-----File: 175.png-----

But, since the triangle $BPC$ is half $ABC$, it is greater
than half the parabolic area $BQPC$;

Therefore (Euclid, Bk.\ \textsc{xii.}) the difference between the
parabolic area and the sum of the triangles can be made
ultimately less than any assignable quantity;

And, the same being true of the outer triangles, it follows
that the area between the curve and $BC$ is double of the
area between the curve and $AB$, $AC$, and is therefore two-thirds
of the triangle $ABC$.

\subparagraph{Cor.} Since $PN$ bisects every chord parallel to $BC$, it
bisects the parabolic area $BPC$; therefore, completing the
parallelogram $PNBU$, the parabolic area $BPN$ is two-thirds
of the parallelogram $UN$.

\carticleplus{160.}{Prop.~IV\@.} \textit{The area of an ellipse is to the area of
the auxiliary circle in the ratio of the conjugate to the transverse
axis.}

Draw a series of ordinates, $QPN$, $Q'P'N'$, $\ldots$ near each
other, and draw $PR$, $QR'$ parallel to $AC$.

\includegraphicsmid{160a}

Then, since
\begin{flalign*}
&                           &PN: QN&:: BC: AC,&&\\
&\text{the area}            &PN': QN'&:: BC: AC,&&\phantom{the area}
\end{flalign*}
and, this being true for all such areas, the sum of the
parallelograms $PN'$ is to the sum of the parallelograms
$QN'$ as $BC$ to $AC$.

But, if the number be increased indefinitely, the sums of
%-----File: 176.png-----
these parallelograms ultimately approximate to the areas of
the ellipse and circle.

Hence the ellipse is to the circle in the ratio of $BC$ to
$AC$.

The student will find in Newton's 2nd and 3rd Lemmas
(\textit{Principia}, Section \textsc{i}.) a formal proof of what we have here
assumed as sufficiently obvious, that the sum of the parallelograms
$PN$ is ultimately equal to the area of the ellipse.

\carticleplus{161.}{Prop.~V.} \textit{If $P$, $Q$ be two points of an hyperbola,
and if $PL$, $QM$ parallel to one asymptote meet the other in $L$
and $M$, the hyperbolic sector $CPQ$ is equal to the hyperbolic
trapezium $PLMQ$.}

\includegraphicsmid{161a}

For the triangles $CPL$, $CQM$ are equal, and, if $PL$ meet
$CQ$ in $R$, it follows that the triangle $CPR =$ the trapezium
$LRQM$; hence, adding to each the area $RPQ$, the theorem
is proved.

\carticleplus{162.}{Prop.~VI.} \textit{If points $L$, $M$, $N$, $K$ be taken in an
asymptote of an hyperbola, such that}
\[
CL: CM:: CN: CK,
\]
\textit{and if $LP$, $MQ$, $NR$, $KS$, parallel to the asymptote, meet the
curve in $P$, $Q$, $R$, $S$, the hyperbolic areas $CPQ$, $CRS$ will be
equal.}

Let $QR$ and $PS$ produced meet the asymptotes in $F$, $F'$,
$G$, $G'$;
%-----File: 177.png-----
\begin{flalign*}
&\text{then} &RF = QF' \text{ and } &SG = PG' \text{ (Art.~\cartref{121}),}&&\\
&                  &\therefore NF = CM &\text{ and } KG = CL.&&\\
&\indent\text{Hence}            &NF : KG &:: CM : CL &\phantom{\indent Hence}&\\
&&                                        &:: CK : CN &&\\
&&                                       &:: RN : SK,&&
\end{flalign*}
and therefore $SP$ is parallel to $QR$.

\includegraphicsmid{162a}

The diameter $CUV$ conjugate to $PS$ bisects all chords
parallel to $PS$, and therefore bisects the area $PQRS$;
\begin{flalign*}
&\text{also the triangle} &CPV &= CSV,& \phantom{also the triangle}&\\
&\text{and } &CQU &= CUR;&&
\end{flalign*}
therefore, taking from $CPV$ and $CSV$ the equal triangles
$CQU$, $CRU$, and the equal areas $PQUV$, $SRUV$, the remaining
areas, which are the hyperbolic sectors $CPQ$, $CRS$,
are equal.

\subparagraph{Cor.} Hence if a series of points, $L$, $M$, $N$, $\ldots$ be taken
such that $CL$, $CM$, $CN$, $CK$, $\ldots$ are in continued proportion, it
follows that the hyperbolic sectors $CPQ$, $CQR$, $CRS$, \&c.\ will
be all equal.

It will be noticed in this case that the tangent at $Q$ will
be parallel to $PR$, the tangent at $R$ parallel to $QS$, and so
also for the rest.
%-----File: 178.png-----

\cisubsection{The Curvature of Conics}

\cheader{CURVATURE}

\carticleplus{163.}{Def.}  If a circle touch a conic at a point $P$, and
pass through another point $Q$ of the conic, and if the point
$Q$ move near to, and ultimately coincide with $P$, the circle
in its ultimate condition is called the circle of curvature
at $P$.

\subparagraph{Prop.~VII\@.}  \textit{The chord of intersection of a conic with the
circle of curvature at any point is inclined to the axis at the
same angle as the tangent at the point.}

It has been shewn that, if a circle intersect a conic in
four points $P$, $Q$, $R$, $V$, the chords $PQ$, $RV$ are equally
inclined to the axis.

Let $P$ and $Q$ coincide with each other; then the tangent
at $P$ and the chord $RV$ are equally inclined to the axis.

Let the point $V$ now approach to and coincide with $P$;
the circle becomes the circle of curvature at $P$, and the
chord $VR$ becomes $PR$ the chord of intersection.

Hence $PR$ and the tangent at $P$ are equally inclined to
the axis.

\carticleplus{164.}{Prop.~VIII\@.}  \textit{If the tangent at any point $P$ of a
parabola meet the axis in $T$, and if the circle of curvature at
$P$ meet the curve in $Q$,}
\[
PQ = 4 \ctrdot PT.
\]

Draw the ordinate $PNP'$; then taking the figure of the
next article, $TP'$ is the tangent at $P'$,\\
and the angle \hfill $P'TF = PTF = PFT$; \hfill \phantom{and the angle}\\
therefore $PQ$ is parallel to $TP'$, and is bisected by the
diameter $P'E$.

Hence \hfill $PQ = 2 \ctrdot PE = 4P'T = 4PT$. \hfill \phantom{\indent Hence}

\carticleplus{165.}{Prop.~IX\@.}  \textit{To find the chord of curvature through
the focus and the diameter of curvature at any point of a
parabola.}

Let the circle meet $PS$ produced in $V$, and the normal
$PG$ produced in~$O$.
%-----File: 179.png-----
\begin{flalign*}
&\text{\indent The angle}     &PFS &= PTS = SPT&\phantom{\indent The angle}&\\
&&                  &= PQV,&&
\end{flalign*}
since $PT$ is a tangent to the circle.

Therefore $QV$ is parallel to the axis,
\begin{flalign*}
&\text{ and }      &PV : SP &:: PQ : PF.&&\\
&\text{\indent Hence}  &  PV &= 4 \ctrdot SP.&\phantom{\indent Hence}&
\end{flalign*}

Again, the angle $POQ = PVQ = PSN$;
\includegraphicsmid{165a}
\begin{flalign*}
&&\therefore PO : PQ &:: SP : PN,&&\\
&\text{or}  &PO : SP &:: 4PT : PN&\phantom{or}&\\
&&                   &:: 4SP : SY,&&
\end{flalign*}
if $SY$ be perpendicular to $PT$.

\subparagraph{Cor.~1.} Since the normal bisects the angle between $SP$
and the diameter through $P$, it follows that the chord of
curvature parallel to the axis is $4SP$.
%-----File: 180.png-----

\subparagraph{Cor.~2.}  The diameter of curvature, $PO$, may also be
expressed as follows:

Let $GL$ be the perpendicular from $G$ on $SP$;\\
then $PL = \text{the semi-latus rectum} = 2AS$.

Also $PVO$ being a right angle,
\begin{flalign*}
&    &PO : PG &:: PV : PL&&\\
&&            &:: 4SP : PL&&\\
&&            &:: 4SP \ctrdot PL : PL^2;&&\\
&\text{but}   &4SP \ctrdot PL = 8SP \ctrdot AS &= 8SY^2 = 2PG^2;&\phantom{but}&\\
&&\therefore PO : PG &:: 2PG^2 : PL^2.&&
\end{flalign*}

\carticleplus{166.}{Prop.~X.}  \textit{If the chord of intersection, $PQ$, of an
ellipse, or hyperbola, with the circle of curvature at P, meet
$CD$, the semi-diameter conjugate to $CP$, in $K$,}
\[
        PQ \ctrdot PK = 2CD^2.
\]

\includegraphicsmid{166a}

Drawing the ordinate $PNP'$, the tangent at $P'$ is parallel
to $PQ$, as in the parabola, and $PQ$ is therefore bisected in $V$,
by the diameter $CP'$.

\includegraphicsright[18]{166b}

Let $PQ$ meet the axes in $U$ and $U'$; then, $U'C$ being
parallel to $PP'$,
%-----File: 181.png-----
\begin{flalign*}
&&PV : PU' &:: VP' : CP'&&&\\
&&         &:: UT: CT,&&&
\end{flalign*}
since $PU$, $P'T$ are parallel.

Also
\begin{flalign*}
&&UT : CT &:: PU : PK;&&\\
&& \therefore PV : PU' &:: PU : PK.&&\\
&\text{Hence}&&&&\\
&&PV \ctrdot PK &= PU \ctrdot PU'&&\\
&&     &= PT \ctrdot PT' = CD^2,&&
\end{flalign*}
observing that $PU = PT$, and
$PU' = PT'$, by the theorem of
Art.~\cartref{163},\\
and\hfill $\therefore PQ \ctrdot PK = 2CD^2$. \hfill\phantom{and}

\carticleplus{167.}{Prop.~XI.} \textit{If the chord of curvature $PQ'$, of an
ellipse or hyperbola in any direction, meet $CD$ in $K'$,}
\[
PQ' \ctrdot PK' = 2CD^2.
\]

Let $PO$ be the diameter of curvature meeting $CD$ in $F$:
%-----File: 182.png-----
then $PQO$, $PQ'O$ are right angles, and a circle can be drawn
through $Q'K'FO$;
\includegraphicsmid{167a}
\begin{align*}
\therefore PQ' \ctrdot PK'&=PF \ctrdot PO\\
&=PK \ctrdot PQ=2 \ctrdot CD^2.
\end{align*}

\subparagraph{Cor.~1.} Hence $PO$ being the diameter of curvature,
\[
PF \ctrdot PO=2 \ctrdot CD^2.
\]

\subparagraph{Cor.~2.} If $PQ'$ pass through the focus,
\begin{flalign*}
&&PK' &= AC,&\\
&\text{and }\quad    &PQ' \ctrdot AC&=2 \ctrdot CD^2.&
\end{flalign*}

\subparagraph{Cor.~3.} If $PQ'$ pass through the centre,
\[
PQ' \ctrdot CP=2 \ctrdot CD^2.
\]

\carticle{168.} We can also express the diameter of curvature as
follows:

$PG$ being the normal, let $GL$ be perpendicular to $SP$,
and let $PR$ be the chord of curvature through $S$.

Then $GL$ is parallel to $OR$,
\begin{flalign*}
&\text{and}         &PO : PG            &:: PR : PL&&\\
&                   &                   &:: PR \ctrdot PL : PL^2.&&\\
&\text{\indent But} &PR \ctrdot AC      &=2 \ctrdot CD^2;&\phantom{\indent But}&\\
&                   &\therefore PR : AC &:: 2 \ctrdot CD^2 : AC^2&&\\
&                   &                   &:: 2 \ctrdot PG^2 : BC^2,&&\\
&\text{and}         &PR \ctrdot PL : AC \ctrdot PL &:: 2 \ctrdot PG^2 : BC^2.&&
\end{flalign*}

But, $PL$ being equal to the semi-latus rectum,
\begin{flalign*}
&           &PL \ctrdot AC            &=BC^2;&&\\
&           &\therefore PR \ctrdot PL &= 2 \ctrdot PG^2,&&\\
&\text{and} &PO : PG                  &:: 2 PG^2 : PL^2.&\phantom{and}&
\end{flalign*}

\textit{Hence, in any conic, the radius of curvature at any point
is to the normal at the point as the square of the normal to the
square of the semi-latus rectum.}
%-----File: 183.png-----

\carticleplus{169.}{Prop.~XII.} \textit{The chord of curvature through the
focus at any point is equal to the focal chord parallel to the
tangent at the point.}
\begin{flalign*}
&\text{\indent Since}     &PQ' \ctrdot AC &= 2CD^2,&&\\
&\text{it follows that}   &PQ' \ctrdot AA'&=DD'^2.&\phantom{it follows that}&
\end{flalign*}

But, if $pp'$ is the focal chord parallel to the tangent at $P$,
\begin{flalign*}
&&pp' \ctrdot AA' &= DD'^2 \text{ (Art.~\cartref{81}),}&&\\
&&\therefore PQ'  &=pp'.&&
\end{flalign*}

\csubsection{EXAMPLES}

\begin{exampleslist}

\item[1.] The radius of curvature at the end of the latus rectum of a
parabola is equal to twice the normal.
\cheader{EXAMPLES}

\item[2.] The circle of curvature at the end of the latus rectum intersects
the parabola on the normal at that point.

\item[3.] If $PV$ is the chord of curvature through the focus, what is the
locus of the point $V$?

\item[4.] An ellipse and a parabola, whose axes are parallel, have the
same curvature at a point $P$ and cut one another in $Q$; if the tangent
at $P$ meets the axis of the parabola in $T$ prove that $PQ=4 \ctrdot PT$.

\item[5.] In a rectangular hyperbola, the radius of curvature at $P$ varies
as $CP^{3}$.

\item[6.] If $P$ be a point of an ellipse equidistant from the axis minor
and one of the directrices, the circle of curvature at $P$ will pass through
one of the foci.

\item[7.] If the normal at a point $P$ of a parabola meet the directrix in
$L$, the radius of curvature at $P$ is equal to $2 \ctrdot PL$.

\item[8.] The normal at any point $P$ of a rectangular hyperbola meets
the curve again in $Q$; shew that $PQ$ is equal to the diameter of curvature
at $P$.

\item[9.] In the rectangular hyperbola, if $CP$ be produced to $Q$, so that
$PQ=CP$, and $QO$ be drawn perpendicular to $CQ$ to intersect the normal
in $O$, $O$ is the centre of curvature at $P$.
%-----File: 184.png-----

\item[10.] At any point of an ellipse the chord of curvature $PV$ through
the centre is to the focal chord $pp'$, parallel to the tangent, as the major
axis is to the diameter through the point.

\item[11.] If the common tangent of an ellipse and its circle of curvature
at $P$ be bisected by their common chord, prove that
\[
CD^2 = AC \ctrdot BC.
\]

\item[12.] The tangent at a point $P$ of an ellipse whose centre is $C$ meets
the axes in $T$ and $t$; if $CP$ produced meet in $L$ the circle described
about the triangle $TCt$, shew that $PL$ is half the chord of curvature at
$P$ in the direction of $C$, and that the rectangle contained by $CP$, $CL$, is
constant.

\item[13.] If $P$ be a point on a conic, $Q$ a point near it, and if $QE$, perpendicular
to $PQ$, meet the normal at $P$ in $E$, then ultimately when $Q$
coincides with $P$, $PE$ is the diameter of curvature at $P$.

\item[14.] If a tangent be drawn from any point of a parabola to the
circle of curvature at the vertex, the length of the tangent will be equal
to the abscissa of the point measured along the axis.

\item[15.] The circle of curvature at a point where the conjugate diameters
are equal, meets the ellipse again at the extremity of the diameter.

\item[16.] The chord of curvature at $P$ perpendicular to the major axis is
to $PM$, the ordinate at $P$, $:: 2 \ctrdot CD^2 : BC^2$.

\item[17.] Prove that there is a point $P$ on an ellipse such that if the
normal at $P$ meet the ellipse in $Q$, $PQ$ is a chord of the circle of curvature
at $P$, and find its position.

\item[18.] The chord of curvature at a point $P$ of a rectangular hyperbola,
perpendicular to an asymptote, is to $CD :: CD : 2 \ctrdot  PN$, where
$PN$ is the distance of $P$ from the asymptote.

\item[19.] If $G$ be the foot of the normal at a point $P$ of an ellipse, and
$GK$, perpendicular to $PG$, meet $CP$ in $K$, then $KE$, parallel to the axis
minor, will meet $PG$ in the centre of curvature at $P$.

\item[20.] The chord of curvature through the vertex at a point of a parabola
is to $4PY :: PY : AP$.

\item[21.] Prove that the locus of the middle points of the common
chords of a given parabola and its circles of curvature is a parabola, and
that the envelope of the chords is also a parabola.

\item[22.] The circles of curvature at the extremities $P$, $D$ of two conjugate
diameters of an ellipse meet the ellipse again in $Q$, $R$, respectively,
shew that $PR$ is parallel to $DQ$.

\item[23.] The tangent at any point $P$ in an ellipse, of which $S$ and $H$
are the foci, meets the axis major in $T$, and $TQR$ bisects $HP$ in $Q$ and
%-----File: 185.png-----
meets $SP$ in $R$; prove that $PR$ is one-fourth of the chord of curvature
at $P$ through $S$.

\item[24.] An ellipse, a parabola, and an hyperbola, have the same vertex
and the same focus; shew that the curvature, at the vertex, of the
parabola is greater than that of the hyperbola, and less than that of
the ellipse.

\item[25.] The circle of curvature at a point of an ellipse cuts the curve
in $Q$; the tangent at P is met by the other common tangent, which
touches the curves at $E$ and $F$, in $T$; if $PQ$ meet $TEF$ in $O$, $TEOF$ is
cut harmonically.

\item[26.] If $E$ is the centre of curvature at the point $P$ of a parabola,
\[
SE^2 + 3 \ctrdot SP^2 = PE^2.
\]

\item[27.] Find the locus of the foci of the parabolas which have a given
circle as circle of curvature, at a given point of that circle.

\item[28.] Two parabolas, whose latera recta have a constant ratio, and
whose foci are two given points $A$, $B$, have a contact of the second
order at $P$. Shew that the locus of $P$ is a circle.

\item[29.] If the fixed straight line $PQ$ is the chord of an ellipse, and is
also the diameter of curvature at $P$, prove that the locus of the
centre of the ellipse is a rectangular hyperbola, the transverse axis of
which is coincident in direction with $PQ$, and equal in length to one-half
of $PQ$.
\end{exampleslist}
%-----File: 186.png-----

\cchapter{CHAPTER VIII}

\csectiontoc{Orthogonal Projections}
\cheader{PROJECTIONS}

\carticleplus{170.}{Def.} The projection of a point on a plane is the
foot of the perpendicular let fall from the point on the plane.

If from all points of a given curve perpendiculars be let
fall on a plane, the curve formed by the feet of the perpendiculars
is the projection of the given curve.

The projection of a straight line is also a straight line,
for it is the line of intersection with the given plane of a
plane through the line perpendicular to the given plane.

Parallel straight lines project into parallel lines, for the
projections are the lines of intersection of parallel planes
with the given plane.

\carticleplus{171.}{Prop.~I.} \textit{Parallel straight lines, of finite lengths,
are projected in the same ratio.}

That is, if $ab$, $pq$ be the projections of the parallel lines
$AB$, $PQ$,
\[
ab: AB:: pq: PQ.
\]

For, drawing $AC$ parallel to $ab$ and meeting $Bb$ in $C$, and
$PR$ parallel to $pq$ and meeting $Qq$ in $R$, $ABC$ and $PQR$ are
similar triangles; therefore
\begin{flalign*}
&&AC: AB&:: PR: PQ,&&\\
&\text{and }&AC = ab&, \: PR=pq.&&\phantom{and }
\end{flalign*}

\carticleplus{172.}{Prop.~II.} \textit{The projection of the tangent to a curve
at any point is the tangent to the projection of the curve at the
projection of the point.}
%-----File: 187.png-----

For if $p$, $q$ be the projections of the two points $P$, $Q$ of a
curve, the line $pq$ is the projection of the line $PQ$, and when
the line $PQ$ turns round $P$ until $Q$ coincides with $P$, $pq$
turns round $p$ until $q$ coincides with $p$, and the ultimate
position of $pq$ is the tangent at $p$.

\carticleplus{173.}{Prop.~III.} \textit{The projection of a circle is an ellipse.}
Let $aba'$ be the projection of a circle $ABA'$.

\includegraphicsmid{173a}

Take a chord $PQ$ parallel to the plane of projection, then
its projection $pq = PQ$.

Let the diameter $ANA'$ perpendicular to $PQ$ meet in $F$
the plane of projection, and let $aa'F$ be the projection of
$AA'F$.

Then $aa'$ bisects $pq$ at right angles in the point $n$, and
\begin{flalign*}
&& an :  AN &:: aF : AF,&&\\
&&a'n : A'N &:: aF : AF;&&\\
&&\therefore   AN \ctrdot NA': an \ctrdot na'  &::  AF^2 : aF^2; \phantom{an \ctrdot na'}&&\\
&\text{but} &AN \ctrdot NA' &= PN^2 = pn^2,&\phantom{but}&\\
&&\therefore  pn2 : an \ctrdot na' &:: AF^2 : aF^2,&&
\end{flalign*}
and the curve $apa'$ is an ellipse, having its axes in the ratio
of
\[
aF : AF, \;\text{ or of }\; aa' : AA'.
\]
%-----File: 188.png-----

Moreover, since we can place the circle so as to make the
ratio of $aa'$ to $AA'$ whatever we please, an ellipse of any
eccentricity can be obtained.

In this demonstration we have assumed only the property
of the principal diameters of an ellipse. Properties of
other diameters can be obtained by help of the preceding
theorems, as in the following instances.

\carticleplus{174.}{Prop.~IV.} \textit{The locus of the middle points of parallel
chords of an ellipse is a straight line.}

For, projecting a circle, the parallel chords of the ellipse
are the projections of parallel chords of the circle, and as the
middle points of these latter lie in a diameter of the circle,
the middle points of the chords of the ellipse lie in the projection
of the diameter, which is a straight line, and is a
diameter of the ellipse.

Moreover, the diameter of the circle is perpendicular to
the chords it bisects; hence

\textit{Perpendicular diameters of a circle project into conjugate
diameters of an ellipse.}

\carticleplus{175.}{Prop.~V\@.} \textit{If two intersecting chords of an ellipse
be parallel to fixed lines, the ratio of the rectangles contained
by their segments is constant.}

Let $OPQ$, $ORS$ be two chords of a circle, parallel to fixed
lines, and $opq, ors$ their projections.

Then $OP \ctrdot OQ$ is to $op \ctrdot oq$ in a constant ratio, and $OR \ctrdot OS$
is to $or \ctrdot os$ in a constant ratio; but
\[
OP \ctrdot OQ=OR \ctrdot OS.
\]

Therefore $op \ctrdot oq$ is to $or \ctrdot os$ in a constant ratio; and
$opq$, $ors$ are parallel to fixed lines.

\carticleplus{176.}{Prop.~VI\@.} \textit{If $qvq'$ be a double ordinate of a diameter
$cp$, and if the tangent at $q$ meet $cp$ produced in $t$,}
\[
cv \ctrdot ct = cp^2.
\]

The lines $qvq'$ and $cp$ are the projections of a chord $QVQ'$
of a circle which is bisected by a diameter $CP$, and $t$ is the
%-----File: 189.png-----
projection of $T$ the point in which the tangent at $Q$ meets
$CP$ produced.

But, in the circle,
\begin{flalign*}
&&CV \ctrdot CT &=CP^2, &&\\
&\text{or }&CV:CP &::CP:CT; &\phantom{or }&
\end{flalign*}
and, these lines being projected in the same ratio, it follows
that
\begin{flalign*}
&&cv:cp&::cp:ct, &&\\
&\text{or }&cv \ctrdot ct&=cp^2. &\phantom{or }&
\end{flalign*}

Hence it follows that tangents to an ellipse at the ends
of any chord meet in the diameter conjugate to the chord.

The preceding articles will shew the utility of the method
in dealing with many of the properties of an ellipse.

The student will find it useful to prove, by orthogonal
projections, the theorems of Arts.~\cartref{58}, \cartref{69},
\cartref{74}, \cartref{75}, \cartref{78}, \cartref{79}, \cartref{80},
\cartref{82}, \cartref{83}, \cartref{89}, \cartref{90}, and~\cartref{92}.

\includegraphicsright[12]{177a}

\carticleplus{177.}{Prop.~VII\@.} \textit{An ellipse can be projected into a
circle.}

This is really the converse of Art.~\cartref{173}, but we give a
construction for the purpose.

Draw a plane through $AA'$,
the transverse axis, perpendicular
to the plane of the ellipse, and in
this plane describe a circle on
$AA'$ as diameter. Also take the
chord $AD$, equal to the conjugate
axis, and join $A'D$, which is perpendicular
to $AD$.

Through $AD$ draw a plane
perpendicular to $A'D$, and project
a principal chord $PNP'$ on
this plane.
\begin{flalign*}
&\indent\text{Then }&PN^2 : AN & \ctrdot  NA' :: BC^2 : AC^2. &&\\
&\text{But }       &&PN=pn,&&\\
%-----File: 190.png-----
&&An : AN &:: AD : AA'&&\\
&&        &:: BC : AC,&&\\
&\text{and }         &Dn : A'N &:: BC : AC.&&\\
&\text{Hence } &An \ctrdot nD : AN \ctrdot &NA' :: BC^2 : AC^2,&&\\
&\text{and therefore }    &pn^2 = &An \ctrdot nD,&\phantom{and therefore }&
\end{flalign*}
and the projection $ApD$ is a circle.

This theorem, in the same manner as that of Art.~\cartref{173},
may be employed in deducing properties of oblique diameters
and oblique chords of an ellipse.

\carticle{178.} \textit{If any figures in one plane be projected on another
plane, the areas of the projections will all be in the same ratio
to the areas of the figures themselves.}

Let $BAD$ be the plane of the figures, and let them be
projected on the plane $CAD, C$ being the projection of the
point $B$, and $BAD$ being a right angle.

Taking a rectangle $EFGH$, the sides of which are parallel
and perpendicular to $AD$, the projection is $efgh$, and it is
clear that the ratio of the areas of these rectangles is that
of $AC$ to $AB$.

\includegraphicsmid{178a}

Now the area of any curvilinear figure in the plane $BAD$
is the sum of the areas of parallelograms such as $EFGH$,
which are inscribed in the figure, if we take the widths,
such as $EF$, infinitesimally small.
%-----File: 191.png-----

It follows that the area of the projection of the figure is
to the area of the figure itself in the ratio of $AC$ to $AB$.

As an illustration, let a square be drawn circumscribing
a circle, and project the figure on any plane. The square
projects into tangents parallel to conjugate diameters of the
ellipse which is the projection of the circle.

The area of the parallelogram thus formed is the same
whatever be the position of the square, and we thus obtain
the theorem of Art.~\cartref{87}.

\carticle{179.} It follows that maxima and minima areas project
into maxima and minima areas. For example, the greatest
triangle which can be inscribed in a circle is an equilateral
triangle.

Projecting this figure we find that the triangle of maximum
area inscribed in an ellipse is such that the tangent at
each angular point is parallel to the opposite side, and that
the centre of the ellipse is the point of intersection of the
lines joining the vertices of the triangle with the middle
points of the opposite sides.

\carticleplus{180.}{Prop.~VII\@.} \textit{The projection of a parabola is a
parabola.}

For if $PNP'$ be a principal chord, bisected by the axis
$AN$, the projection $pnp'$ will be bisected by the projection $an$.

Moreover $pn : PN$ will be a constant ratio, as also will
be $an : AN$.
\begin{flalign*}
&\indent\text{And} &PN^2 = 4AS \ctrdot AN.&&\phantom{\indent\text{And}}
\end{flalign*}

Hence $pn^2$ will be to $4AS \ctrdot an$ in a constant ratio, and
the projection is a parabola, the tangent at $a$ being parallel
to $pn$.

\carticleplus{181.}{Prop.~VIII\@.} \textit{An hyperbola can be always projected
into a rectangular hyperbola.}

For the asymptotes can be projected into two straight
lines $cl$, $cl'$ at right angles, and if $PM$, $PN$ be parallels to
the asymptotes from a point $P$ of the curve, $PM \ctrdot PN$ is
constant.
%-----File: 192.png-----

But $pm : PM$ and $pn : PN$ are constant ratios;
\[
\therefore pm \ctrdot pn \text{ is constant.}
\]

And since $pm$ and $pn$ are perpendicular respectively to
$cl$ and $cl'$, it follows that the projection is a rectangular
hyperbola.

The same proof evidently shews that any projection of
an hyperbola is also an hyperbola.

\csubsection{EXAMPLES}

\begin{exampleslist}

\item[1.] A parallelogram is inscribed in a given ellipse; shew that its
sides are parallel to conjugate diameters, and find its greatest area.
\cheader{EXAMPLES}

\item[2.] {\stretchyspace $TP$, $TQ$ are tangents to an ellipse, and $CP'$, $CQ'$ are parallel
semi-diameters; $PQ$ is parallel to $P'Q'$.}

\item[3.] If a straight line meet two concentric similar and similarly
situated ellipses, the portions intercepted between the curves are equal.

\item[4.] Find the locus of the point of intersection of the tangents at the
extremities of pairs of conjugate diameters of an ellipse.

\item[5.] Find the locus of the middle points of the lines joining the
extremities of conjugate diameters.

\item[6.] If a tangent be drawn at the extremity of the major axis meeting
two equal conjugate diameters $CP$, $CD$ produced in $T$ and $t$; then
$PD^2 = 2AT^2$.

\item[7.] If a chord $AQ$ drawn from the vertex be produced to meet
the minor axis in $O$, and $CP$ be a semi-diameter parallel to it, then
$AQ \ctrdot AO=2CP^2$.

\item[8.] $OQ$, $OQ'$ are tangents to an ellipse from an external point $O$,
and $OR$ is a diagonal of the parallelogram of which $OQ$, $OQ'$ are adjacent
sides; prove that if $R$ be on the ellipse, $O$ will lie on a similar and
similarly situated concentric ellipse.

\item[9.] $AB$ is a given chord of an ellipse, and $C$ any point in the ellipse;
shew that the locus of the point of intersection of lines drawn from
$A$, $B$, $C$ to the middle points of the opposite sides of the triangle $ABC$
is a similar ellipse.

\item[10.] $CP$, $CD$ are conjugate semi-diameters of an ellipse; if an ellipse,
similar and similarly situated to the given ellipse, be described on $PD$
as diameter, it will pass through the centre of the given ellipse.
%-----File: 193.png-----

\item[11.] Parallelograms are inscribed in an ellipse and one pair of
opposite sides constantly touch a similar, similarly situated and concentric
ellipse; shew that the remaining pair of sides are tangents to a
third ellipse and the square on a principal semi-axis of the original
ellipse is equal to the sum of the squares on the corresponding semi-axes
of the other two ellipses.

\item[12.] Find the locus of the middle point of a chord of an ellipse
which cuts off a constant area from the curve.

\item[13.] Find the locus of the middle point of a chord of a parabola
which cuts off a constant area from the curve.

\item[14.] A parallelogram circumscribes an ellipse, touching the curve
at the extremities of conjugate diameters, and another parallelogram
is formed by joining the points where its diagonals meet the ellipse:
prove that the area of the inner parallelogram is half that of the outer
one.

If four similar and similarly situated ellipses be inscribed in the
spaces between the outer parallelogram and the curve, prove that their
centres lie in a similar and similarly situated ellipse.

\item[15.] About a given triangle $PQR$ is circumscribed an ellipse, having
for centre the point of intersection ($C$) of the lines from $P$, $Q$, $R$
bisecting the opposite sides, and $PC$, $QC$, $RC$ are produced to meet the
curve in $P'$, $Q'$, $R'$; shew that, if tangents be drawn at these points,
the triangle so formed will be similar to $PQR$, and four times as great.

\item[16.] The locus of the middle points of all chords of an ellipse which
pass through a fixed point in an ellipse similar and similarly situated
to the given ellipse, and with its centre in the middle point of the line
joining the given point and the centre of the given ellipse.

\item[17.] $PT$, $pt$ are tangents at the extremities of any diameter $Pp$ of
an ellipse; any other diameter meets $PT$ in $T$ and its conjugate meets
$pt$ in $t$; also any tangent meets $PT$ in $T'$ and $pt$ in $t'$; shew that
$PT : PT' :: pt' : pt$.

\item[18.] From the ends $P$, $D$ of conjugate diameters of an ellipse lines
are drawn parallel to any tangent line; from the centre $C$ any line is
drawn cutting these lines and the tangent in $p$, $d$, $t$, respectively; prove
that $Cp^2 + Cd^2 = Ct^2$.

\item[19.] If $CP$, $CD$ be conjugate diameters of an ellipse, and if $BP$, $BD$
be joined, and also $AD$, $A'P$, these latter intersecting in $O$, the figure
$BDOP$ will be a parallelogram.

\item[20.] $T$ is a point on the tangent at a point $P$ of an ellipse, so that
a perpendicular from $T$ on the focal distance $SP$ is of constant length;
shew that the locus of $T$ is a similar, similarly situated and concentric
ellipse.
%-----File: 194.png-----

\item[21.] $Q$ is a point in one asymptote, and $q$ in the other. If $Qq$ move
parallel to itself, find the locus of intersection of tangents to the
hyperbola from $Q$ and $q$.

\item[22.] Tangents are drawn to an ellipse from an external point $T$.
The chord of contact and the major axis, or these produced, intersect
in $K$, and $TN$ is drawn perpendicular to the major axis. Prove that
\[
CN \ctrdot CK = CA^2.
\]

\item[23.] $Q$ is a variable point on the tangent at a fixed point $P$ of an
ellipse and $R$ is taken so that $PQ=QR$. If the other tangent from $Q$
meet the ellipse in $K$, prove that $RK$ passes through a fixed point.

\item[24.] If through any point on an ellipse there be drawn lines conjugate
to the sides of an inscribed triangle they will meet the sides in
three points in a straight line.

\item[25.] $PCP'$ is a diameter of an ellipse, and a chord $PQ$ meets the
tangent at $P'$ in $R$. Prove that $PQ$, $PR$ have the parallel diameter for
a mean proportional.

\item[26.] If $AOA'$, $BOB'$ are conjugate diameters of an ellipse, and if
$AP$ and $BQ$ are parallel chords, $A'Q$ and $B'P$ are parallel to conjugate
diameters.

\item[27.] If the tangents at the ends of a chord of an hyperbola meet in
$T$, and $TM$, $TM'$ be drawn parallel to the asymptotes to meet them in
$M$, $M'$, then $MM'$ is parallel to the chord.

\item[28.] If a windmill in a level field is working uniformly on a sunny
day, the speed of the end of the shadow of one sail varies as the length
of the shadow of the next sail.

\item[29.] Spheres are drawn passing through a fixed point and touching
two fixed planes. Prove that the points of contact lie on two circles,
and that the locus of the centre of the sphere is an ellipse.

If the angle between the planes is the angle of an equilateral
triangle, prove that the distance between the foci of the ellipse is half
the major axis.
\end{exampleslist}
%-----File: 195.png-----

\cchapter{CHAPTER IX}
\csectiontoc{Of Conics in General}

\cheader{OF CONICS IN GENERAL}

\cisubsection{The Construction of a Conic}

\carticle{182.} The method of construction, given in Chapter~\cchapref{I.},
can be extended in the following manner.

Let $fSn$ be any straight line drawn through the focus
$S$, and draw $Ax$ from the vertex parallel to $fS$, and meeting
the directrix in $x$.

\includegraphicsmid{182a}

Divide the line $fSn$ in $a$ and $a'$ so that
\[
  Sa : af :: Sa' : a'f :: SA : Ax;
\]
then $a$ and $a'$ are points on the curve, for, if $ak$ be the
perpendicular on the directrix,
\begin{flalign*}
&&                       ak : af &:: AX : Ax,  &&\\
&\text{and therefore}&  Sa : ak &:: SA : AX. &\phantom{and therefore}&
\end{flalign*}
%-----File: 196.png-----

Take any point $e$ in the directrix, draw the lines $eSl$, $ea$
through $S$ and $a$, and draw $SP$ making the angle $PSl$ equal
to $lSn$.

Through $P$ draw $FPl$ parallel to $fS$, and meeting $eS$
produced in $l$,
\begin{flalign*}
&\text{then} &Pl&=SP,&\phantom{then}&\\
&\text{and}     &Pl : PF &:: Sa : af;&&\\
&&\therefore SP : PF &:: Sa : af,&&\\
&\text{and}        &SP : PK &:: Sa : ak;&&
\end{flalign*}
therefore $P$ is a point in the curve.

\carticle{183.} The construction for the point $a$ gives a simple
proof that the tangent at the vertex is perpendicular to the
axis. For when the angle $ASa$ is diminished, $Sa$ approaches
to equality with $SA$, and therefore the angle $aAS$ is ultimately
a right angle.

\carticleplus{184.}{Prop.~I\@.} \textit{To find the points in which a given
straight line is intersected by a conic of which the focus, the
directrix, and the eccentricity are given.}

\includegraphicsmid{184a}

%-----File: 197.png-----

Let $FPP'$ be the straight line, and draw $Ax$ parallel to
it. Join $FS$, and find the points $D$ and $E$ such that
\[
SD : DF :: SE : EF :: SA : Ax.
\]

Describe the circle on $DE$ as diameter, and let it intersect
the given line in $P$ and $P'$.

Join $DP$, $EP$ and draw $SG$, $FH$ at right angles to $EP$.

Then $DPE$, being the angle in a semicircle, is a right
angle, and $DP$ is parallel to $SG$ and $FH$.
\begin{flalign*}
&\text{Hence}  &SG : FH &:: SE : EF&\phantom{Hence}&\\
&&&:: SD : DF&&\\
&&&:: PG : PH;&&
\end{flalign*}
therefore the angles $SPG$, $FPH$ are equal, and therefore
$PD$ bisects the angle $SPF$.

Hence $SP : PF :: SD : DF :: SA : Ax$,\\
and $P$ is a point in the curve.

Similarly $P'$ is also a point in the curve, and the perpendicular
from $O$, the centre of the circle, on $FPP'$ meets
it in $V$, the middle point of the chord $PP'$.
\begin{flalign*}
&\text{\indent Since}     &SE : EF &:: SA : Ax&\phantom{\indent Since}&\\
&\text{and}      &SD : DF &:: SA : Ax;&&\\
&&\therefore SE-SD : DE &:: SA : Ax,&&\\
&\text{or}       &SO : OD &:: SA : Ax,&&\\
&\rlap{\text{a relation analogous to}}&&&&\\
&&SC : AC &:: SA : AX.&&
\end{flalign*}

We have already shewn, for each conic, that the middle
points of parallel chords lie in a straight line; the following
article contains a proof of the theorem which includes all
the three cases.

\carticleplus{185.}{Prop.~II\@.} \textit{To find the locus of the middle points
of a system of parallel chords.}
%-----File: 198.png-----

Let $P'P$ one of the chords be produced to meet the
directrix in $F$, draw $Ax$ parallel to $FP$, and divide $FS$ so
that
\[
SD : DF :: SE : EF :: SA : Ax;
\]
then, as in the preceding article, the perpendicular $OV$
upon $PP'$ from $O$, the middle point of $DE$, bisects $PP'$.

Draw the parallel focal chord $aSa'$; then $Oc$ parallel to
the directrix bisects $aa'$ in $c$. Also draw $SG$ perpendicular
to the chords, and meeting the directrix in $G$.

\includegraphicsmid{185a}

Then, if $OV$ meet $aa'$ in $n$,
\begin{align*}
Vn : nO &:: SF : SO,\\
&:: Sf : Sc,
\end{align*}
and, since $ncO$, $SGf$ are similar triangles,
\begin{align*}
nO : nc &:: SG : Sf;\\
\therefore Vn: nc &:: SG: Sc,
\end{align*}
and the line $Vc$ passes through $G$.

The straight line $Gc$ is therefore the locus of the middle
points of all chords parallel to $aSa'$.

The ends of the diameter $GC$ may be found by the
construction of the preceding article.
%-----File: 199.png-----
\carticle{186.} When the conic is a parabola, $SA = AX$,
{\allowdisplaybreaks
\begin{flalign*}
&\noindent\text{and}  &Sa : af &:: AX : Ax&&\\
&&&:: SX : Sf.&&\\
&\indent\text{So }   &Sa' : a'f &:: SX : Sf;&&\\
&&\therefore Sc : ac &:: SX: Sf,&&\\
&\noindent\text{and}   &ac : cf &:: SX : Sf.&&\\
&\indent\text{Hence}   &Sc : cf &:: SX^2 : Sf^2&\phantom{\indent Hence}&&\\
&&\phantom{GX \ctrdot Xf : Gf \ctrdot fX}&:: GX \ctrdot Xf : Gf \ctrdot fX&\\
&&&:: GX : Gf;&&
\end{flalign*}}
and therefore $Gc$ is parallel to $SX$, that is, the middle points
of parallel chords of a parabola lie in a straight line parallel
to the axis.

\carticleplus{187.}{Prop.~III\@.} \textit{To find the locus of the middle points
of all focal chords of a conic.}

Taking the case of a central conic, and referring to the
figure of the preceding article, let $Oc$ meet $SC$ in $N$;
\begin{flalign*}
&\text{then}     &cN : NS &:: fX : SX,&\phantom{then}&\\
&\text{and}     &cN : NC &:: GX : CX;&&\\
&&\therefore cN^2 : SN \ctrdot NC &:: fX \ctrdot GX : SX \ctrdot CX&&\\
&&&:: SX^2 : SX \ctrdot CX.&&
\end{flalign*}

Hence it follows that the locus of $c$ is an ellipse of
which $SC$ is the transverse axis, and such that the squares
of its axes are as $SX : CX$, or (Cor.~Art.~\cartref{63}) as $BC^2 : AC^2$.

Hence the locus of $c$ is similar to the conic itself.
%-----File: 200.png-----

\csubsection{EXAMPLES}

\begin{exampleslist}
\cheader{EXAMPLES}

\item[1.] If an ordinate, $PNP'$, to the transverse axis meet the tangent
at the end of the latus rectum in $T$,
\[
SP=TN, \text{ and }TP \ctrdot TP'=SN^2.
\]

\item[2.] A focal chord $PSQ$ of a conic section is produced to meet the
directrix in $K$, and $KM$, $KN$ are drawn through the feet of the
ordinates $PM$, $QN$ of $P$ and $Q$. If $KN$ produced meet $PN$ produced
in $R$, prove that
\[
PR=PM.
\]

\item[3.] The tangents at $P$ and $Q$, two points in a conic, intersect in $T$;
if through $P$, $Q$, chords be drawn parallel to the tangents at $Q$ and $P$,
and intersecting the conic in $p$ and $q$ respectively, and if tangents at
$p$ and $q$ meet in $T$, shew that $Tt$ is a diameter.

\item[4.] Two tangents $TP$, $TP'$ are drawn to a conic intersecting the
directrix in $F$,~$F'$.

If the chord $PQ$ cut the directrix in $R$, prove that
\[
SF : SF' :: RF : RF'.
\]

\item[5.] The chord of a conic $PP'$ meets the directrix in $K$, and the
tangents at $P$ and $P'$ meet in $T$; if $RKR'$, parallel to $ST$, meet the
tangents in $R$ and $R'$,
\[
KR=KR'.
\]

\item[6.] The tangents at $P$ and $P'$, intersecting in $T$, meet the latus
rectum in $D$ and $D'$; prove that the lines through $D$ and $D'$, respectively
perpendicular to $SP$ and $SP'$, intersect in $ST$.

\item[7.] If $P$, $Q$ be two points on a conic, and $p$, $q$ two points on the
directrix such that $pq$ subtends at the focus half the angle subtended
by $PQ$, either $Pp$ and $Qq$ or $Pq$ and $Qp$ meet on the curve.

\item[8.] A chord $PP'$ of a conic meets the directrix in $F$, and from
any point $T$ in $PP'$, $TLL'$ is drawn parallel to $SF$ and meeting $SP$,
$SP'$ in $L$ and $L'$; prove that the ratio of $SL$ or $SL'$ to the distance of
$T$ from the directrix is equal to the ratio of $SA : AX$.

\item[9.] If an ellipse and an hyperbola have their axes coincident and
proportional, points on them equidistant from one axis have the sum
of the squares on their distances from the other axis constant.

\item[10.] If $Q$ be any point in the normal $PG$, $QR$ the perpendicular on
$SP$, and $QM$ the perpendicular on $PN$,
\[
QR : PM :: SA : AX.
\]
%-----File: 201.png-----

\item[11.] Given a focus of a conic section inscribed in a triangle, find
the points where it touches the sides.

\item[12.] $PSQ$ is any focal chord of a conic section; the normals at $P$
and $Q$ intersect in $K$, and $KN$ is drawn perpendicular to $PQ$; prove
that $PN$ is equal to $SQ$, and hence deduce the locus of $N$.

\item[13.] Through the extremity $P$, of the diameter $PQ$ of an ellipse,
the tangent $TPT'$ is drawn meeting two conjugate diameters in $T$, $T'$.
From $P$, $Q$ the lines $PR$, $QR$ are drawn parallel to the same conjugate
diameters. Prove that the rectangle under the semi-axes of the ellipse
is a mean proportional between the triangles $PQR$ and $CTT'$.

\item[14.] Shew that a conic may be drawn touching the sides of a
triangle, having one focus at the centre of the circumscribing circle,
and the other at the orthocentre.

\item[15.] The perpendicular from the focus of a conic on any tangent,
and the central radius to the point of contact, intersect on the
directrix.

\item[16.] $AB$, $AC$ are tangents to a conic at $B$, and $C$, and $DEGF$ is
drawn from a point $D$ in $AC$, parallel to $AB$ and cutting the curve in
$E$ and $F$, and $BC$ in $G$; shew that
\[
DG^2 = DE \ctrdot DF.
\]

\item[17.] A diameter of a parabola, vertex $F$, meets two tangents in $D$
and $E$ and their chord of contact is $G$, shew that
\[
FG^2 = ED \ctrdot FE.
\]

\item[18.] $P$ and $Q$ are two fixed points in a parabola, and from any
other point $R$ in the curve, $RP$, $RQ$ are drawn cutting a fixed diameter,
vertex $E$, in $B$ and $C$; prove that the ratio of $EB$ to $EC$ is constant.

\item[19.] If the normal at $P$ meet the conjugate axis in $g$, and $gk$ be
perpendicular to $SP$, $Pk$ is constant; and if $kl$, parallel to the transverse
axis, meet the normal at $P$ in $l$, $kl$ is constant.

\item[20.] A system of conics is drawn having a common focus $S$ and a
common latus rectum $LSL'$. A fixed straight line through $S$ intersects
the conics, and at the points of intersection normals are drawn. Prove
that the envelope of each of these normals is a parabola whose focus
lies on $LSL'$, and which has the given line as tangent at the vertex.
\end{exampleslist}
%-----File: 202.png-----

\cchapter{CHAPTER X}
\csectiontoc{Ellipses as Roulettes and Glissettes}

\cheader{ELLIPSES AS ROULETTES AND GLISSETTES}

\carticle{188.} \textit{If a circle rolls on the inside of the circumference
of a circle of double its radius, any point in the area of the
rolling circle traces out an ellipse.}

Let $C$ be the centre of the rolling circle, $E$ the point of
contact.

\includegraphicsmid{188a}

Then, if the circle meet in $Q$ a fixed radius $OA$ of the
fixed circle, the angle $ECQ$ is twice the angle $EOA$, and
therefore the arcs $EQ$, $EA$ are equal.

Hence, when the circles touch at $A$, the point $Q$ of the
rolling circle coincides with $A$, and the subsequent path of
$Q$ is the diameter through $A$.

Let $P$ be a given point in the given radius $CQ$, and draw
$RPN$ perpendicular to $OA$, and $PR'$ parallel to $OA$.

Then, $OQE$ being a right angle, $EQ$ is parallel to $RP$
and therefore $CR=CP=CR'$, so that $OR$ and $OR'$ are
constant.
%-----File: 203.png-----
\begin{flalign*}
&Also      &PN : RN :: PQ : OR;&&
\end{flalign*}
therefore, the locus of $R$ being a circle, the locus of $P$ is
an ellipse, whose axes are as $PQ : OR$.

But $OR$ is clearly the length of one semi-axis, and $PQ$
or $OR'$ is therefore the length of the other, $OR$, $OR'$ being
equal to $OC+CP$ and $OC-CP$.

\carticle{189.} Properties of the ellipse are deducible from this
construction.

Thus, as the circle rolls, the point $E$ is instantaneously
at rest, and the motion of $P$ is therefore at right angles to
$EP$, \textit{i.e.} producing $EP$ to $F$, in the direction $FO$.

Therefore, drawing $PT$ parallel to $OF$, $PT$ is the tangent,
and $PF$ the normal.

\includegraphicsmid{189a}

The angles $EPT$, $EQT$ being right angles, the points
$E$, $P$, $Q$, $T$ are concyclic; but the circle through $QPE$ clearly
passes through $R$; therefore the angle $ERT$ and consequently
the angle $ORT$ is a right angle,
\begin{flalign*}
&\text{and}&        ON : OR &:: OR : OT,&&\phantom{\text{and}}\\
&\text{or}&   ON \ctrdot OT &= OR^2,&&
\end{flalign*}
which is the theorem of Art.~\cartref{74}.

Again, since $EQ't$ and $EPt$ are right angles, $E$, $Q'$, $t$, $P$
are concyclic; but the circle through $EQ'P$ clearly passes
%-----File: 204.png-----
through $R'$; therefore the angle $ER't$ and consequently the
angle $OR't$ is a right angle, and
\begin{flalign*}
&& PN : OR' &:: OR' : Ot,\\
&\text{or}& PN \ctrdot Ot &= OR'^2, &&\phantom{\text{or}}
\end{flalign*}
which is the theorem of Art.~\cartref{75}.

\includegraphicsmid{189a} %repeated figure

Further, if $PF$ meet $OQ$ in $G$, the angles $PQG$, $PFQ$ are
equal, being on equal bases $EQ$, $OQ'$;
\begin{flalign*}
&&       \therefore PG : PQ &:: PQ : PF,\\
&\text{or}&   PG \ctrdot PF &= PQ^2 = OR'^2, &&\phantom{\text{or}}
\end{flalign*}
which is the first of theorems of Art.~\cartref{77}.

And again, if $PGF$ produced meet $Q'O$ produced in $g$,
the angles $PQ'g$, $PFQ'$ are equal, being on equal bases $QO$,
$EQ'$; and the angle $Q'Pg$ is common to the two triangles
$PQ'g$, $PFQ'$.

Therefore these triangles are similar, and
\begin{flalign*}
&&                  Pg : PQ' &:: PQ' : PF,\\
&\text{or}&    Pg \ctrdot PF &= PQ'^2.\\
&\indent\text{But}& PQ' = ER'&=OR; && \phantom{\indent\text{But}}\\
&&  \therefore Pg \ctrdot PF &= OR^2,
\end{flalign*}
which is the second theorem of Art.~\cartref{77}.
%-----File: 205.png-----

\carticle{190.} If the carried point $P$ is outside the circle the line
$PNR$, perpendicular to $OA$, will meet $OE$ produced in $R$,
and $CR$ will be equal to $CP$, so that $OR$ will be constant
and the locus of $R$ will be a circle.

Also, the triangles $PQN$, $RON$ being similar, we shall
have
\[
PN : RN :: PQ : OR,
\]
so that the locus of $P$ will be an ellipse, the semi-axes of
which will be $CP + OC$ and $CP-OC$.

\begin{small}
\carticle{191.} The fact that a point on the circumference of the rolling
circle oscillates in a straight line is utilized in the construction of
Wheatstone's Photometer.

By help of machinery a metallic circle, about an inch in diameter,
is made to roll rapidly round the inside of a circle of double this
diameter, and carries a small bright bead which is fastened to its
circumference.

If this machine is held between two candles or other sources of
light, so that the line of oscillation of the bead is equidistant from the
candles, two bright lines will be seen in close contiguity, and it is easy
to form an estimate of their comparative brightnesses.

If bright beads are fastened to points in the area of the rolling
circle not on the circumference, and the machine be held near sources
of light, the appearance, when the circle is made to rotate rapidly, will
be that of a number of bright concentric ellipses.

\end{small}

\carticle{192.} \textit{A given straight line has its ends moveable on two
straight lines at right angles to each other; the path of any
given point in the moving line is an ellipse.}

\includegraphicsmid{192a}
%-----File: 206.png-----

Let $P$ be the point in the moving line $AB$, and $C$ the
middle point of $AB$.

Let the ordinate $NP$, produced if necessary, meet $OC$ in
$Q$; then $CQ=CP$ and $OQ=AP$, so that the locus of $Q$ is a
circle.
\begin{flalign*}
&\indent\text{Also}&    PN : QN &:: PB : OQ &\phantom{\indent\text{Also}}\\
&& &:: PB : PA;
\end{flalign*}
therefore the locus of $P$ is an ellipse, and its semi-axes are
equal to $AP$ and $BP$.

\carticle{193.} The theorem of Art.~\cartref{188} is at once reducible to
this case, for, taking the figure of Art.~\cartref{189}, $QPQ'$ is a diameter
of the rolling circle and is therefore of constant length, and
the points $Q$ and $Q'$ move along fixed straight lines at right
angles to each other; the locus of $P$ is therefore an ellipse
of which $Q'P$ and $PQ$ are the semi-axes.

\carticle{194.} From this construction also properties of the tangent
and normal are deducible.

Complete the rectangle $OAEB$; then, since the directions
of motion of $A$ and $B$ are respectively perpendicular
to $EA$ and $EB$, the state of motion of the line $AB$ may be
represented by supposing that the triangle $EAB$ is turning
round the point $E$.

Hence it follows that $EP$ is the normal to the locus of $P$,
and that $PT$ perpendicular to $EP$ is the tangent.

Let $OF$, parallel to $PT$, meet $EP$ in $F$; then $O$, $F$, $B$, $E$
are concyclic;
\[
\therefore \text{ the angle } PFB = EOB = PBG,
\]
and the triangles $PGB$, $PFB$ are similar.
\begin{flalign*}
&\indent\text{Hence}&  PG : PB &:: PB : PF, &\phantom{\indent\text{Hence}}\\
&\text{or}&     PG \ctrdot P&F = PB^2,
\end{flalign*}
where $PB$ is equal to the semi-conjugate axis.

\includegraphicsmid{192a} %repeated figure

Similarly, by joining $AF$, it can be shewn that
\[
Pg \ctrdot PF=PA^2,
\]
$g$ being the point of intersection of $PG$ and $AO$.
%-----File: 207.png-----

Again, since $EPT$, $EBT$ are right angles, $B$, $T$, $P$, $E$ are
concyclic, and $Q$ is clearly concyclic with $B$, $P$, $E$; so that
$TQE$ is a right angle.

Hence $OQN$ and $OQT$ are similar triangles, and
\begin{flalign*}
&&ON : OQ &:: OQ : OT,&&&\\
&\text{or }     &ON \ctrdot O&T=PA^2,&&&
\end{flalign*}
where $PA$ is equal to the semi-transverse axis.

\carticle{195.} Observing that $F$, $O$, $A$, $E$, $B$ are concyclic, we
have
\[
PF \ctrdot PE=PA \ctrdot PB;
\]
$\therefore PE$ is equal to the semi-diameter conjugate to $OP$.

This suggests a construction for the solution of the
problem,

\textit{Having given a pair of conjugate diameters of an ellipse,
it is required to determine the position and magnitudes of the
principal axes.}

Taking $OP$ and $OD$ as the given semi-conjugate diameters,
draw $PF$ perpendicular to $OD$, and, in $FP$ produced, take
$PE$ equal to $OD$.

Join $OE$, bisect it in $C$, and in $CE$ take $CQ$ equal to $CP$.

Then $OB$, $OA$, drawn perpendicular and parallel to $PQ$,
and meeting $CP$ in $B$ and $A$, will be the directions of the
axes, and their lengths will be $AP$ and $PB$.
%-----File: 208.png-----

\carticle{196.} \textit{If a given triangle $AQB$ move in its own plane so
that the extremities $A$, $B$, of its base $AB$ move on two fixed
straight lines at right angles to each other, the path of the
point $Q$ is an ellipse.}

\includegraphicsleft[9]{196a}

If $O$ be the point of intersection of the
fixed lines, and $C$ the middle point of $AB$,
the angles $COB$, $CBO$ are equal, so that, as
$AB$ slides, the line $CB$, and therefore also
the line $CQ$, turns round as fast as $CO$, but
in the contrary direction.

Produce $OC$ to $P$, making $CP=CQ$;
then the locus of $P$ is a circle the radius of
which is equal to $OC+CQ$.

There is clearly one position of $AB$ for which the points
$O$, $C$, and $Q$ are in one straight line.

\includegraphicsleft[7]{196b}

Let $OX$ be this straight line,
and let $OC$, $CQ$, be any other
corresponding positions of the
lines;

\noindent then, if $CE$ is parallel to $OX$, $CE$
bisects the angle $PCQ$, and, drawing
$PQN$ and $CL$ perpendicular
to $OX$,

\begin{flalign*}
&&QN=CL&-PE, \quad PN=CL + PE,&&\\
&\text{hence}  &QN : PN &:: OC-CP : OC+CP&\phantom{hence}&\\
&&&:: OC-CQ:OC+CQ,&&
\end{flalign*}
and $\therefore$ the locus of $Q$ is an ellipse of which the semi-axes
are $OC+CQ$ and $OC-CQ$.

If the straight lines through $A$ and $B$ perpendicular to
$OA$ and $OB$ meet in $K$, the point $K$ is the instantaneous
centre of rotation. The normal to the path of $Q$ is therefore
$QK$ and the tangent is the straight line through $Q$ perpendicular
to $QK$.

\begin{small}
\carticle{197.} \textit{Elliptic Compasses.} If two fine grooves, at right angles to
each other, be made on the plane surface of a plate of wood or metal,
and if two pegs, fastened to a straight rod, be made to move in these
%-----File: 209.png-----
grooves, then a pencil attached to any point of the rod will trace out an
ellipse.

By fixing the pencil at different points of the rod, we can obtain
ellipses of any eccentricity, but of dimensions limited by the lengths of
the rod and the grooves.

\cisubsection{Burstow's Elliptograph}

$OE$ is a groove in a stand which can be fixed to the paper or
drawing board, and $OA$, $OB$ are rods jointed at $A$, so that the end $B$
can slip along the groove, while $AO$ turns round the fixed end $O$.

\includegraphicsmid{197a}

$C$ is the middle point of $AB$, $CD$ is a rod, the length of which is
half that of $AB$, and the end $D$ can slide along the groove.

It follows that the angle $ADB$ is always a right angle.

A rod $DP$ is taken of any convenient length, and, by means of a
chain round the triangle $ADC$, is made to move so as to be always
parallel to $OA$.

If the end $B$ be moved along the groove, the end $P$ will trace out an
ellipse of which $O$ is the centre, and the lengths of its semi-axes will be
the length of $DP$ and of the difference between the lengths of $OA$ and
$DP$. This can be seen by drawing a line $OF$ perpendicular to $OE$, and
producing $DP$ to meet it in $F$. The motion will be that of a rod of
length $OA$ sliding between $OE$ and $OF$. See Dyck, \textit{Katalog der
mathematischen Instrumente}, München, 1892.

\end{small}
%-----File: 210.png-----

\clearpage
\addtocontents{toc}{\vspace{\baselineskip}}
\csectionuppertoc{\texorpdfstring{Miscellaneous Problems.\quad I}{Miscellaneous Problems. I}}
\cheader{MISCELLANEOUS PROBLEMS}

\begin{exampleslist}

\item[1.] On a plane field the crack of the rifle and the thud of the ball
striking the target are heard at the same instant; find the locus of the
hearer.

\item[2.] $PQ$, $P'Q'$ are two focal chords of a parabola, and $PR$, parallel to
$P'Q'$, meets in $R$ the diameter through $Q$; prove that
\[
PQ \ctrdot P'Q'=PR^2.
\]

\item[3.] $CP$ and $CD$ are conjugate semi-diameters of an ellipse; $PQ$
is a chord parallel to one of the axes; shew that $DQ$ is parallel to one of
the straight lines which join the ends of the axes.

\item[4.] A line cuts two concentric, similar and similarly situated
ellipses in $P$, $Q$, $q$, $p$. If the line move parallel to itself, $PQ \ctrdot Qp$ is
constant.

\item[5.] The portion of a tangent to an hyperbola intersected between
the asymptotes subtends a constant angle at the focus.

\item[6.] If a circle be described passing through any point $P$ of a given
hyperbola and the extremities of the transverse axis, and the ordinate
$NP$ be produced to meet the circle in $Q$, the locus of $Q$ is an hyperbola.

\item[7.] $PQ$ is one of a series of chords inclined at a constant angle
to the diameter $AB$ of a circle; find the locus of the intersection of
$AP$, $BQ$.

\item[8.] If from a point $T$ in the director circle of an ellipse tangents
$TP$, $TP'$ be drawn, the line joining $T$ with the intersection of the
normals at $P$ and $P'$ passes through the centre.

\item[9.] The points, in which the tangents at the extremities of the
transverse axis of an ellipse are cut by the tangent at any point of the
curve, are joined, one with each focus; prove that the point of intersection
of the joining lines lies in the normal at the point.

\item[10.] Having given a focus, the eccentricity, a point of the curve,
and the tangent at the point, shew that in general two conics can be
described.

\item[11.] A parabola is described with its focus at one focus of a given
central conic, and touches the conic; prove that its directrix will
touch a fixed circle.

\item[12.] The extremities of the latera recta of all conics which have a
common transverse axis lie on two parabolas.

\item[13.] The tangent at a moveable point $P$ of a conic intersects a
fixed tangent in $Q$, and from $S$ a straight line is drawn perpendicular
to $SQ$ and meeting in $R$ the tangent at $P$; prove that the locus of $R$ is
a straight line.
%-----File: 211.png-----

\item[14.] On all parallel chords of a circle a series of isosceles triangles
are described, having the same vertical angle, and having their planes
perpendicular to the plane of the circle. Find the locus of their
vertices; and find what the vertical angle must be in order that the
locus may be a circle.

\item[15.] A series of similar ellipses whose major axes are in the same
straight line pass through two given points. Prove that the major
axes subtend right angles at four fixed points.

\item[16.] From the centre of two concentric circles a straight line is
drawn to cut them in $P$ and $Q$; through $P$ and $Q$ straight lines are
drawn parallel to two given lines at right angles to each other. Shew
that the locus of their point of intersection is an ellipse.

\item[17.] A circle always passes through a fixed point, and cuts a given
straight line at a constant angle, prove that the locus of its centre is an
hyperbola.

\item[18.] The area of the triangle formed by three tangents to a parabola
is equal to one half that of the triangle formed by joining the points of
contact.

\item[19.] If a parabola be described with any point on an hyperbola for
focus and passing through the foci of the hyperbola, shew that its axis
will be parallel to one of the asymptotes.

\item[20.] $S$ and $H$ being the foci, $P$ a point in the ellipse, if $HP$ be
bisected in $L$, and $AL$ be drawn from the vertex cutting $SP$ in $Q$, the
locus of $Q$ is an ellipse whose focus is $S$.

\item[21.] If the diagonals of a quadrilateral circumscribing an ellipse
meet in the centre the quadrilateral is a parallelogram.

\item[22.] A series of ellipses pass through the same point, and have a
common focus, and their major axes of the same length; prove that
the locus of their centres is a circle. What are the limits of the
eccentricities of the ellipses, and what does the ellipse become at the
higher limit?

\item[23.] If $S$, $H$ be the foci of an hyperbola, $LL'$ any tangent intercepted
between the asymptotes, $SL \ctrdot HL=CL \ctrdot LL'$.

\item[24.] Tangents are drawn to an ellipse from a point on a similar
and similarly situated concentric ellipse; shew that if $P$, $Q$ be the
points of contact, $A$, $A'$ the ends of the axis of the first ellipse, the
loci of the intersections of $AP$, $A'Q$, and of $AQ$, $A'P$ are two ellipses
similar to the given ellipses.

\item[25.] Draw a parabola which shall touch four given straight lines.
Under what condition is it possible to describe a parabola touching
five given straight lines?

\item[26.] A fixed hyperbola is touched by a concentric ellipse. If the
curvatures at the point of contact are equal the area of the ellipse is
constant.
%-----File: 212.png-----

\item[27.] A circle passes through a fixed point, and cuts off equal chords
$AB$, $CD$ from two given parallel straight lines; prove that the envelope
of each of the chords $AD$, $BC$ is a central conic having the fixed
point for one focus.

\item[28.] A straight line is drawn through the focus parallel to one
asymptote and meeting the other; prove that the part intercepted
between the curve and the asymptote is one-fourth the transverse axis,
and the part between the curve and the focus one-fourth the latus rectum.

\item[29.] $PQ$ is any chord of a parabola, cutting the axis in $L$; $R$, $R'$ are
the two points in the parabola at which this chord subtends a right
angle: if $RR'$ be joined, meeting the axis in $L'$, $LL'$ will be equal to the
latus rectum.

\item[30.] If two equal parabolas have the same focus, tangents at points
angularly equidistant from the vertices meet on the common tangent.

\item[31.] A parabola has its focus at $S$, and $PSQ$ is any focal chord,
while $PP'$, $QQ'$ are two chords drawn at right angles to $PSQ$ at its
extremities; shew that the focal chord drawn parallel to $PP'$ is a mean
proportional between $PP'$ and $QQ'$.

\item[32.] With the orthocentre of a triangle as centre are described two
ellipses, one circumscribing the triangle and the other touching its
sides; prove that these ellipses are similar, and their homologous axes
at right angles.

\item[33.] $ABCD$ is a quadrilateral, the angles at $A$ and $C$ being equal; a
conic is described about $ABCD$ so as to touch the circumscribing
circle of $ABC$ at the point $B$; shew that $BD$ is a diameter of the
conic.

\item[34.] The volume of a cone cut off by a plane bears a constant
ratio to the cube, the edge of which is equal to the minor axis of the
section.

\item[35.] A tangent to an ellipse at $P$ meets the minor axis in $t$, and $tQ$
is perpendicular to $SP$; prove that $SQ$ is of constant length, and that
if $PM$ be the perpendicular on the minor axis, $QM$ will meet the major
axis in a fixed point.

\item[36.] Describe an ellipse with a given focus touching three given
straight lines, no two of which are parallel and on the same side of the
focus.

\item[37.] Prove that the conic which touches the sides of a triangle,
and has its centre at the centre of the nine-point circle, has one focus
at the orthocentre, and the other at the centre of the circumscribing
circle.

\item[38.] From $Q$, the middle point of a chord $PP'$ of an ellipse whose
focus is $S$, $QG$ is drawn perpendicular to $PP'$ to meet the major axis in
$G$; prove that
\[
2 \ctrdot SG:SP+SP'::SA:AX.
\]
%-----File: 213.png-----

\item[39.] A straight rod moves in any manner in a plane; prove that, at
any instant, the directions of motion of all its particles are tangents to
a parabola.

\item[40.] If from a point $T$ on the auxiliary circle, two tangents be
drawn to an ellipse touching it in $P$ and $Q$, and when produced meeting
the circle again in $p$, $q$; shew that the angles $PSp$ and $QSq$ are together
equal to the supplement of $PTQ$.

\item[41.] Tangents at the extremities of a pair of conjugate diameters
of an ellipse meet in $T$; prove that $ST$, $S'T$ meet the conjugate
diameters in four concyclic points.

\item[42.] From the point of intersection of an asymptote and a directrix
of an hyperbola a tangent is drawn to the curve; prove that the line
joining the point of contact with the focus is parallel to the asymptote.

\item[43.] If a string longer than the circumference of an ellipse be
always drawn tight by a pencil, the straight portions being tangents
to the ellipse, the pencil will trace out a confocal ellipse.

\item[44.] $D$ is any point in a rectangular hyperbola from which chords
are drawn at right angles to each other to meet the curve. If $P$, $Q$ be
the middle points of these chords, prove that $P$, $Q$, $D$ and the centre of
the hyperbola are concyclic.

\item[45.] From a point $T$ in the auxiliary circle tangents are drawn to
an ellipse, touching it in $P$ and $Q$, and meeting the auxiliary circle again
in $p$ and $q$; shew that the angle $pCq$ is equal to the sum of the angles
$PSQ$ and $PS'Q$.

\item[46.] The angle between the focal distance and tangent at any point
of an ellipse is half the angle subtended at the focus by the diameter
through the point.

\item[47.] $H$ is a fixed point on the bisector of the exterior angle $A$ of the
triangle $ABC$; a circle is described upon $HA$ as chord cutting the lines
$AB$, $AC$ in $P$ and $Q$; prove that $PQ$ envelopes a parabola which has $H$
for focus, and for tangent at the vertex the straight line joining the feet
of the perpendiculars from $H$ on $AB$ and $AC$.

\item[48.] Tangents to an ellipse, foci $S$ and $H$, at the ends of a focal
chord $PHP'$ meet the further directrix in $Q$, $Q'$. The parabola, whose
focus is $S$, and directrix $PP'$, touches $PQ$, $P'Q'$, in $Q$, $Q'$; it also touches
the normals at $P$, $P'$, and the minor axis, and has for the tangent at its
vertex the diameter parallel to $PP'$.

\item[49.] $S$ is a fixed point, and $E$ a point moving on the arc of a given
circle; prove that the envelope of the straight line through $E$ at right
angles to $SE$ is a conic.

\item[50.] A circle passing through a fixed point $S$ cuts a fixed circle in $P$,
and has its centre at $O$; the lines which bisect the angle $SOP$ all touch
a conic of which $S$ is a focus.
%-----File: 214.png-----

\item[51.] The tangent to an ellipse at $P$ meets the directrix, corresponding
to $S$, in $Z$: through $Z$ a straight line $ZQR$ is drawn cutting the ellipse
in $Q$, $R$; and the tangents at $Q$, $R$ intersect (on $SP$) in $T$. Shew that
a conic can be described with focus $S$, and directrix $PZ$, to pass through
$Q$, $R$ and $T$; and that $TZ$ will be the tangent at $T$.

\item[52.] $TP$, $TQ$ are tangents to an ellipse at $P$ and $Q$; one circle
touches $TP$ at $P$ and meets $TQ$ in $Q$ and $Q'$; another touches $TQ$ at
$Q$ and meets $TP$ in $P$ and $P'$; prove that $PQ'$ and $QP'$ are divided in
the same ratio by the ellipse.

\item[53.] If a chord $RPQV$ meet the directrices of an ellipse in $R$ and $V$,
and the circumference in $P$ and $Q$, then $RP$ and $QV$ subtend, each at
the focus nearer to it, angles of which the sum is equal to the angle
between the tangents at $P$ and $Q$.

\item[54.] Two tangents are drawn to the same branch of a rectangular
hyperbola from an external point; prove that the angles which these
tangents subtend at the centre are respectively equal to the angles
which they make with the chord of contact.

\item[55.] If the normal at a point $P$ of an hyperbola meet the minor
axis in $g$, $Pg$ will be to $Sg$ in a constant ratio.

\item[56.] An ordinate $NP$ of an ellipse is produced to meet the auxiliary
circle in $Q$, and normals to the ellipse and circle at $P$ and $Q$ meet in $R$;
$RK$, $RL$ are drawn perpendicular to the axes; prove that $KPL$ is a
straight line, and also that $KP=BC$ and $LP=AC$.

\item[57.] If the tangent at any point $P$ cut the axes of a conic, produced
if necessary, in $T$ and $T'$, and if $C$ be the centre of the curve, prove that
the area of the triangle $TCT'$ varies inversely as the area of the triangle
$PCN$, where $PN$ is the ordinate of $P$.

\item[58.] The circle of curvature of an ellipse at $P$ passes through the
focus $S$, $SM$ is drawn parallel to the tangent at $P$ to meet the diameter
$PCP'$ in $M$; shew that it divides this diameter in the ratio of $3:1$.

\item[59.] Prove the following construction for a pair of tangents from
any external point $T$ to an ellipse of which the centre is $C$: join $CT$,
let $TPCP'T$ a similar and similarly situated ellipse be drawn, of which
$CT$ is a diameter, and $P$, $P'$ are its points of intersection with the given
ellipse; $TP$, $TP'$ will be tangents to the given ellipse.

\item[60.] Through a fixed point a pair of chords of a circle are drawn at
right angles: prove that each side of the quadrilateral formed by
joining their extremities envelopes a conic of which the fixed point and
the centre of the circle are foci.

\item[61.] Any conic passing through the four points of intersection of
two rectangular hyperbolas will be itself a rectangular hyperbola.

\item[62.] $R$ is the middle point of a chord $PQ$ of a rectangular hyperbola
whose centre is $C$. Through $R$, $RQ'$, $RP'$ are drawn parallel to the
tangents at $P$ and $Q$ respectively, meeting $CQ$, $CP$ in $Q'$, $P'$. Prove
that $C$, $P'$, $R$, $Q'$ are concyclic.
%-----File: 215.png-----

\item[63.] The tangents at two points $Q$, $Q'$ of a parabola meet the
tangent at $P$ in $R$, $R'$ respectively, and the diameter through their
point of intersection $T$ meets it in $K$; prove that $PR=KR'$, and that,
if $QM$, $Q'M'$, $TN$ be the ordinates of $Q$, $Q'$, $T$ respectively to the
diameter through $P$, $PN$ is a mean proportional between $PM$ and $PM'$.

\item[64.] Common tangents are drawn to two parabolas, which have a
common directrix, and intersect in $P$, $Q$: prove that the chords
joining the points of contact in each parabola are parallel to $PQ$, and
the part of each tangent between its points of contact with the two
curves is bisected by $PQ$ produced.

\item[65.] An ellipse has its centre on a given hyperbola and touches the
asymptotes. The area of the ellipse being always a maximum, prove
that its chord of contact with the asymptotes always touches a similar
hyperbola.

\item[66.] A circle and parabola have the same vertex $A$ and a common
axis. $BA'C$ is the double ordinate of the parabola which touches the
circle at $A'$, the other extremity of the diameter which passes through
$A$; $PP'$ is any other ordinate of the parabola parallel to this, meeting
the axis in $N$ and the chord $AB$ produced in $R$: shew that the rectangle
between $RP$ and $RP'$ is proportional to the square on the tangent
drawn from $N$ to the circle.

\item[67.] Tangents are drawn at two points, $P$, $P'$ on an ellipse. If any
tangent be drawn meeting those at $P$, $P'$ in $R$, $R'$, shew that the line
bisecting the angle $RSR'$ intersects $RR'$ on a fixed tangent to the
ellipse. Find the point of contact of this tangent.

\item[68.] Having given a pair of conjugate diameters of an ellipse, $PCP'$,
$DCD'$, let $PF$ be the perpendicular from $P$ on $CD$, in $PF$ take $PE$
equal to $CD$, bisect $CE$ in $O$, and on $CE$ as diameter describe a circle;
prove that $PO$ will meet the circle in two points $Q$ and $R$ such that
$CQ$, $CR$ are the directions of the semi-axes, and $PQ$, $PR$ their lengths.

\item[69.] A straight line is drawn through the angular point $A$ of a
triangle $ABC$ to meet the opposite side in $a$; two points $O$, $O'$ are taken
on $Aa$, and $CO$, $CO'$ meet $AB$ in $c$ and $c'$, and $BO$, $BO'$ meet $CA$ in $b$, $b'$;
shew that a conic passing through $abb'cc'$ will be touched by $BC$.

\item[70.] If $TP$, $TQ$ are two tangents to a parabola, and any other
tangent meets them in $Q$ and $R$, the middle point of $QR$ describes a
straight line.

\item[71.] Lines from the centre to the points of contact of two parallel
tangents to a rectangular hyperbola and concentric circle make equal
angles with either axis of the hyperbola.

\item[72.] A line moves between two lines at right angles so as to subtend
a right angle and a half at a fixed point on the bisector of the
right angle; prove that it touches a rectangular hyperbola.
%-----File: 216.png-----

\item[73.] Two cones, whose vertical angles are supplementary, are placed
with their vertices coincident and their axes at right angles, and are
cut by a plane perpendicular to a common generating line; prove that
the directrices of the section of one cone pass through the foci of the
section of the other.

\item[74.] The normal at a point $P$ of an ellipse meets the curve again in
$P'$, and through $O$, the centre of curvature at $P$, the chord $QOQ'$ is
drawn at right angles to $PP'$; prove that
\[
      QO \ctrdot OQ' : PO \ctrdot OP' :: 2 \ctrdot PO : PP'.
\]

\item[75.] From an external point $T$, tangents are drawn to an ellipse,
the points of contact being on the same side of the major axis. If the
focal distances of these points intersect in $M$ and $N$, $TM$, $TN$ are
tangents to a confocal hyperbola, which passes through $M$ and $N$.

\item[76.] Two tangents to an hyperbola from $T$ meet the directrix in $F$
and $F'$; prove that the circle, centre $T$, which touches $SF$, $SF'$, meets
the directrix in two points the radii to which from the point $T$ are
parallel to the asymptotes.

\item[77.] $QR$, touching the ellipse at $P$, is one side of the parallelogram
formed by tangents at the ends of conjugate diameters; if the normal
at $P$ meet the axes in $G$ and $g$, prove that $QG$ and $Rg$ are at right
angles.

\item[78.] If $PP'$ be a double ordinate of an ellipse, and if the normal at
$P$ meet $CP'$ in $O$, prove that the locus of $O$ is a similar ellipse, and that
its axis is to the axis of the given ellipse in the ratio
\[
      AC^2 - BC^2 : AC^2 + BC^2.
\]

\item[79.] A chord of a conic whose pole is $T$ meets the directrices in
$R$ and $R'$; if $SR$ and $S'R'$ meet in $Q$, prove that the minor axis bisects
$TQ$.

\item[80.] On a parabola, whose focus is $S$, three points $Q$, $P$, $Q'$ are taken
such that the angles $PSQ$, $PSQ'$ are equal; the tangent at $P$ meets the
tangents at $Q$, $Q'$ in $T$, $T'$: shew that
\[
TQ : T'Q' :: SQ : SQ'.
\]

\item[81.] If from any point P of a parabola perpendiculars $PN$, $PL$ are
let fall on the axis and the tangent at the vertex, the line $LN$ always
touches another parabola.

\item[82.] $PQ$ is any diameter of a section of a cone whose vertex is $V$;
prove that  $VP + VQ$  is constant.

\item[83.] If $SY$, $SK$ are the perpendiculars from a focus on the tangent
and normal at any point of a conic, the straight line $YK$ passes through
the centre of the conic.

\item[84.] If the axes of two parabolas are in the same direction, their
common chord bisects their common tangents.
%-----File: 217.png-----

\item[85.] Find the position of the normal chord which cuts off from a
parabola the least segment.

\item[86.] From the point in which the tangent at any point $P$ of an
hyperbola meets either asymptote perpendiculars $PM$, $PN$ are let fall
upon the axes. Prove that $MN$ passes through $P$.

\item[87.] If two parabolas whose latera recta have a constant ratio, and
whose foci are two given points $S$, $S'$, have a contact of the second order
at $P$, the locus of $P$ is a circle.

\item[88.] Find the class of plane curves such that, if from a fixed point
in the plane, perpendiculars are let fall on the tangent and normal at
any point of any one of the curves, the join of the feet of the perpendiculars
will pass through another fixed point.

\item[89.] If two ellipses have one common focus $S$ and equal major axes,
and if one ellipse revolves in its own plane about $S$, the chord of
intersection envelopes a conic confocal with the fixed ellipse.

\item[90.] The tangent at any point $P$ of an ellipse meets the axis minor
in $T$ and the focal distances $SP$, $HP$ meet it in $R$, $r$. Also $ST$, $HT$,
produced if necessary, meet the normal at $P$ in $Q$, $q$, respectively.
Prove that $Qr$ and $qR$ are parallel to the axis major.

\item[91.] Two points describe the circumference of an ellipse, with
velocities which are to one another in the ratio of the squares on the
diameters parallel to their respective directions of motion. Prove that
the locus of the point of intersection of their directions of motion will
be an ellipse, confocal with the given one.

\item[92.] If $AA'$ be the axis major of an elliptic section of a cone, vertex
$O$, and if $AG$, $A'G'$ perpendicular to $AV$, $A'V$ meet the axis of the cone
in $G$ and $G'$, and $GU$, $G'U'$ be the perpendiculars let fall on $AA'$, prove
that $U$ and $U'$ are the centres of curvature at $A$ and $A'$.

\item[93.] By help of the geometry of the cone, or otherwise, prove that
the sum of the tangents from any point of an ellipse to the circles of
curvature at the vertices is constant.

\item[94.] If two tangents be drawn to a section of a cone, and from
their intersection two straight lines be drawn to the points where the
tangent plane to the cone through one of the tangents touches the focal
spheres, prove that the angle contained by these lines is equal to the
angle between the tangents.

\item[95.] If $CP$, $CD$ are conjugate semi-diameters and if through $C$
is drawn a line parallel to either focal distance of $P$, the perpendicular
from $D$ upon this line will be equal to half the minor axis.

\item[96.] The area of the parallelogram formed by the tangents at the
ends of any pair of diameters of a central conic varies inversely as the
area of the parallelogram formed by joining the points of contact.
%-----File: 218.png-----

\item[97.] Shew how to draw through a given point a plane which will
have the given point for (1) focus, (2) centre, of the section it makes
of a given right circular cone: noticing any limitations in the position
of the point which may be necessary.

\item[98.] In the first figure of Art.~\cartref{148}, if a plane be drawn intersecting
the focal spheres in two circles and the cone in an ellipse, the
sum or difference of the tangents from any point of the ellipse to the
circles is constant.

\item[99.] If sections of a right cone be made, perpendicular to a given
plane, such that the distance between a focus of a section and that
vertex which lies on one of the generating lines in the given plane be
constant, prove that the transverse axes, produced if necessary, of all
sections will touch one of two fixed circles.

\item[100.] A sphere rolls in contact with two intersecting straight wires;
prove that its centre describes an ellipse.
\end{exampleslist}
%-----File: 219.png-----

\cchapter{CHAPTER XI}
\csectiontoc{Harmonic Properties, Poles and Polars}

\cheader{HARMONIC PROPERTIES}

\carticleplus{198.}{Def.} \textit{A straight line is harmonically divided in
two points when the whole line is to one of the extreme parts
as the other extreme part is to the middle part.}

Thus $AD$ is harmonically divided in $C$ and $B$, when
\[
AD : AC :: BD : BC.
\]

\includegraphicsmid{198a}

This definition may also be presented in the following
form.

\textit{The straight line $AB$ is harmonically divided in $C$ and $D$,
when it is divided internally in $C$, and externally in $D$, in the
same ratio.}

Under these circumstances the four points $A$, $C$, $B$, $D$
constitute \textit{an Harmonic Range}, and if through any point $O$
four straight lines $OA$, $OC$, $OB$, $OD$ be drawn, these four
lines constitute an \textit{Harmonic Pencil}.

\subparagraph{Prop.~I\@.} \textit{If a straight line be drawn parallel to one of
the rays of an harmonic pencil, its segments made by the other
three will be equal, and any straight line is divided harmonically
by the four rays.}

Let $ACBD$ be the given harmonic range, and draw $ECF$
through $C$ parallel to $OD$, and meeting $OA$, $OB$ in $E$ and $F$.
\begin{flalign*}
&\indent\text{Then }    &AD : AC &:: OD : EC,&\phantom{\indent\text{Then}}&\\
&\text{and }     &BD : BC &:: OD : CF;&&
\end{flalign*}
%-----File: 220.png-----
but from the definition
\begin{flalign*}
&&AD : AC &:: BD : BC;&&\\
&&\therefore EC &= CF,&&
\end{flalign*}
and any other line parallel to $ECF$ is obviously bisected
by $OC$.

Next, let $acbd$ be any straight line cutting the pencil,
and draw $ecf$ parallel to $Od$; so that $ec = cf$.

\includegraphicsmid{198b}
\begin{flalign*}
&\indent\text{Then}&      ad : ac &:: Od : ec, &\phantom{\indent\text{Then}}\\
&\text{and}&       bd : bc &:: Od : cf;\\
&& \therefore  ad : ac &:: bd : bc;
\end{flalign*}
that is, $acbd$ is harmonically divided.

If the line  $c\beta\delta\alpha$ be drawn cutting $AO$ produced,
\begin{flalign*}
&\text{then}&  \alpha\delta : \alpha c &:: O\delta : ec, &\phantom{\text{then}}\\
&\text{and}&   \beta\delta : \beta c &:: O\delta : cf;\\
&& \therefore\alpha\delta : \alpha c &:: \beta\delta : \beta c,\\
&\text{or}&    \alpha c : \alpha\delta &:: \beta c : \beta\delta,
\end{flalign*}
and similarly it may be shewn in all other cases that the
line is harmonically divided.
%-----File: 221.png-----

\carticleplus{199.}{Prop.~II\@.}  \textit{The pencil formed by two straight lines
and the bisectors of the angles between them is an harmonic
pencil.}

For, if $OA$, $OB$ be the lines, and $OC$, $OD$ the bisectors,
draw $KPL$ parallel to $OC$ and meeting $OA$, $OD$, $OB$. Then
the angles $OKL$, $OLK$ are obviously equal,
and the angles at $P$ are right angles; therefore $KP = PL$,
and the pencil is harmonic.

\includegraphicsmid{199a}

\carticleplus{200.}{Prop.~III\@.}  \textit{If $ACBD$, $Acbd$ be harmonic ranges,
the straight lines $Cc$, $Bb$, $Dd$ will meet in a point, as also $Cd$,
$cD$, $Bb$.}

\includegraphicsmid{200a}
%-----File: 222.png-----

For, if $Cc$, $Dd$ meet in $F$, join $Fb$; then the pencil
$F (Acbd)$ is harmonic, and will be cut harmonically by $AD$.

Hence $Fb$ produced will pass through $B$.

Similarly, if $Cd$, $cD$ meet in $E$, \\
$E (Acbd)$ is harmonic, and therefore $bE$ produced will pass
through $B$.

\cisubsection{Harmonic Properties of a Quadrilateral}

In the preceding figure, let $CcdD$ be any quadrilateral;
and let $dc$, $DC$ meet in $A$, $Cd$, $cD$ in $E$, and $Cc$, $Dd$ in $F$.

Then taking $b$ and $B$ so as to divide $Acd$ and $ACD$ harmonically,
the ranges $Acbd$ and $ACBD$ are harmonic, and
therefore $Bb$ passes through both $E$ and $F$.

Similarly it can be shewn that $AF$ is divided harmonically
in $L$ and $M$, by $Dc$ and $dC$.

For $E (Acbd)$ is harmonic and therefore the transversal
$ALFM$ is harmonically divided.

\carticleplus{201.}{Prop.~IV\@.} \textit{If $ACBD$ be an harmonic range, and
$E$ the middle point of $CD$,}
\[
EA \ctrdot EB = EC^2.
\]

\includegraphicsmid{201a}
\begin{flalign*}
&\indent\text{For}&    AD : AC &:: BD : BC, & \phantom{\indent\text{For}}\\
&\text{or}&  AE + EC : AE-EC &:: EC + EB : EC-EB;\\
&& \therefore AE : EC &:: EC : EB,\\
&\text{or}&      AE \ctrdot EB &= EC^2 = ED^2.
\end{flalign*}

Hence also, conversely, if $EC^2 = ED^2 = AE \ctrdot EB$, the
range $ACBD$ is harmonic, $C$ and $D$ being on opposite sides
of $E$.

Hence, if a series of points $A$, $a$, $B$, $b$,~\ldots\ on a straight line
be such that
\begin{align*}
EA \ctrdot Ea = EB \ctrdot Eb &= EC \ctrdot Ec \ldots\\
                &= EP^2,
\end{align*}
%-----File: 223.png-----
and if $EQ = EP$, then the several ranges $(APaQ)$, $(BPbQ)$,
\&c.\ are harmonic.

\carticleplus{202.}{Def.}  A system of pairs of points on a straight
line such that
\[
EA \ctrdot Ea = EB \ctrdot EB \ctrdot Eb = \ldots\quad  = EP^2 = EQ^2
\]
is called a system in \textit{Involution}, the point $E$ being called the
centre and $P$, $Q$ the foci of the system.

Any two corresponding points $A$, $a$, are called \textit{conjugate}
points, and it appears from above that any two conjugate
points form, with the foci of the system, an harmonic range.

It will be noticed that a focus is a point at which conjugate
points coincide, and that the existence of a focus is only
possible when the points $A$ and $a$ are both on the same side
of the centre.

\carticleplus{203.}{Prop.~V\@.}  \textit{Having given two pairs of points, $A$ and
$a$, $B$ and $b$, it is required to find the centre and foci of the
involution.}

If E be the centre,
\begin{flalign*}
&&EA : EB &:: Eb : Ea;&&\\
&&\therefore EA : AB &:: Eb : ab,&&\\
&\text{or}& EA : Eb &:: AB : ab.&\phantom{or}
\end{flalign*}

\includegraphicsmid{203a}

This determines $E$, and the foci $P$ and $Q$ are given by
the relations
\[
EP^2 = EQ^2 = EA \ctrdot EA.
\]

We shall however find the following relation useful.
\begin{flalign*}
&\indent\text{Since} & EA : Eb &:: EB : Ea;&\phantom{\indent\text{Since}}\\
&&\therefore EA : Ab &::  EB : aB,&&\\
&\text{or }& EA : EB &:: Ab : aB;&&\\
&\text{but }& Eb : EA &:: ab : AB;&&\\
&&\therefore Eb : EB &:: Ab \ctrdot ba : AB \ctrdot Ba.&&\\
&\indent\text{Again, }& Qb : Pb &:: QB : PB;&&\\
&&\therefore Qb - Pb : Pb &:: QB - PB : PB,&&\\
%-----File: 224.png-----
&& 2 \ctrdot EP : Pb &:: 2 \ctrdot EB : BP;&&\\
&& \therefore Pb^2 : PB^2 &:: EP^2 : EB^2,&&\\
&& &:: Eb : EB&&\\
&& &:: Ab \ctrdot ba : AB \ctrdot Ba.&&
\end{flalign*}

This determines the ratio in which $Bb$ is divided by $P$.

\carticle{204.} If $QAPa$ be an harmonic range and $E$ the middle
point of $PQ$, and if a circle be described on $PQ$ as diameter,
the lines joining any point $R$ on this circle with $P$ and $Q$
will bisect the angles between $AR$ and $aR$.

\includegraphicsmid{204a}
\begin{flalign*}
&\indent\text{For}& EA \ctrdot Ea &= EP^2 = ER^2;&\phantom{\indent\text{For}}\\
&&             \therefore EA : ER &:: ER : Ea,
\end{flalign*}
and the triangles $ARE$, $aRE$ are similar.
\begin{flalign*}
&\indent\text{Hence}& AR:aR&::EA:ER&\phantom{\indent\text{Hence}}\\
&& &::EA : EP. \\
&\indent\text{But}& Ea : EP &:: EP : EA;\\
&&                  \therefore  aP : EP &:: AP : EA.\\
&\indent\text{Hence}& AR : aR &:: AP : aP,
\end{flalign*}
and $ARa$ is bisected by $RP$.

Hence, if $A$ and $a$, $B$ and $b$ be conjugate points of a
system in involution of which $P$ and $Q$ are the foci, it follows
that $AB$ and $ab$ subtend equal angles at any point of the
circle on $PQ$ as diameter.

This fact also affords a means of obtaining the relations
of Art.~\cartref{203}.
%-----File: 225.png-----

We must observe that if the points $A$, $a$ are on one side
of the centre and $B$, $b$ on the other, the angles subtended by
$AB$, $ab$ are supplementary to each other.

\carticleplus{205.}{Prop.~VI\@.} \textit{If four points form an harmonic range,
their conjugates also form an harmonic range.}

Let $A$, $B$, $C$, $D$ be the four points, $a$, $b$, $c$, $d$ their conjugates.
\includegraphicsmid{205a}

Then, as in the eighth line of Art.~\cartref{203},
\begin{flalign*}
&                 &                   EA : Ed &:: AD: ad,&&\\
&\text{or}        &                   ED : Ea &:: AD: ad;&&\\
&                 & \therefore AD \ctrdot  Ea &= ED \ctrdot ad.&&
\end{flalign*} % to allow page break
\begin{flalign*}
&\text{Similarly} &            AC \ctrdot  Ea &= EC \ctrdot  ac,&\phantom{Similarly}&\\
&                 &            BD \ctrdot  Eb &= ED \ctrdot  bd,&&\\
&                 &            BC \ctrdot  Eb &= EC \ctrdot  bc.&&
\end{flalign*}

But, $ABCD$ being harmonic,
\begin{flalign*}
&&AD : AC &:: BD : BC;&&\\
&&\therefore ED \ctrdot ad:EC \ctrdot ac&::ED \ctrdot bd:EC \ctrdot bc.&&\\
&\text{\indent Hence} &ad:ac&::bd:bc,&\phantom{\indent Hence}&
\end{flalign*}
or the range of the conjugates is harmonic.

\carticleplus{206.}{Prop.~VII\@.} \textit{If a system of conics pass through four
given points, any straight line will be cut by the system in a
series of points in involution.}

The four fixed points being $C$, $D$, $E$, $F$, let the line meet
one of the conics in $A$ and $a$, and the straight lines $CF$, $ED$,
in $B$ and $b$.

Then the rectangles $AB \ctrdot Ba$, $CB \ctrdot BF$ are in the ratio
of the squares on parallel diameters, as also are $Ab \ctrdot ba$ and
$Db \ctrdot bE$.

But the squares on the diameters parallel to $CF$, $ED$
are in the constant ratio $KF \ctrdot KC:KE \ctrdot KD$; and, the
%-----File: 226.png-----
line $Bb$ being given in position, the rectangles $CB \ctrdot BF$ and
$Db \ctrdot bE$ are given; therefore the rectangles $AB \ctrdot Ba$, $Ab \ctrdot ba$
are in a constant ratio.

\includegraphicsmid{206a}

But (Art.~\cartref{203}) this ratio is the same as that of $PB^2$ to
$Pb^2$, if $P$ be a focus of the involution $A$, $a$, $B$, $b$.

Hence $P$ is determined, and all the conics cut the line
$Bb$ in points which form with $B$, $b$ a system in involution.

We may observe that the foci are the points of contact of
the two conics which can be drawn through the four points
touching the line, and that the centre is the intersection of
the line with the conic which has one of its asymptotes
parallel to the line.

\carticleplus{207.}{Prop.~VIII\@.} \textit{If through any point two tangents
be drawn to a conic, any other straight line through the point
will be divided harmonically by the curve and the chord of
contact.}

Let $AB$, $AC$ be the tangents, $ADFE$ the straight line.

Through $D$ and $E$ draw $GDHK$, $LEMN$ parallel to $BC$.

Then the diameter through $A$ bisects $DH$, and $BC$,
and therefore bisects $GK$; hence $GD = HK$, and similarly
$LE = MN$.
\begin{flalign*}
&\indent\text{Also} &    LE : EN &:: GD : DK;&&&\\
&&\therefore LE \ctrdot EN : LE^2 &:: GD \ctrdot DK : GD^2,&&&\\
&\text{or} & LE \ctrdot LM : GD \ctrdot GH &:: LE^2 : GD^2&&&\\
&&&:: LA^2 : GA^2.&&&
\end{flalign*}
%-----File: 227.png-----
\begin{flalign*}
&\text{But }&  LE \ctrdot LM : GD \ctrdot GH &:: LB^2 : BG^2;&&&\\
&\text{hence }&    AL : AG &:: BL : BG,&&&
\end{flalign*}
\begin{flalign*}
&\text{and therefore }&   AE : AD &:: FE : FD,&&&
\end{flalign*}
that is, $ADFE$ is harmonically divided.

\includegraphicsmid{207a}

\carticleplus{208.}{Prop.~IX\@.} \textit{If two tangents be drawn to a conic,
any third tangent is harmonically divided by the two tangents,
the curve, and the chord of contact.}

\includegraphicsmid{208a}
%-----File: 228.png-----

Let $DEFG$ be the third tangent, and through $G$, the
point in which it meets $AC$, draw $GHKL$ parallel to $AB$,
cutting the curve and the chord of contact in $H$, $K$, $L$.
\begin{flalign*}
&\indent\text{Then}& GH \ctrdot GL : GC^2 &:: AB^2 : AC^2&&&\\
&&&:: GK^2 : GC^2;&&&\\
&&\therefore GH \ctrdot GL &= GK^2.&&&\\
&\indent\text{Hence}&DG^2 : DE^2 ::G&K^2 : EB^2&&&\phantom{\indent\text{Hence}}\\
&&:: G&H \ctrdot GL : EB^2&&&\\
&&:: F&G^2 : FE^2;&&&
\end{flalign*}
that is, $DEFG$ is an harmonic range.

\carticleplus{209.}{Prop.~X\@.} \textit{If any straight line meet two tangents to
a conic in $P$ and $Q$, the chord of contact in $T$ and the conic in
$R$ and $V$,}
\[
PR \ctrdot PV : QR \ctrdot QV :: PT^2 : QT^2.
\]

Taking the preceding figure, draw the tangent $DEFG$
parallel to $PQ$.
\begin{flalign*}
&\indent\text{Then}&  PR \ctrdot PV : EF^2 &:: PB^2 : BE^2&&\phantom{\indent\text{Then}}\\
&&&:: PT^2 : DE^2;&&&\\
&\text{and }     &QR \ctrdot QV : GF^2 &:: QC^2 : GC^2 &&&\\
&&&:: QT^2 : DG^2;&&&\\
&\text{but }&       EF : DE &:: GF : DG;&&&\\
&&\therefore PR \ctrdot PV : PT^2 &:: QR \ctrdot QV : QT^2.&&&
\end{flalign*}

\carticleplus{210.}{Prop.~XI\@.} \textit{If chords of a conic be drawn through
a fixed point the pairs of tangents at their extremities will
intersect in a fixed line.}

Let $B$ be the fixed point and $C$ the centre, and let $CB$
meet the curve in~$P$.

Take $A$ in $CP$ such that
\[
CA : CP :: CP : CB;
\]
then $B$ is the middle point of the chord of contact of the
tangents $AQ$, $AR$.
%-----File: 229.png-----

Draw any chord $EBF$, and let the tangents at $E$ and $F$
meet in $G$: also join $CG$ and draw $PN$ parallel to $EF$.

\includegraphicsmid{210a}

Then if $CG$ meet $EF$ in $K$ and the tangent at $P$ in $T$,
\begin{align*}
CK \ctrdot CG &= CN \ctrdot CT;\\
\therefore CG : CT &:: CN : CK\\
&:: CP : CB\\
&:: CA : CP;
\end{align*}
hence $AG$ is parallel to $PT$, and the point $G$ therefore lies
on a fixed line.

If the conic be a parabola, we must take $AP$ equal to
$BP$: then, remembering that $KG$ and $NT$ are bisected by
the curve, the proof is the same as before.

\carticle{211.} If $A$ be the fixed point, let $CA$ meet the curve in
$P$, and take $B$ in $CP$ such that
\[
CB : CP :: CP : CA;
\]
then $B$ is the middle point of the chord of contact of the
tangents $AQ$, $AR$.
%-----File: 230.png-----
Draw any chord $AEF$, and let the tangents at $E$ and $F$
meet in $G$; also join $CG$ and draw $PN$ parallel to $EF$.

\includegraphicsmid{211a}
\begin{flalign*}
&\indent\text{Then}& CK \ctrdot CG&= CN \ctrdot CT;&\phantom{\indent\text{Then}}\\
&&\therefore CG : CT &:: CN : CK \\
&& &:: CP : CA\\
&& &:: CB : CP;
\end{flalign*}
$\therefore BG$ is parallel to $PT$ and coincides with the chord of
contact $QR$.

Hence, conversely, if from points on a straight line pairs
of tangents be drawn to a conic, the chords of contact will
pass through a fixed point.

\cisubsection{Poles and Polars}
\cheader{POLES AND POLARS}

\carticleplus{212.}{Def.} The straight line which is the locus of the
points of intersection of tangents at the extremities of chords
through a fixed point is called the \textit{polar} of the point.

Also, if from points in a straight line pairs of tangents be
drawn to a conic, the point in which all the chords of contact
intersect is called the \textit{pole} of the line.

If the pole be without the curve the polar is the chord of
contact of tangents from the pole.
%-----File: 231.png-----

If the pole be on the curve the polar is the tangent at
the point.

It follows at once from these definitions that the focus of
a conic is the pole of the directrix, and that the foot of the
directrix is the pole of the latus rectum.

\carticleplus{213.}{Prop.~XII\@.} \textit{A straight line drawn through any
point is divided harmonically by the point, the curve, and the
polar of the point.}

If the point be without the conic this is already proved
in Art.~\cartref{207}.

If it be within the conic, as $B$ in the figure of Art.~\cartref{210},
then, drawing any chord $FBEV$ meeting in $V$ the polar of
$B$, which is $AG$, the chord of contact of tangents from $V$
passes through $B$, by Art.~\cartref{211}, and the line $VEBF$ is therefore
harmonically divided.

Hence the polar may be constructed by drawing two
chords through the pole and dividing them harmonically;
the line joining the points of division is the polar.

Or, in the figure of Art.~\cartref{210},
\[
CB \ctrdot CA = CP^2,
\]
so that the polar of $B$ is obtained by taking the point $A$ on
the diameter through $B$, at the distance from $C$ given by the
above relation, and then drawing AG parallel to the diameter
which is conjugate to $CP$.

\subparagraph{Cor.} Hence it follows that \textit{the centre of a conic is the
pole of a line at an infinite distance.}

For, if $CB$ is diminished indefinitely, $CA$ is increased
indefinitely.

\carticleplus{214.}{Prop.~XIII\@.} \textit{The polars of two points intersect in
the pole of the line joining the two points.}

For, if $A$, $B$ be the two points and $O$ the pole of $AB$, the
line $AO$ is divided harmonically by the curve, and therefore
the polar of $A$ passes through the point $O$.
%-----File: 232.png-----

Similarly the polar of $B$ passes through $O$;

That is, the polars of $A$ and $B$ intersect in the pole
of $AB$.

\carticleplus{215.}{Prop.~XIV\@.} \textit{If a quadrilateral be inscribed in a
conic, its opposite sides and diagonals will intersect in three
points such that each is the pole of the line joining the other
two.}

Let $ABCD$ be the quadrilateral, $F$ and $G$ the points of
intersection of $AD$, $BC$, and of $DC$, $AB$.

\includegraphicsmid{215a}

Let $EG$ meet $FA$, $FB$, in $L$ and $M$.

Then (Art.~\cartref{200}) $FDLA$ and $FCMB$ are harmonic ranges;

Therefore $L$ and $M$ are both on the polar of $F$ (Art.~\cartref{213}),
and $EG$ is the polar of $F$.

Similarly, $EF$ is the polar of $G$, and therefore $E$ is the
pole of $FG$ (Art.~\cartref{214}).

\carticleplus{216.}{Def.} If each of the sides of a triangle be the
polar, with regard to a conic, of the opposite angular point,
the triangle is said to be \textit{self-conjugate} with regard to the
conic.

Thus the triangle $EGF$ in the above figure is self-conjugate.
%-----File: 233.png-----

To construct a self-conjugate triangle, take a straight
line $AB$ and find its pole $C$.

Draw through $C$ any straight line $CD$ cutting $AB$ in $D$,
and find the pole $E$ of $CD$, which lies on $AB$: then $CDE$ is
self-conjugate.

\carticleplus{217.}{Prop.~XV\@.} \textit{If a quadrilateral circumscribe a conic,
its three diagonals form a self-conjugate triangle.}

Let the polar of $F$ (that is, the chord of contact $P'P$),
meet $FG$ in $R$; then, since $R$ is on the polar of $F$, it follows
that $F$ is on the polar of $R$.

Now $F (AEBG)$ is harmonic (Art.~\cartref{200}), and, if $FE$ meet
$P'P$ in $T$, $P'TPR$ is an harmonic range; hence, by the
theorem of Art.~\cartref{213}, $FT$, \textit{i.e.}\ $FE$, is the polar of $R$.

\includegraphicsmid{217a}

Similarly, if the other chord of contact $QQ'$ meet $FG$ in
$R'$, $GE$ is the polar of $R'$;
\[
\therefore E \text{ is the pole of }RR', \text{ that is, of } LK.
\]

Again, $DEBK$ is harmonic, and therefore the pencil
$C (QEPK)$ is harmonic.

Hence, if $QP$ meet $AC$ in $S$ and $CK$ in $V$, $QSPV$ is
harmonic, and therefore $S$ is on the polar of $V$.
%-----File: 234.png-----

But $S$ is on the polar of $C$; therefore $CV$, that is, $CK$, is
the polar of $S$.

Similarly, if $P'Q'$ meet $AC$ in $S'$, $AK$ is the polar of $S'$.

Hence it follows that $K$ is the pole of $SS'$, that is, of $EL$;
$ELK$ is therefore a self-conjugate triangle.

\carticleplus{218.}{Prop.~XVI\@.} \textit{If a system of conics have a common
self-conjugate triangle, any straight line passing through one
of the angular points of the triangle is cut in a series of points
in involution.}

For, if $ABC$ be the triangle, and a line $APDQ$ meet $BC$
in $D$, and the conic in $P$ and $Q$, $APDQ$ is an harmonic range,
and all the pairs of points $P$,~$Q$ form with $A$ and $D$ an harmonic
range.

Hence the pairs of points form a system in involution, of
which $A$ and $D$ are the foci.

\carticleplus{219.}{Prop.~XVII\@.} \textit{The pencil formed by the polars of
the four points of an harmonic range is an harmonic pencil.}

Let $ABCD$ be the range, $O$ the pole of $AD$.

Let the polars $Oa$, $Ob$, $Oc$, $Od$ meet $AD$ in $a$, $b$, $c$, $d$, and
let $AD$ meet the conic in $P$ and $Q$.

\includegraphicsmid{219a}

Then $APaQ$, $CPcQ$, \&c.\ are harmonic ranges; and therefore
(Arts.~\cartref{201}, \cartref{202}) $a$, $c$, $b$, $d$ are the conjugates of $A$, $C$, $B$, $D$.

Hence (Art.~\cartref{205}) the range $acbd$ is harmonic, and therefore
the pencil $O$ $(acbd)$ is harmonic.
%-----File: 235.png-----

\csubsection{EXAMPLES}

\begin{exampleslist}

\item[1.] If $PSP'$ is a focal chord of a conic, any other chord through $S$
is divided harmonically by the directrix and the tangents at $P$ and $P'$.
\cheader{EXAMPLES}

\item[2.] If two sections of a right cone be taken, having the same
directrix, the straight line joining the corresponding foci will pass
through the vertex.

\item[3.] If a series of circles pass through the same two points, any
transversal will be cut by the circles in a series of points in involution.

\item[4.] If $O$ be the centre of the circle circumscribing a triangle $ABC$,
and $B'C'$, $C'A'$, $A'B'$, the respective polars with regard to a concentric
circle of the points $A$, $B$, $C$, prove that $O$ is the centre of the circle
inscribed in the triangle $A'B'C'$.

\item[5.] $OA$, $OB$, $OC$ being three straight lines given in position, shew
that there are three other straight lines each of which forms with
$OA$, $OB$, $OC$ an harmonic pencil; and that each of the three $OA$, $OB$, $OC$
forms with the second three an harmonic pencil.

\item[6.] The straight line $ACBD$ is divided harmonically in the points
$C$, $B$; prove that if a circle be described on $CD$ as diameter, any circle
passing through $A$ and $B$ will cut it at right angles.

\item[7.] Three straight lines $AD$, $AE$, $AF$ are drawn through a fixed
point $A$, and fixed points $C$, $B$, $D$ are taken in $AD$, such that $ACBD$ is
an harmonic range. Any straight line through $B$ intersects $AE$ and
$AF$ in $E$ and $F$, and $CE$, $DF$ intersect in $P$; $DE$, $CF$ in $Q$. Shew
that $P$ and $Q$ always lie in a straight line through $A$, forming with
$AD$, $AE$, $AF$ an harmonic pencil.

\item[8.] $CA$, $CB$ are two tangents to a conic section, $O$ a fixed point in
$AB$, $POQ$ any chord of the conic; prove that the intersections of
$AP$, $BQ$, and also of $AQ$, $BP$ lie in a fixed straight line which forms
with $CA$, $CO$, $CB$ an harmonic pencil.

\item[9.] If three conics pass through the same four points, the common
tangent to two of them is divided harmonically by the third.

\item[10.] Two conics intersect in four points, and through the intersection
of two of their common chords a tangent is drawn to one of them;
prove that it is divided harmonically by the other.

\item[11.] Prove that the two tangents through any point to a conic, any
line through the point and the line to the pole of the last line, form an
harmonic pencil.
%-----File: 236.png-----

\item[12.] The locus of the poles, with regard to the auxiliary circle, of
the tangents to an ellipse, is a similar ellipse.

\item[13.] The asymptotes of an hyperbola and any pair of conjugate
diameters form an harmonic pencil.

\item[14.] $PSQ$ and $PS'R$ are two focal chords of an ellipse; two other
ellipses are described having $P$ for a common focus, and touching the
first ellipse at $Q$ and $R$ respectively. The three ellipses have equal
major axes. Prove that the directrices of the last two ellipses pass
through the pole of $QR$.

\item[15.] Tangents from $T$ touch an ellipse in $P$ and $Q$, and $PQ$ meets
the directrices in $R$ and $R'$; shew that $PR$ and $QR'$ subtend equal
angles at $T$.

\item[16.] The poles of a given straight line, with respect to sections
through it of a given cone, all lie upon a straight line passing through
the vertex of the cone.

\item[17.] If from a given point in the axis of a conic a chord be drawn,
the perpendicular from the pole of the chord upon the chord will meet
the axis in a fixed point.

\item[18.] $Q$ is any point in the tangent at a point $P$ of a conic; $QG$ perpendicular
to $CP$ meets the normal at $P$ in $G$, and $QE$ perpendicular
to the polar of $Q$ meets the normal at $P$ in $E$; prove that $EG$ is constant
and equal to the radius of curvature at $P$.

\item[19.] The line joining two fixed points $A$ and $B$ meets the two fixed
lines $OP$, $OQ$ in $P$ and $Q$.

A conic is described so that $OP$ and $OQ$ are the polars of $A$ and $B$
with respect to it. Prove that the locus of its centre is the line $OR$,
where $R$ divides $AB$ so that
\[
AR : RB :: QR: RP.
\]

\item[20.] If from a point $O$ in the normal at a point $R$ of an ellipse
tangents $OP$, $OQ$ are drawn, the angles $PRO$, $QRO$ are equal.

\item[21.] The focal distances of a point on a conic meet the curve again
in $Q$, $R$; shew that the pole of $QR$ will lie upon the normal at the first
point.

\item[22.] The tangent at any point $A$ of a conic is cut by two other
tangents and their chord of contact in $B$, $C$, $D$; shew that ($ABDC$) is
harmonic.

\item[23.] A rectangular hyperbola circumscribes a triangle $ABC$; if
$D$, $E$, $F$ be the feet of the perpendiculars from $A$, $B$, $C$ on the opposite
sides, the loci of the poles of the sides of the triangle $ABC$ are the lines
$EF$, $FD$, $DE$.
%-----File: 237.png-----

\item[24.] Two common chords of a given ellipse and a circle pass through
a given point; shew that the locus of the centres of all such circles is
a straight line through the given point.

\item[25.] If $ABCD$ is a quadrilateral inscribed in a conic, and if $AD$,
$BC$ meet in $P$, and $AC$, $BD$ in $Q$, $PQ$ passes through the pole of $AB$.

\item[26.] $PCP'$ is any diameter of an ellipse. The tangents at the points
$D$, $E$ intersect in $F$, and $PE$, $P'D$ intersect in $G$. Shew that $FG$ is
parallel to $DCD'$.

\item[27.] $PP'$ is a chord of a conic, $QQ'$ any chord through its pole.
Prove that lines drawn from $P$ parallel to the tangents at $Q$ and $Q'$ to
meet $P'Q$, and $P'Q'$ respectively are bisected by $QQ'$.

\item[28.] If the pencil joining four fixed points on a conic to any one
point on the conic is harmonic, the pencil joining the fixed points to
any point on the conic is harmonic.

\item[29.] If $PQ$ is the chord of a conic having its pole on the chord $AB$
or $AB$ produced, and if $Qq$ is the chord parallel to $AB$, then $Pq$ bisects
$AB$.

\item[30.] If a quadrilateral circumscribe a conic, the intersection of the
lines joining opposite points of contact is the same as the intersection
of the diagonals.
\end{exampleslist}
%-----File: 238.png-----

\cchapter{CHAPTER XII}
\csectiontoc{Reciprocal Polars}

\cheader{RECIPROCAL POLARS}

\carticle{220.} The pole of a line with regard to any conic being
a point and the polar of a point a line, it follows that any
system of points and lines can be transformed into a system
of lines and points.

This process is called \textit{reciprocation}, and it is clear that
any theorem relating to the original system will have its
analogue in the system formed by reciprocation.

Thus, if a series of lines be concurrent, the corresponding
points are collinear; and the theorem of Art.~\cartref{219} is an
instance of the effect of reciprocation.

\carticleplus{221.}{Def.} If a point move in a curve ($C$), its polar will
always touch some other curve ($C'$); this latter curve is called
the reciprocal polar of ($C$) with regard to the auxiliary conic.

\subparagraph{Prop.~I\@.} \textit{If a curve $C'$ be the polar of $C$, then will $C$ be
the polar of $C'$.}

For, if $P$, $P'$ be two consecutive points of $C$, the intersection
of the polars of $P$ and $P'$ is a point $Q$, which is the pole
of the line $PP'$.

But the point $Q$ is ultimately, when $P$ and $P'$ coincide,
the point of contact of the curve which is touched by the
polar of $P$.

Hence the polar of any point $Q$ of $C'$ is a tangent to the
curve $C$.
%-----File: 239.png-----

\carticle{222.} So far we have considered poles and polars generally
with regard to any conic; we shall now consider the
case in which a circle is the auxiliary curve.

In this case, if $AB$ be a line, $P$ its pole, and $CY$ the perpendicular
from the centre of the circle on $AB$, the rectangle
$CP$. $CY$ is equal to the square on the radius of the circle.

A simple construction is thus given for the pole of a line,
or the polar of the point.

As an illustration take the theorem of the existence of
the orthocentre in a triangle.

Let $AOD$, $BOE$, $COF$ be the perpendiculars, $O$ being the
orthocentre.

The polar reciprocal of the line $BC$ is a point $A'$, and of
the point $A$ a line $B'C'$.

To the line $AD$ corresponds a point $P$ on $B'C'$, and since
$ADB$ is a right angle, it follows that $PSA'$ is a right angle,
$S$ being the centre of the auxiliary circle.

And, similarly, if $SQ$, $SR$, perpendiculars to $SB'$, $SC'$,
meet $C'A'$ and $A'B'$ in $Q$ and $R$, these points correspond to
$BE$ and $CF$.

But $AD$, $BE$, $CF$ are concurrent;
\[
\therefore P, Q, R \text{ are collinear.}
\]

Hence the reciprocal theorem,

\textit{If from any point $S$ lines be drawn perpendicular respectively
to $SA'$, $SB'$, $SC'$, and meeting $B'C'$, $C'A'$, $A'B'$ in $P$, $Q$,
and $R$, these points are collinear.}

As a second illustration take the theorem,

\textit{If $A$, $B$ be two fixed points, and $AC$, $BC$ at right angles to
each other, the locus of $C$ is a circle.}

Taking $O$, the middle point of $AB$, as the centre of the
auxiliary circle, the reciprocals of $A$ and $B$ are two parallel
straight lines, $PE$, $QF$, perpendicular to $AB$; the reciprocals
of $AC$, $BC$ are points $P$, $Q$ on these lines such that $POQ$ is a
right angle, and $PQ$ is the reciprocal of $C$.
%-----File: 240.png-----

Hence, the locus of $C$ being a circle, it follows that $PQ$
always touches a circle.

The reciprocal theorem therefore is,

\textit{If a straight line $PQ$, bounded by two parallel straight
lines, subtend a right angle at a point $O$, halfway between the
lines, the line $PQ$ always touches a circle, having $O$  for its
centre.}

\carticleplus{223.}{Prop.~II\@.} \textit{The reciprocal polar of a circle with
regard to another circle, called the auxiliary circle, is a conic,
a focus of which is the centre of the auxiliary circle, and the
corresponding directrix the polar of the centre of the reciprocated
circle.}

Let $S$ be the centre of the auxiliary circle, and $KX$ the
polar of $C$, the centre of the reciprocated circle.

\includegraphicsmid{223a}

Then, if $P$ be the pole of a tangent $QY$ to the circle $C$,
$SP$ meeting this tangent in $Y$,
\begin{flalign*}
&&SP \ctrdot SY &= SX \ctrdot SC.\\
\intertext{\indent Therefore, drawing $SL$ parallel to $QY$,}
&&SP : SC &:: SX : QL.\\
%-----File: 241.png-----
\intertext{\indent But, by similar triangles,}
&&SP : SC &:: SN : CL;\\
&&\therefore SP : SC &:: NX: CQ,\\
&\text{or}&SP : PK &:: SC : CQ.&\phantom{\text{or}}
\end{flalign*}

Hence the locus of $P$ is a conic, focus $S$, directrix $KX$
and having for its eccentricity the ratio of $SC$ to $CQ$.

The reciprocal polar of a circle is therefore an ellipse,
parabola, or hyperbola, as the point S is within, upon, or
without the circumference of the circle.

\carticleplus{224.}{Prop.~III\@.} \textit{To find the latus rectum and axes of
the reciprocal conic.}

The ends of the latus rectum are the poles of the tangents
parallel to $SC$.

Hence, if $SR$ be the semi-latus rectum,
\[
SR \ctrdot CQ=SE^2,
\]
$SE$ being the radius of the auxiliary circle.

\includegraphicsmid{224a}

The ends of the transverse axis $A$, $A'$ are the poles of the
tangents at $F$ and $G$;
\begin{flalign*}
&&\therefore SA \ctrdot SG&=SE^2&\\\
&\text{and}& SA' \ctrdot SF&=SE^2.&\phantom{and}
\end{flalign*}
%-----File: 242.png-----

Let $SU$, $SU'$ be the tangents from $S$, then
\begin{flalign*}
&&SG \ctrdot SF &= SU^2,&\\
&&\therefore SA' : SG &:: SE^2 : SU^2&\\
\tag*{\raisetwoline{($\alpha$).}}
&\text{and} &SA  : SF &:: SE^2 : SU^2\;\raisetwoline{$\biggr\}$}&\\
&\text{\indent Hence} &AA' : FG &:: SE^2 : SU^2,& \phantom{Hence\indent}\\
\intertext{or, if $O$ be the centre of the reciprocal,}
&&AO : CQ &:: SE^2 : SU^2.&
\end{flalign*}

Again, if $BOB'$ be the conjugate axis,
\begin{flalign*}
&& BO^2 &= SR \ctrdot AO;\\
&\text{therefore, since}& SE^2 &= SR \ctrdot CQ, &\phantom{\text{therefore, since}}\\
&&BO^2 : SE^2 &:: AO : CQ\\
&&            &:: SE^2 : SU^2\\
&\text{and}& BO \ctrdot SU &= SE^2.
\end{flalign*}

The centre $O$, it may be remarked, is the pole of $UU'$.

For, from the relations ($\alpha$),
\begin{align*}
SE^2 : SU^2 &:: SA + SA' : SF + SG\\
            &:: SO : SC\\
            &:: SO \ctrdot SM : SC \ctrdot SM;
\end{align*}
\[
\therefore SO \ctrdot SM=SE^2.
\]

\carticle{225.} In the figures drawn in the two preceding articles,
the reciprocal conic is an hyperbola; the asymptotes are
therefore the lines through $O$ perpendicular to $SU$ and $SU'$,
the poles of these lines being at an infinite distance.

The semi-conjugate axis is equal to the perpendicular
from the focus on the asymptote (Art.~\cartref{103}), \textit{i.e.}\ if $OD$ be the
asymptote, $SD$ is equal to the semi-conjugate axis.

Further, since $OD$ is perpendicular to $SU$, and $O$ is the
pole of $UU'$, it follows that $D$ is the pole of $CU$, and that
\[
SD \ctrdot SU = SE^2,
\]
as we have already shewn.
%-----File: 243.png-----

Again, $D$, being the intersection of the polars of $C$ and $U$,
is the intersection of $SU$ and the directrix.

\carticle{226.} If the point $S$ be within the circle, so that the
reciprocal is an ellipse, the axes are given by similar relations.

Through $S$ draw $SV$ perpendicular to $FG$, and let $UMU'$
be the polar of $S$ with regard to the circle.

\includegraphicsmid{226a}

Then $SM \ctrdot SC = SC \ctrdot CM - SC^2 = CF^2 - SC^2 = SV^2$;
also, $SE$ being the radius of the auxiliary circle,
\begin{flalign*}
&&    SA \ctrdot SF = SE^2 &= SA' \ctrdot SG,&&\\
&\text{and} &SF \ctrdot SG &= SV^2;&&\\
&&      \therefore SA : SG &:: SE^2 : SV^2&&\\
&&                SA' : SF &:: SE^2 : SV^2\;\raisetwoline{$\biggr\}$.}&&\\
&\text{Hence}     &SO : SC &:: SE^2 : SV^2,&&\\
&\text{and} &SO \ctrdot SM : SC \ctrdot SM &:: SE^2 : SV^2;&&\\
&&\therefore SO \ctrdot SM &= SE^2,&&
\intertext{so that $O$ is the pole of $UU'$.}
&\text{\indent Again}  &SA + SA' : SF + SG &:: SE^2 : SV^2,&\phantom{SA + SA' Again \indent}&\\
&&\therefore AO : CQ &:: SE^2 : SV^2.&&
\end{flalign*}

If $RSR'$ is the latus rectum,
\[
SR \ctrdot CQ = SE^2,
\]
%-----File: 244.png-----
and if $BOB'$ is the minor axis
\begin{flalign*}
&&SR \ctrdot AO &= BO^2;&& \\
&&\therefore BO^2 : SE^2 &:: SE^2 : SV^2,&& \\
&\text{and} &BO \ctrdot SV &= SE^2. &\phantom{and}&
\end{flalign*}

\carticle{227.} The important Theorem we have just considered
enables us to deduce from any property of a circle a corresponding
property of a conic, and we are thus furnished
with a method, which may serve to give easy proofs of
known properties, or to reveal new properties of conics.

In the process of reciprocation we observe that points
become lines and lines points; that a tangent to a curve
reciprocates into a point on the reciprocal, that a curve
inscribed in a triangle becomes a curve circumscribing a
triangle, and that when the auxiliary curve is a circle, the
reciprocal of a circle is a conic, the latus rectum of which
varies inversely as the radius of the circle.

Also, conversely, the reciprocal of a conic with regard to
a circle having its centre at a focus of the conic is a circle
the centre of which is the reciprocal of the directrix of the
conic.

For an ellipse the centre of reciprocation is within the
circle, for a parabola it is upon the circle, and for an
hyperbola it is outside the circle.

\carticle{228.} We give some transformations of theorems as
illustrations of the preceding articles.

{\small
\newcommand{\tableindent}{\hspace{1em}}
\renewcommand{\arraystretch}{1.25}

\begin{longtable}[c]{p{0.45\textwidth}@{\extracolsep{\fill}}p{0.45\textwidth}}

\multicolumn{1}{c}{\textsc{Theorem.}}&
\multicolumn{1}{c}{\textsc{Reciprocal.}\vspace{6pt}}\endhead

\tableindent The line joining the points of contact of parallel tangents of a circle
passes through the centre.&
\tableindent The tangents at the ends of a focal chord intersect in the directrix.\\

\tableindent The angles in the same segment of a circle are equal.&
\tableindent If a moveable tangent of a conic meet two fixed tangents, the
intercepted portion subtends a constant angle at the focus.\\

\tableindent Two of the common tangents of two equal circles are parallel.&
\tableindent If two conics have the same focus, and equal latera recta, the
straight line joining two of their common points passes through the focus.\\
%-----File: 245.png-----

\tableindent The tangent at any point of a circle is perpendicular to the
diameter through the point.&
\tableindent The portion of the tangent to a conic between the point of contact
and the directrix subtends a right angle at the focus.\\

\tableindent A chord of a circle is equally inclined to the tangents at its ends.&
\tableindent \stretchyspace The tangents drawn from any point to a conic subtend equal angles at a focus.\\

\tableindent If a chord of a circle subtend a constant angle at a fixed point
on the curve, the chord always touches a circle.&
\tableindent If two tangents of a conic move so that the intercepted portion of
a fixed tangent subtends a constant angle at the focus, the locus
of the intersection of the moving tangents is a conic having the
same focus and directrix.\\

\tableindent If a chord of a circle pass through a fixed point, the rectangle
contained by the segments is constant.&
\tableindent The rectangle contained by the perpendiculars from the focus on
two parallel tangents is constant.\\

\tableindent If two chords be drawn from a fixed point on a circle at right
angles to each other, the line joining their ends passes through the centre.&
\tableindent If two tangents of a conic move so that the intercepted portion
of a fixed tangent subtends a right angle at the focus, the two
moveable tangents meet in the directrix.\\

\tableindent If a circle be inscribed in a triangle, the lines joining the vertices
with the points of contact meet in a point.&
\tableindent If a triangle be inscribed in a conic the tangents at the vertices
meet the opposite sides in three points lying in a straight line.\\

\tableindent The sum of the reciprocals of the radii of the escribed circles of
a triangle is equal to the reciprocal of the radius of the inscribed circle.&
\tableindent With a given point as focus, four conics can be drawn circumscribing
a triangle, and the latus rectum of one is equal to the sum
of the latera recta of the other three.\\

\tableindent The common chord of two intersecting circles is perpendicular
to the line joining their centres.&
\tableindent If two parabolas have a common focus, the line joining it to
the intersection of the directrices is perpendicular to the common tangent.\\

\tableindent If circles pass through two fixed points, the locus of their
centres is a straight line.&
\tableindent If conics have a fixed focus and a pair of fixed tangents in common,
the corresponding directrices all pass through a fixed point.\\

\tableindent Two tangents to a conic at right angles to each other intersect
on a fixed circle.&
\tableindent Chords of a circle which subtend a right angle at a fixed point
all touch a conic of which that point is a focus.\\

\end{longtable}
} %end \small
%-----File: 246.png-----

\carticleplus{229.}{Prop.~IV\@.} \textit{A system of coaxal
circles can be reciprocated into a system of confocal conics.}

Let $X$ be the point at which the radical axis crosses the
line of centres, and let $E$ and $S$ be the limiting points of the
system.

Then $XE$ is equal to the length of the tangent $XD$ to
any one of the circles, and, therefore, if $A$ is the centre of
this circle, $AD$ is the tangent at $D$ to the circle whose centre
is $X$ and radius $XE$.

Hence it follows that $AE \ctrdot AS = AD^2$, shewing that $UU'$,
the polar of $S$ with regard to the circle $A$, passes through $E$.

Reciprocating with regard to $S$, the centre of the reciprocal
curve is the pole of $UU'$, and is consequently fixed;
and the conics are therefore confocal.

Hence, if we reciprocate with regard to either limiting
point, we obtain confocal conics.

In the particular case in which the circles all touch the
radical axis, we obtain confocal and co-axial parabolas.

\carticleplus{230.}{Prop.~V\@.} \textit{The reciprocal polar of a conic with
regard to a circle, or with regard to any conic, is a conic.}

Taking any two tangents of the conic, their reciprocal
polars are points on the reciprocal curve, and the reciprocal
polar of their point of intersection is the chord joining the
points.

Since only two tangents can be drawn from a point to a
conic, it follows that the reciprocal curve is always intersected
by a straight line in two points only.

It follows therefore that the reciprocal curve is a conic.

In reciprocating a conic with regard to a circle, the
reciprocal polar is an ellipse, parabola, or hyperbola, according
as the centre $S$ of the circle is inside, upon, or outside the
conic.

In the second case the axis of the parabola is parallel to
the normal at the point $S$, and in the third case the
asymptotes are perpendicular to the tangents which can be
drawn from the point $S$ to the conic.
%-----File: 247.png-----

When the auxiliary curve is a conic, centre $S$, the first
of the preceding statements holds good.

When the point $S$ is on the conic, the axis of the parabola
is parallel to the diameter of the auxiliary conic, which is
conjugate to the tangent at $S$.

When the point $S$ is outside the conic, the asymptotes of
the hyperbola are parallel to those diameters of the auxiliary
conic which are conjugate to the straight lines through $S$
touching the conic to be reciprocated.

The following cases will serve to illustrate the theorem
of this article.

\carticle{231.} \textit{The reciprocal polar of a parabola with regard to a
point on the directrix is a rectangular hyperbola.}

For the two tangents from the point are at right angles
to each other, and therefore the asymptotes are at right angles
to each other.

\carticle{232.} \textit{The reciprocal polar of an ellipse or hyperbola, with
regard to its centre, is a similar curve turned through a right
angle about the centre.}

If $CY$ is the perpendicular on the tangent at $P$, and $Q$
the reciprocal of the tangent, $CQ \ctrdot CY$ is constant.

But $CY \ctrdot CD$ is constant;
\[
\therefore CQ \text{ varies as } CD,
\]
and the reciprocal curve is the same as the original curve, or
similar to it.

\carticle{233.} \textit{The chords of a conic which subtend a right angle
at a fixed point $P$ of a conic all pass through a fixed point
in the normal at $P$.}

Reciprocating with regard to $P$, the reciprocal curve is a
parabola, the axis of which is parallel to the normal to the
conic, and the reciprocal of the chord is the point of intersection
of tangents at right angles to each other.

The locus of this point is the directrix of the parabola,
and, being at right angles to the normal, it follows, on
%-----File: 248.png-----
reciprocating backwards, that the chord passes through a
fixed point $E$ in the normal.

\textit{To find the position of the point $E$},\\
let $C$ be the centre of the conic, $CA$, $CB$ its semi-axes, and
$PNP'$ the double ordinate, and let the normal meet the
axes in $G$ and $g$.

Since $CA$ and $CB$ bisect the angle $PCP'$ and its
supplement,
\[
C (BPAP') \text{ is an harmonic pencil};
\]

$\therefore PGEg$ is an harmonic range, so that $PE$ is the
harmonic mean between $PG$ and $Pg$.

In the case of an hyperbola $EGPg$ is an harmonic range.

In the case of a parabola, $E$ is the point of intersection of
the normal with the diameter through $P'$.

\carticle{234.} \textit{The chords of a conic which subtend a right angle
at a fixed point $O$ not on the conic all touch a conic of which
that point is a focus.}

Reciprocating with regard to $O$, the reciprocal of the
envelope of the chords is the director circle of a conic, and
therefore, reciprocating backwards, it follows that the envelope
of the chords is a conic of which $O$ is a focus. This
of course includes the preceding theorem as a particular case,
the fact being that when $O$ is on the conic the envelope of the
chords is a conic, with a vertex and focus at $E$, flattened
into a straight line.

\carticle{235.} \textit{If the sides of a triangle are tangents to a parabola,
the orthocentre of the triangle is on the directrix of the parabola.}

This theorem is at once obtained by reciprocating, with
regard to the orthocentre of the triangle, the theorem, proved
in Art.~\cartref{143}, that, if a rectangular hyperbola passes through
the angular points of a triangle, it also passes through the
orthocentre of the triangle.
%-----File: 249.png-----

\csubsection{EXAMPLES}

\begin{exampleslist}

\item[1.] If any triangle be reciprocated with regard to its orthocentre,
the reciprocal triangle will be similar and similarly situated to the
original one and will have the same orthocentre.
\cheader{EXAMPLES}

\item[2.] If two conics have the same focus and directrix, and a focal
chord be drawn, the four tangents at the points where it meets the
conics intersect in the same point of the directrix.

\item[3.] An ellipse and a parabola have a common focus; prove that the
ellipse either intersects the parabola in two points, and has two common
tangents with it, or else does not cut it.

\item[4.] Prove that the reciprocal polar of the circumscribed circle of a
triangle with regard to the inscribed circle is an ellipse, the major axis
of which is equal in length to the radius of the inscribed circle.

\item[5.] Reciprocate with respect to any point $S$ the theorem that, if
two points on a circle be given, the pole of $PQ$ with respect to that
circle lies on the line bisecting $PQ$ at right angles.

\item[6.] If two parabolas whose axes are at right angles have a common
focus, prove that the part of the common tangent intercepted between
the points of contact subtends a right angle at the focus.

\item[7.] The tangent at a moving point $P$ of a conic intersects a fixed
tangent in $Q$, and from $S$ a straight line is drawn perpendicular to $SQ$
and meeting in $R$ the tangent at $P$; prove that the locus of $R$ is a
straight line.

\item[8.] Four parabolas having a common focus can be described touching
respectively the sides of the triangles formed by four given points.

\item[9.] A triangle $ABC$ circumscribes a parabola, focus $S$; through
$ABC$ lines are drawn respectively perpendicular to $SA$, $SB$, $SC$; shew
that these lines are concurrent.

\item[10.] Prove that the distances, from the centre of a circle, of any
two poles are to one another as their distances from the alternate
polars.

\item[11.] Reciprocate the theorems,
\begin{enumerate}
\item[]\vspace{-0.5em} %to get second level of indentation without excess vertical space
\begin{enumerate}
\item[(1)] The opposite angles of any quadrilateral inscribed in a
circle are equal to two right angles.

\item[(2)] If a line be drawn from the focus of an ellipse making a
constant angle with the tangent, the locus of its intersection
with the tangent is a circle.
\end{enumerate}
\end{enumerate}

%-----File: 250.png-----

\item[12.] The locus of the intersection of two tangents to a parabola
which include a constant angle is an hyperbola, having the same focus
and directrix.

\item[13.] Two ellipses having a common focus cannot intersect in more
than two real points, but two hyperbolas, or an ellipse and hyperbola,
may do so.

\item[14.] $ABC$ is any triangle and $P$ any point: four conic sections
are described with a given focus touching the sides of the triangles
$ABC$, $PBC$, $PCA$, $PAB$ respectively; shew that they all have a common
tangent.

\item[15.] $TP$, $TQ$ are tangents to a parabola cutting the directrix
respectively in $X$ and $Y$; $ESF$ is a straight line drawn through the
focus $S$ perpendicular to $ST$, cutting $TP$, $TQ$ respectively in $E$, $F$;
prove that the lines $EY$, $XF$ are tangents to the parabola.

\item[16.] With the orthocentre of a triangle as focus, two conics are
described touching a side of the triangle and having the other two sides
as directrices respectively; shew that their minor axes are equal.

\item[17.] Two parabolas have a common focus $S$; parallel tangents are
drawn to them at $P$ and $Q$ intersecting the common tangent in $P'$ and
$Q'$; prove that the angle $PSQ$ is equal to the angle between the axes,
and the angle $P'SQ'$ is supplementary.

\item[18.] $ABC$ is a given triangle, $S$ a given point; on $BC$, $CA$, $AB$
respectively, points $A'$, $B'$, $C'$ are taken, such that each of the angles
$ASA'$, $BSB'$, $CSC'$, is a right angle. Prove that $A'$, $B'$, $C'$ lie in the
same straight line, and that the latera recta of the four conics, which
have $S$ for a common focus, and respectively touch the three sides of
the triangles $ABC$, $AB'C'$, $A'BC'$, $A'B'C$ are equal to one another.

\item[19.] A parabola and hyperbola have the same focus and directrix,
and $SPQ$ is a line drawn through the focus $S$ to meet the parabola in $P$,
and the nearer branch of the hyperbola in $Q$; prove that $PQ$ varies as
the rectangle contained by $SP$ and $SQ$.

\item[20.] If two equal parabolas have the same focus, the tangents at
points angularly equidistant from the vertices meet on the common
tangent.

\item[21.] If an ellipse and a parabola have the same focus and directrix,
and if tangents are drawn to the ellipse at the ends of its major axis,
the diagonals of the quadrilateral formed by the four points where
these tangents cut the parabola intersect in the focus.

\item[22.] Find the reciprocals of the theorems of Arts.~\cartref{215} and~\cartref{217}.

\item[23.] If a conic be reciprocated with regard to a point, shew that
there are only two positions of the point, such that the conic may be
similar and similarly situated to the reciprocal.
%-----File: 251.png-----

\item[24.] Conics are described having a common focus and equal latera
recta. Also the corresponding directrices envelope a fixed confocal
conic. Prove that these conics all touch two fixed conics, and that the
reciprocals of the latera recta of these fixed conics are equal to the sum
and difference of the latera recta of the variable conics and of the fixed
confocal.

\item[25.] Given a point, a tangent, and a focus of a conic, prove that the
envelope of the directrix is a conic passing through the given focus.

\item[26.] Two conics have a common focus: their corresponding directrices
will intersect on their common chord, at a point whose focal
distance is at right angles to that of the intersection of their common
tangents.

If the conics are parabolas, the inclination of their axes will be the
angle subtended by the common tangent at the common focus.

\item[27.] If the intercept on a given straight line between two variable
tangents to a conic subtends a right angle at the focus of the conic, the
tangents intersect on a conic.

\item[28.] The tangent at $P$ to an hyperbola meets the directrix in $Q$;
another point $R$ is taken on the directrix such that $QR$ subtends at the
focus an angle equal to that between the transverse axis and an
asymptote; prove that $RP$ envelopes a parabola.

\item[29.] $S$ is the focus of a conic; $P$, $Q$ two points on it such that
the angle $PSQ$ is constant; through $S$, $SR$, $ST$ are drawn meeting the
tangents at $P$, $Q$ in $R$, $T$ respectively, and so that the angles $PSR$, $QST$
are constant; shew that $RT$ always touches a conic having the same
focus and directrix as the original conic.

\item[30.] $OA$, $OB$ are common tangents to two conics having a common
focus $S$, $CA$, $CB$ are tangents at one of their points of intersection,
$BD$, $AE$ tangents intersecting $CA$, $CB$, in $D$, $E$. Prove that $SDE$ is a
straight line.

\item[31.] An hyperbola, of which $S$ is one focus, touches the sides of
a triangle $ABC$; the lines $SA$, $SB$, $SC$ are drawn, and also lines
$SD$, $SE$, $SF$ respectively perpendicular to the former three lines,
and meeting any tangent to the curve in $D$, $E$, $F$; shew that the lines
$AD$, $BE$, $CF$ are concurrent.

\item[32.] If a conic inscribed in a triangle has one focus at the centre of
the circumscribed circle of the triangle, its transverse axis is equal
to the radius of that circle.

\item[33.] If any two diameters of an ellipse at right angles to each other
meet the tangent at a fixed point $P$ in $Q$ and $R$, the other two tangents
through $Q$ and $R$ intersect on a fixed straight line which passes through
a point $T$ on the tangent at $P$, such that $PCT$ is a right angle.
\end{exampleslist}
%-----File: 252.png-----

\cchapter{CHAPTER XIII}
\csectiontoc{The Construction of a Conic from Given Conditions}

\cheader{CONSTRUCTION OF A CONIC FROM GIVEN CONDITIONS}

\carticle{236.} It will be found that, in general, five conditions
are sufficient to determine a conic, but it sometimes happens
that two or more conics can be constructed which will satisfy
the given conditions. We may have, as given conditions,
points and tangents of the curve, the directions of axes or
conjugate diameters, the position of the centre, or any
characteristic or especial property of the curve.

\subparagraph{Prop.~I\@.} \textit{To construct a parabola, passing through three
given points, and having the direction of its axis given.}

In this case the fact that the conic is a parabola is one of
the conditions.

\includegraphicsmid{236a}

Let $P$, $Q$, $R$ be the given points, and let $RE$ parallel to
the given direction meet $PQ$ in $E$.
%-----File: 253.png-----

If $E$ be the middle point of $PQ$, $R$ is the vertex of the
diameter $RE$; but, if not, bisecting $PQ$ in $V$, draw the
diameter through $V$ and take $A$ such that
\[
AV : RE :: QV^2 : QE \ctrdot EP.
\]

Then $A$ is the vertex of the diameter $AV$.

If the point $E$ do not fall between $P$ and $Q$, $A$ must be
taken on the side of $PQ$ which is opposite to $R$.

The focus may then be found by taking $AU$ such that
\[
QV^2 = 4AV \ctrdot AU,
\]

and by then drawing $US$ parallel to $QV$ and taking $AS$
equal to $AU$.

\carticleplus{237.}{Prop.~II\@.} \textit{To describe a parabola through four
given points.}

First, let $ABCD$ be four points in a given parabola, and
let the diameter $CF$ meet $AD$ in $F$.

\includegraphicsmid{237a}

Draw the tangents $PT$, $QT$ parallel to $AD$, $BC$, and the
diameter $QV$ meeting $PT$ in $V$.
%-----File: 254.png-----
\begin{flalign*}
&\indent\text{Then}&   ED \ctrdot EA : EC \ctrdot EB &:: TP^2 : TQ^2 &
    \phantom{\indent\text{Then}}\\
&& &:: TV^2 : TQ^2\\
&& &:: EF^2 : EC^2.
\end{flalign*}

Hence the construction; in $EA$ take $EF$ such that
\[
EF^2 : EC^2 :: ED \ctrdot EA : EC \ctrdot EB,
\]
then $CF$ is the direction of the axis, and the problem is
reduced to the preceding.

If the point $F$ be taken in $AE$ produced, another parabola
can be drawn, so that, in general, two parabolas can be
drawn through four points.

\carticle{238.} This problem may be treated differently by help
of the theorem of Art.~\cartref{52}, viz.;

\textit{If from a point $O$, outside a parabola, a tangent $OM$,
and a chord $OAB$ be drawn, and if the diameter $ME$ meet the
chord in $E$,}
\[
OE^2 = OA \ctrdot OB.
\]

\includegraphicsmid{238a}

Let $A$, $B$, $C$, $D$ be the given points, and let $E$, $E'$,
$F$, $F'$, be so taken that
\begin{flalign*}
&& OE^2 &= OE'^2 = OA \ctrdot OB,\\
&\text{and}&    OF^2 &= OF'^2 = OC \ctrdot OD. & \phantom{\text{and}}
\end{flalign*}
%-----File: 255.png-----

Then $EF$ and $E'F'$ are diameters, and $KL$, the polar
of $O$, will meet $EF$ and $E'F'$ in $M$, $N$, the points of contact
of tangents from $O$.

The second parabola is obtained by taking for diameters
$EF'$ and $E'F$.

\carticleplus{239.}{Prop.~III\@.} \textit{Any conic passing through four points
has a pair of conjugate diameters parallel to the axes of the
two parabolas which can be drawn through the four points.}

\includegraphicsmid{239a}

Let $TP$, $TQ$ be the tangents parallel to $OAB$ and $OCD$,
and such that the angle $PTQ$ is equal to $AOC$.

Then, if $OE^2 = OA \ctrdot OB$, and $OF^2 = OC \ctrdot OD$,
\begin{align*}
OE^2 : OF^2 &:: OA \ctrdot OB : OC \ctrdot OD\\
&:: TP^2 : TQ^2;
\end{align*}

$\therefore EF$ is parallel to $PQ$.

Hence, if $R$ and $V$ be the middle points of $EF$ and $PQ$,
$OR$ is parallel to $TV$;

But, taking $OF'$ equal to $OF$, $OR$ is parallel to $EF'$,\\
$\therefore TV$ and $PQ$ are parallel to $EF'$ and $EF$;

\textit{i.e.}\ the conjugate diameters parallel to $TV$ and $PQ$ are
parallel to the axes of the two parabolas.
%-----File: 256.png-----

\carticleplus{240.}{Prop.~IV\@.} \textit{Having given a pair of conjugate diameters,
$PCP'$, $DCD'$, it is required to construct the ellipse.}

In $CP$ take $E$ such that $PE \ctrdot PC = CD^2$, draw $PF$ perpendicular
to $CD$, and take $FC'$ equal to $FC$.

About $CEC'$ describe a circle, cutting $PF$ in $G$ and $G'$;
then
\[
PG \ctrdot PG' = PE \ctrdot PC = CD^2,
\]
and $GCG'$ is a right angle; therefore $CG$ and $CG'$ are the
directions of the axes and their lengths are given by the
relations,
\begin{align*}
PG \ctrdot PF &= BC^2,\\
PG' \ctrdot PF &= AC^2.
\end{align*}

\includegraphicsmid{240a}

We may observe that, $O$ being the centre of the circle,
\begin{align*}
AC^2 + BC^2 &= PF \ctrdot PG + PF \ctrdot PG'\\
&= 2 \ctrdot PF \ctrdot PO\\
&= 2 \ctrdot PC \ctrdot PN,\\
\intertext{if $N$ be the middle point of $CE$,}
&= PC^2 + PC \ctrdot PE\\
&= CP^2 + CD^2.
\end{align*}
%-----File: 257.png-----

If $PE'$ be taken equal to $PE$ in $CP$ produced, and the
same construction be made, we shall obtain the axes of an
hyperbola having $CP$, $CD$ for a pair of conjugate semi-diameters.

\carticle{241.} This problem may be treated also as follows.

In $PF$, the perpendicular on $CD$, take
\begin{flalign*}
&& PK &= PK' = CD;\\
&\text{then}&      PK^2&=PG \ctrdot PG', & \phantom{\text{then}}
\end{flalign*}
and therefore $K'GKG'$ is an harmonic range; and $GCG'$
being a right angle, it follows (Art.~\cartref{199}), that $CG$ and $CG'$
are the bisectors of the angles between $CK$ and $CK'$.

Hence, knowing $CP$ and $CD$, $G$ and $G'$ are determined.

\carticleplus{242.}{Prop.~V\@.} \textit{Having given the focus and three points
of a conic, to find the directrix.}

Let $A$, $B$, $C$, $S$ be the three points and the focus.

Produce $BA$ to $D$ so that
\begin{align*}
BD : AD &:: SB : SA,
\intertext{and $CB$ to $E$, so that}
BE : CE &:: SB : SC;
\end{align*}
then $DE$ is the directrix.

The lines $BA$, $BC$ may be also divided internally in the
same ratio, so that four solutions are generally possible.

Conversely, if three points $A$, $B$, $C$ and the directrix are
given, let $BA$, $BC$ meet the directrix in $D$ and $E$; then $S$
lies on a circle, the locus of a point, the distances of which
from $A$ and $B$ are in the ratio of $AD$ to $DB$.

$S$ lies also on a circle, similarly constructed with regard
to $BCE$; the intersection of these circles gives two points,
either of which may be the focus.
%-----File: 258.png-----

\carticleplus{243.}{Prop.~VI\@.} \textit{Having given the centre, the directions
of a pair of conjugate diameters, and two points of an ellipse,
to describe the ellipse.}

If $C$ be the centre, $CA$, $CB$ the given directions, and
$P$, $Q$ the points, draw $QMQ'$, $PLP'$ parallel to $CB$ and $CA$,
and make $Q'M = QM$ and $P'L = PL$.

\includegraphicsmid{243a}

Then the ellipse will evidently pass through $P'$ and $Q'$,
and if $CA$, $CB$ be the conjugate radii, their ratio is given by
the relation
\[
CA^2 : CB^2 :: EP \ctrdot EP' : EQ \ctrdot EQ',
\]
$E$ being the point of intersection of $P'P$ and $Q'Q$.

Set up a straight line $ND$ perpendicular to $CA$ and such
that
\[
ND^2 : NP^2 :: EP \ctrdot EP' : EQ \ctrdot EQ',
\]
and describe a circle, radius $CD$ and centre $C$, cutting $CA$ in
$A$, and take
\[
CB : CA :: NP : ND.
\]
\begin{flalign*}
&\indent\text{Then}&     AN \ctrdot NA'& = ND^2, &\phantom{\indent\text{Then}}\\
&\text{and}&    PN^2 : AN \ctrdot NA'& :: CB^2 : CA^2.
\end{flalign*}

Hence $CA$, $CB$ are determined, and the ellipse passes
through $P$ and $Q$.
%-----File: 259.png-----

\carticleplus{244.}{Prop.~VII\@.} \textit{To describe a conic passing through
a given point and touching two given straight lines in given
points.}

Let $OA$, $OB$ be the given tangents, $A$ and $B$ the points of
contact, $N$ the middle point of $AB$.

\includegraphicsmid{244a}

1st. Let the given point $D$ be in $ON$; then, if $ND = OD$,
the curve is a parabola.

But if $ND < OD$, the curve is an ellipse, and, taking $C$
such that $OC \ctrdot CN = CD^2$, the point $C$ is the centre.

If $ND > OD$, the curve is an hyperbola, and its centre
is found in the same manner.

2nd. If the given point be $E$, not in $ON$, draw $GEF$
parallel to $AB$, and make $FL$ equal to $EL$.

Take $K$ such that
\[
GK^2 = GE \ctrdot GF;
\]
then $AK$ produced will meet $ON$ in $D$, and the problem is
reduced to the first case.

To justify this construction, observe that, if $DM$ be the
tangent at $D$,
\begin{flalign*}
&& GE \ctrdot GF : GA^2 &:: DM^2 : MA^2\\
&&                      &:: GK^2 : GA^2,\\
&\text{so that}& GE \ctrdot GF &= GK^2. &\phantom{\text{so that}}
\end{flalign*}
%-----File: 260.png-----

\carticleplus{245.}{Prop.~VIII\@.} \textit{To draw a conic through five given
points.}

Let $A$, $B$, $C$, $D$, $E$ be the five points, and $F$ the intersection
of $DE$, $AB$.

\includegraphicsmid{245a}

Draw $CG$, $CH$, parallel respectively to $AB$ and $ED$, and
meeting $ED$, $AB$ in $G$ and $H$.

If $F$ and $G$ fall between $D$ and $E$, and $F$ and $H$ between
$A$ and $B$, take $GP$ in $CG$ produced and $HQ$ in $CH$ produced,
such that
\begin{flalign*}
&& CG \ctrdot GP : DG \ctrdot GE
    &:: AF \ctrdot FB : DF \ctrdot FE,\\
&\text{and}& CH \ctrdot HQ : AH \ctrdot HB
    &:: DF \ctrdot FE : AF \ctrdot FB; & \phantom{\text{and}}
\end{flalign*}

Then (Arts.~\cartref{92} and~\cartref{134}) $P$ and $Q$ are points in the
conic.

Also $PC$, $AB$ being parallel chords, the line joining their
middle points is a diameter, and another diameter is obtained
from $CQ$ and $DE$.

If these diameters are parallel, the conic is a parabola,
and we fall upon the case of Prop.~II.; but if they intersect
in a point $O$, this point is the centre of the conic, and,
having the centre, the direction of a diameter, and two
ordinates of that diameter, we fall upon the case of Prop.~VI.
%-----File: 261.png-----

The figure is drawn for the case in which the pentagon
$AEBCD$ is not re-entering, in which case the conic may be
an ellipse, a parabola, or an hyperbola.

If any one point fall within the quadrilateral formed by
the other four, the curve is an hyperbola.

In all cases the points $P$, $Q$ must be taken in accordance
with the following rule.

The points $C$, $P$, or $C$, $Q$ must be on the same or different
sides of the points $G$, or $H$, according as the points $D$, $E$, or
$B$, $A$ are on the same or different sides of the points $G$ or $H$.

Thus, if the point $E$ be between $D$ and $F$, and if $G$ be
between $D$ and $E$, and $H$ between $A$ and $B$, the points $P$
and $C$ will be on the same side of $G$, and $C$, $Q$ on the same
side of $H$, but if $H$ do not fall between $A$ and $B$, $C$ and $Q$
will be on opposite sides of $H$.

Remembering that if a straight line meet only one
branch of an hyperbola, any parallel line will meet only one
branch, and that if it meet both branches, any parallel will
meet both branches, the rule may be established by an
examination of the different cases.

\carticle{246.} The above construction depends only on the elementary
properties of Conics, which are given in Chapters
\cchapref{I.}, \cchapref{II.}, \cchapref{III.}, and \cchapref{IV.} For some further constructions we shall
adopt another method depending on harmonic properties.

\subparagraph{Prop.~IX\@.} \textit{Having given two pairs of lines $OA$, $OA'$,
and $OB$, $OB'$, to find a pair of lines $OC$, $OC'$, which shall
make with each of the given pairs an harmonic pencil.}

This is at once effected by help of Art.~\cartref{203}.

For, if any transversal cut the lines in the points $c$, $a$, $b$,
$c'$, $b'$, $a'$, the points $c$, $c'$ are the foci of the involution, in
which $a$, $a'$ are conjugate, and also $b$, $b'$, the centre of the
involution being the middle point of $cc'$.
%-----File: 262.png-----

\carticleplus{247.}{Prop.~X\@.} \textit{If two points and two tangents of a
conic be given, the chord of contact intersects the given chord
in one of two fixed points\symbolfootnote[1]
  {I am indebted to Mr Worthington for much valuable assistance in this
  chapter, and especially for the constructions of Articles \cartref{247},
  \cartref{249}, \cartref{250}, and~\cartref{253}.}%
.}

Let $OP$, $OQ$ be the given tangents, $A$ and $B$ the given
points, and $C$ the intersection of $AB$ and the chord of contact.

\includegraphicsmid{247a}

Let $OC'$ be the polar of $C$, and let $AB$ meet $OC'$ in $D$.

Then $C$ is on the polar of $D$, and therefore $DBCA$ is an
harmonic range.

Also, $C$ being on the polar of $C'$, $C'QCP$ is an harmonic
range.

Hence if two lines $OC$, $OC'$ be found, which are harmonic
with $OA$, $OB$, and also with $OP$, $OQ$, these lines
intersect $AB$ in two points $C$ and $D$, through one of which
the chord of contact must pass.

Or thus, if the tangents meet $AB$ in $a$ and $b$, find the
foci $C$ and $D$ of the involution $AB$, $ab$; the chord of contact
passes through one of these points.
%-----File: 263.png-----

\carticleplus{248.}{Prop.~XI\@.} \textit{Having given three points and two
tangents, to find the chord of contact.}

In the preceding figure let $OP$, $OQ$ be the tangents, and
$A$, $B$, $E$ the points.

Find $OC$, $OC'$ harmonic with $OA$, $OB$, and $OP$, $OQ$; also
find $OF$, $OG$ harmonic with $OA$, $OE$ and $OP$, $OQ$.

Then any one of the four lines joining $C$ or $D$ to $F$ or $G$
is a chord of contact, and the chord of contact and points of
contact being known, the case reduces to that of Art.~\cartref{244}.

Hence four such conics can in general be described.

\carticleplus{249.}{Prop.~XII\@.} \textit{To describe a conic, passing through
two given points, and touching three given straight lines.}

Let $AB$, the line joining the given points, meet the given
tangents $QR$, $RP$, $PQ$, in $N$, $M$, $L$.

\includegraphicsmid{249a}

Find the foci  $C$, $D$ of the involution $A$, $B$ and $L$, $M$;

Then $YZ$, the polar of $P$, passes through $C$ or $D$,
Art.~\cartref{247}.

Also find the foci, $E$, $F$, of the involution $A$, $B$, and
$M$, $N$; then $XY$, the polar of $R$, passes through $F$ or $E$.

Let $ZX$ meet $PR$ in $T$; then $T$ is on the polar of $Q$, and
$QY$ is the polar of $T$.
\begin{flalign*}
&\text{\indent Hence}   &TXUZ \text{ is harmonic;}&\phantom{\indent Hence}&\\
&\text{therefore}    &MEVC \text{ is harmonic.}&&
\end{flalign*}
%-----File: 264.png-----

This determines $V$, and, joining $QV$, we obtain the point
of contact $Y$.

Then, joining $YC$ and $YE$, $Z$ and $X$ are obtained, and
$X$, $Y$, $Z$ being points of contact, we have five points, and
can describe the conic by the construction of Art.~\cartref{245},
or by that of Art.~\cartref{252}.

Since either $C$ or $D$ may be taken with $E$ or $F$, there are
in general four solutions of the problem.

\carticleplus{250.}{Prop.~XIII\@.} \textit{To describe a conic, having given
four points and one tangent.}

Let $A$, $B$, $C$, $D$ be the given points, and complete the
quadrilateral.

\includegraphicsmid{250a}

Then $E$ is the pole of $FG$, and if the given tangent $KL$
meet $FG$ in $K$, $E$ is on the polar of $K$; therefore the other
tangent through $K$ forms an harmonic pencil with $KF$, $KL$,
$KE$.

Hence two tangents being known, and a point $E$ in the
chord of contact, if we find two points $P$, $P'$ in $A$, $B$, such
that $KP$, $KP'$ are harmonic with $KA$, $KB$, and also with
$KL$, $KL'$, we shall have two chords of contact $EP$, $EP'$,
and therefore two points of contact for $KL$ and also for $KL'$.

Hence two conics can be described.

We observe that if two conics pass through four points,
their common tangents meet on one of the sides of the self-conjugate
triangle $EFG$.
%-----File: 265.png-----

\carticleplus{251.}{Prop.~XIV\@.} \textit{Given four tangents and one point, to
construct the conic.}

Let $ABCD$ be the given circumscribing quadrilateral,
and $E$ the given point. Completing the figure, draw $LEF$
through $E$ and $F$, and complete the harmonic range $LEFE'$;
then, since $F$ is the pole of $HG$ (Art.~\cartref{217}), $E'$ is a point in
the conic.

Also, since $K$ is the pole of $FA$ (Art.~\cartref{217}), the chord of
contact of the tangents $AB$, $AD$, passes through $K$.

Hence the construction is the same as that of Art.~\cartref{250},
and there are two solutions of the problem.

\includegraphicsmid{251a}

\carticleplus{252.}{Prop.~XV\@.} \textit{Given five points, to construct the conic.}

Let $A$, $B$, $C$, $D$, $E$ be the five points, and complete the
quadrilateral $ABCD$.

Then $H$ is the pole of $FG$, and $FG$ passes through the
points of contact $P$, $Q$ of the tangents from $H$.

Join $HE$, cutting $FG$ in $K$, and complete the harmonic
range $HEKE'$; then $E'$ is a point in the conic.

Also $AE$, $BE'$ will intersect $FG$ in the same point $F'$,
and $E'A$, $EB$ will also intersect $FG$ in the same point $G'$.
%-----File: 266.png-----

\includegraphicsmid{252a}

But $GPFQ$ and $G'PF'Q$ are both harmonic ranges, therefore
$P$ and $Q$ are the foci of an involution of which $F$, $G$ and
$F'$, $G'$ are pairs of conjugate points.

Hence, finding these foci, $P$ and $Q$, the tangents $HP$, $HQ$
are known, and the case is reduced to that of Prop.~VII.

Hence only one conic can be drawn through five points.

\carticleplus{253.}{Prop.~XVI\@.}  \textit{Given five tangents, to find the points
of contact.}

\includegraphicsmid{253a}

Let $ABCDE$ be the circumscribing pentagon. Considering
the quadrilateral $FBCD$, join $FC$, $BD$, meeting
in $K$.
%-----File: 267.png-----

Then (Art.~\cartref{217}) $K$ is the pole of the line joining the
intersections of $FB$, $CD$, and of $FD$, $BC$; that is, the chords
of contact of $BF$, $CD$, and of $BC$, $FD$ meet in $K$.

Similarly if $BG$, $AC$ meet in $L$, the chords of contact of
$AB$, $CG$, and of $BC$, $AG$ meet in $L$.

Hence $KL$ is the chord of contact of $AB$, $CD$, and therefore
determines $M$, $N$ the points of contact.

Hence it will be seen that only one conic can be drawn
touching five lines.
%-----File: 268.png-----

\cchapter{CHAPTER XIV}
\csectiontoc{The Oblique Cylinder, the Oblique Cone, and the Conoids}

\cheader{OBLIQUE CYLINDER}

\carticleplus{254.}{Def.}  If a straight line, which is not perpendicular
to the plane of a given circle, move parallel to itself, and
always pass through the circumference of the circle, the
surface generated is called an oblique cylinder.

The line through the centre of the circular base, parallel
to the generating lines, is the axis of the cylinder.

It is evident that any section by a plane parallel to the
axis consists of two parallel lines, and that any section by a
plane parallel to the base is a circle.

The plane through the axis perpendicular to the base is
the principal section.

The section of the cylinder by a plane perpendicular to
the principal section, and inclined to the axis at the same
angle as the base, is called a subcontrary section.

\carticleplus{255.}{Prop.~I\@.}  \textit{The subcontrary section of an oblique
cylinder is a circle.}

The plane of the paper being the principal plane and
$APB$ the circular base, a subcontrary section is $DPE$, the
angles $BAE$, $DEA$ being equal.

\includegraphicsmid{255a}

Let $PQ$ be the line of intersection of the two sections;
then
\[
PN \ctrdot NQ \text{ or } PN^2 = BN \ctrdot NA.
\]
%-----File: 269.png-----

But $\qquad NB = ND$, and $NA = NE$;
\[
\therefore PN \ctrdot NQ = DN \ctrdot NE,
\]
and $DPE$ is a circle.

\carticleplus{256.}{Prop.~II\@.} \textit{The section of an oblique cylinder by
a plane which is not parallel to the base or to a subcontrary
section is an ellipse.}

\includegraphicsmid{256a}

Let the plane of the section, $DPE$, meet any circular
section in the line $PQ$, and let $AB$ be that diameter of the
circular section which is perpendicular to $PQ$, and bisect $PQ$
in the point $F$.

Let the plane through the axis and the line $AB$ cut the
section $DPE$ in the line $DPE$.
\begin{flalign*}
&\indent\text{Then } &PF^2 = AF \ctrdot FB.&&
\end{flalign*}
%-----File: 270.png-----

But if $DE$ be bisected in $C$, and $GKC$ be the circular
section through $C$ parallel to $APB$,
\begin{flalign*}
&&AF : FD &:: CG : CD,&&&\\
&\text{and } &FB : FE &:: CG: CD;&&&\\
&&\therefore AF \ctrdot FB : DF & \ctrdot  FE :: CG2 : CD2;&&&
\end{flalign*}
hence, observing that $CG = CK$,
\[
PF^2 : DF \ctrdot FE :: CK^2 : CD^2.
\]

But, if a series of parallel circular sections be drawn, $PQ$
is always parallel to itself and bisected by $DE$;

\includegraphicsmid{256a} %repeated figure

Therefore the curve $DPE$ is an ellipse, of which $CD$, $CK$
are conjugate semi-diameters.

\carticleplus{257.}{Def.} If a straight line pass always through a fixed
point and the circumference of a fixed circle, and if the fixed
point be not in the straight line through the centre of the
circle at right angles to its plane, the surface generated is
called an oblique cone.

The plane containing the vertex and the centre of the
base, and also perpendicular to the base, is called the principal
section.

The section made by a plane not parallel to the base, but
perpendicular to the principal section, and inclined to the
%-----File: 271.png-----
generating lines in that section at the same angle as the
base, is called a subcontrary section.

\carticleplus{258.}{Prop.~III\@.} \textit{The subcontrary section of an oblique
cone is a circle.}

\cheader{OBLIQUE CONE}

\includegraphicsmid{258a}

The plane of the paper being the principal section, let
$APB$ be parallel to the base and $DPE$ a subcontrary section,
so that the angle
\begin{flalign*}
&          &ODE &=OAB,&&\\
&\text{and}&OED &= OBA.&\phantom{and}&
\end{flalign*}

The angles $DBA$, $DEA$ being equal to each other, a
circle can be drawn through $BDAE$.

Hence, if $PNQ$ be the line of intersection of the two
planes $APB$ and $EPD$,
\begin{align*}
DN \ctrdot NE&=BN \ctrdot NA,\\
&= PN \ctrdot NQ;
\end{align*}
therefore $DPE$ is a circle.

And all sections by planes parallel to $DPE$ are circles.

Planes parallel to the base, or to a subcontrary section,
are called also \textit{Cyclic Planes}.
%-----File: 272.png-----

\carticleplus{259.}{Prop.~IV\@.} \textit{The section of a cone by a plane not
parallel to a cyclic plane is an Ellipse, Parabola, or Hyperbola.}

\subparagraph{(1)} Let the section, $DPE$, meet all the generating lines
on one side of the vertex.

\includegraphicsmid{259a}

Let any circular section cut $DPE$ in $PQ$, and take $AB$
the diameter of the circle which bisects $PQ$.

The plane $OAB$ will cut the plane of the section in a line
$DNE$.

Draw $OK$ parallel to $DE$ and meeting in $K$ the plane of
the circular section through $D$ parallel to $APB$, and join
$DK$, meeting $OE$ in $F$.
\begin{flalign*}
&\indent\text{Then } &AN : ND &:: KD : OK,&&\\
&\text{and}     &BN : NE &:: KF : OK;&&\\
&\text{therefore}&AN \ctrdot NB : DN \ctrdot  NE &:: KD \ctrdot KF : OK^2,&\phantom{therefore}&\\
&\text{or}&PN^2 : DN \ctrdot NE &:: KD \ctrdot KF : OK^2.&&
\end{flalign*}

But if a series of circular sections be drawn the lines $PQ$
will always be parallel, and bisected by $DE$;
%-----File: 273.png-----

Therefore the curve $DPE$ is an ellipse, having $DE$ for
a diameter, and the conjugate diameter parallel to $PQ$, and
the squares on these diameters are in the ratio of $KD \ctrdot KF$
to $OK^2$.

\subparagraph{(2)} Let the section be parallel to a tangent plane of the
cone.

\includegraphicsmid{259b}

If $OB$ be the generating line along which the tangent
plane touches the cone, and $BT$ the tangent line at $B$ to a
circular section through $B$, the line of intersection $PQ$ will
be parallel to $BT$, and therefore perpendicular to the diameter
$BA$ through $B$.

Let the plane $BOA$ cut the plane of the section in $DN$.

Then, drawing $DK$ parallel to $AB$,
\begin{flalign*}
&&BN&=KD,&&&\\
&\text{and     }&AN : ND &:: KD : OK;&&&\\
&\text{therefore  }&AN \ctrdot NB : ND & \ctrdot  KD :: KD : OK,&&&\\
&\text{or }     &PN^2 : ND & \ctrdot  KD :: KD : OK,&&&
\end{flalign*}
and $KD$, $OK$ being constant, the curve is a parabola having
the tangent at $D$ parallel to $PQ$.

If the plane of the section meet both branches of the
%-----File: 274.png-----
cone, make the same construction as before, and we shall
obtain, in the same manner as for the ellipse,
\[
PN^2 : DN \ctrdot NE :: DK \ctrdot KF : OK^2,
\]
$OK$ being parallel to $DE$.

\includegraphicsmid{259c}

Therefore, since the point $N$ is not between the points
$D$ and $E$, the curve $DP$ is an hyperbola.

\cisubsection{Conoids}

\cheader{CONOIDS}

\carticleplus{260.}{Def.}  \textit{If a conic revolve about one of its principal
axes, the surface generated is called a conoid.}

If the conic be a circle, the conoid is a sphere.

If the conic be an ellipse, the conoid is an oblate or a
prolate spheroid according as the revolution takes place
about the conjugate or the transverse axis.

If it be an hyperbola the surface is an hyperboloid of one
or two sheets, according as the revolution takes place about
the conjugate or transverse axis, and the surface generated
by the asymptotes is called the asymptotic cone.

If the conic consist of two intersecting straight lines, the
limiting form of an hyperbola, the revolution will be about
one of the lines bisecting the angles between them, and the
conoid will then be a right circular cone.
%-----File: 275.png-----

\carticleplus{261.}{Prop.~V\@.} \textit{A section of a paraboloid by a plane
parallel to the axis is a parabola equal to the generating
parabola, and any other section not perpendicular to the axis
is an ellipse.}

\includegraphicsmid{261a}

Let $PVN$ be a section parallel to the axis, and take the
plane of the paper perpendicular to the section and cutting
it in $VN$.

Take any circular section $DPE$, cutting the section $PVN$
in $PNP'$.

Then $PN$ is perpendicular to $DE$,
\begin{flalign*}
&\text{and     }&PN^2 &= DN \ctrdot NE&&&\\
&&&=DC^2 - NC^2&&&\\
&&&= 4AS \ctrdot AC-4AS \ctrdot An&&&\\
&&&=4AS \ctrdot VN;&&&
\end{flalign*}
therefore the curve $VP$ is a parabola equal to $EAD$.

Again, let $BPF$ be a section not parallel or perpendicular
to the axis, but perpendicular to the plane of the paper;

Then, $BN \ctrdot NF=4SG \ctrdot VN$, $OG$ being the diameter
bisecting $BF$ (Art.~\cartref{51});
\begin{flalign*}
&\text{therefore      }&PN^2 : BN \ctrdot NF :: AS : SG,&&
\end{flalign*}
and the curve $BPN$ is an ellipse.
%-----File: 276.png-----

Moreover if the plane $BF$ move parallel to itself, $SG$
is unaltered, and the \textit{sections by parallel planes are similar
ellipses.}

In exactly the same manner, it may be shewn that the
oblique sections of spheroids are ellipses, and those of hyperboloids
either ellipses or hyperbolas.

\carticleplus{262.}{Prop.~VI\@.}  \textit{The sections of an hyperboloid and its
asymptotic cone by a plane are similar curves.}

Taking the case of an hyperboloid of two sheets, let
$DPF$, $dP'f$, be the sections of the hyperboloid and cone,
$P'PN$ the line in which their plane is cut by a circular section
$GPK$ or $gP'k$.

\includegraphicsmid{262a}

Through $D$ draw $LDl$ perpendicular to the axis; then,
since
\begin{flalign*}
&&    PN^2 = GN \ctrdot NK,  &\text{ and }  P'N^2 = gN \ctrdot Nk,&&&\\
&&    P'N^2 : dN \ctrdot Nf &:: gN \ctrdot Nk : dN \ctrdot Nf,&&&\\
&&                    &:: LD \ctrdot lD : Dd \ctrdot Df,&&&\\
&&                    &:: BC^2 : CE^2&&&
\end{flalign*}
if $CE$ be the semi-diameter parallel to $DF$;
%-----File: 277.png-----
\begin{flalign*}
&\text{and     }&PN^2 : DN \ctrdot NF &:: GN \ctrdot NK : DN \ctrdot NF&\phantom{and}&&\\
&\hfill\ &&:: BC^2 : CE^2 \hfill\text{(Art.~\cartref{134});}&&&
\end{flalign*}
therefore the curves $DPF$, $dP'f$ have their axes in the same
ratio, and are similar ellipses.

In the same manner the theorem can be established if
the sections be hyperbolic, or if the hyperboloid be of one
sheet.

\carticleplus{263.}{Prop.~VII\@.} \textit{If an hyperboloid of one sheet be cut
by a tangent plane of the asymptotic cone, the section will
consist of two parallel straight lines.}

\includegraphicsmid{263a}

Let $AQ$, $A'Q'$ be a section through the axis, $CN$ the
generating line, in the plane $CAQ$, along which the tangent
plane touches the cone; and $PNP'$ the section with this
tangent plane of a circular section $QPQ'$.
\begin{flalign*}
&\text{Then} &PN^2&= QN \ctrdot NQ'&\phantom{Then}&\\
&&&= AC^2 \text{(Art.~\cartref{106})}=BC^2,&&
\end{flalign*}
therefore, if $BCB'$ be the diameter, perpendicular to the
plane $CAQ$, of the principal circular section,
\[
PN = BC \text{ and } P'N = B'C;
\]
therefore $PB$ and $P'B'$ are each parallel to $CN$; that is, the
section consists of two parallel straight lines.
%-----File: 278.png-----

\carticleplus{264.}{Prop.~VIII\@.} \textit{The section of an hyperboloid of one
sheet by a plane parallel to its axis, and touching the central
circular section, consists of two straight lines.}

Let the plane pass through $A$, and be perpendicular to
the radius $CA$ of the central section (fig.~Art.~\cartref{263}).

The plane will cut the circular section $QPQ'$ in a line
$RLR'$, and
\[
      RL^2 = QL \ctrdot LQ' = QM^2 - AC^2,
\]
if $M$ be the middle point of $QQ'$.
\begin{flalign*}
&\indent\text{But}    &QM^2 - AC^2 : CM^2 &:: AC^2 : BC^2;&\\
&\text{therefore}     &RL : AL &:: AC : BC;&
\end{flalign*}
hence it follows that $AR$ is a fixed line; and similarly $AR'$
is also a fixed line.

It will be seen that these lines are parallel to the section
of the cone by the plane through the axis perpendicular
to $CA$.

\carticleplus{265.}{Prop.~IX\@.}  \textit{If a conoid be cut by a plane, and
if spheres be inscribed in the conoid touching the plane, the
points of contact of the spheres with the plane will be the foci
of the section, and the lines of intersection of the planes of
contact with the plane of section will be the directrices.}

In order to establish this statement, we shall first demonstrate
the following theorem;

\textit{If a circle touch a conic in two points, the tangent from
any point of the conic to the circle bears a constant ratio
to its distance from the chord of contact.}

Take the case of an ellipse, the chord of contact being
perpendicular to the transverse axis.

If $EME'$ be this chord, the normal $EG$ is the radius of
%-----File: 279.png-----
the circle, and if $PT$ be a tangent from a point $P$ of the
ellipse,
\begin{flalign*}
&&PT^2 = P&G^2 - GE^2&&&\\
&&    = P&N^2 + NG^2 - EM^2 - MG^2.&&&\\
&\indent\text{But   }&EM^2 - PN^2 &: CN^2 - CM^2 :: BC^2 : AC^2,&&&\\
&\text{and} &CN^2 - &CM^2 = MN(CM + CN).&&&
\end{flalign*}

Let the normal at $P$ meet the axis in $G'$;
\begin{flalign*}
&\text{then}          &NG' : CN  &:: BC^2 : AC^2,&&\\
&\text{and}           &MG : CM   &:: BC^2 : AC^2;&&\\
&\text{therefore}     &NG' + MG : CN &+ CM :: BC^2 : AC^2.&&\\
&\indent \text{Hence} &EM^2 - PN^2 &= MN(NG' + MG).&&\\
&\indent \text{Also}  &NG^2 - MG^2 &= MN(NG + MG);&&\\
&\text{therefore}     &PT^2 = MN &(NG + MG)-MN (NG' + MG)&&\\
&                     &= MN      & \ctrdot GG'.&&\\
&\indent\text{But}    &CG : CM   &:: SC^2 : AC^2,&&\\
&\text{and}           &CG' : CN  &:: SC^2 : AC^2;&&\\
&\text{therefore}     &GG' : MN  &:: SC^2 : AC^2.&&\\
&\indent\text{Hence}  &PT^2 : PL^2 &:: SC^2 : AC^2, & \phantom{\indent Hence}&\\
&             &PL \text{ being } &\text{equal to } MN.
\end{flalign*}

\includegraphicsmid{265a}
%-----File: 280.png-----

This being established let the figure revolve round the
axis $AC$, and let a plane section $ap$ of the conoid, perpendicular
to the plane of the paper, touch the sphere at $S$ and
cut the plane of contact $EE'$ in $lk$.

From a point $p$ of the section let fall the perpendicular
$pm$ on the plane $EE'$, draw $mk$ perpendicular to $lk$, and
join $pk$.

Then $pm : pk$ is a constant ratio.

Also taking the meridian section through $p$, $pS$ is equal
to the tangent from $p$ to the circular section of the sphere,
and is therefore in a constant ratio to $pm$;

Hence $Sp$ is to $pk$ in a constant ratio,\\
and therefore $S$ is the focus and $kl$ the directrix of the
section $ap$.

\carticle{266.} If the curve be a parabola focus $S'$, the proof is as
follows:
\begin{align*}
PT^2&=PG^2 - EG^2\\
&=PN^2 + NG^2 - EM^2 - MG^2\\
&=MN(NG+MG)-4AS' \ctrdot MN\\
&=MN(NG + MG)-2MG \ctrdot MN\\
&=MN^2.
\end{align*}

It will be found that the theorem is also true for an
hyperboloid of two sheets, and for an hyperboloid of one
sheet, but that in the latter case the constant ratio of $PT$ to
$PL$ is not that of $SC$ to $AC$.

\carticle{267.} The geometrical enunciation of the theorem also
requires modification in several cases. To illustrate the
difficulty, take the paraboloid, and observe that if the normal
%-----File: 281.png-----
at $E$ cuts the axis in $G$, and if $O$ be the centre of curvature
at $A$,
\[
AG>AO,
\]
and the radius of the circle is never less than $AO$.

This shews that a circle the radius of which is less than
$AO$ cannot be drawn so as to touch the conic in two points.

We may mention one exceptional case in which the
theorem takes a simple form.

In general
\begin{align*}
EG^2 &= EM^2 + MG^2 = 4AS' (AM+ AS')\\
     &= 4AS' \ctrdot S'G.
\end{align*}

Taking the point $g$ between $S'$ and $O$, describe a circle
centre $g$ and such that the square on its radius $= 4AS' \ctrdot S'g$.

Also take a point $F$ in the axis produced such that
\[
AF=Og;
\]
it will then be found that the tangent from $P$ to the circle
will be equal to $NF$.

When $g$ coincides with $S'$, the circle becomes a point,
\begin{flalign*}
&\text{and}& AF&=AS'; &\phantom{\text{and}}
\end{flalign*}
we thus fall back on the fundamental definition of a parabola.

It will be found that if the plane section of the conoid
pass through $S'$, the point $S'$ is a focus of the section.
%-----File: 282.png-----

\cchapter{CHAPTER XV}
\csectiontoc{Conical Projection}

\cheader{CONICAL PROJECTION}

\carticle{268.} If from any fixed point straight lines are drawn
to all the points of a figure, the section by any plane of the
lines thus drawn is the conical projection of the figure upon
that plane.

The fixed point is called the vertex of projection, and the
plane is called the plane of projection.

Taking the eye as the vertex of projection, the conical
projection of any figure upon a plane is a perspective drawing
of that figure as seen by the eye.

\textit{A straight line is projected into a straight line}, for the
plane through the vertex and the straight line intersects the
plane of projection in a straight line.

\textit{A tangent to a curve is projected into a tangent to the
projection of the curve}, for two consecutive points of a curve
project into two consecutive points.

Hence it follows that \textit{a pole and polar project into a pole
and polar}.

Again, \textit{the degree of a curve is unaltered by projection}, for
any number of collinear points project into the same number
of collinear points.

In particular, the projection of a conic on any plane is a
conic.

\carticle{269.} \textit{Any straight line in a figure can be projected to an
infinite distance.}

This is effected by taking the plane of projection parallel
to the plane through the vertex of projection and the straight
line.
%-----File: 283.png-----

\carticle{270.} \textit{A system of concurrent straight lines in a plane can
be projected into a system of parallel straight lines, and a
system of parallel straight lines can be projected into a system
of concurrent straight lines.}

The first of these is effected by taking for plane of projection
any plane parallel to the straight line joining the
vertex of projection and the point of concurrence.

The second is effected by taking for plane of projection
any plane not parallel to the direction of the parallel straight
lines.

\carticle{271.} \textit{Any angle in a plane can be projected, on any other
plane, into any other angle.}

Let $ACB$ be the angle to be projected, and let $DEF$ be
the plane upon which it is to be projected.

Take any plane parallel to $DEF$, intersecting in $A$ and $B$
the lines forming the angle $ACB$, and take any point $O$ in
the plane.

\includegraphicsmid{271a}

Then, if $CA$, $CB$, $CO$, meet the plane of projection in
$a$, $b$, $c$, the angle $acb$ is the projection of the angle $ACB$ from
the vertex $O$ upon the plane $DEF$.
%-----File: 284.png-----

Now $OA$, $OB$ are parallel to $ca$, $cb$; therefore the angle
$acb$ is equal to the angle $AOB$.

If then we describe on $AB$ an arc of a circle containing
an angle equal to any given angle, and take any point $O$ on
the arc as vertex of projection, the angle $ACB$ will be projected
into the given angle.

It will be seen that the arc of a circle may be described
on the other side of the plane $CAB$, so that the locus of $O$
on the plane $OAB$ consists of two equal arcs on the same
base.

If the plane of projection be assigned, it follows, since
the plane $OAB$ may be taken at any distance from $C$, that
the locus of $O$ consists of portions of two oblique cones
having their common vertex at $C$.

If the plane of projection be not assigned, but if the line
$AB$ be assigned, the locus of $O$ will be the surface generated
by the revolution, about $AB$, of the arc of the circle.

If the angle $ACB$ is to be projected into a right angle,
the locus of $O$ will be the sphere described upon $AB$ as
diameter.

If the assigned plane, $DEF$, be parallel to $CA$, the locus
of $O$ on the plane $OBA$ will be the straight line $BO$ making
with $BA$ the angle $OBA$ equal to the supplement of the
angle into which $ACB$ is to be projected.

In the particular case in which this angle is a right angle
the locus of $O$ will be the straight line $BO$ perpendicular to
$BA$.

If it be required to project two given angles in a plane
into two other given angles in any other plane, we can construct
two arcs of circles in a plane parallel to this other
plane, and, if these arcs intersect, the position of $O$ is determined.

\carticle{272.} \textit{To project a given quadrilateral into a square.}

Let $ABCD$ the quadrilateral, and let $AC$, $BD$ intersect
in $E$, $AD$, $BC$ in $F$, and $BA$, $CD$ in $G$.
%-----File: 285.png-----

Then if $O$ is the vertex of projection, taken anywhere,
the quadrilateral will be projected into a parallelogram on
any plane parallel to $OFG$.

If $O$ be taken on the sphere of which $FG$ is diameter,
the projection on any plane parallel to $OFG$ will be a
rectangle, for the angles subtended by $FG$ at $A$, $B$, $C$, $D$
project into right angles.

If $AC$ and $BD$ meet $FG$ in $L$ and $M$, and if $O$ be taken
on the circle which is the intersection of the spheres on $FG$
and $LM$ as diameters, the angle $LEM$ will be projected into
a right angle, so that the projection of $ABCD$ will be a
rectangle, the diagonals of which are at right angles, and
therefore will be a square.

\carticle{273.} \textit{The projection of an harmonic range is an harmonic
range.}

This is proved in Art.~\cartref{198}.

\textit{The projection of a circle is a conic.}

This is proved in Art.~\cartref{259}.

As an illustration it is easily shown for a circle that, if a
diameter $pP$ passes through an external point $T$ and intersects
in $V$ the polar of $T$, $pVPT$ is an harmonic range.

By projection we at once obtain the theorems of Art.~\cartref{78}
and of Art.~\cartref{117}.

\carticle{274.} \textit{To project a conic into a circle, so that the projection
of a given point inside the conic shall be the centre of the
projection.}

Let $E$ be the given point, $AEB$ the chord bisected at
$E$ and $PEp$ the diameter passing through $E$.

Then, if we project the polar of $E$ to an infinite distance,
and the angles $AEP$, $APB$ into right angles, the projection
of the conic will be a circle, the centre of which is the projection
of the point $E$.

For the centre is the pole of a line at an infinite distance,
and, the projection of $AEP$ being a right angle, the projections
of $AB$ and $Pp$ are the principal axes of the projection.
%-----File: 286.png-----

Also, the projection of $APB$ being a right angle, it follows
that the projection of the conic is a circle.

Another method will be to take points $C$, $C'$, $D$, $D'$ on the
polar of $E$, such that $CED$, $C'ED'$ are self-conjugate triangles,
and then to project $CD$ to an infinite distance and the angles
$CED$, $C'ED'$ into right angles.

The projection will be a conic, having the projection of $E$
for its centre, and also having two pairs of conjugate diameters
at right angles to each other; that is, it will be a circle.

In a subsequent article this question will be treated in a
different manner.

If the point $E$ is outside the conic, we can project the
conic into a rectangular hyperbola, of which the projection of
$E$ is the centre.

For, if $PQ$ is the chord of contact of tangents from $E$, all
we have to do is to project $PQ$ to an infinite distance, and
$PEQ$ into a right angle.

We can also project the conic into an hyperbola of any
given eccentricity.

For, if the eccentricity is given, the angle between the
asymptotes is given, and we can project $PQ$ to an infinite
distance and $PEQ$ into the given angle.

\carticle{275.} \textit{To project a conic on a given plane so that the
projection of a point $S$ inside the conic shall be a focus of the
projection.}

Let the tangent at any point $P$ and any straight line
through $S$ meet the polar of $S$ in $F$ and $X$.

Then, if we project the angles $SXF$, $FSP$ into right
angles, the projections of $S$ and $FX$ are the focus and
directrix of the projection.

If at the same time we project to an infinite distance the
polar of any point $E$ on $XS$, the projection of $E$ will be the
centre of the projection of the conic.
%-----File: 287.png-----

\carticle{276.} \textit{If two conics in different planes have two points
in common, two cones of the second order can be drawn passing
through them, or, in other words, each can be projected into
the other.}

Let $AB$ be the common chord, $F$ and $D$ its poles with
regard to the conics.

Take any point $E$ in $AB$, and let the plane $FED$ meet the
conics in the points $P$, $p$, $Q$, $q$, and let $pq$ intersect $DF$ in $O$.

\includegraphicsmid{276a}

If from $O$ the conic $BPAp$ be projected on to the plane
of the other conic, the projection will be a conic touching the
conic $BQAq$ at $A$ and $B$, so that it will have four points in
common with $BQAq$, and will also have the point $q$ in
common with $BQAq$.

Now it is proved in Art.~\cartref{252}, that only one conic can be
drawn through five points.

Hence the projection, having five points in common with
$BQAq$, coincides with it entirely.

It will be observed that $OPQ$ is a straight line, $Pp$ being
projected into $Qq$.
%-----File: 288.png-----

The point $O$ is therefore the vertex of a quadric cone
which passes through the two conics.

The vertex of another such cone is obtained by producing
$qP$ or $pQ$ to meet $DF$.\symbolfootnote[1]
  {I am indebted to Mr H.~F.~Baker, Fellow and Lecturer of St John's
  College, for having called my attention to this theorem and to the mode of
  proof which is here given. The theorem is given in Poncelet's Treatise, and
  also in the article on Projections in the last edition of the \textit{Encyclopaedia
  Britannica}.}

\carticle{277.} \textit{A conic can be projected into a circle so that the
projection of any point inside the conic shall be the centre of
the circle.}

\includegraphicsmid{277a}

Let $E$ be the point inside the conic and let $AB$ be the
chord of which $E$ is the middle point.

Describe a circle on $AB$ as diameter in any plane passing
through $AB$.

Observing that the pole of $AEB$ with regard to the circle
is at an infinite distance, draw through $F$, the pole of $AB$
with regard to the conic, the line $FL$ parallel to that diameter,
$QEq$, of the circle which is perpendicular to $AB$.
%-----File: 289.png-----

The plane $EFL$ will cut the conic in the diameter $Pp$, and
the circle in the diameter $Qq$.

If $pq$, $qP$ intersect $FL$ in $O$ and $O'$, these two points will
be vertices from which the conic can be projected into a
circle, the centre of which is the projection of the point $E$.
\begin{flalign*}
&\indent\text{Since}&   FO : Fp &:: Eq : Ep :: EA : Ep, &\phantom{\indent\text{Since}}
\end{flalign*}
it follows that, for different positions of the plane through
$AB$, $FO$ is constant, so that $O$ may be taken anywhere on
the circle, centre $F$, in the plane through $F$ perpendicular to
the chord $AEB$.
\begin{flalign*}
&\indent\text{Further,}& FO : FP &:: EQ : EP :: EA : EP,
&\phantom{\indent\text{Further,}}
\end{flalign*}
\[
\therefore FO^2 : FP \ctrdot Fp :: EA^2 : EP \ctrdot Ep :: CD^2 : CP^2,
\]
$DCd$ being the semi-diameter of the conic which is conjugate
to $CP$.

The length $FO$ is thus determined when the position of
the point $E$, inside the conic, is given, and, if we take as the
vertex of projection any point $O$ on the circle, centre $F$, as
described above, the projection of the conic on any plane
parallel to $AEB$ and $FO$ will be a circle.

If the conic is an ellipse, it follows that $FO$ is equal to
the ordinate $FR$, conjugate to $Pp$, of the hyperbola in the
plane of the ellipse which has the same conjugate diameters
$PCp$ and $DCd$.

\includegraphicsmid{277b}

If the conic is an hyperbola, $FO$ is equal to the ordinate
$FR$ of an ellipse in the plane of the hyperbola which has the
same conjugate diameters $PCp$ and $DCd$.

This hyperbola or this ellipse constructed outside the
given conic may be called \textit{the associated conic}.
%-----File: 290.png-----

If the conic is a parabola, the points $O$ and $O'$ are obtained
by drawing lines through $q$ and $Q$ parallel to the axis of the
parabola.

In this case,
\[
FO^2 = Eq^2 = EA^2 = 4SP \ctrdot PE = 4SP \ctrdot PF,
\]
so that the associated conic is a parabola.

If the conic is a circle, the associated conic is a rectangular
hyperbola.

If the conic is an ellipse, the axes of which are indefinitely
small, that is, if it is reduced to a point, the associated conic
lapses into two straight lines, which are at right angles to
each other if the point is the limit of a circle.

\carticle{278.} If the point $E$ be outside the conic, or, in other
words, if the polar of $E$ intersect the conic, it is not possible
to project the conic into a circle, so that the projection of $E$
shall be the centre of the circle.

In this case the conic can be projected into a rectangular
hyperbola, having the projection of the point $E$ for its centre.

Let $RU$ be the chord of contact of the tangents from $E$,
and take any point $O$ on the surface of the sphere of which
$RU$ is a diameter.

Then the projection of the conic from the vertex $O$ on
any plane parallel to $ROU$ will be an hyperbola, and, since
$ROU$ is a right angle, it will be a rectangular hyperbola.

\carticle{279.} \textit{If two conics in a plane are entirely exterior to each
other, they can in general be projected, from the same vertex,
into circles on the same plane.}

Draw four parallel tangents to the conics, and let $F$ be the
point of intersection of the diameters, $PCp$ and $QGq$, joining
the points of contact.

Also, let $FR$, $FR'$ be the ordinates through $F$, parallel to
the tangents, of the associated conics.

If $F$ is so situated that these ordinates are equal, the
locus of the vertices from which the two conics can be projected
%-----File: 291.png-----
into circles will be the same, that is, it will be the
circle of which $F$ is the centre, and $FR$ the length of the
radius, in the plane through $F$ perpendicular to $FR$.

In this case, taking any point $O$ on the circle as the
vertex, the two conics will be projected into circles on any
plane parallel to the plane $OFR$, and the centres of the
circles will be the projections of $E$ and $E'$, the respective
poles of $FR$ with regard to the conics.

\includegraphicsmid{279a}

\carticle{280.} For different directions of the tangents, the points,
$F$, $R$, $R'$, will take up different positions, and for all directions
of the tangents the loci of these points will be continuous
curves.

The loci of $R$ and $R'$ will, in general, intersect each other;
that is to say, there will be, in general, positions of $F$ such
that $FR$ and $FR'$ are equal.

Taking a particular case, let $F$ be so situated that $FR'$ is
greater than $FR$; then taking $F$ at the point where its locus
meets the conic $G$, $FR'$ vanishes, and therefore, between
these two positions of $F$, there must be some position such
that $FR'$ is equal to $FR$.
%-----File: 292.png-----

We may observe that the locus of $F$ passes through $C$ and
$G$, the centres of the two conics.

For, if $CG$ is conjugate to the parallel tangents of the
conic $G$, the point $F$ is at $C$, and, if $CG$ is conjugate to the
parallel tangents of the conic $C$, the point $F$ is at $G$.

When $FR'$ is equal to $FR$, the line thus obtained is called
by \textit{Poncelet} the \textit{Ideal Secant} of the two conics.

\carticle{281.} In a similar manner if one conic is entirely inside
another they can, in general, be projected into circles, one of
which will be inside the other.

Also two conics intersecting in two points may be projected
into two intersecting circles.

Two conics intersecting in four points, or having contact
at two points, cannot be projected into circles, but they can
be projected into rectangular hyperbolas.

\carticle{282.} The method of projections enables us to extend to
conics theorems which have been proved for a circle, and
which involve, amongst other ideas, harmonic ranges, poles
and polars, systems of collinear points, and systems of concurrent
lines.

For instance, the theorems of Arts.~\cartref{208} and~\cartref{210} are easily
proved for a circle, and by this method are at once extended
to conics.

Take as another instance Pascal's theorem, that \textit{the opposite
sides of any hexagon inscribed in a conic intersect in three
collinear points.}

If this be proved for a circle, the method of conical
projection at once shews that it is true for any conic.

The following very elementary proof of the theorem for
a circle is given in \textit{Catalan's Théorèmes et Problèmes de
Géométrie Elémentaire.}

Let $ABCDEF$ be the hexagon, and let $AB$ and $ED$ meet
in $G$, $BC$ and $FE$ in $H$, $FA$ and $DC$ in $K$.
%-----File: 293.png-----

Also let $ED$ meet $BC$ in $M$ and $AF$ in $N$, and let $BC$
meet $AF$ in $L$.

Then we have the relations,
\begin{gather*}
LA \ctrdot LF=LB \ctrdot LC, MC \ctrdot MB=MD \ctrdot ME,\\
NE \ctrdot ND=NF \ctrdot NA.
\end{gather*}
%-----File: 294.png-----

Also, the triangle $LMN$ being cut by the three transversals
$AG$, $DK$, $FH$, we have the relations,
\begin{align*}
LB \ctrdot MG \ctrdot NA&=LA \ctrdot MB \ctrdot NG\\
LC \ctrdot MD \ctrdot NK&=LK \ctrdot MC \ctrdot ND\\
LH \ctrdot ME \ctrdot NF&=LF \ctrdot MH \ctrdot NE.
\end{align*}

Multiplying together these six equalities, taking account
of the relations previously stated, and cutting out the factors
common to the two products, we obtain
\[
LH \ctrdot MG \ctrdot NK=LK \ctrdot MH \ctrdot NG;
\]
$\therefore$ $G$, $H$, $K$ are collinear.

Brianchon's theorem that, \textit{if a hexagon circumscribe a
conic, the three opposite diagonals are concurrent} is proved at
once by observing that it is the reciprocal polar of Pascal's
theorem.

\includegraphicsmid{282a}

\carticle{283.} \textit{Stereographic and Gnomonic Projections.}

If a point on the surface of a sphere be taken as the
vertex of projection, and if the plane of projection be parallel
to the tangent plane at the point, the projection of any
figure drawn on the surface of the sphere is called its
stereographic projection.

If however the centre of the sphere be taken as the
vertex of projection, and any plane be taken as the plane of
projection, the projection of any figure drawn on the surface
of the sphere is called its gnomonic projection.

The stereographic projection of a circle drawn on the
surface of the sphere is a circle; for it can be easily shewn
that it is a subcontrary section of the oblique cone formed by
the vertex of projection and the circle on the sphere.

The gnomonic projection of a circle on the sphere is
obviously a conic.

These projections are sometimes described in treatises on
Astronomy, and in these treatises the vertex for stereographic
projection is taken at the south pole of the earth, and, for
gnomonic projection, at the centre of the earth; and, in both
%-----File: 295.png-----
cases, the plane of projection is taken parallel to the plane of
the equator.

\carticle{284.} It will be seen that the discussions which are given
in this chapter are confined entirely to cases of \textit{real} projection.

The chapter is intended to be simply an introduction to a
large and important subject.

The method of conical projections is due to Poncelet, and
is worked out with great fulness and elaboration in his work
entitled, \textit{Traité des Propriétés Projectives des Figures} (Second
edition, 1865, in two quarto volumes).

In this work Poncelet extends the domain of pure geometry
by the interpretation and use of the law of continuity, and,
as one of its applications, by the introduction of the imaginary
chord of intersection, or, as it is called by Poncelet, the ideal
secant of two conics.

Amongst English writers, the student will find valuable
chapters on projections in \textit{Salmon's Conics}, and in the large
work on the \textit{Geometry of Conics, by Dr C.~Taylor, the Master
of St John's College, Cambridge}.

There is also an important work by \textit{Cremona, on Projective
Geometry}, which has been translated by \textit{Leudesdorf} (Second
edition, 1893).
%-----File: 296.png-----

\clearpage

\addtocontents{toc}{\vspace{\baselineskip}}
\csectionuppertoc{\texorpdfstring{Miscellaneous Problems.\quad II}{Miscellaneous Problems. II}}
\cheader{MISCELLANEOUS PROBLEMS.~II}

\begin{exampleslist}

\item[1.] If two conics have the same directrix, their common points are
concyclic.

\item[2.] If a focal chord of a parabola is bisected in $V$ and the line
perpendicular to it through $V$ meets the axis in $G$, $SG$ is half the
chord.

\item[3.] If the perpendicular to $CP$ from a point $P$ of an ellipse meets
the auxiliary circle in $Q$, $PQ$ varies as $PN$.

\item[4.] $AA'$ and $BB'$ are the axes, and $S$ is one of the foci of an ellipse;
if a parabola is described with $S$ as focus and passing through $B$ and $B'$,
its vertex bisects $SA$ or $SA'$.

\item[5.] Tangents to an ellipse at $P$, $p$ intersect on an axis; if the
perpendicular from $p$ on the tangent at $P$ intersects $CP$ in $L$, the locus
of $L$ is a similar ellipse.

\item[6.] The normal to a hyperbola at $P$ meets the axes in $G$ and $g$
respectively. Prove that the circle circumscribing $SPG$ is touched
$Sg$.

\item[7.] If a tangent to an ellipse meets a pair of conjugate diameters in
points equidistant from the centre, the locus of the points is a circle.

\item[8.] If ellipses are described on $AB$ as diameter, touching $BC$, the
points of contact of tangents from $C$ are on a straight line.

\item[9.] If $Pl$, $Pm$ be drawn perpendicular to $CL$, $CM$ respectively, shew
that the centre of the circle $Plm$ lies on a fixed hyperbola.

\item[10.] $PSQ$, $PHR$ are focal chords of an ellipse, $QT$, $RT$ the tangents
at $Q$ and $R$. Shew that $PT$ is the normal at $P$.

\item[11.] $A$, $B$ are two fixed points. Through them a system of circles
is drawn. Through $A$ draw any two lines meeting the circles in the
points $C_1D_1$, $C_2D_2$, \&c. Shew that the lines $CD$ all touch a parabola,
focus $B$, which also touches the lines $AC$, $AD$.
%-----File: 297.png-----

\item[12.] From any two points $A$, $B$ on an ellipse four lines are drawn to
the foci $S$, $H$. Shew that $SA \ctrdot HB$ and $SB \ctrdot HA$ are to one another as
the squares of the perpendiculars from a focus on the tangents at $A$
and $B$.

\item[13.] If two points of a conic and the angle subtended by these points
at the focus are given, the line joining the focus with the intersection
of the tangents always passes through a fixed point.

\item[14.] If normals to an ellipse are drawn at the extremities of chords
parallel to one of the equi-conjugate diameters, pairs of such normals
intersect on the line through the centre perpendicular to the other
diameter.

\item[15.] From the point in which the tangent at any point $P$ of a
hyperbola cuts either asymptote perpendiculars are dropped upon the
axes. Prove that the line joining the feet of these perpendiculars
passes through $P$.

\item[16.] Tangents are drawn to an ellipse parallel to conjugate diameters
of a second given ellipse. Shew that the locus of their intersection is
an ellipse similar and similarly situated to the second ellipse.

\item[17.] A focus of a conic inscribed in a triangle being given, find the
points of contact.

\item[18.] The normals at $P$ and $Q$, the ends of a focal chord $PSQ$, intersect
in $K$, and $KN$ is perpendicular to $PQ$; prove that $NP$ and $SQ$ are equal.

\item[19.] If $CR$, $SY$, $HZ$ be perpendiculars upon the tangent at a point
$P$ such that $CR=CS$, prove that $R$ lies on the tangent at $B$, and that
the perpendicular from $R$ on $SH$ will divide it into two parts equal to
$SY$, $HZ$ respectively.

\item[20.] If a parabola, having its focus coincident with one of the foci of
an ellipse, touches the conjugate axis of the ellipse, a common tangent
to the ellipse and parabola will subtend a right angle at the focus.

\item[21.] Two tangents $TP$ and $TQ$ are drawn to an ellipse, and any
chord $TRS$ is drawn, $V$ being the middle point of the intercepted part;
$QV$ meets the ellipse in $P'$; prove that $PP'$ is parallel to $ST$.

\item[22.] If $S$, $S'$ are the foci of an ellipse and $SY$, $S'Y'$ the perpendiculars
on any tangent, $XY$, $X'Y'$ meet on the minor axis, and, if $PN$ is the
ordinate of $P$, $NY$ and $NY'$ are perpendicular to $XY$ and $X'Y'$
respectively.

\item[23.] A circle through the centre of a rectangular hyperbola cuts the
curve in the points $A$, $B$, $C$, $D$. Prove that the circle circumscribing
the triangle formed by the tangents at $A$, $B$, $C$ passes through the
centre of the hyperbola.
%-----File: 298.png-----

\item[24.] If the tangent at a point $P$ of an ellipse meets any pair of
parallel tangents in $M$, $N$, and if the circle on $MN$ as diameter meets
the normal at $P$ in $K$, $L$, then $KL$ is equal to $DCD'$, and $CK$, $CL$ are
equal to the sum and difference of the semi-axes.

\item[25.] From a point $O$ two tangents $OA$, $OB$ are drawn to a parabola
meeting any diameter in $P$, $Q$. Prove that the lines $OP$, $OQ$ are similarly
divided by the points of contact, but one internally, the other externally.

\item[26.] If $S$, $H$ be the foci of an ellipse, and $SP$, $HQ$ be parallel radii
vectores drawn towards the same parts, prove that the tangents to the
ellipse at $P$, $Q$ intersect on a fixed circle.

\item[27.] If an ellipse be inscribed in a quadrilateral so that one focus $S$
is equidistant from the four vertices, the other focus must be at the
intersection $H$ of the diagonals.

\item[28.] $P$ is a point on a circle whose centre is $Q$; through $P$ a series
of rectangular hyperbolas are described having $Q$ for their centre of
curvature at $P$. Prove that the locus of their centres is a circle with
diameter of length $PQ$.

\item[29.] Two cones which have a common vertex, their axes at right
angles, and their vertical angles supplementary, are intersected by a
plane at right angles to the plane of their axes. Prove that the distances
of either focus of the elliptic section from the foci of the
hyperbolic section are equal respectively to the distance from the
vertex of the ends of the transverse axis of each, and that the sum of
the squares on the semi-conjugate axes is equal to the rectangle contained
by those distances.

\item[30.] Two plane sections of a cone which are not parallel are such
that a focus of each and the vertex of the cone lie on a straight line.
Shew that the angle included by any pair of focal chords of one section
is equal to that contained by the corresponding focal chords of the other
section, corresponding chords being the projections of each other with
respect to the vertex.

\item[31.] If $PP'$, $QQ'$ be chords normal to a conic at $P$ and $Q$, and also
at right angles to each other, then will $PQ$ be parallel to $P'Q'$.

\item[32.] A system of conics have a common focus $S$ and a common
directrix corresponding to $S$. A fixed straight line through $S$ intersects
the conics, and at the points of intersection normals are drawn. Prove
that these normals are all tangents to a parabola.

\item[33.] If two confocal conics intersect, prove that the centre of
curvature of either curve at a point of intersection is the pole of the
tangent at that point with regard to the other curve.

\item[34.] A chord of a conic whose pole is $O$ meets the directrices in $R$
and $R'$; if $SR$ and $HR'$ meet in $O'$, prove that the minor axis bisects $OO'$.
%-----File: 299.png-----

\item[35.] $TQ$ and $TR$, tangents to a parabola, meet the tangent at $P$ in
$X$ and $Y$, and $TU$ is drawn parallel to the axis, meeting the parabola in
$U$. Prove that the tangent at $U$ passes through the middle point of
$XY$, and that, if $S$ is the focus,
\[
XY^2 = 4SP \ctrdot TU.
\]

\item[36.] The foot of the directrix which corresponds to $S$ is $X$, and $XY$
meets the minor axis in $T$; $CV$ is the perpendicular from the centre on
the tangent at $P$. Prove that, if $CP=CS$, then $CV=VT$.

\item[37.] $A$ is a given point in the plane of a given circle, and $ABC$ a
given angle. If $B$ moves round the circumference of the circle, prove
that, for different values of the angle $ABC$, the envelopes of $BC$ are
similar conics, and that all their directrices pass through one or other
of two fixed points.

\item[38.] If $AA'$ is the transverse axis of an ellipse, and if $Y$, $Y'$ are the
feet of the perpendiculars let fall from the foci on the tangent at any
point of the curve, prove that the locus of the point of intersection of
$AY$ and $A'Y'$ is an ellipse.

\item[39.] The tangent at a point $P$ of an hyperbola cuts the asymptotes
in $L$ and $L'$, and another hyperbola having the same asymptotes bisects
$PL$ and $PL'$. Prove that it intersects $CP$ in a point $p$ such that
\[
Cp^2 : CP^2 :: 3 : 4.
\]

The chord $QR$, joining a point $R$ on an asymptote with a point $Q$ on
the corresponding branch of the first hyperbola, intersects the second
hyperbola in $E$; if $QR$ move off parallel to itself to infinity, prove that,
ultimately $RE:EQ:: 3:1$.

\item[40.] Tangents are drawn to a rectangular hyperbola from a point $T$
in the transverse axis, meeting the tangents at the vertices in $Q$ and $Q$'.
Prove that $QQ'$ touches the auxiliary circle at a point $R$ such that $RT$
bisects the angle $QTQ'$.

\item[41.] Tangents from a point $T$ touch the curve at $P$ and $Q$; if $PQ$
meet the directrices in $R$ and $R'$, $PR$ and $QR'$ subtend equal angles
at $T$.

\item[42.] The straight lines joining any point to the intersections of its
polar with the directrices touch a conic confocal with the given one.

\item[43.] If a point moves in a plane so that the sum or difference of its
distances from two fixed points, one in the given plane and the other
external to it, is constant, it will describe a conic, the section of a right
cone whose vertex is the given external point.

\item[44.] In the construction of Art.~\cartref{241} prove that $CK'$ and $CK$ are
respectively equal to the sum and difference of the semi-axes.

\item[45.] Given a tangent to an ellipse, its point of contact, and the
director circle, construct the ellipse.
%-----File: 300.png-----

\item[46.] If the tangent at any point $P$ of an ellipse meet the auxiliary
circle in $Q'$, $R'$, and if $Q$, $R$ be the corresponding points on the ellipse,
the tangents at $Q$ and $R$ pass through the point $P'$ on the auxiliary
circle corresponding to $P$.

\item[47.] In the ellipse $PDP'D'$, $P'HCSPX$ and $DCD'$ are conjugate
diameters; $CH$ is equal to $CS$, and the polar of $S$ passes through a
point $X$ on $P'P$ produced. If $DX$ is drawn cutting the ellipse in $Q$,
prove that $HD$ is parallel to $SQ$.

\item[48.] If $T$ is the pole of a chord of a conic, and $F$ the intersection of
the chord with the directrix, $TSF$ is a right angle.

\item[49.] The polar of the middle point of a normal chord of a parabola
meets the focal vector to the point of intersection of the chord with the
directrix on the normal at the further end of the chord.

\item[50.] $OP$, $OQ$ touch a parabola at $P$, $Q$; the tangent at $R$ meets
$OP$, $OQ$ in $S$, $T$; if $V$ is the intersection of $PT$, $SQ$, $O$, $R$, $V$ are
collinear.

\item[51.] If from any point $A$ a straight line $AEK$ be drawn parallel to
an asymptote of an hyperbola, and meeting the polar of $A$ in $K$ and the
curve in $E$, shew that $AE=EK$.

\item[52.] If a chord $PQ$ of a parabola, whose pole is $T$, cut the directrix
in $F$, the tangents from $F$ bisect the angle $PFT$ and its supplement.

\item[53.] A parabola, focus $S$, touches the three sides of a triangle $ABC$,
bisecting the base $BC$ in $D$; prove that $AS$ is a fourth proportional to
$AD$, $AB$, and $AC$.

\item[54.] A focal chord $PSQ$ is drawn to a conic of which $C$ is the
centre; the tangents and normals at $P$ and $Q$ intersect in $T$ and $K$
respectively; shew that $ST$, $SP$, $SK$, $SC$ form an harmonic pencil.

\item[55.] $PCP'$ is any diameter of an ellipse. The tangents at any two
points $D$ and $E$ intersect in $F$. $PE$, $P'D$ intersect in $G$. Shew that
$FG$ is parallel to the diameter conjugate to $PCP'$.

\item[56.] A conic section is circumscribed by a quadrilateral $ABCD$: $A$
is joined to the points of contact of $CB$, $CD$; and $C$ to the points of
contact of $AB$, $AD$; prove that $BD$ is a diagonal of the interior quadrilateral
thus formed.

\item[57.] A parabola touches the three lines $CB$, $CA$, $AB$ in $P$, $Q$, $R$, and
through $R$ a line parallel to the axis meets $RQ$ in $E$; shew that $ABEC$
is a parallelogram.

\item[58.] If a series of conics be inscribed in a given quadrilateral, shew
that their centres lie on a fixed straight line.

Shew also that this line passes through the middle points of the
diagonals.
%-----File: 301.png-----

\item[59.] Four points $A$, $B$, $C$, $D$ are taken, no three of which lie in a
straight line, and joined in every possible way; and with another point
as focus four conics are described touching respectively the sides of the
triangles $BCD$, $CDA$, $DAB$, $ABC$; prove that the four conics have a
common tangent.

\item[60.] If the diagonals of a quadrilateral circumscribing a conic
intersect in a focus, they are at right angles to one another, and the
third diagonal is the corresponding directrix.

\item[61.] An ellipse and parabola have the same focus and directrix;
tangents are drawn to the ellipse at the extremities of the major axis;
shew that the diagonals of the quadrilateral formed by the four points
where these tangents cut the parabola intersect in the common focus,
and pass through the extremities of the minor axis of the ellipse.

\item[62.] Three chords of a circle pass through a point on the circumference;
with this point as focus and the chords as axes three parabolas
are described whose parameters are inversely proportional to the chords;
prove that the common tangents to the parabolas, taken two and two,
meet in a point.

\item[63.] A circle is described touching the asymptotes of an hyperbola
and having its centre at the focus. A tangent to this circle cuts the
directrix in $F$, and has its pole with regard to the hyperbola at $T$.
Prove that $TF$ touches the circle.

\item[64.] Two conics have a common focus: their corresponding directrices
will intersect on their common chord, at a point whose focal
distance is at right angles to that of the intersection of their common
tangents. Also the parts into which either common tangent is divided
by their common chord will subtend equal angles at the common focus.

If the conics are parabolas, the inclination of their axes will be the
angle subtended by the common tangent at the common focus.

\item[65.] The tangent at the point $P$ of an hyperbola meets the directrix
in $Q$; another point $R$ is taken on the directrix such that $QR$ subtends
at the focus an angle equal to that between the transverse axis and an
asymptote; prove that the envelope of $RP$ is a parabola.

\item[66.] If an hyperbola passes through the angular points of an
equilateral triangle and has the centre of the circumscribing circle as
focus, its eccentricity is the ratio of 4 to 3, and its latus rectum is one-third
of the diameter of the circle.

\item[67.] An isosceles triangle is circumscribed to a parabola; prove that
the three sides and the three chords of contact intersect the directrix in
five points, such that the distance between any two successive points
subtends the same angle at the focus.

\item[68.] Tangents are drawn at two points $P$, $P'$ on an ellipse. If any
tangent be drawn meeting those at P, P' in $R$, $R'$, shew that the line
bisecting the angle $RSR'$ intersects $RR'$ on a fixed tangent to the
ellipse.
%-----File: 302.png-----

\item[69.] The chords of a conic which subtend the same angle at the
focus all touch another conic having the same focus and directrix.

\item[70.] Two conics have a common focus $S$ and a common directrix,
and tangents $TP$, $TP'$ are drawn to one from any point on the other
and meet the directrix in $F$ and $F'$. Prove that the angles $PSF'$,
$P'SF$ are equal and constant.

\item[71.] A rectangular hyperbola circumscribes a triangle $ABC$; if
$D$, $E$, $F$ are the feet of the perpendiculars from $A$, $B$, $C$ on the opposite
sides, the loci of the poles of the sides of the triangle $ABC$ are the lines
$EF$, $FD$, $DE$.

\item[72.] If two of the sides of a triangle, inscribed in a conic, pass
through fixed points, the envelope of the third side is a conic.

\item[73.] If two circles be inscribed in a conic, and tangents be drawn
to the circles from any point in the conic, the sum or difference of these
tangents is constant, according as the point does or does not lie between
the two chords of contact.

\item[74.] The four common tangents of two conics intersect two and two
on the sides of the common self-conjugate triangle of the conics.

\item[75.] Prove that a right cylinder, upon a given elliptic base, can be
cut in two ways so that the curve of section may be a circle; and that
a sphere can always be drawn through any two circular sections of
opposite systems.

\item[76.] An ellipse revolves about its major axis, and planes are drawn
through a focus cutting the surface thus formed. Prove that the locus
of the centres of the different sections is a surface formed by the
revolution of an ellipse about $CS$ where $C$ or $S$ are respectively the
centre and focus of the original ellipse.

\item[77.] Given five tangents to a conic, find, by aid of Brianchon's
theorem, the points of contact.

\item[78.] The alternate angular points of any pentagon $ABCDE$ are
joined, thus forming another pentagon whose corresponding angular
points are $a$, $b$, $c$, $d$, $e$; $Aa$, $Bb$, $Cc$, $Dd$, $Ee$ are joined and produced to
meet the opposite sides of $ABCDE$ in $\alpha$, $\beta$, $\gamma$, $\epsilon$; shew that if $A$ be
joined with the middle point of $\gamma\delta$, $B$ with the middle point of $\delta\epsilon$, \&c.,
these five lines meet in a point.

\item[79.] If a conic be inscribed in a triangle, the lines joining the
angular points to the points of contact of the opposite sides are concurrent.

\item[80.] If a quadrilateral circumscribe a conic, the intersection of the
lines joining opposite points of contact is the same as the intersection
of the diagonals.
%-----File: 303.png-----

\item[81.] $ABC$ is a triangle, and $D$, $E$, $F$ the middle points of the sides.
Shew that any two similar and similarly situated ellipses one circumscribing
$DEF$ and the other inscribed in $ABC$ will touch each other.

\item[82.] $AB$ is a chord of a conic. The tangents at $A$ and $B$ meet in $T$.
Through $B$ a straight line is drawn meeting the conic in $C$ and $AT$ in
$P$. The tangent to the conic at $C$ meets $AT$ in $Q$. Prove that $TPQA$
is a harmonic range.

\item[83.] $Pp$, $Qq$, $Rr$, $Ss$ are four concurrent chords of a conic; shew that
a conic can be drawn touching $SR$, $RQ$, $QP$, $sr$, $rq$, $qp$.

\item[84.] If two sections of a right cone have a common directrix, the
latera recta are in the ratio of the eccentricities.

\item[85.] $ABCD$ is a parallelogram and a conic is described to touch its
four sides. If $S$ is a focus of this conic and if with $S$ as focus a parabola
is described to touch $AB$ and $BC$, the axis of the parabola passes
through $D$.

\item[86.] If from a point $O$ tangents be drawn to two conics $S$ and $S'$,
and if the tangents to $S$ be conjugate with respect to $S'$, prove that the
tangents to $S'$ are conjugate with respect to $S$.

\item[87.] If a triangle is self-conjugate with respect to each of a series
of parabolas, the lines joining the middle points of its sides will be
tangents; all the directrices will pass through $O$, the centre of the
circumscribing circle; and the focal chords, which are the polars of $O$,
will all touch an ellipse inscribed in the given triangle which has the
nine-point circle for its auxiliary circle.

\item[88.] If a triangle can be drawn so as to be inscribed in one given
conic and circumscribed about another given conic, an infinite number
of such triangles can be drawn.

\item[89.] Prove that the stereographic projection of a series of parallel
circles on a sphere is a series of coaxal circles, the limiting points of
which are the projections of the poles of the circles.

\item[90.] Through the six points of intersection of a conic with the sides
of a triangle straight lines are drawn to the opposite angular points; if
three of these lines are concurrent the other three are also concurrent.

\item[91.] Prove that the asymptotes of an hyperbola, and a pair of conjugate
diameters form an harmonic range, and that the system of pairs
of conjugate diameters is a pencil in involution.

\item[92.] If two concentric conics have the directions of two pairs of
conjugate diameters the same, then the directions are the same for
every pair.

\item[93.] If two concentric conics have all pairs of conjugate diameters
in the same directions, and have a common point, they coincide entirely.
%-----File: 304.png-----

\item[94.] If two conics have two common self-conjugate triangles with
the same vertex, which is interior to both, they cannot intersect in any
point without entirely coinciding.

\item[95.] If two conics in space whose planes intersect in a line which
does not cut either conic, and if on this line there are four points,
$P$, $P'$, $Q$, $Q'$, such that the polars of $P$ with regard to the conics both
pass through $P'$, and that the polars of $Q$ both pass through $Q'$, then
either conic can be projected into the other in two ways.
\end{exampleslist}

\vfill
\begin{center}
\rule{\textwidth}{1pt}
\footnotesize CAMBRIDGE: PRINTED BY J.~AND C.~F. CLAY, AT THE UNIVERSITY PRESS.
\end{center}

%-----File: 305.png-----
%%%
%%% Start of George Bell & Sons' Classified Catalogue of Educational Works
%%%

\cleardoublepage

\cheader{\textit{George Bell \& Sons' Educational Catalogue}}
\pdfbookmark[0]{George Bell \& Sons' Educational Catalogue}{George Bell & Sons' Educational Catalogue}

\setcounter{page}{1} % restart page numbering for the catalogue

\thispagestyle{empty}

\begin{center}

{\large A}

\vspace{1cm}

{\TitleHuge\CatStretch CLASSIFIED CATALOGUE}

\vspace{1cm}

OF

\vspace{1cm}

{\TitleHuge\CatStretch EDUCATIONAL WORKS}

\vspace{1cm}

PUBLISHED BY

\vspace{1cm}

{\TitleHuge\CatStretch GEORGE BELL \& SONS}

\vspace{\stretch{1}}

\includegraphicsmid{catlogo}

\vspace{\stretch{1}}

{\large LONDON: YORK STREET, COVENT GARDEN}

NEW YORK: 66, FIFTH AVENUE; AND BOMBAY

CAMBRIDGE: DEIGHTON, BELL \& CO.

\medskip

\textsc{October}, 1894
\end{center}
%-----File: 306.png-----

\clearpage
\tableofcontentscat

\addtocontents{\tccext}{%
  \protect\begin{flushright}
  \vspace{3ex}
  \textsc{page}
  \vspace{-3ex} %First chapter heading doesn't need extra spacing
  \protect\end{flushright}}

%-----File: 307.png-----
\clearpage

\cchaptercat{GREEK AND LATIN CLASSICS}

\catsectionuppertoc{Annotated And Critical Editions}

\begin{maincatlist}

\mitemspace[AESCHYLUS\@.] Edited by {\smallcapsize F.~A. PALEY, M.A., LL.D}, late Classical Examiner
    to the University of London. 4\textit{th edition, revised}. 8vo, 8\s.
\catright{Bib.\ Class.}

\mitem[---] Edited by {\smallcapsize F.~A. PALEY, M.A., LL.D.}, 6 vols.\ fcap.\ 8vo, 1\s.~6\d.
\catright{Camb.\ Texts with Notes.}

\textbf{%
\begin{tabulartwocol}
Agamemnon.     &  Persae.                \\
Choephoroe.    &  Prometheus Vinctus.    \\
Eumenides.     &  Septem contra Thebas.\\
\end{tabulartwocol}%
}

\mitemspace[ARISTOPHANIS\@.] Comoediae quae supersunt cum perditarum fragmentis
    tertiis curis, recognovit additis adnotatione critica, summariis, descriptione
    metrica, onomastico lexico {\smallcapsize HUBERTUS A.~HOLDEN, LL.D.} [late Fellow of
    Trinity College, Cambridge]. Demy 8vo.

Vol.~I., containing the Text expurgated, with Summaries and Critical
Notes, 18\s.

The Plays sold separately:

\textbf{%
\begin{tabulartwocol}
Acharnenses, 2\s.   &         Aves, 2\s.\\
Equites, 1\s.~6\d.  &         Lysistrata, et\\
Nubes, 2\s.         &         \quad Thesmophoriazusae, 4\s.\\
Vespae, 2\s.        &         Ranae, 2\s.\\
Pax, 2\s.           &         Plutus, 2\s.\\
\end{tabulartwocol}%
}

Vol.~II\@. Onomasticon Aristophaneum continens indicem geographicum
et historicum 5\s.~6\d.

\mitemspace[--- The Peace.] A revised Text with English Notes and a Preface.
By {\smallcapsize F.~A. PALEY, M.A., LL.D\@.} Post 8vo, 4\s.~6\d.
\catright{Pub.\ Sch.\ Ser.}

\mitemspace[--- The Acharnians.] A revised Text with English Notes and a Preface.
By {\smallcapsize F.~A. PALEY, M.A., LL.D\@.} Post 8vo, 4\s.~6\d.
\catright{Pub.\ Sch.\ Ser.}

\mitemspace[--- The Frogs.] A revised Text with English Notes and a Preface.
By {\smallcapsize F.~A. PALEY, M.A., LL.D\@.} Post 8vo, 4\s.~6\d.
\catright{Pub.\ Sch.\ Ser.}

\mitemspace[CAESAR De Bello Gallico.] Edited by {\smallcapsize GEORGE LONG, M.A\@.}
\textit{New edition.} Fcap.\ 8vo, 4\s.

Or in parts, Books I.--III., 1\s.~6\d.; Books IV. and V., 1\s.~6\d.; Books
VI. and VII., 1\s.~6\d.
\catright{Gram.\ Sch.\ Class.}

\mitemspace[--- De Bello Gallico.] Book I\@. Edited by {\smallcapsize GEORGE LONG, M.A\@.} With
Vocabulary by {\smallcapsize W.~F.~R. SHILLETO, M.A\@.} 1\s.~6\d.
\catright{Lower Form Ser.}

\mitemspace[--- De Bello Gallico.] Book II\@. Edited by {\smallcapsize GEORGE LONG, M.A\@.} With
Vocabulary by {\smallcapsize W.~F.~R. SHILLETO, M.A\@.} Fcap.\ 8vo, 1\s.~6\d.
\catright{Lower Form Ser.}

\mitemspace[--- De Bello Gallico.] Book III\@. Edited by {\smallcapsize GEORGE LONG, M.A\@.} With
Vocabulary by {\smallcapsize W.~F.~R. SHILLETO, M.A\@.} Fcap.\ 8vo. 1\s.~6\d.
\catright{Lower Form Ser.}

\mitemspace[--- Seventh Campaign in Gaul.] {\smallcapsize B.C.} 52. De Bello Gallico, Lib.~VII\@.
Edited with Notes, Excursus, and Table of Idioms, by {\smallcapsize REV.~W. COOKWORTHY
COMPTON, M.A.}, Head Master of Dover College. With
Illustrations from Sketches by {\smallcapsize E.~T. COMPTON}, Maps and Plans.
2\textit{nd edition.} Crown 8vo, 2\s.~6\d.\ net.

``A really admirable class book.''---\textit{Spectator.}

``One of the most original and interesting books which have been
published in late years as aids to the study of classical literature. I think
%-----File: 308.png-----
it gives the student a new idea of the way in which a classical book may
be made a living reality.''---\textit{Rev.~J.~E.~C. Welldon}, Harrow.

\mitemspace[--- Easy Selections from the Helvetian War.] Edited by {\smallcapsize A.~M.~M. STEDMAN, M.A\@.}
With Introduction, Notes and Vocabulary. 18mo.  1\s.
\catright{Primary Classics.}

\mitemspace[CALPURNIUS SICULUS and M. AURELIUS OLYMPIUS NE\-MES\-IA\-NUS\@.]
The Eclogues, with Introduction, Commentary, and Appendix.
By {\smallcapsize C.~H. KEENE, M.A\@.} Crown 8vo, 6\s.

\mitemspace[CATULLUS, TIBULLUS, and PROPERTIUS\@.] Selected Poems.
Edited by the {\smallcapsize REV. A.~H. WRATISLAW}, late Head Master of Bury
St.~Edmunds School, and {\smallcapsize F.~N. SUTTON, B.A\@.}
With Biographical Notices of the Poets. Fcap.\ 8vo, 2\s.~6\d.
\catright{Gram.\ Sch.\ Class.}

\mitemspace[CICERO'S Orations.] Edited by {\smallcapsize G.~LONG, M.A\@.} 8vo.
\catright{Bib.\ Class.}

Vol.~I\@. --- In Verrem. 8\s.

Vol.~II\@. --- Pro P. Quintio --- Pro Sex.\ Roscio --- Pro Q. Roscio --- Pro M. Tullio
--- Pro M. Fonteio --- Pro A. Caecina --- De Imperio Cn.\ Pompeii --- Pro A. Cluentio
--- De Lege Agraria --- Pro C. Rabirio. 8\s.

Vols.\ III. and IV\@. \textit{Out of print}.

\mitemspace[--- De Senectute, De Amicitia, and Select Epistles.] Edited by {\smallcapsize GEORGE LONG, M.A\@.}
\textit{New edition}. Fcap.\ 8vo, 3\s.
\catright{Gram.\ Sch.\ Class.}

\mitemspace[--- De Amicitia.] Edited by {\smallcapsize GEORGE LONG, M.A\@.} Fcap.\ 8vo, 1\s.~6\d.
\catright{Camb.\ Texts with Notes.}

\mitemspace[--- De Senectute.] Edited by {\smallcapsize GEORGE LONG, M.A\@.} Fcap.\ 8vo, 1\s.~6\d.
\catright{Camb.\ Texts with Notes.}

\mitemspace[--- Epistolae Selectae.] Edited by {\smallcapsize GEORGE LONG, M.A\@.} Fcap.\ 8vo, 1\s.~6\d.
\catright{Camb.\ Texts with Notes.}

\mitemspace[--- The Letters to Atticus. Book I\@.] With Notes, and an Essay on the
Character of the Writer. By {\smallcapsize A.~PRETOR, M.A.}, late of Trinity College,
Fellow of St.~Catherine's College, Cambridge. 3\textit{rd edition}. Post 8vo, 4\s.~6\d.
\catright{Pub.\ Sch.\ Ser.}

\mitemspace[CORNELIUS NEPOS\@.] Edited by the late {\smallcapsize REV. J.~F. MACMICHAEL},
Head Master of the Grammar School, Ripon. Fcap.\ 8vo, 2\s.
\catright{Gram.\ Sch.\ Class.}

\mitemspace[DEMOSTHENES\@.] Edited by {\smallcapsize R.~WHISTON, M.A.}, late Head Master of
Ro\-ches\-ter Grammar School. 2 vols. 8vo, 8\s.\ each.
\catright{Bib.\ Class.}\\
\indent Vol.~I\@. --- Olynthiacs --- Philippics --- De Pace --- Halonnesus --- Chersonese ---
Letter of Philip --- Duties of the
State --- Symmoriae --- Rhodians --- Megalopolitans --- Treaty with Alexander --- Crown.

Vol.~II\@. --- Embassy --- Leptines --- Meidias --- Androtion --- Aristocrates ---
Timocrates --- Aristogeiton.

\mitemspace[--- De Falsa Legatione.] By the late {\smallcapsize R.~SHILLETO, M.A.}, Fellow of St.\ Peter's
College, Cambridge. 7\textit{th edition}. Post 8vo, 6\s.
\catright{Pub.\ Sch.\ Ser.}

\mitemspace[--- The Oration against the Law of Leptines.] With English Notes.
By the late {\smallcapsize B.~W. BEATSON, M.A.}, Fellow of Pembroke College. 3\textit{rd
edition}. Post 8vo, 3\s.~6\d.
\catright{Pub.\ Sch.\ Ser.}

\mitemspace[EURIPIDES\@.] By {\smallcapsize F.~A. PALEY, M.A., LL.D\@.} 3 vols. 2\textit{nd edition, revised}.
8vo, 8\s.\ each. Vol.~I\@. \textit{Out of print}.
\catright{Bib.\ Class.}

Vol.~II\@. --- Preface --- Ion --- Helena --- Andromache --- Electra --- Bacchae --- Hecuba.
2 Indexes.

Vol.~III\@. --- Preface --- Hercules Furens --- Phoenissae --- Orestes --- Iphigenia in
Tauris --- Iphigenia in Aulide --- Cyclops. 2 Indexes.
%-----File: 309.png-----

\mitemspace[EURIPIDES\@. Electra.] Edited, with Introduction and Notes, by {\smallcapsize C.~H.
KEENE, M.A.}, Dublin, Ex-Scholar and Gold Medallist in Classics. Demy
8vo, 10\s.~6\d.

\mitem[---] Edited by {\smallcapsize F.~A. PALEY, M.A., LL.D\@.} 13 vols. Fcap.\ 8vo, 1\s.~6\d.\ each.
\catright{Camb.\ Texts with Notes.}

\textbf{%
  \begin{tabulartwocol}
    Alcestis.   & Phoenissae.\\
    Medea.      & Troades.\\
    Hippolytus. & Hercules Furens.\\
    Hecuba.     & Andromache.\\
    Bacchae.    & Iphigenia in Tauris.\\
    Ion (2\s.). & Supplices.\\
    Orestes.
  \end{tabulartwocol}}

\mitemspace[HERODOTUS\@.] Edited by {\smallcapsize REV. J.~W. BLAKESLEY, B.D\@.} 2 vols. 8vo, 12\s.
\catright{Bib.\ Class.}

\mitemspace[--- Easy Selections from the Persian Wars.] Edited by {\smallcapsize A.~G. LIDDELL,
M.A\@.} With Introduction, Notes, and Vocabulary. 18mo, 1\s.~6\d.
\catright{Primary Classics.}

\mitemspace[HESIOD\@.] Edited by {\smallcapsize F.~A. PALEY, M.A., LL.D\@.} 2\textit{nd edition, revised.} 8vo, 5\s.
\catright{Bib.\ Class.}

\mitemspace[HOMER\@.] Edited by {\smallcapsize F.~A. PALEY, M.A., LL.D\@.} 2 vols.
2\textit{nd edition, revised.} 14\s. Vol.~II. (Books 13--24) may be had separately, 6\s.
\catright{Bib.\ Class.}

\mitemspace[--- Iliad.] Books I.--XII\@. Edited by {\smallcapsize F.~A. PALEY, M.A., LL.D\@.}
Fcap.\ 8vo, 4\s.~6\d.

Also in 2 Parts. Books I.--VI\@. 2\s.~6\d. Books VII.--XII\@. 2\s.~6\d.
\catright{Gram.\ Sch.\ Class.}

\mitemspace[--- Iliad.] Book I\@. Edited by {\smallcapsize F.~A. PALEY, M.A., LL.D\@.} Fcap.\ 8vo, 1\s.
\catright{Camb.\ Text with Notes.}

\mitemspace[HORACE\@.] Edited by {\smallcapsize REV. A.~J. MACLEANE, M.A\@.} 4\textit{th edition}, revised by
{\smallcapsize GEORGE LONG\@.}  8vo, 8\s.
\catright{Bib.\ Class.}

\mitem[---] Edited by {\smallcapsize A.~J. MACLEANE, M.A\@.} With a short Life. Fcap.\ 8vo, 3\s.~6\d.

Or, Part I., Odes, Carmen Seculare, and Epodes, 2\s.; Part II., Satires,
Epistles, and Art of Poetry, 2\s.
\catright{Gram.\ Sch.\ Class.}

\mitemspace[--- Odes.] Book I\@. Edited by {\smallcapsize A.~J. MACLEANE, M.A\@.} With a Vocabulary
by {\smallcapsize A.~H. DENNIS, M.A\@.} Fcap.\ 8vo, 1\s.~6\d.
\catright{Lower~Form~Ser.}

\mitem[JUVENAL: Sixteen Satires] (expurgated). By {\smallcapsize HERMAN PRIOR, M.A.},
late Scholar of Trinity College, Oxford. Fcap.\ 8vo, 3\s.~6\d.
\catright{Gram.\ Sch.\ Class.}

\mitemspace[LIVY\@.] The first five Books, with English Notes. By {\smallcapsize J.~PRENDEVILLE\@.}
A new edition revised throughout, and the notes in great part re-written,
by {\smallcapsize J.~H. FREESE, M.A.}, late Fellow of St.~John's College, Cambridge.
Books I. II. III. IV. V\@. With Maps and Introductions. Fcap.\ 8vo.
1\s.~6\d.\ each.

\mitemspace[--- Book VI\@.] Edited by {\smallcapsize E.~S. WEYMOUTH, M.A.}, Lond., and
{\smallcapsize G.~F. HAMILTON, B.A\@.}
With Historical Introduction, Life of Livy, Notes, Examination
Questions, Dictionary of Proper Names, and Map. Crown 8vo, 2\s.~6\d.

\mitemspace[--- Book XXI\@.] By the {\smallcapsize REV. L.~D. DOWDALL, M.A.}, late Scholar and University
Student of Trinity College, Dublin, B.D., Ch.\ Ch.\ Oxon. Post
8vo, 3\s.~6\d.
\catright{Pub.\ Sch.\ Ser.}

\mitemspace[--- Book XXII\@.] Edited by the {\smallcapsize REV. L.~D. DOWDALL, M.A., B.D\@.} Post 8vo,
3\s.~6\d.
\catright{Pub.\ Sch.\ Ser.}
%-----File: 310.png-----

\mitemspace[LIVY\@. Easy Selections from the Kings of Rome.] Edited by {\smallcapsize A.~M.~M.
STEDMAN, M.A\@.} With Introduction, Notes, and Vocabulary. 18mo,
1\s.~6\d.
\catright{Primary Class.}

\mitemspace[LUCAN\@. The Pharsalia.] By {\smallcapsize C.~E. HASKINS, M.A.}, Fellow of St.\ John's
College, Cambridge, with an Introduction by {\smallcapsize W.~E. HEITLAND, M.A.},
Fellow and Tutor of St.\ John's College, Cambridge. 8vo, 14\s.

\mitemspace[LUCRETIUS\@. Titi Lucreti Cari De Rerum Natura Libri Sex.] By
the late {\smallcapsize H.~A.~J. MUNRO, M.A.}, Fellow of Trinity College, Cambridge.
4\textit{th edition, finally revised.} 3 vols, demy 8vo. Vols.~I., II., Introduction,
Text, and Notes, 18\s. Vol.~III., Translation, 6\s.

\mitemspace[MARTIAL: Select Epigrams.]  Edited by {\smallcapsize F.~A. PALEY, M.A., LL.D.},
and the late {\smallcapsize W.~H. STONE}, Scholar of Trinity College, Cambridge. With
a Life of the Poet. Fcap.\ 8vo, 4\s.~6\d.
\catright{Gram.\ Sch.\ Class.}

\mitemspace[OVID: Fasti.] Edited by {\smallcapsize F.~A. PALEY, M.A., LL.D\@.} \textit{Second edition.}
Fcap.\ 8vo, 3\s.~6\d.
\catright{Gram.\ Sch.\ Class.}

Or in 3 vols, 1\s.~6\d.\ each [\textit{Grammar School Classics}], or
2\s.\ each [\textit{Camb.\
Texts with Notes}], Books I. and II., Books III. and IV., Books V.
and VI.

\mitemspace[--- Selections from the Amores, Tristia, Heroides, and Metamorphoses.]
By {\smallcapsize A.~J. MACLEANE, M.A\@.} Fcap.\ 8vo, 1\s.~6\d.
\catright{Camb.\ Texts with Notes.}

\mitemspace[--- Ars Amatoria et Amores.] A School Edition. Carefully Revised and
Edited, with some Literary Notes, by {\smallcapsize J.~HERBERT WILLIAMS, M.A.},
late Demy of Magdalen College, Oxford. Fcap.\ 8vo, 3\s.~6\d.

\mitemspace[--- Heroides XIV\@.] Edited, with Introductory Preface and English Notes,
by {\smallcapsize ARTHUR PALMER, M.A.}, Professor of Latin at Trinity College, Dublin.
Demy 8vo, 6\s.

\mitem[--- Metamorphoses,] Book XIII\@.  A School Edition. With Introduction
and Notes, by {\smallcapsize CHARLES HAINES KEENE, M.A.}, Dublin, Ex-Scholar and
Gold Medallist in Classics. 3\textit{rd edition.} Fcap.\ 8vo, 2\s.~6\d.

\mitemspace[--- Epistolarum ex Ponto Liber Primus.] With Introduction and Notes,
by {\smallcapsize CHARLES HAINES KEENE, M.A\@.} Crown 8vo, 3\s.

\mitemspace[PLATO\@. The Apology of Socrates and Crito.] With Notes, critical
and exegetical, by {\smallcapsize WILHELM WAGNER, PH.D\@.} 12\textit{th edition.} Post 8vo,
3\s.~6\d. \textsc{A Cheap Edition.} Limp Cloth. 2\s.~6\d.
\catright{Pub.\ Sch.\ Ser.}

\mitemspace[--- Phaedo.]  With Notes, critical and exegetical, and an Analysis, by
{\smallcapsize WILHELM WAGNER, PH.D\@.} 9\textit{th edition.} Post 8vo, 5\s.~6\d.
\catright{Pub.\ Sch.\ Ser.}

\mitemspace[--- Protagoras.]  The Greek Text revised, with an Analysis and English
Notes, by {\smallcapsize W.~WAYTE, M.A.}, Classical Examiner at University College,
London. 7\textit{th edition.} Post 8vo, 4\s.~6\d.
\catright{Pub.\ Sch.\ Ser.}

\mitemspace[--- Euthyphro.]  With Notes and Introduction by {\smallcapsize G.~H. WELLS, M.A.},
Scholar of St.\ John's College, Oxford; Assistant Master at Merchant
Taylors' School. 3\textit{rd edition.} Post 8vo, 3\s.
\catright{Pub.\ Sch.\ Ser.}

\mitemspace[--- The Republic.]  Books I. and II\@.  With Notes and Introduction by
{\smallcapsize G.~H. WELLS, M.A\@.} 4\textit{th edition}, with the Introduction re-written. Post
8vo, 5\s.
\catright{Pub.\ Sch.\ Ser.}

\mitemspace[--- Euthydemus.] With Notes and Introduction by {\smallcapsize G.~H. WELLS, M.A\@.}
Post 8vo, 4\s.
\catright{Pub.\ Sch.\ Ser.}

\mitemspace[--- Phaedrus.] By the late {\smallcapsize W.~H. THOMPSON, D.D.}, Master of Trinity College,
Cambridge. 8vo, 5\s.
\catright{Bib.\ Class.}

\mitemspace[--- Gorgias.] By the late {\smallcapsize W.~H. THOMPSON, D.D.}, Master of Trinity College,
Cambridge. \textit{New edition.} 6\s.
\catright{Pub.\ Sch.\ Ser.}
%-----File: 311.png-----

\mitemspace[PLAUTUS\@. Aulularia.] With Notes, critical and exegetical, by {\smallcapsize W.~WAGNER,
PH.D\@.} 5\textit{th edition.} Post 8vo, 4\s.~6\d.
\catright{Pub.\ Sch.\ Ser.}

\mitemspace[--- Trinummus.] With Notes, critical and exegetical, by {\smallcapsize WILHELM
WAGNER, PH.D\@.} 5\textit{th edition.} Post 8vo, 4\s.~6\d.
\catright{Pub.\ Sch.\ Ser.}

\mitemspace[--- Menaechmei.] With Notes, critical and exegetical, by {\smallcapsize WILHELM
WAGNER, PH.D\@.} 2\textit{nd edition.} Post 8vo, 4\s.~6\d.
\catright{Pub.\ Sch.\ Ser.}

\mitemspace[--- Mostellaria.] By {\smallcapsize E.~A. SONNENSCHEIN, M.A.}, Professor of Classics at
Mason College, Birmingham. Post 8vo, 5\s.
\catright{Pub.\ Sch.\ Ser.}

\mitemspace[--- Captivi.] Abridged and Edited for the Use of Schools. With Introduction
and Notes by {\smallcapsize J.~H. FREESE, M.A.}, formerly Fellow of St.\ John's
College, Cambridge. Fcap.\ 8vo, 1\s.~6\d.

\mitemspace[PROPERTIUS\@. Sex.\ Aurelii Propertii Carmina.] The Elegies of
Propertius, with English Notes.  By {\smallcapsize F.~A. PALEY, M.A., LL.D\@.} 2\textit{nd
edition.} 8vo, 5\s.

\mitemspace[SALLUST: Catilina and Jugurtha.] Edited, with Notes, by the late
{\smallcapsize GEORGE LONG\@.}  \textit{New edition, revised}, with the addition of the Chief
Fragments of the Histories, by {\smallcapsize J.~G. FRAZER, M.A.}, Fellow of Trinity
College, Cambridge. Fcap.\ 8vo, 3\s.~6\d, or separately, 2\s.\ each.
\catright{Gram.\ Sch.\ Class.}

\mitemspace[SOPHOCLES\@.] Edited by {\smallcapsize REV. F.~H. BLAYDES, M.A\@.} Vol.~I, Oedipus
Tyrannus --- Oedipus Coloneus --- Antigone. 8vo, 8\s.
\catright{Bib.\ Class.}

Vol.~II\@. Philoctetes --- Electra --- Trachiniae --- Ajax.  By {\smallcapsize F.~A. PALEY,
M.A., LL.D\@.} 8vo, 6\s., or the four Plays separately in limp cloth, 2\s.~6\d.\ each.

\mitemspace[--- Trachiniae.] With Notes and Prolegomena. By {\smallcapsize ALFRED PRETOR, M.A.},
Fellow of St.\ Catherine's College, Cambridge. Post 8vo, 4\s.~6\d.
\catright{Pub.\ Sch.\ Ser.}

\mitemspace[--- The Oedipus Tyrannus of Sophocles.] By {\smallcapsize B.~H. KENNEDY, D.D.},
Regius Professor of Greek and Hon.\ Fellow of St.\ John's College, Cambridge.
With a Commentary containing a large number of Notes selected
from the MS. of the late {\smallcapsize T.~H. STEEL, M.A\@.} Crown 8vo, 8\s.

\mitem[---] \textbf{---} \textsc{A School Edition}, post 8vo, 5\s.
\catright{Pub.\ Sch.\ Ser.}

\mitem[---] Edited by {\smallcapsize F.~A. PALEY, M.A., LL.D\@.} 5 vols. Fcap.\ 8vo, 1\s.~6\d.\ each.
\catright{Camb.\ Texts with Notes.}

\textbf{%
  \begin{tabulartwocol}
    Oedipus Tyrannus.  &    Electra.    \\
    Oedipus Coloneus.  &    Ajax.       \\
    Antigone.          &
  \end{tabulartwocol}}

\mitemspace[TACITUS: Germania and Agricola.] Edited by the late {\smallcapsize REV. P.~FROST},
late Fellow of St.\ John's College, Cambridge. Fcap.\ 8vo, 2\s.~6\d.
\catright{Gram.\ Sch.\ Class.}

\mitemspace[--- The Germania.] Edited, with Introduction and Notes, by {\smallcapsize R.~R. DAVIS,
M.A\@.} Fcap.\ 8vo, 1\s.~6\d.

\mitemspace[TERENCE\@.] With Notes, critical and explanatory, by {\smallcapsize WILHELM WAGNER,
PH.D\@.} 3\textit{rd edition.} Post 8vo, 7\s.~6\d.
\catright{Pub.\ Sch.\ Ser.}

\mitem[---] Edited by {\smallcapsize WILHELM WAGNER, PH.D\@.} 4 vols. Fcap.\ 8vo, 1\s.~6\d.\ each.
\catright{Camb.~Texts with Notes.}

\textbf{%
  \begin{tabulartwocol}
    Andria.   &    Hautontimorumenos. \\
    Adelphi.  &    Phormio.           \\
  \end{tabulartwocol}}

\mitemspace[THEOCRITUS\@.] With short, critical and explanatory Latin Notes, by
{\smallcapsize F.~A. PALEY, M.A., LL.D\@.} 2\textit{nd edition, revised.} Post 8vo, 4\s.~6\d.
\catright{Pub.\ Sch.\ Ser.}
%-----File: 312.png-----

\mitem[THUCYDIDES,] Book VI\@. By {\smallcapsize T.~W. DOUGAN, M.A.}, Fellow of St.\ John's
College, Cambridge; Professor of Latin in Queen's College, Belfast.
Edited with English notes. Post 8vo, 3\s.~6\d.  \catright{Pub.\ Sch.\ Ser.}

\mitemspace[--- The History of the Peloponnesian War.] With Notes and a careful
Collation of the two Cambridge Manuscripts, and of the Aldine and
Juntine Editions. By the late {\smallcapsize RICHARD SHILLETO, M.A.}, Fellow of
St.\ Peter's College, Cambridge. 8vo. Book~I. 6\s.~6\d. Book~II. 5\s.~6\d.

\mitemspace[VIRGIL\@.] By the late {\smallcapsize PROFESSOR CONINGTON, M.A\@.} Revised by the late
{\smallcapsize PROFESSOR NETTLESHIP}, Corpus Professor of Latin at Oxford. 8vo.
\catright{Bib.\ Class.}

Vol.~I\@. The Bucolics and Georgics, with new Memoir and three Essays
on Virgil's Commentators, Text, and Critics. 4\textit{th edition}. 10\s.~6\d.

Vol.~II\@. The Aeneid, Books I.--VI\@. 4\textit{th edition}. 10\s.~6\d.

Vol.~III\@. The Aeneid, Books VII.--XII\@. 3\textit{rd edition}. 10\s.~6\d.

\mitem[---] Abridged from {\smallcapsize PROFESSOR CONINGTON'S} Edition,
by the {\smallcapsize REV. J.~G. SHEPPARD, D.C.L.,
H.~NETTLESHIP}, late Corpus Professor of Latin at the
University of Oxford, and {\smallcapsize W.~WAGNER, PH.D\@.} 2~vols.\ fcap.\ 8vo,
4\s.~6\d.\ each.
\catright{Gram.\ Sch.\ Class.}

Vol.~I\@. Bucolics, Georgics, and Aeneid, Books I.--IV.

Vol.~II\@. Aeneid, Books V.--XII.

Also the Bucolics and Georgics, in one vol.\ 3\s.

\begin{samepage}
\begin{center}
\textit{Or in} 9 \textit{separate volumes} (\textit{Grammar School Classics, with Notes at foot of page}),
\textit{price} 1\s.~6\d.\ \textit{each}.
\end{center}

\begin{tabulartwocol}
  \textbf{Bucolics}.                & \textbf{Aeneid}, V. and VI. \\
  \textbf{Georgics}, I. and II.   & \textbf{Aeneid}, VII. and VIII. \\
  \textbf{Georgics}, III. and IV. & \textbf{Aeneid}, IX. and X. \\
  \textbf{Aeneid}, I. and II.     & \textbf{Aeneid}, XI. and XII. \\
  \textbf{Aeneid}, III. and IV.
\end{tabulartwocol}
\end{samepage}

\begin{center}
\textit{Or in} 12 \textit{separate volumes} (\textit{Cambridge Texts with Notes at end}), \textit{price}
1\s.~6\d.\ \textit{each}.
\end{center}

\begin{tabulartwocol}
  \textbf{Bucolics}.                & \textbf{Aeneid}, VII.  \\
  \textbf{Georgics}, I. and II.   & \textbf{Aeneid}, VIII.  \\
  \textbf{Georgics}, III. and IV. & \textbf{Aeneid}, IX.  \\
  \textbf{Aeneid}, I. and II.     & \textbf{Aeneid}, X.  \\
  \textbf{Aeneid}, III. and IV.   & \textbf{Aeneid}, XI.  \\
  \textbf{Aeneid}, V. and VI. (price 2\s.) & \textbf{Aeneid}, XII.
\end{tabulartwocol}

\mitem[--- Aeneid,] Book I\@.  {\smallcapsize CONINGTON'S} Edition abridged. With Vocabulary
by {\smallcapsize W.~F.~R. SHILLETO, M.A\@.} Fcap.\ 8vo, 1\s.~6\d.   \catright{Lower Form Ser.}

\mitemspace[XENOPHON: Anabasis.] With Life, Itinerary, Index, and three Maps.
Edited by the late {\smallcapsize J.~F. MACMICHAEL\@.} \textit{Revised edition.} Fcap.\ 8vo,
3\s.~6\d.
\catright{Gram.\ Sch.\ Class.}
\begin{center}
{\textit{Or in} 4 \textit{separate volumes, price} 1\s.~6\d. \textit{each}.}
\end{center}

Book I. (with Life, Introduction, Itinerary, and three Maps) --- Books
II\@. and III\@. --- Books IV\@. and V\@. --- Books VI\@. and VII.

\mitemspace[--- Anabasis.] {\smallcapsize MACMICHAEL'S} Edition, revised by {\smallcapsize J.~E. MELHUISH, M.A.},
Assistant Master of St.\ Paul's School. In 6 volumes, fcap.\ 8vo. With
Life, Itinerary, and Map to each volume, 1\s.~6\d.\ each.
\catright{Camb.\ Texts with Notes.}

Book I. --- Books II. and III. --- Book IV. --- Book V. --- Book VI. --- Book VII.
%-----File: 313.png-----

\mitemspace[XENOPHON\@. Cyropaedia.] Edited by {\smallcapsize G.~M. GORHAM, M.A.}, late Fellow
of Trinity College, Cambridge. \textit{New edition.} Fcap.\ 8vo, 3\s.~6\d.
\catright{Gram.\ Sch.\ Class.}

Also Books I. and II., 1\s.~6\d.; Books V. and VI., 1\s.~6\d.

\mitemspace[--- Memorabilia.] Edited by {\smallcapsize PERCIVAL FROST, M.A.}, late Fellow of St.\
John's College, Cambridge. Fcap.\ 8vo, 3\s.
\catright{Gram.\ Sch.\ Class.}

\mitemspace[--- Hellenica.] Book I\@. Edited by {\smallcapsize L.~D. DOWDALL, M.A., B.D\@.} Fcap.\ 8vo,
2\s.
\catright{Camb.\ Texts with Notes.}

\mitemspace[--- Hellenica.] Book II\@. By {\smallcapsize L.~D. DOWDALL, M.A., B.D\@.} Fcap.\ 8vo, 2\s.
\catright{Camb.\ Texts with Notes.}

\end{maincatlist}

\catsectionuppertoc{Texts}

\begin{maincatlist}

\mitemspace[AESCHYLUS\@.] Ex novissima recensione {\smallcapsize F.~A. PALEY, A.M., LL.D\@.} Fcap.\ 8vo, 2\s.
\catright{Camb.~Texts.}

\mitemspace[CAESAR De Bello Gallico.] Recognovit {\smallcapsize G.~LONG, A.M\@.} Fcap.\ 8vo,
1\s.~6\d.
\catright{Camb.~Texts.}

\mitemspace[CATULLUS\@.] A New Text, with Critical Notes and an Introduction, by
{\smallcapsize J.~P. POSTGATE, M.A., LITT.D.}, Fellow of Trinity College, Cambridge,
Professor of Comparative Philology at the University of London. Wide
fcap.\ 8vo, 3\s.

\mitemspace[CICERO De Senectute et de Amicitia, et Epistolae Selectae.] Recensuit
{\smallcapsize G.~LONG, A.M\@.} Fcap.\ 8vo, 1\s.~6\d.\catright{Camb. Texts.}

\mitemspace[CICERONIS Orationes in Verrem.] Ex recensione {\smallcapsize G.~LONG, A.M\@.}
Fcap.\ 8vo, 2\s.~6\d.
\catright{Camb.~Texts.}

\mitem[CORPUS POETARUM LATINORUM,] a se aliisque denuo recognitorum
et brevi lectionum varietate instructorum, edidit {\smallcapsize JOHANNES PERCIVAL
POSTGATE\@.} Fasc.~I. quo continentur Ennius, Lucretius, Catullus,
Horatius, Vergil\-ius, Tibullus. Large post 4to, 9\s.\ net.

\threestars To be completed in 4 parts, making 2 volumes. Part II. will be
ready shortly.

\mitemspace[CORPUS POETARUM LATINORUM\@.] Edited by {\smallcapsize WALKER\@.} Con\-tain\-ing:---
Catullus, Lucretius, Virgilius, Tibullus, Propertius, Ovidius,
Horatius, Phaedrus, Lucanus, Persius, Juvenalis, Martialis, Sulpicia,
Statius, Silius Italicus, Valerius Flaccus, Calpurnius Siculus, Ausonius,
and Claudianus. 1 vol. 8vo, cloth, 18\s.

\mitemspace[EURIPIDES\@.] Ex recensione {\smallcapsize F.~A. PALEY, A.M., LL.D\@.} 3 vols. Fcap.\ 8vo, 2\s.\ each.
\catright{Camb.~Texts.}

Vol.~I\@. --- Rhesus --- Medea --- Hippolytus --- Alcestis --- Heraclidae --- Supplices --- Troades.

Vol.~II\@. --- Ion --- Helena --- Andromache --- Electra --- Bacchae --- Hecuba.

Vol.~III\@. --- Hercules Furens --- Phoenissae --- Orestes --- Iphigenia in
Tauris --- Iphigenia in Aulide --- Cyclops.

\mitemspace[HERODOTUS\@.] Recensuit {\smallcapsize J.~G. BLAKESLEY, S.T.B\@.} 2 vols. Fcap.\ 8vo,
2\s.~6\d.\ each.\catright{Camb.~Texts.}

\mitem[HOMERI ILIAS] I.--XII\@. Ex novissima recensione {\smallcapsize F.~A. PASLEY, A.M.,
LL.D\@.} Fcap.\ 8vo, 1\s.~6\d.\catright{Camb.~Texts.}

\mitemspace[HORATIUS\@.] Ex recensione {\smallcapsize A.~J. MACLEANE, A.M\@.} Fcap.\ 8vo, 1\s.~6\d.
\catright{Camb.~Texts.}

\mitemspace[JUVENAL ET PERSIUS\@.] Ex recensione {\smallcapsize A.~J. MACLEANE, A.M\@.}
Fcap.\ 8vo, 1\s.~6\d.\catright{Camb.~Texts.}
%-----File: 314.png-----

\mitemspace[LUCRETIUS\@.] Recognovit {\smallcapsize H.~A.~J. MUNRO, A.M\@.} Fcap.\ 8vo, 2\s.
\catright{Camb.~Texts.}

\mitem[PROPERTIUS\@.  Sex.\ Propertii Elegiarum Libri IV.] recensuit {\smallcapsize A.~PALMER}, collegii sacrosanctae et individuae Trinitatis juxta Dublinum
Socius. Fcap.\ 8vo, 3\s.~6\d.

\mitem[SALLUSTI CRISPI CATILINA ET JUGURTHA,] Recognovit
{\smallcapsize G.~LONG, A.M\@.} Fcap.\ 8vo, 1\s.~6\d.
\catright{Camb.~Texts.}

\mitemspace[SOPHOCLES\@.] Ex recensione {\smallcapsize F.~A. PALEY, A.M., LL.D\@.} Fcap.\ 8vo, 2\s.~6\d.
\catright{Camb.~Texts.}

\mitemspace[TERENTI COMOEDIAE\@.] {\smallcapsize GUL. WAGNER} relegit et emendavit. Fcap.\ 8vo, 2\s.
\catright{Camb.~Texts.}

\mitemspace[THUCYDIDES\@.] Recensuit {\smallcapsize J.~G. DONALDSON, S.T.P\@.} 2 vols. Fcap.\ 8vo, 2\s.\ each.
\catright{Camb.~Texts.}

\mitemspace[VERGILIUS\@.] Ex recensione {\smallcapsize J.~CONINGTON, A.M\@.} Fcap.\ 8vo, 2\s.
\catright{Camb.~Texts.}

\mitemspace[XENOPHONTIS EXPEDITIO CYRI\@.] Recensuit {\smallcapsize J.~F. MACMICHAEL
A.B\@.} Fcap.\ 8vo, 1\s.~6\d.
\catright{Camb.~Texts.}

\end{maincatlist}

\catsectionuppertoc{Translations}

\begin{maincatlist}

\mitemspace[AESCHYLUS, The Tragedies of.] Translated into English Prose. By
{\smallcapsize F.~A. PALEY, M.A., LL.D.}, Editor of the Greek Text. 2\textit{nd edition
revised}, 8vo, 7\s.~6\d.

\mitemspace[--- The Tragedies of.] Translated into English verse by {\smallcapsize ANNA SWANWICK\@.}
4\textit{th edition revised.} Small post 8vo, 5\s.

\mitemspace[--- The Tragedies of.] Literally translated into Prose, by {\smallcapsize T.~A. BUCKLEY, B.A\@.}
Small post 8vo, 3\s.~6\d.

\mitemspace[--- The Tragedies of.] Translated by {\smallcapsize WALTER HEADLAM, M.A.}, Fellow of
King's College, Cambridge.
\catright{Preparing.}

\mitemspace[ANTONINUS (M. Aurelius), The Thoughts of.] Translated by
{\smallcapsize GEORGE LONG, M.A\@.} \textit{Revised edition.} Small post 8vo, 3\s.~6\d.

Fine paper edition on handmade paper. Pott 8vo, 6\s.

\mitemspace[APOLLONIUS RHODIUS\@. The Argonautica.] Translated by {\smallcapsize E.~P.
COLE\-RIDGE\@.} Small post 8vo, 5\s.

\mitemspace[AMMIANUS MARCELLINUS\@.] \textbf{History of Rome} during the
Reigns of Constantius, Julian, Jovianus, Valentinian, and Valens. Translated
by {\smallcapsize PROF. C.~D. YONGE, M.A\@.} With a complete Index.\ Small post
8vo, 7\s.~6\d.

\mitemspace[ARISTOPHANES, The Comedies of.] Literally translated by {\smallcapsize W.~J.
HICKIE\@.} \textit{With Portrait.} 2 vols.\ small post 8vo, 5\s.\ each.

Vol.~I.--Acharnians, Knights, Clouds, Wasps, Peace, and Birds.

Vol.~II.--Lysistrata, Thesmophoriazusae, Frogs, Ecclesiazusae, and Plutus.

\mitemspace[--- The Acharnians.] Translated by {\smallcapsize W.~H. COVINGTON, B.A\@.} With Memoir
and Introduction. Crown 8vo, sewed, 1\s.

\mitemspace[ARISTOTLE on the Athenian Constitution.] Translated, with Notes
and Introduction, by {\smallcapsize F.~G. KENYON, M.A.}, Fellow of Magdalen College,
Oxford. Pott 8vo, printed on handmade paper. 2\textit{nd edition.} 4\s.~6\d.

\mitemspace[--- History of Animals.] Translated by {\smallcapsize RICHARD CRESSWELL, M.A\@.} Small
post 8vo, 5\s.
%-----File: 315.png-----

\mitem[--- Organon:] or, Logical Treatises, and the Introduction of
Porphyry. With Notes, Analysis, Introduction, and Index, by the {\smallcapsize REV.
O.~F. OWEN, M.A\@.} 2 vols.\ small post 8vo, 3\s.~6\d.\ each.

\mitemspace[--- Rhetoric and Poetics.]  Literally Translated, with Hobbes' Analysis,
\&c., by {\smallcapsize T.~BUCKLEY, B.A\@.} Small post 8vo, 5\s.

\mitemspace[--- Nicomachean Ethics.] Literally Translated, with Notes, an Analytical
Introduction, \&c., by the Venerable {\smallcapsize ARCHDEACON BROWN}, late Classical
Professor of King's College. Small post 8vo, 5\s.

\mitemspace[--- Politics and Economics.]  Translated, with Notes, Analyses, and
Index, by {\smallcapsize E.~WALFORD, M.A.}, and an Introductory Essay and a Life by
{\smallcapsize DR. GILLIES\@.} Small post 8vo, 5\s.

\mitemspace[--- Metaphysics.] Literally Translated, with Notes, Analysis, \&c., by the
{\smallcapsize REV. JOHN H.~McMAHON, M.A\@.} Small post 8vo, 5\s.\clabel{note:mc}

\mitem[ARRIAN\@. Anabasis of Alexander,] together with the Indica. Translated
by {\smallcapsize E.~J. CHINNOCK, M.A., LL.D\@.} With Introduction, Notes, Maps,
and Plans. Small post 8vo, 5\s.

\mitem[CAESAR\@. Commentaries on the Gallic and Civil Wars,] with the Supplementary
Books attributed to Hirtius, including the complete Alexandrian,
Af\-rican, and Spanish Wars. Translated by {\smallcapsize W.~A. McDEVITTE, B.A\@.}
Small post 8vo, 5\s.

\mitemspace[--- Gallic War.] Translated by {\smallcapsize W.~A. McDEVITTE, B.A\@.} 2 vols., with Memoir
and Map. Crown 8vo, sewed. Books I. to IV., Books V. to VII.,
1\s.\ each.

\mitemspace[CALPURNIUS SICULUS, The Eclogues of.] The Latin Text, with
English Translation by {\smallcapsize E.~J.~L. SCOTT, M.A\@.} Crown 8vo, 3\s.~6\d.

\mitemspace[CATULLUS, TIBULLUS, and the Vigil of Venus.] Prose Translation.
Small post 8vo, 5\s.

\mitemspace[CICERO, The Orations of.] Translated by {\smallcapsize PROF. C.~D. YONGE, M.A\@.}
With Index. 4 vols.\ small post 8vo, 5\s.\ each.

\mitemspace[--- On Oratory and Orators.] With Letters to Quintus and Brutus. Translated
by the {\smallcapsize REV. J.~S. WATSON, M.A\@.} Small post 8vo, 5\s.

\mitemspace[--- On the Nature of the Gods.] Divination, Fate, Laws, a Republic,
Consulship. Translated by {\smallcapsize PROF. C.~D. YOUNG, M.A.}, and {\smallcapsize FRANCIS
BARHAM\@.} Small post 8vo, 5\s.

\mitem[--- Academics,] De Finibus, and Tusculan Questions. By {\smallcapsize PROF. C.~D.
YOUNG, M.A\@.} Small post 8vo, 5\s.

\mitem[--- Offices;] or, Moral Duties. Cato Major, an Essay on Old Age; Laelius,
an Essay on Friendship; Scipio's Dream; Paradoxes; Letter to Quintus
on Magistrates. Translated by {\smallcapsize C.~R. EDMONDS\@.} \textit{With Portrait}, 3\s.~6\d.

\mitemspace[--- Old Age and Friendship.] Translated, with Memoir and Notes, by
{\smallcapsize G.~H. WELLS, M.A\@.} Crown 8vo, sewed, 1\s.

\mitemspace[DEMOSTHENES, The Orations of.] Translated, with Notes, Arguments,
a Chronological Abstract, Appendices, and Index, by {\smallcapsize C.~RANN KENNEDY\@.}
5~vols.\ small post 8vo.

Vol.~I.---The Olynthiacs, Philippics. 3\s.~6\d.

Vol.~II.---On the Crown and on the Embassy, 5\s.

Vol.~III.---Against Leptines, Midias, Androtion, and Aristocrates. 5\s.

Vols. IV. and V.---Private and Miscellaneous Orations. 5\s.\ each.

\mitemspace[--- On the Crown.] Translated by {\smallcapsize C.~RANN KENNEDY\@.} Small post 8vo,
sewed, 1\s., cloth, 1\s.~6\d.

\mitemspace[DIOGENES LAERTIUS\@.] Translated by {\smallcapsize PROF\@.
C.~D. YOUNG, M.A\@.}
Small post 8vo, 5\s.
%-----File: 316.png-----

\mitemspace[EPICTETUS, The Discourses of.]  With the \textbf{Encheiridion and
Fragments}. Translated by {\smallcapsize GEORGE LONG, M.A\@.} Small post 8vo, 5\s.

Fine Paper Edition, 2 vols.\ Pott 8vo, 10\s.~6\d.

\mitemspace[EURIPIDES\@.] A Prose Translation, from the Text of Paley.  By
{\smallcapsize E.~P. COLERIDGE, B.A\@.} 2 vols., 5\s.\ each.

\indent Vol.~I.---Rhesus, Medea, Hippolytus, Alcestis, Heraclidæ, Supplices,
Tro\-ades, Ion, Helena.

\indent Vol.~II.---Andromache, Electra, Bacchae, Hecuba, Hercules Furens,
Phoenissae, Orestes, Iphigenia in Tauris, Iphigenia in Aulis, Cyclops.

\threestars The plays separately
(except Rhesus, Helena, Electra, Iphigenia in
Aulis, and Cyclops). Crown 8vo, sewed, 1\s.\ each.

---Translated from the Text of Dindorf. By {\smallcapsize T.~A. BUCKLEY, B.A\@.}
2 vols.\ small post 8vo, 5\s.\ each.

\mitemspace[GREEK ANTHOLOGY\@.] Translated by {\smallcapsize GEORGE BURGES, M.A\@.} Small
post 8vo, 5\s.

\mitemspace[HERODOTUS\@.] Translated by the {\smallcapsize REV. HENRY CARY, M.A\@.} Small post
8vo, 3\s.~6\d.

\mitemspace[--- Analysis and Summary of.] By {\smallcapsize J.~T. WHEELER\@.} Small post 8vo, 5\s.

\mitem[HESIOD, CALLIMACHUS,] and \textbf{THEOGNIS\@.} Translated by the
{\smallcapsize REV. J.~BANKS, M.A\@.}  Small post 8vo, 5\s.

\mitemspace[HOMER\@. The Iliad.] Translated by {\smallcapsize T.~A. BUCKLEY, B.A\@.} Small post
8vo, 5\s.

\mitem[--- The Odyssey, Hymns, Epigrams,] and \textbf{Battle of the Frogs and
Mice.} Translated by {\smallcapsize T.~A. BUCKLEY, B.A\@.} Small post 8vo, 5\s.

\mitemspace[--- The Iliad.]  Books I.--IV\@. Translated into English Hexameter Verse,
by {\smallcapsize HENRY SMITH WRIGHT, B.A.}, late Scholar of Trinity College, Cambridge.
Medium 8vo, 5\s.

\mitemspace[HORACE\@.] Translated by Smart. \textit{Revised edition.} By {\smallcapsize T.~A. BUCKLEY,
B.A\@.} Small post 8vo, 3\s.~6\d.

\mitemspace[--- The Odes and Carmen Saeculare.] Translated into English Verse by
the late {\smallcapsize JOHN CONINGTON, M.A.}, Corpus Professor of Latin in the
University of Oxford, 11\textit{th edition.} Fcap.\ 8vo, 3\s.~6\d.

\mitemspace[--- The Satires and Epistles.] Translated into English Verse by {\smallcapsize PROF.
JOHN CONINGTON, M.A\@.} 8\textit{th edition.} Fcap.\ 8vo, 3\s.~6\d.

\mitemspace[--- Odes and Epodes.] Translated by {\smallcapsize SIR STEPHEN E.~DE~VERE, BART\@.}
3\textit{rd edition, enlarged.} Imperial 16mo, 7\s.~6\d.\ net.

\mitemspace[ISOCRATES, The Orations of.] Translated by {\smallcapsize J.~H. FREESE, M.A.}, late
Fellow of St.\ John's College, Cambridge, with Introductions and Notes.
Vol.~I\@. Small post 8vo, 5\s.

\mitemspace[JUSTIN, CORNELIUS NEPOS, \textmd{and} EUTROPIUS\@.] Translated
by the {\smallcapsize REV. J.~S. WATSON, M.A\@.} Small post 8vo, 5\s.

{\stretchyspace
\mitemspace[JUVENAL, PERSIUS, SULPICIA, \textmd{and} LUCILIUS\@.] Translated
by {\smallcapsize L.~EVANS, M.A\@.} Small post 8vo, 5\s.
}

\mitemspace[LIVY\@. The History of Rome.] Translated by {\smallcapsize DR.~SPILLAN, C.~EDMONDS},
and others. 4 vols.\ small post 8vo, 5\s.\ each.

---Books I., II., III., IV\@. A Revised Translation by {\smallcapsize J.~H. FREESE, M.A.},
late Fellow of St.\ John's College, Cambridge. With Memoir, and Maps.
4 vols., crown 8vo, sewed, 1\s.\ each.

---Book V\@. A Revised Translation by {\smallcapsize E.~S. WEYMOUTH, M.A.}, Lond. With
Memoir, and Maps. Crown 8vo, sewed, 1\s.

---Book IX\@. Translated by {\smallcapsize FRANCIS STORR, B.A\@.} With Memoir. Crown
8vo, sewed, 1\s.
%-----File: 317.png-----

\mitemspace[LUCAN\@. The Pharsalia.] Translated into Prose by {\smallcapsize H.~T. RILEY\@.} Small
post 8vo, 5\s.

\mitemspace[--- The Pharsalia.] Book I\@. Translated by {\smallcapsize FREDERICK CONWAY, M.A\@.}
With Memoir and Introduction. Crown 8vo, sewed, 1\s.

\mitemspace[LUCIAN'S Dialogues of the Gods, of the Sea-Gods, and of the Dead.]
Translated by {\smallcapsize HOWARD WILLIAMS, M.A\@.} Small post 8vo, 5\s.

\mitemspace[LUCRETIUS\@.] Translated by the {\smallcapsize REV. J.~S. WATSON, M.A\@.} Small post
8vo, 5\s.

---Literally translated by the late {\smallcapsize H.~A.~J. MUNRO, M.A\@.} 4\textit{th edition.} Demy 8vo, 6\s.

\mitem[MARTIAL'S Epigrams,] complete. Literally translated into Prose, with
the addition of Verse Translations selected from the Works of English
Poets, and other sources. Small post 8vo, 7\s.~6\d.

\mitemspace[OVID, The Works of.] Translated. 3 vols., small post 8vo, 5\s.\ each.

Vol.~I.---Fasti, Tristia, Pontic Epistles, Ibis, and Halieuticon.

Vol.~II.---Metamorphoses.  \textit{With Frontispiece.}

Vol.~III.---Heroides, Amours, Art of Love, Remedy of Love, and
Minor Pieces. \textit{With Frontispiece.}

\mitemspace[PINDAR\@.] Translated by {\smallcapsize DAWSON W.~TURNER\@.} Small post 8vo, 5\s.

\mitemspace[PLATO\@. Gorgias.] Translated by the late {\smallcapsize E.~M. COPE, M.A.}, Fellow
of Trinity College. 2\textit{nd edition.} 8vo, 7\s.

\mitemspace[--- Philebus.] Translated by {\smallcapsize F.~A. PALEY, M.A., LL.D\@.} Small 8vo, 4\s.

\mitemspace[--- Theaetetus.] Translated by {\smallcapsize F.~A. PALEY, M.A., LL.D\@.} Small 8vo, 4\s.

\mitemspace[--- The Works of.] Translated, with Introduction and Notes. 6 vols.\ small
post 8vo, 5\s.\ each.

Vol.~I\@. --- The Apology of Socrates --- Crito --- Phaedo --- Gorgias --- Protagoras ---
Phaedrus --- Theaetetus --- Eutyphron --- Lysis. Translated by the
{\smallcapsize REV. H.~CARY.}

Vol.~II\@. --- The Republic --- Timaeus --- Critias. Translated by {\smallcapsize HENRY
DAVIS.}

Vol.~III\@. --- Meno --- Euthydemus --- The Sophist --- Statesman --- Cratylus ---
Parmenides --- The Banquet. Translated by {\smallcapsize G.~BURGES.}

Vol.~IV\@. --- Philebus --- Charmides --- Laches --- Menexenus --- Hippias --- Ion ---
The Two Alcibiades --- Theages --- Rivals --- Hipparchus --- Minos --- Clitopho ---
Epistles. Translated by {\smallcapsize G.~BURGES.}

Vol.~V\@. --- The Laws. Translated by {\smallcapsize G.~BURGES.}

Vol.~VI\@. --- The Doubtful Works. Edited by {\smallcapsize G.~BURGES\@.} With General
Index to the six volumes.

\mitemspace[--- Apology, Crito, Phaedo, and Protagoras.] Translated by the {\smallcapsize REV.
H. CARY\@.} Small post 8vo, sewed, 1\s., cloth, 1\s.~6\d.

\mitemspace[--- Dialogues.] A Summary and Analysis of. With Analytical Index, giving
references to the Greek text of modern editions and to the above translations.
By {\smallcapsize A.~DAY, LL.D\@.} Small post 8vo, 5\s.

\mitemspace[PLAUTUS, The Comedies of.] Translated by {\smallcapsize H.~T. RILEY, B.A\@.} 2 vols.\
small post 8vo, 5\s.\ each.

Vol.~I\@. --- Trinummus --- Miles Gloriosus --- Bacchides --- Stichus --- Pseudolus ---
Menaechmei --- Aulularia --- Captivi --- Asinaria --- Curculio.

Vol.~II\@. --- Amphitryon --- Rudens --- Mercator --- Cistellaria --- Truculentus ---
Persa --- Casina --- Poenulus --- Epidicus --- Mostellaria --- Fragments.

\mitemspace[--- Trinummus, Menaechmei, Aulularia, and Captivi.] Translated by
{\smallcapsize H.~T. RILEY, B.A\@.} Small post 8vo, sewed, 1\s., cloth, 1\s.~6\d.
%-----File: 318.png-----

\mitemspace[PLINY\@. The Letters of Pliny the Younger.] Melmoth's Translation,
revised, by the {\smallcapsize REV. F.~C.~T. BOSANQUET, M.A\@.} Small post 8vo, 5\s.

\mitemspace[PLUTARCH\@. Lives.] Translated by {\smallcapsize A.~STEWART, M.A.}, late Fellow of
Trinity College, Cambridge, and {\smallcapsize GEORGE LONG, M.A\@.} 4 vols.\ small post
8vo, 3\s.~6\d.\ each.

\mitemspace[--- Morals. Theosophical Essays.] Translated by {\smallcapsize C.~W. KING, M.A.}, late
Fellow of Trinity College, Cambridge. Small post 8vo, 5\s.

\mitemspace[--- Morals. Ethical Essays.] Translated by the {\smallcapsize REV. A.~R. SHILLETO, M.A.}
Small post 8vo, 5\s.

\mitemspace[PROPERTIUS\@.] Translated by {\smallcapsize REV. P.~J.~F. GANTILLON, M.A.}, and
accompanied by Poetical Versions, from various sources. Small post 8vo,
3\s.~6\d.

\mitemspace[PRUDENTIUS, Translations from.] A Selection from his Works, with
a Translation into English Verse, and an Introduction and Notes, by
{\smallcapsize FRANCIS ST. JOHN THACKERAY, M.A., F.S.A.}, Vicar of Mapledurham,
formerly Fellow of Lincoln College, Oxford, and Assistant-Master at
Eton. Wide post 8vo, 7\s.~6\d.

\mitem[QUINTILIAN: Institutes of Oratory,] or, Education of an Orator.
Translated by the {\smallcapsize REV. J.~S. WATSON, M.A\@.} 2 vols.\ small post 8vo,
5\s.\ each.

\mitemspace[SALLUST, FLORUS, and VELLEIUS PATERCULUS\@.] Translated
by {\smallcapsize J.~S. WATSON, M.A\@.} Small post 8vo, 5\s.

\mitemspace[SENECA: On Benefits.] Translated by {\smallcapsize A.~STEWART, M.A.}, late Fellow
of Trinity College, Cambridge. Small post 8vo, 3\s.~6\d.

\mitemspace[--- Minor Essays and On Clemency.] Translated by {\smallcapsize A.~STEWART, M.A.}
Small post 8vo, 5\s.

\mitemspace[SOPHOCLES\@.] Translated, with Memoir, Notes, etc., by {\smallcapsize E.~P. COLERIDGE, B.A.}
Small post 8vo, 5\s.

Or the plays separately, crown 8vo, sewed, 1\s.\ each.

\mitemspace[--- The Tragedies of.] The Oxford Translation, with Notes, Arguments,
and Introduction. Small post 8vo, 5\s.

\mitemspace[--- The Dramas of.] Rendered in English Verse, Dramatic and Lyric, by
{\smallcapsize SIR GEORGE YOUNG, BART., M.A.}, formerly Fellow of Trinity College,
Cambridge. 8vo, 12\s.~6\d.

\mitemspace[--- The {\OE}dipus Tyrannus.] Translated into English Prose. By {\smallcapsize PROF. B.~H.
KEN\-NEDY\@.} Crown 8vo, in paper wrapper, 1\s.

\mitemspace[SUETONIUS\@. Lives of the Twelve Caesars and Lives of the Grammarians.]
Thomson's revised Translation, by {\smallcapsize T.~FORESTER\@.} Small
post 8vo, 5\s.

\mitemspace[TACITUS, The Works of.] Translated, with Notes and Index. 2 vols.
Small post 8vo, 5\s.\ each.

Vol.~I\@. --- The Annals.

Vol.~II\@. --- The History, Germania, Agricola, Oratory, and Index.

\mitemspace[TERENCE and PHAEDRUS\@.] Translated by {\smallcapsize H.~T. RILEY, B.A\@.} Small
post 8vo, 5\s.

\mitemspace[THEOCRITUS, BION, MOSCHUS, and TYRTAEUS\@.] Translated
by the {\smallcapsize REV. J.~BANKS, M.A\@.} Small post 8vo, 5\s.

\mitemspace[THEOCRITUS\@.] Translated into English Verse by {\smallcapsize C.~S. CALVERLEY,
M.A.}, late Fellow of Christ's College, Cambridge. \textit{New edition, revised.}
Crown 8vo, 7\s.~6\d.
%-----File: 319.png-----

\mitemspace[THUCYDIDES\@. The Peloponnesian War.] Translated by the {\smallcapsize REV. H. DALE.}
\textit{With Portrait}. 2 vols., 3\s.~6\d.\ each.

\mitemspace[--- Analysis and Summary of.] By {\smallcapsize J.~T. WHEELER\@.} Small post 8vo, 5\s.

\mitemspace[VIRGIL\@.] Translated by {\smallcapsize A.~HAMILTON BRYCE, LL.D\@.} With Memoir and
Introduction. Small post 8vo, 3\s.~6\d.

Also in 6 vols., crown 8vo, sewed, 1\s.\ each.

\textbf{%
  \begin{tabulartwocol}
    Georgics.       & Æneid IV.--VI.\\
    Bucolics.       & Æneid VII.--IX.\\
    Æneid I.--III.  & Æneid X.--XII.\\
  \end{tabulartwocol}}

\mitem[--- Davidson's Translation,] revised, by {\smallcapsize T.~A. BUCKLEY, B.A\@.} Small post 8vo,
3\s.~6\d.

\mitemspace[XENOPHON\@. The Works of.] In 3 vols. Small post 8vo, 5\s.\ each.

Vol.~I\@. --- The Anabasis, and Memorabilia. Translated by the {\smallcapsize REV. J.~S.
WATSON, M.A\@.} With a Geographical Commentary, by {\smallcapsize W.~F. AINSWORTH,
F.S.A., F.R.G.S.}, etc.

Vol.~II\@. --- Cyropaedia and Hellenics. Translated by the {\smallcapsize REV. J.~S.
WATSON, M.A.}, and the {\smallcapsize REV. H.~DALE.}

Vol.~III\@. --- The Minor Works. Translated by the {\smallcapsize REV. J.~S.
WATSON, M.A.}

\mitem[SABRINAE COROLLA] In Hortulis Regiae Scholae Salopiensis contexuerunt
tres viri floribus legendis. 4\textit{th edition, revised and re-arranged}.
By the late {\smallcapsize BENJAMIN HALL KENNEDY, D.D.}, Regius Professor of Greek
at the University of Cambridge. Large post 8vo, 10\s.~6\d.

\mitem[SERTUM CARTHUSIANUM] Floribus trium Seculorum Contextum.
Cura {\smallcapsize GULIELMI HAIG BROWN}, Scholae Carthusianae Archididascali.
Demy 8vo, 5\s.

\mitemspace[TRANSLATIONS into English and Latin.] By {\smallcapsize C.~S. CALVERLEY, M.A.},
late Fellow of Christ's College, Cambridge, 3\textit{rd edition}. Crown 8vo,
7\s.~6\d.

\mitemspace[TRANSLATIONS from and into the Latin, Greek and English.] By
{\smallcapsize R.~C. JEBB, M.A.}, Regius Professor of Greek in the University of Cambridge,
{\smallcapsize H.~JACKSON, M.A., LITT.D.}, Fellows of Trinity College, Cambridge,
and {\smallcapsize W.~E. CURREY, M.A.}, formerly Fellow of Trinity College,
Cambridge. Crown 8vo. 2\textit{nd edition, revised}. 8\s.
\end{maincatlist}

\catsectionuppertoc{Grammar and Composition}

\begin{maincatlist}

\mitemspace[BADDELEY\@. Auxilia Latina.] A Series of Progressive Latin Exercises.
By {\smallcapsize M.~J.~B. BADDELEY, M.A\@.} Fcap.\ 8vo. Part I., Accidence. 5\textit{th
edition}. 2\s. Part II\@. 5\textit{th edition}. 2\s. Key to Part II\@. 2\s.~6\d.

\mitemspace[BAIRD\@. Greek Verbs.] A Catalogue of Verbs, Irregular and Defective;
their leading formations, tenses in use, and dialectic inflexions, with a
copious Appendix, containing Paradigms for conjugation, Rules for
formation of tenses, \&c., \&c. By {\smallcapsize J.~S. BAIRD, T.C.D\@.} \textit{New edition, revised}.
2\s.~6\d.

\mitemspace[--- Homeric Dialect. Its Leading Forms and Peculiarities.] By {\smallcapsize J.~S. BAIRD,
T.C.D\@.} \textit{New edition, revised}. By the {\smallcapsize REV. W.~GUNION RUTHERFORD,
M.A., LL.D.}, Head Master at Westminster School. 1\s.
%-----File: 320.png-----

\mitemspace[BAKER\@. Latin Prose for London Students.] By {\smallcapsize ARTHUR BAKER,
M.A.}, Classical Master, Independent College, Taunton. Wide fcap.\ 8vo,
2\s.

This book of ninety pages covers systematically the whole ground of the
Latin Sentences included in the Matriculation, Pass Intermediate, and
Pass {\smallcapsize B.A.}\ course of London University.

\mitemspace[BARRY\@.  Notes on Greek Accents.] By the {\smallcapsize RIGHT REV. A.~BARRY,
D.D\@.} \textit{New edition, re-written}. 1\s.

\mitemspace[CHURCH\@. Latin Prose Lessons.] By {\smallcapsize A.~J. CHURCH, M.A.}, Professor of
Latin at University College, London. 9\textit{th edition}. Fcap.\ 8vo, 2\s.~6\d.

\mitemspace[CLAPIN\@. Latin Primer.]  By the {\smallcapsize REV. A.~C. CLAPIN, M.A.}, Assistant
Master at Sherborne School, 3\textit{rd edition}. Fcap.\ 8vo, 1\s.

\mitemspace[COLLINS\@. Latin Exercises and Grammar Papers.] By {\smallcapsize T.~COLLINS,
M.A.}, Head Master of the Latin School, Newport, Salop. 7\textit{th edition}.\
Fcap.\ 8vo, 2\s.~6\d.

\mitem[--- Unseen Papers] in Latin Prose and Verse. With Examination Questions.
6\textit{th edition}.\ Fcap.\ 8vo, 2\s.~6\d.

\mitem[--- Unseen Papers] in Greek Prose and Verse. With Examination Questions.
3\textit{rd edition}.\ Fcap.\ 8vo, 3\s.\

\mitem[--- Easy Translations] from Nepos, Caesar, Cicero, Livy, \&c., for Retranslation
into Latin. With Notes. 2\s.\

\mitemspace[COMPTON\@. Rudiments of Attic Construction and Idiom.]
An Introduction to Greek Syntax for Beginners who have acquired some knowledge
of Latin. By the {\smallcapsize REV. W.~COOKWORTHY COMPTON, M.A.}, Head Master
of Dover College. Crown 8vo, 3\s.\

\mitem[FROST\@. Eclogae Latinae;] or, First Latin Reading Book. With Notes
and Vocabulary by the late {\smallcapsize REV. P.~FROST, M.A\@.} \textit{New edition}.\ Fcap.\ 8vo,
1\s.~6\d.

\mitemspace[--- Analecta Graeca Minora.] With Notes and Dictionary. \textit{New edition}.\
Fcap.\ 8vo, 2\s.\

\mitemspace[--- Materials for Latin Prose Composition.] By the late {\smallcapsize REV. P.~FROST,
M.A\@.} \textit{New edition}.\ Fcap.\ 8vo,\ 2\s.

Key (for tutors only). 4\s.\ net.

\mitemspace[--- A Latin Verse Book.] \textit{New edition}.\ Fcap.\ 8vo, 2\s.\

Key (for tutors only). 5\s.\ net.

\mitemspace[--- Materials for Greek Prose Composition.] \textit{New edition}.\ Fcap.\ 8vo,
2\s.~6\d.\

Key (for tutors only). 5\s.\ net.

\mitemspace[--- Greek Accidence.] \textit{New edition}. 1\s.

\mitemspace[--- Latin Accidence.] 1\s.\

\mitemspace[HARKNESS\@. A Latin Grammar.] By {\smallcapsize ALBERT HARKNESS\@.} Post 8vo, 6\s.\

\mitemspace[KEY\@. A Latin Grammar.] By the late {\smallcapsize T.~H. KEY, M.A., F.R.S\@.} \textit{6th thousand.}
Post 8vo, 8\s.\

\mitemspace[--- A Short Latin Grammar for Schools.] \textsc{16th edition}.\ Post 8vo, 3\s.~6\d.\

\mitemspace[HOLDEN\@.  Foliorum Silvula.] Part I\@. Passages for Translation into
Latin Elegiac and Heroic Verse. By {\smallcapsize H.~A. HOLDEN, LL.D\@.} 11\textit{th edition}.\
Post 8vo, 7\s.~6\d.

\mitemspace[--- Foliorum Silvula.] Part II\@. Select Passages for Translation into Latin
Lyric and Comic Iambic Verse. 3\textit{rd edition}.\ Post 8vo, 5\s.\

\mitemspace[--- Foliorum Centuriae.] Select Passages for Translation into Latin and
Greek Prose. 10\textit{th edition}.\ Post 8vo, 8\s.\
%-----File: 321.png-----

\mitemspace[JEBB, JACKSON, and CURREY\@. Extracts for Translation in Greek, Latin, and English.]
By {\smallcapsize R.~C. JEBB, LITT.D., LL.D.}, Regius Professor
of Greek in the University of Cambridge; {\smallcapsize H.~JACKSON, LITT.D.},
Fellow of Trinity College, Cambridge; and {\smallcapsize W.~E. CURREY, M.A.}, late
Fellow of Trinity College, Cambridge. 4\s.~6\d.

\mitemspace[Latin Syntax, Principles of.] 1\s.\

\mitemspace[Latin Versification.] 1\s.\

\mitemspace[MASON\@. Analytical Latin Exercises.]
By {\smallcapsize C.~P. MASON, B.A\@.} 4\textit{th
edition}.\ Part~I., 1\s.~6\d. Part~II., 2\s.~6\d.

\mitemspace[--- The Analysis of Sentences Applied to Latin.] Post 8vo, 1\s.~6\d.

\mitemspace[NETTLESHIP\@. Passages for Translation into Latin Prose.] Preceded
by Essays on:---I. Political and Social Ideas. II. Range of Metaphorical
Expression. III. Historical Development of Latin Prose Style
in Antiquity. IV. Cautions as to Orthography. By {\smallcapsize H.~NETTLESHIP,
M.A.}, late Corpus Professor of Latin in the University of Oxford. Crown
8vo, 3\s. A Key. 4\s.~6\d.\ net.

{\stretchyspace
``The Introduction ought to be studied by every teacher of Latin.''---\textit{Guardian}.
}

\mitem[Notabilia Quaedam;] or the Principal Tenses of most of the Irregular
Greek Verbs, and Elementary Greek, Latin, and French Constructions.
\textit{New edition}. 1\s.

\mitem[PALEY\@. Greek Particles] and their Combinations according to Attic
Usage. A Short Treatise. By {\smallcapsize F.~A. PALEY, M.A., LL.D\@.} 2\s.~6\d.

\mitem[PENROSE\@.  Latin Elegiac Verse,] Easy Exercises in. By the {\smallcapsize REV. J.\
PENROSE\@.} \textit{New edition}. 2\s. (Key, 3\s.~6\d.\ net.)

\mitemspace[PRESTON\@. Greek Verse Composition.] By {\smallcapsize G.~PRESTON, M.A\@.} 5\textit{th
edition}.\ Crown 8vo, 4\s.~6\d.

\mitemspace[PRUEN\@. Latin Examination Papers.] Comprising Lower, Middle, and
Upper School Papers, and a number of the Woolwich and Sandhurst
Standards. By {\smallcapsize G.~G. PRUEN, M.A.}, Senior Classical Master in the Modern
Department, Cheltenham College. Crown 8vo, 2\s.~6\d.

\mitemspace[SEAGER\@. Faciliora.] An Elementary Latin Book on a New Principle.
By the {\smallcapsize REV. J.~L. SEAGER, M.A\@.} 2\s.~6\d.

\mitemspace[STEDMAN (A.~M.~M.). First Latin Lessons.] By {\smallcapsize A.~M.~M. STEDMAN,
M.A.}, Wadham College, Oxford. 2\textit{nd edition, enlarged}.\ Crown 8vo, 2\s.

\mitemspace[--- Initia Latina.] Easy Lessons on Elementary Accidence.  2\textit{nd edition}.
Fcap.\ 8vo, 1\s.

\mitemspace[--- First Latin Reader.] With Notes adapted to the Shorter Latin Primer
and Vocabulary. Crown 8vo, 1\s.~6\d.

\mitem[--- Easy Latin Passages] for Unseen Translation.  2\textit{nd and enlarged
edition}.\ Fcap.\ 8vo, 1\s.~6\d.

\mitemspace[--- Exempla Latina.] First Exercises in Latin Accidence. With Vocabulary.
Crown 8vo, 1\s.~6\d.

\mitemspace[--- The Latin Compound Sentence; Rules and Exercises.] Crown 8vo,
1\s.~6\d. With Vocabulary, 2\s.

\mitem[--- Easy Latin Exercises] on the Syntax of the Shorter and Revised Latin
Primers. With Vocabulary. 3\textit{rd edition}.\ Crown 8vo, 2\s.~6\d.

\mitem[--- Latin Examination Papers] in Miscellaneous Grammar and Idioms.
3\textit{rd edition}. 2\s.~6\d. Key (for Tutors only), 6\s.\ net.

\mitemspace[--- Notanda Quaedam.]  Miscellaneous Latin Exercises.  On Common
Rules and Idioms. 2\textit{nd edition}.\ Fcap.\ 8vo,
1\s.~6\d. With Vocabulary, 2\s.
%-----File: 322.png-----

\mitemspace[--- Latin Vocabularies for Repetition.] Arranged
according to Subjects. 3\textit{rd edition.} Fcap.\ 8vo, 1\s.~6\d.

\mitemspace[--- First Greek Lessons.]\ \catright{In preparation.}% \space to avoid odd spacing

\mitem[--- Easy Greek Passages] for Unseen Translation. Fcap.\ 8vo, 1\s.~6\d.

\mitem[--- Easy Greek Exercises] on Elementary Syntax.   \catright{In preparation.}

\mitem[--- Greek Vocabularies] for Repetition. Fcap.\ 8vo, 1\s.~6\d.

\mitem[--- Greek Testament Selections] for the Use of Schools,  2\textit{nd edition.}
With Introduction, Notes, and Vocabulary. Fcap.\ 8vo, 2\s.~6\d.

\mitem[--- Greek Examination Papers] in Miscellaneous Grammar and Idioms.
2\textit{nd edition}. 2\s.~6\d. Key (for Tutors only), 6\s.\ net.

\mitemspace[THACKERAY\@. Anthologia Graeca.] A Selection of Greek Poetry,
with Notes. By {\smallcapsize F.~ST. JOHN THACKERAY\@.} 5\textit{th edition.} 16mo, 4\s.~6\d.

\mitemspace[--- Anthologia Latina.]  A Selection of Latin Poetry, from Naevius to
Boëthius, with Notes. By {\smallcapsize REV. F.~ST. JOHN THACKERAY\@.} 6\textit{th edition.}
16mo, 4\s.~6\d.

\mitemspace[--- Hints and Cautions on Attic Greek Prose Composition.] Crown
8vo, 3\s.~6\d.

\mitemspace[--- Exercises on the Irregular and Defective Greek Verbs.] 1\s.~6\d.

\mitemspace[WELLS\@.  Tales for Latin Prose Composition.]  With Notes and
Vocabulary. By {\smallcapsize G.~H. WELLS, M.A.}, Assistant Master at Merchant
Taylor's School. Fcap.\ 8vo, 2\s.
\end{maincatlist}

\catsectionuppertoc{History, Geography, and Reference Books, etc}

\begin{maincatlist}
\mitemspace[TEUFFEL'S History of Roman Literature.] 5\textit{th edition}, revised by
{\smallcapsize DR. SCHWABE}, translated by {\smallcapsize PROFESSOR G.~C.~W. WARR, M.A.}, King's
College, London. Medium 8vo. 2 vols. 30\s. Vol.~I. (The Republican
Period), 15\s. Vol.~II. (The Imperial Period), 15\s.

\mitemspace[KEIGHTLEY'S Mythology of Ancient Greece and Italy.] 4\textit{th edition},
revised by the late {\smallcapsize LEONHARD SCHMITZ, PH.D., LL.D.}, Classical Examiner
to the University of London.
With 12 Plates from the Antique. Small post 8vo, 5\s.

\mitemspace[DONALDSON'S Theatre of the Greeks.] 10\textit{th edition.} Small post 8vo,
5\s.

\mitem[DICTIONARY OF LATIN AND GREEK QUOTATIONS;] including
Pro\-verbs, Maxims, Mottoes, Law Terms and Phrases. With all
the Quantities marked, and English Translations. With Index Verborum.
Small post 8vo, 5\s.

\mitemspace[BENTLEY'S Dissertations upon the Epistles of Phalaris, Themistocles, Socrates, Euripides, and the Fables of Aesop.]
Edited, with
an Introduction and Notes, by the late {\smallcapsize PROF. W.~WAGNER, PH.D\@.} 5\s.

\mitemspace[DOBREE'S Adversaria.]  With Preface by the late {\smallcapsize PROFESSOR W.\
WAGNER\@.}  2~vols.  10\s.

\mitemspace[A GUIDE TO THE CHOICE OF CLASSICAL BOOKS\@.] By {\smallcapsize J.~B.\
MAYOR, M.A.}, Professor of Moral Philosophy at King's College, late
Fellow and Tutor of St.~John's College, Cambridge. 3\textit{rd edition}, with
Supplementary List. Crown 8vo, 4\s.~6\d. Supplement separate, 1\s.~6\d.

\mitemspace[PAUSANIAS' Description of Greece.] Newly translated, with Notes
and Index, by {\smallcapsize A.~R. SHILLETO, M.A\@.} 2 vols. Small post 8vo, 5\s.\ each.

%-----File: 323.png-----

\mitemspace[STRABO'S Geography.] Translated by {\smallcapsize W.~FALCONER, M.A.}, and {\smallcapsize H.~C.\
HAMILTON\@.} 3 vols. Small post 8vo, 5\s.\ each.

\mitemspace[AN ATLAS OF CLASSICAL GEOGRAPHY\@.] By {\smallcapsize W.~HUGHES} and
{\smallcapsize G.~LONG, M.A\@.} Containing Ten selected Maps. Imp.\ 8vo, 3\s.

\mitemspace[AN ATLAS OF CLASSICAL GEOGRAPHY\@.] Twenty-four Maps
by {\smallcapsize W.~HUGHES} and {\smallcapsize GEORGE LONG, M.A\@.} With coloured outlines.
Imperial 8vo, 6\s.

\mitemspace[ATLAS OF CLASSICAL GEOGRAPHY\@.] 22 large Coloured Maps.
With a complete Index.  Imp.\ 8vo, chiefly engraved by the Messrs.\
Walker. 7\s.~6\d.

\end{maincatlist}

\cchaptercat{MATHEMATICS}

\catsectionuppertoc{Arithmetic and Algebra}

\begin{maincatlist}

\mitemspace[BARRACLOUGH (T.). The Eclipse Mental Arithmetic.] By {\smallcapsize TITUS
BARRACLOUGH}, Board School, Halifax. Standards I., II., and III.,
sewed, 6\d.; Standards II., III., and IV., sewed, 6\d.\ net; Book~III.,
Part~A, sewed, 4\d.; Book~III., Part~B, cloth, 1\s.~6\d.

\mitem[BEARD (W.~S.). Graduated Exercises in Addition] (Simple and Compound).
For Candidates for Commercial Certificates and Civil Service
appointments. By {\smallcapsize W.~S. BEARD, F.R.G.S.}, Head Master of the Modern
School, Fareham. 2\textit{nd edition.} Fcap.\ 4to, 1\s.

\mitem[---] \textit{See} \textbf{PENDLEBURY.}

\mitemspace[ELSEE (C.). Arithmetic.] By the {\smallcapsize REV. C.~ELSEE, M.A.}, late Fellow of
St.\ John's College, Cambridge, Senior Mathematical Master at Rugby
School. 14\textit{th edition.} Fcap.\ 8vo, 3\s.~6\d.
\catright{Camb.\ School and College Texts.}

\mitemspace[--- Algebra.] By the {\smallcapsize REV. C.~ELSEE, M.A\@.} 8\textit{th edition.} Fcap.\ 8vo, 4\s.
\catright{Camb.\ S. and C. Texts.}

\mitem[FILIPOWSKI (H.~E.). Anti-Logarithms,] A Table of.  By {\smallcapsize H.~E. FILIPOWSKI},
3\textit{rd edition.} 8vo, 15\s.

\mitemspace[GOUDIE (W.~P.).] See \textbf{Watson.}

\mitemspace[HATHORNTHWAITE (J.~T.). Elementary Algebra for Indian Schools.]
By {\smallcapsize J.~T. HATHORNTHWAITE, M.A.}, Principal and Professor
of Mathematics at Elphinstone College, Bombay. Crown 8vo, 2\s.
\catright{Camb.\ Math.\ Ser.}

\mitem[HUNTER (J.). Supplementary Arithmetic,] with Answers. By {\smallcapsize REV.
J. HUNTER, M.A\@.} Fcap.\ 8vo, 3\s.

\mitemspace[MACMICHAEL (W.~F.) and PROWDE SMITH (R.). Algebra.]
A Progressive Course of Examples. By the {\smallcapsize REV. W.~F. MACMICHAEL},
late Head Master of the Grammar School, Warwick, and {\smallcapsize R.~PROWDE
SMITH, M.A\@.} 4\textit{th edition.} Fcap.\ 8vo, 3\s.~6\d. With answers, 4\s.~6\d.
\catright{Camb.\ S. and C. Texts.}

\mitemspace[MATHEWS (G.~B.). Theory of Numbers.] An account of the Theories
of Congruencies and of Arithmetical Forms. By {\smallcapsize G.~B. MATHEWS, M.A.},
Professor of Mathematics in the University College of North Wales.
Part~I\@. Demy 8vo, 12\s.
%-----File: 324.png-----

\mitemspace[PENDLEBURY (C.). Arithmetic.] With Examination Papers and
8,000 Examples. By {\smallcapsize CHARLES PENDLEBURY, M.A., F.R.A.S.}, Senior
Mathematical Master of St.\ Paul's, Author of ``Lenses and Systems of
Lenses, treated after the manner of Gauss.'' 7\textit{th edition.} Crown 8vo.
Complete, with or without Answers, 4\s.~6\d. In Two Parts, with or
without Answers, 2\s.~6\d.\ each.

Key to Part II., for tutors only, 7\s.~6\d.\ net.   \catright{Camb.\ Math.\ Ser.}

\mitemspace[--- Examples in Arithmetic.] Extracted from Pendlebury's Arithmetic.
With or without Answers, 5\textit{th edition.} Crown 8vo, 3\s., or in Two Parts,
1\s.~6\d.\ and 2\s. \catright{Camb.\ Math.\ Ser.}

\mitemspace[--- Examination Papers in Arithmetic.] Consisting of 140 papers, each
containing 7 questions; and a collection of 357 more difficult problems.
2\textit{nd edition.} Crown 8vo, 2\s.~6\d. Key, for Tutors only, 5\s.\ net.

\mitemspace[PENDLEBURY (C.) and TAIT (T.~S.). Arithmetic for Indian Schools.]
By {\smallcapsize C.~PENDLEBURY, M.A.} and {\smallcapsize T.~S. TAIT, M.A., B.SC.},
Principal of Baroda College. Crown 8vo, 3\s. \catright{Camb.\ Math.\ Ser.}

\mitem[PENDLEBURY (C.) and BEARD (W.~S.).  Arithmetic] for the
Standards. By {\smallcapsize C.~PENDLEBURY, M.A., F.R.A.S.}, and {\smallcapsize W.~S. BEARD,
F.R.G.S\@.} Standards I., II., III., 2\d.\ each; IV., V., VI., 3\d.\ each. VII.,
\textit{in the Press.}

\mitemspace[--- Elementary Arithmetic.] 3\textit{rd edition.} Crown 8vo, 1\s.~6\d.

\mitemspace[POPE (L.~J.). Lessons in Elementary Algebra.] By {\smallcapsize L.~J. POPE, B.A.}
(Lond.), Assistant Master at the Oratory School, Birmingham. First
Series, up to and including Simple Equations and Problems. Crown 8vo,
1\s.~6\d.

\mitemspace[PROWDE SMITH (R.).] \textit{See} \textbf{Macmichael.}

\mitemspace[SHAW (S.~J. D.). Arithmetic Papers.] Set in the Cambridge Higher
Local Examination, from June, 1869, to June, 1887, inclusive, reprinted
by permission of the Syndicate. By {\smallcapsize S.~J.~D. SHAW}, Mathematical
Lecturer of Newnham College. Crown 8vo, 2\s.~6\d.; Key, 4\s.~6\d.\ net.

\mitemspace[TAIT (T.~S.).] \textit{See} \textbf{Pendlebury.}

\mitemspace[WATSON (J.) and GOUDIE (W.~P.). Arithmetic.] A Progressive
Course of Examples With Answers. By {\smallcapsize J.~WATSON, M.A.}, Corpus
Christi College, Cambridge, formerly Senior Mathematical Master of the
Ordnance School, Carshalton. 7\textit{th edition, revised and enlarged.}
By {\smallcapsize W.~P. GOUDIE, B.A.} Lond. Fcap.\ 8vo, 2\s.~6\d. \catright{Camb.\ S. and C. Texts.}

\mitemspace[WHITWORTH (W.~A.). Algebra.] Choice and Chance. An Elementary
Treatise on Permutations, Combinations, and Probability, with
640 Exercises and Answers. By {\smallcapsize W.~A. WHITWORTH, M.A.}, Fellow of
St.\ John's College, Cambridge.  4\textit{th edition, revised and enlarged.}
Crown 8vo, 6\s. \catright{Camb.\ Math.\ Ser.}

\mitemspace[WRIGLEY (A.) Arithmetic.] By {\smallcapsize A.~WRIGLEY, M.A.}, St.\ John's College.
Fcap.\ 8vo, 3\s.~6\d.  \catright{Camb.\ S. and C. Texts.}
\end{maincatlist}

\catsectionuppertoc{Book-keeping}

\begin{maincatlist}
\mitem[CRELLIN (P.). A New Manual of Book-keeping,] combining the
Theory and Practice, with Specimens of a set of Books. By {\smallcapsize PHILLIP
CRELLIN}, Chartered Accountant. Crown 8vo, 3\s.~6\d.

\mitemspace[--- Book-keeping for Teachers and Pupils.] Crown 8vo, 1\s.~6\d. Key,
2\s.\ net.

%-----File: 325.png-----

\mitemspace[FOSTER (B.~W.). Double Entry Elucidated.] By {\smallcapsize B.~W. FOSTER\@.}
14\textit{th edition.} Fcap.\ 4to, 3\s.~6\d.

\mitemspace[MEDHURST (J.~T.) Examination Papers in Book-keeping.] Compiled by {\smallcapsize JOHN T.~MEDHURST, A.K.C., F.S.S.}, Fellow of the Society of
Accountants and Auditors, and Lecturer at the City of London College.
3\textit{rd edition.} Crown 8vo, 3\s.

\mitemspace[THOMSON (A.~W.) A Text-Book of the Principles and Practice of Book-keeping.]
By {\smallcapsize PROFESSOR A.~W. THOMSON, B.SC.}, Royal
Agricultural College, Cirencester. Crown 8vo, 5\s.

\end{maincatlist}

\catsectionuppertoc{Geometry and Euclid}

\begin{maincatlist}

\mitemspace[BESANT (W.~H.). Geometrical Conic Sections.] By {\smallcapsize W.~H. BESANT,
SC.D., F.R.S.}, Fellow of St.\ John's College, Cambridge. 8\textit{th edition.}
Fcap.\ 8vo, 4\s.~6\d. Enunciations, separately, sewed, 1\s.
\catright{Camb.\ S. and C. Texts.}

\mitem[BRASSE (J.). The Enunciations and Figures of Euclid,] prepared for
Students in Geometry.
By the {\smallcapsize REV. J.~BRASSE, D.D\@.} \textit{New edition.}
Fcap.\ 8vo, 1\s. Without the Figures, 6\d.

\mitemspace[DEIGHTON (H.). Euclid.] Books I.--VI., and part of Book XI., newly
translated from the Greek Text, with Supplementary Propositions,
Chapters on Modern Geometry, and numerous Exercises. By {\smallcapsize HORACE
DEIGHTON, M.A.}, Head Master of Harrison College, Barbados. 3\textit{rd
edition. 4\s.~6\d.} Key, for tutors only, 5\s.\ net.   \catright{Camb.\ Math.\ Ser.}

Also issued in parts:---Book I., 1\s.; Books~I. and II., 1\s.~6\d.; Books
I.--III., 2\s.~6\d.; Books III. and IV., 1\s.~6\d.

\mitemspace[DIXON (E.~T.). The Foundations of Geometry.] By {\smallcapsize EDWARD T.\
DIXON}, late Royal Artillery. Demy 8vo, 6\s.

\mitemspace[MASON (C.~P.). Euclid.] The First Two Books Explained to Beginners.
By {\smallcapsize C.~P. MASON, B.A\@.} 2\textit{nd edition.} Fcap.\ 8vo, 2\s.~6\d.

\mitem[McDOWELL (J.) Exercises on Euclid] and in Modern Geometry, containing
Applications of the Principles and Processes of Modern Pure
Geometry. By the late {\smallcapsize J.~McDOWELL, M.A., F.R.A.S.}, Pembroke College,
Cambridge, and Trinity College, Dublin. 4\textit{th edition}. 6\s.
\catright{Camb.\ Math.\ Ser.}

\mitem[TAYLOR (C.). An Introduction to the Ancient and Modern Geometry of Conics,]
with Historical Notes and Prolegomena. 15\s.

\mitemspace[--- The Elementary Geometry of Conics.] By {\smallcapsize C.~TAYLOR, D.D.}, Master
of St.\ John's College. 7\textit{th edition, revised.} With a Chapter on the Line
Infinity, and a new treatment of the Hyperbola. Crown 8vo, 4\s.~6\d.
\catright{Camb.\ Math.\ Ser.}

\mitemspace[WEBB (R.). The Definitions of Euclid.] With Explanations and
Exercises, and an Appendix of Exercises on the First Book by {\smallcapsize R.~WEBB,
M.A\@.} Crown 8vo, 1\s.~6\d.

\mitemspace[WILLIS (H.~G.). Geometrical Conic Sections.] An Elementary
Treatise. By {\smallcapsize H.~G. WILLIS, M.A.}, Clare College, Cambridge, Assistant
Master of Manchester Grammar School. Crown 8vo, 5\s.
\catright{Camb.\ Math.\ Ser.}
\end{maincatlist}

%-----File: 326.png-----
\catsectionuppertoc{Analytical Geometry, etc}

\begin{maincatlist}
\mitem[ALDIS (W.~S.). Solid Geometry,] An Elementary Treatise on. By {\smallcapsize W.~S.\
ALDIS, M.A.}, late Professor of Mathematics in the University College,
Auckland, New Zealand, 4\textit{th edition, revised.} Crown 8vo, 6\s.
\catright{Camb.\ Math.\ Ser.}

\mitemspace[BESANT (W.~H.). Notes on Roulettes and Glissettes.] By {\smallcapsize W.~H.\
BESANT, SC.D., F.R.S\@.} 2\textit{nd edition, enlarged.} Crown 8vo, 5\s.
\catright{Camb.\ Math.\ Ser.}

\mitem[CAYLEY (A.). Elliptic Functions,] An Elementary Treatise on. By
{\smallcapsize ARTHUR CAYLEY}, Sadlerian Professor of Pure Mathematics in the University
of Cambridge. Demy 8vo. \textit{New edition in the Press.}

\mitem[TURNBULL (W.~P.). Analytical Plane Geometry,] An Introduction
to. By {\smallcapsize W.~P. TURNBULL, M.A.}, sometime Fellow of Trinity College.
8vo, 12\s.

\mitemspace[VYVYAN (T.~G.). Analytical Geometry for Schools.] By {\smallcapsize REV. T.\
VYVYAN, M.A.}, Fellow of Gonville and Caius College, and Mathematical
Master of Charterhouse. 6\textit{th edition.} 8vo, 4\s.~6\d.
\catright{Camb.\ S. and C. Texts.}

\mitemspace[--- Analytical Geometry for Beginners.] Part I\@. The Straight Line and
Circle. Crown 8vo, 2\s.~6\d.
\catright{Camb.\ Math.\ Ser.}

\mitem[WHITWORTH (W.~A.). Trilinear Co-ordinates,] and other methods
of Modern Analytical Geometry of Two Dimensions. By {\smallcapsize W.~A. WHITWORTH,
M.A.}, late Professor of Mathematics in Queen's College, Liverpool,
and Scholar of St.\ John's College, Cambridge. 8vo, 16\s.
\end{maincatlist}

\catsectionuppertoc{Trigonometry}

\begin{maincatlist}
\mitemspace[DYER (J.~M.) and WHITCOMBE (R.~H.). Elementary Trigonometry.]\\
By {\smallcapsize J.~M. DYER, M.A\@.} (Senior Mathematical Scholar at Oxford),
and {\smallcapsize REV. R.~H. WHITCOMBE}, Assistant Masters at Eton College. 2\textit{nd
edition.} Crown 8vo, 4\s.~6\d.  \catright{Camb.\ Math.\ Ser.}

\mitemspace[VYVYAN (T.~G.). Introduction to Plane Trigonometry.] By the
{\smallcapsize REV. T.~G. VYVYAN, M.A.}, formerly Fellow of Gonville and Caius College,
Senior Mathematical Master of Charterhouse. 3\textit{rd edition, revised and
augmented.} Crown 8vo, 3\s.~6\d.  \catright{Camb.\ Math.\ Ser.}

\mitemspace[WARD (G.~H.). Examination Papers in Trigonometry.] By {\smallcapsize G.~H.\
WARD, M.A.}, Assistant Master at St.\ Paul's School. Crown 8vo, 2\s.~6\d.
Key, 5\s.\ net.
\end{maincatlist}

\catsectionuppertoc{Mechanics and Natural Philosophy}

\begin{maincatlist}

\mitem[ALDIS (W.~S.). Geometrical Optics,] An Elementary Treatise on. By
{\smallcapsize W.~S. ALDIS, M.A\@.} 4\textit{th edition.} Crown 8vo, 4\s. \catright{Camb.\ Math.\ Ser.}

\mitemspace[--- An Introductory Treatise on Rigid Dynamics.] Crown 8vo, 4\s.
\catright{Camb.\ Math.\ Ser.}

\mitem[--- Fresnel's Theory of Double Refraction,] A Chapter on. 2\textit{nd edition,
revised.} 8vo, 2\s.
%-----File: 327.png-----

\mitem[BASSET (A.~B.). A Treatise on Hydrodynamics,] with numerous
Examples. By {\smallcapsize A.~B. BASSETT, M.A., F.R.S.}, Trinity College, Cambridge.
Demy 8vo. Vol.~I., price 10\s.~6\d.; Vol.~II., 12\s.~6\d.

\mitemspace[--- An Elementary Treatise on Hydrodynamics and Sound.] Demy
8vo, 7\s.~6\d.

\mitemspace[--- A Treatise on Physical Optics.] Demy 8vo, 16\s.

\mitemspace[BESANT (W.~H.). Elementary Hydrostatics.] By {\smallcapsize W.~H. BESANT,
SC.D., F.R.S\@.} 16\textit{th edition.} Crown 8vo, 4\s.~6\d. Solutions, 5\s.
\catright{Camb.\ Math.\ Ser.}

\mitem[--- Hydromechanics,]  A Treatise on. Part I\@. Hydrostatics. 5\textit{th edition,
revised, and enlarged.} Crown 8vo, 5\s. \catright{Camb.\ Math.\ Ser.}

\mitemspace[--- A Treatise on Dynamics.] 2\textit{nd edition.} Crown 8vo, 10\s.~6\d.
\catright{Camb.\ Math.\ Ser.}

\mitemspace[CHALLIS (PROF.). Pure and Applied Calculation.] By the late
{\smallcapsize REV. J.~CHALLIS, M.A., F.R.S.}, \&c. Demy 8vo, 15\s.

\mitem[--- Physics,] The Mathematical Principle of. Demy 8vo, 5\s.

\mitemspace[--- Lectures on Practical Astronomy.] Demy 8vo, 10\s.

\mitem[EVANS (J.~H.) and MAIN (P.~T.). Newton's Principia,] \quad The First
Three Sections of, with an Appendix; and the Ninth and Eleventh
Sections. By {\smallcapsize J.~H. EVANS, M.A.}, St.~John's College. The 5\textit{th edition,}
edited by {\smallcapsize P.~T. MAIN, M.A.}, Lecturer and Fellow of St.~John's College.
Fcap.\ 8vo, 4\s.
\catright{Camb.\ S. and C. Texts.}

\mitem[GALLATLY (W.). Elementary Physics,] Examples and Examination
Papers in. \quad Statics, Dynamics, Hydrostatics, Heat, Light, Chemistry,
Electricity, London Matriculation, Cambridge B.A., Edinburgh, Glasgow,
South Kensington, Cambridge Junior and Senior Papers, and Answers.
By {\smallcapsize W.~GALLATLY, M.A.}, Pembroke College, Cambridge, Assistant
Examiner, London University. Crown 8vo, 4\s. \catright{Camb.\ Math.\ Ser.}

\mitem[GARNETT (W.). Elementary Dynamics] for the use of Colleges and
Schools. By {\smallcapsize WILLIAM GARNETT, M.A., D.C.L.}, Fellow of St.~John's
College, late Principal of the Durham College of Science, Newcastle-upon-Tyne.
5\textit{th edition, revised.} Crown 8vo, 6\s. \catright{Camb.\ Math.\ Ser.}

\mitem[--- Heat,] An Elementary Treatise on. 6\textit{th edition, revised.}  Crown 8vo,
4\s.~6\d.  \catright{Camb.\ Math.\ Ser.}

\mitemspace[GOODWIN (H.). Statics.] By {\smallcapsize H.~GOODWIN, D.D.}, late Bishop of
Carlisle. 2\textit{nd edition.} Fcap.\ 8vo, 3\s. \catright{Camb.\ S. and C. Texts.}

\mitemspace[HOROBIN (J.~C.). Elementary Mechanics.] Stage I. II. and III.,
1\s.~6\d.\ each. By {\smallcapsize J.~C. HOROBIN, M.A.}, Principal of Homerton New
College, Cambridge.

\mitemspace[--- Theoretical Mechanics.] Division I\@. Crown 8vo, 2\s.~6\d.

\threestarsinverted This book covers the ground of the Elementary Stage of Division I.
of Subject VI. of the ``Science Directory,'' and is intended for the
examination of the Science and Art Department.

\mitemspace[JESSOP (C.~M.). The Elements of Applied Mathematics.] Including
Kinetics, Statics and Hydrostatics. By {\smallcapsize C.~M. JESSOP, M.A.}, late
Fellow of Clare College, Cambridge, Lecturer in Mathematics in the
Durham College of Science, Newcastle-on-Tyne. Crown 8vo, 6\s.
\catright{Camb.\ Math.\ Ser.}

\mitem[MAIN (P.~T.). Plane Astronomy,] An Introduction to. By {\smallcapsize P.~T. MAIN,
M.A.}, Lecturer and Fellow of St.\ John's, College. 6\textit{th edition, revised.}
Fcap.\ 8vo, 4\s. \catright{Camb.\ S. and C. Texts.}
%-----File: 328.png-----

\mitemspace[PARKINSON (R.~M.). Structural Mechanics.] By {\smallcapsize R.~M. PARKINSON,
ASSOC. M.I.C.E\@.} Crown 8vo, 4\s.~6\d.

\mitem[PENDLEBURY (C.). Lenses and Systems of Lenses,] Treated after
the Manner of Gauss. By {\smallcapsize CHARLES PENDLEBURY, M.A., F.R.A.S.}, Senior
Mathematical Master of St.~Paul's School, late Scholar of St.~John's
College, Cambridge. Demy 8vo, 5\s.

\mitemspace[STEELE (R.~E.). Natural Science Examination Papers.] By
{\smallcapsize R.~E. STEEL, M.A., F.C.S.}, Chief Natural Science Master, Bradford
Grammar School. Crown 8vo. Part~I., Inorganic Chemistry, 2\s.~6\d.
Part~II., Physics (Sound, Light, Heat, Magnetism, Electricity), 2\s.~6\d.
\catright{School Exam.\ Series.}

\mitem[WALTON (W.). Theoretical Mechanics,] Problems in. By {\smallcapsize W.~WALTON, M.A.},
Fellow and Assistant Tutor of Trinity Hall, Mathematical
Lecturer at Magdalene College, 3\textit{rd edition, revised.} Demy 8vo, 16\s.

\mitem[--- Elementary Mechanics,] Problems in. 2\textit{nd edition.} Crown 8vo, 6\s.\\
\catright{Camb.\ Math.\ Ser.}

\shortrule

\mitemspace[DAVIS (J.~F.). Army Mathematical Papers.] Being Ten Years'
Woolwich and Sandhurst Preliminary Papers. Edited, with Answers, by
{\smallcapsize J.~F. DAVIS, D.LIT., M.A.} Lond. Crown 8vo, 2\s.~6\d.

\mitemspace[DYER (J.~M.) and PROWDE SMITH (R.). Mathematical Examples.]\\
A Collection of Examples in Arithmetic, Algebra, Trigonometry,
Mensuration, Theory of Equations, Analytical Geometry, Statics,
Dynamics, with Answers, \&c. For Army and Indian Civil Service
Candidates. By {\smallcapsize J.~M. DYER, M.A.}, Assistant Master, Eton College
(Senior Mathematical Scholar at Oxford), and {\smallcapsize R.~PROWDE SMITH, M.A.}
Crown 8vo, 6\s.  \catright{Camb.\ Math.\ Ser.}

\mitem[GOODWIN (H.). Problems and Examples,] adapted to ``Goodwin's
Elementary Course of Mathematics.'' By {\smallcapsize T.~G. VYVYAN, M.A\@.} 3\textit{rd
edition.} 8vo, 5\s.; Solutions, 3\textit{rd edition}, 8vo, 9\s.

\mitemspace[SMALLEY (G.~R.). A Compendium of Facts and Formulae in Pure Mathematics and Natural Philosophy.]
By {\smallcapsize G.~R. SMALLEY,
F.R.A.S\@.} \textit{New edition, revised and enlarged.} By {\smallcapsize J.~McDOWELL, M.A.,
F.R.A.S\@.} Fcap.\ 8vo, 2\s.

\mitem[WRIGLEY (A.). Collection of Examples and Problems] in Arithmetic,
Algebra, Geometry, Logarithms, Trigonometry, Conic Sections,
Mechanics, \&c., with Answers and Occasional Hints. By the {\smallcapsize REV. A.
WRIGLEY\@.} 10\textit{th edition, 20th thousand.} Demy 8vo, 8\s.~6\d.

A Key. By {\smallcapsize J.~C. PLATTS, M.A.} and the {\smallcapsize REV. A.~WRIGLEY\@.} 2\textit{nd edition.}
Demy 8vo, 10\s.~6\d.
\end{maincatlist}

\cchaptercat{MODERN LANGUAGES}

\catsectionuppertoc{English}

\begin{maincatlist}
\mitemspace[ADAMS (E.). The Elements of the English Language.] By {\smallcapsize ERNEST
ADAMS, PH.D\@.} 26\textit{th edition.} Revised by {\smallcapsize J.~F. DAVIS, D.LIT., M.A.,
(LOND.)}. Post 8vo, 4\s.~6\d.

\mitemspace[--- The Rudiments of English Grammar and Analysis.] By {\smallcapsize ERNEST
ADAMS, PH.D\@.} 19\textit{th thousand.} Fcap.\ 8vo, 1\s.
%-----File: 329.png-----

\mitemspace[ALFORD (DEAN). The Queen's English: A Manual of Idiom and Usage.]
By the late {\smallcapsize HENRY ALFORD, D.D.}, Dean of Canterbury. 6\textit{th
edition}. Small post 8vo. Sewed, 1\s., cloth, 1\s.~6\d.

\mitemspace[ASCHAM'S Scholemaster.] Edited by {\smallcapsize PROFESSOR J.~E.~E. MAYOR\@.} Small
post 8vo, sewed, s.

\mitemspace[BELL'S ENGLISH CLASSICS\@.] A New Series, Edited for use in
Schools, with Introduction and Notes. Crown 8vo.
\clabel{ser:Classics}

\begin{catsublist}
\mitemspace[JOHNSON'S Life of Addison.] Edited by {\tinycapsize F.~RYLAND}, Author of ``The Students'
Handbook of Psychology,'' etc. 2\s.~6\d.

\mitemspace[--- Life of Swift.] Edited by {\tinycapsize F.~RYLAND, M.A\@.} 2\s.

\mitemspace[--- Life of Pope.] Edited by {\tinycapsize F.~RYLAND, M.A\@.} 2\s.~6\d.

\mitemspace[--- Life of Milton.] Edited by {\tinycapsize F.~RYLAND, M.A\@.} 2\s.~6\d.

\mitemspace[--- Life of Dryden.] Edited by {\tinycapsize F.~RYLAND, M.A.}
\catright{Preparing.}

\mitemspace[LAMB'S Essays.] Selected and Edited by {\tinycapsize K.~DEIGHTON\@.} 3\s.; sewed, 2\s.

\mitemspace[BYRON'S Childe Harold.] Edited by {\tinycapsize H.~G. KEENE, M.A., C.I.E.}, Author of ``A
Manual of French Literature,'' etc. 3\s.~6\d. Also Cantos I. and II. separately;
sewed, 1\s.~9\d.

\mitemspace[--- Siege of Corinth.] Edited by {\tinycapsize P.~HORDERN}, late Director of Public Instruction in
Burma, 1\s.~6\d.; sewed, 1\s.

\mitemspace[MACAULAY'S Lays of Ancient Rome.] Edited by {\tinycapsize P.~HORDERN\@.} 2\s.~6\d.;
sewed, 1\s.~9\d.

\mitemspace[MASSINGER'S A New Way to Pay Old Debts.] Edited by {\tinycapsize K.~DEIGHTON.}
3\s.; sewed, 2\s.

\mitemspace[BURKE'S Letters on a Regicide Peace.] I. and II\@. Edited by {\tinycapsize H.~G. KEENE},
M.A., C.I.E\@. 3\s.: sewed, 2\s.

\mitemspace[MILTON'S Paradise Regained.] Edited by {\tinycapsize K.~DEIGHTON\@.} 2\s.~6\d.; sewed,
1\s.~6\d.

\mitemspace[SELECTIONS FROM POPE\@.] Containing Essay on Criticism, Rape of the
Lock, Temple of Fame, Windsor Forest. Edited by {\tinycapsize K.~DEIGHTON\@.} 2\s.~6\d.;
sewed, 1\s.~9\d.

\mitemspace[GOLDSMITH'S Good-Natured Man and She Stoops to Conquer.] Edited
by {\tinycapsize K.~DEIGHTON\@.} Each, 2\s.\ cloth; 1\s.~6\d.\ sewed.

\mitemspace[DE QUINCEY, Selections from.]  \textbf{The English Mail-Coach and The
Revolt of the Tartars.}  Edited by {\tinycapsize CECIL M. BARROW, M.A.}, Principal of
Victoria College, Palghât.
\catright{In the press.}

\mitemspace[MILTON'S Paradise Lost, Books I and II\@.] Edited by {\tinycapsize R.~G. OXENHAM, M.A.},
Principal of Elphinstone College, Bombay.
\catright{Preparing.}

\mitemspace[--- Books III. and IV\@.] Edited by {\tinycapsize R.~G. OXENHAM.}
\catright{Preparing.}

\mitemspace[SELECTIONS FROM CHAUCER\@.] Edited by {\tinycapsize J.~B. BILDERBECK, B.A.},
Professor of English Literature, Presidency College, Madras.
\catright{Preparing.}

\mitemspace[SHAKESPEARE'S Julius Caesar.] Edited by {\tinycapsize T.~DUFF BARNETT, B.A.} (Lond.).
2\s.

\mitemspace[--- Merchant of Venice.] Edited by {\tinycapsize T.~DUFF BARNETT, B.A.} (Lond.). 2\s.

\mitemspace[--- Tempest.] Edited by {\tinycapsize T.~DUFF BARNETT, B.A.} (Lond.), 2\s.

\begin{center} \textit{Others to follow}.\end{center}
\end{catsublist}

\mitemspace[BELL'S READING BOOKS\@.] Post 8vo, cloth, illustrated.
\clabel{ser:Reading}
\end{maincatlist}

\begin{tabulartwowide}
\tabparbox{%
\tabcenter{\textbf{Infants.}}

\textbf{Infant's Primer.} 3\d.

\textbf{Tot and the Cat.} 6\d.

\textbf{The Old Boathouse.} 6\d.

\textbf{The Cat and the Hen.} 6\d.
\bigskip

\tabcenter{\textbf{Standard I.}}

\textbf{School Primer.} 6\d.

\textbf{The Two Parrots.} 6\d.

\textbf{The Three Monkeys.} 6\d.

\textbf{The New-born Lamb.} 6\d.

\textbf{The Blind Boy.} 6\d.} &

\tabparbox{%
\tabcenter{\textbf{Standard II.}}

\textbf{The Lost Pigs.} 6\d.

\textbf{Story of a Cat.} 6\d.

\textbf{Queen Bee and Busy Bee.} 6\d.

\textbf{Gull's Crag.} 6\d.

\textbf{Great Deeds in English History.} 1\s.}

\tabparbox{%
\tabcenter{\textbf{Standard III.}}

\textbf{Adventures of a Donkey.} 1\s.

\textbf{Grimm's Tales.} 1\s.

\textbf{Great Englishmen.} 1\s.

\textbf{Andersen's Tales.} 1\s.

\textbf{Life of Columbus.} 1\s.}
\end{tabulartwowide}

%allow pagebreak here
%-----File: 330.png-----
\begin{tabulartwowide}

\tabparbox{%
\tabcenter{\textbf{Standard IV.}}

\textbf{Uncle Tom's Cabin.} 1\s.

\textbf{Great Englishwomen.} 1\s.

\textbf{Great Scotsmen.} 1\s.

\textbf{Edgeworth's Tales.} 1\s.

\textbf{Gatty's Parables from Nature.} 1\s.

\textbf{Scott's Talisman.} 1\s.
\bigskip

\tabcenter{\textbf{Standard V.}}

\textbf{Dickens' Oliver Twist.} 1\s.

\textbf{Dickens' Little Nell.} 1\s.

\textbf{Masterman Ready.} 1\s.

\textbf{Marryat's Poor Jack.} 1\s. } &

\tabparbox{%
\textbf{Arabian Nights.} 1\s.

\textbf{Gulliver's Travels.} 1\s.

\textbf{Lyrical Poetry for Boys and Girls.} 1\s.

\textbf{Vicar of Wakefield.} 1\s.
\bigskip

\tabcenter{\textbf{Standards VI. and VII.}}

\textbf{Lamb's Tales from Shakespeare.} 1\s.

\textbf{Robinson Crusoe.} 1\s.

\textbf{Tales of the Coast.} 1\s.

\textbf{Settlers in Canada.} 1\s.

\textbf{Southey's Life of Nelson.} 1\s.

\textbf{Sir Roger de Coverley.} 1\s.}

\end{tabulartwowide}

\begin{maincatlist}
\mitemspace[BELL'S GEOGRAPHICAL READERS\@.] By {\smallcapsize M.~J. BARRINGTON-WARD, M.A\@.} (Worcester College, Oxford).
\clabel{ser:Geographical}
\end{maincatlist}

\begin{tabulartwowide}
\tabparbox{%
\textbf{The Child's Geography.} Illustrated. Stiff paper cover, 6\d.

\textbf{The Map and the Compass.} (Standard I.) Illustrated. Cloth, 8\d.} &
\tabparbox{%
\textbf{The Round World.} (Standard II.) Illustrated. Cloth, 10\d.

\textbf{About England.} (Standard III.) With Illustrations and Coloured Map. Cloth, 1\s.~4\d.}
\end{tabulartwowide}

\begin{maincatlist}
\mitem[EDWARDS (F.). Examples for Analysis] in Verse and Prose from
well-known sources, selected and arranged by {\smallcapsize F.~EDWARDS\@.} \textit{New edition.}
Fcap.\ 8vo, cloth, 1\s.

\mitemspace[GOLDSMITH\@. The Deserted Village.] Edited, with Notes and Life,
by {\smallcapsize C.~P. MASON, B.A., F.C.P\@.} 4\textit{th edition.} Crown 8vo, 1\s.

\mitemspace[HANDBOOKS OF ENGLISH LITERATURE\@.] Edited by {\smallcapsize J.~W.\
HALES, M.A.}, formerly Clark Lecturer in English Literature at Trinity
College, Cambridge, Professor of English Literature at King's College,
London. Crown 8vo, 3\s.~6\d.\ each.
\clabel{ser:EngLit}

\begin{maincatlist}
\mitemspace[The Age of Pope.] By {\smallcapsize JOHN DENNIS.}  \catright{Ready.}
\end{maincatlist}

\begin{center}\textit{In preparation.}\end{center}

\begin{maincatlist}
\mitemspace[The Age of Chaucer.] By {\smallcapsize PROFESSOR HALES.}
\mitemspace[The Age of Shakespeare.] By {\smallcapsize PROFESSOR HALES.}
\mitemspace[The Age of Milton.] By {\smallcapsize J.~BASS MULLINGER, M.A.}
\mitemspace[The Age of Dryden.] By {\smallcapsize W.~GARNETT, LL.D.}
\mitemspace[The Age of Wordsworth.] By {\smallcapsize PROFESSOR C.~H. HERFORD, LITT.D.}
\end{maincatlist}

\begin{center}\textit{Other volumes to follow.}\end{center}

\mitemspace[HAZLITT (W.). Lectures on the Literature of the Age of Elizabeth.]
Small post 8vo, sewed, 1\s.

\mitemspace[--- Lectures on the English Poets.] Small post 8vo, sewed, 1\s.

\mitemspace[--- Lectures on the English Comic Writers.] Small post 8vo, sewed, 1\s.

%-----File: 331.png-----

\mitemspace[LAMB (C.).] Specimens of English Dramatic Poets of the Time of Elizabeth.
With Notes together with the Extracts from the Garrick Plays.

\mitemspace[MASON (C.~P.).] Grammars by {\smallcapsize C.~P. MASON, B.A., F.C.P.}, Fellow of
University College, London.

\mitemspace[--- First Notions of Grammar for Young Learners.] Fcap.\ 8vo. 85\textit{th
thousand.} Cloth, 1\s.

\mitem[--- First Steps in English Grammar,] for Junior Classes. Demy 18mo. 54\textit{th
    thousand}. 1\s.

\mitem[--- Outlines of English Grammar,] for the Use of Junior Classes. 17\textit{th edition}.
    97\textit{th thousand}. Crown 8vo, 2\s.

English Grammar; including the principles of Grammatical Analysis.
   35\textit{th  edition, revised. 148th thousand}. Crown 8vo, green cloth, 3\s.~6\d.

\mitem[--- A Shorter English Grammar,] with copious and carefully graduated
   Exercises, based upon the author's English Grammar. \textit{9th edition. 49th
thousand}. Crown 8vo, brown cloth, 3\s.~6\d.

\mitemspace[--- Practice and Help in the Analysis of Sentences.] Price 2\s. Cloth.

\mitem[--- English Grammar Practice,] consisting of the Exercises of the Shorter
   English Grammar published in a separate form. 3\textit{rd edition}. Crown 8vo,
   1\s.

\mitemspace[--- Remarks on the Subjunctive and the so-called Potential Mood.]
6\d., sewn.

\mitem[--- Blank Sheets] Ruled and headed for Analysis, 1\s.\ per dozen.

\mitemspace[MILTON: Paradise Lost.] Books I., II., and III\@. Edited, with Notes
    on the Analysis and Parsing, and Explanatory Remarks, by
    {\smallcapsize C.~P. MASON, B.A., F.C.P\@.} Crown 8vo.

Book I\@. With Life. 5\textit{th edition}. 1\s.

Book II\@. With Life. 3\textit{rd edition}. 1\s.

Book III\@. With Life. 2\textit{nd edition}. 1\s.

\mitemspace[--- Paradise Lost.] Books V.--VIII\@. With Notes for the Use of Schools.
By C.~M. LUMBY. 2\s.~6\d.

\mitemspace[PRICE (A.~C.). Elements of Comparative Grammar and Philology.]
     For Use in Schools. By {\smallcapsize A.~C. PRICE, M.A.}, Assistant Master at Leeds
     Grammar School; late Scholar of Pembroke College, Oxford. Crown
     8vo, 2\s.~6\d.

\mitemspace[SHAKESPEARE\@. Notes on Shakespeare's Plays.] With Introduction,
     Summary, Notes (Etymological and Explanatory), Prosody, Grammatical
     Peculiarities, etc. By {\smallcapsize T.~DUFF BARNETT, B.A.} Lond., late Second
     Master in the Brighton Grammar School. Specially adapted for the Local
     and Preliminary Examinations. Crown 8vo, 1\s.\ each.

\textbf{Midsummer Night's Dream. --- Julius Cæsar. --- The Tempest. --- Macbeth. --- Henry
V. --- Hamlet. --- Merchant of Venice. --- King
Richard II. --- King John. --- King Lear. --- Coriolanus.}

``The Notes are comprehensive and concise.''---\textit{Educational Times.}

``Comprehensive, practical, and reliable.''---\textit{Schoolmaster.}

\mitemspace[--- Hints for Shakespeare-Study.] Exemplified in an Analytical Study of
    Julius Cæsar. By {\smallcapsize MARY GRAFTON MOBERLY}. 2\textit{nd edition}. Crown 8vo,
    sewed, 1\s.

\mitemspace[--- Coleridge's Lectures and Notes on Shakespeare and other English Poets.]
Edited by {\smallcapsize T.~ASHE, B.A\@.} Small post 8vo, 3\s.~6\d.
%-----File: 332.png-----

\mitemspace[--- Shakespeare's Dramatic Art.] The History and
Character of Shakespeare's Plays. By {\smallcapsize DR. HERMANN ULRICI\@.} Translated
by {\smallcapsize L.~DORA SCHMITZ\@.} 2 vols.\ small post 8vo, 3\s.~6\d.\ each.

\mitemspace[--- William Shakespeare.] A Literary Biography. By {\smallcapsize KARL ELZE, PH.D.,
LL.D\@.} Translated by {\smallcapsize L.~DORA SCHMITZ}. Small post 8\textit{vo}, 5\s.

\mitemspace[--- Hazlitt's Lectures on the Characters of Shakespeare's Plays.] Small
post 8\textit{vo}, 1\s.

\mitemspace[{\normalfont \textit{See}} BELL'S ENGLISH CLASSICS.]

\mitemspace[SKEAT (W.~W.). Questions for Examinations in English Literature.]
With a Preface containing brief hints on the study of English.
Arranged by the {\smallcapsize REV. W.~W. SKEAT, LITT. D.}, Elrington and Bosworth
Professor of Anglo-Saxon in the University of Cambridge, 3\textit{rd edition.}
Crown 8vo, 2\s.~6\d.

\mitemspace[SMITH (C.~J.). Synonyms and Antonyms of the English Language.]
Collected and Contrasted by the {\smallcapsize VEN. C.~J. SMITH, M.A\@.} 2\textit{nd edition,
revised.} Small post 8\textit{vo}, 5\s.

\mitemspace[--- Synonyms Discriminated.] A Dictionary of Synonymous Words in the
English Language. Illustrated with Quotations from Standard Writers.
By the late {\smallcapsize VEN. C.~J. SMITH, M.A\@.} With the Author's latest Corrections
and Additions, edited by the {\smallcapsize REV. H.~PERCY SMITH, M.A.}, of Balliol
College, Oxford, Vicar of Great Barton, Suffolk. 4\textit{th edition.} Demy
8\textit{vo}, 14\s.

\mitemspace[TEN BRINK'S History of English Literature.] Vol.~I\@. Early English
Literature (to Wiclif). Translated into English by {\smallcapsize HORACE M.~KENNEDY},
Professor of German Literature in the Brooklyn Collegiate Institute.
Small post 8vo, 3\s.~6\d.

\mitemspace[--- Vol.~II\@] (Wiclif, Chaucer, Earliest Drama, Renaissance). Translated by
{\smallcapsize W. CLARKE ROBINSON, PH.D\@.} Small post 8vo, 3\s.~6\d.

\mitemspace[THOMSON: Spring.] Edited by {\smallcapsize C.~P. MASON, B.A., F.C.P\@.} With Life.
\textit{2nd edition.} Crown 8vo, 1\s.

\mitemspace[--- Winter.] Edited by {\smallcapsize C.~P. MASON, B.A., F.C.P\@.} With Life. Crown 8vo, 1\s.

{\stretchyspace
\mitem[WEBSTER'S INTERNATIONAL DICTIONARY] of the English
Language.} Including Scientific, Technical, and Biblical Words and
Terms, with their Significations, Pronunciations, Alternative Spellings,
Derivations, Synonyms, and numerous illustrative Quotations, with various
valuable literary Appendices, with 83 extra pages of Illustrations grouped
and classified, rendering the work a \textsc{Complete Literary and Scientific
Reference-book.} \textit{New edition} (1890). Thoroughly revised and enlarged
under the supervision of {\smallcapsize NOAH PORTER, D.D., LL.D\@.} 1 vol.\ (2,118
pages, 3,500 woodcuts), 4to, cloth, 31\s.~6\d.; half calf, \L2 2\s.; half russia,
\L2 5\s.; calf, \L2 8\s.; or in 2 vols.\ cloth, \L1 14\s.

\begin{center}\textit{Prospectuses, with specimen pages, sent post free on application.}\end{center}

\mitemspace[WEBSTERS BRIEF INTERNATIONAL DICTIONARY\@.] A
Pronouncing Dictionary of the English Language, abridged from Webster's
International Dictionary. With a Treatise on Pronunciation, List of
Prefixes and Suffixes, Rules for Spelling, a Pronouncing Vocabulary of
Proper Names in History, Geography, and Mythology, and Tables of
English and Indian Money, Weights, and Measures. With 564 pages
and 800 Illustrations. Demy 8vo, 3\s.
%-----File: 333.png-----

\mitemspace[WRIGHT (T.). Dictionary of Obsolete and Provincial English.]
Containing Words from the English Writers previous to the 19th century,
which are no longer in use, or are not used in the same sense, and Words
which are now used only in the Provincial Dialects. Compiled by {\smallcapsize THOMAS
WRIGHT, M.A., F.S.A.}, etc. 2 vols. 5\s.\ each.
\end{maincatlist}

\catsectionuppertoc{French Class Books}

\begin{maincatlist}

\mitemspace[BOWER (A.~M.). The Public Examination French Reader.] With
a Vocabulary to every extract, suitable for all Students who are preparing
for a French Examination. By {\smallcapsize A.~M. BOWER, F.R.G.S.}, late Master in
University College School, etc. Cloth, 3\s.~6\d.

\mitemspace[BARBIER (PAUL). A Graduated French Examination Course.]
By {\smallcapsize PAUL BARBIER}, Lecturer in the South Wales University College, etc.
Crown 8vo, 3\s.

\mitemspace[BARRÈRE (A.). Junior Graduated French Course.] Affording Materials
for Translation, Grammar, and Conversation. By {\smallcapsize A.~BARRÈRE},
Professor R.M.A., Woolwich, 1\s.~6\d.

\mitem[--- Elements of French Grammar] and \textbf{First Steps in Idioms.} With
numerous Exercises and a Vocabulary. Being an Introduction to the
Précis of Comparative French Grammar. Crown 8vo, 2\s.

\mitemspace[--- Précis of Comparative French Grammar and Idioms and Guide to Examinations.]
4\textit{th edition.} 3\s.~6\d.

\mitemspace[--- Récits Militaires.] From Valmy (1792) to the Siege of Paris (1870).
With English Notes and Biographical Notices. 2nd edition. Crown
8vo, 3\s.

\mitemspace[CLAPIN (A.~C.). French Grammar for Public Schools.] By the
{\smallcapsize REV. A.~C. CLAPIN, M.A.}, St.~John's College, Cambridge, and Bachelierès-lettres
of the University of France. Fcap.\ 8vo. 13\textit{th edition.} 2\s.~6\d.

Key to the Exercises. 3\s.~6\d.\ net.

\mitemspace[--- French Primer.] Elementary French Grammar and Exercises for Junior
Forms in Public and Preparatory Schools. Fcap.\ 8vo. 10\textit{th edition.} 1\s.

\mitemspace[--- Primer of French Philology.] With Exercises for Public Schools.
6\textit{th edition.} Fcap.\ 8vo, 1\s.

\mitemspace[--- English Passages for Translation into French.] Crown 8vo, 2\s.~6\d.
Key (for Tutors only), 4\s.\ net.

\mitemspace[DAVIS (J.~F.).] \textbf{Army Examination Papers in French.} Questions set
at the Preliminary Examinations for Sandhurst and Woolwich, from Nov.,
1876, to June, 1890, with Vocabulary. By {\smallcapsize J.~F. DAVIS, D.LIT., M.A.},
Lond. Crown 8vo, 2\s.~6\d.

\mitemspace[DAVIS (J.~F.) and THOMAS (F.). An Elementary French Reader.]
Compiled, with a Vocabulary, by {\smallcapsize J.~F. DAVIS, M.A., D.LIT.},
and {\smallcapsize FERDINAND THOMAS}, Assistant Examiners in the University of
London. Crown 8vo, 2\s.

\mitemspace[DELILLE'S GRADUATED FRENCH COURSE.]
\end{maincatlist}

\begin{tabulartwowide}
\tabparbox{%
\textbf{The Beginner's own French Book.} 2\s. Key, 2\s.

\textbf{Easy French Poetry for Beginners.} 2\s.

\textbf{French Grammar.} 3\s. Key, 3\s.} &

\tabparbox{%
 \textbf{Repertoire des Prosateurs.} 3\s.~6\d.

 \textbf{Modèles de Poesie.} 3\s.~6\d.

 \textbf{Manuel Etymologique.} 2\s.~6\d.

 \textbf{Synoptical Table of French Verbs.} 6\d.}
\end{tabulartwowide}
%-----File: 334.png-----

\begin{maincatlist}
\mitem[GASC (F.~E.~A.). First French Book;] being a New, Practical, and
Easy Method of Learning the Elements of the French Language. \textit{Reset
and thoroughly revised. 116th thousand.} Crown 8vo, 1\s.

\mitem[--- Second French Book;] being a Grammar and Exercise Book, on a new
and practical plan, and intended as a sequel to the ``First French Book.''
52\textit{nd thousand.} Fcap.\ 8vo, 1\s.~6\d.

\mitem[--- Key] to First and Second French Books, 5\textit{th edition}, Fcap.\ 8vo, 3\s.~6\d.\ net.

\mitem[--- French Fables,] for Beginners, in Prose, with an Index of all the Words
at the end of the work. 16\textit{th thousand.} 12mo, 1\s.~6\d.

\mitemspace[--- Select Fables of La Fontaine.] 19\textit{th thousand.} Fcap.\ 8vo, 1\s.~6\d.

\mitem[--- Histoires Amusantes et Instructives;] or, Selections of Complete
Stories from the best French modern authors, who have written for the
young. With English notes, 17\textit{th thousand.} Fcap.\ 8vo, 2\s.

\mitem[--- Practical Guide to Modern French Conversation,] containing:---I.
The most current and useful Phrases in Everyday Talk. II\@. Everybody's
necessary Questions and Answers in Travel-Talk. 19\textit{th edition.}
Fcap.\ 8vo, 1\s.~6\d.

\mitemspace[--- French Poetry for the Young.] With Notes, and preceded by a few
plain Rules of French Prosody. 5\textit{th edition, revised.} Fcap.\ 8vo, 1\s.~6\d.

\mitemspace[--- French Prose Composition, Materials for.] With copious footnotes, and
hints for idiomatic renderings. 21\textit{st thousand.} Fcap.\ 8vo, 3\s.

Key. 2\textit{nd edition.} 6\s.\ net.

\mitem[--- Prosateurs Contemporains;] or, Selections in Prose chiefly from contemporary
French literature. With notes. 11\textit{th edition.} 12mo, 3\s.~6\d.

\mitem[--- Le Petit Compagnon;] a French Talk-Book for Little Children, 14\textit{th
edition.} 16mo, 1\s.~6\d.

\mitem[--- French and English Dictionary,] with upwards of Fifteen Thousand
new words, senses, \&c., hitherto unpublished. 5\textit{th edition, with numerous
additions and corrections.} In one vol. 8vo, cloth, 10\s.~6\d. \textbf{In use at
Harrow, Rugby, Shrewsbury, \&c.}

\mitem[--- Pocket Dictionary] of the French and English Languages; for the everyday
purposes of Travellers and Students. Containing more than Five
Thousand modern and current words, senses, and idiomatic phrases
and renderings, not found in any other dictionary of the two languages.
\textit{New edition}. 51\textit{st thousand.} 16mo, cloth, 2\s.~6\d.

\mitem[GOSSET (A.). Manual of French Prosody] for the use of English
Students. By {\smallcapsize ARTHUR GOSSET, M.A.}, Fellow of New College, Oxford.
Crown 8vo, 3\s.

``This is the very book we have been looking for. We hailed the title
with delight, and were not disappointed by the perusal. The reader who
has mastered the contents will know, what not one in a thousand of
Englishmen who read French knows, the rules of French poetry.''---\textit{Journal
of Education.}

\mitem[LE NOUVEAU TRÉSOR;] designed to facilitate the Translation of
English into French at Sight. By {\smallcapsize M.~E.~S\@.} 18\textit{th edition.} Fcap.\ 8vo,
1\s.~6\d.

\mitemspace[STEDMAN (A.~M.~M.).] French Examination Papers in Miscellaneous
Grammar and Idioms. Compiled by {\smallcapsize A.~M.~M. STEDMAN, M.A.}
5\textit{th edition.} Crown 8vo, 2\s.~6\d.

A Key. By {\smallcapsize G.~A. SCHRUMPF\@.} For Tutors only. 6\s.\ net.

\mitemspace[--- Easy French Passages for Unseen Translation.] Fcap.\ 8vo, 1\s.~6\d.

%-----File: 335.png-----

\mitem[--- Easy French Exercises] on Elementary
Syntax. Crown 8vo, 2\s.~6\d.

\mitemspace[--- First French Lessons.] Crown 8vo, 1\s.

\mitemspace[--- French Vocabularies for Repetition.] Fcap.\ 8vo, 1\s.

\mitemspace[--- Steps to French.] 12mo, 8\d.
\end{maincatlist}

\catsectionuppertoc{French Annotated Editions}

\begin{maincatlist}

\mitemspace[BALZAC\@. Ursule Mirouët.] By {\smallcapsize HONORÉ DE BALZAC\@.} Edited, with
Introduction and Notes, by {\smallcapsize JAMES BOÏELLE, B.-ès-L.}, Senior French
Master, Dulwich College. 3\s.

\mitemspace[CLARÉTIE\@. Pierrille.] By {\smallcapsize JULES CLARÉTIE\@.} With 27 Illustrations.
Edited, with Introduction and Notes, by {\smallcapsize JAMES BOÏELLE, B.-ès-L\@.} 2\s.~6\d.

\mitemspace[DAUDET\@. La Belle Nivernaise.] Histoire d'un vieux bateau et de son
équipage. By {\smallcapsize ALPHONSE DAUDET\@.} Edited, with Introduction and
Notes, by {\smallcapsize JAMES BOÏELLE, B.-ès-L\@.} With Six Illustrations. 2\s.

\mitemspace[FÉNELON\@. Aventures de Télémaque.] Edited by {\smallcapsize C.~J. DELILLE\@.}
4\textit{th edition.} Fcap.\ 8vo, 2\s.~6\d.

\clabel{ser:Gombert}
\mitemspace[GOMBERT'S FRENCH DRAMA\@.] Re-edited, with Notes, by {\smallcapsize F.~E.~A.\
GASC\@.} Sewed, 6\d.\ each.
\end{maincatlist}

\begin{tabulartwowide}
\multicolumn{2}{c}{MOLIÈRE}\\
\tabparbox{%
  \textbf{Le Misanthrope.}

  \textbf{L'Avare.}

  \textbf{Le Bourgeois Gentilhomme.}

  \textbf{Le Tartuffe.}

  \textbf{Le Malade Imaginaire.}

  \textbf{Les Femmes Savantes.} } &
\tabparbox{%
  \textbf{Les Fourberies de Scapin.}

  \textbf{Les Précieuses Ridicules.}

  \textbf{L'Ecole des Femmes.}

  \textbf{L'Ecole des Maris.}

  \textbf{Le Médecin Malgré Lui.} }
\end{tabulartwowide}

\begin{tabulartwowide}
\multicolumn{2}{c}{RACINE}\\
\tabparbox{%
  \textbf{La Thébaïde, ou Les Frères Ennemis.}

  \textbf{Andromaque.}

  \textbf{Les Plaideurs.}

  \textbf{Iphigénie.} } &
\tabparbox{%
  \textbf{Britannicus.}

  \textbf{Phèdre.}

  \textbf{Esther.}

  \textbf{Athalie.} }
\end{tabulartwowide}

\begin{tabulartwowide}
\multicolumn{2}{c}{CORNEILLE.}  \\
\tabparbox{%
  \textbf{Le Cid.}

  \textbf{Horace.} } &
\tabparbox{%
  \textbf{Cinna.}

  \textbf{Polyeucte.} }
\end{tabulartwowide}

{\centering VOLTAIRE.---\textbf{Zaïre.}\par}

\begin{maincatlist}

\mitemspace[GREVILLE\@. Le Moulin Frappier.] By {\smallcapsize HENRY GREVILLE\@.} Edited,
with Introduction and Notes, by {\smallcapsize JAMES BOÏELLE, B.-ès-L\@.}  3\s.

\mitemspace[HUGO\@. Bug Jargal.] Edited, with Introduction and Notes, by {\smallcapsize JAMES
BOÏELLE, B.-ès-L\@.} 3\s.

\mitemspace[LA FONTAINE\@. Select Fables.] Edited by {\smallcapsize F.~E.~A. GASC\@.} 19\textit{th
thousand}. Fcap.\ 8vo, 1\s.~6\d.

\mitemspace[LAMARTINE\@. Le Tailleur de Pierres de Saint-Point.] Edited with
Notes by {\smallcapsize JAMES BOÏELLE, B.-ès-L\@.} 6\textit{th thousand.} Fcap.\ 8vo, 1\s.~6\d.

\mitemspace[SAINTINE\@. Picciola.] Edited by {\smallcapsize DR. DUBUC\@.} 16\textit{th thousand.} Fcap.\
8vo, 1\s.~6\d.

\mitemspace[VOLTAIRE\@. Charles XII\@.] Edited by {\smallcapsize L.~DIREY\@.} 7\textit{th edition.} Fcap.\
8vo, 1\s.~6\d.
\end{maincatlist}

%-----File: 336.png-----

\catsectionuppertoc{German Class Books}

\begin{maincatlist}

\mitemspace[BUCHHEIM (DR. C.~A.). German Prose Composition.] Consisting
of Selections from Modern English Writers. With grammatical notes,
idiomatic renderings, and general introduction. By {\smallcapsize C.~A.~BUCHHEIM, PH.D.},
Professor of the German Language and Literature in King's College, and
Examiner in German to the London University. 14\textit{th edition, enlarged and
revised.} With a list of subjects for original composition. Fcap.\ 8vo, 4\s.~6\d.
A Key to the 1st and 2nd parts. 3\textit{rd edition.} 3\s.\ net. To the 3rd and
4th parts. 4\s.\ net.

\mitemspace[--- First Book of German Prose.] Being Parts I. and II. of the above.
With Vocabulary. Fcap.\ 8vo, 1\s.~6\d.

\mitemspace[CLAPIN (A.~C.). A German Grammar for Public Schools.] By the
{\smallcapsize REV. A.~C.~CLAPIN}, and {\smallcapsize F.~HOLL-MÜLLER}, Assistant Master at the Bruton
Grammar School. 6\textit{th edition.} Fcap.\ 8vo, 2\s.~6\d.

\mitemspace[--- A German Primer.] With Exercises. 2\textit{nd edition.} Fcap.\ 8vo, 1\s.

\mitemspace[German.  The Candidate's Vade Mecum.]  Five Hundred Easy
Sentences and Idioms. By an Army Tutor. Cloth, 1\s. For Army
Prelim. Exam.

\mitemspace[LANGE (F.). A Complete German Course for Use in Public Schools.]
By {\smallcapsize F.~LANGE, PH.D.}, Professor R.M.A. Woolwich, Examiner in German
to the College of Preceptors, London; Examiner in German at the Victoria
University, Manchester. Crown 8vo.

\begin{catsublist}

\mitemspace[Concise German Grammar.] With special reference to Phonology,
Comparative Philology, English and German Equivalents and Idioms.
Comprising Materials for Translation, Grammar, and Conversation.
Elementary, 2\s.; Intermediate, 2\s.; Advanced, 3\s.~6\d.

\mitemspace[Progressive German Examination Course.] Comprising the Elements
of German Grammar, an Historic Sketch of the Teutonic Languages,
English and German Equivalents, Materials for Translation, Dictation,
Extempore Conversation, and Complete Vocabularies. I. Elementary
Course, 2\s. II. Intermediate Course, 2\s. III. Advanced Course.
\textit{Second revised edition.} 1\s.~6\d.

\mitemspace[Elementary German Reader.] A Graduated Collection of Readings in
Prose and Poetry. With English Notes and a Vocabulary. 4\textit{th
edition.} 1\s.~6\d.

\mitemspace[Advanced German Reader.] A Graduated Collection of Readings in
Prose and Poetry. With English Notes by {\tinycapsize F.~LANGE, PH.D.}; and
{\tinycapsize J.~F. DAVIS, D.LIT\@.} 2\textit{nd edition.} 3\s.

\end{catsublist}

\mitem[MORICH (R.~J.).  German Examination Papers] in Miscellaneous
Grammar and Idioms. By {\smallcapsize R.~J. MORICH}, Manchester Grammar School.
2\textit{nd edition.} Crown 8vo, 2\s.~6\d. A Key, for Tutors only. 5\s.\ net.

\mitem[STOCK (DR.). Wortfolge,] or Rules and Exercises on the order of Words
in German Sentences. With a Vocabulary. By the late {\smallcapsize FREDERICK
STOCK, D.LIT., M.A\@.}  Fcap.\ 8vo, 1\s.~6\d.

\shortrule

\mitemspace[KLUGE'S Etymological Dictionary of the German Language.]
Translated by {\smallcapsize J.~F. DAVIS, D.LIT.} (Lond.). Crown 4to, 18\s.
\end{maincatlist}

%-----File: 337.png-----

\catsectionuppertoc{German Annotated Editions}

\begin{maincatlist}

\mitemspace[AUERBACH (B.). Auf Wache.] Novelle von {\smallcapsize BERTHOLD AUERBACH.}
Der Gefrorene Kuss. Novelle von {\smallcapsize OTTO ROQUETTE\@.} Edited by {\smallcapsize A.~A.
MACDONELL, M.A., PH.D\@.} 2\textit{nd edition.} Crown 8vo, 2\s.

\mitemspace[BENEDIX (J.~R.). Doktor Wespe.] Lustspiel in fünf Aufzügen von
{\smallcapsize JULIUS RODERICH BENEDIX\@.}  Edited by {\smallcapsize PROFESSOR F.~LANGE, PH.D.}
Crown 8vo, 2\s.~6\d.

\mitemspace[EBERS (G.). Eine Frage.] Idyll von {\smallcapsize GEORG EBERS\@.} Edited by {\smallcapsize F.~STORR,
B.A.}, Chief Master of Modern Subjects in Merchant Taylors' School.
Crown 8vo, 2\s.

\mitemspace[FREYTAG (G.). Die Journalisten.] Lustspiel von {\smallcapsize GUSTAV FREYTAG\@.}
Edited by {\smallcapsize PROFESSOR F.~LANGE, PH.D\@.} 4th revised edition. Crown 8vo,
2\s.~6\d.

\mitemspace[--- SOLL UND HABEN\@.] Roman von {\smallcapsize GUSTAV FREYTAG\@.} Edited by
{\smallcapsize W.~HANBY CRUMP, M.A\@.}  Crown 8vo, 2\s.~6\d.

\mitem[GERMAN BALLADS] from Uhland, Goethe, and Schiller. With Introductions,
Copious and Biographical Notices. Edited by {\smallcapsize C.~L. BIELEFELD\@.}
4th edition. Fcap.\ 8vo, 1\s.~6\d.

\mitemspace[GERMAN EPIC TALES IN PROSE\@.] I. Die Nibelungen, von
{\smallcapsize A.~F.~C. VILMAR\@.} II. Walther und Hildegund, von {\smallcapsize ALBERT RICHTER\@.}
Edited by {\smallcapsize KARL NEUHAUS, PH.D.}, the International College, Isleworth.
Crown 8vo, 2\s.~6\d.

\mitemspace[GOETHE\@. Hermann und Dorothea.] With Introduction, Notes, and
Arguments. By {\smallcapsize E.~BELL, M.A.}, and {\smallcapsize E. WÖLFEL\@.} 2\textit{nd edition.} Fcap.
8vo, 1\s.~6\d.

\mitemspace[GOETHE\@. FAUST\@.] Part I. German Text with Hayward's Prose
Translation and Notes. Revised, With Introduction by {\smallcapsize C.~A. BUCHHEIM,
PH.D.}, Professor of German Language and Literature at King's College,
London. Small post 8vo, 5\s.

\mitemspace[GUTZKOW (K.). Zopf und Schwert.] Lustspiel von {\smallcapsize KARL GUTZKOW.}
Edited by {\smallcapsize PROFESSOR F.~LANGE, PH.D\@.} Crown 8vo, 2\s.~6\d.

\mitemspace[HEY'S FABELN FÜR KINDER\@.] Illustrated by {\smallcapsize O.~SPECKTER.}
Edited, with an Introduction, Grammatical Summary, Words, and a complete
Vocabulary, by {\smallcapsize PROFESSOR F.~LANGE, PH.D\@.} Crown 8vo, 1\s.~6\d.

\mitem[---] The same. With a Phonetic Introduction, and Phonetic Transcription of
the Text. By {\smallcapsize PROFESSOR F.~LANGE, PH.D\@.} Crown 8vo, 2\s.

\mitemspace[HEYSE (P.). Hans Lange.] Schauspiel von {\smallcapsize PAUL HEYSE\@.} Edited by
{\smallcapsize A.~A. MACDONELL, M.A., PH.D.}, Taylorian Teacher, Oxford University.
Crown 8vo, 2\s.

\mitem[HOFFMANN. (E.~T.~A.). Meister Martin,] der Küfner. Erzählung
von {\smallcapsize E.~T.~A. HOFFMANN\@.} Edited by {\smallcapsize F.~LANGE, PH.D\@.} 2nd edition.
Crown 8vo, 1\s.~6\d.

\mitemspace[MOSER (G. VON). Der Bibliothekar.] Lustspiel von {\smallcapsize G.~VON MOSER.}
Edited by {\smallcapsize F.~LANGE, PH.D\@.} 4th edition. Crown 8vo, 2\s.

\mitemspace[ROQUETTE (O.).] \textit{See} \textbf{Auerbach.}

\mitemspace[SCHEFFEL (V. VON). Ekkehard.] Erzählung des zehnten Jahrhunderts,
von {\smallcapsize VICTOR VON SCHEFFEL\@.} Abridged edition, with Introduction
and Notes by {\smallcapsize HERMAN HAGER, PH.D.}, Lecturer in the German
Language and Literature in The Owens College, Victoria University,
Manchester. Crown 8vo, 3\s.
%-----File: 338.png-----

\mitemspace[SCHILLER'S Wallenstein.] Complete Text, comprising the Weimar
Prologue, Lager, Piccolomini, and Wallenstein's Tod. Edited by {\smallcapsize DR.
BUCHHEIM}, Professor of German in King's College, London. 6\textit{th edition.}
Fcap.\ 8vo, 5\s. Or the Lager and Piccolomini, 2\s.~6\d. Wallenstein's
Tod, 2\s.~6\d.

\mitemspace[--- Maid of Orleans.] With English Notes by {\smallcapsize DR. WILHELM WAGNER\@.} 3\textit{rd
edition.} Fcap.\ 8vo, 1\s.~6\d.

\mitemspace[--- Maria Stuart.] Edited by {\smallcapsize V.~KASTNER}, B.-ès-L., Lecturer on French
Language and Literature at Victoria University, Manchester. 3\textit{rd edition.}
Fcap.\ 8vo, 1\s.~6\d.
\end{maincatlist}

\catsectionuppertoc{Italian}

\begin{maincatlist}

\mitemspace[CLAPIN (A.~C.). Italian Primer.] With Exercises. By the {\smallcapsize REV. A.~C.
CLAPIN, M.A.}, B.-ès-L. 3\textit{rd edition.} Fcap.\ 8vo, 1\s.

\mitemspace[DANTE\@. The Inferno.] A Literal Prose Translation, with the Text of the
Original collated with the best editions, printed on the same page, and
Explanatory Notes. By {\smallcapsize JOHN A.~CARLYLE, M.D\@.} With Portrait. 2\textit{nd
edition.} Small post 8vo, 5\s.

\mitemspace[--- The Purgatorio.] A Literal Prose Translation, with the Text of Bianchi
printed on the same page, and Explanatory Notes. By {\smallcapsize W.~S. DUGDALE.}
Small post 8vo, 5\s.
\end{maincatlist}

\catsectionuppertoc{Bell's Modern Translations}
\clabel{ser:Translations}

\begin{center}
\textit{A Series of Translations from Modern Languages, with Memoirs,
Introductions, etc. Crown 8vo, 1\s.\ each.}
\end{center}

\begin{maincatlist}

\mitemspace[GOETHE\@. Egmont.] Translated by {\smallcapsize ANNA SWANWICK\@.} With Memoir.

\mitemspace[--- Iphigenia in Tauris.] Translated by {\smallcapsize ANNA SWANWICK\@.} With Memoir.

\mitemspace[HAUFF\@. The Caravan.] Translated by {\smallcapsize S.~MENDEL\@.} With Memoir.

\mitemspace[LESSING\@. Laokoon.] Translated by {\smallcapsize E.~C. BEASLEY\@.} With Memoir.

\mitemspace[--- Nathan the Wise.] Translated by {\smallcapsize R.~DILLON BOYLAN\@.} With Memoir.

\mitemspace[--- Minna von Barnhelm.] Translated by {\smallcapsize ERNEST BELL, M.A\@.} With
Memoir.

\mitemspace[MOLIÈRE\@. The Misanthrope.] Translated by {\smallcapsize C.~HERON WALL\@.} With
Memoir.

\mitemspace[--- The Doctor in Spite of Himself.] (Le Médecin malgré lui). Translated
by {\smallcapsize C.~HERON WALL\@.} With Memoir.

\mitemspace[--- Tartuffe; or, The Impostor.] Translated by {\smallcapsize C.~HERON WALL\@.} With
Memoir.

\mitemspace[--- The Miser.] (L'Avare). Translated by {\smallcapsize C.~HERON WALL\@.} With Memoir.

\mitemspace[--- The Shopkeeper turned Gentleman.] (Le Bourgeois Gentilhomme).
Translated by {\smallcapsize C.~HERON WALL\@.} With Memoir.

\mitemspace[RACINE\@. Athalie.] Translated by {\smallcapsize R.~BRUCE BOSWELL, M.A\@.} With
Memoir.

\mitemspace[--- Esther.] Translated by {\smallcapsize R.~BRUCE BOSWELL, M.A\@.} With Memoir.

%-----File: 339.png-----

\mitemspace[SCHILLER\@. William Tell.] Translated by {\smallcapsize SIR THEODORE MARTIN,
K.C.B., LL.D\@.} \textit{New edition, entirely revised.} With Memoir.

\mitemspace[--- The Maid of Orleans.] Translated by {\smallcapsize ANNA SWANWICK\@.} With Memoir.

\mitemspace[--- Mary Stuart.] Translated by {\smallcapsize J.~MELLISH\@.} With Memoir.

\begin{center}
\threestars For other Translations of Modern Languages, \textit{see} the Catalogue of
Bohn's Libraries, which will be forwarded on application.
\end{center}

\end{maincatlist}

\cchaptercat{SCIENCE, TECHNOLOGY, AND ART}

\catsectionuppertoc{Chemistry}

\begin{maincatlist}

\mitemspace[STÖCKHARDT (J.~A.). Experimental Chemistry.] Founded on the
work of {\smallcapsize J.~A. STÖCKHARDT\@.} A Handbook for the Study of Science by
Simple Experiments. By {\smallcapsize C.~W. HEATON, F.I.C., F.C.S.}, Lecturer in
Chemistry in the Medical School of Charing Cross Hospital, Examiner in
Chemistry to the Royal College of Physicians, etc. New revised edition.
5\s.

\mitemspace[WILLIAMS (W.~M.). The Framework of Chemistry.] Part I. Typical
Facts and Elementary Theory. By {\smallcapsize W.~M. WILLIAMS, M.A.}, St.~John's
College, Oxford; Science Master, King Henry VIII.'s School, Coventry.
Crown 8vo, paper boards, 9\d.\ net.
\end{maincatlist}

\catsectionuppertoc{Botany}

\begin{maincatlist}

{\stretchyspace
\mitemspace[EGERTON-WARBURTON (G.). Names and Synonyms of British Plants.]
By the {\smallcapsize REV. G.~EGERTON-WARBURTON\@.} Fcap.\ 8vo, 3\s.~6\d.
(\textit{Uniform with Hayward's Botanist's Pocket Book.})
}

\mitemspace[HAYWARD (W. R.). The Botanist's Pocket-Book.] Containing in
a tabulated form, the chief characteristics of British Plants, with the
botanical names, soil, or situation, colour, growth, and time of flowering
of every plant, arranged under its own order; with a copious Index.
By {\smallcapsize W.~R. HAYWARD\@.} 6\textit{th edition, revised.} Fcap.\ 8vo, cloth limp, 4\s.~6\d.

\mitemspace[MASSEE (G.). British Fungus-Flora.] A Classified Text-Book of
Mycology. By {\smallcapsize GEORGE MASSEE}, Author of ``The Plant World.'' With
numerous Illustrations. 3 vols.\ post 8vo. Vols. I., II., and III. ready,
7\s.~6\d.\ each. Vol.~IV. in the Press.

\mitemspace[SOWERBY'S English Botany.] Containing a Description and Life-size
Drawing of every British Plant. Edited and brought up to the present
standard of scientific knowledge, by {\smallcapsize T.~BOSWELL} (late {\smallcapsize SYME}), {\smallcapsize LL.D.,
F.L.S.}, etc. 3\textit{rd edition, entirely revised.} With Descriptions of all the
Species by the Editor, assisted by {\smallcapsize N.~E. BROWN\@.} 12 vols., with 1,937
\textit{coloured plates}, \L24 3\s.\ in cloth, \L26 11\s.\ in half-morocco, and \L30 9\s.\ in
whole morocco. Also in 89 parts, 5\s., except Part 89, containing
an Index to the whole work, 7\s.~6\d.

\threestars A Supplement, to be completed in 8 or 9 parts, is now publishing.
Parts I., II., and III. ready, 5\s.\ each, or bound together, making
Vol.~XIII. of the complete work, 17\s.
%-----File: 340.png-----

\mitem[TURNBULL (R.). Index of British Plants,] according to the London
Catalogue (Eighth Edition), including the Synonyms used by the principal
authors, an Alphabetical List of English Names, etc. By {\smallcapsize ROBERT
TURNBULL\@.} Paper cover, 2\s.~6\d., cloth, 3\s.
\end{maincatlist}

\catsectionuppertoc{Geology}

\begin{maincatlist}

\mitemspace[JUKES-BROWNE (A.~J.). Student's Handbook of Physical Geology.]
By {\smallcapsize A.~J. JUKES-BROWNE, B.A., F.G.S.}, of the Geological Survey of
England and Wales. With numerous Diagrams and Illustrations. 2\textit{nd
edition, much enlarged}, 7\s.~6\d.

``Should be in the hands of every teacher of geology.''---\textit{Journal of
Education.}

``A very useful book dealing with geology from its physical side.''---\textit{Ath\-e\-næum.}

\mitemspace[--- Student's Handbook of Historical Geology.] With numerous Diagrams
and Illustrations. 6\s.

``An admirably planned and well executed `Handbook of Historical
Geology.'\;''---\textit{Journal of Education.}

\mitemspace[--- The Building of the British Isles.] A Study in Geographical Evolution.
With Maps. 2\textit{nd edition revised.} 7\s.~6\d.
\end{maincatlist}

\catsectionuppertoc{Medicine}

\begin{maincatlist}

\mitemspace[CARRINGTON (R.~E.), and LANE (W.~A.). A Manual of Dissections of the Human Body.]
For the use of Students, and particularly
for those preparing for the Higher Examinations in Anatomy. By the
late {\smallcapsize R.~E. CARRINGTON, M.D.} (Lond.), {\smallcapsize F.R.C.P.}, Senior Assistant Physician,
Guy's Hospital. 2\textit{nd edition.} Revised and enlarged by {\smallcapsize W.
ARBUTHNOT LANE, M.S., F.R.C.S.}, Assistant Surgeon to Guy's Hospital,
and the Hospital for Sick Children, Great Ormond Street, etc. Crown
8vo, 9\s.

``As solid a piece of work as ever was put into a book; accurate from
beginning to end, and unique of its kind.''---\textit{British Medical Journal.}

\mitemspace[HILTON'S Rest and Pain.] Lectures on the Influence of Mechanical and
Physiological Rest in the Treatment of Accidents and Surgical Diseases,
and the Diagnostic Value of Pain. By the late {\smallcapsize JOHN HILTON, F.R.S.,
F.R.C.S.}, etc. Edited by {\smallcapsize W.~H.~A. JACOBSON, M.A., M.CH.} (Oxon.),
{\smallcapsize F.R.C.S\@.} 5\textit{th edition.} 9\s.

``Mr.~Hilton's work is \textit{facile princeps} of its kind in our own or any
other language.''---\textit{Lancet.}

\mitemspace[HOBLYN'S Dictionary of Terms used in Medicine and the Collateral Sciences.]
12\textit{th edition.} Revised and enlarged by {\smallcapsize J.~A.~P. PRICE, B.A.,
M.D.} (Oxon.). 10\s.~6\d.

\mitemspace[LANE (W.~A.). Manual of Operative Surgery.] For Practitioners and
Students. By {\smallcapsize W.~ARBUTHNOT LANE, M.B., M.S., F.R.C.S.}, Assistant
Surgeon to Guy's Hospital, and to the Hospital for Sick Children. Crown
8vo, 8\s.~6\d.

\mitemspace[SHARP (W.) Therapeutics founded on Antipraxy.] By {\smallcapsize WILLIAM
SHARP, M.D., F.R.S\@.} Demy 8vo, 6\s.
\end{maincatlist}
%-----File: 341.png-----

\catsectionuppertoc{Bell's Agricultural Series}
\clabel{ser:Agricultural}

\begin{center}
\textit{In crown} 8\textit{vo, Illustrated,} 160 \textit{pages, cloth,} 2\s.~6\d. \textit{each}.
\end{center}

\begin{maincatlist}

\mitemspace[CHEAL (J.). Fruit Culture.] A Treatise on Planting, Growing, Storage
of Hardy Fruits for Market and Private Growers. By {\smallcapsize J.~CHEAL, F.R.H.S.},
Member of Fruit Committee, Royal Hort. Society, etc.

\mitemspace[FREAM (DR.). Soils and their Properties.] By {\smallcapsize DR. WILLIAM FREAM,
B.SC.} (Lond.)., {\smallcapsize F.L.S., F.G.S., F.S.S.}, Associate of the Surveyor's Institution,
Consulting Botanist to the British Dairy Farmers' Association and
the Royal Counties Agricultural Society; Prof.\ of Nat.\ Hist.\ in Downton
College, and formerly in the Royal Agric. Coll., Cirencester.

\mitemspace[GRIFFITHS (DR.). Manures and their Uses.] By {\smallcapsize DR. A.~B. GRIFFITHS,
F.R.S.E., F.C.S.}, late Principal of the School of Science, Lincoln; Membre
de la Société Chimique de Paris; Author of ``A Treatise on Manures,''
etc., etc. \textit{In use at Downton College.}

\mitemspace[--- The Diseases of Crops and their Remedies.]

\mitemspace[MALDEN (W.~J.). Tillage and Implements.] By {\smallcapsize W.~J. MALDEN},
Prof.\ of Agriculture in the College, Downton.

\mitemspace[SHELDON (PROF.). The Farm and the Dairy.] By {\smallcapsize PROFESSOR
J.~P. SHELDON}, formerly of the Royal Agricultural College, and of the
Downton College of Agriculture, late Special Commissioner of the
Canadian Government; Author of ``Dairy Farming,'' etc. \textit{In use at
Downton College.}
\end{maincatlist}

\shortrule

{\small\centering Specially adapted for Agricultural Classes. Crown 8vo. Illustrated, 1\s.\ each.\par}

\begin{maincatlist}
\mitemspace[Practical Dairy Farming.] By {\smallcapsize PROFESSOR SHELDON\@.} Reprinted from the
Author's larger work entitled ``The Farm and the Dairy.''

\mitemspace[Practical Fruit Growing.] By {\smallcapsize J.~CHEAL, F.R.H.S\@.} Reprinted from the
Author's larger work entitled ``Fruit Culture.''
\end{maincatlist}

\catsectionuppertoc{Technological Handbooks}
\clabel{ser:Technological}

\begin{center}
\textit{Edited by Sir H. Trueman Wood.}
\end{center}

{\small
{\centering Specially adapted for candidates in the examinations of the City Guilds
Institute. Illustrated and uniformly printed in small post 8vo.\par}

``The excellent series of technical handbooks.''---\textit{Textile Manufacturer.}

``The admirable series of technological handbooks.''---\textit{British Journal of
Commerce.}

``Messrs.~Bell's excellent technical series.''---\textit{Manchester Guardian.}
}

\begin{maincatlist}

\mitemspace[BEAUMONT (R.). Woollen and Worsted Cloth Manufacture.] {\stretchyspace By
{\smallcapsize ROBERTS BEAUMONT}, Professor of Textile Industry, Yorkshire College,
Leeds;} Examiner in Cloth Weaving to the City and Guilds of London
Institute. 2\textit{nd edition.} 7\s.~6\d.

\mitem[BENEDIKT (R.), and KNECHT (E.). Coal-tar Colours,] The
Chemistry of. With special reference to their application to Dyeing, etc.
By {\smallcapsize DR. R.~BENEDIKT}, Professor of Chemistry in the University of Vienna.
Translated by {\smallcapsize E.~KNECHT, PH.D\@.} of the Technical College, Bradford.
2\textit{nd and enlarged edition,} 6\s.~6\d.
%-----File: 342.png-----

\mitemspace[CROOKES (W.). Dyeing and Tissue-Printing.] By {\smallcapsize WILLIAM CROOKES,
F.R.S., V.P.C.S\@.}  5\s.

\mitemspace[GADD (W.~L.). Soap Manufacture.] By {\smallcapsize W.~LAWRENCE GADD, F.I.C.,
F.C.S.}, Registered Lecturer on Soap-Making and the Technology of Oils
and Fats, also on Bleaching, Dyeing, and Calico Printing, to the City and
Guilds of London Institute. 5\s.

{\stretchyspace
\mitemspace[HELLYER (S.~S.). Plumbing: Its Principles and Practice.] By
{\smallcapsize S.~STEVENS HELLYER\@.} With numerous Illustrations. 5\s.
}

\mitemspace[HORNBY (J.). Gas Manufacture.] By {\smallcapsize J.~HORNBY, F.I.C.}, Lecturer
under the City and Guilds of London Institute.      [\textit{Preparing.}

\mitemspace[HURST (G.~H.). Silk-Dyeing and Finishing.] By {\smallcapsize G.~H. HURST, F.C.S.},
Lecturer at the Manchester Technical School, Silver Medallist, City and
Guilds of London Institute. With Illustrations and numerous Coloured
Patterns. 7\s.~6\d.

\mitemspace[JACOBI (C.~T.). Printing. A Practical Treatise.] By {\smallcapsize C.~T. JACOBI},
Manager of the Chiswick Press, Examiner in Typography to the City and
Guilds of London Institute. With numerous Illustrations. 5\s.

\mitemspace[MARSDEN (R.). Cotton Spinning: Its Development, Principles, and Practice,]
with Appendix on Steam Boilers and Engines. By {\smallcapsize R.
MARSDEN}, Editor of the ``Textile Manufacturer.''  4\textit{th edition.}~6\s.~6\d.

\mitemspace[--- Cotton Weaving.] With numerous Illustrations. [\textit{In the press.}

\mitem[POWELL (H.), CHANCE (H.), and HARRIS (H.~G.). Glass Manufacture. Introductory Essay,]
by {\smallcapsize H.~POWELL, B.A.} (Whitefriars
Glass Works); \textbf{Sheet Glass}, by {\smallcapsize HENRY CHANCE, M.A.} (Chance Bros.,
Birmingham); \textbf{Plate Glass}, by {\smallcapsize H.~G. HARRIS}, Assoc.\ Memb.\ Inst.\ C.E. 3\s.~6\d.

\mitemspace[ZAEHNSDORF (J.~W.). Bookbinding.]  By {\smallcapsize J.~W. ZAEHNSDORF},
Examiner in Bookbinding to the City and Guilds of London Institute.
With 8 Coloured Plates and numerous Diagrams. 2\textit{nd edition, revised
and enlarged.} 5\s.
\end{maincatlist}

\begin{center}
\threestarsinverted \textit{Complete List of Technical Books on Application.}
\end{center}

\catsectionuppertoc{Music}

\begin{maincatlist}

\mitem[BANISTER (H.~C.). A Text Book of Music:] By {\smallcapsize H.~C. BANISTER},
Professor of Harmony and Composition at the R.A. of Music, at the Guildhall
School of Music, and at the Royal Normal Coll.\ and Acad.\ of Music
for the Blind. 15\textit{th edition.} Fcap.\ 8vo. 5\s.

This Manual contains chapters on Notation, Harmony, and Counterpoint;
Modulation, Rhythm, Canon, Fugue, Voices, and Instruments; together
with exercises on Harmony, an Appendix of Examination Papers, and a
copious Index and Glossary of Musical Terms.

\mitemspace[--- Lectures on Musical Analysis.] Embracing Sonata Form, Fugue,
etc., Illustrated by the Works of the Classical Masters. 2\textit{nd edition,
revised.} Crown 8vo, 7\s.~6\d.

``It is beyond comparison the best book on the subject in our language.''---\textit{Athenæum.}

\mitem[--- Musical Art and Study:] Papers for Musicians. Fcap.\ 8vo, 2\s.

\mitemspace[CHATER (THOMAS). Scientific Voice, Artistic Singing, and Effective Speaking.]
A Treatise on the Organs of the Voice, their
Natural Functions, Scientific Development, Proper Training, and Artistic
Use. By {\smallcapsize THOMAS CHATER\@.} With Diagrams. Wide fcap.\ 2\s.~6\d.
%-----File: 343.png-----

\mitem[HUNT (H.~G. BONAVIA). A Concise History of Music,] from the
Commencement of the Christian era to the present time. For the use of
Students. By {\smallcapsize REV. H.~G. BONAVIA HUNT}, Mus.\ Doc.\ Dublin; Warden
of Trinity College, London; and Lecturer on Musical History in the same
College. 12\textit{th edition, revised to date} (1893). Fcap.\ 8vo, 3\s.~6\d.
\end{maincatlist}

\catsectionuppertoc{Art}

\begin{maincatlist}

\mitemspace[BARTER (S.). Manual Instruction---Woodwork.] By {\smallcapsize S.~BARTER}
Organizer and Instructor for the London School Board, and to the Joint
Committee on Manual Training of the School Board for London, the City
and Guilds of London Institute, and the Worshipful Company of Drapers.
With over 300 Illustrations. Fcap.\ 4to, cloth. 7\s.~6\d.

\mitemspace[BELL (SIR CHARLES).] The Anatomy and Philosophy of Expression,
as connected with the Fine Arts. By {\smallcapsize SIR CHARLES BELL, K.H.}
7\textit{th edition, revised.} 5\s.

\mitemspace[BRYAN'S Biographical and Critical Dictionary of Painters and Engravers.]
With a List of Ciphers, Monograms, and Marks. A new
Edition, thoroughly Revised and Enlarged. By {\smallcapsize R.~E. GRAVES}, British
Museum, and {\smallcapsize WALTER ARMSTRONG\@.} 2 volumes. Imp.\ 8vo, buckram,
3\textit{l.}~3\s.

\mitemspace[CHEVREUL on Colour.] Containing the Principles of Harmony and Contrast
of Colours, and their Application to the Arts. 3\textit{rd edition,} with
Introduction. Index and several Plates. 5\s.---With an additional series
of 16 Plates in Colours, 7\s.~6\d.

\mitemspace[DELAMOTTE (P.~H.).] The Art of Sketching from Nature. By {\smallcapsize P.~H.
DELAMOTTE}, Professor of Drawing at King's College, London. Illustrated
by Twenty-four Woodcuts and Twenty Coloured Plates, arranged
progressively, from Water-colour Drawings by {\smallcapsize PROUT, E.~W. COOKE, R.A.,
GIRTIN, VARLEY, DE WINT}, and the Author. \textit{New edition.} Imperial
4to, 21\s.

\mitem[FLAXMAN'S CLASSICAL COMPOSITIONS,] reprinted in a cheap
form for the use of Art Students. Oblong paper covers, 2\s.~6\d.\ each.

\begin{catsublist}
\mitemspace[The Iliad of Homer.] 39 Designs.

\mitemspace[The Odyssey of Homer.] 34 Designs.

\mitemspace[The Tragedies of Æschylus.] 36 Designs.

\mitemspace[The ``Works and Days'' and ``Theogony'' of Hesiod.] 37 Designs.

\mitemspace[Select Compositions from Dante's Divine Drama.] 37 Designs.
\end{catsublist}

\mitem[FLAXMAN'S Lectures on Sculpture,] as delivered before the President
and Members of the Royal Academy. With Portrait and 53 plates. 6\s.

\mitemspace[HEATON (MRS.). A Concise History of Painting.] By the late {\smallcapsize MRS.\
CHARLES HEATON\@.} New edition. Revised by {\smallcapsize COSMO MONKHOUSE\@.} 5\s.

\mitemspace[LELAND (C.~G.). Drawing and Designing.] In a series of Lessons
for School use and Self Instruction. By {\smallcapsize CHARLES G.~LELAND, M.A.,
F.R.L.S\@.} Paper cover, 1\s.; or in cloth, 1\s.~6\d.

\mitem[--- Leather Work:] Stamped, Moulded, and Cut, Cuir-Bouillé, Sewn, etc.
With numerous Illustrations. Fcap.\ 4to, 5\s.

\mitemspace[--- Manual of Wood Carving.] By {\smallcapsize CHARLES G.~LELAND, M.A., F.R.L.S\@.}
Revised by {\smallcapsize J.~J. HOLTZAPFFEL, A.M. INST. C.E\@.} With numerous Illustrations.
Fcap.\ 4to, 5\s.

\mitemspace[--- Metal Work.] With numerous Illustrations. Fcap.\ 4to, 5\s.

%-----File: 344.png-----

\mitemspace[LEONARDO DA VINCI'S Treatise on Painting.] Translated from
the Italian by {\smallcapsize J.~F. RIGAUD, R.A\@.}  With a Life of Leonardo and an
Account of his Works, by J.~W. BROWN. \textit{New edition.} With numerous
Plates. 5\s.

\mitemspace[MOODY (F.~W.). Lectures and Lessons on Art.] By the late {\smallcapsize F.~W.
MOODY}, Instructor in Decorative Art at South Kensington Museum. With
Diagrams to illustrate Composition and other matters. \textit{A new and cheaper
edition.} Demy 8vo, sewed, 4\s.~6\d.

``There are few books that we can more thoroughly recommend to
every student and thinker on Art than these Lectures.''---\textit{Art Journal.}

\mitem[WHITE (GLEESON). Practical Designing:] A Handbook on the
Preparation of Working Drawings, showing the Technical Methods employed
in preparing them for the Manufacturer and the Limits imposed on
the Design by the Mechanism of Reproduction and the Materials employed.
Edited by {\smallcapsize GLEESON WHITE\@.} Freely Illustrated. Crown 8vo, 6\s.\ net.

Contents:---Bookbinding, by H.~ORRINSMITH---Carpets, by {\smallcapsize ALEXANDER
MILLAR}---Drawing for Reproduction, by the Editor---Pottery, by {\smallcapsize W.~P.
RIX}---Metal Work, by {\smallcapsize R.~LL. RATHBONE}---Stained Glass, by {\smallcapsize SELWYN
IMAGE}---Tiles, by {\smallcapsize OWEN CARTER}---Woven Fabrics, Printed Fabrics, and
Floorcloths, by {\smallcapsize ARTHUR SILVER}---Wall Papers, by G.~C. HAITÉ.
\end{maincatlist}

\cchaptercat{MENTAL, MORAL, AND SOCIAL SCIENCES}

\catsectionuppertoc{Psychology and Ethics}

\begin{maincatlist}

\mitemspace[ANTONINUS (M. Aurelius). The Thoughts of.] Translated literally,
with Notes, Biographical Sketch, Introductory Essay on the Philosophy,
and Index, by {\smallcapsize GEORGE LONG, M.A\@.} Revised edition. Small post 8vo,
3\s.~6\d., \textit{or new edition on Handmade paper, buckram}, 6\s.

\mitemspace[BACON'S Novum Organum and Advancement of Learning.] Edited,
with Notes, by {\smallcapsize J.~DEVEY, M.A\@.} Small post 8vo, 5\s.

\mitemspace[EPICTETUS\@. The Discourses of.] With the Encheiridion and Fragments.
Translated with Notes, a Life of Epictetus, a View of his Philosophy,
and Index, by {\smallcapsize GEORGE LONG, M.A\@.} Small post 8vo, 5\s., \textit{or new
edition on Handmade paper}, 2 vols., buckram, 10\s.~6\d.

\mitemspace[KANT'S Critique of Pure Reason.] Translated by {\smallcapsize J.~M.~D. MEIKLEJOHN},
Professor of Education at St.~Andrew's University. Small post 8vo, 5\s.

\mitemspace[--- Prolegomena and Metaphysical Foundations of Science.] With
Life. Translated by {\smallcapsize E.~BELFORT BAX\@.} Small post 8vo, 5\s.

\mitem[LOCKE'S Philosophical Works,] containing Essay on the Human Understanding,
Controversy with Bishop of Worcester, Examination of
Malebranche's Opinions, Elements of Natural Philosophy, Thoughts
concerning Reading and Study. Edited by {\smallcapsize J.~A. ST. JOHN\@.} 2 vols.
Small post 8vo, 3\s.~6\d.\ each.

%-----File: 345.png-----

\mitem[RYLAND (F.). The Student's Manual of Psychology and Ethics,]
    designed chiefly for the London B.A. and B.Sc. By {\smallcapsize F.~RYLAND, M.A.},
    late Scholar of St.~John's College, Cambridge. Cloth, red edges. 5\textit{th
    edition, revised and enlarged.} With lists of books for Students, and
    Examination Papers set at London University. Crown 8vo, 3\s.~6\d.

\mitem[--- Ethics:] An Introductory Manual for the use of University Students.
    With an Appendix containing List of Books recommended, and Examination
    Questions. Crown 8vo, 3\s.~6\d.

\mitem[SCHOPENHAUER on the Fourfold Root of the Principle of Sufficient Reason, \textmd{and} On the Will in Nature.]
    Translated by {\smallcapsize MADAME
    HILLEBRAND\@.} Small post 8vo, 5\s.

\mitemspace[--- Essays.] Selected and Translated. With a Biographical Introduction
    and Sketch of his Philosophy, by {\smallcapsize E.~BELFORT BAX\@.} Small post 8vo, 5\s.

\mitemspace[SMITH (Adam). Theory of Moral Sentiments.] With Memoir of the
    Author by {\smallcapsize DUGALD STEWART\@.} Small post 8vo, 3\s.~6\d.

\mitemspace[SPINOZA'S Chief Works.] Translated with Introduction, by {\smallcapsize R.~H.~M.
    ELWES\@.} 2 vols. Small post 8vo, 5\s.\ each.

       Vol.~I.---Tractatus Theologico-Politicus---Political Treatise.

            II.---Improvement of the Understanding---Ethics---Letters.
\end{maincatlist}

\catsectionuppertoc{History of Philosophy}

\begin{maincatlist}

\mitemspace[BAX (E.~B.). Handbook of the History of Philosophy.] By {\smallcapsize E.~BELFORT
    BAX\@.} 2\textit{nd edition, revised.} Small post 8vo, 5\s.

\mitemspace[DRAPER (J.~W.). A History of the Intellectual Development of Europe.]
    By {\smallcapsize JOHN WILLIAM DRAPER, M.D., LL.D\@.} With Index. 2
    vols. Small post 8vo, 5\s.\ each.

\mitemspace[HEGEL'S Lectures on the Philosophy of History.] \quad Translated by
    {\smallcapsize J.~SIBREE, M.A\@.} Small post 8vo, 5\s.
\end{maincatlist}

\catsectionuppertoc{Law and Political Economy}

\begin{maincatlist}

\mitemspace[KENT'S Commentary on International Law.] Edited by {\smallcapsize J.~T. ABDY,
    LL.D.}, Judge of County Courts and Law Professor at Gresham College,
    late Regius Professor of Laws in the University of Cambridge. 2\textit{nd
    edition, revised and brought down to a recent date.} Crown 8vo, 10\s.~6\d.

\mitemspace[LAWRENCE (T.~J.). Essays on some Disputed Questions in Modern International Law.]
    By {\smallcapsize T.~J. LAWRENCE, M.A.}, LL.M. 2\textit{nd
    edition, revised and enlarged.} Crown 8vo, 6\s.

\mitemspace[--- Handbook of Public International Law.] 2nd edition. Fcap.\ 8vo, 3\s.

\mitemspace[MONTESQUIEU'S Spirit of Laws.] A New Edition, revised and
    corrected, with D'Alembert's Analysis, Additional Notes, and a Memoir,
    by {\smallcapsize J.~V. PRITCHARD, A.M\@.} 2 vols. Small post 8vo, 3\s.~6\d.\ each.

\mitemspace[RICARDO on the Principles of Political Economy and Taxation.]
    Edited by {\smallcapsize E.~C.~K. GONNER, M.A.}, Lecturer in University College,
    Liverpool. Small post 8vo, 5\s.

\mitemspace[SMITH (Adam). The Wealth of Nations.] An Inquiry into the Nature
    and Causes of. Reprinted from the Sixth Edition, with an Introduction
    by {\smallcapsize ERNEST BELFORT BAX\@.} 2 vols. Small post 8vo, 3\s.~6\d.\ each.
\end{maincatlist}

%-----File: 346.png-----

\catsectionuppertoc{History}

\begin{maincatlist}

\mitem[BOWES (A.). A Practical Synopsis of English History;] or, A
General Summary of Dates and Events for the use of Schools, Families,
and Candidates for Public Examinations. By {\smallcapsize ARTHUR BOWES\@.} 10\textit{th
edition.} Revised and brought down to the present time. Demy 8vo, 1\s.

\mitemspace[COXE (W.).  History of the House of Austria, 1218--1792.]  By
{\smallcapsize ARCHDN. COXE, M.A., F.R.S\@.} Together with a Continuation from the
Accession of Francis I. to the Revolution of 1848. 4 vols. Small post
8vo. 3\s.~6\d.\ each.

\mitemspace[DENTON (W.). England in the Fifteenth Century.] By the late
{\smallcapsize REV. W.~DENTON, M.A.}, Worcester College, Oxford. Demy 8vo, 12\s.

\mitem[DYER (Dr.~T.~H.). History of Modern Europe,] from the Taking of
Constantinople to the Establishment of the German Empire, A.D. 1453--1871.
{\smallcapsize By DR. T.~H. DYER\@.} \textit{A new edition,} revised by the Author, and
brought down to the close of the Franco-German War. In 5 vols.
\L2 12\s.~6\d.

\mitemspace[GIBBON'S Decline and Fall of the Roman Empire.] Complete and
Un\-abridged, with Variorum Notes. Edited by an English Churchman.
With 2 Maps. 7 vols. Small post 8vo, 3\s.~6\d.\ each.

\mitemspace[GUIZOT'S History of the English Revolution of 1640.] Translated by
WILLIAM HAZLITT\@. Small post 8vo, 3\s.~6\d.

\mitem[--- History of Civilization,] from the Fall of the Roman Empire to the
French Revolution. Translated by {\smallcapsize WILLIAM HAZLITT\@.} 3 vols. Small
post 8vo, 3\s.~6\d.\ each.

\mitemspace[HENDERSON (E.~F.). Select Historical Documents of the Middle Ages.]
Including the most famous Charters relating to England, the
Empire, the Church, etc., from the sixth to the fourteenth centuries.
Translated and edited, with Introductions, by {\smallcapsize ERNEST F. HENDERSON},
A.B., A.M., PH.D. Small post 8vo, 5\s.

\mitemspace[--- A History of Germany in the Middle Ages.] Post 8vo, 7\s.~6\d.\ net.

\mitem[HOOPER (George). The Campaign of Sedan:] The Downfall of the
Second Empire, August-September, 1870. By {\smallcapsize GEORGE HOOPER\@.} With
General Map and Six Plans of Battle. Demy 8vo, 14\s.

\mitem[--- Waterloo: The Downfall of the First Napoleon:] a History of the
Campaign of 1815. With Maps and Plans. Small post 8vo, 3\s.~6\d.

\mitemspace[LAMARTINE'S History of the Girondists.] Translated by {\smallcapsize H.~T. RYDE.}
3 vols. Small post 8vo, 3\s.~6\d.\ each.

\mitem[--- History of the Restoration of Monarchy in France] (a Sequel to his
History of the Girondists). 4 vols. Small post 8vo, 3\s.~6\d.\ each.

\mitemspace[--- History of the French Revolution of 1848.] Small post 8vo, 3\s.~6\d.

\mitem[LAPPENBERG'S History of England under the Anglo-Saxon Kings.]
Translated by the late {\smallcapsize B.~THORPE, F.S.A\@.} \textit{New edition,} revised
by {\smallcapsize E.~C. OTTÉ}. 2 vols. Small post 8vo, 3\s.~6\d.\ each.

\mitem[LONG (G.).  The Decline of the Roman Republic:] From the
Destruction of Carthage to the Death of Cæsar. By the late {\smallcapsize GEORGE
LONG, M.A\@.} Demy 8vo. In 5 vols. 5\s.\ each.

\mitem[MACHIAVELLI'S History of Florence,] and of the Affairs of Italy
from the Earliest Times to the Death of Lorenzo the Magnificent: together
with the Prince, Savonarola, various Historical Tracts, and a Memoir of
Machiavelli. Small post 8vo, 3\s.~6\d.

%-----File: 347.png-----

\mitemspace[MARTINEAU (H.). History of England from 1800--15.] By {\smallcapsize HARRIET
MARTINEAU\@.} Small post 8vo, 3\s.~6\d.

\mitemspace[--- History of the Thirty Years' Peace, 1815--46.] 4 vols. Small post
    8vo, 3\s.~6\d.\ each.

\mitemspace[MAURICE (C.~E.). The Revolutionary Movement of 1848--9 in Italy, Austria, Hungary, and Germany.]
    With some Examination
    of the previous Thirty-three Years. By {\smallcapsize C.~EDMUND MAURICE\@.} With an
    engraved Frontispiece and other Illustrations. Demy 8vo, 16\s.

\mitem[MENZEL'S History of Germany,] from the Earliest Period to 1842.
    3 vols. Small post 8vo, 3\s.~6\d.\ each.

\mitem[MICHELET'S History of the French Revolution] from its earliest
    indications to the flight of the King in 1791. Small post 8vo, 3\s.~6\d.

\mitem[MIGNET'S History of the French Revolution,] from 1789 to 1814.
    Small post 8vo, 3\s.~6\d.

\mitemspace[PARNELL (A.). The War of the Succession in Spain during the Reign of Queen Anne, 1702--1711.]
    Based on Original Manuscripts
    and Contemporary Records. By {\smallcapsize COL. THE HON. ARTHUR PARNELL,
    R.E\@.} Demy 8vo, 14\s. With Map, etc.

\mitemspace[RANKE (L.). History of the Latin and Teutonic Nations, 1494--1514.]
Translated by {\smallcapsize P.~A. ASHWORTH\@.} Small post 8vo, 3\s.~6\d.

\mitem[--- History of the Popes,] their Church and State, and especially of their
    conflicts with Protestantism in the 16th and 17th centuries. Translated
    by {\smallcapsize E.~FOSTER\@.} 3 vols. Small post 8vo, 3\s.~6\d.\ each.

\mitemspace[--- History of Servia and the Servian Revolution.] Translated by {\smallcapsize MRS.
    KERR\@.} Small post 8vo, 3\s.~6\d.

\mitem[SIX OLD ENGLISH CHRONICLES:] viz., Asser's Life of Alfred
    and the Chronicles of Ethelwerd, Gildas, Nennius, Geoffrey of Monmouth,
    and Richard of Cirencester. Edited, with Notes and Index, by {\smallcapsize J.~A.
    GILES, D.C.L\@.} Small post 8vo, 5\s.

\mitem[STRICKLAND (Agnes). The Lives of the Queens of England;]
    from the Norman Conquest to the Reign of Queen Anne. By {\smallcapsize AGNES
    STRICKLAND\@.} 6 vols. 5\s.\ each.

\mitemspace[--- The Lives of the Queens of England.] Abridged edition for the use
    of Schools and Families, Post 8vo, 6\s.~6\d.

\mitem[THIERRY'S History of the Conquest of England by the Normans;]
    its Causes, and its Consequences in England, Scotland, Ireland, and the
    Continent. Translated from the 7th Paris edition by {\smallcapsize WILLIAM HAZLITT.}
    2 vols. Small post 8vo, 3\s.~6\d.\ each.

\mitem[WRIGHT (H.~F.). The Intermediate History of England,] with
    Notes, Supplements, Glossary, and a Mnemonic System. For Army
    and Civil Service Candidates. By {\smallcapsize H.~F. WRIGHT, M.A., LL.M\@.} Crown
    8vo, 6\s.
\end{maincatlist}

\begin{center}
\small \textbf{For other Works of value to Students of History, see Catalogue of\\
Bohn's Libraries, sent post-free on application.}
\end{center}
%-----File: 348.png-----

\catsectionuppertoc{Divinity, etc}

\begin{maincatlist}

\mitemspace[ALFORD (DEAN). Greek Testament.] With a Critically revised Text,
    a digest of Various Readings, Marginal References to verbal and idiomatic
    usage, Prolegomena, and a Critical and Exegetical Commentary.
    For the use of theological students and ministers. By the late {\smallcapsize HENRY
    ALFORD, D.D.}, Dean of Canterbury. 4 vols. 8vo. Sold separately.

       Vol.~I., 7\textit{th edition,} the Four Gospels. \L1 8\s.

       Vol.~II., 8\textit{th edition,} the Acts of the Apostles, Epistles to the Romans
    and Corinthians. \L1 4\s.

       Vol.~III., 10\textit{th edition,} the Epistles to the Galatians, Ephesians, Philippians,
    Colossians, Thessalonians,---to Timotheus, Titus, and Philemon.
    18\s.

       Vol.~IV. Part I, 5\textit{th edition,} the Epistle to the Hebrews, the Catholic
    Epistles of St.~James and St.~Peter. 18\s.

       Vol.~IV. Part 2, 4\textit{th edition,} the Epistles of St.~John and St.~Jude, and
    the Revelation. 14\s.

       Vol.~IV. in one Vol. 32\s.

\mitemspace[--- The New Testament for English Readers.] Containing the Authorized
    Version, with additional Corrections of Readings and Renderings, Marginal
    References, and a Critical and Explanatory Commentary. In 2 vols.
     \L2 14\s.~6\d.

      Vol.~I. Part 1. The first three Gospels. 3\textit{rd edition.} 12\s.

      Vol.~I. Part 2. St.~John and the Acts. 2\textit{nd edition.} 10\s.~6\d.

      Vol.~II. Part 1. The Epistles of St.~Paul. 2\textit{nd edition.} 16\s.

      Vol.~II. Part 2. The Epistles to the Hebrews, the Catholic Epistles,
    and the Revelation. 2\textit{nd edition.} 16\s.

\mitemspace[AUGUSTINE de Civitate Dei.] Books XI. and XII\@. By the {\smallcapsize REV.
    HENRY D.~GEE, B.D., F.S.A\@.} I. Text only. 2\s. II. Introduction and
    Translation. 3\s.

\mitemspace[BARRETT (A.~C.). Companion to the Greek Testament.] For the
    use of Theological Students and the Upper Forms in Schools. By the
    late {\smallcapsize A.~C. BARRETT, M.A.}, Caius College, Cambridge. \textit{5th edition.}
    Fcap.\ 8vo, 5\s.

\mitemspace[BARRY (BP.). Notes on the Catechism.] For the use of Schools. By
    the {\smallcapsize RT. REV. BISHOP BARRY, D.D\@.} 10\textit{th edition.} Fcap.\ 2\s.

\mitemspace[BLEEK\@. Introduction to the Old Testament.] By {\smallcapsize FRIEDRICH BLEEK.}
    Edited by {\smallcapsize JOHANN BLEEK} and {\smallcapsize ADOLF KAMPHAUSEN\@.} Translated from
    the second edition of the German by {\smallcapsize G.~H. VENABLES}, under the supervision
    of the {\smallcapsize REV. E.~VENABLES}, Residentiary Canon of Lincoln. 2\textit{nd
    edition,} with Corrections. With Index. 2 vols.\ small post 8vo, 5\s.\ each.

\mitemspace[BUTLER (BP.). Analogy of Religion.] With Analytical Introduction
    and copious Index, by the late {\smallcapsize RT. REV. DR. STEERE}, Bishop in Central
    Africa. Fcap.\ \textit{New edition,} 3\s.~6\d.

\mitemspace[EUSEBIUS\@. Ecclesiastical History of Eusebius Pamphilus, Bishop of Cæsarea.]
    Translated from the Greek by {\smallcapsize REV. C.~F. CRUSE, M.A.}
    With Notes, a Life of Eusebius, a Chronological Table of Persons and
    Events mentioned in the History, and an Index. Small post 8vo, 5\s.

\mitemspace[GREGORY (DR.). Letters on the Evidences, Doctrines, and Duties of the Christian Religion.]
    By {\smallcapsize DR. OLINTHUS GREGORY, F.R.A.S.}
    Small post 8vo, 3\s.~6\d.
%-----File: 349.png-----

\mitemspace[HUMPHRY (W.~G.). Book of Common Prayer.] An Historical and
Explanatory Treatise on the. By {\smallcapsize W.~G. HUMPHRY, B.D.}, late Fellow of
Trinity College, Cambridge, Prebendary of St.~Paul's, and Vicar of St.
Martin's-in-the-Fields, Westminster. 6\textit{th edition.} Fcap.\ 8vo, 2\s.~6\d.

Cheap Edition, for Sunday School Teachers. 1\s.

\mitemspace[JOSEPHUS (FLAVIUS). The Works of.] {\smallcapsize WHISTON'S} Translation.
Revised by {\smallcapsize REV. A.~R. SHILLETO, M.A\@.} With Topographical and Geographical
Notes by {\smallcapsize COLONEL SIR C.~W. WILSON, K.C.B\@.} 5 vols.\ 3\s.~6\d.\ each.

\mitemspace[LUMBY (DR.). The History of the Creeds.] I. Ante-Nicene. II.
Nicene and Constantinopolitan.  III. The Apostolic Creed. IV. The
Quicunque, commonly called the Creed of St.~Athanasius. By {\smallcapsize J.~RAWSON
LUMBY, D.D.}, Norrisian Professor of Divinity, Fellow of St.\ Catherine's
College, and late Fellow of Magdalene College, Cambridge. 3\textit{rd edition,
revised.} Crown 8vo, 7\s.~6\d.

\mitemspace[--- Compendium of English Church History, from 1688--1830.] With a
Pre\-face by {\smallcapsize J.~RAWSON LUMBY, D.D.}, Norrisian Professor of Divinity.
Crown 8vo, 6\s.

\mitemspace[MACMICHAEL (J.~F.). The New Testament: In Greek.] With
English Notes and Preface, Synopsis, and Chronological Tables. By the
late {\smallcapsize REV. J.~F. MACMICHAEL\@.} Fcap.\ 8vo (730 pp.), 4\s.~6\d.

Also the Four Gospels, and the Acts of the Apostles, separately.
In paper wrappers, 6\d.\ each.

\mitemspace[MILLER (E.). Guide to the Textual Criticism of the New Testament.]
By {\smallcapsize REV. E.~MILLER, M.A.}, Oxon, Rector of Bucknell, Bicester.
Crown 8vo, 4\s.

\mitem[NEANDER (DR. A.). History of the Christian Religion] and
Church. Translated by {\smallcapsize J.~TORREY\@.} 10 vols.\ small post 8vo, 3\s.~6\d.\ each.

\mitemspace[--- Life of Jesus Christ.] Translated by {\smallcapsize J.~M{\scriptsize C}CLINTOCK} and {\smallcapsize C.~BLUMENTHAL.}
Small post 8vo, 3\s.~6\d.

\mitemspace[--- History of the Planting and Training of the Christian Church by the Apostles.]
Translated by {\smallcapsize J.~E. RYLAND\@.} 2 vols.\ small post 8vo,
3\s.~6\d.\ each.

\mitemspace[--- Lectures on the History of Christian Dogmas.] Edited by {\smallcapsize DR.\
JACOBI\@.} Translated by {\smallcapsize J.~E. RYLAND\@.} 2 vols.\ small post 8vo, 3\s.~6\d.\ each.

\mitemspace[--- Memorials of Christian Life in the Early and Middle Ages.] Translated
by {\smallcapsize J.~E. RYLAND\@.} Small post 8vo, 3\s.~6\d.

\mitemspace[PEARSON (BP.). On the Creed.] Carefully printed from an Early
Edition. With Analysis and Index. Edited by {\smallcapsize E.~WALFORD, M.A.}
Post 8vo, 5\s.

\mitemspace[PEROWNE (BP.). The Book of Psalms.] A New Translation, with
Introductions and Notes, Critical and Explanatory. By the {\smallcapsize RIGHT
REV. J.~J. STEWART PEROWNE, D.D.}, Bishop of Worcester. 8vo. Vol.~I.
8\textit{th edition, revised}. 18\s. Vol.~II. 7\textit{th edition, revised}. 16\s.

\mitemspace[--- The Book of Psalms.] Abridged Edition for Schools.
Crown 8vo. 7\textit{th edition}. 10\s.~6\d.

\mitem[SADLER (M.~F.). The Church Teacher's Manual] of Christian Instruction.
Being the Church Catechism, Expanded and Explained in Question
and Answer. For the use of the Clergyman, Parent, and Teacher. By the
{\smallcapsize REV. M.~F. SADLER}, Prebendary of Wells, and Rector of Honiton. 43\textit{rd
thousand}. 2\s.~6\d.

\begin{center}
\hspace{-1cm}
\threestars A Complete List of Prebendary Sadler's Works will be sent on
application.
\end{center}

%-----File: 350.png-----
\mitemspace[SCRIVENER (DR.). A Plain Introduction to the Criticism of the New Testament.]
    With Forty-four Facsimiles from Ancient Manuscripts. For
    the use of Biblical Students. By the late {\smallcapsize F.~H. SCRIVENER, M.A., D.C.L.},
    LL.D., Prebendary of Exeter. 4\textit{th edition,} thoroughly revised, by the {\smallcapsize REV.
    E.~MILLER}, formerly Fellow and Tutor of New College, Oxford. 2 vols.\ demy 8vo, 32\s.

\mitem[--- Novum Testamentum  Græce,] Textus Stephanici, 1550. Accedunt
    variae lectiones editionum Bezae, Elzeviri, Lachmanni, Tischendorfii,
    Tregellesii, curante {\smallcapsize F.~H.~A. SCRIVENER, A.M., D.C.L., LL.D\@.} \textit{Revised
    edition,} giving all the readings of Tregelles and of Tischendorf's eighth
    edition. 4\s.~6\d.

\mitem[--- Novum Testamentum Græce [Editio Major]] textus Stephanici,
    A.D. 1556. Cum variis lectionibus editionum Bezae, Elzeviri, Lachmanni,
    Tischendorfii, Tregellesii, Westcott-Hortii, versionis Anglicanæ emendatorum
    curante {\smallcapsize F.~H.~A. SCRIVENER, A.M., D.C.L., LL.D.}, accedunt parallela
    s.\ scripturæ loca. Small post 8vo. 2\textit{nd edition.} 7\s.~6\d.

       An Edition on writing-paper, with wide margin for notes. 4to, half bound,
    12\s.

      This is an enlarged edition of Dr.~Scrivener's well-known Greek Testament.
    It contains the readings approved by Bishop Westcott and Dr.
    Hort, and also those adopted by the Revisers---the Eusebian Canons, and
    the Capitula are included. An enlarged and revised series of references is
    also added, so that the volume affords a sufficient apparatus for the Critical
    Study of the Text.

{\stretchyspace
\mitemspace[WHEATLEY\@. A Rational Illustration of the Book of Common Prayer.]
    Being the Substance of everything Liturgical in Bishop Sparrow,
    Mr.~L'Estrange, Dr.~Comber, Dr.~Nicholls, and all former Ritualist
    Commentators upon the same subject. Small post 8vo, 3\s.~6\d.
}

\mitemspace[WHITAKER (C.). Rufinus and His Times.] With the Text of his
    \textbf{Commentary on the Apostles' Creed} (with various readings), and a
    Translation of the same. To which is added a Condensed History of the
    Creeds and Councils. By the {\smallcapsize REV. CHARLES WHITAKER, B.A.}, Vicar of
    Natland, Kendal. Demy 8vo, 5\s.

      Or in separate Parts.---1. Latin Text, with Various Readings, 2\s.~6\d.
    2. Summary of the History of the Creeds, 1\s.~6\d. 3. Charts of the
    Heresies of the Times preceding Rufinus, and the First Four General
    Councils, 6\d.\ each.

\mitem[--- St.~Augustine:] De Fide et Symbolo---Sermo ad Catechumenos. St.~Leo
    ad Flavianum Epistola---Latin Text, with Literal Translation, Notes, and
    History of Creeds and Councils. 5\s.

    Also separately, Literal Translation. 2\s.

\mitemspace[--- Student's Help to the Prayer-Book.] 3\s.
\end{maincatlist}

%-----File: 351.png-----
%Start of Summary of Series
\clearpage

\tableofcontentssum

\addtocontents{\tcsext}{%
  \protect\begin{flushright}
  \textsc{page}\vspace{-12pt}
  \protect\end{flushright}}

% Add this summary of series section to the main toc for the catalogue
\addtocontents{\tccext}{\vspace*{6pt}}
\addcontentsline{\tccext}{section}{\hspace{-\parindent}\textsc{Summary of Series}}

\bigskip

\cseriestoc{Bibliotheca Classica}

\begin{seriescatlist}
\mitemspace[AESCHYLUS\@.] By {\smallcapsize DR. PALEY\@.} 8\s.
\mitemspace[CICERO\@.] By {\smallcapsize G.~LONG\@.} Vols.\ I. AND II\@. 8\s.\ each.
\mitemspace[DEMOSTHENES\@.] By {\smallcapsize R. WHISTON\@.} 2 Vols. 8\s.\ each.
\mitemspace[EURIPIDES\@.] By {\smallcapsize DR. PALEY\@.} Vols.\ II. and III\@. 8\s.\ each.
\mitemspace[HERODOTUS\@.] By {\smallcapsize DR. BLAKESLEY\@.} 2 Vols. 12\s.
\mitemspace[HESIOD\@.] By {\smallcapsize DR. PALEY\@.} 5\s.
\mitemspace[HOMER\@.] By {\smallcapsize DR. PALEY\@.} 2 Vols. 14\s.
\mitemspace[HORACE\@.] By {\smallcapsize A.~G. MACLEANE\@.} 8\s.
\mitemspace[PLATO\@.] \textbf{Phaedrus.} By {\smallcapsize DR. THOMPSON\@.} 5\s.
\mitemspace[SOPHOCLES\@. Vol.~I\@.] By {\smallcapsize F.~H. BLAYDES\@.} 5\s.
\mitemspace[--- Vol.~II\@.] By {\smallcapsize DR. PALEY\@.} 6\s.
\mitemspace[VIRGIL\@.] By {\smallcapsize CONINGTON AND NETTLESHIP\@.} 3 Vols. 10\s.~6\d.\ each.
\end{seriescatlist}

\cseriestoc{Public School Series}

\begin{seriescatlist}
\mitemspace[ARISTOPHANES\@. Peace.] By {\smallcapsize DR. PALEY\@.} 4\s.~6\d.
\mitemspace[--- Acharnians.] By {\smallcapsize DR. PALEY\@.} 4\s.~6\d.
\mitemspace[--- Frogs.] By {\smallcapsize DR. PALEY\@.} 4\s.~6\d.
\mitemspace[CICERO\@. Letters To Atticus. Book I\@.] By {\smallcapsize A.~PRETOR\@.} 4\s.~6\d.
\mitemspace[DEMOSTHENES\@. De Falsa Legatione.] By {\smallcapsize R.~SHILLETO\@.} 6\s.
\mitemspace[--- Adv. Leptinem.] By {\smallcapsize B.~W. BEATSON\@.} 3\s.~6\d.
\mitem[LIVY\@. Books XXI. and XXII.] By {\smallcapsize L.~D. DOWDALL\@.} 3\s.~6\d.\ each.
\mitemspace[PLATO\@. Apology of Socrates and Crito.] By {\smallcapsize DR. W.~WAGNER\@.} 3\s.~6\d.\ and 2\s.~6\d.
\mitemspace[--- Phaedo.] By {\smallcapsize DR. W.~WAGNER\@.} 5\s.~6\d.
\mitemspace[--- Protagoras.] By {\smallcapsize W.~WAYTE\@.} 4\s.~6\d.
\mitemspace[--- Gorgias.] By {\smallcapsize DR. THOMPSON\@.} 6\s.
\mitemspace[--- Euthyphro.] By {\smallcapsize G.~H. WELLS\@.} 3\s.
\mitemspace[--- Euthydemus.] By {\smallcapsize G.~H. WELLS\@.} 4\s.
\mitemspace[--- Republic.] By {\smallcapsize G.~H. WELLS\@.} 5\s.
%-----File: 352.png-----
\mitemspace[PLAUTUS\@. Aulularia.] By {\smallcapsize DR. W.~WAGNER\@.} 4\s.~6\d.
\mitemspace[--- Trinummus.] By {\smallcapsize DR. W.~WAGNER\@.} 4\s.~6\d.
\mitemspace[--- Menaechmei.] By {\smallcapsize DR. W.~WAGNER\@.} 4\s.~6\d.
\mitemspace[--- Mostellaria.] By {\smallcapsize E.~A. SONNENSCHEIN\@.} 5\s.
\mitemspace[SOPHOCLES\@. Trachiniae.] By {\smallcapsize A.~PRETOR\@.} 4\s.~6\d.
\mitemspace[--- Oedipus Tyrannus.] By {\smallcapsize B.~H. KENNEDY\@.} 5\s.
\mitemspace[TERENCE\@.] By {\smallcapsize DR. W.~WAGNER\@.} 7\s.~6\d.
\mitemspace[THEOCRITUS\@.] By {\smallcapsize DR. PALEY\@.} 4\s.~6\d.
\mitemspace[THUCYDIDES\@. Book VI\@.] By {\smallcapsize T.~W. DOUGAN\@.} 3\s.~6\d.
\end{seriescatlist}

\cseriestoc{Cambridge Greek and Latin Texts}

\begin{seriescatlist}
\mitemspace[AESCHYLUS\@.] By {\smallcapsize DR. PALEY\@.} 2\s.
\mitemspace[CAESAR\@.] By {\smallcapsize G.~LONG\@.} 1\s.~6\d.
\mitemspace[CICERO\@.] De Senectute, de Amicitia, et Epistolae Selectae. By {\smallcapsize G.~LONG\@.} 1\s.~6\d.
\mitemspace[--- Orationes in Verrem.] By {\smallcapsize G.~LONG\@.}  2\s.~6\d.
\mitemspace[EURIPIDES\@.] By {\smallcapsize DR. PALEY\@.} 3 Vols.  2\s.\ each.
\mitemspace[HERODOTUS\@.] By {\smallcapsize DR. BLAKESLEY\@.}  2 Vols.  2\s.~6\d.\ each.
\mitemspace[HOMER'S Iliad.] By {\smallcapsize DR. PALEY\@.} 1\s.~6\d.
\mitemspace[HORACE\@.] By {\smallcapsize A.~J. MACLEANE\@.} 1\s.~6\d.
\mitemspace[JUVENAL AND PERSIUS\@.] By {\smallcapsize A.~J. MACLEANE\@.} 1\s.~6\d.
\mitemspace[LUCRETIUS\@.] By {\smallcapsize H.~A.~J. MUNRO\@.} 2\s.
\mitemspace[SOPHOCLES\@.] By {\smallcapsize DR. PALEY\@.} 2\s.~6\d.
\mitemspace[TERENCE\@.] By DR. {\smallcapsize W.~WAGNER\@.} 2\s.
\mitemspace[THUCYDIDES\@.] By {\smallcapsize DR. DONALDSON\@.} 2 Vols. 2\s.\ each.
\mitemspace[VIRGIL\@.] By {\smallcapsize PROF. CONINGTON\@.} 2\s.
\mitemspace[XENOPHON\@.] By {\smallcapsize J.~F. MACMICHAEL\@.} 1\s.~6\d.
\mitemspace[NOVUM TESTAMENTUM GRAECE\@.] By {\smallcapsize DR. SCRIVENER\@.} 4\s.~6\d.
\end{seriescatlist}

\cseriestoc{Cambridge Texts with Notes}

\begin{seriescatlist}
\mitemspace[AESCHYLUS\@.] By {\smallcapsize DR. PALEY\@.} 6 Vols. 1\s.~6\d.\ each.
\mitemspace[EURIPIDES\@.] By {\smallcapsize DR. PALEY\@.} 13 Vols. (Ion, 2\s.) 1\s.~6\d.\ each.
\mitemspace[HOMER'S Iliad.] By {\smallcapsize DR. PALEY\@.} 1\s.
\mitemspace[SOPHOCLES\@.] By {\smallcapsize DR. PALEY\@.} 5 Vols. 1\s.~6\d.\ each.
\mitemspace[XENOPHON\@. Hellenica.] By {\smallcapsize REV. L.~D. DOWDALL\@.} Books I. and II\@. 2\s.\ each.
\mitemspace[--- Anabasis.] By {\smallcapsize J.~F. MACMICHAEL\@.} 6 Vols. 1\s.~6\d.\ each.
\mitemspace[CICERO\@. De Senectute, de Amicitia, et Epistolae Selectae.]
By {\smallcapsize G.~LONG\@.} 3 Vols. 1\s.~6\d.\ each.
\mitemspace[OVID\@. Selections.] By {\smallcapsize A.~J. MACLEANE\@.} 1\s.~6\d.
\mitemspace[--- Fasti.] By {\smallcapsize DR. PALEY\@.} 3 Vols. 2\s.\ each.
\mitemspace[TERENCE\@.] By {\smallcapsize DR. W.~WAGNER\@.} 4 Vols. 1\s.~6\d.\ each.
\mitemspace[VIRGIL\@.] By {\smallcapsize PROF. CONINGTON\@.} 12 Vols. 1\s.~6\d.\ each.
\end{seriescatlist}

\cseriestoc{Grammar School Classics}

\begin{seriescatlist}
\mitemspace[CAESAR, De Bello Gallico.] By {\smallcapsize G.~LONG\@.} 4\s., or in 3 parts, 1\s.~6\d.\ each.
\mitemspace[CATULLUS, TIBULLUS, and PROPERTIUS\@.] By {\smallcapsize A.~H. WRATISLAW}, and {\smallcapsize F.~N. SUTTON\@.} 2\s.~6\d.
\mitemspace[CORNELIUS NEPOS\@.] By {\smallcapsize J.~F. MACMICHAEL\@.} 2\s.
\mitemspace[CICERO\@. De Senectute, De Amicitia, and Select Epistles.]
By. {\smallcapsize G LONG\@.} 3\s.
\mitemspace[HOMER\@. Iliad.] By {\smallcapsize DR. PALEY\@.} Books I.--XII\@. 4\s.~6\d., or in 2 Parts, 2\s.~6\d.\ each.
\mitemspace[HORACE\@.] By {\smallcapsize A.~J. MACLEANE\@.} 3\s.~6\d., or in 2 Parts, 2\s.\ each.
\mitemspace[JUVENAL\@.] By {\smallcapsize HERMAN PRIOR\@.} 3\s.~6\d.
\mitemspace[MARTIAL\@.] By {\smallcapsize DR. PALEY} and {\smallcapsize W.~H. STONE\@.} 4\s.~6\d.
\mitemspace[OVID\@. Fasti.] By {\smallcapsize DR. PALEY\@.} 3\s.~6\d., or in 3 Parts, 1\s.~6\d.\ each.
\mitemspace[SALLUST\@. Catilina and Jugurtha.] By {\smallcapsize G.~LONG} and {\smallcapsize J.~G. FRAZER\@.} 3\s.~6\d.,
or in 2 Parts, 2\s.\ each.
\mitemspace[TACITUS\@. Germania and Agricola.] By {\smallcapsize P.~FROST\@.} 2\s.~6\d.
\mitemspace[VIRGIL\@.] {\smallcapsize CONINGTON'S} edition abridged. 2 Vols. 4\s.~6\d.\ each, or in 9 Parts,
             1\s.~6\d.\ each.
\mitemspace[--- Bucolics and Georgics.] {\smallcapsize CONINGTON'S} edition abridged. 3\s.
\mitemspace[XENOPHON\@.] By {\smallcapsize J.~F. MACMICHAEL\@.} 3\s.~6\d., or in 4 Parts, 1\s.~6\d.\ each.
\mitemspace[--- Cyropaedia.] By {\smallcapsize G.~M. GORHAM\@.} 3\s.~6\d., or in 2 Parts, 1\s.~6\d.\ each.
\mitemspace[--- Memorabilia.] By {\smallcapsize PERCIVAL FROST\@.} 3\s.
\end{seriescatlist}
%-----File: 353.png-----

\cseriestoc{Lower Form Series}

\begin{seriescatlist}
\mitemspace[VIRGIL'S Aeneid.] Book I. {\smallcapsize CONINGTON'S} edition abridged, with Vocabulary.
1\s.~6\d.
\mitem[CAESAR,] De Bello Gallico. By {\smallcapsize G.~LONG\@.} Books I., II. and III., with
Vocabulary, 1\s.~6\d.\ each.
\mitemspace[HORACE\@.] Book I\@. By {\smallcapsize A.~J. MACLEANE\@.} with Vocabulary. 1\s.~6\d.
\mitemspace[ECLOGAE LATINAE\@.] By {\smallcapsize REV. P.~FROST\@.} 1\s.~6\d.
\mitemspace[LATIN VERSE BOOK\@.] By {\smallcapsize REV. P.~FROST\@.} 2\s.
\mitemspace[ANALECTA GRAECA MINORA\@.] By {\smallcapsize REV. P.~FROST\@.} 2\s.
\mitemspace[TALES FOR LATIN PROSE COMPOSITION\@.] By {\smallcapsize G.~H. WELLS\@.} 2\s.
\mitemspace[LATIN VOCABULARIES FOR REPETITION\@.] By {\smallcapsize A.~M.~M. STEDMAN.}
1\s.~6\d.
\mitemspace[EASY LATIN PASSAGES\@.] By {\smallcapsize A.~M.~M. STEDMAN\@.} 1\s.~6\d.
\mitemspace[GREEK TESTAMENT SELECTIONS\@.] By {\smallcapsize A.~M.~M. STEDMAN\@.} 2\s.~6\d.
\end{seriescatlist}

\cseriestoc{Primary Classics}
\begin{seriescatlist}
\mitemspace[EASY SELECTIONS FROM CAESAR\@.] By {\smallcapsize A.~M.~M. STEDMAN\@.} 1\s.
\mitemspace[EASY SELECTIONS FROM LIVY\@.] By {\smallcapsize A.~M.~M. STEDMAN\@.} 1\s.~6\d.
\mitemspace[EASY SELECTIONS FROM HERODOTUS\@.] By {\smallcapsize A.~G. LIDDELL\@.} 1\s.~6\d.
\end{seriescatlist}

\cseriestoc{Classical Tables}

\begin{seriescatlist}
\mitemspace[NOTABILIA QUAEDUM\@.] 1\s.
\mitemspace[GREEK VERBS\@.] By {\smallcapsize J.~S. BAIRD\@.} 2\s.~6\d.
{\stretchyspace
\mitemspace[EXERCISES ON THE IRREGULAR AND DEFECTIVE GREEK VERBS\@.]}
By {\smallcapsize F.~ST. JOHN THACKERAY\@.} 1\s.~6\d.
\mitemspace[NOTES ON GREEK ACCENTS\@.] By {\smallcapsize DR. BARRY\@.} 1\s.
\mitemspace[HOMERIC DIALECT\@.] By {\smallcapsize J.~S. BAIRD\@.} 1\s.
\mitemspace[GREEK ACCIDENCE\@.] By {\smallcapsize P.~FROST\@.} 1\s.
\mitemspace[LATIN ACCIDENCE\@.] By {\smallcapsize P.~FROST\@.} 1\s.
\mitemspace[LATIN VERSIFICATION\@.] 1\s.
\mitemspace[PRINCIPLES OF LATIN SYNTAX\@.] 1\s.
\end{seriescatlist}

\cseriestoc{Bell's Classical Translations}

\begin{seriescatlist}
\mitemspace[AESCHYLUS\@.] By {\smallcapsize WALTER HEADLAM\@.} 6 Vols.   \catright{In the press.}
\mitemspace[ARISTOPHANES\@.] Acharnians. By {\smallcapsize W.~H. COVINGTON\@.} 1\s.
\mitemspace[CAESAR'S Gallic War.] By {\smallcapsize W.~A. McDEVITTE\@.} 2 Vols. 1\s.\ each.
\mitemspace[CICERO\@.] Friendship and Old Age. By {\smallcapsize G.~H. WELLS\@.} 1\s.
\mitemspace[EURIPIDES\@.] 14 Vols. By {\smallcapsize E.~P. COLERIDGE\@.} 1\s.\ each.
\mitemspace[LIVY\@.] Books I.--IV\@. By {\smallcapsize J.~H. FREESE\@.} 1\s.\ each.
\mitemspace[--- Book V\@.] By {\smallcapsize E.~S. WEYMOUTH\@.} 1\s.
\mitemspace[--- Book IX\@.] By {\smallcapsize F.~STORR\@.} 1\s.
\mitem[LUCAN]: The Pharsalia. Book I\@. By {\smallcapsize F.~CONWAY\@.} 1\s.
\mitemspace[SOPHOCLES\@.] 7 Vols. By {\smallcapsize E.~P. COLERIDGE\@.} 1\s.\ each.
\mitemspace[VIRGIL\@.] 6 Vols. By {\smallcapsize A.~HAMILTON BRYCE\@.} 1\s.\ each.
\end{seriescatlist}

\cseriestoc{Cambridge Mathematical Series}

\begin{seriescatlist}
\mitemspace[ARITHMETIC\@.] By {\smallcapsize C.~PENDLEBURY\@.} 4\s.~6\d., or in 2 Parts, 2\s.~6\d.\ each.
Key to Part II\@. 7\s.~6\d.\ net.

\mitemspace[EXAMPLES IN ARITHMETIC\@.] By {\smallcapsize C.~PENDLEBURY\@.} 3\s., or in 2 Parts,
1\s.~6\d.\ and 2\s.
\mitemspace[ARITHMETIC FOR INDIAN SCHOOLS\@.] By {\smallcapsize PENDLEBURY AND TAIT\@.} 3\s.
\mitemspace[ELEMENTARY ALGEBRA\@.] By {\smallcapsize J.~T. HATHORNTHWAITE\@.} 2\s.
\mitemspace[CHOICE AND CHANCE\@.] By {\smallcapsize W.~A. WHITWORTH\@.} 6\s.
\mitemspace[EUCLID\@.] By {\smallcapsize H.~DEIGHTON\@.} 4\s.~6\d., or Book I., 1\s.; Books I. and II., 1\s.~6\d.;
Books I.--III., 2\s.~6\d.; Books III. and IV., 1\s.~6\d.
\mitemspace[--- Key.] 5\s.\ net.
\mitemspace[EXERCISES ON EUCLID, \&c.] By {\smallcapsize J.~McDOWELL\@.} 6\s.
\mitemspace[ELEMENTARY TRIGONOMETRY\@.] By {\smallcapsize DYER AND WHITCOMBE\@.} 4\s.~6\d.
\mitemspace[PLANE TRIGONOMETRY\@.] By {\smallcapsize T.~G. VYVYAN\@.} 3\s.~6\d.
\mitemspace[ANALYTICAL GEOMETRY FOR BEGINNERS\@.] Part I\@. By {\smallcapsize T.~G.
VYVYAN\@.} 2\s.~6\d.
\mitemspace[ELEMENTARY GEOMETRY OF CONICS\@.] By {\smallcapsize DR. TAYLOR\@.} 4\s.~6\d.
%-----File: 354.png-----
\mitemspace[GEOMETRICAL CONIC SECTIONS\@.] By {\smallcapsize H.~G. WILLIS}, 5\s.
\mitemspace[SOLID GEOMETRY\@.] By {\smallcapsize W.~S. ALDIS\@.} 6\s.
\mitemspace[GEOMETRICAL OPTICS\@.] By {\smallcapsize W.~S. ALDIS\@.} 4\s.
\mitemspace[ROULETTES AND GLISSETTES\@.] By {\smallcapsize DR. W.~H. BESANT\@.} 5\s.
\mitemspace[ELEMENTARY HYDROSTATICS\@.] By {\smallcapsize DR. W.~H. BESANT\@.} 4\s.~6\d.
         Solutions. 5s
\mitemspace[HYDROMECHANICS\@.] Part I\@. Hydrostatics. By {\smallcapsize DR. W.~H. BESANT\@.} 5\s.
\mitemspace[DYNAMICS\@.] By {\smallcapsize DR. W.~H. BESANT\@.}  10\s.~6\d.
\mitemspace[RIGID DYNAMICS\@.] By {\smallcapsize W.~S. ALDIS\@.} 4\s.
\mitemspace[ELEMENTARY DYNAMICS\@.] By {\smallcapsize DR. W.~GARNETT\@.} 6\s.
\mitemspace[ELEMENTARY TREATISE ON HEAT\@.] By {\smallcapsize DR. W.~GARNETT\@.} 4\s.~6\d.
\mitemspace[ELEMENTS OF APPLIED MATHEMATICS\@.] By {\smallcapsize C.~M. JESSOP\@.} 6\s.
\mitemspace[PROBLEMS IN ELEMENTARY MECHANICS\@.] By {\smallcapsize W.~WALTON\@.} 6\s.
\mitemspace[EXAMPLES IN ELEMENTARY PHYSICS\@.] By {\smallcapsize W.~GALLATLY\@.} 4\s.
\mitemspace[MATHEMATICAL EXAMPLES\@.] By {\smallcapsize DYER} and {\smallcapsize PROWDE SMITH\@.} 6\s.
\end{seriescatlist}

\cseriestoc{Cambridge School and College Text Books}
\begin{seriescatlist}
\mitemspace[ARITHMETIC\@.] By {\smallcapsize C.~ELSEE\@.} 3\s.~6\d.
\mitem[---] By A.~WRIGLEY.  3\s.~6\d.
\mitemspace[EXAMPLES IN ARITHMETIC\@.] By {\smallcapsize WATSON and GOUDIE\@.} 2\s.~6\d.
\mitemspace[ALGEBRA\@.] By C.~ELSEE. 4\s.
\mitemspace[EXAMPLES IN ALGEBRA\@.] By {\smallcapsize MACMICHAEL} and {\smallcapsize PROWDE SMITH\@.}
3\s.~6\d.\ and 4\s.~6\d.
\mitemspace[PLANE ASTRONOMY\@.] By {\smallcapsize P.~T. MAIN\@.} 4\s.
\mitemspace[GEOMETRICAL CONIC SECTIONS\@.] By {\smallcapsize DR. W.~H. BESANT\@.} 4\s.~6\d.
\mitemspace[STATICS\@.] By {\smallcapsize BISHOP GOODWIN\@.} 3\s.
\mitemspace[NEWTON'S Principia.] By {\smallcapsize EVANS and MAIN\@.} 4\s.
\mitemspace[ANALYTICAL GEOMETRY\@.] By {\smallcapsize T.~G. VYVYAN\@.} 4\s.~6\d.
\mitemspace[COMPANION TO THE GREEK TESTAMENT\@.] By {\smallcapsize A.~C. BARRETT}, 5\s.
\mitemspace[TREATISE ON THE BOOK OF COMMON PRAYER\@.] By {\smallcapsize W.~G.
           HUMPHRY\@.}  2\s.~6\d.
\mitemspace[TEXT BOOK OF MUSIC\@.] By {\smallcapsize H.~C. BANISTER\@.} 5\s.
\mitemspace[CONCISE HISTORY OF MUSIC\@.] By {\smallcapsize DR. H.~G. BONAVIA HUNT\@.} 3\s.~6\d.
\end{seriescatlist}

\cseriestoc{Foreign Classics}
\begin{seriescatlist}
\mitemspace[FÉNELON'S Télémaque.] By {\smallcapsize C.~J. DELILLE\@.} 2\s.~6\d.
\mitemspace[LA FONTAINE'S Select Fables.] By {\smallcapsize F.~E.~A. GASC\@.} 1\s.~6\d.
\mitemspace[LAMARTINE'S Le Tailleur de Pierres de Saint-Point.] By {\smallcapsize J.~BOÏELLE\@.} 1\s.~6\d.
\mitemspace[SAINTINE'S Picciola.] By {\smallcapsize DR. DUBEC\@.} 1\s.~6\d.
\mitemspace[VOLTAIRE'S Charles XII\@.] By {\smallcapsize L.~DIRY\@.} 1\s.~6\d.
\mitemspace[GERMAN BALLADS\@.] By {\smallcapsize C.~L. BIELEFELD\@.} 1\s.~6\d.
\mitemspace[GOETHE'S Hermann und Dorothea.] By {\smallcapsize E.~BELL} and {\smallcapsize E. WÖLFEL\@.} 1\s.~6\d.
\mitemspace[SCHILLER'S Wallenstein.] By {\smallcapsize DR. BUCHHEIM\@.} 5\s, or in 2 Parts, 2\s.~6\d.\ each.
\mitemspace[--- Maid of Orleans.] By {\smallcapsize DR. W.~WAGNER\@.} 1\s.~6\d.
\mitemspace[--- Maria Stuart.] By {\smallcapsize V.~KASTNER\@.} 1\s.~6\d.
\end{seriescatlist}

\cseriestoc{Modern French Authors}

\begin{seriescatlist}
\mitemspace[BALZAC'S Ursule Mirouët.] By {\smallcapsize J.~BOÏELLE\@.} 3\s.
\mitemspace[CLARÉTIE'S Pierrille.] By {\smallcapsize J.~BOÏELLE\@.} 2\s.~6\d.
\mitemspace[DAUDET'S La Belle Modernaise.] By {\smallcapsize J.~BOÏELLE\@.} 2\s.
\mitemspace[GREVILLE'S Le Moulin Frappier.] By {\smallcapsize J.~BOÏELLE\@.} 3\s.
\mitemspace[HUGO'S Bug Jargal.] By J.~BOÏELLE. 3\s.
\end{seriescatlist}

\cseriestoc{Modern German Authors}

{\centering\scshape For Beginners.\par}
\begin{seriescatlist}
\mitemspace[HEY'S Fabeln für Kinder.] By {\smallcapsize PROF. LANGE\@.} 1\s.~6\d.
\mitem[--- ---] with Phonetic Transcription of Text, \&c. 2\s.
\mitemspace[FREYTAG'S Soll und Haben.] By {\smallcapsize W.~H. CRUMP\@.} 2\s.~6\d.
\end{seriescatlist}
%-----File: 355.png-----
{\smallskip\centering\scshape For Intermediate Students.\par}
\begin{seriescatlist}
\mitemspace[BENEDIX'S Doktor Wespe.] By {\smallcapsize PROF. LANGE\@.} 2\s.~6\d.
\end{seriescatlist}

{\smallskip\centering\scshape For Advanced Students.\par}
\begin{seriescatlist}
\mitemspace[HOFFMANN'S Meister Martin.] By {\smallcapsize PROF. LANGE\@.} 1\s.~6\d.
\mitemspace[HEYSE'S Hans Lange.] By {\smallcapsize A.~A. MACDONELL\@.} 2\s.
\mitemspace[ANERBACH'S Auf Wache, and Roquette's Der Gefrorene Kuss.] By
{\smallcapsize A.~A. MACDONELL\@.} 2.
\mitemspace[MOSER'S Der Bibliothekar.] By {\smallcapsize PROF. LANGE\@.} 2\s.
\mitemspace[EBERS' Eine Frage.] By {\smallcapsize F.~STORR\@.} 2\s.
\mitemspace[FREYTAG'S Die Journalisten.] By {\smallcapsize PROF. LANGE\@.} 2\s.~6\d.
\mitemspace[GUTZKOW'S Zopf und Schwert.] By {\smallcapsize PROF. LANGE\@.} 2\s.~6\d.
\mitemspace[GERMAN EPIC TALES\@.] By {\smallcapsize DR. KARL NEUHAUS\@.} 2\s.~6\d.
\mitemspace[SCHEFFEL'S Ekkehard.] By {\smallcapsize DR. H.~HAGER\@.} 3\s.
\end{seriescatlist}

\smallskip
% Each of these is also in the Summary of Series table of contents
\textbf{The following Series are given in full in the body of the Catalogue.}

\begin{seriescatlist}
\mitemspace[GOMBERT'S French Drama.] {\textit{See page} \pageref{ser:Gombert}.}
\ctoc{\tcsext}{Gombert's French Drama}
\mitemspace[BELL'S Modern Translations.] {\textit{See page} \pageref{ser:Translations}.}
\ctoc{\tcsext}{Bell's Modern Translations}
\mitemspace[BELL'S English Classics.] {\textit{See page} \pageref{ser:Classics}.}
\ctoc{\tcsext}{Bell's English Classics}
\mitemspace[HANDBOOKS OF ENGLISH LITERATURE\@.] {\textit{See page} \pageref{ser:EngLit}.}
\ctoc{\tcsext}{Handbooks of English Literature}
\mitemspace[TECHNOLOGICAL HANDBOOKS\@.] {\textit{See page} \pageref{ser:Technological}.}
\ctoc{\tcsext}{Technological Handbooks}
\mitemspace[BELL'S Agricultural Series.] {\textit{See page} \pageref{ser:Agricultural}.}
\ctoc{\tcsext}{Bell's Agricultural Series}
\mitemspace[BELL'S Reading Books and Geographical Readers.] {\textit{See pp}.\ \pageref{ser:Reading}, \pageref{ser:Geographical}.}
\ctoc{\tcsext}{Bell's Reading Books and Geographical Readers}
\end{seriescatlist}

\clearpage

\pagestyle{empty}
\vspace*{\fill}
\pdfbookmark[0]{Transcriber's Notes}{Transcriber's Notes}

{%
  \setlength{\parindent}{0pt}
	\setlength{\parskip}{6pt plus 2pt minus 1pt}
  \subsection*{{\normalsize\centering\itshape Transcriber's Notes}}
  \clabel{section:tn}
  \small

  Spelling has been made consistent for the words \textit{encyclopaedia} and \textit{hypotenuse}.

  Hyphenation has been made consistent for the words \textit{co-axial}, \textit{equi-conjugate},
      \textit{semi-axes}, \textit{semi-axis}, \textit{semi-diameter}, \textit{semi-diameters},
      \textit{sub-tangent} and \textit{book-keeping}.

  Hyphenation and capitalisation have been made consistent for the words \textit{latus rectum}
      and \textit{semi-latus rectum}, retaining capitals only where the term is defined.

  Table of contents entries have been altered to match chapter and section headings.

  The \textit{æ} ligature and \textit{ae} as separate letters have been retained as
      found in the catalogue.

  Minor inconsistencies in punctuation, particularly in the catalogue, have been
      silently corrected.

  In the original, figures are sometimes repeated during long articles; these duplicates have
      been retained.

  Article \cartref{36}: Original text reads \textit{``If from a point $Q$ tangents $QP$, $QP$ be drawn\ldots''}
      The second $QP$ has been changed to $QP'$.

  Catalogue: \textit{M'Mahon} and \textit{M'Devitte} (p.\pageref{note:mc}) have been
      changed to \textit{McMahon} and \textit{McDevitte} for consistency.

  Note that the author refers to \textit{\textbf{coaxal} circles} and
      \textit{\textbf{co-axial} parabolas}---these are intentionally different.
      Also \textit{radii vectores} is used for the plural of \textit{radius vector},
      and \textit{latera recta} for the plural of \textit{latus rectum}.
}

\cleardoublepage

\backmatter

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\cheader{LICENSING}
\pagenumbering{Alph}

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% End of the Project Gutenberg EBook of Conic Sections Treated Geometrically, by 
% W.H. Besant                                                             %
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% *** END OF THIS PROJECT GUTENBERG EBOOK CONIC SECTIONS ***              %
%                                                                         %
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[53] [54] [55] [56] [57] [58] (./29913-t.tcs)
\tf@tcs=\write6
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[8] [9] (./29913-t.aux)

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 29913-t.out
 29913-t.out
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 ***********

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Here is how much of TeX's memory you used:
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 20644 words of font info for 77 fonts, out of 1200000 for 2000
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Output written on 29913-t.pdf (367 pages, 2076062 bytes).
PDF statistics:
 2735 PDF objects out of 2984 (max. 8388607)
 522 named destinations out of 1000 (max. 131072)
 1140 words of extra memory for PDF output out of 10000 (max. 10000000)

