% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
%                                                                         %
% Project Gutenberg's Elements of Plane Trigonometry, by Hugh Blackburn   %
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% 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     %
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%                                                                         %
% Title: Elements of Plane Trigonometry                                   %
%        For the use of the junior class of mathematics in the            %
%        University of Glasgow                                            %
%                                                                         %
% Author: Hugh Blackburn                                                  %
%                                                                         %
% Release Date: June 25, 2010 [EBook #32973]                              %
%                                                                         %
% Language: English                                                       %
%                                                                         %
% Character set encoding: ISO-8859-1                                      %
%                                                                         %
% *** START OF THIS PROJECT GUTENBERG EBOOK ELEMENTS OF PLANE TRIGONOMETRY ***
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%%%%%%%%%%%%%%%%%%%%%%%% START OF DOCUMENT %%%%%%%%%%%%%%%%%%%%%%%%%%

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Project Gutenberg's Elements of Plane Trigonometry, by Hugh Blackburn

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: Elements of Plane Trigonometry
       For the use of the junior class of mathematics in the
       University of Glasgow

Author: Hugh Blackburn

Release Date: June 25, 2010 [EBook #32973]

Language: English

Character set encoding: ISO-8859-1

*** START OF THIS PROJECT GUTENBERG EBOOK ELEMENTS OF PLANE TRIGONOMETRY ***
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Produced by Andrew D. Hwang, Laura Wisewell and the Online
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University of Glasgow.)
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%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%%
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\LARGE PLANE TRIGONOMETRY \\[2\TmpLen]
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MATHEMATICS IN THE UNIVERSITY \\[\TmpLen]
OF GLASGOW.
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\Large HUGH BLACKBURN, M.A. \\[\TmpLen]
\scriptsize PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF GLASGOW, \\[\TmpLen]
LATE FELLOW OF TRINITY COLLEGE, CAMBRIDGE.
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PRINTED BY C. J. CLAY, M.A. \\[2\TmpLen]
AT THE UNIVERSITY PRESS.
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\Preface

\First{Some} apology is required for adding another to the long
list of books on Trigonometry. My excuse is that during
twenty years' experience I have not found any published
book exactly suiting the wants of my Students. In conducting
a Junior Class by regular progressive steps from
Euclid and Elementary Algebra to Trigonometry, I have
had to fill up by oral instruction the gap between the Sixth
Book of Euclid and the circular measurement of Angles;
which is not satisfactorily bridged by the propositions of
Euclid's Tenth and Twelfth Books usually \emph{supposed to be}
learned; nor yet by demonstrations in the modern books
on Trigonometry, which mostly follow Woodhouse; while
the Appendices to Professor Robert Simson's Euclid in the
editions of Professors Playfair and Wallace of Edinburgh,
and of Professor James Thomson of Glasgow, seemed to me
defective for modern requirements, as not sufficiently connected
with Analytical Trigonometry.

What I felt the want of was a short Treatise, to be used
as a Text Book after the Sixth Book of Euclid had been
learned and some knowledge of Algebra acquired, which
should contain satisfactory demonstrations of the propositions
to be used in teaching Junior Students the Solution of
Triangles, and should at the same time lay a solid foundation
for the study of Analytical Trigonometry.

This want I have attempted to supply by applying, in
the first Chapter, Newton's Method of Limits to the mensuration
of circular arcs and areas; choosing that method
\DPPageSep{004.png}{iv}%
both because it is the strictest and the easiest, and because
I think the Mathematical Student should be early introduced
to the method.

The succeeding Chapters are devoted to an exposition
of the nature of the Trigonometrical ratios, and to the
demonstration by geometrical constructions of the principal
propositions required for the Solution of Triangles. To
these I have added a general explanation of the applications
of these propositions in Trigonometrical Surveying:
and I have concluded with a proof of the formulæ for the
sine and cosine of the sum of two angles treated (as it
seems to me they should be) as examples of the Elementary
Theory of Projection. Having learned thus much the
Student has gained a knowledge of Trigonometry as originally
understood, and may apply his knowledge in Surveying;
and he has also reached a point from which he may advance
into Analytical Trigonometry and its use in Natural Philosophy.

Thinking that others may have felt the same want as
myself, I have published the Tract instead of merely printing
it for the use of my Class.
\bigskip
\begin{flushright}
H. B.\hspace*{1.5em}
\end{flushright}
\vfill
\DPPageSep{005.png}{1}%

%%%% MAIN MATTER %%%%
\BookTitle%[** TN: Macro prints text below]
% ELEMENTS
% OF
% PLANE TRIGONOMETRY.

\First{Trigonometry} (from \textgreek{tr'igwnon}, triangle, and \textgreek{metr'ew}, I measure)
is the science of the numerical relations between the
sides and angles of triangles.

This Treatise is intended to demonstrate, to those who
have learned the principal propositions in the first six books
of Euclid, so much of Trigonometry as was originally implied
in the term, that is, how from given values of some of
the sides and angles of a triangle to calculate, in the most
convenient way, all the others.

A few propositions supplementary to Euclid are premised
as introductory to the propositions of Trigonometry as usually
understood.


\Chapter{I.}{OF THE MENSURATION OF THE CIRCLE.}

\Def{1} A magnitude or ratio, which is fixed in value
by the conditions of the question, is called a \Emph{Constant}.

\Def{2} A magnitude or ratio, which is not fixed in
\DPPageSep{006.png}{2}%
value by the conditions of the question and which is conceived
to change its value by lapse of time, or otherwise,
is called a \Emph{Variable}.

\Def{3} If a variable shall be always less than a given
constant, but shall in time become greater than any less
constant, the given constant is the \Emph{Superior Limit} of the
variable: and if the variable shall be always greater than a
given constant but in time shall become less than any
greater constant, the given constant is the \Emph{Inferior Limit}
of the variable.

\Lemma{If two variables are at every instant equal their
limits are equal.}

For if the limits be not equal, the one variable shall
necessarily in time become greater than the one limit and
less than the other, while at the same instant the other
variable shall be greater than both limits or less than both
limits, which is impossible, since the variables are always
equal.

\Def{4} Curvilinear segments are \emph{similar} when, if on
the chord of the one as base any triangle be described with
its vertex in the arc, a similar triangle with its vertex in the
other arc can always be described on its chord as base; and
the arcs are \Emph{Similar Curves}.

\Cor{1} Arcs of circles subtending equal angles at the
centres are similar curves.

\Cor{2} If a polygon of any number of sides be inscribed
in one of two similar curves, a similar polygon can
be inscribed in the other.

\Def{5} Let a number of points be taken in a terminated
curve line, and let straight lines be drawn from each
\DPPageSep{007.png}{3}%
point to the next, then if the number of points be conceived
to increase and the distance between each two to diminish
continually, the extremities remaining fixed, the limit of the
sum of the straight lines is called the \Emph{Length of the
Curve}.

\Prop{I}{The lengths of similar arcs are proportional to
their chords.}

For let any number of points be taken in the one and the
points be joined by straight lines so as to inscribe a polygon
in it, and let a similar polygon be inscribed in the other, the
perimeters of the two polygons are proportional to the
chords, or the ratio of the perimeter of the one to its chord
is equal to the ratio of the perimeter of the other to its
chord. Then if the number of sides of the polygons increase
these two ratios vary but remain always equal to each other,
therefore (Lemma) their limits are equal. But the limit of
the ratio of the perimeter of the polygon to the chord is
(Def.~\osf{5}) the ratio of the length of the curve to its chord,
therefore the ratio of the length of the one curve to its
chord is equal to the ratio of the length of the other curve
to its chord, or the lengths of similar finite curve lines are
proportional to their chords.

\Cor{1} Since semicircles are similar curves and the
diameters are their chords, the ratio of the semi-circumference
to the diameter is the same for all circles.

If this ratio be denoted, as is customary, by $\dfrac{\pi}{2}$, then
numerically
\begin{gather*}
\text{the circumference $÷$ the diameter $= \pi$,} \\
\text{and the circumference $= 2\pi R$.}
\end{gather*}

%[Illustration]
\begin{wrapfigure}{r}{2in}
  \Input[2in]{008a}
\end{wrapfigure}
\Cor{2} The angle subtended at the centre of a circle
by an arc equal to the radius is the same for all circles. For
\DPPageSep{008.png}{4}%
if $AC$~be the arc equal to the radius,
and $AB$~the arc subtending a right
angle, then by \Euclid{VI.}{33}
\[
AOC : AOB :: AC : AB.
\]

But $AB$ is a fourth of the circumference
$ = \dfrac{\pi R}{2}$;
\begin{gather*}
\text{therefore }AOC:\text{a right angle}::
  R : \frac{\pi R}{2} :: 2 : \pi \\
\text{or numerically }
  AOC = \frac{2}{\pi} × \text{a right angle},
\end{gather*}
that is the angle subtended by an arc equal to the radius is
a fixed fraction of a right angle.

\Prop{II}{The areas of similar segments are proportional
to the squares on their chords.}

For, if similar polygons of any number of sides be inscribed
in the similar segments, they are to one another in
the duplicate ratio of the chords, or, alternately, the ratio of
the polygon inscribed in the one segment to the square on
its chord is the same as the ratio of the similar polygon in
the other segment to the square on its chord. Now conceive
the polygons to vary by the number of sides increasing
continually while the two polygons remain always similar,
then the variable ratios of the polygons to the squares on
the chords always remain equal, and therefore their limits
are equal (Lemma); and these limits are obviously the
ratios of the areas of the segments to the squares on the
chords, which ratios are therefore equal.

\Cor{}Circles are to one another as the squares of their
diameters.
\DPPageSep{009.png}{5}%

\Note From Prop.\ II.~and~III. it is obvious that ``The
corresponding sides, whether straight or curved, of similar
figures, are proportionals; and their areas are in the duplicate
ratio of the sides.'' (Newton, \textit{Princip.}~I. Sect.~\Rom{I}.
Lemma~\Rom{V}.)

\Prop{III}{The area of any circular sector is half the
rectangle contained by its arc and the radius of the circle.}

%[Illustration]
\begin{figure}[hbt]
  \begin{center}
    \Input[3in]{009a}
  \end{center}
\end{figure}
Let $AOB$ be a sector. In the arc~$AB$ take any number
of equidistant points $A_1, A_2\Add{,} \Dots A_n$, and join $AA_1,
A_1A_2, \Dots A_nB$. Produce $AA_1$, and along it take parts $A_1A_2',
A_2'A_3', \Dots A_n'B'$ equal to $A_1A_2, A_2A_3, \Dots A_nB$ respectively:
so that $AB'$~is equal to the polygonal perimeter $AA_1A_2\Dots A_nB$;
then if the number of points $A_1, A_2$,~\&c., be conceived
to increase continually, the limit of~$AB'$ is the arc~$AB$.

Now through $A$ draw the line~$AT$ at right angles to~$OA$,
then as the number of points increases continually,
\DPPageSep{010.png}{6}%
the angle~$TAB'$ shall diminish continually, and shall in time
become less than any finite angle, and the limit of the
position of~$AB'$ shall be~$AB''$ measured along~$AT$, where
$AB''$~is equal in length to the arc~$AB$.

Join $OA_1'$, $OA_2'$, $OA_3', \Dots OA_n'$ and the triangles
$OA_1A_2$, $OA_2A_3$, $OA_3A_4, \Dots OA_nB$ are equal, each to each,
to $OA_1A_2'$, $OA_2'A_3'$, $OA_3'A_4', \Dots OA_n'B$, for the perpendiculars
from~$O$ on the sides of the polygon are all equal to the
perpendicular on~$AB'$; therefore the variable triangle~$OAB'$
is always equal to the variable polygon $OAA_1A_2 \Dots A_nB$;
therefore their limits are equal. But the limit of the triangle~$OAB'$
is~$OAB''$ and the limit of the polygon is the
sector~$OAB$; therefore the sector~$AOB$ is equal to the
triangle~$OAB''$, which is half the rectangle $OA$,~$AB''$, or
half the rectangle contained by the radius and the arc.

Hence the area of a circle $= \dfrac{1}{2}\, R × \text{circumference} = \pi R^2$
and the ratio of the circle to the square on its diameter
is $= \dfrac{\pi}{4}$.

\Prop{IV}{Any line, whether curved or polygonal, which
is convex throughout \(that is, which can be cut by a straight
line in only two points\), is less than any line, curved or polygonal,
which envelopes it from one extremity to the other\footnotemark.}
\footnotetext{This enunciation is taken from Legendre, \textit{Elements de Geometrie},
  12\textsuperscript{me}~ed.\ Liv.~\Rom{IV}. Prop.~\Rom{IX}., but the demonstration is different.}%

For the enveloping line is obviously greater than the
sum of any number of straight lines drawn as in Def.~\osf{5},
and therefore is greater than the limit of that sum, that is,
than the length of the curve.

\Cor{}Hence two straight lines, touching at its extremities
\DPPageSep{011.png}{7}%
any circular arc less than a semicircle, are together
greater than the arc.

\Prop{V}{If circles be inscribed in and described about
two regular polygons of the same perimeter, the second of which
has twice as many sides as the first, then \Inum{1}~the radius of the
circle inscribed in the second is an arithmetic mean between
\(i.e.\ is half the sum of\) the radius of the circle inscribed in and
the radius of the circle described about the first; and \Inum{2}~the
radius of the circle described about the second is a mean proportional
between the radius of the circle inscribed in the
second, and the radius of the circle described about the first.}

Let $BB'$ be a side of the first polygon, $C$~the centre
of the circle described about it.

From $C$ as centre with $CB$~as radius describe the
circle~$BB'E$.

Draw $ECA$ a diameter perpendicular to~$BB'$ and therefore
bisecting it in~$D$.

%[Illustration]
\begin{figure}[hbt]
  \begin{center}
    \Input{011a}
  \end{center}
\end{figure}

Join $EB$,~$EB'$. Draw $CF$ perpendicular to~$EB$, and
$FGH$ perpendicular to~$EA$.
\DPPageSep{012.png}{8}%

Then, because the angle~$BEB'$ is half of~$BCB'$, and
$FH$~is half of~$BB'$, for $FH$ bisects $EB$~and~$EB'$; therefore
$FH =$ the side of the second polygon, and $FEH =$ the
angle it subtends at the centre.

Therefore $EF$ is the radius of the circle described about
the second polygon, and $EG$~the radius of the circle
inscribed in it.

And $CD$, $CB$ are the radii of the circles inscribed in
and described about the first polygon.

But $EG$ is half of~$ED$, that is, half of $EC$ (or~$CB$)
and~$CD$ together, that is the radius of the circle inscribed
in the second polygon is the arithmetic mean of the radii
of the circles inscribed in and described about the first
polygon.

Again, because the triangles $EFG$,~$ECF$ are similar,
\[
EC: EF:: EF: EG,
\]
that is, the radius of the circle described about the second
polygon is a mean proportional to that of the circle described
about the first and that of the circle inscribed in
the second.

%[** TN: Italicized in original]
\Cor{}Hence the ratio of the circumference of a circle to its
diameter (or $\pi$) can be calculated to any degree of accuracy.

For let $R$, $R'$ be the radii of the circles described about,
and $r$,~$r'$ of those inscribed in, the first and second polygon
respectively, then
\[
r' = \frac{R + r}{2};\quad R' = \sqrt{r' · R}.
\]

From these it will be easy to calculate successively the
radii of circles inscribed in and described about isoperimetrical
polygons of $2$,~$4$, $8$,~$16$, $32$,~\&c.\ times the number
of sides of a given regular polygon.
\DPPageSep{013.png}{9}%

Then, if the radii and perimeter of a regular polygon
of any number of sides be known, by making it the first
polygon of the series and calculating the radii for a sufficient
number of succeeding polygons, we can calculate the value
of~$\pi$ (the ratio of the circumference of a circle to its
diameter) to any degree of accuracy. For since the perimeter
of each polygon will lie between the circumference
of its inscribed and circumscribed circles if $R$~and~$r$ be the
radii for any polygon of the series, we shall have $2\pi R$
greater, and $2\pi r$ less than~$p$, the common perimeter of
all the polygons. Therefore $\pi$~is intermediate to $\dfrac{p}{2R}$ and
$\dfrac{p}{2r}$, and, by doubling the number of sides of the polygon
sufficiently, $R$~and~$r$ can be made to differ as little as
we please, and therefore $\pi$~can be calculated as accurately
as desired.

The calculation is not very laborious. Thus, if we begin
from a square, each side of which is the unit, we have
$r_1 = 0.5$ and
\[
R_1 = \sqrt{.5} = 0.7071067812.
\]

Then
\[
r_2 = \frac{0.5 + 0.7071067812}{2} = 0.6035533906,
\]
and
\begin{gather*}
  R_2 = \sqrt{.7071067812 × .6035533906} \\
  = 0.6532814824.
\end{gather*}

In like manner the radii of circles inscribed in and
described about polygons of $16$,~$32$, $64$, $128$,~\&c.\ sides with
the same perimeter (viz.~4) are successively found by alternately
taking arithmetic and geometric means.
\DPPageSep{014.png}{10}%
\begin{table}[htb]
\[
\begin{array}{|r<{\quad\ }|c|c|}
\hline
\multicolumn{1}{|c|}{%
\settowidth{\TmpLen}{No.\ of sides }%
\parbox[c]{\TmpLen}{\medskip\centering No.\ of sides of the Polygon.\medskip}} &
\multicolumn{1}{c|}{%
\settowidth{\TmpLen}{Radius of Inscribed}%
\parbox[c]{\TmpLen}{\centering Radius of Inscribed Circle $= r$.}} &
\multicolumn{1}{c|}{%
\settowidth{\TmpLen}{Radius of Circum-}%
\parbox[c]{\TmpLen}{\centering Radius of Circumscribing Circle $= R$.}} \\
\hline\Strut
   4 & .5000000000 & .7071067812 \\
   8 & .6035533906 & .6532814824 \\
  16 & .6284174365 & .6407288619 \\
  32 & .6345731492 & .6376435773 \\
  64 & .6361083633 & .6368755077 \\
 128 & .6364919355 & .6366836927 \\
 256 & .6365878141 & .6366357516 \\
 512 & .6366117828 & .6366237671 \\
1024 & .6366177750 & .6366207710 \\
\&c. &     \&c.    &      \&c.   \\
\hline
\end{array}
\]
\end{table}

Stopping at the polygon of $1024$~sides, it appears that
\[
\DPchg{\pi < \frac{2000000}{636617} > \frac{2000000}{636621}}
      {\frac{2000000}{636621} < \pi < \frac{2000000}{636617}},
\]
i.e.\ $\DPchg{\pi < 3.14160 > 3.14158}{3.14158 < \pi < 3.14160}$.

It may however be shewn (see \Appref) that, when the
difference between $R$~and~$r$ is small, $\dfrac{1}{3} (r + 2R)$ is a very
near approximation to the limit of both radii, and that
therefore $\pi$~may be taken $= \dfrac{\dfrac{1}{2}\, p}{\dfrac{1}{3}(r + 2R)}$ with great accuracy.

Taking the radii for $1024$~sides
\[
\frac{r + 2R}{3}
  = \frac{1}{3} \left\{
    \begin{array}{r}
       .6366177750 \\
      1.2732415420
    \end{array}
  \right\}
  = \frac{1.9098593170}{3}
  = .6366197723,
\]
which will be found to be ten decimals of the radius of
the circle inscribed in a polygon of~$262144$ and every greater
number of sides if the table be continued.
\DPPageSep{015.png}{11}%

Thus we may take
\begin{align*}
  \pi &= \frac{20000000000}{6366197723}, \\
\text{or}\quad
      &= 3.141592654.
\end{align*}

By the method of ``continued fractions'' it will be found
that $\dfrac{22}{7}$~and~$\dfrac{355}{113}$ are nearer approximations to the value
of~$\pi$ than any simpler fractions.

Of these $\dfrac{22}{7}$ ($=3.14$) is the approximation discovered by
Archimedes (killed, it is said, at the siege of Syracuse,
\BC~\Date{212}); and the approximation $\dfrac{355}{113}$ ($=3.14159$) was given
by Adrian Metius of Alkmaer (died \AD~\Date{1636})\footnotemark.
\footnotetext{This simple and elegant elementary method of approximating to~$\pi$
  is taken from Leslie's \textit{Geometry}, \Rom{V}.~20; compare Legendre, \textit{Geometrie},
  \Rom{IV}.~14~and~16.}%


\Chapter[OF THE AREA OF A TRIANGLE]{II.}{OF THE AREA OF A TRIANGLE AND OF THE INSCRIBED
CIRCLE.}

\Prop{I}{A triangle is equal to the rectangle contained
by its semi-perimeter and the radius of the inscribed circle.}

Let $ABC$ be the triangle. Bisect the angles by the
lines $AO$, $BO$, $CO$, meeting (\Euclid{IV.}{4}) in~$O$, the centre
of the inscribed circle.

Then the triangle $ABC$ is made up of the triangles
$BOC$, $COA$, $AOB$, each of which stands on one of the
sides, as base, with its altitude equal to the radius of the
inscribed circle. Therefore the whole triangle $ABC$ is
\DPPageSep{016.png}{12}%
equal to a triangle having the sum of the three sides (or
the perimeter) for base and the radius of the inscribed circle
for altitude; or to the rectangle having the semi-perimeter
for base and the radius of the inscribed circle for altitude.

\Paragraph{Scholium.} The two tangents from each angle to the
inscribed circle are equal: hence, if three tangents, one from
each angle, be taken, their sum is the semi-perimeter, and
therefore a tangent from one of the angles, together with
the side opposite that angle, is equal to the semi-perimeter.

Let the sides opposite the angles $A$, $B$, $C$ be represented
numerically by $a$,~$b$,~$c$; the semi-perimeter by~$s$, and
the radius of the inscribed circle by~$r$.
\begin{gather*}
\text{Then, numerically, the Area} = rs. \\
\text{And } Ab = Ac = s - a,\quad Bc = Ba = s-b,\quad Ca = Cb = s - c.
\end{gather*}

\Def{} Let two of the sides of the triangle $ABC$ be
produced, and a circle described touching the two produced
sides and the third side. The circle is said to be \emph{excribed}\footnote
  {This word is often spelled ``\emph{escribed}'' improperly. The Latin
  word is exscribo, but the English usage is to elide the~\textit{s} in such cases, as
  \emph{expect} from exspecto, \emph{expatiate} from exspatior, \emph{extinguish} from exstinguo.
  No one ever proposed to emend these words into \emph{espect}, \emph{espatiate}, and
  \emph{estinguish}. Why then \emph{escribe?}}
on the third side.

\Prop{II}{A triangle is equal to the rectangle contained
by the radius of the circle excribed on one of its sides and
the tangent from the opposite angle to the inscribed circle.}

%[Illustration]
\begin{figure}[hbt]
  \begin{center}
    \Input{017a}
  \end{center}
\end{figure}
Let $ABC$ be the triangle. Bisect the angle~$A$ and the
exterior angles at $B$~and~$C$ by the lines $AO'$, $BO'$, $CO'$,
which will meet in the centre of the excribed circle\footnotemark.
\footnotetext{The proof of this is left to the reader, or he may consult Thomson's
  \textit{Euclid},~\Rom{IV}.~4.}%
Draw perpendiculars $O'a'$, $Ob'$, $O'c'$ on the three sides.
\DPPageSep{017.png}{13}%
Then these perpendiculars are all equal and each of them is
a radius of the excribed circle. Also the two tangents from
each angle to the excribed circle are equal; and therefore
$Ba'$~is equal to~$Bc'$, $Ca'$~to~$Cb'$, and $Ab'$~to~$Ac'$. Hence $Ab'$~and~$Ac'$
are together equal to the perimeter of the triangle
and each of them to the semi-perimeter. But because the
two triangles $AcO$~and~$Ac'O'$ are similar therefore $Ac : Oc
:: Ac': O'c'$ and (\Euclid{VI.}{16}) the rectangle $Ac$,~$O'c'$ is
equal to the rectangle $Oc$,~$Ac'$, which by Prop.~I. is the
area of the triangle.

Numerically, if the radius of the circle excribed on the
side~$BC$ be represented by~$\alpha$, this may be written
\[
(s - a) \alpha = r s = \text{the area}.
\]

\Prop{III\footnotemark}{A triangle is a mean proportional to the
\DPPageSep{018.png}{14}%
rectangle contained by the semi-perimeter and its excess over
one of the sides, and the rectangle contained by the excess of
the semi-perimeter over each of the other sides.}
\footnotetext{This most useful proposition was known to the Greeks of Alexandria,
  and by them communicated to the Arabians, but seems to have
  ``been reinvented in Europe about the latter part of the \osf{15}th~century.''
  Leslie's \textit{Geometry}, \Date{1828},~\Rom{V.}~\osf{19}, where the above demonstration (nearly)
  will be found.}%

With the same figure as before the right-angled triangles
$BOc$ and $BO'c'$ have the angles $BOc$ and $OBc$ equal to the
angles $O'Bc'$ and $BO'c'$ each to each. Therefore these triangles
are similar and (\Euclid{VI.}{4}) $Bc : cO :: c'O' :: c'B$.

Now, on the first and second of these lines let rectangles
of altitude~$Ac'$ be constructed, and on the third and
fourth rectangles of altitude~$Ac$. Then (\Euclid{VI.}{1})
$Ac', Bc : Ac', cO :: Ac, c'O' : Ac, c'B$.

But the rectangle $Ac'$,~$cO$ is equal to the triangle $ABC$
by Prop.~I. and the rectangle $Ac, c'O'$ is equal to $ABC$ by
Prop.~II\@.

Therefore the triangle is a mean proportional between
the rectangles $Ac'$,~$Bc$ and $Ac$,~$c'B$, that is between the
rectangle contained by the semi-perimeter and its excess
over the side~$CA$, and the rectangle contained by its excess
over the sides $BC$~and~$AB$ respectively.

Writing this numerically, and supposing the area to be
represented in square units by the number~$\Delta$, it becomes
\[
s(s-b) : \Delta :: \Delta : (s-a)(s-c)
\]
or
\[
\Delta^2 = s(s-a)(s-b)(s-c),
\]
whence the area can be calculated in square units when
the lengths of the sides are given numerically in units.

Also by Prop.~II. $r^2 = \dfrac{(s-a)(s-b)(s-c)}{s}$.
\DPPageSep{019.png}{15}%


\Chapter{III.}{OF SYMBOLS OF QUANTITY.}

\Paragraph{Angles not limited in magnitude.} In Euclid an angle is
not defined as a magnitude but as the inclination of two
lines, which never exceeds two right angles: and in most
of the propositions in Euclid it is not necessary to treat an
angle otherwise than as a change of direction of a line.
But where an angle is treated as a magnitude (notably in
\Euclid{VI.}{33} and consequently in \Euclid[]{III.}{7}, on which it depends)
any multiple whatever of an angle is termed an angle. So
also in Trigonometry, where angular magnitude in general
is treated numerically, it is desirable to use the term angle
for the sum of a number of angles, which may be greater
than two or than any number of right angles. In the same
way an arc of a circle may be greater than a circumference
or any number of circumferences.

\Paragraph{Negative quantities.} Again, when magnitudes are represented
numerically by algebraic symbols, the values of which
are defined but not specified, it is often desirable to express a
difference without limiting the generality of the expression by
stating which of the symbols stands for the greater number.
For instance, if a distance $a$~miles be measured from a fixed
point~$O$ along (or parallel to) a given line in a standard
direction, say east, to~$A$; and a line~$AB$ be cut off from~$OA$
by measuring from~$A$ in the opposite direction, or
westward, a distance $b$~miles, the distance~$OB$ may be
said to be $= (a - b)$ miles east of~$O$ not only in the case
where $a > b$, but also when $a < b$, if it be agreed to interpret
the result as meaning $a - b$ east of~$O$ (or in the standard
\DPPageSep{020.png}{16}%
direction) if $a - b$ is a $+$~number, and $b - a$ miles west of~$O$
(or in the contrary direction), when $a - b$ is a $-$~number.

Then the standard direction may be called the~$+$ (or
positive) direction, and the contrary direction the~$-$ (or
negative) direction.

In the same way, if $A$ be $a$~feet above~$O$ and $B$ be $b$~feet
below~$A$, $B$~is $a - b$ feet above~$O$ if $a > b$, and $b - a$ feet
below~$O$ if $a < b$. To express the result by one formula we
may say that $B$~is $a - b$ feet above~$O$ in both cases, if we
interpret the $+$~sign as meaning upwards from~$O$ and the
$-$~sign as meaning downwards (or in the contrary direction)
from~$O$.

Thus, in the case of lines measured along (or parallel
to) a specified line from a given point (or \emph{origin}) the sign~$+$
is conveniently prefixed (or understood) before lines measured
in a standard direction and the sign~$-$ before those
measured in the contrary direction.

Again, with reference to angles, if the hand of a going
clock be put back through an angle~$\theta$, then, after the time
during which the hand moves through an angle~$\phi$, the hand
will make an angle $\theta - \phi$ with its present position, the angle
being~$+$ and measured in the opposite way to that in which
the hand of the clock moves, if $\theta > \phi$; or $-$~and measured
in the contrary direction, if $\theta < \phi$.

In what follows, an angle will be considered as if produced
by the revolution of a radius of a circle, the direction
of revolution from an initiatory position being considered as
$-$~or~$+$ according as it takes place in the direction of the
motion of the hand of a clock or the reverse.
\DPPageSep{021.png}{17}%


\Chapter[UNIT OF ANGULAR MAGNITUDE]{IV.}{OF THE UNIT OF ANGULAR MAGNITUDE.}

\First{In} order to treat angular magnitude numerically it is
necessary to use some fixed angle as a standard of comparison,
by reference to which the magnitudes of angles
under consideration may be denoted.

The angle of easiest construction is the angle of an
equilateral triangle, which is also two-thirds of a right
angle.

For the purpose of expressing simply fractions of the
standard, the sexagesimal division (or division into $60$ths)
of the standard is probably the most convenient (because
the third, fourth, fifth, sixth, tenth, twelfth, fifteenth, twentieth
and thirtieth are all exact numbers of sixtieths).

For such reasons perhaps the sexagesimal scale, which
has prevailed since the time of Ptolemy\footnotemark, was originally
adopted. It is still employed, and we have the following \\
\emph{Notation for angles in aliquot parts of a right angle}:~---
\footnotetext{See article ``Arithmetic'' in the \textit{Encyclopædia Metropolitana},
  p.~\osf{401},~§\;\osf{39}.}%

The $90$th part of a right angle (or the $60$th of the angle
of an equilateral triangle) is called a degree.

One degree is denoted by~$1°$; so that a right angle is~$90°$;
the angle of an equilateral triangle is~$60°$.

The $60$th part of a degree is a minute, denoted by~$1'$,
$\therefore 1° = 60'$.

The $60$th part of a minute is a second, denoted by~$1''$,
$\therefore 1' =  60''$.
\DPPageSep{022.png}{18}%

Fractions of a second are now usually denoted by decimals,
but in older books, as for instance in Newton's \textit{Principia},
the sexagesimal division is carried farther, so that
\[
1'' = 60''',\quad
1''' = 60^\text{iv},\quad
1^\text{iv} = 60^\text{v}.
\]

This notation is used for many practical purposes\footnotemark.
\footnotetext{The centesimal division of the right angle into $100$~grades,~\&c.\
  proposed at the French Revolution, though adopted in the \textit{Mécanique
  Céleste} of Laplace, has been abandoned even in France.}%

\Paragraph{Circular Measure.} In formulæ, involving explicitly the
numerical value of an angle, it is more suitable and it is
usual to represent the angle by its ratio to the angle subtended
at the centre of a circle by an arc equal to the
radius. This angle (Chap.~\DPtypo{\osf{1}}{I}.\ Prop.~\DPtypo{\osf{1}}{I}.\ Cor.~\osf{2}) is invariable,
that is, is the same whatever radius be taken, and can
therefore be used with propriety for a standard of comparison.
It is called the \emph{unit of circular measure}, and the
ratio of any angle to this unit is called the \emph{circular measure}
of the angle.

\emph{The circular measure} of an angle is also the ratio of
the arc subtending the angle at the centre of any circle to
the radius. For let $AB$ be the arc subtending the angle~$AOB$,
%[Illustration]
\begin{figure}[hbt]
  \begin{center}
    \Input[2.75in]{022a}
  \end{center}
\end{figure}
of which the circular measure is~$\theta$, at the centre of
the circle of which the radius $OA = R$. And let $AOC$ be
\DPPageSep{023.png}{19}%
subtended by the arc $AC = R$. Then $AOC$ is the unit;
and (\Euclid{VI.}{33}) $AOB: AOC:: AB: AC$,
\begin{gather*}
\text{or } \theta = \frac{AB}{AC} = \frac{AB}{R}, \\
\text{and the arc } AB = R\theta.
\end{gather*}

The ratio of the arc to the radius is obviously the same
whatever radius be taken. For if $A'B'$ be the arc subtending
the same angle at the centre of the circle of radius
$OA'=R'$, we have $AB:A'B':: R:R'$,
\[
\text{or } \frac{AB}{R} = \frac{A'B'}{R'}.
\]

The circular measure of a right angle
\[
= \frac{\frac{1}{4} \text{ circumference}}{R} = \frac{\pi}{2}.
\]
And the number of degrees in the unit of circular measure
\[
= \frac{180°}{\pi}.
\]

\Paragraph{Complementary angles.} Two angles are said to be complements,
each of the other, when their sum is a right angle.
Hence the complement of an angle~$A°$ contains $90° - A°$,
and the circular measure of the complement of~$\theta$ is $\dfrac{\pi}{2} - \theta$.
If $A° > 90°$, or $\theta > \dfrac{\pi}{2}$, the complement is~$-$ and to be measured
in the negative direction.

\Paragraph{Supplementary angles.} Two angles are supplements, each
of the other, when their sum is two right angles. Hence the
supplement of~$A°$ is $180° - A°$; and the circular measure
of the supplement of~$\theta$ is $\pi - \theta$. If $A° > 180°$, or $\theta > \pi$,
the supplement is~$-$ and to be measured in the negative
direction.
\DPPageSep{024.png}{20}%


\Chapter[CIRCULAR FUNCTIONS]{V.}{CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS.}

\Paragraph{Function.} When one magnitude or ratio is so connected
with another that the former changes with the latter, but is
determinable for any given value of the latter, the former
is said to be a function of the latter.

Hence certain ratios, which depend on the value of
the angle, or its circular measure, are called circular functions.

%[Illustration]
\begin{figure}[hbt]
  \begin{center}
    \Input{024a}
  \end{center}
\end{figure}

Let $ACA'C'$ be any circle, $O$~its centre, $A'OA$, $COC'$
two diameters at right angles dividing the circle into four
quadrants.

Let $OA$, $OC$ be the $+$~directions for lines measured from~$O$
along these lines respectively. Let $OA$ be taken as the
\DPPageSep{025.png}{21}%
initial line for angles, and $ABC$ the $+$~direction for angles
measured from~$OA$. Let $R$~be the radius of the circle,
and $\theta\left(\dfrac{AB}{R}\right)$ be the circular measure of~$AOB$.

\Paragraph[\No{1}]{Sin~$\theta$.} The sine of the angle $AOB$ is the ratio to
the radius of the perpendicular from the end of the arc subtending
the angle on the initial line~$OA$, $\therefore \sin\theta = \dfrac{BD}{R}$.

If the position of $B$ be above the line $AOA'$, $\sin\theta$ is~$+$;
if below, $\sin\theta$ is~$-$.

\Paragraph[\No{2}]{Cos~$\theta$.} The cosine of an angle is the sine of its
complement, or
\[
\cos\theta = \sin\left(\frac{\pi}{2} - \theta\right).
\]

Conversely
\[
\sin\theta = \cos\left(\frac{\pi}{2} - \theta\right).
\]

The angle $AOB$ being~$\theta$, and $COA = \dfrac{\pi}{2}$,
\[
COB = \frac{\pi}{2} - \theta;
\]
therefore
\[
\cos\theta = \sin COB = \frac{BE}{R} = \frac{OD}{R}.
\]

Hence, if $OD$ is measured towards~$A$, or if $B$~lie to that
side of~$COC'$, $\cos\theta$ is~$+$; if $OD$~is measured towards~$A'$, or
if $B$~lies on the same side of~$COC'$ as~$A'$, $\cos\theta$ is~$-$.

\Paragraph{Variation of the Sine and Cosine.} We may examine how
the values of these two functions change with the variation
of the angle.

When $AB$ is small, $BD$~is small and~$+$; $OD =$ the
radius nearly and~$+$. As $\theta$~increases from $0$ to~$\dfrac{\pi}{2}$, $BD$~and~$OD$
\DPPageSep{026.png}{22}%
remain both~$+$, but $BD$~increases from $0$ to~$R$, and
$OD$ decreases from $R$ to~$0$; therefore
\begin{gather*}
\sin\theta \text{ increases from } +0 \text{ to } +1; \\
\cos\theta \text{ decreases from } +1 \text{ to }  0.
\end{gather*}

As $\theta$ increases from $\dfrac{\pi}{2}$ to~$\pi$, $BD$ remains~$+$, but diminishes
from $+R$ to~$0$; also $OD$ becomes~$-$, and $-OD$ increases
from $0$ to~$+ R$; therefore
\begin{gather*}
+\sin\theta \text{ decreases from } +1 \text{ to }  0; \\
-\cos\theta \text{ increases from }  0 \text{ to } +1.
\end{gather*}

As $\theta$ increases from $\pi$ to~$\dfrac{3\pi}{2}$, $BD$ becomes~$-$, and $-BD$
increases from $0$ to~$+R$; also $OD$ is~$-$, and $-OD$ decreases
to~$0$; so that
\begin{gather*}
-\sin\theta \text{ increases from }  0 \text{ to } +1; \\
-\cos\theta \text{ decreases from } +1 \text{ to }  0.
\end{gather*}

As $\theta$ increases from $\dfrac{3\pi}{2}$ to~$2\pi$, $-BD$ decreases to~$0$;
and $+OD$ increases to~$+R$, so that
\begin{gather*}
-\sin\theta \text{ diminishes from } +1 \text{ to }  0; \\
+\cos\theta \text{ increases from }   0 \text{ to } +1.
\end{gather*}

After this the values from $2\pi$~to~$4\pi$ are the same as from
$0$~to~$2\pi$: and if $m$~be any whole number
\begin{align*}
\sin\theta &= \sin(2m\pi + \theta); \\
\cos\theta &= \cos(2m\pi + \theta).
\end{align*}

If equal angles, $+\theta$~and~$-\theta$, be measured in opposite
directions from~$OA$, and the other ends of the arcs subtending
them be joined by the straight line~$BB'''$, $AOA$ will
bisect~$BB'''$ at right angles in the point~$D$.

Therefore the sines of these two angles will be equal but
\DPPageSep{027.png}{23}%
of opposite sign, and their cosines will be equal in sign as
well as magnitude; or
\begin{align*}
\sin(-\theta) &= -\sin\theta;\\
\cos(-\theta) &= +\cos\theta.
\end{align*}

If $A' B'$, $A' B''$ be arcs, each $= AB = R\theta$, but measured
from~$A'$; the circular measure of the angles subtended by
$AB'$, $AB''$ are respectively $\pi - \theta$, $\pi + \theta$; and the perpendicular
$B'D' = + BD$, $B''D = - BD$, and $OD' = - OD$.
Consequently
\begin{align*}
\sin (\pi - \theta) &=  \sin\theta, & \cos(\pi - \theta) &= -\cos\theta;\\
\sin (\pi + \theta) &= -\sin\theta, & \cos(\pi + \theta) &= -\cos\theta.
\end{align*}

And, since both sine and cosine remain unchanged when
the angle is increased by a multiple of~$2\pi$ (say by~$2m\pi$),
we have
\begin{align*}
%[** TN: Replaced vinicula over 2m + 1 with parentheses]
&\sin(2m\pi + \theta)= +\sin\theta; \\
&\sin(-\theta) = \sin(2m\pi - \theta) = -\sin\theta; \\
&\sin(\pi - \theta) = \sin\bigl((2m + 1) · \pi - \theta\bigr) = +\sin\theta; \\
&\sin(\pi + \theta) = \sin\bigl((2m + 1) · \pi + \theta\bigr) = -\sin\theta. \\
\intertext{\indent And}
&\cos(2m\pi + \theta) = +\cos\theta; \\
&\cos(-\theta) = \cos(2m\pi - \theta) = + \cos\theta;\\
&\cos(\pi - \theta) = \cos\bigl((2m + 1) · \pi - \theta\bigr) = -\cos\theta; \\
&\cos(\pi + \theta) = \cos\bigl((2m + 1) · \pi + \theta\bigr) = -\cos\theta.
\end{align*}

By these the sine and cosine of an angle of any magnitude
can be obtained from the sine or cosine of an acute
angle.

It should be observed that, while the number~$\theta$ continuously
increases, the numbers $\sin\theta$, $\cos\theta$ pass through a
series of values between $+1$~and~$-1$, and return to the
same values again for every increase of~$2\pi$ in the value of~$\theta$.
They are therefore said to be \emph{periodic functions} of~$\theta$, of
which the period is~$2\pi$.
\DPPageSep{028.png}{24}%

Since the perpendicular from the centre of a circle on
any chord bisects it at right angles, the ratio of the chord to
the radius is twice the sine of half the angle subtended by
the chord.

Hence, if we can calculate the ratio to the radius of the
side of an inscribed polygon, the sine of half the angle subtended
by the side is at once known.

For instance, the side of a square inscribed in a circle
$= R\sqrt{2}$, and it subtends an angle of~$\dfrac{\pi}{2}$ or~$90°$. Therefore
\Pagelabel{24}%
\[
\sin \frac{\pi}{4}
  = \sin 45°
  = \frac{1}{2}\sqrt{2}
  = \cos 45°
  = \cos \frac{\pi}{4}.
\]

The side of the hexagon inscribed is $= R$, and it subtends
$\dfrac{\pi}{3}$ or~$60°$. Therefore
\[
\sin \frac{\pi}{6}
  = \sin 30°
  = \frac{1}{2}
  = \cos 60°
  = \cos \frac{\pi}{3}.
\]

The side of the inscribed equilateral triangle $= R\sqrt{3}$,
and it subtends $\dfrac{2\pi}{3}$ or~$120°$. Hence
\[
\sin \frac{\pi}{3}
  = \sin 60°
  = \frac{1}{2}\sqrt{3}
  = \cos 30°
  = \cos \frac{\pi}{6}.
\]

The side of the regular decagon inscribed is (\Euclid[Euc.]{IV.}{10},
and \Euclid[]{II.}{11}) $\dfrac{\sqrt{5} - 1}{2}R$, and it subtends an angle
$\dfrac{\pi}{5}$ or~$36°$. Hence
\[
\sin \frac{\pi}{10}
  = \sin 18°
  = \dfrac{\sqrt{5} - 1}{4}
  = \cos \frac{4\pi}{10}
  = \cos 72°.
\]

\Paragraph[\No{3}]{Tan~$\theta$.} The tangent of an angle is the ratio to the
radius of the part of the line, touching at the initial end the
\DPPageSep{029.png}{25}%
%[Illustration]
\begin{figure}[hbt]
  \begin{center}
    \Input{029a}
  \end{center}
\end{figure}
arc subtending the angle, intercepted between the point of
contact and the radius produced through the other end of
that arc. In the diagram
\[
\tan\theta = \frac{AF}{R} \text{ and } AF = R\tan\theta.
\]

If $AF$ be measured towards~$T$, it is to be considered~$+$,
and $-$,~if in the contrary direction towards~$T'$.

\Paragraph[\No{4}]{Cot~$\theta$.} The cotangent of an angle is the tangent
of the complement of the angle. Whence
\[
\cot\theta = \tan \left(\frac{\pi}{2} - \theta\right) \text{ and }
\tan\theta = \cot \left(\frac{\pi}{2} - \theta\right).
\]

The cotangent may hence be geometrically defined as
follows. Let $OC$~be drawn perpendicular to~$OA$. Then
\DPPageSep{030.png}{26}%
$COB$ is the complement of $AOB$, or $COB = \dfrac{\pi}{2} - \theta$. If
$AOB$ be greater than $AOC$, or $\theta > \dfrac{\pi}{2}$, $COB$ will be measured
from $CO$ in the other direction, and $\dfrac{\pi}{2} - \theta$ will be~$-$ to
correspond.

%[Illustration]
\begin{figure}[hbt]
  \begin{center}
    \Input{030a}
  \end{center}
\end{figure}
Draw $H'CH$ touching the arc subtending the complement
of~$\theta$ at the point~$C$, which is always an extremity of
that arc, and through the other extremity of the arc produce
the radius to meet it; the tangent of the complement,
or cotangent, is the ratio to the radius of the part of $HCH'$
intercepted. In the diagram
\[
\cot\theta = \frac{CG}{R} \text{ and } CG = R\cot\theta.
\]
\DPPageSep{031.png}{27}%

If $CG$ is measured in the direction from $C$~to~$H'$ the
cotangent is~$-$.

\Paragraph{Variation of the Tangent and Cotangent.} In the first
quadrant, when the angle is indefinitely small, the tangent
is indefinitely small, and the cotangent indefinitely great;
since $AF$ diminishes, and $CG$ increases without limit, as
$AOB$ decreases. Hence it is usually said that $\tan 0 = 0$,
and $\cot 0 = \infty$.

As the angle increases to~$90°$, $AF$ increases and $CG$
decreases, both drawn in the $+$~direction; so that $\tan\theta$
increases and $\cot\theta$ decreases, as $\theta$~increases from $0$ to~$\dfrac{\pi}{2}$,
and when $\theta = \dfrac{\pi}{2}$, $\tan\theta = \infty$, $\cot\theta = 0$.

In the second quadrant, where $\theta$~increases from $\dfrac{\pi}{2}$ to~$\pi$,
$AF'$ is measured in the $-$~direction $AT'$, and $CG'$ in the
$-$~direction $CH'$ for an angle $AOB'$: so that $\tan\theta$ and
$\cot\theta$ are both~$-$.

Also, $AF'$ decreases from an indefinitely great distance
in the direction $AT'$, when $\theta$~barely exceeds $\dfrac{\pi}{2}$, to $0$ when
$\theta = \pi$; while $CG'$ increases from $0$ to an indefinitely great
distance in the direction $CH'$. Hence, as $\theta$ increases
from $\dfrac{\pi}{2}$ to~$\pi$, $-\tan\theta$ decreases from $\infty$ to~$0$ and $-\cot\theta$ increases
from $0$ to~$\infty$.

In the third quadrant, or when $\theta$ increases from $\pi$ to~$\dfrac{3\pi}{2}$,
the lines $AF$~and~$CG$ are again measured in the positive
direction, and the tangent and cotangent are therefore both~$+$,
and vary in magnitude as in the first quadrant.
\DPPageSep{032.png}{28}%

In the fourth quadrant, where $\theta$ is greater than $\dfrac{3 \pi}{2}$, and
less than $2 \pi$, the lines are again drawn in the $-$~direction,
and the tangent and cotangent vary just as in the second
quadrant.

If two equal angles, as $AOB$ and $AOB'''$, or $AOB'$ and
$AOB''$, be measured in opposite directions from $OA$, it is
obvious that their tangents and cotangents are equal in
magnitude, but opposite in sign; or
\[
\tan(-\theta) = -\tan\theta; \text{ and } \cot(-\theta) = -\cot\theta.
\]

Also if $A'OB'$ and $AOB$ are equal angles, so that
$AOB'$ and $AOB$ are supplementary, their tangents and cotangents
are equal, but of opposite signs; that is
\[
\tan(\pi - \theta) = -\tan\theta; \quad
\cot(\pi - \theta) = -\cot\theta.
\]

If $m$ be any whole number, it is clear that the angle
$2m\pi + \theta$ begins and terminates at the same point as the
angle~$\theta$. Therefore,
\begin{gather*}
\begin{aligned}
%[** TN: Replaced vinicula over 2m + 1 with parentheses]
&\tan(2m\pi + \theta) = \tan\theta; &
&\cot(2m\pi + \theta) = \cot\theta\DPtypo{,}{;} \\
%
&\tan(2m\pi - \theta) = -\tan\theta; &
&\cot(2m\pi - \theta) = -\cot\theta\DPtypo{.}{;}
\end{aligned} \\
%
\begin{aligned}
&\tan\bigl((2m + 1) · \pi - \theta\bigr) = -\tan\theta; &
&\cot\bigl((2m + 1) · \pi - \theta\bigr) = -\cot\theta\DPtypo{,}{;} \\
%
&\tan\bigl((2m + 1) · \pi + \theta\bigr) =  \tan\theta; &
&\cot\bigl((2m + 1) · \pi + \theta\bigr) =  \cot\theta.
\end{aligned}
\end{gather*}

Whence the tangent and cotangent of any angle can be
found from those of an acute angle.

\Paragraph[\No{5}]{Sec~$\theta$.} The secant of an angle is the ratio to the
radius of the initial radius produced to meet the tangent at
the other end of the arc subtending the angle.
\DPPageSep{033.png}{29}%

\Paragraph[\No{6}]{Cosec~$\theta$.} The cosecant is the secant of the complement.
%[Illustration]
\begin{figure}[hbt]
  \begin{center}
    \Input[4in]{033a}
  \end{center}
\end{figure}

In the diagram, $AOB$ being the angle, and $COB$ the
complement, $KS$ the tangent at~$B$;
\begin{gather*}
\sec\theta = \frac{OS}{R};\quad \cosec\theta = \frac{OK}{R}; \\
OS = R\sec\theta;\quad OK = R\cosec\theta.
\end{gather*}

Then $\sec 0 = 1$; and the secant increases till $\sec \dfrac{\pi}{2} = \infty$.
When $\theta$~is between $\dfrac{\pi}{2}$ and $\dfrac{3\pi}{2}$, $\sec\theta$ is~$-$, and $-\sec\theta$ decreases
\DPPageSep{034.png}{30}%
from~$\infty$ (when $\theta = \dfrac{\pi}{2}$) till $-\sec\pi = 1$, and then
increases again till $-\sec\dfrac{3\pi}{2} = \infty$; after which, while $\theta$~increases
from $\dfrac{3\pi}{2}$ to~$2\pi$, $+\sec\theta$ decreases from $\infty$ to~$1$.

Similarly $\cosec\theta$ is~$+$ in the first and second quadrants
diminishing from $\cosec 0 = \infty$ till $\cosec\dfrac{\pi}{2} = +1$ and then
increasing till $\cosec\pi = +\infty$.

In the third and fourth quadrants $\cosec\theta$ is~$-$, and
$-\cosec\theta$ decreases from $\infty$ at the beginning of the
third quadrant till $-\cosec\dfrac{3\pi}{2} = 1$, and then increases till
$-\cosec 2\pi = \infty$.

\Paragraph[\No{7}]{Versin~$\theta$.} The versed sine of an angle is the ratio
%[Illustration]
\begin{figure}[hbt]
  \begin{center}
    \Input{034a}
  \end{center}
\end{figure}
to the radius of the part of the initial radius (produced if
necessary) which is intercepted between its extremity $A$ and
the perpendicular from the other end~$B$ of the arc subtending
\DPPageSep{035.png}{31}%
the angle. In the diagram, we have
\[
\versin\theta = \frac{AD}{R}; \text{ and }
AD = R\versin\theta.
\]

The line $AD$ is always drawn in one direction and
$\versin\theta$ is always~$+$.

As $\theta$ increases from $0$ to~$\pi$, $\versin\theta$ increases from $0$
to~$2$; and as $\theta$~increases from $\pi$ to~$2\pi$, $\versin\theta$ diminishes
from $2$~to~$0$.

The seven ratios defined above are altogether independent
of the size of the circle described, and depend only
on the angle. They are therefore \emph{functions} of the circular
measure~$\theta$, remaining the same for the same value of~$\theta$ whatever
radius be taken, but changing with the value of~$\theta$;
and they are numbers calculable from the number~$\theta$. The
lines, to which their ratios correspond, depend partly on
the value of~$\theta$ and partly on the radius, and are expressed
by the radius multiplied by the corresponding circular
function.

Thus if the angle $AOB = \theta$ in the diagrams of this Chapter,
\begin{gather*}
\begin{aligned}
BD &= OB \sin\theta = OD \tan\theta, \\
EB &= OB \cos\theta = EO \cot\theta, \\
AF &= OA \tan\theta = OF \sin\theta, \\
CG &= OC \cot\theta = OG \cos\theta,
\end{aligned} \\
OS = OB \sec\theta,\quad OK = OB \cosec\theta.
\end{gather*}

\Paragraph{Relations between the circular functions of the same angle.}
If the angle $AOB$ and the radius of the circle be the same
in these diagrams, the triangles $BOD$, $FAO$, $SOB$ are
right-angled and similar. Hence expressing their sides
\DPPageSep{036.png}{32}%
in terms of the radius and the circular function corresponding
to each, we have (\Euclid[Euc.]{I.}{47})
\begin{gather*}
R^2 = R^2 \sin^2\theta + R^2 \cos^2\theta, \\
\text{or } \sin^2\theta + \cos^2\theta = 1\Add{.}
\Tag{1} \\
R^2 \sec^2\theta = R^2 + R^2 \tan^2\theta, \\
\text{or } \sec^2\theta = 1 + \tan^2\theta\Add{.}
\Tag{2}
\end{gather*}
And putting $\dfrac{\pi}{2} - \theta$ for $\theta$ in this
\[
\cosec^2\theta = 1 + \cot^2\theta\Add{.}
\Tag{3}
\]
Also (\Euclid[Euc.]{VI.}{4}) $R \tan\theta: R :: R \sin\theta: R \cos\theta$,
\begin{gather*}
\text{or } \tan\theta = \frac{\sin\theta}{\cos\theta}
\Tag{4} \\
R \sec\theta : R :: R : R \cos\theta, \\
\text{or } \sec\theta = \frac{1}{\cos\theta}\Add{.}
\Tag{5}
\end{gather*}
From which putting $\dfrac{\pi}{2} - \theta$ for~$\theta$,
\begin{align*}
\cotan\theta = \frac{\cos\theta}{\sin\theta} &= \frac{1}{\tan\theta}
\Tag{6} \\
\cosec\theta &= \frac{1}{\sin\theta}
\Tag{7}
\end{align*}
And $R\versin\theta = R - R \cos\theta$
\[
\text{whence } \versin\theta = 1 - \cos\theta\Add{.}
\Tag{8}
\]

From these equations all the functions can be found,
when one has been given. For instance, from the values
of the sine and cosine given above (\Pageref{24}) we get
\begin{align*}
\tan 45° &= \cot 45° = 1, \\
\sec 45° &= \cosec 45° = \sqrt{2}, \\
\DPPageSep{037.png}{33}%
\tan 30° &= \cotan 60° = \frac{1}{\sqrt{3}} = \frac{1}{3}\sqrt{3}, \\
\sec 30° &= \cosec 60° = \frac{2}{\sqrt{3}} = \frac{2}{3}\sqrt{3}, \\
\cot 30° &= \tan 60° = \sqrt{3} = \sqrt{3}, \\
\cosec 30° &= \sec 60° = 2, \\
\cos 18° &= \frac{1}{4}\sqrt{\Strut[10pt]\smash[t]{10 + 2\sqrt{5}}}, \text{ \&c.}
\end{align*}


\Chapter{VI.}{OF LOGARITHMIC TABLES.}

\First{Tables} have been formed of the values of the circular
functions of angles at intervals of~$1'$, or in more minute
tables at intervals of~$10''$, from $0°$~to~$45°$. It is unnecessary
to calculate those of greater angles, as the angles between
$45°$ and $90°$ are complements of the angles between $0°$~and~$45°$.
Methods of calculation are given in Higher Trigonometry.
Such tables are called Tables of Natural Sines,
Cosines,~\&c. For practical calculations they are superseded
by logarithmic tables, the use of which may be shortly
explained.

Logarithms of ordinary numbers may be defined to be
numbers, so calculated from the ordinary numbers, that the
sum of the logarithms of two numbers is the logarithm of
their product. Thus, if $m$~and~$n$ be two numbers,
\begin{gather*}
\log m + \log n = \log (m × n).
\Tag{1} \\
\intertext{\indent Now}
\log (m ÷ n) + \log n = \log n (m ÷ n) = \log m; \\
\therefore \log m - \log n = \log(m ÷ n).
\Tag{2}
\end{gather*}
\DPPageSep{038.png}{34}%

From~\Eq{2} it follows, putting $m = n$, that
\[
\log 1 = 0,
\Tag{3}
\]
and that the logarithm of a proper fraction is~$-$.

Also from~\Eq{1}
\begin{align*}
\log n^2 &= 2 \log n, \\
\log n^3 &= 3 \log n,\quad \text{\&c.}
\end{align*}
and, generally, $\log n^p = p \log n$, if $p$ is an integer, \\
and therefore $q \log n^{\tfrac{p}{q}} = p \log n$,
\[
\text{or } \log n^{\tfrac{p}{q}} = \frac{p}{q} × \log n.
\]

So that whether $x$ is an integer, or a fraction,
\[
\log n^x = x \log n.
\Tag{4}
\]

If $a =$ the number of which the logarithm is~$1$, $a$~is said
to be the base of the system of logarithms: then denoting
logarithms in the system of which the base is~$a$ by~$\log_a$, \Eq{4}~becomes
\[
\log_a n = x, \text{ when } a^x = n.
\Tag{5}
\]

Common logarithms are those for which the base is~$10$.

Hence in \Eq{3}~and~\Eq{5} denoting common logarithms by~$\log$;
$\log 1 = 0$; $\log 10 = 1$; $\log 100 = 2$; $\log 1000 = 3$; and so on.

Equation~\Eq{5} shews that the logarithm increases with the
number: therefore it appears that the common logarithm of
a number of one integral digit is a proper fraction; that of
a number of $2$~digits is $1 + \text{a fraction}$; of $3$~digits $2 + \text{a
fraction}$, and so on. The integral part, or \emph{characteristic}, of
the logarithm, therefore, is one less than the number of
integral digits and is known by inspection, so that it is
usual in the tables to give only the decimal part of the
logarithm, or the \emph{mantissa}.
\DPPageSep{039.png}{35}%

Since also by \Eq{1}~and~\Eq{2},
\begin{align*}
\log (10^m × n) &= m + \log n,\\
\text{and }\log (n \div 10^m) &= -m + \log n,
\end{align*}
it appears that multiplication or division by any power of~$10$
will affect only the characteristic of the logarithm, and
that the decimal part of the logarithm, or mantissa, will be
the same for numbers consisting of the same succession of
digits followed or preceded by any number of ciphers, and
with a decimal point occurring between any two digits.

It will be seen that all the arithmetical operations for
calculation are simplified by the use of logarithmic tables.

Thus, with tables, the result of multiplication is calculated
by adding logarithms, of division by subtracting logarithms,
of raising to a power by multiplying a logarithm
by a number, of extracting a root by dividing a logarithm
by a number.

Calculations are thus so much abbreviated\footnote
  {Laplace has remarked with reference to the value of the invention,
  that ``by reducing to a few hours what otherwise is the labour of several
  months, it doubles, so to speak, the life of the astronomer, and spares
  him the errors and annoyance inseparable from long calculations.''}
that it is
very desirable in establishing formulæ for the calculation of
some of the sides and angles of triangles from given values
of others, that the formulæ should be adapted to logarithmic
calculation, as is done in the next chapter.

It is also necessary to have logarithmic tables of the
Trigonometrical ratios.

Since the sine of an angle is always less than 1, the
common logarithm of the sine is a $-$~number. The same is
true of the cosine, of the tangent of an angle~$< 45°$,~\&c. To
avoid the use of negative numbers, the tabular logarithms
\DPPageSep{040.png}{36}%
of the circular functions are the common logarithms of
these ratios increased by~$10$: which must be remembered
in using the tables in calculations.

Where, as is generally the case in practice, the logarithm
of a given number, or, conversely, the number for a given
logarithm is not actually to be found in the tables, these
can be found approximately from the principle that, for
considerable numbers of small difference, the difference of
two numbers is nearly proportional to the difference of
their logarithms. Whence the difference between a number
and the nearest number in the tables being known, the
difference to be added to the nearest logarithm in the tables
can be approximately found, and the converse.

The same method applies also to the tabular logarithms
of the circular functions, it being generally approximately
true that the difference of two tabular logarithms of a
circular function is proportional to the difference of the
corresponding angles, when that difference is small, and the
angles are neither very small nor very near~$90°$\footnotemark.
\footnotetext{Logarithms were invented by John Napier, who was born in \Date{1550}
  and died \osf{4}th~April, \Date{1617}. He was descended from and claimed to
  represent the elder branch of the Earls of Lennox. From him are
  descended the Baronet of Milliken and the Lord Napier. But the
  inventor of Logarithms was not a Peer, and should not be styled Baron
  Napier as is often done. The mistake arises from his having been
  ``Baron'' in the sense of proprietor of the Barony of Merchistoun, near
  Edinburgh. In a paper dated \osf{23}rd~April, \Date{1584}, at Gartness on the
  Endrick (where were his Lennox Estates and where it is a local tradition
  that he calculated his Canon of Logarithms), he signs himself ``Jhone
  Neper \textit{Fear} of Merchiston.'' The word fear is the Celtic equivalent of
  Baron.

  His ``Canon,'' or table of Logarithms, was published in~\Date{1614}.}%
\DPPageSep{041.png}{37}%


\Chapter{VII.}{SOLUTION OF TRIANGLES.}

\First{A triangle} is said to be solved when the sides and
angles are calculated from the data.

\Paragraph{Right-Angled Triangles.} Let $ABC$ be a triangle. Let
$C = 90°$; and let $BC = a$, $CA = b$,
$AB = c$.

%[Illustration]
\begin{wrapfigure}[8]{r}{1.75in}
  \hspace*{-0.375in}\Input[2.125in]{041a}
\end{wrapfigure}

Then if the side $AB$ be taken
as radius, the ratio of $BC$ to $AB$
is the sine of~$A$; and of $AC$ to
$AB$ is its cosine; or
\begin{align*}
a &= c \sin A;\quad b = c \cos A. \\
\intertext{Similarly}
b &= c \sin B;\quad a = c \cos B.
\end{align*}

If $AC$ be taken as radius, since $CB$ touches the arc
described, the ratio of $CB$ to $CA$ is the tangent, and of $AB$
to $CA$ is secant of~$A$; so that
\begin{align*}
a &= b \tan A; \quad c = b \sec A. \\
\intertext{\DPtypo{\indent}{}And similarly }
b &= a \tan B; \quad c = a \sec B.
\end{align*}

From these equations, it is easy logarithmically to solve
a right-angled triangle when a side and an angle, or when
two sides are given.

\Paragraph{Given $c$ and $A$}, we have $B = 90° - A$,
\begin{align*}
\log a &= \log c + \tab\log \sin A - 10, \\
\log b &= \log c + \tab\log \cos A - 10;
\end{align*}
whence $B$, $a$, $b$ are found.

\Paragraph{Given $a$ and $A$}, we have $B = 90°- A$,
\begin{align*}
\log b &= \log a + \tab\log \tan B - 10, \\
\log c &= \log a + \tab\log \sec B - 10;
\end{align*}
whence $B$, $c$, $b$ are found.
\DPPageSep{042.png}{38}%

\Paragraph{Given $c$ and $a$}, we have
\begin{align*}
&\begin{aligned}
  \tab\log \sin A &= 10 + \log a - \log c, \\
                B &= 90° - A,
\end{aligned} \\
&\log b = \log c + \tab \log \cos A - 10;
\end{align*}
whence $A$, $B$, $b$ are found.

\Paragraph{Triangles not Right-Angled.}

Case~I\@. Let $ABC$ be an acute-angled triangle, $ABC'$
%[Illustration]
\begin{figure}[hbt]
  \begin{center}
    \Input[\linewidth]{042a}
  \end{center}
\end{figure}
a triangle having an obtuse angle at~$C'$. In both triangles
draw $AD$ perpendicular to $BC$ produced if necessary.
Then in both triangles, if $AB$ be taken as radius,
\[
AD = AB \sin B = c \sin B;
\]
and if $CA$ be taken as radius,
\[
AD = CA \sin C = b \sin C.
\]

Hence
\begin{gather*}
  b \sin C = c \sin B, \\
  \text{or } \frac{b}{\sin B} = \frac{c}{\sin C}.
\end{gather*}

If the perpendicular had been drawn from $B$ to $CA$ we
should similarly have got
\begin{gather*}
  \frac{a}{\sin A} = \frac{c}{\sin C}\;, \\
\text{so that }
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\;,
\end{gather*}
\DPPageSep{043.png}{39}%
or \emph{the sides are proportional to the sines of the opposite angles}.
This is true of all triangles. Hence we can logarithmically
solve a triangle when a side and the opposite angle are
given, and likewise another side or another angle.

\Paragraph{Given $C$ and $c$, and $b$.}
\[
\Tab\log\sin B = (\tab\log\sin C - \log c) + \log b,
\]
a logarithmic equation for the determination of~$B$.

There is but one acute angle which can have its sine
of a particular value: so that, if we know $B$ to be acute,
there is but one value of~$B$ to be taken. And if $b < c$ the
angle~$B$ must be acute; and we find it by means of the
equation.

Then
\[
A = 180° - (B + C),
\]
and
\[
\log a = \tab\log\sin A - (\tab\log\sin C - \log c),
\]
so that the triangle is solved.

If however $b > c$, we do not know that $B$~is acute, and
as the sine of the supplement of an acute angle is the same
as the sine of the angle, there are two values of~$B$, the one
less than~$90°$, the other its supplement, which satisfy the
above logarithmic relation. That is, there is an \emph{ambiguity}
as to the value of~$B$. There will always be this ambiguity
in determining the angle of a triangle from its sine, unless
there be something to point out whether the angle is acute
or obtuse: and therefore it is generally inconvenient to use
a method involving the determination of the angle from its
sine. But in this instance there is a real \emph{geometrical} ambiguity.
For if we construct a triangle with its angle
$ACB = C$, its side $CA = b$, and the side opposite $C = c$, it is
evident that, if $c$ be $< b$, there are two triangles $AB_1C$ and
$AB_2C$ corresponding to the data. The angle opposite $CA$
in the one of these is acute $= B_1$; in the other it is obtuse
\DPPageSep{044.png}{40}%
$= B_2$. But since $AB_1 = AB_2$, $B_2 = 180° - B_1$. In this case
%[Illustration]
\begin{figure}[hbt]
  \begin{center}
    \Input{044a}
  \end{center}
\end{figure}
therefore, \emph{where two sides and the angle opposite the \DPtypo{less}{lesser} are
given}, which is called the \emph{ambiguous case}, there are two
triangles. The acute angle~$B_1$ can be found from the
formula
\[
\tab\log\sin B_1 = \log b + (\tab\log\sin C - \log c),
\]
and then $B_2 = 180° - B$.

If $A_1$ and $A_2$ are the corresponding values of the third
angle $a_1$,~$a_2$ of the third side,
\[
A_1 = 180° - (B_1 + C);\quad A_2 = 180° - (B_2 + C),
\]
and
\begin{gather*}
\log a_1 = \tab\log\sin A_1 - (\tab\log\sin C - \log c), \\
\log a_2 = \tab\log\sin A_2 - (\tab\log\sin C - \log c).
\end{gather*}

Case~II\@. Let $ABC$ be a triangle: and let $CB$ be
greater than~$CA$.

%[Illustration]
\begin{figure}[hbt]
  \begin{center}
    \Input{045a}
  \end{center}
\end{figure}
From $C$ as centre with $CB$~as radius, describe the circle
$DBE$. Produce $BA$ to meet its circumference in~$F$, and
$CA$ both ways to meet it in $E$~and~$D$.

Join $FC$, $BD$, $BE$, $BF$.

Then $DBE$ is a right angle; and
\[
DCB = A + B = 2BED;
\]
also
\[
A = ECF + CFA = ECF + B,
\]
and
\[
ECF = A - B = 2FBE.
\]
\DPPageSep{045.png}{41}%

So that $BED = \dfrac{1}{2}(A + B)$; $FBE = \dfrac{1}{2}(A - B)$.

And $BDA$ is the complement of $BED$, $ABD$ of $FBE$.

Now in the triangle $DBA$,
\[
\frac{DA}{\sin ABD} = \frac{BA}{\sin BDA},\quad\text{or}\quad
\frac{a+b}{\cos\dfrac{1}{2}(A - B)} = \frac{c}{\cos\dfrac{1}{2}(A + B)}.
\]

And in the triangle $BAE$,
\[
\frac{EA}{\sin FBE} = \frac{AB}{\sin BED},\quad\text{or}\quad
\frac{a-b}{\sin\dfrac{1}{2}(A - B)} = \frac{c}{\sin\dfrac{1}{2}(A + B)}.
\]

By combining these and remembering that $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$
we get
\[
\frac{\tan\dfrac{1}{2}(A - B)}{\tan\dfrac{1}{2}(A + B)}
  = \frac{a - b}{a + b}.
\]
\DPPageSep{046.png}{42}%

These relations give the simplest logarithmic solution
of a triangle, when two sides and the contained angle are
given\footnotemark.
\footnotetext{These formulæ are given in p.~\osf{13} of Thacker's \textit{Miscellany}, \Date{1743}.
  Their application to the solution of triangles was pointed out by
  Professor Wallace, of Edinburgh, in the \textit{Transactions, R.S.E.}, \Date{1823}.}%

\Paragraph{Given $a$,~$b$, and~$C$}, we have
\[
A + B = 180° - C, \text{ so that } \frac{1}{2} (A + B) \text{ is known}.
\]

Then
\begin{multline*}
\tab\log\tan \frac{1}{2} (A - B) = \tab\log\tan \frac{1}{2} (A + B) \\
+ \log (a - b) - \log (a + b),
\end{multline*}
which determines the acute angle $\dfrac{1}{2} (A - B)$.

Whence
\begin{align*}%[** TN: Discarded condensed intertext]
  A &= \frac{1}{2} (A + B) + \frac{1}{2} (A - B) \\
\intertext{and}
  B &= \frac{1}{2} (A + B) - \frac{1}{2} (A - B)
\end{align*}
are found.

The third side $c$ could now be found from the relation
$\log c = \tab\log\sin C - (\tab\log\sin A - \log a)$, but it is shorter
in practice to determine~$c$ either from
\[
\log c
  = \tab\log\cos \frac{1}{2} (A + B)
  - \tab\log\cos \frac{1}{2} (A - B) + \log (a + b),
\]
or from
\[
\log c
  = \tab\log\sin \frac{1}{2} (A + B)
  - \tab\log\sin \frac{1}{2} (A - B) + \log (a - b),
\]

Thus when two unequal sides and the angle between
them are given, the other side and other angles are found.

If an angle and two equal sides containing it are given,
the remaining angles are each the complement of half
the given angle, and the third side can be found as in
Case~I\@.
\DPPageSep{047.png}{43}%

Case~III\@. Let $ABC$ be the triangle, $O$~the centre of
the inscribed circle, so that $AO, BO, CO$ bisect the angles,
%[Illustration]
\begin{figure}[hbt]
  \begin{center}
    \Input{047a}
  \end{center}
\end{figure}
and the perpendiculars $Oa, Ob, Oc$ are each of them a radius
of the inscribed circle.

Let this radius $= r$, let the sum of the sides $a + b + c = 2s$;
then (Chapter~\Rom{II}.)
\[
Ab = Ac = s - a,\quad
Bc = Ba = s - b,\quad
Ca = Cb = s - c.
\]

And in the right-angled triangles $AOb$, $BOc$, $COa$,
we have
\begin{gather*}
r = (s - a) \tan \frac{1}{2} A
  = (s - b) \tan \frac{1}{2} B
  = (s - c) \tan \frac{1}{2} C, \\
\text{where } r^2 = \frac{(s - a)(s - b)(s - c)}{s}.
\end{gather*}

From these the angles may be found when the sides are
given.

\Paragraph{Given $a, b, c$}, we get the logarithmic relations
\[
\log r = \frac{1}{2} \bigl\{
  \log (s - a) + \log (s - b) + \log (s - c) - \log s
\bigr\},
\]
\DPPageSep{048.png}{44}%
and
\begin{align*}
\tab\log\tan \frac{1}{2} A &= 10 + \log r - \log (s - a), \\
\tab\log\tan \frac{1}{2} B &= 10 + \log r - \log (s - b), \\
\tab\log\tan \frac{1}{2} C &= 10 + \log r - \log (s - c),
\end{align*}
whence the three angles are easily found. There is also a
simple verification of the process afforded by taking the sum
of the three angles, which ought to be~$180°$\footnotemark.
\footnotetext{The formulæ used here were discovered by William Purser of Dublin,
  in \Date{1632}. Consult Wallace's \textit{Geometrical Theorems}.}%


\Chapter{VIII.}{OF TRIGONOMETRICAL SURVEYING.}

\First{In} \DPtypo{}{the} last Chapter methods have been given of calculating
the remaining parts of a triangle when one side and two
other parts of the triangle are previously known.

In Geodesy, or the application of this part of Trigonometry
to surveying, a line, called \emph{the Base}, is measured
between two convenient stations; and the angles between
lines from these stations to points visible from the stations
are measured with appropriate instruments. With these
data calculations are made of the distances between new
stations, where additional angles may be observed and the
operations repeated as often as required. The angles measured
by instruments are of three classes.
\DPPageSep{049.png}{45}%

\No{1} Horizontal angles, or angles in the horizontal plane
through the station.

The horizontal angle between two objects is the angle
between the vertical planes through the station and each
object: it is also called the angle in azimuth.

\No{2} Vertical angles, or angles in a vertical plane through
the station.

The angle of elevation (or depression) of an object, also
called its altitude, is the vertical angle between the horizontal
line and the line from the station to the object.

\No{3} Angles in planes neither horizontal nor vertical.

The angle between two objects is the angle between
the lines from the station to each object. This angle is in
the plane through the station and the two objects.

In land surveying the angles measured belong to the
first or second class, and are measured with a Theodolite
or similar instrument.

Angles of the third class are usually measured with a
Sextant or similar instrument.

For details of the measurement of a base, and a description
of the instruments principally employed in practical
surveying, the reader is referred to Professor Rankine's
\textit{Manual of Civil Engineering}, Part~\osf{1}. \textit{Of Engineering Geodesy},
and for the methods employed in a great Trigonometrical
survey to the Ordnance Survey \textit{Account of the
Principal Triangulation}, published by the Board of Ordnance,
\Date{1858}. Assuming, however, that a base line and angles can
be accurately measured by appropriate instruments, it seems
within the scope of this elementary treatise to give instances
of the determination of heights and distances from such
measurements.
\DPPageSep{050.png}{46}%

I\@. The distance between two stations $A$~and~$B$, visible
from each other, being ascertained, and the angles between
a visible object~$C$ and the other station being measured at
each station, the distances $AC$~and~$BC$ can be found by
Case~I\@.

In this instance the angles measured are in the plane
through $A$,~$B$,~$C$.

If the angle of elevation of $C$ be likewise measured at
either station, the height of~$C$ above that station can be
found by solving a right-angled triangle of which the hypotenuse
and one of the acute angles are known.

When the horizontal distance of the stations is given
and is in a vertical plane through~$C$, only vertical angles
are required and the work is simplified; and when in addition
one station is in the vertical line through~$C$ (which
is then said to be ``accessible'') the measurement of one
vertical angle and the solution of one right-angled triangle
determine the height of~$C$.

II\@. From each of two stations $A$~and~$B$, visible from
each other, and at a known distance, the angles between
the other station and two objects $C$~and~$D$ are measured.
Then the triangles $ABC$ and $ABD$ can be solved (by
Case~I.), and the distances $AC$, $BC$, $AD$, $BD$ are found.

If in addition the angles between $C$~and~$D$ are measured
at either station, the distance $CD$ can be found by Case~II\@.

If the survey extends over so small a fraction of the
earth's surface that the vertical lines over the whole may
be considered perpendicular to one horizontal plane, and if
the distances measured or calculated are the \emph{horizontal distances}
of the points (or the ``projections'' of the actual distances
on the horizontal plane), which is the case when
\DPPageSep{051.png}{47}%
the survey is conducted for the purpose of making a plan;
then the angles to be measured will all be horizontal angles,
and the work will be simpler. And if in addition the angle
of elevation of any point be observed from a known horizontal
distance, the height of the point above the horizontal
plane of reference (which may be ``the level of the sea'')
can be found by the solution of a right-angled triangle.

Where the extent of the earth's surface surveyed is too
great for the above supposition, the calculations have to be
corrected for the sphericity of the earth (see books on
Spherical Trigonometry), and in very great and very accurate
surveys also for the earth's deviation from a perfect
sphere. (See the \textit{Principal Triangulation} of the Ordnance
Survey.)


\Chapter{IX.}{OF PROJECTIONS.}

\First{The} point where the perpendicular from a given point
on a given plane or a given line meets the plane or line
is called the projection (or more precisely the orthogonal
projection) of the point on the plane or line.

The part of one line intercepted between the projections
on it of the extremities of another line, or of a broken line,
is called the projection of the second line, or broken line,
on the first.

It is convenient to take one direction of the line of projection
(say from left to right) as the $+$ direction and the
opposite as the $-$~direction; also to consider one end of
the projected line as its beginning; and to measure the
inclination of the two lines by the angle between the projected
line and a line drawn from its beginning parallel to
the line of projection and in the $+$~direction.
\DPPageSep{052.png}{48}%

\Prop{I}{The projection of a line on a given line is equal
to the projected line multiplied by the cosine of the angle between
the lines.}

%[Illustration]
\begin{figure}[hbt]
  \begin{center}
    \Input[\linewidth]{052a}
  \end{center}
\end{figure}
Let $AB$ be the projected line, $X'X$~the line of projection;
then in each case, $A'B'$~being the projection, we have
\[
A'B'= AB \cos BAC.
\]

If $BAC$ be acute, $\cos BAC$ is~$+$, and $A'B'$ is~$+$ and is
measured from~$A'$ in the $+$~direction; but if $BAC$ be obtuse,
$\cos BAC$ is~$-$, and $A'B'$ is~$-$ and is measured from~$A'$
in the $-$~direction.

\Prop{II}{The projection of a broken line is the algebraic
sum of the projections of the parts of which it is made up.}

%[Illustration]
\begin{figure}[hbt]
  \begin{center}
    \Input{052b}
  \end{center}
\end{figure}

Let $ABCD$ be a broken line, made up of the parts $AB$,
$BC$, $CD$. Let $A'B'$, $B'C'$, $C'D'$ be the projections of these
parts on $X'X$. Thus $A'D'$ is the projection of the broken
line. And
\[
A'D' = A'B' + B'C' + C'D'
\]
both in the case where all these projections are~$+$ and also
where, as in the figure, $B'C'$~is~$-$.
\DPPageSep{053.png}{49}%


%[** TN: Unit needs no running head in original; extrapolated.]
\Chapter[SUM AND DIFFERENCE OF ANGLES]
{X.}{THE SINE AND COSINE OF THE SUM AND DIFFERENCE
OF TWO ANGLES.}

\Paragraph[\No{1}]{To find the sine and cosine of the sum of two angles.}

%[Illustration]
\begin{wrapfigure}[9]{r}{2.25in}
  \Input[2.25in]{053a}
\end{wrapfigure}
Let $C$ be the centre of a circle of which the radius
$CA = R$.

Let the circular measure of
$ACB = \theta$, and of $BCD = \phi$.

Then $\theta + \phi$ is the circular
measure of~$ACD$.

Draw $DE$, $DF$ perpendicular
to $CA$,~$CB$.

Then
\begin{align*}
&&DE &= R \sin (\theta + \phi); & CE &= R \cos (\theta + \phi); &&\\
&&DF &= R \sin \phi;            & CF &= R \cos \phi. &&
\end{align*}

But $DE$ is the projection of the broken line $CFD$ on~$DE$;
therefore
\begin{gather*}
\begin{aligned}
DE &= \text{projection of } CF + \text{projection of } FD \\
   &= CF \cos \left(\frac{\pi}{2} - \theta\right) + FD \cos\theta \\
   &= CF \sin\theta + FD \cos\theta,                  \\
\end{aligned} \\
\begin{aligned}
\text{i.e.\ }
R \sin (\theta + \phi) &= R \cos\phi \sin\theta + R \sin\phi \cos\theta, \\
\text{or\ }
\sin (\theta + \phi) &= \sin\theta \cos\phi + \cos\theta \sin\phi.
\end{aligned}
\end{gather*}

Again, $CE$ is the projection of the broken line $CFD$
on~$CA$,
\begin{gather*}
\begin{aligned}
\therefore CE &= \text{projection of } CF + \text{projection of } FD \\
  &= CF \cos\theta + FD \cos \left(\frac{\pi}{2} + \theta\right) \\
  &= CF \cos\theta - FD \sin\theta;
\end{aligned} \\
\begin{aligned}
\text{i.e.\ }
R \cos (\theta + \phi) &= R \cos\phi \cos\theta - R \sin\phi \sin\theta, \\
\text{or }
\cos (\theta + \phi)   &= \cos\theta \cos\phi - \sin\theta \sin\phi.
\end{aligned}
\end{gather*}
\DPPageSep{054.png}{50}%

\Paragraph[\No{2}]{To find the sine and cosine of the difference of two
angles.}

%[Illustration]
\begin{wrapfigure}[9]{r}{2.25in}
  \Input[2.25in]{054a}
\end{wrapfigure}
Let
\begin{align*}
\theta &= \text{circular measure of $ACB$}, \\
\phi   &= \text{\PadTxt{circular measure}{\dotfill} of $BCD$.}
\end{align*}

Then $\theta - \phi =$ circular measure
of $ACD$.

Draw $DE$, $DF$ perpendicular
to $CA$,~$CB$.

Then
\DPchg{%
\begin{align*}%[** Duplicates the original book]
&&DE &= R \sin (\theta - \phi); &&&&\\
&&CE &= R \cos (\theta - \phi); &&&&\\
&&CF &= R \cos \phi; & FD &= R \sin \phi.&&
\end{align*}
}{%
\begin{align*}%[** Consistent with parallel construction on previous page]
&&DE &= R \sin (\theta - \phi); & CE &= R \cos (\theta - \phi); && \\
&&CF &= R \cos \phi; & FD &= R \sin \phi. &&
\end{align*}
}

But $DE$ is the projection of $CFD$ on~$DE$
\begin{gather*}
\begin{aligned}
  &= \text{projection of } CF + \text{projection of } FD \\
  &= CF \cos \left(\frac{\pi}{2} - \theta\right) + FD \cos (\pi - \theta) \\
  &= CF \sin \theta - FD \cos \theta;
\end{aligned} \\
\begin{aligned}
\text{i.e.\ }
R \sin (\theta - \phi) &= R \cos\phi \sin\theta - R \sin\phi \cos\theta, \\
\text{or }
\sin (\theta - \phi) &= \sin\theta \cos\phi - \cos\theta \sin\phi.
\end{aligned}
\end{gather*}

Again, $CE$ is the projection of $CFD$ on~$CA$
\begin{gather*}
\begin{aligned}
  &= \text{projection of } CF + \text{projection of } FD \\
  &= CF \cos \theta + FD \cos \left(\theta - \frac{\pi}{2}\right) \\
  &= CF \cos \theta + FD \sin \theta;
\end{aligned} \\
\begin{aligned}
\text{i.e.\ }
R \cos (\theta - \phi) &= R \cos\phi \cos\theta + R \sin\phi \sin\theta, \\
\text{or }
\cos (\theta - \phi) &= \cos\phi \cos\theta + \sin\phi \sin\theta.
\end{aligned}
\end{gather*}

The four expressions for $\sin (\theta ± \phi)$ and $\cos (\theta ± \phi)$ are
true for all values of $\theta$~and~$\phi$, though the diagrams suppose
the angles all acute. They form the fundamental formulæ
of Analytical Trigonometry.
\DPPageSep{055.png}{51}%


\Appendix{(Note to \Pageref{10}.)}

To shew that, when the difference between $R$~and~$r$ is
small, the limit of each radius is very nearly $= \dfrac{1}{3}(r + 2R)$.

Let $r_1, r_2, \dots$ be the radii of the circles inscribed in the
successive polygons, and $R_1, R_2, \dots$ of those circumscribed
about the same. Let also the limit of both be called~$r_\infty$.

Let $R - r = 2\delta$; then $r_1 = \dfrac{R + r}{2} = r + \delta = R - \delta$.

Let
\begin{align*}
  r_2 &= r_1 + \delta_1;\quad r_3 = r_2 + \delta_2;\quad \text{\&c.} \\
  \intertext{\indent Then}
  r_\infty &= r + \delta + \delta_1 + \delta_2 + \text{\&c.\ \textit{ad inf.}} \\
  \intertext{\indent Also}
  R &= r + 2\delta;\quad R_1 = r_1 + 2\delta_1;\quad \text{\&c.} \\
  \intertext{and}
  R &= r_1 + \delta;\quad R_1 = r_2 + \delta_1;\quad \text{\&c.} \\
  \intertext{\indent But}
  R_1^2 &= r_1R;\quad \text{or } (r_1 + 2\delta_1)^2 = r_1(r_1 + \delta); \\
\end{align*}
\begin{gather*}
\therefore 4r_1\delta_1 + 4\delta_1^2 = r_1\delta; \\
%[** TN: The original writes b < c > a instead of a < b < c.]
\DPchg{%
\therefore \delta_1
  = \frac{\delta}{4} - \frac{\delta_1^2}{r_1}
  < \frac{\delta}{4} > \frac{\delta}{4} - \frac{\delta_1^2}{r}; \\
\therefore \delta_1
  < \frac{\delta}{4}
  > \frac{\delta}{4} - \frac{1}{r}\left(\frac{\delta}{4}\right)^2. \\
\intertext{\indent Similarly}
\delta_2
  < \frac{\delta_1}{4}
  > \frac{\delta_1}{4} - \frac{1}{r}\left(\frac{\delta_1}{4}\right)^2
  > \frac{\delta_1}{4} - \frac{1}{r}\left(\frac{\delta}{4^2}\right)^2, \\
\delta_3
  < \frac{\delta_2}{4}
  > \frac{\delta_2}{4} - \frac{1}{r}\left(\frac{\delta_2}{4}\right)^2
  > \frac{\delta_2}{4} - \frac{1}{r}\left(\frac{\delta}{4^3}\right)^2, \\
\delta_4
  < \frac{\delta_3}{4}
  > \frac{\delta_3}{4} - \frac{1}{r}\left(\frac{\delta_3}{4}\right)^2
  > \frac{\delta_3}{4} - \frac{1}{r}\left(\frac{\delta}{4^4}\right)^2, \\
\text{\&c.\qquad\qquad\qquad \&c.}
}{% [** TN: Preceding code mimics original, succeeding code modernizes]
\therefore
 \frac{\delta}{4} - \frac{\delta_1^2}{r}
  < \delta_1 = \frac{\delta}{4} - \frac{\delta_1^2}{r_1}
  < \frac{\delta}{4}; \\
\therefore \frac{\delta}{4} - \frac{1}{r}\left(\frac{\delta}{4}\right)^2
  < \delta_1 < \frac{\delta}{4}. \\
\intertext{\indent Similarly}
\frac{\delta_1}{4} - \frac{1}{r}\left(\frac{\delta}{4^2}\right)^2
  < \frac{\delta_1}{4} - \frac{1}{r}\left(\frac{\delta_1}{4}\right)^2
  < \delta_2 < \frac{\delta_1}{4}, \\
\frac{\delta_2}{4} - \frac{1}{r}\left(\frac{\delta}{4^3}\right)^2
  < \frac{\delta_2}{4} - \frac{1}{r}\left(\frac{\delta_2}{4}\right)^2
  < \delta_3 < \frac{\delta_2}{4}, \\
\frac{\delta_3}{4} - \frac{1}{r}\left(\frac{\delta}{4^4}\right)^2
  < \frac{\delta_3}{4} - \frac{1}{r}\left(\frac{\delta_3}{4}\right)^2
  < \delta_4 < \frac{\delta_3}{4}, \\
\text{\&c.\qquad\qquad\qquad \&c.}} \\ %[** TN: End of modernized code]
\DPPageSep{056.png}{52}%
\therefore \delta + \delta_1 + \delta_2 + \dots
\text{ \textit{ad inf.}}
< \delta + \frac{\delta}{4} + \frac{\delta_1}{4} + \frac{\delta_2}{4} + \dots
\text{ \textit{ad inf.}}
\end{gather*}
and
\[
\therefore \frac{3}{4}\{\delta + \delta_1 + \delta_2 + \dots
\text{ \textit{ad inf.}}\} < \delta,
\]
or
\[
\delta + \delta_1 + \delta_2 + \dots < \frac{4}{3}\, \delta.
\]

And
\begin{multline*}
\delta + \delta_1 + \delta_2 + \dots
  > \delta + \frac{\delta}{4} + \frac{\delta_1}{4} +
  \text{ \&c.\ \textit{ad inf.}} \\
- \frac{1}{r} \left\{
  \left(\frac{\delta}{4}\right)^2
  + \left(\frac{\delta}{4^2}\right)^2 + \dots
  \text{ \textit{ad inf.}}\right\};
\end{multline*}
\begin{gather*}
\therefore \frac{3}{4}(\delta + \delta_1 + \delta_2 + \dots)
  > \delta - \frac{\delta^2}{16r} × \frac{16}{15}
  > \delta - \frac{\delta^2}{15r}; \\
\therefore \delta + \delta_1 + \delta_2 + \dots \text{ \textit{ad inf.}}
  > \frac{4}{3}\, \delta - \frac{4}{45}\DPchg{·}{\,} \frac{\delta^2}{r};
\end{gather*}
\DPchg{%[** TN: See notes above re: b < c > a]
\begin{align*}
\therefore r_\infty
  &< r + \frac{4}{3} \delta &
  &> r + \frac{4}{3}\, \delta - \frac{4}{45}\, \frac{\delta^2}{r}, \\
  &< \frac{1}{3} (r + 2R)   &
  &> \frac{1}{3} (r + 2R) - \frac{(R - r)^2}{45r}.
\end{align*}
}{%
\begin{align*}%[** TN: Rearranged and line-broken for legibility]
\therefore
\frac{1}{3} (r + 2R) - \frac{(R-r)^2}{45r}
  &< r + \frac{4}{3}\, \delta - \frac{4}{45}\, \frac{\delta^2}{r} \\
  &< r_\infty \\
  &< r + \frac{4}{3}\, \delta
   < \frac{1}{3} (r + 2R).
\end{align*}
}%

So that when, as in the text, $R - r < .00003$ the error
in taking the ultimate radius $= \dfrac{1}{3}(r + 2R)$ is $< \dfrac{.0000000009}{45r}$
which does not affect the tenth decimal place.
\vfill

\begin{center}
\rule{4.25in}{0.5pt} \\
\scriptsize CAMBRIDGE: PRINTED BY C. J. CLAY, M.A. AT THE UNIVERSITY PRESS.
\end{center}
\DPPageSep{057.png}{53}%


% [** TN: \Catalog macro prints commented material below]
% March, 1871.
% A Catalogue of Educational Books
% with a Short Account of their
% Character and Aim,
%
% Published by \\
% MACMILLAN AND CO \\
% 16, Bedford Street, Covent Garden, London.
\Catalog

\Csection{Classical.}

\Book{\Title{ÆSCHYLI EUMENIDES.} The Greek Text, with English Notes
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%
{The Greek text adopted in this Edition is based upon that of Wellauer,
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\DPPageSep{058.png}{54}%

\Book{\Title{ARISTOTLE ON FALLACIES; OR, THE SOPHISTICI
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%
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\Book{\Author{Aristotle.}---\Title{AN INTRODUCTION TO ARISTOTLE'S
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%
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%
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\DPPageSep{059.png}{55}%
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%
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\DPtypo{to ether}{together}
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%
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\DPPageSep{060.png}{56}%
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\DPPageSep{061.png}{57}%

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\DPPageSep{062.png}{58}%
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\DPPageSep{063.png}{59}%

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\noindent The \Title{JUGURTHA} and the \Title{CATILINA} may be had separately, price
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%
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\DPPageSep{064.png}{60}%

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Sir Thomas Overbury, and others. The notes are for the most part
selected from ancient sources.}

\Book{\Author{Thring.}---Works by the Rev.\ \Name{E. THRING}, M.A., Head Master
of Uppingham School.}{}

\Book{\Title{A LATIN GRADUAL.} A First Latin Construing Book for
Beginners. By \Name{Edward Thring}, M.A. New Edition, enlarged,
with Coloured Sentence Maps. Fcap.~8vo. \Price{2s. 6d.}}
%
{The Head Master of Uppingham has here sought to supply by easy steps
a knowledge of grammar, combined with a good Vocabulary. Passages
have been selected from the best Latin authors in prose and verse. These
\DPPageSep{065.png}{61}%
passages are gradually built up in their grammatical structure, and
finally printed in full. A short practical manual of common mood constructions,
with their English equivalents, forms a second part.}

\Book{\Title{A MANUAL OF MOOD CONSTRUCTIONS.} Fcap.~8vo. \Price{1s. 6d.}}
%
{Treats of the ordinary mood constructions, as found in the Latin, Greek,
and English languages.}

\Book{\Title{A CONSTRUING BOOK.} Fcap.~8vo. \Price{2s. 6d.}}{}

\Book{\Author{Thucydides.}---\Title{THE SICILIAN EXPEDITION.} Being Books
VI.~and~VII. of Thucydides, with Notes. A New Edition, revised
and enlarged, with a Map. By the Rev.\ \Name{Percival Frost}, M.A.,
late Fellow of St.~John's College, Cambridge. Fcap.~8vo. \Price{5s.}}
%
{This edition is mainly a grammatical one. Attention is called to the
force of compound verbs, and the exact meaning of the various tenses
employed.}

\Book{\Author{Virgil.}---\Title{THE WORKS OF VIRGIL RENDERED INTO
ENGLISH PROSE}, with Introductions, Running Analysis, and
an Index, by \Name{James Lonsdale}, M.A. and \Name{Samuel Lee}, M.A.
Globe 8vo. \Price{3s. 6d.}; gilt edges, \Price{4s. 6d.}}
%
{The preface of this new volume informs us that ``the original has been
faithfully rendered, and paraphrase altogether avoided. At the same time,
the translators have endeavoured to adapt the book to the use of the
English reader. Some amount of rhythm in the structure of the sentence
has been generally maintained; and, when in the Latin the sound of the
words is an echo to the sense \(as so frequently happens in Virgil\), an
attempt has been made to produce the same result in English.''

The general introduction gives us whatever is known of the poet's life,
an estimate of his genius, an account of the principal editions and translations
of his works, and a brief view of the influence he has had on
modern poets; special introductory essays are prefixed to the ``Eclogues,''
``Georgics,'' and ``Æneid.'' The text is divided into sections, each of
which is headed by a concise analysis of the subject; the index contains
references to all the characters and events of any importance.}
\DPPageSep{066.png}{62}%

\Book{\Author{Wright.}---Works by \Name{J. Wright}, M.A., late Head Master of
Sutton Coldfield School.

\Title{HELLENICA; OR, A HISTORY OF GREECE IN GREEK}, as
related by Diodorus and Thucydides; being a First Greek Reading
Book, with explanatory Notes, Critical and Historical. Third
Edition, with a Vocabulary. 12mo. \Price{3s. 6d.}}
%
{In the last twenty chapters of this volume, Thucydides sketches the rise
and progress of the Athenian Empire in so clear a style and in such simple
language, that the editor has doubts whether any easier or more instructive
passages can be selected for the use of the pupil who is commencing
Greek. This book includes a chronological table of the events recorded.}

\Book{\Title{A HELP TO LATIN GRAMMAR}; or, The Form and Use of Words
in Latin, with Progressive Exercises. Crown 8vo. \Price{4s. 6d.}}
%
{This book is not intended as a rival to any of the excellent Grammars
now in use; but as a help to enable the beginner to understand them.}

\Book{\Title{THE SEVEN KINGS OF ROME.} An Easy Narrative, abridged
from the First Book of Livy by the omission of Difficult Passages;
being a First Latin Reading Book, with Grammatical Notes.
With Vocabulary and Exercises. Fourth Edition. Fcap.~8vo. \Price{5s.}}
%
{This work is intended to supply the pupil with an easy construing book,
which may at the same time be made the vehicle for instructing him in the
rules of grammar and principles of composition. The notes profess to
teach what is commonly taught in grammars. It is conceived that the
pupil will learn the rules of construction of the language much more
easily from separate examples, which are pointed out to him in the course
of his reading, and which he may himself set down in his note-book after
some scheme of his own, than from a heap of quotations amassed for him
by others.}

{\centering Or, separately,\par}
\Book{\Title{SEVEN KINGS OF ROME.} \Price{3s.}}{}

\Book{\Title{VOCABULARY AND EXERCISES TO ``THE SEVEN KINGS.\Add{''}}
\Price{2s. 6d.}}{}
\DPPageSep{067.png}{63}%

\Csubsection{CLASSIC VERSIONS OF ENGLISH BOOKS,
AND LATIN HYMNS.}

\noindent\First{The} following works are, as the heading indicates,
classic renderings of English books. For scholars, and
particularly for writers of Latin Verse, the series has a
special value. The Hymni Ecclesiæ are here inserted, as
partly falling under the same class.

\Book{\Author{Church (A. J., A.M.)}---\Title{HORÆ TENNYSONIANÆ}, sive
Eclogae e Tennysono. Latine redditæ. Cura \Name{A.~J. Church},
A.M. Extra fcap.~8vo. \Price{6s.}}
%
{Latin versions of Selections from Tennyson. Among the authors are
the Editor, the late Professor Conington, Professor Seeley, Dr.~Hessey,
Mr.~Kebbel, and other gentlemen.}

\Book{\Author{Latham.}---\Title{SERTUM SHAKSPERIANUM}, Subnexis aliquot
aliunde excerptis floribus. Latine reddidit Rev.\ \Name{H. Latham}, M.A.
Extra fcap.~8vo. \Price{5s.}}
%
{Besides versions of \DPtypo{Shakspeare}{Shakespeare} this volume contains, among other pieces,
Gray's ``Elegy,'' Campbell's ``Hohenlinden,'' Wolfe's ``Burial of Sir
John Moore,'' and selections from Cowper and George Herbert.}

\Book{\Author{Lyttelton.}---\Title{THE COMUS OF MILTON}, rendered into Greek
Verse. By \Name{Lord Lyttelton}. Extra fcap.~8vo. \Price{5s.}}{}

\Book{\Title{THE SAMSON AGONISTES OF MILTON}, rendered into Greek
Verse. By \Name{Lord Lyttelton}. Extra fcap.~8vo. \Price{6s. 6d.}}{}

\Book{\Author{Merivale.}---\Title{KEATS' HYPERION}, rendered into Latin Verse.
By \Name{C. Merivale}, B.D. Second Edit. Extra fcap.~8vo. \Price{3s. 6d.}}{}
\DPPageSep{068.png}{64}%

\Book{\Author{Hymni Ecclesiæ.}---Edited by Rev.\ \Name{Dr.~Newman.} Extra
fcap.~8vo. \Price{7s. 6d.}}
%
{Hymns of the Mediæval Church. The first Part contains selections
from the Parisian Breviary; the second from those of Rome, Salisbury,
and York.}

\Book{\Author{Trench (Archbishop).}---\Title{SACRED LATIN POETRY},
chiefly Lyrical, selected and arranged for Use; with Notes and
Introduction. Fcap.~8vo. \Price{7s.}}
%
{In this work the editor has selected hymns of a catholic religious
sentiment that are common to Christendom, while rejecting those of a
distinctively Romish character.}
\DPPageSep{069.png}{65}%


\Csection{Mathematics.}

\Book{\Author{Airy.}---Works by G.~B. AIRY, Astronomer Royal:---}{}

\Book{\Title{ELEMENTARY TREATISE ON PARTIAL DIFFERENTIAL
EQUATIONS.} Designed for the Use of Students in the Universities.
With Diagrams. Crown 8vo.\ cloth. \Price{5s. 6d.}}
%
{It is hoped that the methods of solution here explained, and the instances
exhibited, will be found sufficient for application to nearly all the important
problems of Physical Science, which require for their complete investigation
the aid of Partial Differential Equations.}

\Book{\Title{ON THE ALGEBRAICAL AND NUMERICAL THEORY OF
ERRORS OF OBSERVATIONS AND THE COMBINATION
OF OBSERVATIONS.} Crown 8vo.\ cloth. \Price{6s. 6d.}}
%
{In order to spare astronomers and observers in natural philosophy the
confusion and loss of time which are produced by referring to the ordinary
treatises embracing \emph{both} branches of probabilities \(the first relating to
chances which can be altered only by the changes of entire units or integral
multiples of units in the fundamental conditions of the problem;
the other concerning those chances which have respect to insensible gradations
in the value of the element measured\) the present tract has been drawn
up. It relates only to errors of observation, and to the rules, derivable
from the consideration of these errors, for the combination of the results
of observations.}
\DPPageSep{070.png}{66}%

\Book{\Title{UNDULATORY THEORY OF OPTICS.} Designed for the Use of
Students in the University. New Edition. Crown 8vo.\ cloth.
\Price{6s. 6d.}}
%
{The undulatory theory of optics is presented to the reader as having the
same claims to his attention as the theory of gravitation: namely, that it is
certainly true, and that, by mathematical operations of general elegance, it
leads to results of great interest. This theory explains with accuracy a
vast variety of phenomena of the most complicated kind. The plan of this
tract has been to include those phenomena only which admit of calculation,
and the investigations are applied only to phenomena which actually have
been observed.}

\Book{\Title{ON SOUND AND ATMOSPHERIC VIBRATIONS.} With the
Mathematical Elements of Music. Designed for the Use of Students
of the University. Crown 8vo. \Price{9s.}}
%
{This volume consists of sections, which again are divided into numbered
articles, on the following topics: General recognition of the air as the
medium which conveys sound; Properties of the air on which the formation
and transmission of sound depend; Theory of undulations as applied
to sound,~\&c.; Investigation of the motion of a wave of air through the
atmosphere; Transmission of waves of soniferous vibrations through different
gases, solids, and fluids; Experiments on the velocity of sound,~\&c.;
On musical sounds, and the manner of producing them; On the
elements of musical harmony and melody, and of simple musical composition;
On instrumental music; On the human organs of speech and
hearing.}

\Book{\Title{A TREATISE ON MAGNETISM.} Designed for the use of
Students in the University. Crown 8vo. \Price{9s. 6d.}}{}

\Book{\Author{Airy (Osmund.)}---\Title{A TREATISE ON GEOMETRICAL
OPTICS.} Adapted for the use of the Higher Classes in Schools.
By \Name{Osmund Airy}, B.A., one of the Mathematical Masters in
Wellington College. Extra fcap.~8vo. \Price{3s. 6d.}}
%
{``This is, I imagine, the first time that any attempt has been made to
adapt the subject of Geometrical Optics, to the reading of the higher
classes in our good schools. That this should be so is the more a matter
\DPPageSep{071.png}{67}%
for remark, since the subject would appear to be peculiarly fitted for such
an adaptation\ldots. I have endeavoured, as much as possible, to avoid
the example of those popular lecturers who explain difficulties by ignoring
them. But as the nature of my design necessitated brevity, I have omitted
entirely one or two portions of the subject which I considered unnecessary
to a clear understanding of the rest, and which appear to me better learnt
at a more advanced stage.''---\Cite{Author's Preface}.}

\Book{\Author{Bayma.}---\Title{THE ELEMENTS OF MOLECULAR MECHANICS.}
By \Name{Joseph Bayma}, S.J., Professor of Philosophy,
Stonyhurst College. Demy 8vo.\ cloth. \Price{10s. 6d.}}
%
{Of the twelve Books into which the present treatise is divided, the first
and second give the demonstration of the principles which bear directly on
the constitution and the properties of matter. The next three books contain
a series of theorems and of problems on the laws of motion of elementary
substances. In the sixth and seventh, the mechanical constitution of molecules
is investigated and determined: and by it the general properties of
bodies are explained. The eighth book treats of luminiferous æther. The
ninth explains some special properties of bodies. The tenth and eleventh
contain a radical and lengthy investigation of chemical principles and
relations, which may lead to practical results of high importance. The
twelfth and last book treats of molecular masses, distances, and powers.}

\Book{\Author{Beasley.}---\Title{AN ELEMENTARY TREATISE ON PLANE
TRIGONOMETRY.} With Examples. By \Name{R.~D. Beasley},
M.A., Head Master of Grantham Grammar School. Second
Edition, revised and enlarged. Crown 8vo.\ cloth. \Price{3s. 6d.}}
%
{This treatise is specially intended for use in schools. The choice of matter
has been chiefly guided by the requirements of the three days' examination
at Cambridge. About four hundred examples have been added to this edition,
mainly collected from the Examination Papers of the last ten years.}

\Book{\Author{Boole.}---Works by G. BOOLE, D.C.L., F.R.S., Professor of
Mathematics in the Queen's University, Ireland.}{}

\Book{\Title{A TREATISE ON DIFFERENTIAL EQUATIONS.} New and
Revised Edition. Edited by \Name{I. Todhunter}. Crown 8vo.\ cloth.
\Price{14s.}%
\DPPageSep{072.png}{68}%
}
%
{Professor Boole has endeavoured in this treatise to convey as complete an
account of the present state of knowledge on the subject of Differential Equations,
as was consistent with the idea of a work intended, primarily, for
elementary instruction. The earlier sections of each chapter contain that
kind of matter which has usually been thought suitable for the beginner,
while the latter ones are devoted either to an account of recent discovery, or
the discussion of such deeper questions of principle as are likely to present
themselves to the reflective student in connexion with the methods and
processes of his previous course.}

\Book{\Title{A TREATISE ON DIFFERENTIAL EQUATIONS.} Supplementary
Volume. Edited by \Name{I. Todhunter}. Crown 8vo.\ cloth.
\Price{8s. 6d.}}
%
{This volume contains all that Professor Boole wrote for the purpose of
enlarging his treatise on Differential Equations.}

\Book{\Title{THE CALCULUS OF FINITE DIFFERENCES.} Crown 8vo.\
cloth. \Price{10s. 6d.}}
%
{In this exposition of the Calculus of Finite Differences, particular attention
has been paid to the connexion of its methods with those of the Differential
Calculus---a connexion which in some instances involves far more than a
merely formal analogy. The work is in some measure designed as a
sequel to Professor Boole's Treatise on Differential Equations.}

\Book{\Title{CAMBRIDGE SENATE-HOUSE PROBLEMS AND RIDERS,
WITH SOLUTIONS}:---

\SubBook{\Date{1848}--\Date{1851}.---\Title{PROBLEMS.} By \Name{Ferrers} and \Name{Jackson}. 8vo.\
cloth. \Price{15s. 6d.}}

\SubBook{\Date{1848}--\Date{1851}.---\Title{RIDERS.} By \Name{Jameson}. 8vo.\ cloth. \Price{7s. 6d.}}

\SubBook{\Date{1854}.---\Title{PROBLEMS AND RIDERS.} By \Name{Walton} and
\Name{Mackenzie}. 8vo.\ cloth. \Price{10s. 6d.}}

\SubBook{\Date{1857}.---\Title{PROBLEMS AND RIDERS.} By \Name{Campion} and
\Name{Walton}. 8vo.\ cloth. \Price{8s. 6d.}}

\SubBook{\Date{1860}.---\Title{PROBLEMS AND RIDERS.} By \Name{Watson} and \Name{Routh}.
Crown 8vo.\ cloth. \Price{7s. 6d.}}

\SubBook{\Date{1864}.---\Title{PROBLEMS AND RIDERS.} By \Name{Walton} and \Name{Wilkinson}.
8vo.\ cloth. \Price{10s. 6d.}}\par%
\DPPageSep{073.png}{69}%
}
%
{These volumes will be found of great value to Teachers and Students, as
indicating the style and range of mathematical study in the University of
Cambridge.}

\Book{\Title{CAMBRIDGE COURSE OF ELEMENTARY NATURAL
PHILOSOPHY}, for the Degree of B.A. Originally compiled by
\Name{J.~C. Snowball}, M.A., late Fellow of St.~John's College.
Fifth Edition, revised and enlarged, and adapted for the Middle-Class
Examinations by \Name{Thomas Lund}, B.D., Late Fellow and
Lecturer of St.~John's College, Editor of Wood's Algebra,~\&c.
Crown 8vo.\ cloth. \Price{5s.}}
%
{This work will be found adapted to the wants, not only of University
Students, but also of many others who require a short course of Mechanics
and Hydrostatics, and especially of the candidates at our Middle Class
Examinations. At the end of each chapter a series of easy questions is
added for the exercise of the student.}

\Book{\Title{CAMBRIDGE AND DUBLIN MATHEMATICAL JOURNAL.}
The Complete Work, in Nine Vols. 8vo.\ cloth, \Price{7l. 4s.}}
%
{Only a few copies remain on hand. Among Contributors to this
work will be found Sir W.~Thomson, Stokes, Adams, Boole, Sir W.~R.
Hamilton, De~Morgan, Cayley, Sylvester, Jellett, and other distinguished
mathematicians.}

\Book{\Author{Candler.}---\Title{HELP TO ARITHMETIC.} Designed for the use of
Schools. By \Name{H. Candler}, M.A. Mathematical Master of
Uppingham School. Extra fcap.~8vo. \Price{2s. 6d.}}
%
{This work is intended as a companion to any text book that may be
in use.}

\Book{\Author{Cheyne.}---\Title{AN ELEMENTARY TREATISE ON THE
PLANETARY THEORY.} With a Collection of Problems.
By \Name{C.~H.~H. Cheyne}, M.A., F.R.A.S. Second Edition. Crown
8vo.\ cloth\Add{.} \Price{6s. 6d.}}
%
{In this volume, an attempt has been made to produce a treatise on the
Planetary theory, which, being elementary in character, should be so far
complete, as to contain all that is usually required by students in the
University of Cambridge. This Edition has been carefully revised. The
stability of the Planetary System has been more fully treated, and an
\DPPageSep{074.png}{70}%
elegant geometrical explanation of the formulæ for the secular variation of
the node and inclination, due to Mr.~H.~M. Taylor, has been introduced.}

\Book{\Title{THE EARTH'S MOTION OF ROTATION.} By \Name{C.~H.~H.
Cheyne}, M.A., F.R.A.S. Crown 8vo. \Price{3s. 6d.}}
%
{The first part of this work consists of an application of the method of the
variation of elements to the general problem of rotation. In the second
part the general rotation formulæ are applied to the particular case of
the earth.}

\Book{\Author{Childe.}---\Title{THE SINGULAR PROPERTIES OF THE ELLIPSOID
AND ASSOCIATED SURFACES OF THE Nth
DEGREE.} By the Rev.\ \Name{G.~F. Childe}, M.A., Author of
``Ray Surfaces,'' ``Related Caustics,''~\&c. 8vo. \Price{10s. 6d.}}
%
{The object of this volume is to develop peculiarities in the Ellipsoid;
and, further, to establish analogous properties in the unlimited congeneric
series of which this remarkable surface is a constituent.}

\Book{\Author{Christie.}---\Title{A COLLECTION OF ELEMENTARY TEST-QUESTIONS
IN PURE AND MIXED MATHEMATICS};
with Answers and Appendices on Synthetic Division, and on the
Solution of Numerical Equations by Horner's Method. By \Name{James
R. Christie}, F.R.S., late First Mathematical Master at the
Royal Military Academy, Woolwich. Crown 8vo.\ cloth. \Price{8s. 6d.}}
%
{The series of Mathematical exercises here offered to the public is collected
from those which the author has, from time to time, proposed for solution
by his pupils during a long career at the Royal Military Academy. A
student who finds that he is able to solve the larger portion of these exercises,
may consider that he is thoroughly well grounded in the elementary principles
of pure and mixed Mathematics.}

\Book{\Author{Dalton.}---\Title{ARITHMETICAL EXAMPLES.} Progressively
arranged, with Exercises and Examination Papers. By the Rev.\
\Name{T. Dalton}, M.A., Assistant Master of Eton College, 18mo.\
cloth. \Price{2s. 6d.} \textit{Answers to the Examples are appended.}}{}
\DPPageSep{075.png}{71}%

\Book{\Author{Day.}---\Title{PROPERTIES OF CONIC SECTIONS PROVED
GEOMETRICALLY\@. PART~I., THE ELLIPSE}, with
Problems. By the Rev.\ \Name{H.~G. Day}, M.A., Head Master of
Sedburgh Grammar School. Crown 8vo. \Price{3s. 6d.}}
%
{The object of this book is the introduction of a treatment of Conic
Sections which should be simple and natural, and lead by an easy transition
to the analytical methods, without departing from the strict geometry
of Euclid.}

\Book{\Author{Dodgson.}---\Title{AN ELEMENTARY TREATISE ON DETERMINANTS},
with their Application to Simultaneous Linear
Equations and Algebraical Geometry. By \Name{Charles L\Add{.} Dodgson},
M.A., Student and Mathematical Lecturer of Christ Church,
Oxford. Small 4to.\ cloth. \Price{10s. 6d.}}
%
{The object of the author is to present the subject as a continuous chain of
argument, separated from all accessories of explanation or illustration.
All such explanation and illustration as seemed necessary for a beginner
are introduced either in the form of foot-notes, or, where that would have
occupied too much room, of Appendices.}

\Book{\Author{Drew.}---\Title{GEOMETRICAL TREATISE ON CONIC SECTIONS.}
By \Name{W.~H. Drew}, M.A., St.~John's College, Cambridge.
Fourth Edition. Crown 8vo.\ cloth. \Price{4s. 6d.}}
%
{In this work the subject of Conic Sections has been placed before the student
in such a form that, it is hoped, after mastering the elements of Euclid, he
may find it an easy and interesting continuation of his geometrical studies.
With a view, also, of rendering the work a complete manual of what is
required at the Universities, there have either been embodied into the text or
inserted among the examples, every book-work question, problem, and rider,
which has been proposed in the Cambridge examinations up to the present
time.}

\Book{\Title{SOLUTIONS TO THE PROBLEMS IN DREW'S CONIC
SECTIONS.} Crown 8vo.\ cloth. \Price{4s. 6d.}}{}
\DPPageSep{076.png}{72}%

\Book{\Author{Edgar (J. H.)}---\Title{NOTE-BOOK ON PRACTICAL SOLID
GEOMETRY.} Containing Problems with help for Solutions. By
\Name{J.~H. Edgar}, M.A. Lecturer on Mechanical Drawing at the
Royal School of Mines. 4to. \Price{2s.}}
%
{In teaching a large class, if the method of lecturing and demonstrating
from the black board only is pursued, the more intelligent students have
generally to be kept back, from the necessity of frequent repetition, for the
sake of the less promising; if the plan of setting problems to each pupil is
adopted, the teacher finds a difficulty in giving to each sufficient attention.
A judicious combination of both methods is doubtless the best; and it is
hoped that this result may be arrived at in some degree by the use of this
book, which is simply a collection of examples, with helps for solution,
arranged in progressive sections.}

\Book{\Author{Ferrers.}---\Title{AN ELEMENTARY TREATISE ON TRILINEAR
CO-ORDINATES}, the Method of Reciprocal Polars, and the
Theory of Projectors. By the Rev.\ \Name{N.~M. Ferrers}, M.A., Fellow
and Tutor of Gonville and Caius College, Cambridge. Second
Edition. Crown 8vo. \Price{6s. 6d.}}
%
{The object of the author in writing on this subject has mainly been to
place it on a basis altogether independent of the ordinary Cartesian system,
instead of regarding it as only a special form of Abridged Notation.
A short chapter on Determinants has been introduced.}

\Book{\Author{Frost.}---\Title{THE FIRST THREE SECTIONS OF NEWTON'S
PRINCIPIA.} With Notes and Illustrations. Also a collection of
Problems, principally intended as Examples of Newton's Methods.
By \Name{Percival Frost}, M.A., late Fellow of St.~John's College,
Mathematical Lecturer of King's College, Cambridge. Second
Edition. 8vo.\ cloth. \Price{10s. 6d.}}
%
{The author's principal intention is to explain difficulties which may be
encountered by the student on first reading the \emph{Principia}, and to illustrate
the advantages of a careful study of the methods employed by Newton, by
showing the extent to which they may be applied in the solution of problems;
\DPPageSep{077.png}{73}%
he has also endeavoured to give assistance to the student who is engaged in
the study of the higher branches of mathematics, by representing in a
geometrical form several of the processes employed in the Differential and
Integral Calculus, and in the analytical investigations of Dynamics.}

\Book{\Author{Frost and Wolstenholme.}---\Title{A TREATISE ON SOLID
GEOMETRY.} By \Name{Percival Frost}, M.A., and the Rev.\ \Name{J.
Wolstenholme}, M.A., Fellow and Assistant Tutor of Christ's
College. 8vo.\ cloth. \Price{18s.}}
%
{The authors have endeavoured to present before students as comprehensive
a view of the subject as possible. Intending to make the subject accessible,
at least in the earlier portion, to all classes of students, they have endeavoured
to explain completely all the processes which are most useful in
dealing with ordinary theorems and problems, thus directing the student
to the selection of methods which are best adapted to the exigencies of each
problem. In the more difficult portions of the subject, they have considered
themselves to be addressing a higher class of students; and they have there
tried to lay a good foundation on which to build, if any reader should
wish to pursue the science beyond the limits to which the work extends.}

\Book{\Author{Godfray.}---\Title{A TREATISE ON ASTRONOMY}, for the Use of
Colleges and Schools. By \Name{Hugh Godfray}, M.A., Mathematical
Lecturer at Pembroke College, Cambridge. 8vo.\ cloth. \Price{12s. 6d.}}
%
{This book embraces all those branches of Astronomy which have, from
time to time, been recommended by the Cambridge Board of Mathematical
Studies: but by far the larger and easier portion, adapted to the first three
days of the Examination for Honours, may be read by the more
advanced pupils in many of our schools. The author's aim has been to
convey clear and distinct ideas of the celestial phenomena.}

\Book{\Title{AN ELEMENTARY TREATISE ON THE LUNAR THEORY},
with a Brief Sketch of the Problem up to the time of Newton.
By \Name{Hugh Godfray}, M.A. Second Edition, revised. Crown
8vo.\ cloth. \Price{5s. 6d.}%
\DPPageSep{078.png}{74}%
}
%
{These pages will, it is hoped, form an introduction to more recondite
works. Difficulties have been discussed at considerable length. The
selection of the method followed with regard to analytical solutions,
which is the same as that of Airy, Herschel,~\&c.\ was made on account
of its simplicity; it is, moreover, the method which has obtained in the
University of Cambridge.}

\Book{\Author{Hemming.}---\Title{AN ELEMENTARY TREATISE ON THE
DIFFERENTIAL AND INTEGRAL CALCULUS}, for the
Use of Colleges and Schools. By \Name{G.~W. Hemming}, M.A.,
Fellow of St.~John's College, Cambridge. Second Edition, with
Corrections and Additions. 8vo.\ cloth. \Price{9s.}}{}

\Book{\Author{Jones and Cheyne.}---\Title{ALGEBRAICAL EXERCISES.} Progressively
arranged. By the Rev.\ \Name{C.~A. Jones}, M.A., and \Name{C.~H.
Cheyne}, M.A., F.R.A.S., Mathematical Masters of Westminster
School. New Edition. 18mo.\ cloth. \Price{2s. 6d.}}
%
{This little book is intended to meet a difficulty which is probably felt more
or less by all engaged in teaching Algebra to beginners. It is, that while
new ideas are being acquired, old ones are forgotten. In the belief that
constant practice is the only remedy for this, the present series of miscellaneous
exercises has been prepared. Their peculiarity consists in this,
that though miscellaneous they are yet progressive, and may be used by
the pupil almost from the commencement of his studies. They are not
intended to supersede the systematically arranged examples to be found in
ordinary treatises on Algebra, but rather to supplement them.

The book being intended chiefly for Schools and Junior Students, the
higher parts of Algebra have not been included.}

\Book{\Author{Kitchener.}---\Title{A GEOMETRICAL NOTE-BOOK}, containing
Easy Problems in Geometrical Drawing preparatory to the Study
of Geometry. For the Use of Schools. By \Name{F.~E. Kitchener},
M.A., Mathematical Master at Rugby. 4to. \Price{2s.}%
\DPPageSep{079.png}{75}%
}
%
{It is the object of this book to make some way in overcoming the difficulties
of Geometrical conception, before the mind is called to the attack of
Geometrical theorems. A few simple methods of construction are given;
and space is left on each page, in order that the learner may draw in the
figures.}

\Book{\Author{Morgan.}---\Title{A COLLECTION OF PROBLEMS AND EXAMPLES
IN MATHEMATICS.} With Answers. By \Name{H. A.
Morgan}, M.A., Sadlerian and Mathematical Lecturer of Jesus
College, Cambridge. Crown 8vo.\ cloth. \Price{6s. 6d.}}
%
{This book contains a number of problems, chiefly elementary, in the
Mathematical subjects usually read at Cambridge. They have been
selected from the papers set during late years at Jesus College. Very few
of them are to be met with in other collections, and by far the larger
number are due to some of the most distinguished Mathematicians in the
University.}

\Book{\Author{Parkinson.}---Works by \Name{S. Parkinson}, D.D., F.R.S., Fellow and
Tutor of St.~John's College, Cambridge.}{}

\Book{\Title{AN ELEMENTARY TREATISE ON MECHANICS.} For the
Use of the Junior Classes at the University and the Higher Classes
in Schools. With a Collection of Examples. Fourth edition, revised.
Crown 8vo.\ cloth. \Price{9s. 6d.}}
%
{In preparing a fourth edition of this work the author has kept the same
object in view as he had in the former editions---namely, to include in it
such portions of Theoretical Mechanics as can be conveniently investigated
without the use of the Differential Calculus, and so render it suitable as
a manual for the junior classes in the University and the higher classes
in Schools. With one or two short exceptions, the student is not presumed
to require a knowledge of any branches of Mathematics beyond the elements
of Algebra, Geometry, and Trigonometry. Several additional propositions
have been incorporated in the work for the purpose of rendering it more
complete; and the collection of Examples and Problems has been largely
increased.}
\DPPageSep{080.png}{76}%

\Book{\Title{A TREATISE ON OPTICS.} Third Edition, revised and enlarged.
Crown 8vo.\ cloth. \Price{10s. 6d.}}
%
{A collection of examples and problems has been appended to this work,
which are sufficiently numerous and varied in character to afford useful
exercise for the student. For the greater part of them, recourse has been
had to the Examination Papers set in the University and the several
Colleges during the last twenty years.}

\Book{\Author{Phear.}---\Title{ELEMENTARY HYDROSTATICS.} With Numerous
Examples. By \Name{J.~B. Phear}, M.A., Fellow and late Assistant
Tutor of Clare College, Cambridge. Fourth Edition. Crown
8vo.\ cloth. \Price{5s. 6d.}}
%
{This edition has been carefully revised throughout, and many new
illustrations and examples added, which it is hoped will increase its
usefulness to students at the Universities and in Schools. In accordance
with suggestions from many engaged in tuition, answers to all the
Examples have been given at the end of the book.}

\Book{\Author{Pratt.}---\Title{A TREATISE ON ATTRACTIONS, LAPLACE'S
FUNCTIONS, AND THE FIGURE OF THE EARTH.}
By \Name{John H. Pratt}, M.A., Archdeacon of Calcutta, Author of
``The Mathematical Principles of Mechanical Philosophy.'' Third
Edition. Crown 8vo.\ cloth. \Price{6s. 6d.}}
%
{The author's chief design in this treatise is to give an answer to the
question, ``How the Earth acquired its present form from being originally
in a fluid state?'' This Edition is a complete revision of the former ones.}

\Book{\Author{Puckle.}---\Title{AN ELEMENTARY TREATISE ON CONIC SECTIONS
AND ALGEBRAIC GEOMETRY.} With Numerous
Examples and Hints for their Solution; especially designed for the
Use of Beginners. By \Name{G.~H. Puckle}, M.A., Head Master of
Windermere College. New Edition, revised and enlarged. Crown
8vo.\ cloth. \Price{7s. 6d.}%
\DPPageSep{081.png}{77}%
}
%
{This work is recommended by the Syndicate of the Cambridge Local
Examinations, and is the text-book in Harvard University, U.S.}

\Book{\Author{Rawlinson.}---\Title{ELEMENTARY STATICS}, by the Rev.\ \Name{GEORGE
RAWLINSON}, M.A. Edited by the Rev.\ \Name{Edward Sturges}, M.A.,
of Emmanuel College, Cambridge, and late Professor of the Applied
Sciences, Elphinstone College, Bombay. Crown 8vo.\ cloth. \Price{4s. 6d.}}
%
{Published under the authority of Her Majesty's Secretary of State for
India, for use in the Government Schools and Colleges in India.}

\Book{\Author{Reynolds.}---\Title{MODERN METHODS IN ELEMENTARY
GEOMETRY.} By \Name{E.~M. Reynolds}, M.A., Mathematical
Master in Clifton College. Crown 8vo. \Price{3s. 6d.}}
%
{Some change, it is evident, in our English ways of teaching can now no
longer be postponed, and this little book, mainly derived from French and
German sources, has been written in the hope of facilitating that change.
It has been constructed on one plan throughout, that of always giving in
the simplest possible form the direct proof from the nature of the case. The
axioms necessary to this simplicity have been assumed without hesitation,
and no scruple has been felt as to the increase of their number, or the
acceptance of as many elementary notions as common experience places
past all doubt.

The book differs most from established teaching in its constructions, and
in its early application of Arithmetic to Geometry.}

\Book{\Author{Routh.}---\Title{AN ELEMENTARY TREATISE ON THE DYNAMICS
OF THE SYSTEM OF RIGID BODIES.} With
Numerous Examples. By \Name{Edward John Routh}, M.A., late
Fellow and Assistant Tutor of St.~Peter's College, Cambridge;
Examiner in the University of London. Second Edition, enlarged.
Crown 8vo.\ cloth. \Price{14s.}%
\DPPageSep{082.png}{78}%
}
%
{In this edition the author has made several additions to each chapter.
He has tried, even at the risk of some little repetition, to make each
chapter, as far as possible, complete in itself, so that all that relates to any
one part of the subject may be found in the same place. This arrangement
will enable every student to select his own order in which to read the
subject. The Examples which will be found at the end of each chapter
have been chiefly selected from the Examination Papers which have been
set in the University and the Colleges in the last few years.}

\Book{\Author{Smith (Barnard).}---Works by BARNARD SMITH, M.A.,
Rector of Glaston, Rutlandshire, late Fellow and Senior Bursar
of St.~Peter's College, Cambridge.}{}

\Book{\Title{ARITHMETIC AND ALGEBRA}, in their Principles and Application;
with numerous systematically arranged Examples taken from
the Cambridge Examination Papers, with especial reference to the
Ordinary Examination for the B.A. Degree. Tenth Edition.
Crown 8vo.\ cloth. \Price{10s. 6d.}}
%
{This manual is now extensively used in Schools and Colleges, both in
England and in the Colonies. It has also been found of great service for
students preparing for the Middle Class and Civil and Military Service
Examinations, from the care that has been taken to elucidate the principles
of all the rules. The present edition has been carefully revised. ``To
all those whose minds are sufficiently developed to comprehend the simplest
mathematical reasoning, and who have not yet thoroughly mastered the
principles of Arithmetic and Algebra, it is calculated to be of great
advantage.''---\Cite{Athenæum}.

Of this work, also, one of the highest possible authorities, the late Dean
Peacock, writes: ``Mr.~Smith's work is a most useful publication. The
rules are stated with great clearness. The examples are well selected, and
worked out with just sufficient detail, without being encumbered by too
minute explanations; and there prevails throughout it that just proportion
of theory and practice, which is the crowning excellence of an elementary
work.''}
\DPPageSep{083.png}{79}%

\Book{\Title{ARITHMETIC FOR SCHOOLS.} New Edition. Crown 8vo.\
cloth. \Price{4s. 6d.}}
%
{Adapted from the author's work on ``Arithmetic and Algebra,'' by the
omission of the algebraic portion, and by the introduction of new exercises.
The reason of each arithmetical process is fully exhibited. The system of
Decimal Coinage is explained; and answers to the exercises are appended
at the end. This Arithmetic is characterised as ``admirably adapted for
instruction, combining just sufficient theory with a large and well-selected
collection of exercises for practice.''---\Cite{Journal of Education}.}

\Book{\Title{COMPANION TO ARITHMETIC FOR SCHOOLS.}\InPress[Preparing.]}{}

\Book{\Title{A KEY TO THE ARITHMETIC FOR SCHOOLS.} Seventh
Edition. Crown 8vo.\ cloth. \Price{8s. 6d.}}{}

\Book{\Title{EXERCISES IN ARITHMETIC.} With Answers. Crown 8vo.\ limp
cloth. \Price{2s. 6d.}

Or sold separately, Part~I\@. \Price{1s.}; Part~II\@. \Price{1s.}; Answers, \Price{6d.}}
%
{These Exercises have been published in order to give the pupil examples
in every rule of Arithmetic. The greater number have been carefully
compiled from the latest University and School Examination Papers.}

\Book{\Title{SCHOOL CLASS-BOOK OF ARITHMETIC.} 18mo.\ cloth. \Price{3s.}

Or sold separately, Parts I.~and~II\@. \Price{10d.} each; Part~III\@. \Price{1s.}}
%
{This manual, published at the request of many schoolmasters, and
chiefly intended for National and Elementary Schools, has been prepared
on the same plan as that adopted in the author's School Arithmetic, which
is in extensive circulation in England and abroad. The Metrical Tables
have been introduced, from the conviction on the part of the author, that
the knowledge of such tables, and the mode of applying them, will be of
great use to the rising generation.}

\Book{\Title{KEYS TO SCHOOL CLASS-BOOK OF ARITHMETIC.} Complete
in one volume, 18mo.\ cloth, \Price{6s. 6d.}; or Parts I.~II. and~III\@.
\Price{2s. 6d.} each.}{}
\DPPageSep{084.png}{80}%

\Book{\Title{SHILLING BOOK OF ARITHMETIC FOR NATIONAL AND
ELEMENTARY SCHOOLS.} 18mo.\ cloth. Or separately,
Part~I\@. \Price{2d.}; Part~II\@. \Price{3d.}; Part~III\@. \Price{7d.} Answers, \Price{6d.}}{}

\Book{\Title{THE SAME}, with Answers complete. 18mo.\ cloth. \Price{1s. 6d.}}
%
{This Shilling Book of Arithmetic has been prepared for the use of
National and other schools at the urgent request of numerous masters of
schools both at home and abroad. The Explanations of the Rules, and
the Examples will, it is hoped, be found suited to the most elementary
classes.}

\Book{\Title{KEY TO SHILLING BOOK OF ARITHMETIC.} 18mo.\ cloth.
\Price{4s. 6d.}}{}

\Book{\Title{EXAMINATION PAPERS IN ARITHMETIC.} 18mo.\ cloth.
\Price{1s. 6d.} The same, with Answers, 18mo. \Price{1s. 9d.}}
%
{The object of these Examination Papers is to test students both in the
theory and practice of Arithmetic. It is hoped that the method adopted
will lead students to deduce results from general principles rather than
to apply stated rules. The author believes that the practice of giving
examples under particular rules makes the working of Arithmetic quite
mechanical, and tends to throw all but very clever boys off their balance
when a general paper on the subject is put before them.}

\Book{\Title{KEY TO EXAMINATION PAPERS IN ARITHMETIC.}
18mo.\ cloth. \Price{4s. 6d.}}{}

\Book{\Author{Smith (J. Brook).}---\Title{ARITHMETIC IN THEORY AND
PRACTICE, FOR ADVANCED PUPILS.} By \Name{J. Brook
Smith}, M.A. Part~I\@. Crown 8vo. \Price{3s. 6d.}}
%
{The following pages form the first part of a Treatise on Arithmetic, in
which the Author has endeavoured from very simple principles to explain,
in a full and satisfactory manner, all the more important processes in
that subject. The proofs have in all cases been given in a form entirely
\DPPageSep{085.png}{81}%
arithmetical, and at the end of every chapter several examples have been
worked out at length, and the best practical method of operation carefully
pointed out.}

\Book{\Author{Snowball.}---\Title{THE ELEMENTS OF PLANE AND SPHERICAL
TRIGONOMETRY}; with the Construction and Use of
Tables of Logarithms. By \Name{J.~C. Snowball}, M.A. Tenth Edition.
Crown 8vo.\ cloth. \Price{7s. 6d.}}
%
{In preparing the present edition for the press, the text has been
subjected to a careful revision; the proofs of some of the more important
propositions have been rendered more strict and general; and a
considerable addition of more than two hundred examples, taken principally
from the questions set of late years in the public examinations of the
University and of individual Colleges, has been made to the collection of
Examples and Problems for practice.}

\Book{\Author{Tait and Steele.}---\Title{A TREATISE ON DYNAMICS OF A
PARTICLE.} With numerous Examples. By Professor \Name{Tait} and
Mr.~\Name{Steele}. New Edition Enlarged. Crown 8vo.\ cloth. \Price{10s. 6d.}}
%
{In this treatise will be found all the ordinary propositions, connected
with the Dynamics of Particles, which can be conveniently deduced without
the use of D'Alembert's Principle. Throughout the book will be found a
number of illustrative examples introduced in the text, and for the most
part completely worked out; others with occasional solutions or hints to
assist the student are appended to each chapter. For by far the greater
portion of these, the Cambridge Senate-House and College Examination
Papers have been applied to.}

\Book{\Author{Taylor.}---\Title{GEOMETRICAL CONICS}; including Anharmonic
Ratio and Projection, with numerous Examples. By \Name{C. Taylor},
B.A., Scholar of St.~John's College, Cambridge. Crown 8vo.\ cloth.
\Price{7s. 6d.}}
%
{This work contains elementary proofs of the principal properties of Conic
Sections, together with chapters on Projection and Anharmonic Ratio.}
\DPPageSep{086.png}{82}%

\Book{\Author{Tebay.}---\Title{ELEMENTARY MENSURATION FOR SCHOOLS.}
With numerous Examples. By \Name{Septimus Tebay}, B.A., Head
Master of Queen Elizabeth's Grammar School, Rivington. Extra
fcap.~8vo. \Price{3s. 6d.}}
%
{The object of the present work is to enable boys to acquire a moderate
knowledge of Mensuration in a reasonable time. All difficult and useless
matter has been avoided. The examples for the most part are easy, and
the rules are concise.}

\Book{\Author{Todhunter.}---Works by \Name{I. Todhunter}, M.A., F.R.S.,
of St.~John's College, Cambridge.}{}

\Book{\Title{THE ELEMENTS OF EUCLID.} For the Use of Colleges and
Schools. New Edition. 18mo.\ cloth. \Price{3s. 6d.}}
%
{As the elements of Euclid are usually placed in the hands of young
students, it is important to exhibit the work in such a form as will assist
them in overcoming the difficulties which they experience on their first introduction
to processes of continuous argument. No method appears to be
so useful as that of breaking up the demonstrations into their constituent
parts; a plan strongly recommended by Professor De Morgan. In the
present Edition each distinct assertion in the argument begins a new line:
and at the ends of the lines are placed the necessary references to the
preceding principles on which the assertions depend. The longer propositions
are distributed into subordinate parts, which are distinguished by
breaks at the beginning of the lines. Notes, appendix, and a collection of
exercises are added.}

\Book{\Title{MENSURATION FOR BEGINNERS.} With Numerous Examples.
18mo.\ cloth. \Price{2s. 6d.}}
%
{The subjects included in the present work are those which have usually
found a place in Elementary Treatises on Mensuration. The mode of
treatment has been determined by the fact that the work is intended for the
use of beginners. Accordingly it is divided into short independent chapters,
which are followed by appropriate examples. A knowledge of the elements
of Arithmetic is all that is assumed; and in connexion with most of the
Rules of Mensuration it has been found practicable to give such explanations
and illustrations as will supply the place of formal mathematical
\DPPageSep{087.png}{83}%
demonstrations, which would have been unsuitable to the character of the
work.}

\Book{\Title{ALGEBRA FOR BEGINNERS.} With numerous Examples. New
Edition. 18mo.\ cloth. \Price{2s. 6d.}}
%
{Great pains have been taken to render this work intelligible to young
students, by the use of simple language and by copious explanations. In
determining the subjects to be included and the space to be assigned to each,
the Author has been guided by the papers given at the various examinations
in elementary Algebra which are now carried on in this country. The
book may be said to consist of three parts. The first part contains the
elementary operations in integral and fractional expressions; the second
the solution of equations and problems; the third treats of various subjects
which are introduced but rarely into examination papers, and are more
briefly discussed. Provision has at the same time been made for the
introduction of easy equations and problems at an early stage---for those
who prefer such a course.}

\Book{\Title{KEY TO ALGEBRA FOR BEGINNERS.} Crown 8vo.\ cloth.
\Price{6s. 6d.}}{}

\Book{\Title{TRIGONOMETRY FOR BEGINNERS.} With numerous Examples.
New Edition. 18mo.\ cloth. \Price{2s. 6d.}}
%
{Intended to serve as an introduction to the larger treatise on Plane
Trigonometry, published by the Author. The same plan has been adopted
as in the \emph{Algebra for Beginners}: the subject is discussed in short chapters,
and a collection of examples is attached to each chapter. The first fourteen
chapters present the geometrical part of Plane Trigonometry; and contain
all that is necessary for practical purposes. The range of matter included
is such as seems required by the various examinations in elementary Trigonometry
which are now carried on in the country. Answers are appended
at the end.}

\Book{\Title{MECHANICS FOR BEGINNERS.} With numerous Examples.
Second Edition. 18mo.\ cloth. \Price{4s. 6d.}}
%
{Intended as a companion to the two preceding books. The work forms
an elementary treatise on demonstrative mechanics. It may be true that
this part of mixed mathematics has been sometimes made too abstract and
speculative; but it can hardly be doubted that a knowledge of the elements
\DPPageSep{088.png}{84}%
at least of the theory of the subject is extremely valuable even for those
who are mainly concerned with practical results. The Author has accordingly
endeavoured to provide a suitable introduction to the study of applied
as well as of theoretical mechanics. The work consists of two parts,
namely, Statics and Dynamics. It will be found to contain all that is
usually comprised in elementary treatises on Mechanics, together with some
additions.}

\Book{\Title{ALGEBRA.} For the Use of Colleges and Schools. Fifth Edition.
Crown 8vo.\ cloth. \Price{7s. 6d.}}
%
{This work contains all the propositions which are usually included in
elementary treatises on Algebra, and a large number of \emph{Examples for
Exercise}. The author has sought to render the work easily intelligible to
students, without impairing the accuracy of the demonstrations, or contracting
the limits of the subject. The Examples, about \emph{Sixteen hundred
and fifty} in number, have been selected with a view to illustrate every part
of the subject. Each chapter is complete in itself; and the work will be
found peculiarly adapted to the wants of students who are without the aid
of a teacher. The Answers to the examples, with hints for the solution of
some in which assistance may be needed, are given at the end of the book.
In the present edition two New Chapters and \emph{Three hundred} miscellaneous
Examples have been added. The latter are arranged in sets, each set
containing ten examples.}

\Book{\Title{KEY TO ALGEBRA FOR THE USE OF COLLEGES AND
SCHOOLS.} Crown 8vo. \Price{10s. 6d.}}{}

\Book{\Title{AN ELEMENTARY TREATISE ON THE THEORY OF
EQUATIONS.} Second Edition, revised. Crown 8vo.\ cloth.
\Price{7s. 6d.}}
%
{This treatise contains all the propositions which are usually included
in elementary treatises on the theory of Equations, together with Examples
for exercise\Add{.} These have been selected from the College and University
Examination Papers, and the results have been given when it appeared
necessary. In order to exhibit a comprehensive view of the subject, the
treatise includes investigations which are not found in all the preceding
elementary treatises, and also some investigations which are not to be found
in any of them. For the second edition the work has been revised and
\DPPageSep{089.png}{85}%
some additions have been made, the most important being an account of
the researches of Professor Sylvester respecting Newton's Rule.}

\Book{\Title{PLANE TRIGONOMETRY.} For Schools and Colleges. Fourth
Edition. Crown 8vo.\ cloth. \Price{5s.}}
%
{The design of this work has been to render the subject intelligible to
beginners, and at the same time to afford the student the opportunity of
obtaining all the information which he will require on this branch of
Mathematics. Each chapter is followed by a set of Examples: those
which are entitled \emph{Miscellaneous Examples}, together with a few in some
of the other sets, may be advantageously reserved by the student for exercise
after he has made some progress in the subject. In the Second Edition
the hints for the solution of the Examples have been considerably increased.}

\Book{\Title{A TREATISE ON SPHERICAL TRIGONOMETRY.} Second
Edition, enlarged. Crown 8vo.\ cloth. \Price{4s. 6d.}}
%
{The present work is constructed on the same plan as the treatise on
Plane Trigonometry, to which it is intended as a sequel. In the account
of Napier's Rules of Circular Parts, an explanation has been given of a
method of proof devised by Napier, which seems to have been overlooked
by most modern writers on the subject. Considerable labour has been
bestowed on the text in order to render it comprehensive and accurate, and
the Examples (selected chiefly from College Examination Papers) have
all been carefully verified.}

\Book{\Title{PLANE CO-ORDINATE GEOMETRY}, as applied to the Straight
Line and the Conic Sections. With numerous Examples. Fourth
Edition, revised and enlarged. Crown 8vo.\ cloth. \Price{7s. 6d.}}
%
{The Author has here endeavoured to exhibit the subject in a simple
manner for the benefit of beginners, and at the same time to include in one
volume all that students usually require. In addition, therefore, to the
propositions which have always appeared in such treatises, he has introduced
the methods of \emph{abridged notation}, which are of more recent origin;
these methods, which are of a less elementary character than the rest of the
work, are placed \DPtypo{n}{in} separate chapters, and may be omitted by the student
at first.}
\DPPageSep{090.png}{86}%

\Book{\Title{A TREATISE ON THE DIFFERENTIAL CALCULUS.} With
numerous Examples. Fourth Edition. Crown 8vo.\ cloth. \Price{10s. 6d.}}
%
{The Author has endeavoured in the present work to exhibit a comprehensive
view of the Differential Calculus on the method of limits. In the
more elementary portions he has entered into considerable detail in the
explanations, with the hope that a reader who is without the assistance of a
tutor may be enabled to acquire a competent acquaintance with the subject.
The method adopted is that of Differential Coefficients. To the different
chapters are appended examples sufficiently numerous to render another
book unnecessary; these examples being mostly selected from College Examination
Papers.}

\Book{\Title{A TREATISE ON THE INTEGRAL CALCULUS AND ITS
APPLICATIONS.} With numerous Examples. Third Edition,
revised and enlarged. Crown 8vo.\ cloth. \Price{10s. 6d.}}
%
{This is designed as a work at once elementary and complete, adapted
for the use of beginners, and sufficient for the wants of advanced students.
In the selection of the propositions, and in the mode of establishing them,
it has been sought to exhibit the principles clearly, and to illustrate
all their most important results. The process of summation has been
repeatedly brought forward, with the view of securing the attention of
the student to the notions which form the true foundation of the Calculus
itself, as well as of its most valuable applications. Every attempt has been
made to explain those difficulties which usually perplex beginners, especially
with reference to the \emph{limits} of integrations. A new method has been adopted
in regard to the transformation of multiple integrals. The last chapter
deals with the Calculus of Variations. A large collection of exercises,
selected from College Examination Papers, has been appended to the several
chapters.}

\Book{\Title{EXAMPLES OF ANALYTICAL GEOMETRY OF THREE
DIMENSIONS.} Second Edition, revised. Crown 8vo.\ cloth\Add{.} \Price{4s.}}{}
\DPPageSep{091.png}{87}%

\Book{\Title{A TREATISE ON ANALYTICAL STATICS.} With numerous
Examples. Third Edition, revised and enlarged. Crown 8vo.\
cloth. \Price{10s. 6d.}}
%
{In this work on statics \(treating of the laws of the equilibrium of bodies\)
will be found all the propositions which usually appear in treatises on
Theoretical Statics. To the different chapters examples are appended,
which have been principally selected from University Examination Papers.
In the Third Edition many additions have been made, in order to illustrate
the application of the principles of the subject to the solution of
problems.}

\Book{\Author{Wilson (J. M.)}---\Title{ELEMENTARY GEOMETRY.} Angles,
Parallels, Triangles, Equivalent Figures, the Circle, and Proportion.
By \Name{J. M. Wilson}, M.A., Fellow of St.~John's College,
Cambridge, and Mathematical Master in Rugby School. Second
Edition. Extra fcap.~8vo. \Price{3s. 6d.}}
%
{The distinctive features of this work are intended to be the following.
The classification of Theorems according to their subjects; the separation
of Theorems and Problems; the use of hypothetical constructions; the
adoption of independent proofs where they are possible and simple; the
introduction of the terms \emph{locus, projection,~\&c.}; the importance given to
the notion of direction as the property of a straight line; the intermixing
of exercises, classified according to the methods adopted for their solution;
the diminution of the number of Theorems; the compression of proofs,
especially in the later parts of the book; the tacit, instead of the explicit,
reference to axioms; and the treatment of parallels.}

\Book{\Title{ELEMENTARY GEOMETRY\@. PART~II.} (separately). The
Circle and Proportion. By \Name{J.~M. Wilson}, M.A. Extra fcap.~8vo.\
\Price{2s. 6d.}}{}
\DPPageSep{092.png}{88}%

\Book{\Author{Wilson (W. P.)}---\Title{A TREATISE ON DYNAMICS.} By
\Name{W.~P. Wilson}, M.A., Fellow of St.~John's College, Cambridge,
and Professor of Mathematics in Queen's College, Belfast. 8vo.
\Price{9s. 6d.}}{}

\Book{\Author{Wolstenholme.}---\Title{A BOOK OF MATHEMATICAL
PROBLEMS}, on Subjects included in the Cambridge Course.
By \Name{Joseph Wolstenholme}, Fellow of Christ's College, sometime
Fellow of St.~John's College, and lately Lecturer in Mathematics
at Christ's College. Crown 8vo.\ cloth. \Price{8s. 6d.}}
%
{\textsc{Contents}: --- Geometry \(Euclid\) --- Algebra --- Plane Trigonometry --- Geometrical
Conic Sections --- Analytical Conic Sections --- Theory of Equations --- Differential
Calculus --- Integral Calculus --- Solid Geometry --- Statics --- Elementary
Dynamics --- Newton --- Dynamics of a Point --- Dynamics of
a Rigid Body --- Hydrostatics --- Geometrical Optics --- Spherical Trigonometry
and Plane Astronomy.}
\DPPageSep{093.png}{89}%


\Csection{Science.}

\Csubsection{ELEMENTARY CLASS-BOOKS.}

\noindent\First{The} importance of Science as an element of sound education
is now generally acknowledged; and accordingly it
is obtaining a prominent place in the ordinary course of
school instruction. It is the intention of the Publishers to
produce a complete series of Scientific Manuals, affording
full and accurate elementary information, conveyed in clear
and lucid English. The authors are well known as among
the foremost men of their several departments; and their
names form a ready guarantee for the high character of the
books. Subjoined is a list of those Manuals that have
already appeared, with a short account of each. Others
are in active preparation; and the whole will constitute a
standard series specially adapted to the requirements of beginners,
whether for private study or for school instruction.

\Book{\Banner{ASTRONOMY, by the Astronomer Royal.}
\Title{POPULAR ASTRONOMY}. With Illustrations. By \Name{G.~B.
Airy}, Astronomer Royal. Sixth and cheaper Edition. 18mo.\
cloth. \Price{4s. 6d.}}
%
{This work consists of six lectures, which are intended ``to explain to
intelligent persons the principles on which the instruments of an Observatory
are constructed \(omitting all details, so far as they are merely subsidiary\),
\DPPageSep{094.png}{90}%
and the principles on which the observations made with these
instruments are treated for deduction of the distances and weights of the
bodies of the Solar System, and of a few stars, omitting all minutiæ of
formulæ, and all troublesome details of calculation.'' The speciality of this
volume is the direct reference of every step to the Observatory, and the full
description of the methods and instruments of observation.}

\Book{\Banner{ASTRONOMY.}
\Title{MR.\ LOCKYER'S ELEMENTARY LESSONS IN ASTRONOMY.}
With Coloured Diagram of the Spectra of the Sun,
Stars, and Nebulæ, and numerous Illustrations. By \Name{J. Norman
Lockyer}, F.R.S. Seventh Thousand. 18mo. \Price{5s. 6d.}}
%
{The author has here aimed to give a connected view of the whole subject,
and to supply facts, and ideas founded on the facts, to serve as a basis for
subsequent study and discussion. The chapters treat of the Stars and
Nebulæ; the Sun; the Solar System; Apparent movements of the Heavenly
Bodies; the Measurement of Time; Light; the Telescope and Spectroscope;
Apparent Places of the Heavenly Bodies; the Real Distances and Dimensions;
Universal Gravitation. The most recent astronomical discoveries
are incorporated. Mr.~Lockyer's work supplements that of the Astronomer
Royal mentioned in the previous article.}

\Book{\Title{QUESTIONS ON LOCKYER'S ELEMENTARY LESSONS
IN ASTRONOMY.} For the use of Schools. By \Name{John Forbes-Robertson}.
18mo.\ cloth limp. \Price{1s. 6d.}}{}


\Book{\Banner{PHYSIOLOGY.}
\Title{PROFESSOR HUXLEY'S LESSONS IN ELEMENTARY
PHYSIOLOGY.} With numerous Illustrations. By \Name{T.~H.
Huxley}, F.R.S. Professor of Natural History in the Royal School
of Mines. Sixteenth Thousand. 18mo.\ cloth. \Price{4s. 6d.}%
\DPPageSep{095.png}{91}%
}
%
{This book describes and explains, in a series of graduated lessons, the
principles of Human Physiology; or the Structure and Functions of the
Human Body. The first lesson supplies a general view of the subject.
This is followed by sections on the Vascular or Veinous System, and the
Circulation; the Blood and the Lymph; Respiration; Sources of Loss
and of Gain to the Blood; the Function of Alimentation; Motion and
Locomotion; Sensations and Sensory Organs; the Organ of Sight; the
Coalescence of Sensations with one another and with other States of Consciousness;
the Nervous System and Innervation; Histology, or the
Minute Structure of the Tissues. A Table of Anatomical and Physiological
Constants is appended. The lessons are fully illustrated by
numerous engravings. The manual is primarily intended to serve as a
text-book for teachers and learners in boys' and girls' schools.}

%[** Awful spacing]
\Book{\Title{QUESTIONS ON HUXLEY'S PHYSIOLOGY FOR SCHOOLS.}
By \Name{T. Alcock}, M.D. 18mo. \Price{1s. 6d.}}
%
{These Questions were drawn up as aids to the instruction of a class \DPtypo{o}{of}
young people in Physiology.}


\Book{\Banner{BOTANY.}
\Title{PROFESSOR OLIVER'S LESSONS IN ELEMENTARY
BOTANY.} With nearly Two Hundred Illustrations. Tenth
Thousand. 18mo.\ cloth. \Price{4s. 6d.}}
%
{This book is designed to teach the Elements of Botany on Professor
Henslow's plan of selected Types and by the use of Schedules. The earlier
chapters, embracing the elements of Structural and Physiological Botany,
introduce us to the methodical study of the Ordinal Types. The concluding
chapters are entitled, ``How to dry Plants'' and ``How to
describe Plants.'' A valuable Glossary is appended to the volume. In
the preparation of this work free use has been made of the manuscript
materials of the late Professor Henslow.}
\DPPageSep{096.png}{92}%

\Book{\Banner{CHEMISTRY.}
\Title{PROFESSOR ROSCOE'S LESSONS IN ELEMENTARY
CHEMISTRY, INORGANIC AND ORGANIC.} By \Name{Henry
E. Roscoe}, F.R.S., Professor of Chemistry in Owens College,
Manchester. With numerous Illustrations and Chromo-Litho.\ of
the Solar Spectrum, and of the Alkalies and Alkaline Earths,
\emph{New Edition}. Twenty-sixth Thousand. 18mo.\ cloth. \Price{4s. 6d.}}
%
{It has been the endeavour of the author to arrange the most important
facts and principles of Modern Chemistry in a plain but concise and
scientific form, suited to the present requirements of elementary instruction.
For the purpose of facilitating the attainment of exactitude in the knowledge
of the subject, a series of exercises and questions upon the lessons have been
added. The metric system of weights and measures, and the centigrade
thermometric scale, are used throughout the work. The new Edition,
besides new wood-cuts, contains many additions and improvements, and
includes the most important of the latest discoveries.}

\Book{\Banner{POLITICAL ECONOMY.}
\Title{POLITICAL ECONOMY FOR BEGINNERS.} By \Name{Millicent
G. Fawcett}. 18mo. \Price{2s. 6d.}}
%
{The following pages have been written mainly with the hope that a short
and elementary book might help to make Political Economy a more popular
study in boys' and girls' schools. In order to adapt the book especially for
school use, questions have been added at the end of each chapter.}

\Book{\Banner{LOGIC.}
\Title{ELEMENTARY LESSONS IN LOGIC}; Deductive and Inductive,
with copious Questions and Examples, and a Vocabulary of
Logical Terms. By \Name{W. Stanley Jevons}, M.A., Professor of
Logic in Owens College, Manchester. 18mo. \Price{3s. 6d.}}
%
{In preparing these Lessons the author has attempted to show that Logic,
even in its traditional form, can be made a highly useful subject of study,
and a powerful means of mental exercise. With this view he has avoided
the use of superfluous technical terms, and has abstained from entering
\DPPageSep{097.png}{93}%
into questions of a purely speculative or metaphysical character. For the
puerile illustrations too often found in works on Logic, examples drawn
from the distinct objects and ideas treated in the natural and experimental
sciences have been generally substituted. At the end of almost every
Lesson will be found references to the works in which the student will most
profitably continue his reading of the subject treated, so that this little
volume may serve as a guide to a more extended course of study.}

\Book{\Banner{PHYSICS.}
\Title{LESSONS IN ELEMENTARY PHYSICS.} By \Name{Balfour
Stewart}, F.R.S., Professor of Natural Philosophy in Owens
College, Manchester. With numerous Illustrations and Chromoliths
of the Spectra of the Sun, Stars, and Nebulæ. 18mo. \Price{4s. 6d.}}
%
{A description, in an elementary manner, of the most important of those
laws which regulate the phenomena of nature. The active agents, heat,
light, electricity, etc., are regarded as varieties of energy, and the work is
so arranged that their relation to one another, looked at in this light, and
the paramount importance of the laws of energy are clearly brought out.
The volume contains all the necessary illustrations, and a plate representing
the Spectra of Sun, Stars, and Nebulæ\DPtypo{,}{} forms a frontispiece.}

\Csubsection{MANUALS FOR STUDENTS.}

\Book{\Author{Flower (W. H.)}---\Title{AN INTRODUCTION TO THE OSTEOLOGY
OF THE MAMMALIA.} Being the substance of
the Course of Lectures delivered at the Royal College of Surgeons
of England in \Date{1870}. By \Name{W.~H. Flower}, F.R.S., F.R.C.S.,
Hunterian Professor of Comparative Anatomy and Physiology.
With numerous Illustrations. Globe 8vo. \Price{7s. 6d.}}
%
{Although the present work contains the substance of a Course of Lectures,
the form has been changed, so as the better to adapt it as a handbook for
students. Theoretical views have been almost entirely excluded; and while
\DPPageSep{098.png}{94}%
it is impossible in a scientific treatise to avoid the employment of technical
terms, it has been the author's endeavour to use no more than absolutely
necessary, and to exercise due care in selecting only those that seem most
appropriate, or which have received the sanction of general adoption. With
a very few exceptions the illustrations have been drawn expressly for this
work from specimens in the Museum of the Royal College of Surgeons.}

\Book{\Author{Hooker (Dr.)}---\Title{THE STUDENT'S FLORA OF THE
BRITISH ISLANDS.} By \Name{J.~D. Hooker}, C.B., F.R.S.,
M.D., D.C.L., Director of the Royal Gardens, Kew. Globe
8vo. \Price{10s. 6d.}}
%
{The object of this work is to supply students and field-botanists with a
fuller account of the Plants of the British Islands than the manuals
hitherto in use aim at giving. The Ordinal, Generic, and Specific
characters have been re-written, and are to a great extent original, and
drawn from living or dried specimens, or both.}

\Book{\Author{Oliver (Professor).}---\Title{FIRST BOOK OF INDIAN BOTANY.}
By \Name{Daniel Oliver}, F.R.S., F.L.S., Keeper of the Herbarium
and Library of the Royal Gardens, Kew, and Professor of Botany
in University College, London. With numerous Illustrations.
Extra fcap.~8vo. \Price{6s. 6d.}}
%
{This manual is, in substance, the author's ``Lessons in Elementary
Botany,'' adapted for use in India. In preparing it he has had in view
the want, often felt, of some handy résumé of Indian Botany, which might
be serviceable not only to residents of India, but also to any one about to
proceed thither, desirous of getting some preliminary idea of the Botany of
that country.}
\begin{center}
  \textit{Other volumes of these Manuals will follow.}
\end{center}
\DPPageSep{099.png}{95}%

\Book{\Author{Cooke (Josiah P., Jun.)}---\Title{FIRST PRINCIPLES OF
CHEMICAL PHILOSOPHY.} By \Name{Josiah P. Cooke}, Jun.,
Ervine Professor of Chemistry and Mineralogy in Harvard College.
Crown 8vo. \Price{12s.}}
%
{The object of the author in this book is to present the philosophy of
Chemistry in such a form that it can be made with profit the subject of
College recitations, and furnish the teacher with the means of testing the
student's faithfulness and ability. With this view the subject has been
developed in a logical order, and the principles of the science are taught
independently of the experimental evidence on which they rest.}

\Book{\Author{Johnson (S. W., M.A.)}---\Title{HOW CROPS GROW}: A
Treatise on the Chemical Composition, Structure, and Life of the
Plant, for Agricultural Students. By \Name{S.~W. Johnson}, M.A.,
Professor of Analytical and Agricultural Chemistry in Yale College.
With Illustrations and Tables of Analyses. Revised, with Numerous
Additions, and adapted for English use by \Name{A.~H. Church},
M.A. and \Name{W.~T. Dyer}, B.A., Professors at the Royal Agricultural
College, Cirencester. Crown 8vo. \Price{8s. 6d.}}
%
{In order that this book may be complete in itself, so far as its special scope
is concerned, not only have the rudiments of Chemistry and structural
Botany been introduced, but a series of Experiments has been described, by
which the student, who has access to chemical apparatus and tests, may
become conversant with the most salient properties of the elements, and of
those of their chief natural compounds, which constitute the food or the
materials of plants.

It has also been attempted to adapt the work in form and contents to the
wants of the class-room by a strictly systematic arrangement of topics, and
by division of the matter into convenient paragraphs.}
\DPPageSep{100.png}{96}%

\Book{\Author{Roscoe (H. E.)}---\Title{SPECTRUM ANALYSIS.} Six Lectures,
with Appendices, Engravings, Maps, and Chromolithographs.
By \Name{H.~E. Roscoe}, F.R.S., Professor of Chemistry in Owens
College, Manchester. Royal 8vo. \Price{21s.}}
%
{A Second Edition of these popular Lectures, containing all the most
recent discoveries and several additional Illustrations.

``The lectures themselves furnish a most admirable elementary treatise
on the subject, whilst by the insertion in appendices to each lecture of
extracts from the most important published memoirs, the author has
rendered it equally valuable as a text book for advanced students.''---%
\Cite{Westminster Review.}}

\Book{\Author{Thorpe (T. E.)}---\Title{A SERIES OF CHEMICAL PROBLEMS},
for use in Colleges and Schools. Adapted for the preparation of
Students for the Government, Science, and Society of Arts Examinations.
With a Preface by Professor \Name{Roscoe}. 18mo.\
cloth. \Price{1s.}}
%
{In the Preface Dr.~Roscoe says---``My experience has led me to feel more
and more strongly that by no method can accuracy in a knowledge of
chemistry be more surely secured than by attention to the working of well-selected
problems, and Dr.~Thorpe's thorough acquaintance with the wants
of the student is a sufficient guarantee that this selection has been carefully
made. I intend largely to use these questions in my own classes, and I can
confidently recommend them to all teachers and students of the science.''}

\Book{\Author{Wurtz.}---\Title{A HISTORY OF CHEMICAL THEORY}, from the
Age of Lavoisier down to the present time. By \Name{Ad. Wurtz}.
Translated by HENRY WATTS, F.R.S. Crown 8vo. \Price{6s.}}{}
\DPPageSep{101.png}{97}%


\Csection{Miscellaneous.}

\Book{\Author{Abbott.}---\Title{A SHAKESPEARIAN GRAMMAR.} An Attempt to
illustrate some of the Differences between Elizabethan and Modern
English. By the Rev.\ \Name{E.~A. Abbott}, M.A., Head Master of the
City of London School. For the Use of Schools. New and Enlarged
Edition. Extra fcap.~8vo. \Price{6s.}}
%
{The object of this work is to furnish students of Shakespeare and Bacon
with a short systematic account of some points of difference between Elizabethan
syntax and our own. A section on Prosody is added, and Notes
and Questions.

The success which has attended the First and Second Editions of the
``\textsc{Shakespearian Grammar},'' and the demand for a Third Edition
within a year of the publication of the First, has encouraged the Author to
endeavour to make the work somewhat more useful, and to render it, as
far as possible, a complete book of reference for all difficulties of Shakespearian
syntax or prosody. For this purpose the whole of Shakespeare has
been re-read, and an attempt has been made to include within this Edition
the explanation of every idiomatic difficulty \(where the text is not confessedly
corrupt\) that comes within the province of a grammar as distinct
from a glossary.

The great object being to make a useful book of reference for students,
and especially for classes in schools, several Plays have been indexed so
fully that with the aid of a glossary and historical notes the references will
serve for a complete commentary.}
\DPPageSep{102.png}{98}%

\Book{\Title{ATLAS OF EUROPE\@. GLOBE EDITION.} Uniform in size
with Macmillan's Globe Series, containing 45 Coloured Maps, on
a uniform scale and projection: with Plans of London and Paris,
and a copious Index. Strongly bound in half-morocco, with flexible
back, \Price{9s.}}
%
{This Atlas includes all the countries of Europe in a series of 48 Maps,
drawn on the same scale, with an Alphabetical Index to the situation of
more than ten thousand places; and the relation of the various maps and
countries to each other is defined in a general Key-map. The identity of
scale in all the maps facilitates the comparison of extent and distance, and
conveys a just impression of the magnitude of different countries. The
size suffices to show the provincial divisions, the railways and main roads,
the principal rivers and mountain ranges. ``This Atlas,'' writes the
\emph{British Quarterly}, ``will be an invaluable boon for the school, the desk, or
the traveller's portmanteau.''}

\Book{\Author{Bates \& Lockyer.}---\Title{A CLASS-BOOK OF GEOGRAPHY.}
Adapted to the recent Programme of the Royal Geographical
Society. By \Name{H.~W. Bates}, Assistant Secretary to the Royal
Geographical Society, and \Name{J.~N. Lockyer}, F.R.A.S.
\InPress}{}

\Book{\Title{CAMEOS FROM ENGLISH HISTORY.} From Rollo to Edward~II\@.
By the Author of ``The Heir of Redclyffe.'' Extra fcap.~8vo.
Second Edition, enlarged. \Price{5s.} \\
\textsc{A Second Series} nearly ready.}
%
{The endeavour has not been to chronicle facts, but to put together a series
of pictures of persons and events, so as to arrest the attention, and give
some individuality and distinctness to the recollection, by gathering together
details at the most memorable moments. The ``Cameos'' are intended as
a book for young people just beyond the elementary histories of England,
and able to enter in some degree into the real spirit of events, and to be
struck with characters and scenes presented in some relief. ``Instead of
dry details,'' says the \emph{Nonconformist}, ``we have living pictures, faithful,
vivid, and striking.''}
\DPPageSep{103.png}{99}%

\Book{\Author{Delamotte.}---\Title{A BEGINNER'S DRAWING BOOK.} By \Name{P.~H.
Delamotte}, F.S.A. Progressively arranged, with upwards of
Fifty Plates. Crown 8vo.\ Stiff covers. \Price{2s. 6d.}}
%
{This work is intended to give such instruction to Beginners in Drawing,
and to place before them copies so easy, that they may not find any obstacle
in making the first step. Thenceforward the lessons are gradually
progressive. Mechanical improvements too have lent their aid. The whole
of the Plates have been engraved by a new process, by means of which a
varying depth of tone---up to the present time the distinguishing characteristic
of pencil drawing---has been imparted to \DPchg{woodcuts}{wood-cuts}.}

\Book{\Author{D'Oursy and Feillet.}---\Title{A FRENCH GRAMMAR AT
SIGHT}, on an entirely new method. By \Name{A. D'Oursy} and
\Name{A. Feillet}. Especially adapted for Pupils preparing for Examination.
Fcap.~8vo.\ cloth extra. \Price{2s. 6d.}}
%
{The method followed in this volume consists in presenting the grammar
as much as possible by synoptical tables, which, striking the eye at once, and
following throughout the same order---\emph{``used---not used;'' ``changes---does
not change''}---are easily remembered. The parsing tables will enable
the pupil to parse easily from the beginning. The exercises consist of
translations from French into English, and from English into French;
and of a number of grammatical questions.}

\Book{\Title{EUROPEAN HISTORY.} Narrated in a Series of Historical Selections
from the Best Authorities. Edited and arranged by \Name{E.~M.
Sewell} and \Name{C.~M. Yonge}. First Series, crown 8vo. \Price{6s.}
Second Series, \Date{1088}--\Date{1228}. Crown 8vo. \Price{6s.}}
%
{When young children have acquired the outlines of History from abridgments
and catechisms, and it becomes desirable to give a more enlarged
view of the subject, in order to render it really useful and interesting, a
difficulty often arises as to the choice of books. Two courses are open, either
to take a general and consequently dry history of facts, such as Russel's
Modern Europe, or to choose some work treating of a particular period or
subject, such as the works of Macaulay and Froude. The former course
\DPPageSep{104.png}{100}%
usually renders history uninteresting; the latter is unsatisfactory, because
it is not sufficiently comprehensive. To remedy this difficulty selections,
continuous and chronological, have, in the present volume, been taken from
the larger works of Freeman, Milman, Palgrave, and others, which may
serve as distinct landmarks of historical reading. ``We know of scarcely
anything,'' says the \emph{Guardian}, of this volume, ``which is so likely to raise
to a higher level the average standard of English education.''}

\Book{\Author{Freeman (Edward A.)}---\Title{OLD-ENGLISH HISTORY.}
By \Name{Edward A. Freeman}, D.C.L., late Fellow of Trinity
College, Oxford. With Five Coloured Maps. New Edition.
Extra fcap.~8vo.\ half-bound. \Price{6s.}}
%
{``Its object is to show that clear, accurate, and scientific views of history,
or indeed of any subject, may be easily given to children from the very first\ldots.
I have, I hope, shown that it is perfectly easy to teach children,
from the very first, to distinguish true history alike from legend and from
wilful invention, and also to understand the nature of historical authorities
and to weigh one statement against another\ldots. I have throughout
striven to connect the history of England with the general history of
civilized Europe, and I have especially tried to make the book serve as an
incentive to a more accurate study of historical geography.''---\Cite{Preface}.}

\Book{\Author{Helfenstein (James).}---\Title{A COMPARATIVE GRAMMAR
OF THE TEUTONIC LANGUAGES.} Being at the same
time a Historical Grammar of the English Language, and comprising
Gothic, Anglo-Saxon, Early English, Modern English, Icelandic
(Old Norse), Danish, Swedish, Old High German, Middle High
German, Modern German, Old Saxon, Old Frisian, and Dutch.
By \Name{James Helfenstein}, Ph.D. 8vo. \Price{18s.}}
%
{This work traces the different stages of development through which
the various Teutonic languages have passed, and the laws which have
regulated their growth. The reader is thus enabled to study the relation
which these languages bear to one another, and to the English language in
particular, to which special attention is devoted throughout. In the
chapters on Ancient and Middle Teutonic languages no grammatical form
\DPPageSep{105.png}{101}%
is omitted the knowledge of which is required for the study of ancient
literature, whether Gothic or Anglo-Saxon or Early English. To each
chapter is prefixed a sketch showing the relation of the Teutonic to the
cognate languages, Greek, Latin, and Sanskrit. Those who have mastered
the book will be in a position to proceed with intelligence to the more
elaborate works of Grimm, Bopp, Pott, Schleicher, and others.}

\Book{\Author{Hole.}---\Title{A GENEALOGICAL STEMMA OF THE KINGS OF
ENGLAND AND FRANCE.} By the Rev.\ \Name{C. Hole}. On
Sheet. \Price{1s.}}
%
{The different families are printed in distinguishing colours, thus
facilitating reference.}

\Book{\Title{A BRIEF BIOGRAPHICAL DICTIONARY.} Compiled and
Arranged by \Name{Charles Hole}, M.A., Trinity College, Cambridge.
Second Edition, 18mo.\ neatly and strongly bound in cloth. \Price{4s. 6d.}}
%
{The inquiry is frequently made concerning an eminent man, when did
he live, or for what was he celebrated, or what biographies have we about
him? Such information is concisely supplied in this Dictionary. It contains
more than {\upshape18,000}~names. Extreme care has been bestowed on the verification
of the dates, and thus numerous errors current in previous works have
been corrected. Its size adapts it for the desk, portmanteau, or pocket.

``An invaluable addition to our manuals of reference, and from its
moderate price cannot fail to become as popular as it is useful.''---\Cite{Times}.}

\Book{\Author{Jephson.}---\Title{SHAKESPEARE'S ``TEMPEST.''} With Glossarial
and Explanatory Notes. By the Rev.\ \Name{J.~M. Jephson}. 18mo.
\Price{1s. 6d.}}
%
{It is important to find some substitute for classical study, and it is
believed that such a substitute may be found in the Plays of Shakespeare.
Each sentence of Shakespeare becomes, like a sentence in Thucydides or
Cicero, a lesson in the origin and derivation of words, and in the fundamental
rules of grammatical construction. On this principle the present
edition of the ``Tempest'' has been prepared. The text is taken from the
``Cambridge Shakespeare.''}
\DPPageSep{106.png}{102}%

\Book{\Author{M`Cosh (Rev.\ Principal).}---\Title{THE LAWS OF DISCURSIVE
THOUGHT.} Being a Text-Book of Formal Logic. By
\Name{James M`Cosh}, D.D., LL.D. 8vo. \Price{5s.}}
%
{In this treatise the \emph{Notion} \(with the Term and the Relation of Thought
to Language,\) will be found to occupy a larger relative place than in any
logical work written since the time of the famous ``Art of Thinking.''}

\Book{\Author{Oppen.}---\Title{FRENCH READER.} For the Use of Colleges and
Schools. Containing a graduated Selection from modern Authors
in Prose and Verse; and copious Notes, chiefly Etymological. By
\Name{Edward A. Oppen}. Fcap.~8vo.\ cloth. \Price{4s. 6d.}}
%
{This is a Selection from the best modern authors of France. Its distinctive
feature consists in its etymological notes, connecting French with
the classical and modern languages, including the Celtic. This subject
has hitherto been little discussed even by the best-educated teachers.}

\Book{\Title{A SHILLING BOOK OF GOLDEN DEEDS.} A Reading Book
for Schools and General Readers. By the Author of ``The Heir
of Redclyffe.'' 18mo.\ cloth.}
%
{A record of some of the good and great deeds of all time, abridged from
the larger work of the same author in the Golden Treasury Series.}

\Book{\Author{Sonnenschein and Meiklejohn.}---\Title{THE ENGLISH
METHOD OF TEACHING TO READ.} By \Name{A. Sonnenschein}
and \Name{J.~M.~D. Meiklejohn}, M.A. Fcap.~8vo.

{\centering \textsc{Comprising.}\hspace*{\parindent}\par}

\SubBook{\Title{The Nursery Book}, containing all the Two-Letter Words in the
Language. \Price{1d.}}

\SubBook{\Title{The First Course}, consisting of Short Vowels with Single
Consonants. \Price{3d.}}

\SubBook{\Title{The Second Course}, with Combinations and Bridges, consisting
of Short Vowels with Double Consonants. \Price{4d.}}

\SubBook{\Title{The Third And Fourth Courses}, consisting of Long
Vowels, and all the Double Vowels in the Language. \Price{6d.}}\par%
\DPPageSep{107.png}{103}%
}
%
{A Series of Books in which an attempt is made to place the process of
learning to read English on a scientific basis. This has been done by
separating the perfectly regular parts of the language from the irregular,
and by giving the regular parts to the learner in the exact order of their
difficulty. The child begins with the smallest possible element, and adds to
that element one letter---in only one of its functions---at one time. Thus
the sequence is natural and complete.}

\Book{\Author{Vaughan (C.~M.)}---\Title{A SHILLING BOOK OF WORDS
FROM THE POETS.} By \Name{C.~M. Vaughan}. 18mo.\ cloth.}
%
{It has been felt of late years that the children of our parochial schools,
and those classes of our countrymen which they commonly represent, are
capable of being interested, and therefore benefited also, by something higher
in the scale of poetical composition than those brief and somewhat puerile
fragments to which their knowledge was formerly restricted. An attempt
has here been made to supply the want by forming a selection at once
various and unambitious; healthy in tone, just in sentiment, elevating in
thought, and beautiful in expression.}

\Book{\Author{Thring.}---Works by \Name{Edward Thring}, M.A., Head Master of
Uppingham.}{}

\Book{\Title{THE ELEMENTS OF GRAMMAR TAUGHT IN ENGLISH},
with Questions. Fourth Edition. 18mo. \Price{2s.}}
%
{This little work is chiefly intended for teachers and learners. It took its
rise from questionings in National Schools, and the whole of the first part
is merely the writing out in order the answers to questions which have been
used already with success. A chapter on Learning Language is especially
addressed to teachers.}

\Book{\Title{THE CHILD'S GRAMMAR.} Being the Substance of ``The
Elements of Grammar taught in English,'' adapted for the Use of
Junior Classes. A New Edition. 18mo. \Price{1s.}}{}
\DPPageSep{108.png}{104}%

\Book{\Title{SCHOOL SONGS.} A Collection of Songs for Schools. With the
Music arranged for four Voices. Edited by the Rev.\ \Name{E. Thring}
and \Name{H. Riccius}. Folio. \Price{7s. 6d.}}
%
{There is a tendency in schools to stereotype the forms of life. Any genial
solvent is valuable. Games do much; but games do not penetrate to
domestic life, and are much limited by age. Music supplies the want.
The collection includes the ``Agnus Dei,'' Tennyson's ``Light Brigade,''
Macaulay's ``Ivry,''~\&c.\ among other pieces.}

\Book{\Author{Trench (Archbishop).}---\Title{HOUSEHOLD BOOK OF ENGLISH
POETRY.} Selected and Arranged, with Notes, by
\Name{R.~C. Trench}, D.D., Archbishop of Dublin. Extra fcap.~8vo.
\Price{5s. 6d.} Second \DPtypo{Edtion}{Edition}.}
%
{This volume is called a ``Household Book,'' by this name implying that
it is a book for all---that there is nothing in it to prevent it from being
confidently placed in the hands of every member of the household. Specimens
of all classes of poetry are given, including selections from living
authors. The Editor has aimed to produce a book ``which the emigrant,
finding room for little not absolutely necessary, might yet find room for it
in his trunk, and the traveller in his knapsack, and that on some narrow
shelves where there are few books this might be one.''

``The Archbishop has conferred in this delightful volume an important
gift on the whole English-speaking population of the world.''---\Cite{Pall Mall
Gazette}.}

\Book{\Author{Yonge (Charlotte M.).}---\Title{A PARALLEL HISTORY OF
FRANCE AND ENGLAND}: consisting of Outlines and Dates.
By \Name{Charlotte M. Yonge}, Author of ``The Heir of Redclyffe,''
``Cameos of English History,''~\&c.,~\&c. Oblong 4to. \Price{3s. 6d.}}
%
{This tabular history has been drawn up to supply a want felt by many
teachers of some means of making their pupils realize what events in the
two countries were contemporary. A skeleton narrative has been constructed
of the chief transactions in either country, placing a column
between for what affected both alike, by which means it is hoped that young
people may be assisted in grasping the mutual relation of events.}
\DPPageSep{109.png}{105}%


\Csection{Divinity.}

\Book{\Author{Abbott (Rev.\ E. A.)}---\Title{BIBLE LESSONS.} By the Rev.\
\Name{E.~A. Abbott}, M.A., Head Master of the City of London
School. Second Edition, crown 8vo. \Price{4s. 6d.}}
%
{This book is written in the form of dialogues carried on between a
teacher and pupil, and its main object is to make the scholar think for
himself. The great bulk of the dialogues represents in the spirit, and
often in the words, the religious instruction which the author has been
in the habit of giving to the Fifth and Sixth Forms of the City of London
School.}

\Book{\Author{Cheyne (T. K.)}---\Title{THE BOOK OF ISAIAH CHRONOLOGICALLY
ARRANGED.} An Amended Version, with
Historical and Critical Introductions and Explanatory Notes. By
\Name{T.~K. Cheyne}, M.A., fellow of Balliol College, Oxford.
Crown 8vo. \Price{7s. 6d.}}
%
{The object of this edition is simply to restore the probable meaning of
Isaiah, so far as this can be expressed in modern English. The basis of
the version is the revised translation of \Date{1611}, but no scruple has been felt
in introducing alterations, wherever the true sense of the prophecies
appeared to require it.}

\Book{\Author{Eastwood.}---\Title{THE BIBLE WORD-BOOK.} A Glossary of
Old English Bible Words. By \Name{J. Eastwood}, M.A., of St.~John's
College, and \Name{W. Aldis Wright}, M.A., Trinity College, Cambridge.
18mo. \Price{5s. 6d.}%
\DPPageSep{110.png}{106}%
}
%
{It is the object of this Glossary to explain and illustrate all such words,
phrases, and constructions, in the Authorized Version of the Old and New
Testaments and the Apocrypha, and in the Book of Common Prayer, as
are either obsolete or archaic. Full explanations are supplied, and these
illustrated by numerous citations from the elder writers. An index of
editions quoted is appended. Apart from its immediate subject, this work
serves to illustrate a well-marked period in the history of the English
language. It is thus of distinct philological value.}

\Book{\Title{GOLDEN TREASURY PSALTER.} Students' Edition. Being an
Edition of ``The Psalms Chronologically Arranged, by Four
Friends,'' with briefer Notes. 18mo. \Price{3s. 6d.}}
%
{In making this abridgment of ``The Psalms Chronologically Arranged,''
the editors have endeavoured to meet the requirements of readers of a
different class from those for whom the larger edition was intended. Some
who found the large book useful for private reading, have asked for an
edition of a smaller size and at a lower price, for family use, while at the
same time some Teachers in Public Schools have suggested that it would be
convenient for them to have a simpler book, which they could put into the
hands of younger pupils.}

\Book{\Author{Hardwick.}---\Title{A HISTORY OF THE CHRISTIAN CHURCH.}
Middle Age. From Gregory the Great to the Excommunication
of Luther. By \Name{Archdeacon Hardwick}. Edited by \Name{Francis
Procter}, M.A. With Four Maps constructed for this work by
\Name{A. Keith Johnston}. Second Edition. Crown 8vo. \Price{10s. 6d.}}
%
{The ground-plan of this treatise coincides in many points with one
adopted at the close of the last century in the colossal work of Schröckh, and
since that time by others of his thoughtful countrymen; but in arranging
the materials a very different course has frequently been pursued. With
regard to the opinions of the author, he is willing to avow distinctly that he
has construed history with the specific prepossessions of an Englishman and
a member of the English Church. The reader is constantly referred to
the authorities, both original and critical, on which the statements are
founded.}
\DPPageSep{111.png}{107}%

\Book{\Title{A HISTORY OF THE CHRISTIAN CHURCH DURING THE
REFORMATION.} By \Name{Archdeacon Hardwick}. Revised by
\Name{Francis Procter}, M.A. Second Edition. Crown 8vo. \Price{10s. 6d.}}
%
{This volume is intended as a sequel and companion to the ``History of
the Christian Church during the Middle Age.'' The author's earnest
wish has been to give the reader a trustworthy version of those stirring
incidents which mark the Reformation period, without relinquishing his
former claim to characterise peculiar systems, persons, and events according
to the shades and colours they assume, when contemplated from an English
point of view, and by a member of the Church of England.}

\Book{\Author{Maclear.}---Works by the Rev.\ \Name{G.~F. Maclear}, B.D., Head
Master of King's College School, and Preacher at the Temple
Church.}{}

\Book{\Title{A CLASS-BOOK OF OLD TESTAMENT HISTORY.} Fifth
Edition, with Four Maps. 18mo.\ cloth. \Price{4s. 6d.}}
%
{This volume forms a Class-book of Old Testament History from the
earliest times to those of Ezra and Nehemiah. In its preparation the
most recent authorities have been consulted, and wherever it has appeared
useful, Notes have been subjoined illustrative of the Text, and, for the sake
of more advanced students, references added to larger works. The Index
has been so arranged as to form a concise dictionary of the persons and
places mentioned in the course of the narrative; while the maps, which have
been prepared with considerable care at Stanford's Geographical Establishment,
will, it is hoped, materially add to the value and usefulness of the
Book.}

\Book{\Title{A CLASS-BOOK OF NEW TESTAMENT HISTORY}, including
the Connexion of the Old and New Testament. With Four Maps.
Third Edition. 18mo.\ cloth. \Price{5s. 6d.}}
%
{A sequel to the author's Class-book of Old Testament History, continuing
the narrative from the point at which it there ends, and carrying it on to
the close of St.~Paul's second imprisonment at Rome. In its preparation,
as in that of the former volume, the most recent and trustworthy authorities
\DPPageSep{112.png}{108}%
have been consulted, notes subjoined, and references to larger works added.
It is thus hoped that it may prove at once an useful class-book and a
convenient companion to the study of the Greek Testament.}

\Book{\Title{A SHILLING BOOK OF OLD TESTAMENT HISTORY}, for
National and Elementary Schools. With Map. 18mo.\ cloth.}{}

\Book{\Title{A SHILLING BOOK OF NEW TESTAMENT HISTORY}, for
National and Elementary Schools. With Map. 18mo.\ cloth.}
%
{These works have been carefully abridged from the author's larger
manuals.}

\Book{\Title{CLASS-BOOK OF THE CATECHISM OF THE CHURCH OF
ENGLAND.} Second Edition. 18mo.\ cloth. \Price{2s. 6d.}}
%
{This may be regarded as a sequel to the Class-books of Old and New
Testament History. Like them, it is furnished with notes and references
to larger works, and it is hoped that it may be found, especially in the
higher forms of our Public Schools, to supply a suitable manual of
instruction in the chief doctrines of the English Church, and a useful
help in the preparation of Candidates for Confirmation.}

\Book{\Title{A FIRST CLASS-BOOK OF THE CATECHISM OF THE
CHURCH OF ENGLAND}, with Scripture Proofs, for Junior
Classes and Schools. 18mo. \Price{6d.}}{}

\Book{\Title{THE ORDER OF CONFIRMATION.} A Sequel to the \DPtypo{Class
Book}{Class-book} of the Catechism. For the use of Candidates for Confirmation.
With Prayers and Collects. 18mo. \Price{3d.}}{}

\Book{\Author{Maurice.}---\Title{THE LORD'S PRAYER, THE CREED, AND
THE COMMANDMENTS.} A Manual for Parents and Schoolmasters.
By the Rev.\ \Name{F.~D. Maurice}. To which is added the
Order of the Scriptures. 18mo. \Price{1s.}}{}

\Book{\Author{Procter.}---\Title{A HISTORY OF THE BOOK OF COMMON
PRAYER}, with a Rationale of its Offices. By \Name{Francis Procter},
M.A. Ninth Edition, revised and enlarged. Crown 8vo.
\Price{10s. 6d.}%
\DPPageSep{113.png}{109}%
}
%
{In the course of the last twenty years the whole question of Liturgical
knowledge has been reopened with great learning and accurate research;
and it is mainly with the view of epitomizing extensive publications, and
correcting the errors and misconceptions which had obtained currency,
that the present volume has been put together.}

\Book{\Author{Procter and Maclear.}---\Title{AN ELEMENTARY INTRODUCTION
TO THE BOOK OF COMMON PRAYER.}
Re-arranged and supplemented by an Explanation of the Morning
and Evening Prayer and the Litany. By the Rev.\ \Name{F. Procter}
and the Rev.\ \Name{G.~F. Maclear}. Fourth Edition. 18mo. \Price{2s. 6d.}}
%
{As in the other Class-books of the series, notes have also been subjoined,
and references given to larger works, and it is hoped that the volume will
be found adapted for use in the higher forms by our Public Schools, and a
suitable manual for those preparing for the Oxford and Cambridge local
examinations. This new Edition has been considerably altered, and
several important additions have been made. Besides a re-arrangement
of the work generally, the Historical Portion has been supplemented by an
Explanation of the Morning and Evening Prayer and of the Litany.}

\Book{\Title{PSALMS OF DAVID CHRONOLOGICALLY ARRANGED.}
\DPtypo{BY \Name{FOUR FRIENDS}}{By \Name{Four Friends}}. An Amended Version, with Historical
Introduction and Explanatory Notes. Second Edition, with
Additions and Corrections. Crown 8vo. \Price{8s. 6d.}}
%
{To restore the Psalter as far as possible to the order in which the Psalms
were written,---to give the division of each Psalm into strophes, of each
strophe into the lines which composed it,---to amend the errors of translation,
is the object of the present Edition. Professor Ewald's works, especially
that on the Psalms, have been extensively consulted.

This book has been used with satisfaction by masters for private work in
higher classes in schools.}

\Book{\Author{Ramsay.}---\Title{THE CATECHISER'S MANUAL}; or, the Church
Catechism illustrated and explained, for the use of Clergymen,
Schoolmasters, and Teachers. By the Rev.\ \Name{Arthur Ramsay},
M.A. Second Edition. 18mo. \Price{1s. 6d.}}
%
{A clear explanation of the Catechism, by way of Question and Answer.}
\DPPageSep{114.png}{110}%

\Book{\Author{Simpson.}---\Title{AN EPITOME OF THE HISTORY OF THE
CHRISTIAN CHURCH.} By \Name{William Simpson}, M.A.
Fifth Edition. Fcap.~8vo. \Price{3s. 6d.}}
%
{A compendious summary of Church History.}

\Book{\Author{Swainson.}---\Title{A HANDBOOK to BUTLER'S ANALOGY.} By
\Name{C.~A. Swainson}, D.D., Canon of Chichester. Crown 8vo. \Price{1s. 6d.}}
%
{This manual is designed to serve as a handbook or road-book to the
Student in reading the Analogy, to give the Student a sketch or outline map
of the country on which he is entering, and to point out to him matters of
interest as he passes along.}

\Book{\Author{Westcott.}---\Title{A GENERAL SURVEY OF THE HISTORY
OF THE CANON OF THE NEW TESTAMENT DURING
THE FIRST FOUR CENTURIES.} By \Name{Brooke Foss Westcott},
B.D., Canon of Peterborough. Third Edition, revised.
Crown 8vo. \Price{10s. 6d.}}
%
{The Author has endeavoured to connect the history of the New Testament
Canon with the growth and consolidation of the Church, and to point out
the relation existing between the amount of evidence for the authenticity of
its component parts, and the whole mass of Christian literature. Such a
method of inquiry will convey both the truest notion of the connexion of the
written Word with the living Body of Christ, and the surest conviction of
its divine authority.

Of this work the \emph{Saturday Review} writes: ``Theological students, and
not they only, but the general public, owe a deep debt of gratitude to
Mr.~Westcott for bringing this subject fairly before them in this candid and
comprehensive essay\ldots. As a theological work it is at once perfectly fair
and impartial, and imbued with a thoroughly religious spirit; and as a
manual it exhibits, in a lucid form and in a narrow compass, the results
of extensive research and accurate thought. We cordially recommend it.''}
\DPPageSep{115.png}{111}%

\Book{\Title{INTRODUCTION TO THE STUDY OF THE FOUR GOSPELS.}
By \Name{Brooke Foss Westcott}, B.D. Third Edition. Crown 8vo.
\Price{10s. 6d.}}
%
{This book is intended to be an introduction to the Study of the Gospels.
The author has made it a point carefully to study the researches of the great
writers, and consciously to neglect none. There is an elaborate discussion
appended ``On the Primitive Doctrine of Inspiration.''}

\Book{\Title{A GENERAL VIEW OF THE HISTORY OF THE ENGLISH
BIBLE.} By \Name{Brooke Foss Westcott}, B.D. Crown 8vo. \Price{10s. 6d.}}
%
{``The first trustworthy account we have had of that unique and marvellous
monument of the piety of our ancestors.''---\Cite{Daily News}.}{}

\Book{\Title{THE BIBLE IN THE CHURCH.} A Popular Account of the
Collection and Reception of the Holy Scriptures in the Christian
Churches. Third Edition. By \Name{Brooke Foss Westcott}, B.D.
18mo.\ cloth\DPtypo{,}{.} \Price{4s. 6d.}}
%
{The present book is an attempt to answer a request, which has been made
from time to time, to place in a simple form, for the use of general readers,
the substance of the author's ``History of the Canon of the New Testament.''
An elaborate and comprehensive Introduction is followed by chapters on
the Bible of the Apostolic Age; on the Growth of the New Testament; the
Apostolic Fathers; the Age of the Apologists; the First Christian Bible;
the Bible Proscribed and Restored; the Age of Jerome and Augustine;
the Bible of the Middle Ages in the West and in the East, and in the
Sixteenth Century. Two appendices on the History of the Old Testament
Canon before the Christian Era, and on the Contents of the most ancient
MSS. of the Christian Bible, complete the volume.}

\Book{\Title{THE GOSPEL OF THE RESURRECTION.} Thoughts on its
Relation to Reason and History. By \Name{Brooke Foss Westcott},
B.D. New Edition. Fcap.~8vo. \Price{4s. 6d.}%
\DPPageSep{116.png}{112}%
}
%
{This Essay is an endeavour to consider some of the elementary truths
of Christianity as a miraculous Revelation, from the side of History and
Reason. If the arguments which are here adduced are valid, they will go
far to prove that the Resurrection, with all that it includes, is the key to
the history of man, and the complement of reason.}

\Book{\Author{Wilson.}---\Title{AN ENGLISH, HEBREW, AND CHALDEE
LEXICON AND CONCORDANCE}, to the more Correct
Understanding of the English translation of the Old Testament,
by reference to the Original Hebrew. By \Name{William Wilson},
D.D., Canon of Winchester, late Fellow of Queen's College,
Oxford. Second Edition, carefully Revised. 4to.\ cloth. \Price{25s.}}
%
{The aim of this work is, that it should be useful to clergymen and all
persons engaged in the study of the Bible, even when they do not possess a
knowledge of Hebrew; while able Hebrew scholars have borne testimony to
the help that they themselves have found in it.}
\DPPageSep{117.png}{113}%


\Csection[Education.]{Books On Education.}

\Book{\Author{Arnold.}---\Title{A FRENCH ETON; OR, MIDDLE CLASS
EDUCATION AND THE STATE.} By \Name{Matthew Arnold}.
Fcap.~8vo.\ cloth. \Price{2s. 6d.}}
%
{``A very interesting dissertation on the system of secondary instruction
in France, and on the advisability of copying the system in England.''---\Cite{Saturday
Review}.}{}

\Book{\Title{SCHOOLS AND UNIVERSITIES ON THE CONTINENT.}
8vo. \Price{10s. 6d.}}
%
{The Author was in \Date{1865} charged by the Schools Inquiry Commissioners
with the task of investigating the system of education for the middle and
upper classes in France, Italy, Germany, and Switzerland. In the discharge
of this task he was on the Continent nearly seven months, and
during that time he visited the four countries named and made a careful
study of the matters to which the Commissioners had directed his attention.
The present volume contains the report which he made to them. It is here
adapted to the use of the general reader.}

\Book{\Title{ESSAYS ON A LIBERAL EDUCATION.} Edited by the Rev.\
\Name{F.~W. Farrar}, M.A., F.R.S., Assistant Master at Harrow,
late Fellow of Trinity College, Cambridge, and Hon.\ Fellow of
King's College, London. Second Edition. 8vo.\ cloth. \Price{10s. 6d.}}
%
{\textsc{Contents}:---History of Classical Education, by Charles S. Parker,
M.A.; Theory of Classical Education, by Henry Sedgwick, M.A.;
\DPPageSep{118.png}{114}%
Liberal Education in Universities, by John Seeley, M.A.; Teaching by
means of Grammar, by E. E. Bowen, M.A.; Greek and Latin Verse-Composition,
by the Rev.\ F. W. Farrar; Natural Science in Schools, by
J. M. Wilson, M.A., F.G.S.; The Teaching of English, by J. W. Hales,
M.A.; Education of the Reasoning Faculties, by W. Johnson, M.A.;
The present Social Results of Classical Education, by Lord Houghton.

The Authors have sought to hasten the expansion and improvement of
liberal education by showing in what light some of the most interesting
questions of Educational Reform are viewed by men who have had
opportunities for forming a judgment respecting them, and several of
whom have been for some time engaged in the work of education at our
Universities and Schools.}

\Book{\Author{Farrar.}---\Title{ON SOME DEFECTS IN PUBLIC SCHOOL
EDUCATION.} A Lecture delivered at the Royal Institution.
With Notes and Appendices. Crown 8vo. \Price{1s.}}{}

\Book{\Author{Jex-Blake.}---\Title{A VISIT TO SOME AMERICAN SCHOOLS
AND COLLEGES.} By \Name{Sophia Jex-Blake}. Crown 8vo.\ cloth.
\Price{6s.}}
%
{``In the following pages I have endeavoured to give a simple and accurate
account of what I saw during a series of visits to some of the Schools and
Colleges in the United States\ldots. I wish simply to give other teachers an
opportunity of seeing through my eyes what they cannot perhaps see for
themselves, and to this end I have recorded just such particulars as I should
myself care to know.''---\Cite{Author's Preface}.

``Miss Blake gives a living picture of the Schools and Colleges themselves
in which that education is carried on.''---\Cite{Pall Mall Gazette}.}

\Book{\Author{Quain (Richard, F.R.S.)}---\Title{ON SOME DEFECTS IN
GENERAL EDUCATION.} By \Name{Richard Quain}, F.R.S.
Crown 8vo. \Price{3s. 6d.}%
\DPPageSep{119.png}{115}%
}
%
{Having been charged by the College of Surgeons with the delivery of the
Hunterian Oration for \Date{1869}, the Author has availed himself of the
occasion to bring under notice some defects in the general education of the
country, which, in his opinion, effect injuriously all classes of the people,
and not least the members of his own profession. The earlier pages of the
address contain a short notice of the genius and labours of John Hunter,
but the subject of education will be found to occupy the larger part---from
page twelve to the end.}

\Book{\Author{Thring.}---\Title{EDUCATION AND SCHOOL.} By the Rev.\ \Name{Edward
Thring}, M.A., Head Master of Uppingham. Second Edition.
Crown 8vo.\ cloth. \Price{5s. 6d.}}{}

\Book{\Author{Youmans.}---\Title{MODERN CULTURE}: its True Aims and Requirements.
A Series of Addresses and Arguments on the Claims of
Scientific Education. Edited by \Name{Edward L. Youmans}, M.D.
Crown 8vo. \Price{8s. 6d.}}
%
{\textsc{Contents}:---Professor Tyndall on the Study of Physics; Dr.~Daub\-eny
on the Study of Chemistry; Professor Henfrey on the Study of Botany;
Professor Huxley on the Study of Zoology; Dr.~J. Paget on the Study of
Physiology; Dr.~Whewell on the Educational History of Science; Dr.
Faraday on the Education of the Judgment; Dr.~Hodgson on the Study
of Economic Science; Mr.~Herbert Spencer on Political Education;
Professor Masson on College Education and Self Education; Dr.~Youmans
on the Scientific Study of Human Nature. An Appendix contains extracts
from distinguished authors, and from the Scientific Evidence given before
the Public Schools Commission.}
\DPPageSep{120.png}{116}%
\clearpage
\thispagestyle{empty}
\null\vfill
\begin{center}
\footnotesize
LONDON: \\
R. CLAY, SONS, AND TAYLOR, PRINTERS, \\
BREAD STREET HILL.
\end{center}
\vfill
\cleardoublepage
\backmatter

%%%% LICENSE %%%%
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% End of Project Gutenberg's Elements of Plane Trigonometry, by Hugh Blackburn
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% *** END OF THIS PROJECT GUTENBERG EBOOK ELEMENTS OF PLANE TRIGONOMETRY ***
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% ***** This file should be named 32973-t.tex or 32973-t.zip *****        %
% This and all associated files of various formats will be found in:      %
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