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| author | Roger Frank <rfrank@pglaf.org> | 2025-10-14 19:58:36 -0700 |
|---|---|---|
| committer | Roger Frank <rfrank@pglaf.org> | 2025-10-14 19:58:36 -0700 |
| commit | e9e8372ef2ebbb69ce06aba5292335b8c179dd53 (patch) | |
| tree | 2e861a5ed35e225cfa70102bc869ed2a53200846 | |
63 files changed, 15734 insertions, 0 deletions
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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] % +% Most recently updated: June 11, 2021 % +% % +% Language: English % +% % +% Character set encoding: UTF-8 % +% % +% *** START OF THIS PROJECT GUTENBERG EBOOK ELEMENTS OF PLANE TRIGONOMETRY *** +% % +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % + +\def\ebook{32973} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %% +%% Packages and substitutions: %% +%% %% +%% book: Required. %% +%% inputenc: Standard DP encoding. Required. %% +%% babel: Greek snippets. Required. %% +%% %% +%% calc: Infix arithmetic. 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Sectioning: Chapter, Preface, Section, Article +\newcommand{\ChapterHead}[1]{% + \centering\normalfont\MakeUppercase{#1}% +} + +\newcommand{\Paragraph}[2][]{% + \ifthenelse{\not\equal{#1}{}}{#1}{}% + \textit{#2}% +} + +\newcommand{\BookTitle}{% + \mainmatter + \renewcommand{\thepage}{\texorpdfstring{\osf{\arabic{page}}}{\arabic{page}}} + \phantomsection\pdfbookmark[-1]{Main Matter.}{Main Matter}% + \thispagestyle{plain} + \begin{center} + \setlength{\TmpLen}{6pt}% + \Large ELEMENTS \\[\TmpLen] + \normalsize OF \\[\TmpLen] + \Large PLANE TRIGONOMETRY. + \end{center} + \bigskip +} + +% \Chapter{N}{Title} -- for numbered chapters +\newcommand{\Chapter}[3][]{% + \ifthenelse{\not\equal{#2}{I.}}{% + \ifthenelse{\boolean{ForPrinting}}{\fancyhead[RE]{}}{}% + \clearpage\fancyhf{}\cleardoublepage + \thispagestyle{plain}% + }{} + \phantomsection\pdfbookmark[0]{Chapter #2}{Chapter #2}% + + \section*{\ChapterHead{\Large Chapter #2}} + \subsection*{\ChapterHead{\small #3}} + 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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] +Most recently updated: June 11, 2021 + +Language: English + +Character set encoding: UTF-8 + +*** START OF THIS PROJECT GUTENBERG EBOOK ELEMENTS OF PLANE TRIGONOMETRY *** +\end{PGtext} +\end{minipage} +\end{center} + + +%%%% Credits and transcriber's note %%%% +\clearpage +\thispagestyle{empty} + +\begin{center} +\begin{minipage}{\textwidth} +\begin{PGtext} +Produced by Andrew D. Hwang, Laura Wisewell and the Online +Distributed Proofreading Team at http://www.pgdp.net (The +original copy of this book was generously made available +for scanning by the Department of Mathematics at the +University of Glasgow.) +\end{PGtext} +\end{minipage} +\end{center} +\vfill + +\begin{minipage}{0.85\textwidth} +\small +\pdfbookmark[0]{Transcriber's Note.}{Transcriber's Note} +\subsection*{\centering\normalfont\scshape% +\normalsize\MakeLowercase{\TransNote}}% + +\raggedright +\TransNoteText +\end{minipage} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%% +\cleardoublepage +\frontmatter +\normalsize + +\DPPageSep{001.png}{i}% +% title page +\setlength{\TmpLen}{6pt}% +\thispagestyle{empty}% +{\stretchyspace% +\begin{center} +\Large ELEMENTS \\[2\TmpLen] +\small OF \\[2\TmpLen] +\LARGE PLANE TRIGONOMETRY \\[2\TmpLen] +\normalsize FOR THE USE OF THE JUNIOR CLASS OF \\[\TmpLen] +MATHEMATICS IN THE UNIVERSITY \\[\TmpLen] +OF GLASGOW. +\vfill + +\small BY \\[\TmpLen] +\Large HUGH BLACKBURN, M.A. \\[\TmpLen] +\scriptsize PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF GLASGOW, \\[\TmpLen] +LATE FELLOW OF TRINITY COLLEGE, CAMBRIDGE. +\vfill + +\normalsize% +\textgoth{\Large London and New York:} \\[\TmpLen] +\Large MACMILLAN AND CO. \\[\TmpLen] +\Date{1871}. \\[\TmpLen] +\normalsize [\textit{All Rights reserved}.] +\end{center} +\DPPageSep{002.png}{ii}% +\newpage\null +% half title page +\thispagestyle{empty}% +\begin{center} +\vfill +\textgoth{\Large Cambridge}: \\[2\TmpLen] +PRINTED BY C. J. CLAY, M.A. \\[2\TmpLen] +AT THE UNIVERSITY PRESS. +\vfill +\end{center}}%[** TN: End of \stretchyspace] +\DPPageSep{003.png}{iii}% + + +\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 +and English Verse, Translation, and an Introduction. By \Name{Bernard +Drake}, M.A., late Fellow of King's College, Cambridge. +8vo. \Price{3s. 6d.}} +% +{The Greek text adopted in this Edition is based upon that of Wellauer, +which may be said, in general terms, to represent that of the best manuscripts. +But in correcting the Text, and in the Notes, advantage has been +taken of the suggestions of Hermann, Paley, Linwood, and other commentators. +In the Translation, the simple character of the æschylean +dialogues has generally enabled the author to render them without any +material deviation from the construction and idioms of the original Greek.} +\DPPageSep{058.png}{54}% + +\Book{\Title{ARISTOTLE ON FALLACIES; OR, THE SOPHISTICI +ELENCHI.} With a Translation and Notes by \Name{Edward Poste}, +M.A., Fellow of Oriel College, Oxford. 8vo. \Price{8s. 6d.}} +% +{Besides the doctrine of Fallacies, Aristotle offers, either in this treatise +or in other passages quoted in the commentary, various glances over the +world of science and opinion, various suggestions or problems which are +still agitated, and a vivid picture of the ancient system of dialectics, which +it is hoped may be found both interesting and instructive.} + +\Book{\Author{Aristotle.}---\Title{AN INTRODUCTION TO ARISTOTLE'S +RHETORIC.} With Analysis, Notes, and Appendices. By \Name{E.~M. +Cope}, Senior Fellow and Tutor of Trinity College, Cambridge. +8vo. \Price{14s.}} +% +{This work is introductory to an edition of the Greek Text of Aristotle's +Rhetoric, which is in course of preparation. Its object is to render that +treatise thoroughly intelligible. The author has aimed to illustrate, as +preparatory to the detailed explanation of the work, the general bearings +and relations of the Art of Rhetoric in itself, as well as the special mode of +treating it adopted by Aristotle in his peculiar system. The evidence upon +obscure or doubtful questions connected with the subject is examined; and +the relations which Rhetoric bears, in Aristotle's view, to the kindred art +of Logic are fully considered. A connected Analysis of the work is given, +sometimes in the form of paraphrase; and a few important matters are +separately discussed in Appendices. There is added, as a general Appendix, +by way of specimen of the antagonistic system of Isocrates and others, a +complete analysis of the treatise called \textgreek{<Rhtoriq`h pr`os >Al'exandron}, with a +discussion of its authorship and of the probable results of its teaching.} + +\Book{\Author{Cicero.}---\Title{THE SECOND PHILIPPIC ORATION.} With an +Introduction and Notes, translated from the German of \Name{Karl +Halm}. Edited, with Corrections and Additions, by \Name{John E.~B. +Mayor}, M.A., Fellow and Classical Lecturer of St.~John's +College, Cambridge. Third Edition, revised. Fcap.~8vo. \Price{5s.}} +% +{This volume opens with a List of Books useful to the Student of Cicero, +including History, Chronology, Lexicons, and some account of various +editions, mostly German, of the works of Cicero. The Introduction is +\DPPageSep{059.png}{55}% +based on Halm: where Halm gives a reference to a classic, the passage has +been commonly printed at length; where the reference is to Halm's notes +on other Ciceronian speeches, or to modern books, the additional matter has +been incorporated: and the numerous Greek quotations have been rendered +into English. The English editor has further illustrated the work by +additions drawn, for the most part, \Inum{1}~from the ancient authorities; \Inum{2}~from +his own private marginal references, and from collections; \Inum{3}~from +the notes of previous commentators. A copious `argument' is also given.} + +\Book{\Title{THE ORATIONS OF CICERO AGAINST CATILINA.} With +Notes translated chiefly from Halm. By \Name{A.~S. Wilkins}, M.A. +Fcap.~8vo. \Price{3s. 6d.}} +% +{This edition is a reprint of the one prepared by Professor Halm for +\DPtypo{Orellli's}{Orelli's} Cicero. The historical introduction of Mr.\ Wilkins brings +\DPtypo{to ether}{together} +all the details which are known respecting Catiline and his +relations with the great orator. A list of passages where conjectures +have been admitted into the text, and also of all variations from the text +of Kayser \(\Date{1862}\) is added at the end. Finally the English Editor has +subjoined a large number of notes, both original \(distinguished by a +square bracket\) and selected from Curtius, Schleischer, Corssen, and +other well-known critics, an analysis of the orations, and an index.} + +\Book{\Title{DEMOSTHENES ON THE CROWN.} The Greek Text with +English Notes. By \Name{B. Drake}, M.A., late Fellow of King's +College, Cambridge. Fourth Edition, to which is prefixed +ÆSCHINES AGAINST CTESIPHON, with English Notes. +Fcap.~8vo. \Price{5s.}} +% +{An Introduction discusses the immediate causes of the two orations, and +their general character. The Notes contain frequent references to the best +authorities. Among the appendices at the end is a chronological table of +the life and public career of Æschines and Demosthenes.} + +\Book{\Author{Hodgson.}---\Title{MYTHOLOGY FOR LATIN VERSIFICATION.} +A brief Sketch of the Fables of the Ancients, prepared to be +rendered into Latin Verse for Schools. By \Name{F. Hodgson}, B.D., +late Provost of Eton. New Edition, revised by \Name{F.~C. Hodgson}, +M.A. 18mo. \Price{3s}.% +\DPPageSep{060.png}{56}% +} +% +{The late Provost of Eton has here supplied a help to the composition of +Latin Verse, combined with a brief introduction to Classical Mythology. +In this new edition a few mistakes have been rectified; rules have been +added to the Prosody; and a more uniform system has been adopted with +regard to the help afforded.} + +\Book{\Author{Juvenal.}---Thirteen Satires of \Title{JUVENAL}. With a Commentary. +By \Name{John E.~B. Mayor}, M.A., Fellow of St.~John's College, +Cambridge. Second Edition, enlarged. Part~I\@. Crown 8vo.\ sewed. +\Price{3s. 6d.}} +% +{The text is accompanied by a copious Commentary. For various notes +the author is indebted to Professors Munro and Conington. All the +citations have been taken anew from the original authors.} + +\Book{\Author{Marshall.}---\Title{A TABLE OF IRREGULAR GREEK VERBS} +classified according to the arrangement of Curtius' Greek Grammar. +By \Name{J.~M. Marshall}, M.A., Fellow and late Lecturer of Brasenose +College, Oxford; one of the Masters in Clifton College. 8vo. +cloth. \Price{1s.}} +% +{The system of this table has been borrowed from the excellent Greek +Grammar of Dr. Curtius.} + +\Book{\Author{Mayor (John E.~B.)}---\Title{FIRST GREEK READER.} Edited +after \Name{Karl Halm}, with Corrections and large Additions by \Name{John +E.~B. Mayor}, M.A. Fellow and Classical Lecturer of St.~John's +College, Cambridge. Second and Cheaper Edition. Fcap.~8vo. +\Price{4s. 6d.}} +% +{A selection of short passages, serving to illustrate especially the Greek +Accidence. A good deal of syntax is incidentally taught, and Madvig and +other books are cited, for the use of masters: but no learner is expected to +know more of syntax than is contained in the Notes and Vocabulary. +A preface ``To the Reader,'' not only explains the aim and method of +the volume, but also deals with classical instruction generally. The +extracts are uniformly in the Attic dialect, and any Hellenistic forms +occurring in the original classic authors, such as Ælian and Polybius, +have been discarded in favour of the corresponding Attic expressions. +This book may be used in connexion with Mayor's ``Greek for Beginners.''} +\DPPageSep{061.png}{57}% + +\Book{\Author{Mayor (Joseph B.)}---\Title{GREEK FOR BEGINNERS.} By the +Rev.\ \Name{J.~B. Mayor}, M.A., Professor of Classical Literature in +King's College, London. Part~I., with Vocabulary, \Price{1s. 6d.}; +Parts II.~and~III., with Vocabulary and Index, \Price{3s. 6d.}; complete +in one vol., fcap.~8vo.\ cloth\DPtypo{,}{.} \Price{4s. 6d.}} +% +{The distinctive method of this book consists in building up a boy's +knowledge of Greek upon the foundation of his knowledge of English and +Latin, instead of trusting everything to the unassisted memory. The +forms and constructions of Greek have been thoroughly compared with +those of Latin, and no Greek words have been used in the earlier part of +the book except such as have connexions either in English or Latin. Each +step leads naturally on to its successor, grammatical forms and rules are +at once applied in a series of graduated exercises, accompanied by ample +vocabularies. Thus the book serves as Grammar, Exercise book, and +Vocabulary. Where possible, the Grammar has been simplified; the +ordinary ten declensions are reduced to three, which correspond to the +first three in Latin; and the system of stems is adopted. A general +Vocabulary, and Index of Greek words, completes the work.} + +\Book{\Author{Peile (John, M.A.)}---\Title{AN INTRODUCTION TO GREEK +AND LATIN ETYMOLOGY.} By \Name{John Peile}, M.A., Fellow +and Assistant Tutor of Christ's College, Cambridge, formerly +Teacher of Sanskrit in the University of Cambridge. 8vo. \Price{10s. 6d.}} +% +{These Philological Lectures are the result of Notes made during the +author's reading during the last three or four years. These Notes were +put into the shape of lectures, delivered at Christ's College, during the last +May term, as one set in the ``Intercollegiate'' list. They are now printed +with some additions and modifications, but substantially as they were +delivered.} + +\Book{\Author{Plato.}---\Title{THE REPUBLIC OF PLATO.} Translated into English, +with an Analysis and Notes, by \Name{J. Ll. Davies}, M.A., and \Name{D.~J. +Vaughan}, M.A\@. Third Edition, with Vignette Portraits of \DPtypo{Plate}{Plato} +and Socrates, engraved by \Name{Jeens} from an Antique Gem. 18mo. +\Price{4s. 6d.}% +\DPPageSep{062.png}{58}% +} +% +{An introductory notice supplies some account of the life of Plato, and +the translation is preceded by an elaborate analysis. ``The translators +have,'' in the Judgment of the \emph{Saturday Review}, ``produced a book which +any reader, whether acquainted with the original or not, can peruse with +pleasure as well as profit.''} + +\Book{\Author{Plautus (Ramsay).}---\Title{THE MOSTELLARIA OF PLAUTUS.} +With Notes Critical and Explanatory, Prolegomena, and +Excursus. By \Name{William Ramsay}, M.A., formerly Professor of +Humanity in the University of Glasgow. Edited by Professor +\Name{George G. Ramsay}, M.A., of the University of Glasgow. 8vo. +\Price{14s.}} +% +{``The fruits of that exhaustive research and that ripe and well-digested +scholarship which its author brought to bear upon everything that he +undertook are visible throughout it. It is furnished with a complete +apparatus of prolegomena, notes, and excursus; and for the use of veteran +scholars it probably leaves nothing to be desired.''---\Cite{Pall Mall Gazette}.} + +\Book{\Author{Potts (Alex.\ W., M.A.)}---\Title{HINTS TOWARDS LATIN +PROSE COMPOSITION.} By \Name{Alex.~W. Potts}, M.A., late +Fellow of St.~John's College, Cambridge; Assistant Master in +Rugby School; and Head Master of the Fettes College, Edinburgh. +Second Edition, enlarged. Extra fcap.~8vo.\ cloth. \Price{3s.}} +% +{Those engaged in Classical teaching seem to be unanimously of the +opinion that Composition in Latin Prose is not only the most efficient +method of acquiring a mastery of the Latin language, but is in itself +a valuable means of mental training, and an admirable corrective of some +of the worst features in English writing. An attempt is here made to +give students, after they have mastered ordinary syntactical rules, some idea +of the characteristics of Latin Prose and the means to be employed to +reproduce them. Some notion of the treatment of the subject may be +gathered from the `Contents.' \textsc{Chap.~I.}---Characteristics of Classical +Latin, Hints on turning English into Latin; \textsc{Chap.~II.}---Arrangement +of Words in a Sentence; \textsc{Chap.~III.}---Unity in Latin Prose, Subject and +Object; \textsc{Chap.~IV.}---On the Period in Latin Prose; \textsc{Chap.~V.}---On the +position of the Relative and Relative Clauses.} +\DPPageSep{063.png}{59}% + +\Book{\Author{Roby.}---\Title{A LATIN GRAMMAR} for the Higher Classes in Grammar +Schools. By \Name{H.~J. Roby}, M.A.\InPress}{} + +\Book{\Author{Sallust.}---\Title{CAII SALLUSTII CRISPI CATILINA ET JUGURTHA.} +For Use in Schools. With copious Notes. By \Name{C. +Merivale}, B.D. (In the present Edition the Notes have been +carefully revised, and a few remarks and explanations added.) +Second Edition. Fcap.~8vo. \Price{4s. 6d.} + +\noindent The \Title{JUGURTHA} and the \Title{CATILINA} may be had separately, price +\Price{2s. 6d.} each.} +% +{This edition of Sallust, prepared by the distinguished historian of Rome, +contains an introduction, concerning the life and works of Sallust, lists +of the Consuls, and elaborate notes.} + +\Book{\Author{Tacitus.}---\Title{THE HISTORY OF TACITUS TRANSLATED +INTO ENGLISH.} By \Name{A.~J. Church}, M.A., and \Name{W.~J. +Brodribb}, M.A. With Notes and a Map. 8vo. \Price{10s. 6d.}} +% +{The translators have endeavoured to adhere as closely to the original as +was thought consistent with a proper observance of English idiom. At +the same time, it has been their aim to reproduce the precise expressions of +the author. The campaign of Civilis is elucidated in a note of some length, +which is illustrated by a map, containing the names of places and of tribes +occurring in the work. There is also a complete account of the Roman army +as it was constituted in the time of Tacitus. This work is characterised +by the Spectator as ``a scholarly and faithful translation.''} + +\Book{\Title{THE AGRICOLA AND GERMANIA OF TACITUS.} A Revised +Text, English Notes, and Maps. By \Name{Alfred J. Church}, M.A., +and \Name{W.~J. Brodribb}, M.A. Fcap.~8vo. \Price{3s. 6d.}} +% +{``We have endeavoured, with the aid of recent editions, thoroughly to +elucidate the text, explaining the various difficulties, critical and grammatical, +which occur to the student. We have consulted throughout, besides +the older commentators, the editions of Ritter and Orelli, but we are +under special obligations to the labours of the recent German editors, Wex +and Kritz.'' Two Indexes are appended, \Inum{1}~Proper Names, \Inum{2}~of +Words and Phrases explained\Add{.}} +\DPPageSep{064.png}{60}% + +\Book{\Title{THE AGRICOLA} and \Title{GERMANIA} may be had separately, price +\Price{2s.} each. + +\noindent\Title{THE AGRICOLA AND GERMANIA.} Translated into English +by \Name{A.~J. Church}, M.A., and \Name{W.~J. Brodribb}, M.A. With +Maps and Notes. Extra fcap.~8vo. \Price{2s. 6d.}} +% +{The translators have sought to produce such a version as may satisfy +scholars who demand a faithful rendering of the original, and English +readers who are offended by the baldness and frigidity which commonly +disfigure translations. The treatises are accompanied by introductions, +notes, maps, and a chronological summary. The \emph{Athenæum} says of this +work that it is ``a version at once readable and exact, which may be perused +with pleasure by all, and consulted with advantage by the classical student.''} + +\Book{\Author{Theophrastus.}---\Title{THE CHARACTERS OF THEO\-PHRAS\-TUS.} +An English Translation from a Revised Text. +With Introduction and Notes. By \Name{R.~C. Jebb}, M.A., Public +Orator in the University of Cambridge. Extra fcap.~8vo. \Price{6s. 6d.}} +% +{To the average English reader Theophrastus is little known. At the +present time, when there is a general desire to see ancient life more vividly +on every side from which it can illustrate our own, it seems possible that +the characters of Theophrastus may possess some potent interest. The text +has undergone careful revision. An Introduction supplies an account of +the origin of the book, and of writers who have imitated it: as Hall, +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 %%%% +\License + +\begin{PGtext} +End of Project Gutenberg's Elements of Plane Trigonometry, by Hugh Blackburn + +*** END OF THIS PROJECT GUTENBERG EBOOK ELEMENTS OF PLANE TRIGONOMETRY *** + +***** This file should be named 32973-pdf.pdf or 32973-pdf.zip ***** +This and all associated files of various formats will be found in: + http://www.gutenberg.org/3/2/9/7/32973/ + +Produced by Andrew D. 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new file mode 100644 index 0000000..bb3eedc --- /dev/null +++ b/32973-t/images/sources/009a.xp @@ -0,0 +1,71 @@ +/* -*-ePiX-*- */ +#include "epix.h" +using namespace ePiX; + +int N(6); +double th(80), dth(th/N); + +std::string mylab(int i) +{ + std::stringstream buf; + buf << "$A_{" << i << "}$"; + return buf.str(); +} + +std::string mylab2(int i) +{ + std::stringstream buf; + buf << "$A_{" << i << "}'$"; + return buf.str(); +} + +int main() +{ + picture(P(0,0), P(1,1.5), "3 x 4.5in"); + + begin(); + degrees(); + bold(); + P O(0,0), pA(1,0), pB(cis(th)), AA1(cis(dth) - pA); + + P curr(pA), pB1(pA + N*AA1), pB2(pA + P(0,th*M_PI/180)); + + arc(O, 1, 0, th); + + line(O,pB); + line(pA, pB1); + line(pA, pB2); + + for (int i=1; i<=N; ++i) + { + P next(cis(i*dth)); + line(O, curr); + line(O, next); + line(curr, next); + + if (i != 6) + label(next, P(2*(i - 4), -6), mylab(i), b); + + if (i != 1 && i != 6) + label(pA + i*AA1, P(-2, 6*(3.5 - i)), mylab2(i), l); + + curr = next; + } + + line(O, pB2); + + dashed(); + dash_size(6); + line(O, pB1); + + label(O, P(-2,-4), "$O$", bl); + label(pA, P(0,-4), "$A$", b); + label(pB, P(-2,0), "$B$", tl); + label(pB1, P(-2,0), "$B'$", tl); + label(pB2, P(2,0), "$B''$", r); + + label(pB2, P(-2,2), "$T$", tl); + + tikz_format(); + end(); +} diff --git a/32973-t/images/sources/011a.eepic b/32973-t/images/sources/011a.eepic new file mode 100644 index 0000000..601e451 --- /dev/null +++ b/32973-t/images/sources/011a.eepic @@ -0,0 +1,245 @@ +%% Generated from 011a.xp on Wed Jun 23 15:35:30 EDT 2010 by +%% ePiX-1.2.4 +%% +%% Cartesian bounding box: [-1,1] x [-1,1] +%% Actual size: 4 x 4in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% +\xdefinecolor{rgb_000000}{rgb}{0,0,0}% +\begin{tikzpicture} +\pgfsetlinewidth{0.4pt} +\useasboundingbox (0in,0in) rectangle (4in,4in); +\pgfsetlinewidth{0.8pt} +\draw (4in,2in)--(3.99726in,2.10467in)--(3.98904in,2.20906in)-- + (3.97538in,2.31287in)--(3.9563in,2.41582in)-- + 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+\pgftext[at={\pgfpoint{3.55976in}{0.686751in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}$B'$}}} +\pgftext[at={\pgfpoint{1.97233in}{1.94465in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}$C$}}} +\pgftext[at={\pgfpoint{3.50441in}{1.94465in}}] {\makebox(0,0)[tr]{\hbox{\color{rgb_000000}$D$}}} +\pgftext[at={\pgfpoint{-0.055348in}{2in}}] {\makebox(0,0)[r]{\hbox{\color{rgb_000000}$E$}}} +\pgftext[at={\pgfpoint{1.73837in}{2.67046in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}$F$}}} +\pgftext[at={\pgfpoint{1.7107in}{1.94465in}}] {\makebox(0,0)[tr]{\hbox{\color{rgb_000000}$G$}}} +\pgftext[at={\pgfpoint{1.73837in}{1.32954in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}$H$}}} +\end{tikzpicture} diff --git a/32973-t/images/sources/011a.xp b/32973-t/images/sources/011a.xp new file mode 100644 index 0000000..2556b6d --- /dev/null +++ b/32973-t/images/sources/011a.xp @@ -0,0 +1,54 @@ +/* -*-ePiX-*- */ +#include "epix.h" +using namespace ePiX; + +P O(0,0); +double th(40); + +int main() +{ + picture(P(-1,-1), P(1,1), "4 x 4in"); + + begin(); + degrees(); + bold(); + P pA(1,0), pB(cis(th)), pC(O), pD(Cos(th),0), pE(-1,0); + P pB2(cis(-th)), pF(0.5*(pB + pE)), pG(pF.x1(),0), pH(0.5*(pB2 + pE)); + circle(); + + line(pE, pA); + + line(pC, pB); + line(pC, pB2); + line(pC, pF); + + line(pE, pB); + line(pE, pB2); + + line(pB, pB2); + line(pF, pH); + + double CD(norm(pD - pC)), EF(norm(pF - pE)), EG(norm(pG - pE)); + + dashed(); + dash_size(6); + ellipse(pC, CD*E_1, CD*E_2, -th, th, 20); + + ellipse(pE, EF*E_1, EF*E_2, -0.5*th, 0.5*th, 10); + ellipse(pE, EG*E_1, EG*E_2, -0.5*th, 0.5*th, 10); + + label(pA, P( 2, 0), "$A$", r); + label(pB, P( 2, 4), "$B$", r); + label(pB2, P(2,-2), "$B'$", r); + label(pC, P(-2,-4), "$C$", b); + + label(pD, P(-2,-4), "$D$", bl); + label(pE, P(-4, 0), "$E$", l); + + label(pF, P(-2, 2), "$F$", t); + label(pG, P(-4,-4), "$G$", bl); + label(pH, P(-2,-2), "$H$", b); + + tikz_format(); + end(); +} diff --git a/32973-t/images/sources/017a.eepic b/32973-t/images/sources/017a.eepic new file mode 100644 index 0000000..86c9308 --- /dev/null +++ b/32973-t/images/sources/017a.eepic @@ -0,0 +1,281 @@ +%% Generated from 017a.xp on Wed Jun 23 16:55:47 EDT 2010 by +%% ePiX-1.2.4 +%% +%% Cartesian bounding box: [-1,4] x [-4,2] +%% Actual size: 3.5 x 4.2in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% +\xdefinecolor{rgb_000000}{rgb}{0,0,0}% +\begin{tikzpicture} +\pgfsetlinewidth{0.4pt} +\useasboundingbox (0in,0in) rectangle (3.5in,4.2in); +\pgfsetlinewidth{0.8pt} +\draw (0in,4.01244in)--(0in,2.03608in)--(2.97268in,2.29616in)--(0in,4.01244in)--cycle; +\draw (0.7in,2.8in)--(0.730505in,2.45133in)--(0.761009in,2.10266in); +\draw (0.7in,2.8in)--(0.875in,3.10311in)--(1.05in,3.40622in); +\draw (0.7in,2.8in)--(0.35in,2.8in)--(0in,2.8in); +\draw (0.7in,2.8in)--(0.35in,3.40622in)--(0in,4.01244in); +\draw (0.7in,2.8in)--(0.35in,2.41804in)--(0in,2.03608in); +\draw 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{\makebox(0,0)[r]{\hbox{\color{rgb_000000}$B$}}} +\pgftext[at={\pgfpoint{3.00035in}{2.32383in}}] {\makebox(0,0)[bl]{\hbox{\color{rgb_000000}$C$}}} +\pgftext[at={\pgfpoint{0.761009in}{2.04732in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}$a$}}} +\pgftext[at={\pgfpoint{1.07767in}{3.43389in}}] {\makebox(0,0)[bl]{\hbox{\color{rgb_000000}$b$}}} +\pgftext[at={\pgfpoint{-0.055348in}{2.8in}}] {\makebox(0,0)[r]{\hbox{\color{rgb_000000}$c$}}} +\pgftext[at={\pgfpoint{2.23934in}{2.20191in}}] {\makebox(0,0)[tl]{\hbox{\color{rgb_000000}$a'$}}} +\pgftext[at={\pgfpoint{3.66192in}{1.94188in}}] {\makebox(0,0)[bl]{\hbox{\color{rgb_000000}$b'$}}} +\pgftext[at={\pgfpoint{-0.027674in}{-0.184032in}}] {\makebox(0,0)[r]{\hbox{\color{rgb_000000}$c'$}}} +\pgftext[at={\pgfpoint{0.769185in}{2.86919in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}$O$}}} +\pgftext[at={\pgfpoint{2.42283in}{-0.211706in}}] {\makebox(0,0)[tl]{\hbox{\color{rgb_000000}$O'$}}} +\end{tikzpicture} diff --git a/32973-t/images/sources/017a.xp b/32973-t/images/sources/017a.xp new file mode 100644 index 0000000..f1eb137 --- /dev/null +++ b/32973-t/images/sources/017a.xp @@ -0,0 +1,82 @@ +/* -*-ePiX-*- */ +#include "epix.h" +using namespace ePiX; + +P O(0,0); + +void ray(const P& tail, const P& head, double t) +{ + line(tail, head + t*(head - tail)); +} + +int main() +{ + picture(P(-1,-4), P(4,2), "3.5 x 4.2in"); + + begin(); + degrees(); + bold(); + + double th1(-85), th2(60), th3(180); + P pa(cis(th1)), pb(cis(th2)), pc(cis(th3)); + + Segment AB(pc, pc + cis(th3 + 90)); + Segment BC(pa, pa + cis(th1 + 90)); + Segment CA(pb, pb + cis(th2 + 90)); + + P pA(AB*CA), pB(AB*BC), pC(CA*BC); + + Segment BO(pB, pB+cis(90 + 0.5*(th1 + th3))); + Segment CO(pC, pC+cis(90 + 0.5*(th1 + th2))); + + P O2(BO*CO); + + triangle(pA, pB, pC); + + P pA2(O2 + (pB - O2)%(pB - pC)); + P pB2(O2 + (pC - O2)%(pC - pA)); + P pC2(O2 + (pB - O2)%(pB - pA)); + + double rad(norm(pC2 - O2)); + + line(O, pa); + line(O, pb); + line(O, pc); + + line(O, pA); + line(O, pB); + line(O, pC); + + line(O2, pA2); + line(O2, pB2); + line(O2, pC2); + + line(pA, O2); + line(pB, O2); + line(pC, O2); + + ray(pA, pB2, 0.1); + ray(pA, pC2, 0.1); + + dashed(); + ellipse(O, E_1, E_2, 0, 360, 60); + ellipse(O2, rad*E_1, rad*E_2, th2 - 10, th3 + 10, 60); + + label(pA, P(-2, 2), "$A$", tl); + label(pB, P(-2, 0), "$B$", l); + label(pC, P( 2, 2), "$C$", tr); + + label(pa, P( 0,-4), "$a$", b); + label(pb, P( 2, 2), "$b$", tr); + label(pc, P(-4, 0), "$c$", l); + + label(pA2, P( 2,-2), "$a'$", br); + label(pB2, P( 2, 2), "$b'$", tr); + label(pC2, P(-2, 0), "$c'$", l); + + label(O, P(5,5), "$O$", r); + label(O2, P(0,-2), "$O'$", br); + + tikz_format(); + end(); +} diff --git a/32973-t/images/sources/022a.eepic b/32973-t/images/sources/022a.eepic new file mode 100644 index 0000000..9cf33e2 --- /dev/null +++ b/32973-t/images/sources/022a.eepic @@ -0,0 +1,140 @@ +%% Generated from 022a.xp on Wed Jun 23 16:56:17 EDT 2010 by +%% ePiX-1.2.4 +%% +%% Cartesian bounding box: 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{\makebox(0,0)[b]{\hbox{\color{rgb_000000}$B$}}} +\pgftext[at={\pgfpoint{3.07957in}{0.989933in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}$B'$}}} +\pgftext[at={\pgfpoint{0.972326in}{1.75972in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}$C$}}} +\end{tikzpicture} diff --git a/32973-t/images/sources/022a.xp b/32973-t/images/sources/022a.xp new file mode 100644 index 0000000..2590b7f --- /dev/null +++ b/32973-t/images/sources/022a.xp @@ -0,0 +1,35 @@ +/* -*-ePiX-*- */ +#include "epix.h" +using namespace ePiX; + +P O(0,0); +double th(60), th2(17.5), rad(1.6); + +int main() +{ + picture(P(0,0), P(rad,1), "3.2 x 2in"); + + begin(); + degrees(); + bold(); + + P pA(1,0), pA2(rad,0), pB(cis(th2)), pB2(polar(rad, th2)), pC(cis(th)); + + arc(O, 1, 0, th); + arc(O, rad, 0, th2); + line(O, pA2); + line(O, pB2); + line(O, pC); + + label(O, P(-2,-2), "$O$", bl); + label(pA, P(0,-4), "$A$", b); + label(pA2, P(2,0), "$A'$", r); + + label(pB, P(3,4), "$B$", t); + label(pB2, P(2,2), "$B'$", t); + + label(pC, P(-2,2), "$C$", t); + + tikz_format(); + end(); +} diff --git a/32973-t/images/sources/024a.eepic b/32973-t/images/sources/024a.eepic new file mode 100644 index 0000000..80c5715 --- /dev/null +++ b/32973-t/images/sources/024a.eepic @@ -0,0 +1,87 @@ +%% Generated from 024a.xp on Wed Jun 23 17:13:53 EDT 2010 by +%% ePiX-1.2.4 +%% +%% Cartesian bounding box: [-1,1] x [-1,1] +%% Actual size: 4 x 4in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% +\xdefinecolor{rgb_000000}{rgb}{0,0,0}% +\begin{tikzpicture} +\pgfsetlinewidth{0.4pt} +\useasboundingbox (0in,0in) rectangle (4in,4in); +\pgfsetlinewidth{0.8pt} +\draw (4in,2in)--(3.99726in,2.10467in)--(3.98904in,2.20906in)-- + (3.97538in,2.31287in)--(3.9563in,2.41582in)-- + (3.93185in,2.51764in)--(3.90211in,2.61803in)-- + (3.86716in,2.71674in)--(3.82709in,2.81347in)-- + (3.78201in,2.90798in)--(3.73205in,3in)--(3.67734in,3.08928in)-- + (3.61803in,3.17557in)--(3.55429in,3.25864in)-- + 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(0.585786in,3.41421in)--(0.585786in,2in)--(0.585786in,0.585786in); +\draw (3.41421in,3.41421in)--(2.70711in,3.41421in)--(2in,3.41421in); +\pgftext[at={\pgfpoint{3.97233in}{2.02767in}}] {\makebox(0,0)[br]{\hbox{\color{rgb_000000}$A$}}} +\pgftext[at={\pgfpoint{0.027674in}{1.97233in}}] {\makebox(0,0)[tl]{\hbox{\color{rgb_000000}$A'$}}} +\pgftext[at={\pgfpoint{3.44189in}{3.44189in}}] {\makebox(0,0)[bl]{\hbox{\color{rgb_000000}$B$}}} +\pgftext[at={\pgfpoint{0.558112in}{3.44189in}}] {\makebox(0,0)[br]{\hbox{\color{rgb_000000}$B'$}}} +\pgftext[at={\pgfpoint{0.558112in}{0.558112in}}] {\makebox(0,0)[tr]{\hbox{\color{rgb_000000}$B''$}}} +\pgftext[at={\pgfpoint{3.44189in}{0.558112in}}] {\makebox(0,0)[tl]{\hbox{\color{rgb_000000}$B'''$}}} +\pgftext[at={\pgfpoint{1.97233in}{3.97233in}}] {\makebox(0,0)[tr]{\hbox{\color{rgb_000000}$C$}}} +\pgftext[at={\pgfpoint{1.97233in}{0.027674in}}] {\makebox(0,0)[br]{\hbox{\color{rgb_000000}$C'$}}} +\pgftext[at={\pgfpoint{3.44189in}{1.97233in}}] {\makebox(0,0)[tl]{\hbox{\color{rgb_000000}$D$}}} +\pgftext[at={\pgfpoint{0.61346in}{1.97233in}}] {\makebox(0,0)[tl]{\hbox{\color{rgb_000000}$D'$}}} +\pgftext[at={\pgfpoint{2.02767in}{3.38654in}}] {\makebox(0,0)[tl]{\hbox{\color{rgb_000000}$E$}}} +\pgftext[at={\pgfpoint{2.13837in}{2.08302in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}$O$}}} +\end{tikzpicture} diff --git a/32973-t/images/sources/024a.xp b/32973-t/images/sources/024a.xp new file mode 100644 index 0000000..727eff4 --- /dev/null +++ b/32973-t/images/sources/024a.xp @@ -0,0 +1,49 @@ +/* -*-ePiX-*- */ +#include "epix.h" +using namespace ePiX; + +P O(0,0); +double th(45); + +int main() +{ + picture(P(-1,-1), P(1,1), "4 x 4in"); + + begin(); + degrees(); + bold(); + P pA(1,0), pA2(-1,0), pB(cis(th)), pB1(cis(180-th)), pB2(-pB), pB3(-pB1); + P pC(0,1), pC2(0,-1), pD(Cos(th),0), pD2(-pD), pE(0,Sin(th)), pE2(-pE); + + circle(); + line(pA, pA2); + line(pB, pB2); + line(pC, pC2); + line(pB1, pB3); + + line(pB, pB3); + line(pB1, pB2); + + line(pB, pE); + + label(pA, P(-2,2), "$A$", tl); + label(pA2, P(2,-2), "$A'$", br); + + label(pB, P(2,2), "$B$", tr); + label(pB1, P(-2,2), "$B'$", tl); + label(pB2, P(-2,-2), "$B''$", bl); + label(pB3, P(2,-2), "$B'''$", br); + + label(pC, P(-2,-2), "$C$", bl); + label(pC2, P(-2,2), "$C'$", tl); + + label(pD, P(2,-2), "$D$", br); + label(pD2, P(2,-2), "$D'$", br); + + label(pE, P(2,-2), "$E$", br); + + label(O, P(10, 6), "$O$", r); + + tikz_format(); + end(); +} diff --git a/32973-t/images/sources/029a.eepic b/32973-t/images/sources/029a.eepic new file mode 100644 index 0000000..2cbd7b6 --- /dev/null +++ b/32973-t/images/sources/029a.eepic @@ -0,0 +1,94 @@ +%% Generated from 029a.xp on Wed Jun 23 17:22:23 EDT 2010 by +%% ePiX-1.2.4 +%% +%% Cartesian bounding box: [-1.25,1.25] x [-1.25,1.25] +%% Actual size: 4.5 x 4.5in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% +\xdefinecolor{rgb_000000}{rgb}{0,0,0}% +\begin{tikzpicture} +\pgfsetlinewidth{0.4pt} +\useasboundingbox (0in,0in) rectangle (4.5in,4.5in); +\pgfsetlinewidth{0.8pt} +\draw (4.05in,2.25in)--(4.04753in,2.3442in)-- + (4.04014in,2.43815in)--(4.02784in,2.53158in)-- + (4.01067in,2.62424in)--(3.98867in,2.71587in)-- + (3.9619in,2.80623in)--(3.93044in,2.89506in)-- + (3.89438in,2.98213in)--(3.85381in,3.06718in)--(3.80885in,3.15in)-- + (3.75961in,3.23035in)--(3.70623in,3.30801in)-- + (3.64886in,3.38278in)--(3.58766in,3.45444in)-- + (3.52279in,3.52279in)--(3.45444in,3.58766in)-- + (3.38278in,3.64886in)--(3.30801in,3.70623in)-- + (3.23035in,3.75961in)--(3.15in,3.80885in)--(3.06718in,3.85381in)-- + (2.98213in,3.89438in)--(2.89506in,3.93044in)-- + (2.80623in,3.9619in)--(2.71587in,3.98867in)-- + (2.62424in,4.01067in)--(2.53158in,4.02784in)-- + (2.43815in,4.04014in)--(2.3442in,4.04753in)--(2.25in,4.05in)-- + (2.1558in,4.04753in)--(2.06185in,4.04014in)-- + (1.96842in,4.02784in)--(1.87576in,4.01067in)-- + (1.78413in,3.98867in)--(1.69377in,3.9619in)-- + (1.60494in,3.93044in)--(1.51787in,3.89438in)-- + (1.43282in,3.85381in)--(1.35in,3.80885in)--(1.26965in,3.75961in)-- + (1.19199in,3.70623in)--(1.11722in,3.64886in)-- + (1.04556in,3.58766in)--(0.977208in,3.52279in)-- + (0.912339in,3.45444in)--(0.851137in,3.38278in)-- + (0.793769in,3.30801in)--(0.740393in,3.23035in)-- + (0.691154in,3.15in)--(0.646188in,3.06718in)-- + (0.605618in,2.98213in)--(0.569555in,2.89506in)-- + (0.538098in,2.80623in)--(0.511334in,2.71587in)-- + (0.489334in,2.62424in)--(0.472161in,2.53158in)-- + (0.459861in,2.43815in)--(0.452467in,2.3442in)--(0.45in,2.25in)-- + (0.452467in,2.1558in)--(0.459861in,2.06185in)-- + (0.472161in,1.96842in)--(0.489334in,1.87576in)-- + (0.511334in,1.78413in)--(0.538098in,1.69377in)-- + (0.569555in,1.60494in)--(0.605618in,1.51787in)-- + (0.646188in,1.43282in)--(0.691154in,1.35in)-- + (0.740393in,1.26965in)--(0.793769in,1.19199in)-- + (0.851137in,1.11722in)--(0.912339in,1.04556in)-- + (0.977208in,0.977208in)--(1.04556in,0.912339in)-- + (1.11722in,0.851137in)--(1.19199in,0.793769in)-- + (1.26965in,0.740393in)--(1.35in,0.691154in)-- + (1.43282in,0.646188in)--(1.51787in,0.605618in)-- + (1.60494in,0.569555in)--(1.69377in,0.538098in)-- + (1.78413in,0.511334in)--(1.87576in,0.489334in)-- + (1.96842in,0.472161in)--(2.06185in,0.459861in)-- + (2.1558in,0.452467in)--(2.25in,0.45in)--(2.3442in,0.452467in)-- + (2.43815in,0.459861in)--(2.53158in,0.472161in)-- + (2.62424in,0.489334in)--(2.71587in,0.511334in)-- + (2.80623in,0.538098in)--(2.89506in,0.569555in)-- + (2.98213in,0.605618in)--(3.06718in,0.646188in)-- + (3.15in,0.691154in)--(3.23035in,0.740393in)-- + (3.30801in,0.793769in)--(3.38278in,0.851137in)-- + (3.45444in,0.912339in)--(3.52279in,0.977208in)-- + (3.58766in,1.04556in)--(3.64886in,1.11722in)-- + (3.70623in,1.19199in)--(3.75961in,1.26965in)--(3.80885in,1.35in)-- + (3.85381in,1.43282in)--(3.89438in,1.51787in)-- + (3.93044in,1.60494in)--(3.9619in,1.69377in)-- + (3.98867in,1.78413in)--(4.01067in,1.87576in)-- + (4.02784in,1.96842in)--(4.04014in,2.06185in)-- + (4.04753in,2.1558in)--(4.05in,2.25in)--cycle; +\draw (4.05in,2.25in)--(2.25in,2.25in)--(0.45in,2.25in); +\draw (4.05in,4.39516in)--(2.57149in,2.63314in)--(1.09298in,0.87112in); +\draw (2.25in,2.25in)--(2.25in,3.15in)--(2.25in,4.05in); +\draw (0.104844in,4.05in)--(2.07742in,2.39481in)--(4.05in,0.739621in); +\draw (4.5in,4.05in)--(2.25in,4.05in)--(0in,4.05in); +\draw (4.05in,4.725in)--(4.05in,2.475in)--(4.05in,0.225in); +\pgftext[at={\pgfpoint{4.02233in}{2.27767in}}] {\makebox(0,0)[br]{\hbox{\color{rgb_000000}$A$}}} +\pgftext[at={\pgfpoint{0.477674in}{2.27767in}}] {\makebox(0,0)[bl]{\hbox{\color{rgb_000000}$A'$}}} +\pgftext[at={\pgfpoint{3.40702in}{3.54586in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}$B$}}} +\pgftext[at={\pgfpoint{0.926468in}{3.43469in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}$B'$}}} +\pgftext[at={\pgfpoint{1.10682in}{0.967979in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}$B''$}}} +\pgftext[at={\pgfpoint{3.57353in}{1.05147in}}] {\makebox(0,0)[r]{\hbox{\color{rgb_000000}$B'''$}}} +\pgftext[at={\pgfpoint{2.25in}{4.07767in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}$C$}}} +\pgftext[at={\pgfpoint{4.02233in}{4.42283in}}] {\makebox(0,0)[r]{\hbox{\color{rgb_000000}$F$}}} +\pgftext[at={\pgfpoint{4.07767in}{0.739621in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}$F'$}}} +\pgftext[at={\pgfpoint{3.78805in}{4.02233in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}$G$}}} +\pgftext[at={\pgfpoint{0.104844in}{4.07767in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}$G'$}}} +\pgftext[at={\pgfpoint{4.5in}{4.07767in}}] {\makebox(0,0)[br]{\hbox{\color{rgb_000000}$H$}}} +\pgftext[at={\pgfpoint{0in}{4.02233in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}$H'$}}} +\pgftext[at={\pgfpoint{4.02233in}{4.725in}}] {\makebox(0,0)[tr]{\hbox{\color{rgb_000000}$T$}}} +\pgftext[at={\pgfpoint{4.07767in}{0.225in}}] {\makebox(0,0)[bl]{\hbox{\color{rgb_000000}$T'$}}} +\pgftext[at={\pgfpoint{2.25in}{2.16698in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}$O$}}} +\end{tikzpicture} diff --git a/32973-t/images/sources/029a.xp b/32973-t/images/sources/029a.xp new file mode 100644 index 0000000..be6f224 --- /dev/null +++ b/32973-t/images/sources/029a.xp @@ -0,0 +1,54 @@ +/* -*-ePiX-*- */ +#include "epix.h" +using namespace ePiX; + +P O(0,0); +double th(50), MAX(1.25); + +int main() +{ + picture(P(-MAX,-MAX), P(MAX,MAX), "4.5 x 4.5in"); + + begin(); + degrees(); + bold(); + P pA(1,0), pA2(-1,0), pB(cis(th)), pB1(cis(th+90)), pB2(-pB), pB3(-pB1); + P pC(0,1), pF(1, Tan(th)), pG(Cot(th), 1), pG2(-Tan(th), 1), pF2(1,-Cot(th)); + P pH(MAX, 1), pH2(-MAX, 1), pT(1,1.1*MAX), pT2(1,-0.9*MAX); + + circle(); + line(pA, pA2); + line(pF, pB2); + line(O, pC); + line(pG2, pF2); + + line(pH, pH2); + line(pT, pT2); + + label(pA, P(-2,2), "$A$", tl); + label(pA2, P(2,2), "$A'$", tr); + + label(pB, P(0, -6), "$B$", b); + label(pB1, P(4, 2), "$B'$", r); + label(pB2, P(1, 7), "$B''$", t); + label(pB3, P(-4,-3), "$B'''$", l); + + label(pC, P(0,2), "$C$", t); + + label(pF, P(-2,2), "$F$", l); + label(pF2, P(2,0), "$F'$", r); + + label(pG, P(2,-2), "$G$", b); + label(pG2, P(0,2), "$G'$", t); + + label(pH, P(0,2), "$H$", tl); + label(pH2, P(0,-2), "$H'$", b); + + label(pT, P(-2,0), "$T$", bl); + label(pT2, P(2,0), "$T'$", tr); + + label(O, P(0, -6), "$O$", b); + + tikz_format(); + end(); +} diff --git a/32973-t/images/sources/033a.eepic b/32973-t/images/sources/033a.eepic new file mode 100644 index 0000000..335e37d --- /dev/null +++ b/32973-t/images/sources/033a.eepic @@ -0,0 +1,88 @@ +%% Generated from 033a.xp on Wed Jun 23 17:40:21 EDT 2010 by +%% ePiX-1.2.4 +%% +%% Cartesian bounding box: [-1.5,1.5] x [-1.25,1.25] +%% Actual size: 4.5 x 3.75in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% +\xdefinecolor{rgb_000000}{rgb}{0,0,0}% +\begin{tikzpicture} +\pgfsetlinewidth{0.4pt} +\useasboundingbox (0in,0in) rectangle (4.5in,3.75in); +\pgfsetlinewidth{0.8pt} +\draw (3.75in,1.875in)--(3.74794in,1.9535in)-- + (3.74178in,2.03179in)--(3.73153in,2.10965in)-- + (3.71722in,2.18687in)--(3.69889in,2.26323in)-- + (3.67658in,2.33853in)--(3.65037in,2.41255in)-- + (3.62032in,2.4851in)--(3.58651in,2.55599in)--(3.54904in,2.625in)-- + (3.50801in,2.69196in)--(3.46353in,2.75668in)-- + (3.41572in,2.81898in)--(3.36472in,2.8787in)-- + (3.31066in,2.93566in)--(3.2537in,2.98972in)-- + (3.19398in,3.04072in)--(3.13168in,3.08853in)-- + (3.06696in,3.13301in)--(3in,3.17404in)--(2.93099in,3.21151in)-- + (2.8601in,3.24532in)--(2.78755in,3.27537in)-- + (2.71353in,3.30158in)--(2.63823in,3.32389in)-- + (2.56187in,3.34222in)--(2.48465in,3.35653in)-- + (2.40679in,3.36678in)--(2.3285in,3.37294in)--(2.25in,3.375in)-- + (2.1715in,3.37294in)--(2.09321in,3.36678in)-- + (2.01535in,3.35653in)--(1.93813in,3.34222in)-- + (1.86177in,3.32389in)--(1.78647in,3.30158in)-- + (1.71245in,3.27537in)--(1.6399in,3.24532in)-- + (1.56901in,3.21151in)--(1.5in,3.17404in)--(1.43304in,3.13301in)-- + (1.36832in,3.08853in)--(1.30602in,3.04072in)-- + (1.2463in,2.98972in)--(1.18934in,2.93566in)--(1.13528in,2.8787in)-- + (1.08428in,2.81898in)--(1.03647in,2.75668in)-- + (0.991994in,2.69196in)--(0.950962in,2.625in)-- + (0.91349in,2.55599in)--(0.879682in,2.4851in)-- + (0.849629in,2.41255in)--(0.823415in,2.33853in)-- + (0.801111in,2.26323in)--(0.782779in,2.18687in)-- + (0.768467in,2.10965in)--(0.758217in,2.03179in)-- + (0.752056in,1.9535in)--(0.75in,1.875in)--(0.752056in,1.7965in)-- + (0.758217in,1.71821in)--(0.768467in,1.64035in)-- + (0.782779in,1.56313in)--(0.801111in,1.48677in)-- + (0.823415in,1.41147in)--(0.849629in,1.33745in)-- + (0.879682in,1.2649in)--(0.91349in,1.19401in)-- + (0.950962in,1.125in)--(0.991994in,1.05804in)-- + (1.03647in,0.993322in)--(1.08428in,0.931019in)-- + (1.13528in,0.871304in)--(1.18934in,0.81434in)-- + (1.2463in,0.760283in)--(1.30602in,0.709281in)-- + (1.36832in,0.661475in)--(1.43304in,0.616994in)-- + (1.5in,0.575962in)--(1.56901in,0.53849in)--(1.6399in,0.504682in)-- + (1.71245in,0.474629in)--(1.78647in,0.448415in)-- + (1.86177in,0.426111in)--(1.93813in,0.407779in)-- + (2.01535in,0.393467in)--(2.09321in,0.383217in)-- + (2.1715in,0.377056in)--(2.25in,0.375in)--(2.3285in,0.377056in)-- + (2.40679in,0.383217in)--(2.48465in,0.393467in)-- + (2.56187in,0.407779in)--(2.63823in,0.426111in)-- + (2.71353in,0.448415in)--(2.78755in,0.474629in)-- + (2.8601in,0.504682in)--(2.93099in,0.53849in)--(3in,0.575962in)-- + (3.06696in,0.616994in)--(3.13168in,0.661475in)-- + (3.19398in,0.709281in)--(3.2537in,0.760283in)-- + (3.31066in,0.81434in)--(3.36472in,0.871304in)-- + (3.41572in,0.931019in)--(3.46353in,0.993322in)-- + (3.50801in,1.05804in)--(3.54904in,1.125in)--(3.58651in,1.19401in)-- + (3.62032in,1.2649in)--(3.65037in,1.33745in)-- + (3.67658in,1.41147in)--(3.69889in,1.48677in)-- + (3.71722in,1.56313in)--(3.73153in,1.64035in)-- + (3.74178in,1.71821in)--(3.74794in,1.7965in)--(3.75in,1.875in)--cycle; +\draw (4.58359in,1.875in)--(2.25in,1.875in)--(-0.0835857in,1.875in); +\draw (3.21418in,3.02407in)--(2.25in,1.875in)--(1.28582in,0.725933in); +\draw (1.28582in,3.02407in)--(2.25in,1.875in)--(3.21418in,0.725933in); +\draw (2.25in,3.83311in)--(2.25in,1.875in)--(2.25in,-0.0831109in); +\draw (4.58359in,1.875in)--(2.25in,3.83311in)-- + (-0.0835857in,1.875in)--(2.25in,-0.0831109in)--(4.58359in,1.875in)--cycle; +\pgftext[at={\pgfpoint{3.72233in}{1.84733in}}] {\makebox(0,0)[tr]{\hbox{\color{rgb_000000}$A$}}} +\pgftext[at={\pgfpoint{0.777674in}{1.84733in}}] {\makebox(0,0)[tl]{\hbox{\color{rgb_000000}$A'$}}} +\pgftext[at={\pgfpoint{3.24186in}{3.05174in}}] {\makebox(0,0)[bl]{\hbox{\color{rgb_000000}$B$}}} +\pgftext[at={\pgfpoint{1.25814in}{3.05174in}}] {\makebox(0,0)[br]{\hbox{\color{rgb_000000}$B'$}}} +\pgftext[at={\pgfpoint{1.25814in}{0.698259in}}] {\makebox(0,0)[tr]{\hbox{\color{rgb_000000}$B''$}}} +\pgftext[at={\pgfpoint{3.21418in}{0.725933in}}] {\makebox(0,0)[tl]{\hbox{\color{rgb_000000}$B'''$}}} +\pgftext[at={\pgfpoint{2.27767in}{3.34733in}}] {\makebox(0,0)[tl]{\hbox{\color{rgb_000000}$C$}}} +\pgftext[at={\pgfpoint{2.25in}{3.86078in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}$K$}}} +\pgftext[at={\pgfpoint{2.25in}{-0.110785in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}$K'$}}} +\pgftext[at={\pgfpoint{4.61126in}{1.875in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}$S$}}} +\pgftext[at={\pgfpoint{-0.11126in}{1.875in}}] {\makebox(0,0)[r]{\hbox{\color{rgb_000000}$S'$}}} +\pgftext[at={\pgfpoint{2.41604in}{1.90267in}}] {\makebox(0,0)[bl]{\hbox{\color{rgb_000000}$O$}}} +\end{tikzpicture} diff --git a/32973-t/images/sources/033a.xp b/32973-t/images/sources/033a.xp new file mode 100644 index 0000000..8bff9f9 --- /dev/null +++ b/32973-t/images/sources/033a.xp @@ -0,0 +1,46 @@ +/* -*-ePiX-*- */ +#include "epix.h" +using namespace ePiX; + +P O(0,0); +double th(50); + +int main() +{ + picture(P(-1.5,-1.25), P(1.5,1.25), "4.5 x 3.75in"); + + begin(); + degrees(); + bold(); + P pA(1,0), pA2(-1,0), pB(cis(th)), pB1(cis(180-th)), pB2(-pB), pB3(-pB1); + P pC(0,1), pK(0, Csc(th)), pK2(-pK), pS(Sec(th),0), pS2(-pS); + + circle(); + line(pS, pS2); + line(pB, pB2); + line(pB1, pB3); + line(pK, pK2); + + quad(pS, pK, pS2, pK2); + + label(pA, P(-2,-2), "$A$", bl); + label(pA2, P(2,-2), "$A'$", br); + + label(pB, P(2, 2), "$B$", tr); + label(pB1, P(-2,2), "$B'$", tl); + label(pB2, P(-2,-2), "$B''$", bl); + label(pB3, P(0,0), "$B'''$", br); + + label(pC, P(2,-2), "$C$", br); + + label(pK, P(0,2), "$K$", t); + label(pK2, P(0,-2), "$K'$", b); + + label(pS, P(2,0), "$S$", r); + label(pS2, P(-2,0), "$S'$", l); + + label(O, P(12, 2), "$O$", tr); + + tikz_format(); + end(); +} diff --git a/32973-t/images/sources/041a.eepic b/32973-t/images/sources/041a.eepic new file mode 100644 index 0000000..8b520ad --- /dev/null +++ b/32973-t/images/sources/041a.eepic @@ -0,0 +1,81 @@ +%% Generated 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+\pgftext[at={\pgfpoint{1.98338in}{1.54958in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}$B$}}} +\pgftext[at={\pgfpoint{1.98338in}{-0.027674in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}$C$}}} +\end{tikzpicture} diff --git a/32973-t/images/sources/041a.xp b/32973-t/images/sources/041a.xp new file mode 100644 index 0000000..9164ff4 --- /dev/null +++ b/32973-t/images/sources/041a.xp @@ -0,0 +1,32 @@ +/* -*-ePiX-*- */ +#include "epix.h" +using namespace ePiX; + +P O(0,0); +double th(37.5); + +int main() +{ + picture(P(0,0), P(1,0.6), "2.5 x 1.5in"); + + begin(); + degrees(); + bold(); + double rad(Cos(th)); + P pB(cis(th)), pC(rad,0); + + line(O, P(1,0)); + line(O, pB); + line(pB, pC); + + dashed(); + ellipse(O, E_1, E_2, 0, th, 15); + ellipse(O, rad*E_1, rad*E_2, 0, th, 15); + + label(O, P(-2,-2), "$A$", b); + label(pB, P(0, 2), "$B$", t); + label(pC, P(0,-2), "$C$", b); + + tikz_format(); + end(); +} diff --git a/32973-t/images/sources/042a.eepic b/32973-t/images/sources/042a.eepic 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-*-ePiX-*- */ +#include "epix.h" +using namespace ePiX; + +P O(0,0); +int N(16); + +double theta(const P& tail, const P& head) +{ + P tmp(head - tail); + return Atan2(tmp.x2(), tmp.x1()); +} + +int main() +{ + picture(P(0,0), P(5,1.5), "6 x 1.8in"); + + begin(); + degrees(); + bold(); + + P pA1(0,1.5), pA2(0,1), + pB1(-1.75,0), pB2(-2,0), + pC1(1.25,0), pC2(-1,0); + + screen scr1(pB1, pA1 + pC1); + screen scr2(pB2, P(1, 1)); + + activate(scr1); + line(pA1, pB1); + line(pA1, pC1); + line(pB1, pC1); + line(O, pA1); + + double rad1a(norm(pB1 - pA1)), rad1b(norm(pC1 - pA1)); + dashed(); + ellipse(pB1, rad1a*E_1, rad1a*E_2, 0, theta(pB1, pA1), N); + ellipse(pC1, rad1b*E_1, rad1b*E_2, theta(pC1, pA1), 180, N); + solid(); + + label(O, P(0,-2), "$D$", b); + label(pA1, P(0, 2), "$A$", t); + label(pB1, P(0,-2), "$B$", b); + label(pC1, P(0,-2), "$C$", b); + deactivate(scr1); + inset(scr1, P(0,0), P(2,1)); + + + + activate(scr2); + line(pA2, pB2); + line(pA2, pC2); + line(pB2, pC2 + P(norm(pC2 - pA2), 0)); + line(O, pA2); + + double rad2a(norm(pB2 - pA2)), rad2b(norm(pC2 - pA2)); + dashed(); + ellipse(pB2, rad2a*E_1, rad2a*E_2, 0, theta(pB2, pA2), N); + ellipse(pC2, rad2b*E_1, rad2b*E_2, 0, theta(pC2, pA2), N); + solid(); + + label(O, P(0,-2), "$D$", b); + label(pA2, P(0, 2), "$A$", t); + label(pB2, P(0,-2), "$B$", b); + label(pC2, P(0,-2), "$C'$", b); + deactivate(scr2); + inset(scr2, P(2.5,0), P(5.5,1)); + + tikz_format(); + end(); +} diff --git a/32973-t/images/sources/044a.eepic b/32973-t/images/sources/044a.eepic new file mode 100644 index 0000000..8c5b72d --- /dev/null +++ b/32973-t/images/sources/044a.eepic @@ -0,0 +1,83 @@ +%% Generated from 044a.xp on Wed Jun 23 18:26:48 EDT 2010 by +%% ePiX-1.2.4 +%% +%% Cartesian bounding box: [-0.5,2.5] x [0,1] +%% Actual size: 4.5 x 1.5in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% +\xdefinecolor{rgb_000000}{rgb}{0,0,0}% +\begin{tikzpicture} +\pgfsetlinewidth{0.4pt} +\useasboundingbox (0in,0in) 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+\pgftext[at={\pgfpoint{4.5in}{-0.027674in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}$C$}}} +\end{tikzpicture} diff --git a/32973-t/images/sources/044a.xp b/32973-t/images/sources/044a.xp new file mode 100644 index 0000000..fbea300 --- /dev/null +++ b/32973-t/images/sources/044a.xp @@ -0,0 +1,33 @@ +/* -*-ePiX-*- */ +#include "epix.h" +using namespace ePiX; + +P O(0,0); +double th(20); + +int main() +{ + picture(P(-0.5,0), P(2.5,1), "4.5 x 1.5in"); + + begin(); + degrees(); + bold(); + + P pA(0,Cos(th)), pB1(-Sin(th), 0), pB2(Sin(th), 0), pC(xmax(),0); + + line(pA, pB1); + line(pA, pB2); + line(pA, pC); + line(pB1, pC); + + arc(pA, 1, -90 - th - 10, -90 + th + 10); + + label(pA, P(0,2), "$A$", t); + label(pB1, P(-2,-3), "$B_1$", b); + label(pB2, P(2,-3), "$B_2$", b); + + label(pC, P(0,-2), "$C$", b); + + tikz_format(); + end(); +} diff --git a/32973-t/images/sources/045a.eepic b/32973-t/images/sources/045a.eepic new file mode 100644 index 0000000..7a7f26e --- /dev/null +++ b/32973-t/images/sources/045a.eepic @@ -0,0 +1,80 @@ +%% Generated from 045a.xp on Wed Jun 23 18:35:31 EDT 2010 by +%% ePiX-1.2.4 +%% +%% Cartesian bounding box: [-1,1] x [-1,1] +%% Actual size: 4 x 4in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% +\xdefinecolor{rgb_000000}{rgb}{0,0,0}% +\begin{tikzpicture} +\pgfsetlinewidth{0.4pt} +\useasboundingbox (0in,0in) rectangle (4in,4in); +\pgfsetlinewidth{0.8pt} +\draw (4in,2in)--(3.99726in,2.10467in)--(3.98904in,2.20906in)-- + (3.97538in,2.31287in)--(3.9563in,2.41582in)-- + (3.93185in,2.51764in)--(3.90211in,2.61803in)-- + (3.86716in,2.71674in)--(3.82709in,2.81347in)-- + (3.78201in,2.90798in)--(3.73205in,3in)--(3.67734in,3.08928in)-- + (3.61803in,3.17557in)--(3.55429in,3.25864in)-- + (3.48629in,3.33826in)--(3.41421in,3.41421in)-- + (3.33826in,3.48629in)--(3.25864in,3.55429in)-- + (3.17557in,3.61803in)--(3.08928in,3.67734in)--(3in,3.73205in)-- + (2.90798in,3.78201in)--(2.81347in,3.82709in)-- + 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+\pgftext[at={\pgfpoint{1.97233in}{1.95849in}}] {\makebox(0,0)[tr]{\hbox{\color{rgb_000000}$C$}}} +\pgftext[at={\pgfpoint{0.027674in}{1.95849in}}] {\makebox(0,0)[tl]{\hbox{\color{rgb_000000}$D$}}} +\pgftext[at={\pgfpoint{4.02767in}{2in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}$E$}}} +\pgftext[at={\pgfpoint{3.73205in}{0.972326in}}] {\makebox(0,0)[tl]{\hbox{\color{rgb_000000}$F$}}} +\end{tikzpicture} diff --git a/32973-t/images/sources/045a.xp b/32973-t/images/sources/045a.xp new file mode 100644 index 0000000..ac643da --- /dev/null +++ b/32973-t/images/sources/045a.xp @@ -0,0 +1,41 @@ +/* -*-ePiX-*- */ +#include "epix.h" +using namespace ePiX; + +P O(0,0); +double th(55), th2(-30); + +int main() +{ + picture(P(-1,-1), P(1,1), "4 x 4in"); + + begin(); + degrees(); + bold(); + + P pB(cis(th)), pE(1,0), pD(-1,0), pF(cis(th2)); + Segment DE(pD, pE), BF(pB, pF); + P pA(DE*BF); + + circle(); + line(O, pB); + line(O, pF); + + line(pD, pE); + + line(pB, pD); + line(pB, pE); + line(pB, pF); + + label(pA, P(0,-3), "$A$", bl); + label(pB, P(0, 2), "$B$", tr); + + label(O, P(-2,-3), "$C$", bl); + label(pD, P( 2,-3), "$D$", br); + + label(pE, P( 2, 0), "$E$", r); + label(pF, P( 0,-2), "$F$", br); + + tikz_format(); + end(); +} diff --git a/32973-t/images/sources/047a.eepic b/32973-t/images/sources/047a.eepic new file mode 100644 index 0000000..138656c --- /dev/null +++ b/32973-t/images/sources/047a.eepic @@ -0,0 +1,82 @@ +%% Generated from 047a.xp on Wed Jun 23 18:49:42 EDT 2010 by +%% ePiX-1.2.4 +%% +%% Cartesian bounding box: [-2,2] x [-1,1.5] +%% Actual size: 3.6 x 2.25in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% +\xdefinecolor{rgb_000000}{rgb}{0,0,0}% +\begin{tikzpicture} +\pgfsetlinewidth{0.4pt} +\useasboundingbox (0in,0in) rectangle (3.6in,2.25in); +\pgfsetlinewidth{0.8pt} +\draw (2.7in,0.9in)--(2.69877in,0.947102in)-- + (2.69507in,0.994076in)--(2.68892in,1.04079in)-- + (2.68033in,1.08712in)--(2.66933in,1.13294in)-- + 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{\makebox(0,0)[br]{\hbox{\color{rgb_000000}$c$}}} +\end{tikzpicture} diff --git a/32973-t/images/sources/047a.xp b/32973-t/images/sources/047a.xp new file mode 100644 index 0000000..d331854 --- /dev/null +++ b/32973-t/images/sources/047a.xp @@ -0,0 +1,43 @@ +/* -*-ePiX-*- */ +#include "epix.h" +using namespace ePiX; + +P O(0,0); +double th1(-90), th2(35), th3(135); + +int main() +{ + picture(P(-2,-1), P(2,1.5), "3.6 x 2.25in"); + + begin(); + degrees(); + bold(); + + P pa(cis(th1)), pb(cis(th2)), pc(cis(th3)); + + Segment BC(pa, pa + J(pa)), CA(pb, pb + J(pb)), AB(pc, pc + J(pc)); + + P pA(CA*AB), pB(AB*BC), pC(BC*CA); + + circle(); + triangle(pA, pB, pC); + line(O, pa); + line(O, pb); + line(O, pc); + + line(O, pA); + line(O, pB); + line(O, pC); + + label(O, P(-8,2), "$O$", l); + label(pA, P(0,2), "$A$", t); + label(pB, P(0,-2), "$B$", b); + label(pC, P(0,-2), "$C$", b); + + label(pa, P(0,-4), "$a$", b); + label(pb, P(2, 2), "$b$", tr); + label(pc, P(-2,2), "$c$", tl); + + tikz_format(); + end(); +} diff --git a/32973-t/images/sources/052a.eepic b/32973-t/images/sources/052a.eepic new file mode 100644 index 0000000..7d20677 --- /dev/null +++ b/32973-t/images/sources/052a.eepic @@ -0,0 +1,249 @@ +%% Generated from 052a.xp on Wed Jun 23 21:20:31 EDT 2010 by +%% ePiX-1.2.4 +%% +%% Cartesian bounding box: [0,5] x [-1,1.6] +%% Actual size: 5 x 2.6in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% +\xdefinecolor{rgb_000000}{rgb}{0,0,0}% +\begin{tikzpicture} +\pgfsetlinewidth{0.4pt} +\useasboundingbox (0in,0in) rectangle (5in,2.6in); +\pgfsetlinewidth{0.8pt} +\draw (-0.179434in,1in)--(2.5in,1in)--(5.17943in,1in); +\draw (0.5in,1in)--(0.5in,1.25in)--(0.5in,1.5in); +\draw (0.5in,1.5in)--(1in,2in)--(1.5in,2.5in); +\draw (1.5in,2.5in)--(1.5in,1.75in)--(1.5in,1in); +\draw (2in,1in)--(2in,0.625in)--(2in,0.25in); +\draw (2in,0.25in)--(2.5in,1.25in)--(3in,2.25in); +\draw (3in,2.25in)--(3in,1.625in)--(3in,1in); +\draw 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{\makebox(0,0)[bl]{\hbox{\color{rgb_000000}$C$}}} +\pgftext[at={\pgfpoint{3.02767in}{0.222326in}}] {\makebox(0,0)[tl]{\hbox{\color{rgb_000000}$C$}}} +\end{tikzpicture} diff --git a/32973-t/images/sources/052a.xp b/32973-t/images/sources/052a.xp new file mode 100644 index 0000000..bbb06cf --- /dev/null +++ b/32973-t/images/sources/052a.xp @@ -0,0 +1,72 @@ +/* -*-ePiX-*- */ +#include "epix.h" +using namespace ePiX; + +P O(0,0); +double rad(0.4); + +double theta(const P& tail, const P& head) +{ + P tmp(head - tail); + return Atan2(tmp.x2(), tmp.x1()); +} + +void ray(const P& tail, const P& head, double t) +{ + line(tail, head + t*(head - tail)); +} + +int main() +{ + picture(P(0,-1), P(5,1.6), "5 x 2.6in"); + + begin(); + degrees(); + bold(); + + P A1(0.5,0.5), B1(1.5,1.5), C1(B1.x1(),A1.x2()), A1p(A1&E_1), B1p(B1&E_1); + P A2(2,-0.75), B2(3, 1.25), C2(B2.x1(),A2.x2()), A2p(A2&E_1), B2p(B2&E_1); + P A3(4.5,0.6),B3(3.5,1.6), C3(B3.x1(),A3.x2()), A3p(A3&E_1), B3p(B3&E_1); + + line(P(xmin(),0), P(xmax(),0), 10); + line(A1p, A1); line(A1, B1); line(B1, B1p); + line(A2p, A2); line(A2, B2); line(B2, B2p); + line(A3p, A3); line(A3, B3); line(B3, B3p); + + dashed(); + dash_size(6); + ray(A1, C1, 0.3); + ray(A2, C2, 0); + ray(C3, A3, 0.6); + + line(B2, C2); + + ellipse(A1, rad*E_1, rad*E_2, 0, theta(A1, B1), 4); + ellipse(A2, rad*E_1, rad*E_2, 0, theta(A2, B2), 4); + ellipse(A3, rad*E_1, rad*E_2, 0, theta(A3, B3), 8); + + label(P(xmin(),0), P(0,-4), "$X'$", b); + label(P(xmax(),0), P(0,-4), "$X$", b); + + label(A1p, P(0,-4), "$A'$", b); + label(A2p, P(0, 2), "$A'$", t); + label(A3p, P(0,-4), "$A'$", b); + + label(B1p, P(0,-4), "$B'$", b); + label(B2p, P(0,-4), "$B'$", br); + label(B3p, P(0,-4), "$B'$", b); + + label(A1, P(-2,2), "$A$", tl); + label(A2, P(0,-4), "$A$", b); + label(A3, P(2,-4), "$A$", br); + + label(B1, P(0,2), "$B$", t); + label(B2, P(0,2), "$B$", t); + label(B3, P(0,2), "$B$", t); + + label(C1, P(2, 2), "$C$", tr); + label(C2, P(2,-2), "$C$", br); + + tikz_format(); + end(); +} diff --git a/32973-t/images/sources/052b.eepic b/32973-t/images/sources/052b.eepic new file mode 100644 index 0000000..83ca01a --- /dev/null +++ b/32973-t/images/sources/052b.eepic @@ -0,0 +1,61 @@ +%% Generated from 052b.xp on Wed Jun 23 21:31:41 EDT 2010 by +%% ePiX-1.2.4 +%% +%% Cartesian bounding box: [0,5] x [-1,1.6] +%% Actual size: 5 x 2.6in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% +\xdefinecolor{rgb_000000}{rgb}{0,0,0}% +\begin{tikzpicture} +\pgfsetlinewidth{0.4pt} +\useasboundingbox (0in,0in) rectangle (5in,2.6in); +\pgfsetlinewidth{0.8pt} +\draw (-0.179434in,1in)--(2.5in,1in)--(5.17943in,1in); +\draw (0.5in,0.5in)--(2.125in,0.915in)--(3.75in,1.33in); +\draw (3.75in,1.33in)--(2.875in,1.665in)--(2in,2in); +\draw (2in,2in)--(3.25in,2.25in)--(4.5in,2.5in); +\draw (0.5in,1in)--(0.5in,0.75in)--(0.5in,0.5in); +\draw (3.75in,1in)--(3.75in,1.165in)--(3.75in,1.33in); +\draw (2in,1in)--(2in,1.5in)--(2in,2in); +\draw 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+\pgftext[at={\pgfpoint{3.75in}{1.35767in}}] {\makebox(0,0)[bl]{\hbox{\color{rgb_000000}$B$}}} +\pgftext[at={\pgfpoint{1.97233in}{2.02767in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}$C$}}} +\pgftext[at={\pgfpoint{4.5in}{2.52767in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}$D$}}} +\end{tikzpicture} diff --git a/32973-t/images/sources/052b.xp b/32973-t/images/sources/052b.xp new file mode 100644 index 0000000..9ad7e2a --- /dev/null +++ b/32973-t/images/sources/052b.xp @@ -0,0 +1,50 @@ +/* -*-ePiX-*- */ +#include "epix.h" +using namespace ePiX; + +void ray(const P& tail, const P& head, double t) +{ + line(tail, head + t*(head - tail)); +} + +int main() +{ + picture(P(0,-1), P(5,1.6), "5 x 2.6in"); + + begin(); + degrees(); + bold(); + + P pA(0.5,-0.5), pB(3.75,0.33), pC(2, 1), pD(4.5, 1.5); + P pAp(pA&E_1), pBp(pB&E_1), pCp(pC&E_1), pDp(pD&E_1); + + line(P(xmin(),0), P(xmax(),0), 10); + line(pA, pB); + line(pB, pC); + line(pC, pD); + + line(pAp, pA); + line(pBp, pB); + line(pCp, pC); + line(pDp, pD); + + dashed(); + dash_size(6); + line(pA, pA+E_1); + + label(P(xmin(),0), P(0,2), "$X'$", t); + label(P(xmax(),0), P(0,2), "$X$", t); + + label(pAp, P(0, 2), "$A'$", t); + label(pBp, P(0,-4), "$B'$", b); + label(pCp, P(-4,2), "$C'$", tl); + label(pDp, P(0,-4), "$D'$", b); + + label(pA, P(0,-4), "$A$", b); + label(pB, P(0, 2), "$B$", tr); + label(pC, P(-2,2), "$C$", t); + label(pD, P(0, 2), "$D$", t); + + tikz_format(); + end(); +} diff --git a/32973-t/images/sources/053a.eepic b/32973-t/images/sources/053a.eepic new file mode 100644 index 0000000..338ee53 --- /dev/null +++ b/32973-t/images/sources/053a.eepic @@ -0,0 +1,115 @@ +%% Generated from 053a.xp on Wed Jun 23 19:20:38 EDT 2010 by +%% ePiX-1.2.4 +%% +%% Cartesian bounding box: [0,1] x [0,1] +%% Actual size: 2.5 x 2.5in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% +\xdefinecolor{rgb_000000}{rgb}{0,0,0}% +\begin{tikzpicture} +\pgfsetlinewidth{0.4pt} +\useasboundingbox (0in,0in) rectangle 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+1,41 @@ +/* -*-ePiX-*- */ +#include "epix.h" +using namespace ePiX; + +P O(0,0); +double th1(20), th2(35); + +int main() +{ + picture(P(0,0), P(1,1), "2.5 x 2.5in"); + + begin(); + degrees(); + bold(); + + P pA(1,0), pB(cis(th1)), pD(cis(th1 + th2)), pE(pD&E_1); + P pF(pD - (pD % pB)); + + line(O, pA); + line(O, pB); + line(O, pD); + + line(pD, pE); + line(pD, pF); + arc(O, 1, 0, th1 + th2); + + dashed(); + dash_size(6); + line(pF, pF&E_1); + line(pF, P(pE.x1(), pF.x2())); + + label(pA, P(2,0), "$A$", r); + label(pB, P(2,0), "$B$", tr); + label(O, P(-2,-2), "$C$", b); + label(pD, P( 0,0), "$D$", tl); + label(pE, P( 0,-2), "$E$", b); + label(pF, P( 4, 4), "$F$", t); + + tikz_format(); + end(); +} diff --git a/32973-t/images/sources/054a.eepic b/32973-t/images/sources/054a.eepic new file mode 100644 index 0000000..3d0385f --- /dev/null +++ b/32973-t/images/sources/054a.eepic @@ -0,0 +1,163 @@ +%% Generated from 054a.xp on Wed Jun 23 19:23:36 EDT 2010 by +%% ePiX-1.2.4 +%% +%% Cartesian 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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 *** +% % +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % + +\def\ebook{32973} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %% +%% Packages and substitutions: %% +%% %% +%% book: Required. %% +%% inputenc: Standard DP encoding. Required. %% +%% babel: Greek snippets. Required. %% +%% %% +%% calc: Infix arithmetic. Required. %% +%% %% +%% ifthen: Logical conditionals. 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Hwang, Laura Wisewell, + Project Gutenberg Online Distributed Proofreading Team, + University of Glasgow Department of Mathematics}, + pdfstartview=Fit, % default value + pdfstartpage=1, % default value + pdfpagemode=UseNone, % default value + bookmarks=true, % default value + linktocpage=false, % default value + pdfpagelayout=\PDFPageLayout, + pdfdisplaydoctitle, + pdfpagelabels=true, + bookmarksopen=true, + bookmarksopenlevel=0, + colorlinks=true, + linkcolor=\HLinkColor]{hyperref}[2007/02/07] + +%%%% Re-crop, squash screen-formatted version, omit blank verso pages +\ifthenelse{\not\boolean{ForPrinting}}{% + \hypersetup{pdfpagescrop= 0 15 378 566}% +% \raggedbottom% + \renewcommand{\cleardoublepage}{\clearpage} +}{}% Else do nothing + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%% COMMANDS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%%%% Fixed-width environment to format PG boilerplate %%%% +% 9.2pt leaves no overfull hbox at 80 char line width +\newenvironment{PGtext}{% +\begin{alltt} 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Sectioning: Chapter, Preface, Section, Article +\newcommand{\ChapterHead}[1]{% + \centering\normalfont\MakeUppercase{#1}% +} + +\newcommand{\Paragraph}[2][]{% + \ifthenelse{\not\equal{#1}{}}{#1}{}% + \textit{#2}% +} + +\newcommand{\BookTitle}{% + \mainmatter + \renewcommand{\thepage}{\texorpdfstring{\osf{\arabic{page}}}{\arabic{page}}} + \phantomsection\pdfbookmark[-1]{Main Matter.}{Main Matter}% + \thispagestyle{plain} + \begin{center} + \setlength{\TmpLen}{6pt}% + \Large ELEMENTS \\[\TmpLen] + \normalsize OF \\[\TmpLen] + \Large PLANE TRIGONOMETRY. + \end{center} + \bigskip +} + +% \Chapter{N}{Title} -- for numbered chapters +\newcommand{\Chapter}[3][]{% + \ifthenelse{\not\equal{#2}{I.}}{% + \ifthenelse{\boolean{ForPrinting}}{\fancyhead[RE]{}}{}% + \clearpage\fancyhf{}\cleardoublepage + \thispagestyle{plain}% + }{} + \phantomsection\pdfbookmark[0]{Chapter #2}{Chapter #2}% + + \section*{\ChapterHead{\Large Chapter #2}} + \subsection*{\ChapterHead{\small #3}} + 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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 *** +\end{PGtext} +\end{minipage} +\end{center} + + +%%%% Credits and transcriber's note %%%% +\clearpage +\thispagestyle{empty} + +\begin{center} +\begin{minipage}{\textwidth} +\begin{PGtext} +Produced by Andrew D. Hwang, Laura Wisewell and the Online +Distributed Proofreading Team at http://www.pgdp.net (The +original copy of this book was generously made available +for scanning by the Department of Mathematics at the +University of Glasgow.) +\end{PGtext} +\end{minipage} +\end{center} +\vfill + +\begin{minipage}{0.85\textwidth} +\small +\pdfbookmark[0]{Transcriber's Note.}{Transcriber's Note} +\subsection*{\centering\normalfont\scshape% +\normalsize\MakeLowercase{\TransNote}}% + +\raggedright +\TransNoteText +\end{minipage} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%% +\cleardoublepage +\frontmatter +\normalsize + +\DPPageSep{001.png}{i}% +% title page +\setlength{\TmpLen}{6pt}% +\thispagestyle{empty}% +{\stretchyspace% +\begin{center} +\Large ELEMENTS \\[2\TmpLen] +\small OF \\[2\TmpLen] +\LARGE PLANE TRIGONOMETRY \\[2\TmpLen] +\normalsize FOR THE USE OF THE JUNIOR CLASS OF \\[\TmpLen] +MATHEMATICS IN THE UNIVERSITY \\[\TmpLen] +OF GLASGOW. +\vfill + +\small BY \\[\TmpLen] +\Large HUGH BLACKBURN, M.A. \\[\TmpLen] +\scriptsize PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF GLASGOW, \\[\TmpLen] +LATE FELLOW OF TRINITY COLLEGE, CAMBRIDGE. +\vfill + +\normalsize% +\textgoth{\Large London and New York:} \\[\TmpLen] +\Large MACMILLAN AND CO. \\[\TmpLen] +\Date{1871}. \\[\TmpLen] +\normalsize [\textit{All Rights reserved}.] +\end{center} +\DPPageSep{002.png}{ii}% +\newpage\null +% half title page +\thispagestyle{empty}% +\begin{center} +\vfill +\textgoth{\Large Cambridge}: \\[2\TmpLen] +PRINTED BY C. J. CLAY, M.A. \\[2\TmpLen] +AT THE UNIVERSITY PRESS. +\vfill +\end{center}}%[** TN: End of \stretchyspace] +\DPPageSep{003.png}{iii}% + + +\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 +and English Verse, Translation, and an Introduction. By \Name{Bernard +Drake}, M.A., late Fellow of King's College, Cambridge. +8vo. \Price{3s. 6d.}} +% +{The Greek text adopted in this Edition is based upon that of Wellauer, +which may be said, in general terms, to represent that of the best manuscripts. +But in correcting the Text, and in the Notes, advantage has been +taken of the suggestions of Hermann, Paley, Linwood, and other commentators. +In the Translation, the simple character of the æschylean +dialogues has generally enabled the author to render them without any +material deviation from the construction and idioms of the original Greek.} +\DPPageSep{058.png}{54}% + +\Book{\Title{ARISTOTLE ON FALLACIES; OR, THE SOPHISTICI +ELENCHI.} With a Translation and Notes by \Name{Edward Poste}, +M.A., Fellow of Oriel College, Oxford. 8vo. \Price{8s. 6d.}} +% +{Besides the doctrine of Fallacies, Aristotle offers, either in this treatise +or in other passages quoted in the commentary, various glances over the +world of science and opinion, various suggestions or problems which are +still agitated, and a vivid picture of the ancient system of dialectics, which +it is hoped may be found both interesting and instructive.} + +\Book{\Author{Aristotle.}---\Title{AN INTRODUCTION TO ARISTOTLE'S +RHETORIC.} With Analysis, Notes, and Appendices. By \Name{E.~M. +Cope}, Senior Fellow and Tutor of Trinity College, Cambridge. +8vo. \Price{14s.}} +% +{This work is introductory to an edition of the Greek Text of Aristotle's +Rhetoric, which is in course of preparation. Its object is to render that +treatise thoroughly intelligible. The author has aimed to illustrate, as +preparatory to the detailed explanation of the work, the general bearings +and relations of the Art of Rhetoric in itself, as well as the special mode of +treating it adopted by Aristotle in his peculiar system. The evidence upon +obscure or doubtful questions connected with the subject is examined; and +the relations which Rhetoric bears, in Aristotle's view, to the kindred art +of Logic are fully considered. A connected Analysis of the work is given, +sometimes in the form of paraphrase; and a few important matters are +separately discussed in Appendices. There is added, as a general Appendix, +by way of specimen of the antagonistic system of Isocrates and others, a +complete analysis of the treatise called \textgreek{<Rhtoriq`h pr`os >Al'exandron}, with a +discussion of its authorship and of the probable results of its teaching.} + +\Book{\Author{Cicero.}---\Title{THE SECOND PHILIPPIC ORATION.} With an +Introduction and Notes, translated from the German of \Name{Karl +Halm}. Edited, with Corrections and Additions, by \Name{John E.~B. +Mayor}, M.A., Fellow and Classical Lecturer of St.~John's +College, Cambridge. Third Edition, revised. Fcap.~8vo. \Price{5s.}} +% +{This volume opens with a List of Books useful to the Student of Cicero, +including History, Chronology, Lexicons, and some account of various +editions, mostly German, of the works of Cicero. The Introduction is +\DPPageSep{059.png}{55}% +based on Halm: where Halm gives a reference to a classic, the passage has +been commonly printed at length; where the reference is to Halm's notes +on other Ciceronian speeches, or to modern books, the additional matter has +been incorporated: and the numerous Greek quotations have been rendered +into English. The English editor has further illustrated the work by +additions drawn, for the most part, \Inum{1}~from the ancient authorities; \Inum{2}~from +his own private marginal references, and from collections; \Inum{3}~from +the notes of previous commentators. A copious `argument' is also given.} + +\Book{\Title{THE ORATIONS OF CICERO AGAINST CATILINA.} With +Notes translated chiefly from Halm. By \Name{A.~S. Wilkins}, M.A. +Fcap.~8vo. \Price{3s. 6d.}} +% +{This edition is a reprint of the one prepared by Professor Halm for +\DPtypo{Orellli's}{Orelli's} Cicero. The historical introduction of Mr.\ Wilkins brings +\DPtypo{to ether}{together} +all the details which are known respecting Catiline and his +relations with the great orator. A list of passages where conjectures +have been admitted into the text, and also of all variations from the text +of Kayser \(\Date{1862}\) is added at the end. Finally the English Editor has +subjoined a large number of notes, both original \(distinguished by a +square bracket\) and selected from Curtius, Schleischer, Corssen, and +other well-known critics, an analysis of the orations, and an index.} + +\Book{\Title{DEMOSTHENES ON THE CROWN.} The Greek Text with +English Notes. By \Name{B. Drake}, M.A., late Fellow of King's +College, Cambridge. Fourth Edition, to which is prefixed +ÆSCHINES AGAINST CTESIPHON, with English Notes. +Fcap.~8vo. \Price{5s.}} +% +{An Introduction discusses the immediate causes of the two orations, and +their general character. The Notes contain frequent references to the best +authorities. Among the appendices at the end is a chronological table of +the life and public career of Æschines and Demosthenes.} + +\Book{\Author{Hodgson.}---\Title{MYTHOLOGY FOR LATIN VERSIFICATION.} +A brief Sketch of the Fables of the Ancients, prepared to be +rendered into Latin Verse for Schools. By \Name{F. Hodgson}, B.D., +late Provost of Eton. New Edition, revised by \Name{F.~C. Hodgson}, +M.A. 18mo. \Price{3s}.% +\DPPageSep{060.png}{56}% +} +% +{The late Provost of Eton has here supplied a help to the composition of +Latin Verse, combined with a brief introduction to Classical Mythology. +In this new edition a few mistakes have been rectified; rules have been +added to the Prosody; and a more uniform system has been adopted with +regard to the help afforded.} + +\Book{\Author{Juvenal.}---Thirteen Satires of \Title{JUVENAL}. With a Commentary. +By \Name{John E.~B. Mayor}, M.A., Fellow of St.~John's College, +Cambridge. Second Edition, enlarged. Part~I\@. Crown 8vo.\ sewed. +\Price{3s. 6d.}} +% +{The text is accompanied by a copious Commentary. For various notes +the author is indebted to Professors Munro and Conington. All the +citations have been taken anew from the original authors.} + +\Book{\Author{Marshall.}---\Title{A TABLE OF IRREGULAR GREEK VERBS} +classified according to the arrangement of Curtius' Greek Grammar. +By \Name{J.~M. Marshall}, M.A., Fellow and late Lecturer of Brasenose +College, Oxford; one of the Masters in Clifton College. 8vo. +cloth. \Price{1s.}} +% +{The system of this table has been borrowed from the excellent Greek +Grammar of Dr. Curtius.} + +\Book{\Author{Mayor (John E.~B.)}---\Title{FIRST GREEK READER.} Edited +after \Name{Karl Halm}, with Corrections and large Additions by \Name{John +E.~B. Mayor}, M.A. Fellow and Classical Lecturer of St.~John's +College, Cambridge. Second and Cheaper Edition. Fcap.~8vo. +\Price{4s. 6d.}} +% +{A selection of short passages, serving to illustrate especially the Greek +Accidence. A good deal of syntax is incidentally taught, and Madvig and +other books are cited, for the use of masters: but no learner is expected to +know more of syntax than is contained in the Notes and Vocabulary. +A preface ``To the Reader,'' not only explains the aim and method of +the volume, but also deals with classical instruction generally. The +extracts are uniformly in the Attic dialect, and any Hellenistic forms +occurring in the original classic authors, such as Ælian and Polybius, +have been discarded in favour of the corresponding Attic expressions. +This book may be used in connexion with Mayor's ``Greek for Beginners.''} +\DPPageSep{061.png}{57}% + +\Book{\Author{Mayor (Joseph B.)}---\Title{GREEK FOR BEGINNERS.} By the +Rev.\ \Name{J.~B. Mayor}, M.A., Professor of Classical Literature in +King's College, London. Part~I., with Vocabulary, \Price{1s. 6d.}; +Parts II.~and~III., with Vocabulary and Index, \Price{3s. 6d.}; complete +in one vol., fcap.~8vo.\ cloth\DPtypo{,}{.} \Price{4s. 6d.}} +% +{The distinctive method of this book consists in building up a boy's +knowledge of Greek upon the foundation of his knowledge of English and +Latin, instead of trusting everything to the unassisted memory. The +forms and constructions of Greek have been thoroughly compared with +those of Latin, and no Greek words have been used in the earlier part of +the book except such as have connexions either in English or Latin. Each +step leads naturally on to its successor, grammatical forms and rules are +at once applied in a series of graduated exercises, accompanied by ample +vocabularies. Thus the book serves as Grammar, Exercise book, and +Vocabulary. Where possible, the Grammar has been simplified; the +ordinary ten declensions are reduced to three, which correspond to the +first three in Latin; and the system of stems is adopted. A general +Vocabulary, and Index of Greek words, completes the work.} + +\Book{\Author{Peile (John, M.A.)}---\Title{AN INTRODUCTION TO GREEK +AND LATIN ETYMOLOGY.} By \Name{John Peile}, M.A., Fellow +and Assistant Tutor of Christ's College, Cambridge, formerly +Teacher of Sanskrit in the University of Cambridge. 8vo. \Price{10s. 6d.}} +% +{These Philological Lectures are the result of Notes made during the +author's reading during the last three or four years. These Notes were +put into the shape of lectures, delivered at Christ's College, during the last +May term, as one set in the ``Intercollegiate'' list. They are now printed +with some additions and modifications, but substantially as they were +delivered.} + +\Book{\Author{Plato.}---\Title{THE REPUBLIC OF PLATO.} Translated into English, +with an Analysis and Notes, by \Name{J. Ll. Davies}, M.A., and \Name{D.~J. +Vaughan}, M.A\@. Third Edition, with Vignette Portraits of \DPtypo{Plate}{Plato} +and Socrates, engraved by \Name{Jeens} from an Antique Gem. 18mo. +\Price{4s. 6d.}% +\DPPageSep{062.png}{58}% +} +% +{An introductory notice supplies some account of the life of Plato, and +the translation is preceded by an elaborate analysis. ``The translators +have,'' in the Judgment of the \emph{Saturday Review}, ``produced a book which +any reader, whether acquainted with the original or not, can peruse with +pleasure as well as profit.''} + +\Book{\Author{Plautus (Ramsay).}---\Title{THE MOSTELLARIA OF PLAUTUS.} +With Notes Critical and Explanatory, Prolegomena, and +Excursus. By \Name{William Ramsay}, M.A., formerly Professor of +Humanity in the University of Glasgow. Edited by Professor +\Name{George G. Ramsay}, M.A., of the University of Glasgow. 8vo. +\Price{14s.}} +% +{``The fruits of that exhaustive research and that ripe and well-digested +scholarship which its author brought to bear upon everything that he +undertook are visible throughout it. It is furnished with a complete +apparatus of prolegomena, notes, and excursus; and for the use of veteran +scholars it probably leaves nothing to be desired.''---\Cite{Pall Mall Gazette}.} + +\Book{\Author{Potts (Alex.\ W., M.A.)}---\Title{HINTS TOWARDS LATIN +PROSE COMPOSITION.} By \Name{Alex.~W. Potts}, M.A., late +Fellow of St.~John's College, Cambridge; Assistant Master in +Rugby School; and Head Master of the Fettes College, Edinburgh. +Second Edition, enlarged. Extra fcap.~8vo.\ cloth. \Price{3s.}} +% +{Those engaged in Classical teaching seem to be unanimously of the +opinion that Composition in Latin Prose is not only the most efficient +method of acquiring a mastery of the Latin language, but is in itself +a valuable means of mental training, and an admirable corrective of some +of the worst features in English writing. An attempt is here made to +give students, after they have mastered ordinary syntactical rules, some idea +of the characteristics of Latin Prose and the means to be employed to +reproduce them. Some notion of the treatment of the subject may be +gathered from the `Contents.' \textsc{Chap.~I.}---Characteristics of Classical +Latin, Hints on turning English into Latin; \textsc{Chap.~II.}---Arrangement +of Words in a Sentence; \textsc{Chap.~III.}---Unity in Latin Prose, Subject and +Object; \textsc{Chap.~IV.}---On the Period in Latin Prose; \textsc{Chap.~V.}---On the +position of the Relative and Relative Clauses.} +\DPPageSep{063.png}{59}% + +\Book{\Author{Roby.}---\Title{A LATIN GRAMMAR} for the Higher Classes in Grammar +Schools. By \Name{H.~J. Roby}, M.A.\InPress}{} + +\Book{\Author{Sallust.}---\Title{CAII SALLUSTII CRISPI CATILINA ET JUGURTHA.} +For Use in Schools. With copious Notes. By \Name{C. +Merivale}, B.D. (In the present Edition the Notes have been +carefully revised, and a few remarks and explanations added.) +Second Edition. Fcap.~8vo. \Price{4s. 6d.} + +\noindent The \Title{JUGURTHA} and the \Title{CATILINA} may be had separately, price +\Price{2s. 6d.} each.} +% +{This edition of Sallust, prepared by the distinguished historian of Rome, +contains an introduction, concerning the life and works of Sallust, lists +of the Consuls, and elaborate notes.} + +\Book{\Author{Tacitus.}---\Title{THE HISTORY OF TACITUS TRANSLATED +INTO ENGLISH.} By \Name{A.~J. Church}, M.A., and \Name{W.~J. +Brodribb}, M.A. With Notes and a Map. 8vo. \Price{10s. 6d.}} +% +{The translators have endeavoured to adhere as closely to the original as +was thought consistent with a proper observance of English idiom. At +the same time, it has been their aim to reproduce the precise expressions of +the author. The campaign of Civilis is elucidated in a note of some length, +which is illustrated by a map, containing the names of places and of tribes +occurring in the work. There is also a complete account of the Roman army +as it was constituted in the time of Tacitus. This work is characterised +by the Spectator as ``a scholarly and faithful translation.''} + +\Book{\Title{THE AGRICOLA AND GERMANIA OF TACITUS.} A Revised +Text, English Notes, and Maps. By \Name{Alfred J. Church}, M.A., +and \Name{W.~J. Brodribb}, M.A. Fcap.~8vo. \Price{3s. 6d.}} +% +{``We have endeavoured, with the aid of recent editions, thoroughly to +elucidate the text, explaining the various difficulties, critical and grammatical, +which occur to the student. We have consulted throughout, besides +the older commentators, the editions of Ritter and Orelli, but we are +under special obligations to the labours of the recent German editors, Wex +and Kritz.'' Two Indexes are appended, \Inum{1}~Proper Names, \Inum{2}~of +Words and Phrases explained\Add{.}} +\DPPageSep{064.png}{60}% + +\Book{\Title{THE AGRICOLA} and \Title{GERMANIA} may be had separately, price +\Price{2s.} each. + +\noindent\Title{THE AGRICOLA AND GERMANIA.} Translated into English +by \Name{A.~J. Church}, M.A., and \Name{W.~J. Brodribb}, M.A. With +Maps and Notes. Extra fcap.~8vo. \Price{2s. 6d.}} +% +{The translators have sought to produce such a version as may satisfy +scholars who demand a faithful rendering of the original, and English +readers who are offended by the baldness and frigidity which commonly +disfigure translations. The treatises are accompanied by introductions, +notes, maps, and a chronological summary. The \emph{Athenæum} says of this +work that it is ``a version at once readable and exact, which may be perused +with pleasure by all, and consulted with advantage by the classical student.''} + +\Book{\Author{Theophrastus.}---\Title{THE CHARACTERS OF THEO\-PHRAS\-TUS.} +An English Translation from a Revised Text. +With Introduction and Notes. By \Name{R.~C. Jebb}, M.A., Public +Orator in the University of Cambridge. Extra fcap.~8vo. \Price{6s. 6d.}} +% +{To the average English reader Theophrastus is little known. At the +present time, when there is a general desire to see ancient life more vividly +on every side from which it can illustrate our own, it seems possible that +the characters of Theophrastus may possess some potent interest. The text +has undergone careful revision. An Introduction supplies an account of +the origin of the book, and of writers who have imitated it: as Hall, +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 %%%% +\License + +\begin{PGtext} +End of Project Gutenberg's Elements of Plane Trigonometry, by Hugh Blackburn + +*** END OF THIS PROJECT GUTENBERG EBOOK ELEMENTS OF PLANE TRIGONOMETRY *** + +***** This file should be named 32973-pdf.pdf or 32973-pdf.zip ***** +This and all associated files of various formats will be found in: + http://www.gutenberg.org/3/2/9/7/32973/ + +Produced by Andrew D. 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/dev/null +++ b/LICENSE.txt @@ -0,0 +1,11 @@ +This eBook, including all associated images, markup, improvements, +metadata, and any other content or labor, has been confirmed to be +in the PUBLIC DOMAIN IN THE UNITED STATES. + +Procedures for determining public domain status are described in +the "Copyright How-To" at https://www.gutenberg.org. + +No investigation has been made concerning possible copyrights in +jurisdictions other than the United States. 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