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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.net % +% % +% % +% Title: Elementary Illustrations of the Differential and Integral Calculus +% % +% Author: Augustus De Morgan % +% % +% Release Date: March 3, 2012 [EBook #39041] % +% Most recently updated: June 11, 2021 % +% % +% Language: English % +% % +% Character set encoding: UTF-8 % +% % +% *** START OF THIS PROJECT GUTENBERG EBOOK DIFFERENTIAL, INTEGRAL CALCULUS *** +% % +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % + +\def\ebook{39041} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %% +%% Packages and substitutions: %% +%% %% +%% book: Required. %% +%% inputenc: Latin-1 text encoding. Required. %% +%% %% +%% ifthen: Logical conditionals. Required. %% +%% %% +%% amsmath: AMS mathematics enhancements. Required. %% +%% amssymb: Additional mathematical symbols. 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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.net + + +Title: Elementary Illustrations of the Differential and Integral Calculus + +Author: Augustus De Morgan + +Release Date: March 3, 2012 [EBook #39041] +Most recently updated: June 11, 2021 + +Language: English + +Character set encoding: UTF-8 + +*** START OF THIS PROJECT GUTENBERG EBOOK DIFFERENTIAL, INTEGRAL CALCULUS *** +\end{PGtext} +\end{minipage} +\end{center} +\newpage +%% Credits and transcriber's note %% +\begin{center} +\begin{minipage}{\textwidth} +\begin{PGtext} +Produced by Andrew D. Hwang. +\end{PGtext} +\end{minipage} +\vfill +\end{center} + +\begin{minipage}{0.85\textwidth} +\small +\BookMark{0}{Transcriber's Note.} +\subsection*{\centering\normalfont\scshape% +\normalsize\MakeLowercase{\TransNote}}% + +\raggedright +\TransNoteText +\end{minipage} +%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%% +\PageSep{ii} +\FrontMatter + + +\Section{In the Same Series.} + +\Book{ON THE STUDY AND DIFFICULTIES OF MATHEMATICS.} +By \Author{Augustus De~Morgan}. Entirely new edition, +with portrait of the author, index, and annotations, +bibliographies of modern works on algebra, the philosophy +of mathematics, pan-geometry,~etc. Pp.,~288. Cloth, \$1.25 +net~(\Price{5s.}). + +\Book{LECTURES ON ELEMENTARY MATHEMATICS.} By +\Author{Joseph Louis Lagrange}. Translated from the French by +\Translator{Thomas~J. McCormack}. With photogravure portrait of +Lagrange, notes, biography, marginal analyses,~etc. Only +separate edition in French or English, Pages,~172. Cloth, +\$1.00 net~(\Price{5s.}). + +\Book{ELEMENTARY ILLUSTRATIONS OF THE DIFFERENTIAL +AND INTEGRAL CALCULUS.} By \Author{Augustus De~Morgan}. +New reprint edition. With sub-headings, and +a brief bibliography of English, French, and German text-books +of the Calculus. Pp.,~144. Price, \$1.00 net~(\Price{5s.}). + +\Book{MATHEMATICAL ESSAYS AND RECREATIONS.} By +\Author{Hermann Schubert}, Professor of Mathematics in the +Johanneum, Hamburg, Germany. Translated from the +German by \Translator{Thomas~J. McCormack}. Containing essays on +the Notion and Definition of Number, Monism in Arithmetic, +On the Nature of Mathematical Knowledge, The +Magic Square, The Fourth Dimension, The Squaring of +the Circle. Pages,~149. Cuts,~37. Price, Cloth,~75c net~(\Price{3s. 6d.}). + +\Book{HISTORY OF ELEMENTARY MATHEMATICS.} By \Author{Dr.\ +Karl Fink}, late Professor in Tübingen. Translated from +the German by Prof.\ \Translator{Wooster Woodruff Beman} and Prof.\ +\Translator{David Eugene Smith}. (Nearly Ready.) +\vfill +\begin{center} +\small +THE OPEN COURT PUBLISHING CO. \\ +\footnotesize +324 DEARBORN ST., CHICAGO. +\end{center} +\PageSep{iii} +\cleardoublepage +%[** Title page] +\begin{center} +ELEMENTARY ILLUSTRATIONS +\vfil +\footnotesize OF THE +\vfil +\textsc{\LARGE Differential and Integral \\[4pt] +Calculus} +\vfil\vfil\vfil + +\footnotesize BY \\ +\normalsize AUGUSTUS DE MORGAN +\vfil\vfil\vfil + +\textit{\small NEW EDITION} +\vfil\vfil\vfil\vfil + +\footnotesize CHICAGO \\ +THE OPEN COURT PUBLISHING COMPANY \\ +\scriptsize FOR SALE BY \\ +\footnotesize\textsc{Kegan Paul, Trench, Trübner \&~Co., Ltd., London} \\ +1899 +\end{center} +\PageSep{iv} +% [Blank page] +\PageSep{v} + + +\Section{Editor's Preface.} + +\First{The} publication of the present reprint of De~Morgan's \Title{Elementary +Illustrations of the Differential and Integral Calculus} +forms, quite independently of its interest to professional +students of mathematics, an integral portion of the general educational +plan which the Open Court Publishing Company has been +systematically pursuing since its inception,---which is the dissemination +among the public at large of sound views of science and of +an adequate and correct appreciation of the methods by which +truth generally is reached. Of these methods, mathematics, by +its simplicity, has always formed the type and ideal, and it is +nothing less than imperative that its ways of procedure, both in +the discovery of new truth and in the demonstration of the necessity +and universality of old truth, should be laid at the foundation +of every philosophical education. The greatest achievements in +the history of thought---Plato, Descartes, Kant---are associated +with the recognition of this principle. + +But it is precisely mathematics, and the pure sciences generally, +from which the general educated public and independent +students have been debarred, and into which they have only rarely +attained more than a very meagre insight. The reason of this is +twofold. In the first place, the ascendant and consecutive character +of mathematical knowledge renders its results absolutely unsusceptible +of presentation to persons who are unacquainted with +what has gone before, and so necessitates on the part of its devotees +a thorough and patient exploration of the field from the very +beginning, as distinguished from those sciences which may, so to +speak, be begun at the end, and which are consequently cultivated +with the greatest zeal. The second reason is that, partly through +the exigencies of academic instruction, but mainly through the +martinet traditions of antiquity and the influence of mediæval +\PageSep{vi} +logic-mongers, the great bulk of the elementary text-books of +mathematics have unconsciously assumed a very repellent form,---something +similar to what is termed in the theory of protective +mimicry in biology ``the terrifying form.'' And it is mainly to +this formidableness and touch-me-not character of exterior, concealing +withal a harmless body, that the undue neglect of typical +mathematical studies is to be attributed. + +To this class of books the present work forms a notable exception. +It was originally issued as numbers 135 and 140 of the +Library of Useful Knowledge (1832), and is usually bound up with +De~Morgan's large \Title{Treatise on the Differential and Integral +Calculus} (1842). Its style is fluent and familiar; the treatment +continuous and undogmatic. The main difficulties which encompass +the early study of the Calculus are analysed and discussed in +connexion with practical and historical illustrations which in point +of simplicity and clearness leave little to be desired. No one who +will read the book through, pencil in hand, will rise from its perusal +without a clear perception of the aim and the simpler fundamental +principles of the Calculus, or without finding that the profounder +study of the science in the more advanced and more +methodical treatises has been greatly facilitated. + +The book has been reprinted substantially as it stood in its +original form; but the typography has been greatly improved, and +in order to render the subject-matter more synoptic in form and +more capable of survey, the text has been re-paragraphed and a +great number of descriptive sub-headings have been introduced, a +list of which will be found in the Contents of the book. An index +also has been added. + +Persons desirous of continuing their studies in this branch of +mathematics, will find at the end of the text a bibliography of the +principal English, French, and German works on the subject, as +well as of the main Collections of Examples. From the information +there given, they may be able to select what will suit their +special needs. + +\Signature{Thomas J. McCormack.} +{\textsc{La Salle}, Ill., August, 1899.} +\PageSep{vii} + + +\TableofContents +\iffalse +CONTENTS: + +PAGE + +On the Ratio or Proportion of Two Magnitudes 2 +On the Ratio of Magnitudes that Vanish Together.... 4 +On the Ratios of Continuously Increasing or Decreasing Quantities 7 +The Notion of Infinitely Small Quantities 11 +On Functions 14 +Infinite Series 15 +Convergent and Divergent Series 17 +Taylor's Theorem. Derived Functions 19 +Differential Coefficients 22 +The Notation of the Differential Calculus 25 +Algebraical Geometry.... 29 +On the Connexion of the Signs of Algebraical and the Directions + of Geometrical Magnitudes 31 +The Drawing of a Tangent to a Curve 36 +Rational Explanation of the Language of Leibnitz.... 38 +Orders of Infinity 42 +A Geometrical Illustration: Limit of the Intersections of Two + Coinciding Straight Lines 45 +The Same Problem Solved by the Principles of Leibnitz. . 48 +An Illustration from Dynamics; Velocity, Acceleration, etc.. 52 +Simple Harmonic Motion 57 +The Method of Fluxions 60 +Accelerated Motion 60 +Limiting Ratios of Magnitudes that Increase Without Limit. 65 +Recapitulation of Results Reached in the Theory of Functions. 74 +Approximations by the Differential Calculus 74 +Solution of Equations by the Differential Calculus.... 77 +Partial and Total Differentials 78 +\PageSep{viii} +Application of the Theorem for Total Differentials to the + Determination of Total Resultant Errors 84 +Rules for Differentiation 85 +Illustration of the Rules for Differentiation 86 +Differential Coefficients of Differential Coefficients .... 88 +Calculus of Finite Differences. Successive Differentiation . 88 +Total and Partial Differential Coefficients. Implicit Differentiation 94 +Applications of the Theorem for Implicit Differentiation .. 101 +Inverse Functions 102 +Implicit Functions 106 +Fluxions, and the Idea of Time 110 +The Differential Coefficient Considered with Respect to Its + Magnitude 112 +The Integral Calculus 115 +Connexion of the Integral with the Differential Calculus.. 120 +Nature of Integration 122 +Determination of Curvilinear Areas. The Parabola... 124 +Method of Indivisibles 125 +Concluding Remarks on the Study of the Calculus.... 132 +Bibliography of Standard Text-books and Works of Reference + on the Calculus 133 +Index 143 +\fi +\PageSep{1} +\MainMatter + + +\Section{Differential and Integral Calculus.} + +\SubSectHead{Elementary Illustrations.} + +\First{The} Differential and Integral Calculus, or, as it +was formerly called in this country [England], +the Doctrine of Fluxions, has always been supposed +to present remarkable obstacles to the beginner. It +is matter of common observation, that any one who +commences this study, even with the best elementary +works, finds himself in the dark as to the real meaning +of the processes which he learns, until, at a certain +stage of his progress, depending upon his capacity, +some accidental combination of his own ideas throws +light upon the subject. The reason of this may be, that +it is usual to introduce him at the same time to new +principles, processes, and symbols, thus preventing +his attention from being exclusively directed to one +new thing at a time. It is our belief that this should +be avoided; and we propose, therefore, to try the experiment, +whether by undertaking the solution of +some problems by common algebraical methods, without +calling for the reception of more than one new +symbol at once, or lessening the immediate evidence +of each investigation by reference to general rules, the +study of more methodical treatises may not be somewhat +\PageSep{2} +facilitated. We would not, nevertheless, that +the student should imagine we can remove all obstacles; +we must introduce notions, the consideration +of which has not hitherto occupied his mind; and +shall therefore consider our object as gained, if we +can succeed in so placing the subject before him, that +two independent difficulties shall never occupy his +mind at once. + + +\Subsection{On the Ratio or Proportion of Two Magnitudes.} + +The ratio or proportion of two magnitudes is best +\index{Proportion|EtSeq}% +\index{Ratio!defined|EtSeq}% +conceived by expressing them in numbers of some +unit when they are commensurable; or, when this is +not the case, the same may still be done as nearly as +we please by means of numbers. Thus, the ratio of +the diagonal of a square to its side is that of $\sqrt{2}$ to~$1$, +which is very nearly that of $14142$ to~$10000$, and is +certainly between this and that of $14143$ to~$10000$. +Again, any ratio, whatever numbers express it, may +be the ratio of two magnitudes, each of which is as +small as we please; by which we mean, that if we +take any given magnitude, however small, such as the +line~$A$, we may find two other lines $B$~and~$C$, each +less than~$A$, whose ratio shall be whatever we please. +Let the given ratio be that of the numbers $m$~and~$n$. +Then, $P$~being a line, $mP$~and~$nP$ are in the proportion +of $m$ to~$n$; and it is evident, that let $m$,~$n$, and~$A$ +be what they may, $P$~can be so taken that $mP$~shall be +less than~$A$. This is only saying that $P$~can be taken +less than the $m$\th~part of~$A$, which is obvious, since~$A$, +however small it may be, has its tenth, its hundredth, +its thousandth part,~etc., as certainly as if it were +larger. We are not, therefore, entitled to say that +because two magnitudes are diminished, their ratio is +\PageSep{3} +diminished; it is possible that~$B$, which we will suppose +to be at first a hundredth part of~$C$, may, after +a diminution of both, be its tenth or thousandth, or +may still remain its hundredth, as the following example +will show: +\begin{alignat*}{5} +&C && 3600 && 1800 && 36 && 90 \\ +&B && 36 && 1\tfrac{8}{10} && \tfrac{36}{100} && 9 \\ +&B &{}={}& \frac{1}{100} C\qquad + B &{}={}& \frac{1}{1000} C\qquad + B &{}={}& \frac{1}{100} C\qquad + B &{}={}& \frac{1}{10} C. +\end{alignat*} +Here the values of $B$~and~$C$ in the second, third, and +fourth column are less than those in the first; nevertheless, +the ratio of $B$ to~$C$ is less in the second column +than it was in the first, remains the same in the +third, and is greater in the fourth. + +In estimating the approach to, or departure from +equality, which two magnitudes undergo in consequence +of a change in their values, we must not look +at their differences, but at the proportions which those +differences bear to the whole magnitudes. For example, +if a geometrical figure, two of whose sides are +$3$~and $4$~inches now, be altered in dimensions, so that +the corresponding sides are $100$~and $101$~inches, they +are nearer to equality in the second case than in the +first; because, though the difference is the same in +both, namely one inch, it is one third of the least side +in the first case, and only one hundredth in the second. +This corresponds to the common usage, which +rejects quantities, not merely because they are small, +but because they are small in proportion to those of +which they are considered as parts. Thus, twenty +miles would be a material error in talking of a day's +journey, but would not be considered worth mentioning +in one of three months, and would be called totally +\PageSep{4} +insensible in stating the distance between the +earth and sun. More generally, if in the two quantities +$x$~and~$x + a$, an increase of~$m$ be given to~$x$, +the two resulting quantities $x + m$~and $x + m + a$ are +nearer to equality as to their \emph{ratio} than $x$~and~$x + a$, +\index{Equality}% +though they continue the same as to their \emph{difference}; for +\index{Differences!arithmetical}% +$\dfrac{x + a}{x} = 1 + \dfrac{a}{x}$ and $\dfrac{x + m + a}{x + m} = 1 + \dfrac{a}{x + m}$ of which +$\dfrac{a}{x + m}$~is less than~$\dfrac{a}{x}$, and therefore $1 + \dfrac{a}{x + m}$ is nearer +to unity than $1 + \dfrac{a}{x}$. In future, when we talk of an +approach towards equality, we mean that the ratio is +made more nearly equal to unity, not that the difference +is more nearly equal to nothing. The second +may follow from the first, but not necessarily; still +less does the first follow from the second. + + +\Subsection{On the Ratio of Magnitudes that Vanish Together.} + +It is conceivable that two magnitudes should decrease +\index{Time, idea of}% +simultaneously,\footnote + {In introducing the notion of time, we consult only simplicity. It would + do equally well to write any number of successive values of the two quantities, + and place them in two columns.} +so as to vanish or become +nothing, together. For example, let a point~$A$ move +on a circle towards a fixed point~$B$. The arc~$AB$ will +then diminish, as also the chord~$AB$, and by bringing +the point~$A$ sufficiently near to~$B$, we may obtain an +arc and its chord, both of which shall be smaller than +a given line, however small this last may be. But +while the magnitudes diminish, we may not assume +either that their ratio increases, diminishes, or remains +the same, for we have shown that a diminution +of two magnitudes is consistent with either of these. +\PageSep{5} +\index{Increase without limit|EtSeq}% +We must, therefore, look to each particular case for +the change, if any, which is made in the ratio by the +diminution of its terms. + +Now two suppositions are possible in every increase +or diminution of the ratio, as follows: Let $M$~and~$N$ +be two quantities which we suppose in a state +of decrease. The first possible case is that the ratio +of $M$ to~$N$ may decrease without limit, that is, $M$~may +be a smaller fraction of~$N$ after a decrease than it was +before, and a still smaller after a further decrease, +and so on; in such a way, that there is no fraction so +small, to which $\dfrac{M}{N}$~shall not be equal or inferior, if the +decrease of $M$~and~$N$ be carried sufficiently far. As +an instance, form two sets of numbers as in the adjoining +table: +\[ +\begin{array}{*{7}{c}} +\PadTxt{Ratio of }{} M \PadTxt{ to~$N$}{} + & 1 & \dfrac{1}{20} & \dfrac{1}{400} & \dfrac{1}{8000} & \dfrac{1}{160000} & \etc. \\ +\PadTxt{Ratio of }{} N \PadTxt{ to~$N$}{} + & 1 & \dfrac{1}{2} & \dfrac{1}{4} & \dfrac{1}{8} & \dfrac{1}{16} & \etc. \\ +\text{Ratio of~$M$ to~$N$} + & 1 & \dfrac{1}{10} & \dfrac{1}{100} & \dfrac{1}{1000} & \dfrac{1}{10000} & \etc. +\end{array} +\] +Here both $M$~and~$N$ decrease at every step, but $M$~loses +at each step a larger fraction of itself than~$N$, +and their ratio continually diminishes. To show that +this decrease is without limit, observe that $M$~is at +first equal to~$N$, next it is one tenth, then one hundredth, +then one thousandth of~$N$, and so on; by continuing +the values of $M$ and~$N$ according to the same +law, we should arrive at a value of~$M$ which is a +smaller part of~$N$ than any which we choose to name; +for example,~$.000003$. The second value of~$M$ beyond +our table is only one millionth of the corresponding +value of~$N$; the ratio is therefore expressed by~$.000001$ +\PageSep{6} +which is less than~$.000003$. In the same law of formation, +the ratio of $N$ to~$M$ is also \emph{increased} without limit. + +The second possible case is that in which the ratio +of $M$ to~$N$, though it increases or decreases, does not +increase or decrease without limit, that is, continually +approaches to some ratio, which it never will exactly +reach, however far the diminution of $M$ and~$N$ may +be carried. The following is an example: +\[ +\begin{array}{*{9}{c}} +\PadTxt{Ratio of }{} M \PadTxt{ to~$N$}{} + & 1 & \dfrac{1}{3} & \dfrac{1}{6} & \dfrac{1}{10} & \dfrac{1}{15} & \dfrac{1}{21} & \dfrac{1}{28} & \etc. \\ +\PadTxt{Ratio of }{} N \PadTxt{ to~$N$}{} + & 1 & \dfrac{1}{4} & \dfrac{1}{9} & \dfrac{1}{16} & \dfrac{1}{25} & \dfrac{1}{36} & \dfrac{1}{49} & \etc. \\ +\text{Ratio of~$M$ to~$N$} + & 1 & \dfrac{4}{3} & \dfrac{9}{6} & \dfrac{16}{10} & \dfrac{25}{15} & \dfrac{36}{21} & \dfrac{49}{28} & \etc. +\end{array} +\] +The ratio here increases at each step, for $\dfrac{4}{3}$~is greater +than~$1$, $\dfrac{9}{6}$~than~$\dfrac{4}{3}$, and so on. The difference between +this case and the last is, that the ratio of $M$ to~$N$, +though perpetually increasing, does not increase without +limit; it is never so great as~$2$, though it may be +brought as near to~$2$ as we please. + +To show this, observe that in the successive values +of~$M$, the denominator of the second is~$1 + 2$, that of +the third $1 + 2 + 3$, and so on; whence the denominator +of the $x$\th~value of~$M$ is +\[ +1 + 2 + 3 + \dots + x,\quad\text{or}\quad \frac{x(x + 1)}{2}\Add{.} +\] +Therefore the $x$\th~value of~$M$ is~$\dfrac{2}{x(x + 1)}$, and it is +evident that the $x$\th~value of~$N$ is~$\dfrac{1}{x^{2}}$, which gives the +$x$\th~value of the ratio $\dfrac{M}{N} = \dfrac{2x^{2}}{x(x + 1)}$, or~$\dfrac{2x}{x + 1}$, or +\PageSep{7} +$\dfrac{x}{x + 1} × 2$. If $x$~be made sufficiently great, $\dfrac{x}{x + 1}$~may +be brought as near as we please to~$1$, since, being +$1 - \dfrac{1}{x + 1}$, it differs from~$1$ by~$\dfrac{1}{x + 1}$, which may be +made as small as we please. But as $\dfrac{x}{x + 1}$, however +great $x$~may be, is always less than~$1$, $\dfrac{2x}{x + 1}$~is always +less than~$2$. Therefore (1)~$\dfrac{M}{N}$~continually increases; +(2)~may be brought as near to~$2$ as we please; (3)~can +never be greater than~$2$. This is what we mean by +saying that $\dfrac{M}{N}$~is an increasing ratio, the limit of +which is~$2$. Similarly of~$\dfrac{N}{M}$, which is the reciprocal +of~$\dfrac{M}{N}$, we may show (1)~that it continually decreases; +(2)~that it can be brought as near as we please to~$\frac{1}{2}$; +(3)~that it can never be less than~$\frac{1}{2}$. This we express +by saying that $\dfrac{N}{M}$~is a decreasing ratio, whose limit +is~$\frac{1}{2}$. + + +\Subsection{On the Ratios of Continuously Increasing or +Decreasing Quantities.} + +To the fractions here introduced, there are intermediate +\index{Continuous quantities|EtSeq}% +\index{Quantities, continuous|EtSeq}% +fractions, which we have not considered. +Thus, in the last instance, $M$~passed from $1$ to~$\frac{1}{2}$ without +any intermediate change. In geometry and mechanics, +it is necessary to consider quantities as +increasing or decreasing \emph{continuously}; that is, a magnitude +does not pass from one value to another without +passing through every intermediate value. Thus +if one point move towards another on a circle, both +the arc and its chord decrease continuously. Let $AB$ +\index{Arc and its chord, a continuously decreasing|EtSeq}% +(\Fig{1}) be an arc of a circle, the centre of which is~$O$. +\PageSep{8} +Let $A$ remain fixed, but let $B$, and with it the radius~$OB$, +move towards~$A$, the point~$B$ always remaining +on the circle. At every position of~$B$, suppose +the following figure. Draw $AT$ touching the circle at~$A$, +produce $OB$ to meet~$AT$ in~$T$, draw $BM$~and~$BN$ +perpendicular and parallel to~$OA$, and join~$BA$. Bisect +the arc~$AB$ in~$C$, and draw~$OC$ meeting the chord in~$D$ +and bisecting it. The right-angled triangles $ODA$ +and $BMA$ having a common angle, and also right +angles, are similar, as are also $BOM$ and~$TBN$. If +now we suppose $B$ to move towards~$A$, before $B$ +\Figure{1} +reaches~$A$, we shall have the following results: The +arc and chord~$BA$, the lines $BM$,~$MA$, $BT$,~$TN$, the +angles $BOA$,~$COA$,~$MBA$, and~$TBN$, will diminish +without limit; that is, assign a line and an angle, +however small, $B$~can be placed so near to~$A$ that the +lines and angles above alluded to shall be severally +less than the assigned line and angle. Again, $OT$~diminishes +and $OM$~increases, but neither without limit, +for the first is never less, nor the second greater, than +the radius. The angles $OBM$,~$MAB$, and~$BTN$, increase, +but not without limit, each being always less +than the right angle, but capable of being made as +\PageSep{9} +near to it as we please, by bringing~$B$ sufficiently near +to~$A$. + +So much for the magnitudes which compose the +figure: we proceed to consider their ratios, premising +that the arc~$AB$ is greater than the chord~$AB$, and +less than $BN + NA$. The triangle~$BMA$ being always +similar to~$ODA$, their sides change always in the same +proportion; and the sides of the first decrease without +limit, which is the case with only one side of the +second. And since $OA$~and~$OD$ differ by~$DC$, which +diminishes without limit as compared with~$OA$, the +ratio $OD ÷ OA$ is an increasing ratio whose limit is~$1$. +But $OD ÷ OA = BM ÷ BA$. We can therefore bring~$B$ +so near to~$A$ that $BM$~and~$BA$ shall differ by as +small a fraction of either of them as we please. + +To illustrate this result from the trigonometrical +tables, observe that if the radius~$OA$ be the linear +unit, and $\angle BOA = \theta$, $BM$~and~$BA$ are respectively +$\sin\theta$ and $2\sin\frac{1}{2}\theta$. Let $\theta = 1°$; then $\sin\theta = .0174524$ +and $2\sin\frac{1}{2}\theta = .0174530$; whence $2\sin\frac{1}{2}\theta ÷ \sin\theta = 1.00003$ very nearly, so that $BM$~differs from~$BA$ by +less than four of its own hundred-thousandth parts. +If $\angle BOA = 4'$, the same ratio is~$1.0000002$, differing +from unity by less than the hundredth part of the +difference in the last example. + +Again, since $DA$~diminishes continually and without +limit, which is not the case either with $OD$ or~$OA$, +the ratios $OD ÷ DA$ and $OA ÷ DA$ increase without +limit. These are respectively equal to $BM ÷ MA$ +and $BA ÷ MA$; whence it appears that, let a number +be ever so great, $B$~can be brought so near to~$A$, that +$BM$ and $BA$ shall each contain~$MA$ more times than +there are units in that number. Thus if $\angle BOA = 1°$, +$BM ÷ MA = 114.589$ and $BA ÷ MA = 114.593$ very +\PageSep{10} +nearly; that is, $BM$ and $BA$ both contain~$MA$ +more than $114$~times. If $\angle BOA = 4'$, $BM ÷ MA = 1718.8732$, +and $BA ÷ MA = 1718.8375$ very nearly; +or $BM$ and $BA$ both contain~$MA$ more than $1718$~times. + +No difficulty can arise in conceiving this result, if +the student recollect that the degree of greatness or +smallness of two magnitudes determines nothing as +to their ratio; since every quantity~$N$, however small, +can be divided into as many parts as we please, and +has therefore another small quantity which is its millionth +\Figure[nolabel]{1} +or hundred-millionth part, as certainly as if it +had been greater. There is another instance in the +line~$TN$, which, since $TBN$~is similar to~$BOM$, decreases +continually with respect to~$TB$, in the same +manner as does $BM$ with respect to~$OB$. + +The arc~$BA$ always lies between $BA$ and $BN + NA$, +or $BM + MA$; hence $\dfrac{\arc BA}{\chord BA}$ lies between $1$ and +$\dfrac{BM}{BA} + \dfrac{MA}{BA}$. But $\dfrac{BM}{BA}$~has been shown to approach +continually towards~$1$, and $\dfrac{MA}{BA}$~to decrease without +limit; hence $\dfrac{\arc BA}{\chord BA}$ continually approaches towards~$1$. +\PageSep{11} +If $\angle BOA = 1°$, $\dfrac{\arc BA}{\chord BA} = .0174533 ÷ .0174530 = 1.00002$, +very nearly. If $\angle BOA = 4'$, it is less than +$1.0000001$. + +We now proceed to illustrate the various phrases +which have been used in enunciating these and similar +propositions. + + +\Subsection{The Notion of Infinitely Small Quantities.} + +It appears that it is possible for two quantities $m$ +and $m + n$ to decrease together in such a way, that $n$~continually +decreases with respect to~$m$, that is, becomes +a less and less part of~$m$, so that $\dfrac{n}{m}$~also decreases +when $n$~and~$m$ decrease. Leibnitz,\footnote + {Leibnitz was a native of Leipsic, and died in 1716, aged~70. His dispute +\index{Leibnitz}% + with Newton, or rather with the English mathematicians in general, about +\index{Newton}% + the invention of Fluxions, and the virulence with which it was carried on, +\index{Fluxions}% + are well known. The decision of modern times appears to be that both Newton + and Leibnitz were independent inventors of this method. It has, perhaps, + not been sufficiently remarked how nearly several of their predecessors approached + the same ground; and it is a question worthy of discussion, whether + either Newton or Leibnitz might not have found broader hints in writings + accessible to both, than the latter was ever asserted to have received from + the former.} +in introducing +the Differential Calculus, presumed that in +such a case, $n$~might be taken so small as to be utterly +inconsiderable when compared with~$m$, so that $m + n$ +might be put for~$m$, or \textit{vice versa}, without any error at +all. In this case he used the phrase that $n$~is \emph{infinitely} +small with respect to~$m$. + +The following example will illustrate this term. +Since $(a + h)^{2} = a^{2} + 2ah + h^{2}$, it appears that if $a$~be +increased by~$h$, $a^{2}$~is increased by~$2ah + h^{2}$. But if $h$~be +taken very small, $h^{2}$~is very small with respect to~$h$, +for since $1:h :: h:h^{2}$, as many times as $1$~contains~$h$, +so many times does $h$~contain~$h^{2}$; so that by taking +\PageSep{12} +$h$~sufficiently small, $h$~may be made to be as many +times~$h^{2}$ as we please. Hence, in the words of Leibnitz, +if $h$~be taken \emph{infinitely} small, $h^{2}$~is \emph{infinitely} small +\index{Infinitely small, the notion of}% +with respect to~$h$, and therefore $2ah + h^{2}$ is the same +as~$2ah$; or if $a$~be increased by an infinitely small +quantity~$h$, $a^{2}$~is increased by another infinitely small +quantity~$2ah$, which is to~$h$ in the proportion of $2a$ +to~$1$. + +In this reasoning there is evidently an absolute +error; for it is impossible that $h$~can be so small, that +$2ah + h^{2}$ and $2ah$ shall be the same. The word \emph{small} +itself has no precise meaning; though the word \emph{smaller}, +\index{Small, has no precise meaning}% +or \emph{less}, as applied in comparing one of two magnitudes +with another, is perfectly intelligible. Nothing is +either small or great in itself, these terms only implying +a relation to some other magnitude of the same +kind, and even then varying their meaning with the +subject in talking of which the magnitude occurs, so +that both terms may be applied to the same magnitude: +thus a large field is a very small part of the +earth. Even in such cases there is no natural point +at which smallness or greatness commences. The +thousandth part of an inch may be called a small distance, +a mile moderate, and a thousand leagues great, +but no one can fix, even for himself, the precise mean +between any of these two, at which the one quality +ceases and the other begins. These terms are not +therefore a fit subject for mathematical discussion, +until some more precise sense can be given to them, +which shall prevent the danger of carrying away with +the words, some of the confusion attending their use +in ordinary language. It has been usual to say that +when $h$~decreases from any given value towards nothing, +$h^{2}$~will become \emph{small} as compared with~$h$, because, +\PageSep{13} +let a number be ever so great, $h$~will, before it becomes +nothing, contain $h^{2}$~more than that number of +times. Here all dispute about a standard of smallness +is avoided, because, be the standard whatever it may, +the proportion of~$h^{2}$ to~$h$ may be brought under it. It +is indifferent whether the thousandth, ten-thousandth, +or hundred-millionth part of a quantity is to be considered +small enough to be rejected by the side of the +whole, for let $h$~be $\dfrac{1}{1000}$, $\dfrac{1}{10,000}$, or $\dfrac{1}{100,000,000}$ of the +unit, and $h$~will contain~$h^{2}$, $1000$, $10,000$, or $100,000,000$ +of times. + +The proposition, therefore, that $h$~can be taken so +small that $2ah + h^{2}$ and~$2ah$ are rigorously equal, +though not true, and therefore entailing error upon +all its subsequent consequences, yet is of this character, +that, by taking $h$ sufficiently small, all errors may +be made as small as we please. The desire of combining +simplicity with the appearance of rigorous +demonstration, probably introduced the notion of infinitely +small quantities; which was further established +by observing that their careful use never led to +any error. The method of stating the above-mentioned +proposition in strict and rational terms is as follows: +If $a$~be increased by~$h$, $a^{2}$~is increased by $2ah + h^{2}$, +which, whatever may be the value of~$h$, is to~$h$ in the +proportion of $2a + h$ to~$1$. The smaller $h$~is made, +the more near does this proportion diminish towards +that of $2a$ to~$1$, to which it may be made to approach +within any quantity, if it be allowable to take $h$ as +small as we please. Hence the ratio, $\emph{increment of } a^{2} ÷ \emph{increment of } a$, is a decreasing ratio, whose limit is~$2a$. + +In further illustration of the language of Leibnitz, +\index{Leibnitz}% +we observe, that according to his phraseology, if $AB$~be +\PageSep{14} +an \emph{infinitely} small arc, the chord and arc~$AB$ are +equal, or the circle is a polygon of an \emph{infinite} number +of \emph{infinitely} small rectilinear sides. This should +be considered as an abbreviation of the proposition +proved (\PageRef{10}), and of the following: If a polygon +be inscribed in a circle, the greater the number of its +sides, and the smaller their lengths, the more nearly +will the perimeters of the polygon and circle be equal +to one another; and further, if any straight line be +given, however small, the difference between the perimeters +of the polygon and circle may be made less +than that line, by sufficient increase of the number of +sides and diminution of their lengths. Again, it would +be said (\Fig{1}) that if $AB$~be infinitely small, $MA$~is +infinitely less than~$BM$. What we have proved is, +that $MA$ may be made as small a part of~$BM$ as we +please, by sufficiently diminishing the arc~$BA$. + + +\Subsection{On Functions.} + +An algebraical expression which contains~$x$ in any +\index{Functions!definition of|EtSeq}% +way, is called a \emph{function} of~$x$. Such are $x^{2} + a^{2}$, +$\dfrac{a + x}{a - x}$, $\log(x + y)$, $\sin 2x$. An expression may be a +function of more quantities than one, but it is usual +only to name those quantities of which it is necessary +to consider a change in the value. Thus if in $x^{2} + a^{2}$ +$x$~only is considered as changing its value, this is +called a function of~$x$; if $x$~and~$a$ both change, it is +called a function of $x$~and~$a$. Quantities which change +their values during a process, are called \emph{variables}, and +\index{Variables!independent and dependent}% +those which remain the same, \emph{constants}; and variables +\index{Constants}% +which we change at pleasure are called \emph{independent}, +while those whose changes necessarily follow from +\PageSep{15} +\index{Variables!independent and dependent}% +the changes of others are called \emph{dependent}. Thus in +\Fig{1}, the length of the radius~$OB$ is a constant, the +arc~$AB$ is the independent variable, while $BM$,~$MA$, +the chord~$AB$,~etc., are dependent. And, as in algebra +we reason on numbers by means of general symbols, +each of which may afterwards be particularised +as standing for any number we please, unless specially +prevented by the conditions of the problem, so, in +treating of functions, we use general symbols, which +may, under the restrictions of the problem, stand for +any function whatever. The symbols used are the letters +$F$,~$f$, $\Phi$,~$\phi$,~$\psi$; $\phi(x)$~and~$\psi(x)$, or $\phi x$~and~$\psi x$, may +represent any functions of~$x$, just as $x$~may represent +any number. Here it must be borne in mind that $\phi$~and~$\psi$ +do not represent numbers which multiply~$x$, but +are \emph{the abbreviated directions to perform certain operations +with $x$ and constant quantities}. Thus, if $\phi x = x + x^{2}$, +$\phi$~is equivalent to a direction to add~$x$ to its +square, and the whole~$\phi x$ stands for the result of this +operation. Thus, in this case, $\phi(1) = 2$; $\phi(2) = 6$; +$\phi a = a + a^{2}$; $\phi(x + h) = x + h + (x + h)^{2}$; $\phi \sin x = \sin x + (\sin x)^{2}$. +It may be easily conceived that this +notion is useless, unless there are propositions which +are generally true of all functions, and which may be +made the foundation of general reasoning. + + +\Subsection{Infinite Series.} + +To exercise the student in this notation, we proceed +\index{Series|EtSeq}% +\index{Taylor's Theorem|EtSeq}% +to explain one of these functions which is of +most extensive application and is known by the name +of \emph{Taylor's Theorem}. If in~$\phi x$, any function of~$x$, the +value of~$x$ be increased by~$h$, or $x + h$~be substituted +instead of~$x$, the result is denoted by~$\phi(x + h)$. It +\PageSep{16} +will generally\footnote + {This word is used in making assertions which are for the most part +\index{Generally@\emph{Generally}, the word}% + true, but admit of exceptions, few in number when compared with the other + cases. Thus it generally happens that $x^{2} - 10x + 40$ is greater than~$15$, with + the exception only of the case where $x = 5$. It is generally true that a line + which meets a circle in a given point meets it again, with the exception only + of the tangent.} +happen that this is either greater or +less than~$\phi x$, and $h$~is called the \emph{increment} of~$x$, and +\index{Increment}% +$\phi(x + h) - \phi x$ is called the \emph{increment} of~$\phi x$, which is +negative when $\phi(x + h) < \phi x$. It may be proved +that $\phi(x + h)$ can generally be expanded in a series +of the form +\[ +\phi x + ph + qh^{2} + rh^{3} + \etc.,\quad \textit{ad infinitum}, +\] +which contains none but whole and positive powers +of~$h$. It will happen, however, in many functions, +that one or more values can be given to~$x$ for which +it is impossible to expand $f(x + h)$ without introducing +negative or fractional powers. These cases are +considered by themselves, and the values of~$x$ which +produce them are called \emph{singular} values. +\index{Singular values}% + +As the notion of a series which has no end of its +terms, may be new to the student, we will now proceed +to show that there may be series so constructed, +that the addition of any number of their terms, however +great, will always give a result less than some +determinate quantity. Take the series +\[ +1 + x + x^{2} + x^{3} + x^{4} + \etc., +\] +in which $x$~is supposed to be less than unity. The +first two terms of this series may be obtained by dividing +$1 - x^{2}$ by $1 - x$; the first three by dividing +$1 - x^{3}$ by $1 - x$; and the first $n$~terms by dividing +$1 - x^{n}$ by $1 - x$. If $x$~be less than unity, its successive +powers decrease without limit;\footnote + {This may be proved by means of the proposition established in \Title{Study + of Mathematics} (Chicago: The Open Court Publishing~Co., Reprint Edition), + page~247. For $\dfrac{m}{n} × \dfrac{n}{m}$ is formed (if $m$~be less than~$n$) by dividing $\dfrac{m}{n}$ into $n$~parts, + and taking away $n - m$ of them.} +that is, there is +\PageSep{17} +no quantity so small, that a power of~$x$ cannot be +found which shall be smaller. Hence by taking $n$~sufficiently +great, $\dfrac{1 - x^{n}}{1 - x}$ or $\dfrac{1}{1 - x} - \dfrac{x^{n}}{1 - x}$ may be +brought as near to~$\dfrac{1}{1 - x}$ as we please, than which, +however, it must always be less, since $\dfrac{x^{n}}{1 - x}$ can never +entirely vanish, whatever value $n$~may have, and therefore +there is always something subtracted from $\dfrac{1}{1 - x}$. +It follows, nevertheless, that $1 + x + x^{2} + \etc.$, if we +are at liberty to take as many terms as we please, can +be brought as near as we please to~$\dfrac{1}{1 - x}$, and in this +sense we say that +\[ +\frac{1}{1 - x} = 1 + x + x^{2} + x^{3} + \etc.,\quad\textit{ad infinitum}. +\] + + +\Subsection{Convergent and Divergent Series.} + +A series is said to be \emph{convergent} when the sum of +its terms tends towards some limit; that is, when, by +taking any number of terms, however great, we shall +never exceed some certain quantity. On the other +hand, a series is said to be \emph{divergent} when the sum of +a number of terms may be made to surpass any quantity, +however great. Thus of the two series, +\[ +1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \etc. +\] +and +\[ +1 + 2 + 4 + 8 + \etc.\Add{,} +\] +the first is convergent, by what has been shown, and +the second is evidently divergent. A series cannot be +convergent, unless its separate terms decrease, so as, +\PageSep{18} +at last, to become less than any given quantity. And +the terms of a series may at first increase and afterwards +decrease, being apparently divergent for a finite +number of terms, and convergent afterwards. It will +only be necessary to consider the latter part of the +series. + +Let the following series consist of terms decreasing +without limit: +\[ +a + b + c + d + \dots + k + l + m + \dots, +\] +which may be put under the form +\[ +%[** TN: Small parentheses in the original here and below, as noted] +a\left(1 + \frac{b}{a} + + \frac{c}{b}\, \frac{b}{a} + + \frac{d}{c}\, \frac{c}{b}\, \frac{b}{a} + \etc.\right); +\] +the same change of form may be made, beginning +from any term of the series, thus: +\[ +%[** TN: Small ()] +k + l + m + \etc. + = k\left(1 + \frac{l}{k} + \frac{m}{l}\, \frac{l}{k} + \etc.\right). +\] +We have introduced the new terms, $\dfrac{b}{a}$,~$\dfrac{c}{b}$,~etc., or the +ratios which the several terms of the original series +bear to those immediately preceding. It may be shown +(1)~that if the terms of the series $\dfrac{b}{a}$,~$\dfrac{c}{b}$,~$\dfrac{d}{c}$, etc., come +at last to be less than unity, and afterwards either +continue to approximate to a limit which is less than +unity, or decrease without limit, the series $a + b + c + \etc.$, +is convergent; (2)~if the limit of the terms +$\dfrac{b}{a}$,~$\dfrac{c}{b}$,~etc., is either greater than unity, or if they increase +without limit, the series is divergent. + +(1\ia). Let $\dfrac{l}{k}$~be the first which is less than unity, +and let the succeeding ratios $\dfrac{m}{l}$,~etc., decrease, either +with or without limit, so that $\dfrac{l}{k} > \dfrac{m}{l} > \dfrac{n}{m}$, etc.; +whence it follows, that of the two series, +\PageSep{19} +\begin{align*} +%[** TN: Small ()] +k\biggl(1 &+ \frac{l}{k} + + \frac{l}{k}\, \PadTo{\dfrac{m}{l}}{\dfrac{l}{k}} + + \frac{l}{k}\, + \PadTo{\dfrac{m}{l}}{\dfrac{l}{k}}\, + \PadTo{\dfrac{n}{m}}{\dfrac{l}{k}} + + \etc.\biggr), \\ +k\biggl(1 &+ \frac{l}{k} + + \frac{l}{k}\, \frac{m}{l} + + \frac{l}{k}\, \frac{m}{l}\, \frac{n}{m} + + \etc.\biggr), +\end{align*} +the first is greater than the second. But since $\dfrac{l}{k}$~is +less than unity, the first can never surpass $k × \dfrac{1}{1 - \dfrac{l}{k}}$, +or~$\dfrac{k^{2}}{k - l}$, and is convergent; the second is therefore +convergent. But the second is no other than $k + l + m + \etc.$; +therefore the series $a + b + c + \etc.$, is convergent +from the term~$k$. + +\Chg{(1\ib.)}{(1\ib).} Let $\dfrac{l}{k}$~be less than unity, and let the successive +ratios $\dfrac{l}{k}$,~$\dfrac{m}{l}$,~etc., increase, never surpassing a +limit~$A$, which is less than unity. Hence of the two +series, +\begin{align*} +%[** TN: Small ()] +k(1 &+ \PadTo{\dfrac{l}{k}}{A} + + \PadTo{\dfrac{l}{k}}{A}\,\PadTo{\dfrac{m}{l}}{A} + + \PadTo{\dfrac{l}{k}}{A}\,\PadTo{\dfrac{m}{l}}{A}\,\PadTo{\dfrac{n}{m}}{A} + + \etc.), \\ +k\biggl(1 &+ \frac{l}{k} + + \frac{l}{k}\, \frac{m}{l} + + \frac{l}{k}\, \frac{m}{l}\, \frac{n}{m} + + \etc.\biggr), +\end{align*} +the first is the greater. But since $A$~is less than unity, +the first is convergent; whence, as before, $a + b + c + \etc.$, +converges from the term~$k$. + +(2) The second theorem on the divergence of series +we leave to the student's consideration, as it is not +immediately connected with our object. + + +\Subsection{Taylor's Theorem. Derived Functions.} + +We now proceed to the series +\index{Derivatives}% +\index{Derived Functions|EtSeq}% +\index{Functions!derived|EtSeq}% +\index{Taylor's Theorem|EtSeq}% +\[ +ph + qh^{2} + rh^{3} + sh^{4} + \etc., +\] +in which we are at liberty to suppose $h$ as small as +we please. The successive ratios of the terms to those +\PageSep{20} +immediately preceding are $\dfrac{qh^{2}}{ph}$ or~$\dfrac{q}{p}h$, $\dfrac{rh^{3}}{qh^{2}}$ or~$\dfrac{r}{q}h$, +$\dfrac{sh^{4}}{rh^{3}}$ or $\dfrac{s}{r}h$,~etc. If, then, the terms $\dfrac{q}{p}$,~$\dfrac{r}{q}$,~$\dfrac{s}{r}$, etc., +are always less than a finite limit~$A$, or become so after +a definite number of terms, $\dfrac{q}{p}h$,~$\dfrac{r}{q}h$,~etc., will always +be, or will at length become, less than~$Ah$. And since $h$~may +be what we please, it may be so chosen that $Ah$~shall +be less than unity, for which $h$~must be less than~$\dfrac{1}{A}$. +In this case, by theorem~(1\ib), the series is convergent; +it follows, therefore, that a value of~$h$ can +always be found so small that $ph + qh^{2} + rh^{3} + \etc.$, +shall be convergent, at least unless the coefficients +$p$,~$q$,~$r$,~etc., be such that the ratio of any one to the +preceding increases without limit, as we take more +distant terms of the series. This never happens in +the developments which we shall be required to consider +in the Differential Calculus. + +We now return to $\phi(x +h)$, which we have asserted +(\PageRef{16}) can be expanded (with the exception +of some particular values of~$x$) in a series of the form +$\phi x + ph + qh^{2} + \etc$. The following are some instances +of this development derived from the Differential +Calculus, most of which are also to be found in +treatises on algebra: +\index{Logarithms}% + +{\scriptsize +\begin{alignat*}{4} +(x + h)^{n} &= x^{n} + &+ nx^{n-1}h + &&+ n(n - 1)x^{n-2} \frac{h^{2}}{2} + &&+ n(n - 1)(n - 2)x^{n-3} \frac{h^{3}}{2·3} &\ \etc.\Add{,} \\ +% +a^{x + h} &= a^{x} + &+ ka^{x} h\rlap{\normalsize\footnotemark[1]} + &&+ k^{2} a^{x} \frac{h^{2}}{2} + &&+ k^{3} a^{x} \frac{h^{3}}{2·3} &\ \etc.\Add{,} \\ +% +\log(x + h) &= \log x + &+ \frac{1}{x}\, h + &&- \frac{1}{x^{2}}\, \frac{h^{2}}{2} + &&+ \frac{2}{x^{3}}\, \frac{h^{3}}{2·3} &\ \etc.\Add{,} \\ +% +\sin(x + h) &= \sin x + &+ \cos x\, h + &&- \sin x\, \frac{h^{2}}{2}\rlap{\normalsize\footnotemark[2]} + &&- \cos x\, \frac{h^{3}}{2·3} &\ \etc.\Add{,} \\ +% +%[** TN: Moved up from top of page21] +\cos(x + h) &= \cos x + &- \sin x\, h + &&- \cos x\, \frac{h^{2}}{2} + &&+ \sin x\, \frac{h^{3}}{2·3} &\ \etc. +\end{alignat*}}% +\footnotetext[1]{Here $k$~is the Naperian or hyperbolic logarithm of~$a$; that is, the common + logarithm of~$a$ divided by~$.434294482$.}% +\footnotetext[2]{In the last two series the terms are positive and negative in pairs.} +\PageSep{21} + +It appears, then, that the development of~$\phi(x + h)$ +consists of certain functions of~$x$, the first of which is +$\phi x$~itself, and the remainder of which are multiplied +by $h$,~$\dfrac{h^{2}}{2}$, $\dfrac{h^{3}}{2·3}$, $\dfrac{h^{4}}{2·3·4}$, and so on. It is usual to denote +the coefficients of these divided powers of~$h$ by $\phi' x$, +$\phi'' x$, $\phi''' x$,\footnote + {Called \emph{derived functions} or \emph{derivatives}.---\Ed.} +\index{Derivatives}% +\index{Derived Functions}% +\index{Functions!derived}% +etc., where $\phi'$,~$\phi''$,~etc., are merely functional +symbols, as is $\phi$~itself; but it must be recollected +that $\phi' x$,~$\phi'' x$,~etc., are rarely, if ever, employed +to signify anything except the coefficients of~$h$, $\dfrac{h^{2}}{2}$,~etc., +in the development of~$\phi(x + h)$. Hence this development +is usually expressed as follows: +\[ +\phi(x + h) + = \phi x + \phi' x\, h + \phi''x\, \frac{h^{2}}{2} + \phi''' x\, \frac{h^{3}}{2·3} + \etc. +\] + +{\Loosen Thus, when $\phi x = x^{n}$, $\phi' x = nx^{n-1}$, $\phi'' x = n(n - 1)x^{n-2}$, etc.; +when $\phi x = \sin x$, $\phi' x = \cos x$, $\phi'' x = -\sin x$,~etc. +In the first case $\phi'(x + h) = n(x + h)^{n-1}$, +$\phi''(x + h) = n(n - 1)(x + h)^{n-2}$; and in the second +$\phi'(x + h) = \cos (x + h)$, $\phi''(x + h) = -\sin(x + h)$.} + +The following relation exists between $\phi x$,~$\phi' x$, +$\phi'' x$,~etc. In the same manner as $\phi' x$~is the coefficient +of~$h$ in the development of~$\phi(x + h)$, so $\phi'' x$~is the coefficient +of~$h$ in the development of~$\phi'(x + h)$, and +$\phi''' x$~is the coefficient of~$h$ in the development of~$\phi''(x + h)$; +$\phi^{\text{iv}} x$~is the coefficient of~$h$ in the development +of $\phi'''(x + h)$, and so on. + +The proof of this is equivalent to \emph{Taylor's Theorem} +already alluded to (\PageRef{15}); and the fact may be +verified in the examples already given. When $\phi x = a^{x}$, +$\phi' x = ka^{x}$, and $\phi'(x + h) = ka^{x+h} = k(a^{x} + ka^{x}\, h + \etc.)$. +The coefficient of~$h$ is here~$k^{2} a^{x}$, which is the +\PageSep{22} +same as~$\phi'' x$. (See the second example of the preceding +table.) Again, $\phi''(x + h) = k^{2}a^{x+h} = k^{2}(a^{x} + ka^{x}\, h + \etc.)$, +in which the coefficient of~$h$ is~$k^{3}a^{x}$, the +same as~$\phi''' x$. Again, if $\phi x = \log x$, $\phi' x = \dfrac{1}{x}$, and +$\phi'(x + h) = \dfrac{1}{x + h} = \dfrac{1}{x} - \dfrac{h}{x^{2}} + \etc.$, as appears by +common division. Here the coefficient of~$h$ is~$-\dfrac{1}{x^{2}}$, +which is the same as $\phi'' x$~in the third example. Also +$\phi''(x + h) = -\dfrac{1}{(x + h)^{2}} = -(x + h)^{-2}$, which by the +Binomial Theorem is $-(x^{-2} - 2x^{-3}\, h + \etc.)$. The +coefficient of~$h$ is~$2x^{-3}$ or~$\dfrac{2}{x^{3}}$, which is~$\phi''' x$ in the +same example. + + +\Subsection{Differential Coefficients.} + +It appears, then, that if we are able to obtain the +\index{Coefficients, differential|EtSeq}% +\index{Differential coefficients|EtSeq}% +coefficient of~$h$ in the development of \emph{any} function +whatever of~$x + h$, we can obtain all the other coefficients, +since we can thus deduce $\phi' x$ from~$\phi x$, $\phi'' x$ +from~$\phi' x$, and so on. It is usual to call~$\phi' x$ the first +differential coefficient of~$\phi x$, $\phi'' x$~the second differential +coefficient of~$\phi x$, or the first differential coefficient +of~$\phi' x$; $\phi''' x$~the third differential coefficient of~$\phi x$, +or the second of~$\phi' x$, or the first of~$\phi'' x$; and so on.\footnote + {The first, second, third, etc., differential coefficients, as thus obtained, + are also called the first, second, third, etc., \emph{derivatives}.---\Ed.} +\index{Derivatives}% +The name is derived from a method of obtaining~$\phi' x$, +etc., which we now proceed to explain. + +Let there be any function of~$x$, which we call~$\phi x$, +in which $x$~is increased by an increment~$h$; the function +then becomes +\[ +\phi x + \phi' x\, h + + \phi'' x\, \frac{h^{2}}{2} + + \phi''' x\, \frac{h^{3}}{2·3} + \etc. +\] +\PageSep{23} +The original value~$\phi x$ is increased by the increment +\[ +\phi' x\, h + \phi'' x\, \frac{h^{2}}{2} x + + \phi''' x\, \frac{h^{3}}{2·3} + \etc.; +\] +whence ($h$~being the increment of~$x$) +\[ +\frac{\emph{increment of } \phi x}{\emph{increment of } x} + = \phi' x + \phi'' x\, \frac{h}{2} x + + \phi''' x\, \frac{h^{2}}{2·3} + \etc., +\] +which is an expression for the ratio which the increment +of a function bears to the increment of its variable. +It consists of two parts. The one,~$\phi' x$, into +which $h$~does not enter, depends on $x$~only; the remainder +is a series, every term of which is multiplied +by some power of~$h$, and which therefore diminishes +as $h$~diminishes, and may be made as small as we +please by making $h$~sufficiently small. + +To make this last assertion clear, observe that all +the ratio, except its first term~$\phi' x$, may be written as +follows: +\[ +%[** TN: Small () in the original] +h\left(\phi'' x\, \frac{1}{2} + \phi''' x\, \frac{h}{2·3} + \etc.\right); +\] +the second factor of which (\PageRef{19}) is a convergent +series whenever $h$~is taken less than~$\dfrac{1}{A}$, where $A$~is +the limit towards which we approximate by taking +the coefficients $\phi'' x × \dfrac{1}{2}$, $\phi''' x × \dfrac{1}{2·3}$,~etc., and forming +the ratio of each to the one immediately preceding. +This limit, as has been observed, is finite in +every series which we have occasion to use; and +therefore a value for~$h$ can be chosen so small, that +for it the series in the last-named formula is convergent; +still more will it be so for every smaller value +of~$h$. Let the series be called~$P$. If $P$~be a finite quantity, +which decreases when $h$~decreases, $Ph$~can be +made as small as we please by sufficiently diminishing~$h$; +\PageSep{24} +whence $\phi' x + Ph$ can be brought as near as we +please to~$\phi' x$. Hence the ratio of the increments of +$\phi x$ and~$x$, produced by changing $x$ into~$x + h$, though +never equal to~$\phi' x$, approaches towards it as $h$~is diminished, +and may be brought as near as we please +to it, by sufficiently diminishing~$h$. Therefore to find +the coefficient of~$h$ in the development of~$\phi(x + h)$, +find $\phi(x + h) - \phi x$, divide it by~$h$, and find the limit +towards which it tends as $h$~is diminished. + +In any series such as +\index{Series|EtSeq}% +\[ +a + bh + ch^{2} + \dots + kh^{n} + lh^{n+1} + mh^{n+2} + \etc. +\] +which is such that some given value of~$h$ will make it +convergent, it may be shown that $h$~can be taken so +small that any one term shall contain all the succeeding +ones as often as we please. Take any one term, +as~$kh^{n}$. It is evident that, be $h$ what it may, +\[ +kh^{n} : lh^{n+1} + mh^{n+2} + \etc.,\ ::\ k : lh + mh^{2} + \etc., +\] +the last term of which is $h(l + mh + \etc.)$. By reasoning +similar to that in the last paragraph, we can +show that this may be made as small as we please, +since one factor is a series which is always finite when +$h$~is less than~$\dfrac{1}{A}$, and the other factor~$h$ can be made +as small as we please. Hence, since $k$~is a given +quantity, independent of~$h$, and which therefore remains +the same during all the changes of~$h$, the series +$h(l + mh + \etc.)$ can be made as small a part of~$k$ as +we please, since the first diminishes without limit, +and the second remains the same. By the proportion +above established, it follows then that $lh^{n+1} + mh^{n+2} + \etc.$, +can be made as small a part as we please of~$kh^{n}$. +It follows, therefore, that if, instead of the full +development of~$\phi(x + h)$, we use only its two first +\PageSep{25} +terms $\phi x + \phi' x\, h$, the error thereby introduced may, +by taking $h$ sufficiently small, be made as small a portion +as we please of the small term~$\phi' x\, h$. + + +\Subsection{The Notation of the Differential Calculus.} + +The first step usually made in the Differential Calculus +\index{Calculus, notation of}% +\index{Notation!of the Differential Calculus}% +is the determination of~$\phi' x$ for all possible values +of~$\phi x$, and the construction of general rules for +that purpose. Without entering into these we proceed +to explain the notation which is used, and to apply +the principles already established to the solution +of some of those problems which are the peculiar +province of the Differential Calculus. + +When any quantity is increased by an increment, +which, consistently with the conditions of the problem, +may be supposed as small as we please, this increment +is denoted, not by a separate letter, but by +prefixing the letter~$d$, either followed by a full stop or +not, to that already used to signify the quantity. For +example, the increment of~$x$ is denoted under these +circumstances by~$dx$; that of~$\phi x$ by~$d.\phi x$; that of~$x^{n}$ +by~$d.x^{n}$. If instead of an increment a decrement +be used, the sign of~$dx$, etc., must be changed in all +expressions which have been obtained on the supposition +of an increment; and if an increment obtained +by calculation proves to be negative, it is a sign that +a quantity which we imagined was increased by our +previous changes, was in fact diminished. Thus, if +$x$~becomes $x + dx$, $x^{2}$~becomes $x^{2} + d.x^{2}$. But this is +also $(x + dx)^{2}$ or $x^{2} + 2x\, dx + (dx)^{2}$; whence $d.x^{2} = 2x\, dx + (dx)^{2}$. +Care must be taken not to confound +$d.x^{2}$, the increment of~$x^{2}$, with~$(dx)^{2}$, or, as it is often +written,~$dx^{2}$, the square of the increment of~$x$. Again, +\PageSep{26} +if $x$~becomes $x + dx$, $\dfrac{1}{x}$ becomes $\dfrac{1}{x} + d.\dfrac{1}{x}$ and the +change of~$\dfrac{1}{x}$ is $\dfrac{1}{x + dx} - \dfrac{1}{x}$ or $-\dfrac{dx}{x^{2} + x\, dx}$; showing +that an increment of~$x$ produces a decrement in~$\dfrac{1}{x}$. + +It must not be imagined that because $x$~occurs in +the symbol~$dx$, the value of the latter in any way depends +upon that of the former: both the first value of~$x$, +and the quantity by which it is made to differ from +its first value, are at our pleasure, and the letter~$d$ must +merely be regarded as an abbreviation of the words +``\emph{difference of}.'' In the first example, if we divide +\index{Differences!of increments}% +both sides of the resulting equation by~$dx$, we have +$\dfrac{d.x^{2}}{dx} = 2x + dx$. The smaller $dx$~is supposed to be, +the more nearly will this equation assume the form +$\dfrac{d.x^{2}}{dx} = 2x$, and the ratio of $2x$ to~$1$ is the limit of the +\index{Limits|EtSeq}% +ratio of the increment of~$x^{2}$ to that of~$x$; to which +this ratio may be made to approximate as nearly as +we please, but which it can never actually reach. In +the Differential Calculus, the limit of the ratio only is +retained, to the exclusion of the rest, which may be +explained in either of the two following ways: + +(1) The fraction $\dfrac{d.x^{2}}{dx}$ may be considered as standing, +not for any value which it can actually have as +long as $dx$~has a real value, but for the limit of all +those values which it assumes while $dx$~diminishes. +In this sense the equation $\dfrac{d.x^{2}}{dx} = 2x$ is strictly true. +But here it must be observed that the algebraical +meaning of the sign of division is altered, in such a +way that it is no longer allowable to use the numerator +and denominator separately, or even at all to consider +\PageSep{27} +them as quantities. If $\dfrac{dy}{dx}$~stands, not for the +ratio of two quantities, but for the limit of that ratio, +which cannot be obtained by taking any real value of~$dx$, +however small, the whole $\dfrac{dy}{dx}$ may, by convention, +have a meaning, but the separate parts $dy$ and~$dx$ +have none, and can no more be considered as separate +quantities whose ratio is~$\dfrac{dy}{dx}$, than the two loops +of the figure~$8$ can be considered as separate numbers +whose sum is eight. This would be productive of no +great inconvenience if it were never required to separate +the two; but since all books on the Differential +Calculus and its applications are full of examples in +which deductions equivalent to assuming $dy = 2x\, dx$ +are drawn from such an equation as $\dfrac{dy}{dx} = 2x$, it becomes +necessary that the first should be explained, independently +of the meaning first given to the second. +It may be said, indeed, that if $y = x^{2}$, it follows that +$\dfrac{dy}{dx} = 2x + dx$, in which, \emph{if we make $dx = 0$}, the result +is $\dfrac{dy}{dx} = 2x$. But if $dx = 0$, $dy$~also~$= 0$, and this +equation should be written $\dfrac{0}{0} = 2x$, as is actually done +in some treatises on the Differential Calculus,\footnote + {This practice was far more common in the early part of the century + than now, and was due to the precedent of Euler (1755). For the sense in +\index{Euler}% + which Euler's view was correct, see the \Title{Encyclopedia Britannica}, art.\ \Title{Infinitesimal + Calculus}, Vol.~XII, p.~14, 2nd~column.---\Ed.} +to the +great confusion of the learner. Passing over the difficulties\footnote + {See \Title{Study of Mathematics} (Reprint Edition, Chicago: The Open Court + Publishing~Co., 1898), page~126.} +of the fraction~$\dfrac{0}{0}$, still the former objection +recurs, that the equation $dy = 2x\, dx$ cannot be used +\PageSep{28} +(and it \emph{is} used even by those who adopt this explanation) +without supposing that~$0$, which merely implies +an absence of all magnitude, can be used in different +senses, so that one~$0$ may be contained in another a +certain number of times. This, even if it can be considered +as intelligible, is a notion of much too refined +a nature for a beginner. + +(2) The presence of the letter~$d$ is an indication, +not only of an increment, but of an increment which +we are at liberty to suppose as small as we please. +The processes of the Differential Calculus are intended +to deduce relations, not between the ratios of different +increments, but between the limits to which those ratios +approximate, when the increments are decreased. +And it may be true of some parts of an equation, that +though the taking of them away would alter the relation +between $dy$ and~$dx$, it would not alter the limit +towards which their ratio approximates, when $dx$ +and~$dy$ are diminished. For example, $dy = 2x\, dx + (dx)^{2}$. +If $x = 4$ and $dx = .01$, then $dy = .0801$ and +$\dfrac{dy}{dx} = 8.01$. If $dx = .0001$, $dy = .00080001$ and $\dfrac{dy}{dx} = 8.0001$. +The limit of this ratio, to which we shall +come still nearer by making $dx$ still smaller, is~$8$. The +term~$(dx)^{2}$, though its presence affects the value of~$dy$ +and the ratio~$\dfrac{dy}{dx}$, does not affect the limit of the latter, +for in $\dfrac{dy}{dx}$ or $2x + dx$, the latter term~$dx$, which arose +from the term~$(dx)^{2}$, diminishes continually and without +limit. If, then, we throw away the term~$(dx)^{2}$, +the consequence is that, make $dx$ what we may, we +never obtain~$dy$ as it would be if correctly deduced +from the equation $y = x^{2}$, but we obtain the limit of +the ratio of~$dy$ to~$dx$. If we throw away all powers of~$dx$ +\PageSep{29} +above the first, and use the equations so obtained, +all ratios formed from these last, or their consequences, +are themselves the limiting ratios of which we are in +search. \emph{The equations which we thus use are not absolutely +true in any case, but may be brought as near as we +please to the truth}, by making $dy$~and~$dx$ sufficiently +small. If the student at first, instead of using $dy = 2x\, dx$, +were to write it thus, $dy = 2x\, dx + \etc.$, the \emph{etc.}\ +would remind him that there are other terms; \emph{necessary}, +if the value of~$dy$ corresponding to any value of~$dx$ +is to be obtained; \emph{unnecessary}, if the \emph{limit} of the +ratio of $dy$ to~$dx$ is all that is required. + +We must adopt the first of these explanations when +$dy$ and $dx$ appear in a fraction, and the second when +they are on opposite sides of an equation. + + +\Subsection{Algebraical Geometry.} + +If two straight lines be drawn at right angles to +each other, dividing the whole of their plane into four +parts, one lying in each right angle, the situation +of any point is determined when we know, (1)~in +which angle it lies, and (2)~its perpendicular distances +from the two right lines. Thus (\Fig{2}) the point~$P$ +lying in the angle~$AOB$, is known when $PM$~and~$PN$, +or when $OM$~and~$PM$ are known; for, though there +is an infinite number of points whose distance from~$OA$ +only is the same as that of~$P$, and an infinite number +of others, whose distance from~$OB$ is the same as +that of~$P$, there is no other point whose distances +from both lines are the same as those of~$P$. The line~$OA$ +is called the axis of~$x$, because it is usual to denote +any variable distance measured on or parallel to~$OA$ +by the letter~$x$. For a similar reason, $OB$~is called +\PageSep{30} +\index{Co-ordinates}% +the axis of~$y$. The \emph{co-ordinates}\footnote + {The distances $OM$ and~$MP$ are called the \emph{co-ordinates} of the point~$P$. It + is moreover usual to call the co-ordinate~$OM$, the \emph{abscissa}, and $MP$, the \emph{ordinate}, + of the point~$P$.} +or perpendicular distances +of a point~$P$ which is supposed to vary its position, +are thus denoted by $x$~and~$y$; hence $OM$ or~$PN$ +is~$x$, and $PM$ or~$ON$ is~$y$. Let a linear unit be chosen, +so that any number may be represented by a straight +line. Let the point~$M$, setting out from~$O$, move in +the direction~$OA$, always carrying with it the indefinitely +extended line~$MP$ perpendicular to~$OA$. While +this goes on, let $P$~move upon the line~$MP$ in such a +way, that $MP$ or~$y$ is always equal to a given function +of~$OM$ or~$x$; for example, let $y = x^{2}$, or let the number +\index{Parabola, the}% +\Figure{2} +of units in~$PM$ be the square of the number of +units in~$OM$. As $O$~moves towards~$A$, the point~$P$ +will, by its motion on~$MP$, compounded with the motion +of the line $MP$ itself, describe a curve~$OP$, in +which $PM$~is less than, equal to, or greater than,~$OM$, +according as $OM$~is less than, equal to, or greater +than the linear unit. It only remains to show how +the other branch of this curve is deduced from the +equation $y = x^{2}$. And to this end we shall first have +to interpolate a few remarks. +\PageSep{31} + + +\Subsection{On the Connexion of the Signs of Algebraical and +the Directions of Geometrical Magnitudes.} + +It is shown in algebra, that if, through misapprehension +\index{Signs|EtSeq}% +of a problem, we measure in one direction, a +line which ought to lie in the exactly opposite direction, +or if such a mistake be a consequence of some +previous misconstruction of the figure, any attempt +to deduce the length of that line by algebraical reasoning, +will give a negative quantity as the result. +And conversely it may be proved by any number of +examples, that when an equation in which $a$~occurs +has been deduced strictly on the supposition that $a$~is +a line measured in one direction, a change of sign in~$a$ +will turn the equation into that which would have +been deduced by the same reasoning, had we begun +by measuring the line~$a$ in the contrary direction. +Hence the change of~$+a$ into~$-a$, or of~$-a$ into~$+a$, +corresponds in geometry to a change of direction of +the line represented by~$a$, and \textit{vice versa}. + +In illustration of this general fact, the following +\index{Circle, equation of|EtSeq}% +\index{Circle cut by straight line, investigated|EtSeq}% +problem may be useful. Having a circle of given radius, +whose centre is in the intersection of the axes +of $x$~and~$y$, and also a straight line cutting the axes in +two given points, required the co-ordinates of the +points (if any) in which the straight line cuts the circle. +Let $OA$, the radius of the circle~$= r$, $OE = a$, +$OF = b$, and let the co-ordinates of~$P$, one of the +points of intersection required, be $OM = x$, $MP = y$. +(\Fig{3}.) The point~$P$ being in the circle whose radius +is~$r$, we have from the right-angled triangle~$OMP$, +$x^{2} + y^{2} = r^{2}$, which equation belongs to the co-ordinates +of every point in the circle, and is called +\PageSep{32} +the equation of the circle. Again, $EM : MP :: EO : OF$ +by similar triangles; or $a - x : y :: a : b$, whence $ay + bx = ab$, +which is true, by similar reasoning, for every +point of the line~$EF$. But for a point~$P'$ lying in~$EF$ +produced, we have $EM' : M'P' :: EO : OF$, or $x + a : y :: a : b$, +whence $ay - bx = ab$, an equation which may +be obtained from the former by changing the sign of~$x$; +and it is evident that the direction of~$x$, in the +\Figure{3} +second case, is opposite to that in the first. Again, +for a point~$P''$ in $FE$ produced, we have $EM'' : M''P'' :: EO : OF$, +or $x - a : y :: a : b$, whence $bx - ay = ab$, which +may be deduced from the first by changing the sign +of~$y$; and it is evident that $y$~is measured in different +directions in the first and third cases. Hence the +equation $ay + bx = ab$ belongs to all parts of the +straight line~$EF$, if we agree to consider $M''P''$ as +negative, when $MP$~is positive, and $OM'$~as negative +\PageSep{33} +when $OM$~is positive. Thus, if $OE = 4$, and $OF = 5$, +and $OM = 1$, we can determine~$MP$ from the equation +$ay + bx = ab$, or $4y + 5 = 20$, which gives $y$~or $MP = 3\frac{3}{4}$. +But if $OM'$~be $1$ in length, we can determine~$M'P'$ +either by calling $OM'$,~$1$, and using the equation +$ay - bx = ab$, or calling $OM'$,~$-1$, and using the equation +$ay + bx = ab$, as before. Either gives $M'P' = 6\frac{1}{4}$. +The latter method is preferable, inasmuch as it enables +us to contain, in one investigation, all the different +cases of a problem. + +We shall proceed to show that this may be done +in the present instance. We have to determine the +co-ordinates of the point~$P$, from the following equations: +\begin{align*} +ay + bx &= ab, \\ +x^{2} + y^{2} &= r^{2}. +\end{align*} +Substituting in the second the value of~$y$ derived from +the first, or $b\left(\dfrac{a - x}{b}\right)$, we have +\[ +x^{2} + b^{2}\, \frac{(a - x)^{2}}{a^{2}} = r^{2}, +\] +or +\[ +(a^{2} + b^{2}) x^{2} - 2ab^{2}x + a^{2}(b^{2} - r^{2}) = 0; +\] +and proceeding in a similar manner to find~$y$, we have +\[ +(a^{2} + b^{2}) y^{2} - 2a^{2}by + b^{2}(a^{2} - r^{2}) = 0, +\] +which \Typo{give}{gives} +\begin{align*} +x &= a\, \frac{b^{2} ± \sqrt{(a^{2} + b^{2})r^{2} - a^{2}b^{2}}}{a^{2} + b^{2}}, \\ +y &= b\, \frac{a^{2} \mp \sqrt{(a^{2} + b^{2})r^{2} - a^{2}b^{2}}}{a^{2} + b^{2}}; +\end{align*} +the upper or the lower sign to be taken in both. +Hence when $(a^{2} + b^{2})r^{2} > a^{2}b^{2}$, that is, when $r$~is greater +than the perpendicular let fall from~$O$ upon~$EF$, which +perpendicular is +\PageSep{34} +\[ +\frac{ab}{\sqrt{a^{2} + b^{2}}}, +\] +there are two points of intersection. When $(a^{2} + b^{2})r^{2} = a^{2}b^{2}$, +the two values of~$x$ become equal, and also +those of~$y$, and there is only one point in which the +straight line meets the circle; in this case $EF$~is a +tangent to the circle. And if $(a^{2} + b^{2})r^{2} < a^{2}b^{2}$, the +values of $x$~and~$y$ are impossible, and the straight line +does not meet the circle. + +Of these three cases, we confine ourselves to the +first, in which there are two points of intersection. +The product of the values of~$x$, with their proper +sign, is\footnote + {See \Title{Study of Mathematics} (Chicago: The Open Court Pub.~Co.), page~136.} +\[ +a^{2}\, \frac{b^{2} - r^{2}}{a^{2} + b^{2}}, +\] +and of~$y$, +\[ +b^{2}\, \frac{a^{2} - r^{2}}{a^{2} + b^{2}}, +\] +the signs of which are the same as those of~$b^{2} - r^{2}$, +and $a^{2} - r^{2}$. If $b$~and~$a$ be both $> r$, the two values +of~$x$ have the same sign; and it will appear from the +figure, that the lines they represent are measured in +the same direction. And this whether $b$~and~$a$ be positive +or negative, since $b^{2} - r^{2}$ and $a^{2} - r^{2}$ are both +positive when $a$~and~$b$ are numerically greater than~$r$, +whatever their signs may be. That is, if our rule, +connecting the signs of algebraical and the directions +of geometrical magnitudes, be true, let the directions +of $OE$ and $OF$ be altered in any way, so long as $OE$ +and $OF$ are both greater than~$OA$, the two values of~$OM$ +will have the same direction, and also those of~$MP$. +This result may easily be verified from the +figure. +\PageSep{35} + +Again, the values of $x$~and~$y$ having the same sign, +that sign will be (see the equations) the same as that +of $2ab^{2}$ for~$x$, and of $2a^{2}b$ for~$y$, or the same as that of +$a$~for~$x$ and of $b$~for~$y$. That is, when $OE$~and~$OF$ are +both greater than~$OA$, the direction of each set of co-ordinates +will be the same as those of $OE$ and~$OF$, +which may also be readily verified from the figure. + +Many other verifications might thus be obtained of +the same principle, viz., that any equation which corresponds +to, and is true for, all points in the angle~$AOB$, +may be used without error for all points lying +in the other three angles, by substituting the proper +numerical values, with a negative sign, for those co-ordinates +whose directions are opposite to those of +the co-ordinates in the angle~$AOB$. In this manner, +if four points be taken similarly situated in the four +angles, the numerical values of whose co-ordinates +are $x = 4$ and $y = 6$, and if the co-ordinates of that +point which lies in the angle~$AOB$, are called $+4$ and~$+6$; +those of the points lying in the angle~$BOC$ will +be $-4$~and~$+6$; in the angle~$COD$ $-4$~and~$-6$; +and in the angle~$DOE$ $+4$~and~$-6$. + +To return to \Fig{2}, if, after having completed the +branch of the curve which lies on the right of~$BC$, +and whose equation is $y = x^{2}$, we seek that which lies +on the left of~$BC$, we must, by the principles established, +substitute $-x$ instead of~$x$, when the numerical +value obtained for~$(-x)^{2}$ will be that of~$y$, and the +sign will show whether $y$~is to be measured in a similar +or contrary direction to that of~$MP$. Since $(-x)^{2} = x^{2}$, +the direction and value of~$y$, for a given value +of~$x$, remains the same as on the right of~$BC$; whence +the remaining branch of the curve is similar and equal +in all respects to~$OP$, only lying in the angle~$BOD$. +\PageSep{36} +And thus, if $y$ be any function of~$x$, we can obtain a +geometrical representation of the same, by making $y$ +the ordinate, and $x$~the abscissa of a curve, every ordinate +of which shall be the linear representation of +the numerical value of the given function corresponding +to the numerical value of the abscissa, the linear +unit being a given line. + + +\Subsection{The Drawing of a Tangent to a Curve.} + +If the point~$P$ (\Fig{2}), setting out from~$O$, move +along the branch~$OP$, it will continually change the +\Figure[nolabel]{2} +\emph{direction} of its motion, never moving, at one point, in +\index{Direction}% +the direction which it had at any previous point. Let +the moving point have reached~$P$, and let $OM = x$, +$MP = y$. Let $x$~receive the increment $MM' = dx$, in +consequence of which $y$ or $MP$ becomes~$M'P'$, and +receives the increment $QP' = dy$; so that $x + dx$ and +$y + dy$ are the co-ordinates of the moving point~$P$, +when it arrives at~$P'$. Join~$PP'$, which makes, with +$PQ$ or~$OM$, an angle, whose tangent is $\dfrac{P'Q}{PQ}$ or~$\dfrac{dy}{dx}$. +Since the relation $y = x^{2}$ is true for the co-ordinates of +every point in the curve, we have $y + dy = (x + dx)^{2}$, +\PageSep{37} +the subtraction of the former equation from which +gives $dy = 2x\, dx + (dx)^{2}$, or $\dfrac{dy}{dx} = 2x + dx$. If the +point~$P'$ be now supposed to move backwards towards~$P$, +the chord~$PP'$ will diminish without limit, and the +inclination of $PP'$ to $PQ$ will also diminish, but not +without limit, since the tangent of the angle~$P'PQ$, or~$\dfrac{dy}{dx}$, +\index{Tangent}% +is always greater than the limit~$2x$. If, therefore, +a line~$PV$ be drawn through~$P$, making with~$PQ$ an +angle whose tangent is~$2x$, the chord~$PP'$ will, as $P'$~approaches +towards~$P$, or as $dx$~is diminished, continually +approximate towards~$PV$, so that the angle~$P'PV$ +may be made smaller than any given angle, by +sufficiently diminishing~$dx$. And the line~$PV$ cannot +again meet the curve on the side of~$PP'$, nor can any +straight line be drawn between it and the curve, the +proof of which we leave to the student. + +Again, if $P'$~be placed on the other side of~$P$, so that +its co-ordinates are $x - dx$ and $y - dy$, we have $y - dy = (x - dx)^{2}$, +which, subtracted from $y = x^{2}$, gives $dy = 2x\, dx - (dx)^{2}$, +or $\dfrac{dy}{dx} = 2x - dx$. By similar reasoning, +if the straight line~$PT$ be drawn in continuation +of~$PV$, making with~$PN$ an angle, whose tangent is~$2x$, +the chord~$PP'$ will continually approach to this +line, as before. + +The line~$TPV$ indicates the direction in which the +point~$P$ is proceeding, and is called the \emph{tangent} of the +curve at the point~$P$. If the curve were the interior +of a small solid tube, in which an atom of matter were +made to move, being projected into it at~$O$, and if all +the tube above~$P$ were removed, the line~$PV$ is in the +direction which the atom would take on emerging at~$P$, +and is the line which it would describe. The angle +\PageSep{38} +which the tangent makes with the axis of~$x$ in any +\index{Tangent}% +curve, may be found by giving $x$ an increment, finding +the ratio which the corresponding increment of~$y$ +bears to that of~$x$, and determining the limit of that +ratio, or the \emph{differential coefficient}. This limit is the +\index{Coefficients, differential}% +\index{Differential coefficients}% +trigonometrical tangent\footnote + {There is some confusion between these different uses of the word tangent. + The geometrical tangent is, as already defined, the line between which + and a curve no straight line can be drawn; the trigonometrical tangent has + reference to an angle, and is the ratio which, in any right-angled triangle, + the side opposite the angle bears to that which is adjacent.} +of the angle which the geometrical +tangent makes with the axis of~$x$. If $y = \phi x$, +$\phi' x$~is this trigonometrical tangent. Thus, if the curve +be such that the ordinates are the Naperian logarithms\footnote + {It may be well to notice that in analysis the Naperian logarithms are +\index{Logarithms}% + the only ones used; while in practice the common, or Briggs's logarithms, + are always preferred.} +of the abscissæ, or $y = \log x$, and $y + dy = +\log x + \dfrac{1}{x}\, dx - \dfrac{1}{2x^{2}}\, dx^{2}$, etc., the geometrical tangent +of any point whose abscissa is~$x$, makes with the axis +an angle whose trigonometrical tangent is~$\dfrac{1}{x}$. + +This problem, of drawing a tangent to any curve, +was one, the consideration of which gave rise to the +methods of the Differential Calculus. + + +\Subsection{Rational Explanation of the Language of Leibnitz.} + +As the peculiar language of the theory of infinitely +\index{Infinitely small, the notion of|EtSeq}% +\index{Leibnitz}% +small quantities is extensively used, especially in +works of natural philosophy, it has appeared right to +us to introduce it, in order to show how the terms +which are used may be made to refer to some natural +and rational mode of explanation. In applying this +language to \Fig{2}, it would be said that the curve~$OP$ +is a polygon consisting of an infinite number of +\index{Polygon}% +\PageSep{39} +infinitely small sides, each of which produced is a +tangent to the curve; also that if $MM'$ be taken infinitely +small, the chord and arc~$PP'$ coincide with +\index{Arc and its chord, a continuously decreasing|EtSeq}% +one of these rectilinear elements; and that an infinitely +small arc coincides with its chord. All which +must be interpreted to mean that, the chord and arc +being diminished, approach more and more nearly to +a ratio of equality as to their lengths; and also that +the greatest separation between an arc and its chord +may be made as small a part as we please of the whole +chord or arc, by sufficiently diminishing the chord. + +We shall proceed to a strict proof of this; but in +the meanwhile, as a familiar illustration, imagine a +small arc to be cut off from a curve, and its extremities +joined by a chord, thus forming an arch, of which +the chord is the base. From the middle point of the +chord, erect a perpendicular to it, meeting the arc, +which will thus represent the height of the arch. +Imagine this figure to be magnified, without distortion +or alteration of its proportions, so that the larger figure +may be, as it is expressed, a true picture of the +smaller one. However the original arc may be diminished, +let the magnified base continue of a given +length. This is possible, since on any line a figure +may be constructed similar to a given figure. If the +original curve could be such that the height of the +arch could never be reduced below a certain part of +the chord, say one thousandth, the height of the magnified +arch could never be reduced below one thousandth +of the magnified chord, since the proportions +of the two figures are the same. But if, in the original +curve, an arc can be taken so small that the height +of the arch is as small a part as we please of the +chord, it will follow that in the magnified figure where +\PageSep{40} +the chord is always of one length, the height of the +arch can be made as small as we please, seeing that +it can be made as small a part as we please of a given +line. It is possible in this way to conceive a whole +curve so magnified, that a given arc, however small, +shall be represented by an arc of any given length, +however great; and the proposition amounts to this, +that let the dimensions of the magnified curve be any +\index{Curve, magnified}% +\index{Magnified curve}% +given number of times the original, however great, an +arch can be taken upon the original curve so small, +that the height of the corresponding arch in the magnified +figure shall be as small as we please. +\Figure{4} + +Let $PP'$ (\Fig{4}) be a part of a curve, whose equation +is $y = \phi(x)$, that is, $PM$~may always be found by +substituting the numerical value of~$OM$ in a given +function of~$x$. Let $OM = x$ receive the increment +$MM' = dx$, which we may afterwards suppose as small +as we please, but which, in order to render the figure +more distinct, is here considerable. The value of $PM$ +or~$y$ is~$\phi x$, and that of $P'M'$ or $y + dy$ is~$\phi(x + dx)$. + +Draw $PV$, the tangent at~$P$, which, as has been +\index{Tangent}% +shown, makes, with~$PQ$, an angle, whose trigonometrical +tangent is the limit of the ratio~$\dfrac{dy}{dx}$, when $x$~is decreased, +or~$\phi' x$. Draw the chord~$PP'$, and from any +\PageSep{41} +point in it, for example, its middle point~$p$, draw~$pv$ +parallel to~$PM$, cutting the curve in~$a$. The value of~$P'Q$, +or~$dy$, or $\phi(x + dx) - \phi x$ is %[** TN: This line displayed in the orig] +\[ +P'Q = \phi' x\, dx + + \phi'' x\, \frac{(dx)^{2}}{2} + + \phi''' x\, \frac{(dx)^{3}}{2·3} + \etc. +\] +But $\phi' x\, dx$~is $\tan VPQ·PQ = VQ$. Hence $VQ$~is the +first term of this series, and $P'V$~the aggregate of the +rest. But it has been shown that $dx$~can be taken so +small, that any one term of the above series shall contain +the rest, as often as we please. Hence $PQ$~can +be taken so small that $VQ$~shall contain~$VP'$ as often +as we please, or the ratio of $VQ$ to~$VP'$ shall be as +great as we please. And the ratio $VQ$ to~$PQ$ continues +finite, being always~$\phi' x$; hence $P'V$~also decreases +without limit as compared with~$PQ$. + +Next, the chord~$PP'$ or $\sqrt{(dx)^{2} + (dy)^{2}}$, or +\[ +dx \sqrt{1 + \left(\frac{dy}{dx}\right)^{2}} +\] +is to~$PQ$ or~$dx$ in the ratio of $\sqrt{1 + \left(\dfrac{dy}{dx}\right)^{2}} : 1$, which, +as $PQ$~is diminished, continually approximates to that +of $\sqrt{1 + (\phi' x)^{2}} : 1$, which is the ratio of~$PV : PQ$. +Hence the ratio of~$PP' : PV$ continually approaches to +unity, or $PQ$~may be taken so small that the difference +of $PP'$~and~$PV$ shall be as small a part of either +of them as we please. + +Finally, the arc~$PP'$ is greater than the chord~$PP'$ +and less than $PV + VP'$. Hence $\dfrac{\arc PP'}{\chord PP'}$ lies between +$1$~and $\dfrac{PV}{PP'} + \dfrac{VP'}{PP'}$, the former of which two +fractions can be brought as near as we please to unity, +and the latter can be made as small as we please; for +\PageSep{42} +since $P'V$~can be made as small a part of~$PQ$ as we +please, still more can it be made as small a part as we +please of~$PP'$, which is greater than~$PQ$. Therefore +the arc and chord~$PP'$ may be made to have a ratio as +nearly equal to unity as we please. And because $pa$~is +less than~$pv$, and therefore less than~$P'V$, it follows +that $pa$~may be made as small a part as we please of~$PQ$, +and still more of~$PP'$. + +In these propositions is contained the rational explanation +of the proposition of Leibnitz, that ``an infinitely +\index{Leibnitz}% +small arc is equal to, and coincides with, its +chord.'' + + +\Subsection{Orders of Infinity.} + +Let there be any number of series, arranged in +\index{Infinity, orders of|EtSeq}% +\index{Orders of infinity|EtSeq}% +powers of~$h$, so that the lowest power is first; let +them contain none but whole powers, and let them all +be such, that each will be convergent, on giving to~$h$ +a sufficiently small value: as follows, +\begin{alignat*}{4} +Ah + Bh^{2} &{}+{}& Ch^{3} &{}+{}& Dh^{4} &{}+{}& Eh^{5} &+ \etc. +\Tag{(1)} \\ + B'h^{2} &{}+{}& C'h^{3} &{}+{}& D'h^{4} &{}+{}& E'h^{5} &+ \etc. +\Tag{(2)} \\ + && C''h^{3} &{}+{}& D''h^{4} &{}+{}& E''h^{5} &+ \etc. +\Tag{(3)} \\ + &&&& D'''h^{4} &{}+{}& E'''h^{5} &+ \etc. +\Tag{(4)} \\ + &&&&&& \etc. & +\tag*{\etc.} +\end{alignat*} + +As $h$~is diminished, all these expressions decrease +without limit; but the first \emph{increases} with respect to +the second, that is, contains it more times after a decrease +of~$h$ than it did before. For the ratio of \Eq{(1)} +to~\Eq{(2)} is that of $A + Bh + Ch^{2} + \etc.$ to $B'h + C'h^{2} + \etc.$, +the ratio of the two not being changed by dividing +both by~$h$. The first term of the latter ratio +approximates continually to~$A$, as $h$~is diminished, +and the second can be made as small as we please, +and therefore can be contained in the first as often as +\PageSep{43} +we please. Hence the ratio \Eq{(1)}~to~\Eq{(2)} can be made +as great as we please. By similar reasoning, the ratio +\Eq{(2)}~to~\Eq{(3)}, of \Eq{(3)}~to~\Eq{(4)}, etc., can be made as great as +we please. We have, then, a series of quantities, +each of which, by making $h$ sufficiently small, can be +made as small as we please. Nevertheless this decrease +increases the ratio of the first to the second, of +the second to the third, and so on, and the increase is +without limit. + +Again, if we take \Eq{(1)}~and~$h$, the ratio of \Eq{(1)}~to~$h$ is +that of $A + Bh + Ch^{2} + \etc.$ to~$1$, which, by a sufficient +decrease of~$h$, may be brought as near as we +please to that of $A$~to~$1$. But if we take \Eq{(1)}~and~$h^{2}$, +the ratio of \Eq{(1)}~to~$h^{2}$ is that of $A + Bh + \etc.$ to~$h$, +which, by previous reasoning, may be increased without +limit; and the same for any higher power of~$h$. +Hence \Eq{(1)}~is said to be \emph{comparable} to the first power +of~$h$, or \emph{of the first order}, since this is the only power +of~$h$ whose ratio to~\Eq{(1)} tends towards a finite limit. +By the same reasoning, the ratio of \Eq{(2)}~to~$h^{2}$, which is +that of $B' + C'h + \etc.$ to~$1$, continually approaches +that of $B'$~to~$1$; but the ratio \Eq{(2)}~to~$h$, which is that +of $B'h + C'h^{2} + \etc.$ to~$1$, diminishes without limit, as +$h$~is decreased, while the ratio of \Eq{(2)}~to~$h^{2}$, or of $B' + C'h + \etc.$ +to~$h$, increases without limit. Hence \Eq{(2)}~is +said to be \emph{comparable} to the second power of~$h$, or \emph{of +the second order}, since this is the only power of~$h$ whose +ratio to~\Eq{(2)} tends towards a finite limit. In the language +of Leibnitz if $h$~be an infinitely small quantity, +\Eq{(1)}~is an infinitely small quantity of the first order, +\Eq{(2)}~is an infinitely small quantity of the second +order, and so on. + +We may also add that the ratio of two series of +the same order continually approximates to the ratio +\PageSep{44} +of their lowest terms. For example, the ratio of $Ah^{3} + Bh^{4} + \etc.$ +to $A'h^{3} + B'h^{4} + \etc.$ is that of $A + Bh + \etc.$ +to $A' + B'h + \etc.$, which, as $h$~is diminished, +continually approximates to the ratio of $A$ to~$A'$, which +is also that of $Ah^{3}$ to~$A'h^{3}$, or the ratio of the lowest +terms. In \Fig{4}, $PQ$~or $dx$ being put in place of~$h$, +$QP'$, or $\phi' x\, dx + \phi'' x\, \dfrac{(dx)^{2}}{2}$, etc., is of the first order, +as are~$PV$, and the chord~$PP'$; while $P'V$, or +$\phi'' x\, \dfrac{(dx)^{2}}{2} + \etc.$, is of the second order. + +The converse proposition is readily shown, that if +the ratio of two series arranged in powers of~$h$ continually +approaches to some finite limit as $h$~is diminished, +the two series are of the same order, or the exponent +of the lowest power of~$h$ is the same in both. +Let $Ah^{a}$ and $Bh^{b}$ be the lowest powers of~$h$, whose ratio, +as has just been shown, continually approximates +to the actual ratio of the two series, as $h$~is diminished. +The hypothesis is that the ratio of the two series, and +therefore that of $Ah^{a}$ to~$Bh^{b}$, has a finite limit. This +cannot be if $a > b$, for then the ratio of $Ah^{a}$ to $Bh^{b}$ is +that of $Ah^{a-b}$ to~$B$, which diminishes without limit; +neither can it be when $a < b$, for then the same ratio +is that of $A$ to~$Bh^{b-a}$, which increases without limit; +hence $a$~must be equal to~$b$. + +We leave it to the student to prove strictly a proposition +assumed in the preceding; viz., that if the +ratio of $P$~to~$Q$ has unity for its limit, when $h$~is diminished, +the limiting ratio of $P$~to~$R$ will be the same +as the limiting ratio of $Q$~to~$R$. We proceed further +to illustrate the Differential Calculus as applied to +Geometry. +\PageSep{45} + + +\Subsection[A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines.] +{A Geometrical Illustration.} + +Let $OC$ and~$OD$ (\Fig{5}) be two axes at right angles +to one another, and let a line~$AB$ of given length +be placed with one extremity in each axis. Let this +line move from its first position into that of~$A'B'$ on +one side, and afterwards into that of~$A''B''$ on the +other side, always preserving its first length. The +motion of a ladder, one end of which is against a wall, +and the other on the ground, is an instance. + +Let $A'B'$ and $A''B''$ intersect~$AB$ in $P'$~and~$P''$. If +\index{Ladder against wall|EtSeq}% +$A''B''$~were gradually moved from its present position +into that of~$A'B'$, the point~$P''$ would also move gradually +\Figure{5} +from its present position into that of~$P'$, passing, +in its course, through every point in the line~$P'P''$. +But here it is necessary to remark that $AB$~is itself +one of the positions intermediate between $A'B'$ and~$A''B''$, +and when two lines are, by the motion of one +of them, brought into one and the same straight line, +they intersect one another (if this phrase can be here +applied at all) in every point, and all idea of one distinct +point of intersection is lost. Nevertheless $P''$~describes +one part of~$P''P'$ before $A''B''$~has come into +the position~$AB$, and the rest afterwards, when it is +between $AB$ and~$A'B'$. +\PageSep{46} + +Let $P$~be the point of separation; then every point +of~$P'P''$, except~$P$, is a real point of intersection of~$AB$, +with one of the positions of~$A''B''$, and when +$A''B''$~has moved very near to~$AB$, the point~$P''$ will +be very near to~$P$; and there is no point so near to~$P$, +that it may not be made the intersection of $A''B''$ and~$AB$, +by bringing the former sufficiently near to the +latter. This point~$P$ is, therefore, the \emph{limit} of the intersections +\index{Intersections, limit of|EtSeq}% +\index{Limit of intersections|EtSeq}% +of $A''B''$~and~$AB$, and cannot be found by +the ordinary application of algebra to geometry, but +may be made the subject of an inquiry similar to those +\Figure[nolabel]{5} +which have hitherto occupied us, in the following +manner: + +Let $OA = a$, $OB = b$, $AB = A'B' = A''B'' = l$. Let +$AA' = da$, $BB' = db$, whence $OA' = a + da$, $OB' = b - db$. +We have then $a^{2} + b^{2} = l^{2}$, and $(a + da)^{2} + (b - db)^{2} = l^{2}$; +subtracting the former of which from +the development of the latter, we have +\[ +2a\, da + (da)^{2} - 2b\, db + (db)^{2} = 0\Add{,} +\] +or +\[ +\frac{db}{da} = \frac{2a + da}{2b - db}\Add{.} +\Tag{(1)} +\] +As $A'B'$ moves towards~$AB$, $da$~and~$db$ are diminished +without limit, $a$~and~$b$ remaining the same; hence the +limit of the ratio~$\dfrac{db}{da}$ is $\dfrac{2a}{2b}$ or~$\dfrac{a}{b}$. +\PageSep{47} + +Let the co-ordinates\footnote + {The lines $OM'$ and $M'P'$ are omitted, to avoid crowding the figure.} +of~$P'$ be $OM' = x$ and $M'P = y$. +Then (\PageRef{32}) the co-ordinates of any point in~$AB$ +have the equation +\[ +ay + bx = ab\Add{.} +\Tag{(2)} +\] +The point~$P'$ is in this line, and also in the one which +cuts off $a + da$ and $b - db$ from the axes, whence +\[ +(a + da)y + (b - db)x = (a + da)(b - db)\Add{;} +\Tag{(3)} +\] +subtract \Eq{(2)} from~\Eq{(3)} after developing the latter, which +gives +\[ +y\, da - x\, db = b\, da - a\, db - da\, db\Add{.} +\Tag{(4)} +\] +If we now suppose $A'B'$~to move towards~$AB$, equation~\Eq{(4)} +gives no result, since each of its terms diminishes +without limit. If, however, we divide~\Eq{(4)} by~$da$, +and substitute in the result the value of~$\dfrac{db}{da}$ obtained +from~\Eq{(1)} we have +\[ +y - x\, \frac{2a + da}{2b - db} + = b - a\, \frac{2a + da}{2b - db} - db\Add{.} +\Tag{(5)} +\] +From this and~\Eq{(2)} we might deduce the values of $y$ +and~$x$, for the point~$P'$, as the figure actually stands. +Then by diminishing $db$~and $da$ without limit, and +observing the limit towards which $x$~and~$y$ tend, we +might deduce the co-ordinates of~$P$, the limit of the +intersections. + +The same result may be more simply obtained, by +diminishing $da$~and~$db$ in equation~\Eq{(5)}, before obtaining +the values of $y$~and~$x$. This gives +\[ +y - \frac{a}{b}\, x = b - \frac{a^{2}}{b} \quad\text{or}\quad +by - ax = b^{2} - a^{2}\Add{.} +\Tag{(6)} +\] +From \Eq{(6)}~and~\Eq{(2)} we find (\Fig{6}) +\[ +x = OM = \frac{a^{3}}{a^{2} + b^{2}} = \frac{a^{3}}{l^{2}} \quad\text{and}\quad +y = MP = \frac{b^{3}}{a^{2} + b^{2}} = \frac{b^{3}}{l^{2}}. +\] +\PageSep{48} + +This limit of the intersections is different for every +different position of the line~$AB$, but may be determined, +in every case, by the following simple construction. + +Since (\Fig{6}) $BP: PN$, or $OM :: BA : AO$, we +have $BP = OM\, \dfrac{BA}{AO} = \dfrac{a^{3}}{l^{2}}\, \dfrac{l}{a} = \dfrac{a^{2}}{l}$; and, similarly, +$PA = \dfrac{b^{2}}{l}$. Let $OQ$~be drawn perpendicular to~$BA$; +then since $OA$~is a mean proportional between $AQ$ +and~$AB$, we have $AQ = \dfrac{a^{2}}{l}$, and similarly $BQ = \dfrac{b^{2}}{l}$. +Hence $BP = AQ$ and $AP = BQ$, or the point~$P$ is +as far from either extremity of~$AB$ as $Q$~is from the +other. +\Figure{6} + + +\Subsection{The Same Problem Solved by the Principles of +Leibnitz.} + +We proceed to solve the same problem, using the +\index{Leibnitz}% +principles of Leibnitz, that is, supposing magnitudes +can be taken so small, that those proportions may be +regarded as absolutely correct, which are not so in +reality, but which only approach more nearly to the +truth, the smaller the magnitudes are taken. The inaccuracy +of this supposition has been already pointed +out; yet it must be confessed that this once got over, +\PageSep{49} +the results are deduced with a degree of simplicity +and consequent clearness, not to be found in any other +method. The following cannot be regarded as a demonstration, +except by a mind so accustomed to the +subject that it can readily convert the various inaccuracies +into their corresponding truths, and see, at one +glance, how far any proposition will affect the final +result. The beginner will be struck with the extraordinary +assertions which follow, given in their most +naked form, without any attempt at a less startling +mode of expression. +\Figure{7} + +Let $A'B'$ (\Fig{7}) be a position of~$AB$ infinitely +\index{Infinitely small, the notion of}% +near to it; that is, let $A'PA$~be an infinitely small +angle. With the centre~$P$, and the radii $PA'$ and~$PB$, +describe the infinitely small arcs $A'a$,~$Bb$. An infinitely +small arc of a circle is a straight line perpendicular +to its radius; hence $A'aA$~and~$BbB'$ are right-angled +triangles, the first similar to~$BOA$, the two +having the angle~$A$ in common, and the second similar +to~$B'OA'$. Again, since the angles of~$BOA$, which +are finite, only differ from those of~$B'OA'$ by the infinitely +small angle~$A'PA$, they may be regarded as +\PageSep{50} +equal; whence $A'aA$~and~$B'bB$ are similar to~$BOA$, +and to one another. Also $P$~is the point of which we +are in search, or infinitely near to it; and since $BA = B'A'$, +of which $BP = bP$ and $aP = A'P$, the remainders +$B'b$~and~$Aa$ are equal. Moreover, $Bb$~and~$A'a$ +being arcs of circles subtending equal angles, are in +the proportion of the radii $BP$~and~$PA'$. + +Hence we have the following proportions: +\begin{gather*} +Aa : A'a :: OA : OB :: a : b \\ +Bb : B'b :: OA : OB :: a : b\rlap{.} +\end{gather*} +The composition of which gives, since $Aa = B'b$: +\[ +\PadTo[r]{BP + Pa}{Bb} : A'a :: \PadTo[l]{a^{2} + b^{2}}{a^{2}} : b^{2}. +\] +Also +\[ +\PadTo[r]{BP + Pa}{Bb} : A'a :: \PadTo[l]{a^{2} + b^{2}}{BP} : Pa, +\] +whence +\[ +\PadTo[r]{BP + Pa}{BP} : Pa :: \PadTo[l]{a^{2} + b^{2}}{a^{2}} : b^{2}, +\] +and +\[ +BP + Pa : Pa :: a^{2} + b^{2} : b^{2}. +\] +But $Pa$~only differs from~$PA$ by the infinitely small +quantity~$Aa$, and $BP + PA = l$, and $a^{2} + b^{2} = l^{2}$; +whence +\[ +l : PA :: l^{2} : b^{2},\quad\text{or}\quad PA = \frac{b^{2}}{l}, +\] +which is the result already obtained. + +In this reasoning we observe four independent +errors, from which others follow: (1)~that $Bb$~and~$A'a$ +are straight lines at right-angles to~$Pa$; (2)~that $BOA$\Add{,}~$B'OA'$ +are similar triangles; (3)~that $P$~is really the +point of which we are in search; (4)~that $PA$~and~$Pa$ +are equal. But at the same time we observe that +every one of these assumptions approaches the truth, +as we diminish the angle~$A'PA$, so that there is no +magnitude, line or angle, so small that the linear or +angular errors, arising from the above-mentioned suppositions, +may not be made smaller. + +We now proceed to put the same demonstration +\PageSep{51} +in a stricter form, so as to neglect no quantity during +the process. This should always be done by the beginner, +until he is so far master of the subject as to be +able to annex to the inaccurate terms the ideas necessary +for their rational explanation. To the former figure +add $B\beta$ and~$A\alpha$, the real perpendiculars, with +which the arcs have been confounded. Let $\angle A'PA = d\theta$, +\index{Angle, unit employed in measuring an}% +$PA = p$, $Aa = dp$, $BP = q$, $B'b = dq$; and $OA = a$, +$OB = b$, and $AB = l$. Then\footnote + {For the unit employed in measuring an angle, see \Title{Study of Mathematics} + (Chicago, 1898), pages 273--277.} +$A'a = (p - dp)\, d\theta$, $Bb = q\, d\theta$, +and the triangles $A'A\alpha$ and $B'B\beta$ are similar to +\Figure[nolabel]{7} +$BOA$~and~$B'OA'$. The perpendiculars $A'\alpha$ and~$B\beta$ +are equal to $PA' \sin d\theta$ and $PB \sin d\theta$, or $(p - dp) \sin d\theta$ +and $q \sin d\theta$. Let $a\alpha = \mu$ and $b\beta = \nu$. These +(\PageRef[p.]{9}) will diminish without limit as compared with +$A'\alpha$ and~$B\beta$; and since the ratios of $A'\alpha$ to~$\alpha A$ and $B\beta$ +to~$\beta B'$ continue finite (these being sides of triangles +similar to $AOB$ and~$A'OB'$), $a\alpha$~and~$b\beta$ will diminish +indefinitely with respect to $\alpha A$~and~$\beta B'$. Hence the +ratio $A\alpha$ to~$\beta B'$ or $dp + \mu$ to $dq + \nu$ will continually +approximate to that of $dp$ to~$dq$, or a ratio of equality. +\PageSep{52} + +The exact proportions, to which those in the last +page are approximations, are as follows: +\begin{alignat*}{3} +dp + \mu &: (p - dp) \sin d\theta &&:: a &&: b, \\ +q \sin d\theta &: \PadTo{(p - dp) \sin d\theta}{dq + \nu} &&:: a - da &&: b + db; +\end{alignat*} +by composition of which, recollecting that $dp = dq$ +(which is rigorously true) and dividing the two first +terms of the resulting proportion by~$dp$, we have +\[ +q\left(1 + \frac{\mu}{dp}\right) : (p - dp)\left(1 + \frac{\nu}{dp}\right) + :: a(a - da) : b(b + db). +\] + +If $d\theta$ be diminished without limit, the quantities +$da$,~$db$, and~$dp$, and also the ratios $\dfrac{\mu}{dp}$ and~$\dfrac{\nu}{dp}$, as +above-mentioned, are diminished without limit, so +that the limit of the proportion just obtained, or the +proportion which gives the limits of the lines into +which $P$~divides~$AB$, is +\begin{alignat*}{3} +q &: p &&:: a^{2} &&: b^{2}, \\ +\intertext{hence} +q + p = l &: p &&:: a^{2} + b^{2} = l^{2} &&: b^{2}, +\end{alignat*} +the same as before. + + +\Subsection[An Illustration from Dynamics: Velocity, Acceleration, etc.] +{An Illustration from Dynamics.} + +We proceed to apply the preceding principles to +dynamics, or the theory of motion. + +Suppose a point moving along a straight line uniformly; +that is, if the whole length described be divided +into any number of equal parts, however great, +each of those parts is described in the same time. +Thus, whatever length is described in the first second +of time, or in any part of the first second, the same +is described in any other second, or in the same part +of any other second. The number of units of length +described in a unit of time is called the \emph{velocity}; thus +\index{Velocity!linear|EtSeq}% +a velocity of $3.01$~feet in a second means that the +\PageSep{53} +point describes three feet and one hundredth in each +second, and a proportional part of the same in any +part of a second. Hence, if $v$~be the velocity, and +$t$~the units of time elapsed from the beginning of the +motion, $vt$~is the length described; and if any length +described be known, the velocity can be determined +by dividing that length by the time of describing it. +Thus, a point which moves uniformly through $3$~feet +in $1\frac{1}{2}$~second, moves with a velocity of $3 ÷ 1\frac{1}{2}$, or $2$~feet +per~second. + +Let the point not move uniformly; that is, let different +\index{Continuous quantities}% +\index{Quantities, continuous}% +parts of the line, having the same length, be +described in different times; at the same time let the +motion be \emph{continuous}, that is, not suddenly increased +or decreased, as it would be if the point were composed +of some hard matter, and received a blow while +it was moving. This will be the case if its motion be +represented by some algebraical function of the time, +or if, $t$~being the number of units of time during which +the point has moved, the number of units of length +described can be represented by~$\phi t$. This, for example, +we will suppose to be~$t + t^{2}$, the unit of time +being one second, and the unit of length one inch; +so that $\frac{1}{2} + \frac{1}{4}$, or $\frac{3}{4}$~of an inch, is described in the first +half second; $1 + 1$, or two inches, in the first second; +$2 + 4$, or six inches, in the first two seconds, and so on. + +Here we have no longer an evident measure of the +velocity of the point; we can only say that it obviously +increases from the beginning of the motion to +the end, and is different at every two different points. +Let the time~$t$ elapse, during which the point will describe +the distance $t + t^{2}$; let a further time~$dt$ elapse, +during which the point will increase its distance to +$t + dt + (t + dt)^{2}$, which, diminished by~$t + t^{2}$, gives +\PageSep{54} +$dt + 2t\, dt + (dt)^{2}$ for the length described during the +increment of time~$dt$. This varies with the value of~$t$; +thus, in the interval~$dt$ after the first second, the +length described is $3\, dt + dt^{2}$; after the second second, +it is $5\, dt + (dt)^{2}$, and so on. Nor can we, as in the +case of uniform motion, divide the length described, +by the time, and call the result the velocity with which +that length is described; for no length, however small, +is here uniformly described. If we were to divide a +length by the time in which it is described, and also +its first and second halves by the times in which they +are respectively described, the three results would be +all different from one another. + +Here a difficulty arises, similar to that already noticed, +when a point moves along a curve; in which, +as we have seen, it is improper to say that it is moving +in any one direction through an arc, however +small. Nevertheless a straight line was found at every +point, which did, more nearly than any other straight +line, represent the direction of the motion. So, in +this case, though it is incorrect to say that there is +any uniform velocity with which the point continues +to move for any portion of time, however small, we +can, at the end of every time, assign a uniform velocity, +which shall represent, more nearly than any +other, the rate at which the point is moving. If we +say that, at the end of the time~$t$, the point is moving +with a velocity~$v$, we must not now say that the length~$v\, dt$ +is described in the succeeding interval of time~$dt$; +but we mean that $dt$~may be taken so small, that $v\, dt$~shall +bear to the distance actually described a ratio as +near to equality as we please. + +Let the point have moved during the time~$t$, after +which let successive intervals of time elapse, each +\PageSep{55} +\index{Coefficients, differential}% +\index{Differential coefficients}% +equal to~$dt$. At the end of the times, $t$,~$t + dt$, $t + 2\, dt$, +$t + 3\, dt$,~etc., the whole lengths described will be $t + t^{2}$, +$t + dt + (t + dt)^{2}$, $t + 2\, dt + (t + 2\, dt)^{2}$, $t + 3\, dt + (t + 3\, dt)^{2}$, +etc.; the differences of which, or $dt + 2t\, dt + (dt)^{2}$, +$dt + 2t\, dt + 3(dt)^{2}$, $dt + 2t\, dt + 5(dt)^{2}$, etc., +are the lengths described in the first, second, third, +etc., intervals~$dt$. These are not equal to one another, +as would be the case if the velocity were uniform; but +by making $dt$ sufficiently small, their ratio may be +brought as near to equality as we please, since the +terms $(dt)^{2}$,~$3(dt)^{2}$,~etc., by which they all differ from +the common part $(1 + 2t)\, dt$, may be made as small as +we please, in comparison of this common part. If we +divide the above-mentioned lengths by~$dt$, which does +not alter their ratio, they become $1 + 2t + dt$, $1 + 2t + 3\, dt$, +$1 + 2t + 5\, dt$, etc., which may be brought as +near as we please to equality, by sufficient diminution +of~$dt$. Hence $1 + 2t$ is said to be the velocity of the +point after the time~$t$; and if we take a succession of +equal intervals of time, each equal to~$dt$, and sufficiently +small, the lengths described in those intervals +will bear to $(1 + 2t)\, dt$, the length which would be described +in the same interval with the uniform velocity +$1 + 2t$, a ratio as near to equality as we please. And +observe, that if $\phi t$ is $t + t^{2}$, $\phi' t$~is $1 + 2t$, or the coefficient +of~$h$ in $(t + h) + (t + h)^{2}$. + +In the same way it may be shown, that if the point +moves so that $\phi t$~always represents the length described +in the time~$t$, the differential coefficient of~$\phi t$\Add{,} +or~$\phi' t$, is the velocity with which the point is moving +at the end of the time~$t$. For the time~$t$ having elapsed, +the whole lengths described at the end of the times $t$ +and $t + dt$ are $\phi t$ and $\phi(t + dt)$; whence the length +described during the time~$dt$ is +\PageSep{56} +\[ +\phi(t + dt) - \phi t, \quad\text{or}\quad +\phi't\, dt + \phi'' t\, \frac{(dt)^{2}}{2} + \etc. +\] +Similarly, the length described in the next interval +$dt$ is +\begin{gather*} +\phi(t + 2\, dt) - \phi(t + dt); \quad\text{or}, \displaybreak[0] \\ + \phi t + \phi' t\, 2\, dt + \phi'' t\, \frac{(2\, dt)^{2}}{2} + \etc. \displaybreak[0] \\ +-(\phi t + \phi' t\, dt + \phi'' t\, \frac{(dt)^{2}}{2} + \etc.), \displaybreak[0] \\ +\intertext{which is} +\phi' t\, dt + 3\phi'' t\, \frac{(dt)^{2}}{2} + \etc.; +\end{gather*} +the length described in the third interval~$dt$ is +$\phi' t\, dt + 5\phi'' t\, \dfrac{(dt)^{2}}{2} + \etc.$,~etc. + +Now, it has been shown for each of these, that the +first term can be made to contain the aggregate of all +the rest as often as we please, by making $dt$ sufficiently +small; this first term is $\phi' t\, dt$ in all, or the length +which would be described in the time~$dt$ by the velocity +$\phi' t$ continued uniformly: it is possible, therefore, +to take $dt$ so small, that the lengths actually described +in a succession of intervals equal to~$dt$, shall be as +nearly as we please in a ratio of equality with those +described in the same intervals of time by the velocity~$\phi' t$. +For example, it is observed in bodies which fall +\index{Falling bodies}% +to the earth from a height above it, when the resistance +of the air is removed, that if the time be taken +in seconds, and the distance in feet, the number of +feet fallen through in $t$~seconds is always~$at^{2}$, where +$a = 16\frac{1}{12}$ very nearly; what is the velocity of a body +which has fallen \textit{in~vacuo} for four seconds? Here $\phi t$ +being~$at^{2}$, we find, by substituting $t + h$, or $t + dt$, instead +of~$t$, that $\phi' t$~is~$2at$, or $2 × 16\frac{1}{12} × t$; which, at +\PageSep{57} +the end of four seconds, is $32\frac{1}{6} × 4$, or $128\frac{2}{3}$~feet. That +is, at the end of four seconds a falling body moves at +the rate of $128\frac{2}{3}$~feet per~second. By which we do +not mean that it continues to move with this velocity +for any appreciable time, since the rate is always +varying; but that the length described in the interval~$dt$ +after the fourth second, may be made as nearly as +we please in a ratio of equality with $128\frac{2}{3} × dt$, by +taking $dt$ sufficiently small. This velocity~$2at$ is said +to be \emph{uniformly} accelerated; since in each second the +\index{Accelerated motion}% +\index{Uniformly accelerated}% +same velocity~$2a$ is gained. And since, when $x$~is the +space described, $\phi' t$~is the limit of~$\dfrac{dx}{dt}$, the velocity is +also this limit; that is, when a point does not move +uniformly, the velocity is not represented by any increment +of length divided by its increment of time, +but by the limit to which that ratio continually tends, +as the increment of time is diminished. + + +\Subsection{Simple Harmonic Motion.} + +We now propose the following problem: A point +\index{Motion!simple harmonic}% +\index{Simple harmonic motion}% +moves uniformly round a circle; with what velocities +do the abscissa and ordinate increase or decrease, at +any given point? (\Fig{8}.) + +Let the point~$P$, setting out from~$A$, describe the +arc~$AP$, etc., with the uniform velocity of $a$~inches +per~second. Let $OA = r$, $\angle A0P = \theta$, $\angle POP' = d\theta$, +$0M = x$, $MP = y$, $MM' = dx$, $QP' = dy$. + +From the first principles of trigonometry +\begin{alignat*}{4}%[** TN: Re-aligned from the original] +&x &&= r \cos\theta\Add{,} \\ +&x - dx &&= r \cos(\theta + d\theta) + &&= r \cos\theta \cos d\theta - r \sin\theta \sin d\theta\Add{,} \displaybreak[0] \\ +&y &&= r \sin\theta\Add{,} \\ +&y + dy &&= r \sin(\theta + d\theta) + &&= r \sin\theta \cos d\theta + r \cos\theta \sin d\theta. +\end{alignat*} +\PageSep{58} +Subtracting the second from the first, and the third +from the fourth, we have +\begin{alignat*}{2} +dx &= r \sin\theta \sin d\theta + r \cos\theta(1 - \cos d\theta)\Add{,} +\Tag{(1)} \\ +dy &= r \cos\theta \sin d\theta + r \sin\theta(1 - \cos d\theta)\Add{.} +\Tag{(2)} \\ +\end{alignat*} +But if $d\theta$ be taken sufficiently small, $\sin d\theta$, and~$d\theta$, +may be made as nearly in a ratio of equality as we +please, and $1 - \cos d\theta$ may be made as small a part +as we please, either of $d\theta$ or $\sin d\theta$. These follow from +\Fig{1}, in which it was shown that $BM$ and the arc~$BA$, +or (if $OA = r$ and $AOB = d\theta$), $r \sin d\theta$ and~$r\, d\theta$, +may be brought as near to a ratio of equality as we +\Figure{8} +please, which is therefore true of $\sin d\theta$ and~$d\theta$. Again, +it was shown that~$AM$, or $r - r \cos d\theta$, can be made +as small a part as we please, either of~$BM$ or the arc~$BA$, +that is, either of $r \sin d\theta$, or~$r\, d\theta$; the same is +therefore true of $1 - \cos d\theta$, and either $\sin d\theta$ or~$d\theta$. +Hence, if we write equations \Eq{(1)}~and~\Eq{(2)} thus, +\[ +dx = r \sin\theta\, d\theta\quad (1)\qquad\qquad +dy = r \cos\theta\, d\theta\quad (2), +\] +we have equations, which, though never exactly true, +are such that by making $d\theta$ sufficiently small, the +errors may be made as small parts of~$d\theta$ as we please. +Again, since the arc~$AP$ is uniformly described, so +also is the angle~$POA$; and since an arc~$a$ is described +\PageSep{59} +in one second, the angle~$\dfrac{a}{r}$ is described in the same +\index{Velocity!angular}% +time; this is, therefore, the \emph{angular velocity}.\footnote + {The same considerations of velocity which have been applied to the + motion of a point along a line may also be applied to the motion of a line + round a point. If the angle so described be always increased by equal angles + in equal portions of time, the angular velocity is said to be uniform, and is + measured by the number of angular units described in a unit of time. By + similar reasoning to that already described, if the velocity with which the + angle increases be not uniform, so that at the end of the time~$t$ the angle described + is $\theta = \phi t$, the angular velocity is~$\phi' t$, or the limit of the ratio~$\dfrac{d\theta}{dt}$.} +If we +divide equations \Eq{(1)}~and~\Eq{(2)} by~$dt$, we have +\[ +%[** TN: Signs OK; De Morgan absorbs the - in dx/dt at the bottom of p. 57] +\frac{dx}{dt} = r \sin\theta\, \frac{d\theta}{dt}\qquad +\frac{dy}{dt} = r \cos\theta\, \frac{d\theta}{dt}; +\] +these become more nearly true as $dt$~and~$d\theta$ are diminished, +so that if for $\dfrac{dx}{dt}$,~etc., the limits of these ratios +be substituted, the equations will become rigorously +true. But these limits are the velocities of $x$,~$y$, and~$\theta$, +the last of which is also~$\dfrac{a}{r}$; hence +\begin{alignat*}{2} +\text{velocity of~$x$} &= r \sin\theta × \frac{a}{r} &&= a \sin\theta, \\ +\text{velocity of~$y$} &= r \cos\theta × \frac{a}{r} &&= a \cos\theta; +\end{alignat*} +that is, the point~$M$ moves towards~$O$ with a variable +velocity, which is always such a part of the velocity +of~$P$, as $\sin\theta$~is of unity, or as $PM$~is of~$OB$; and the +distance~$PM$ increases, or the point~$N$ moves from~$O$, +with a velocity which is such a part of the velocity of~$P$ +as $\cos\theta$~is of unity, or as $OM$~is of~$OA$. [The motion +of the point~$M$ or the point~$N$ is called in physics +a \emph{simple harmonic motion}.] + +In the language of Leibnitz, the results of the two +\index{Leibnitz}% +foregoing sections would be expressed thus: If a +point move, but not uniformly, it may still be considered +as moving uniformly for any infinitely small +\index{Infinitely small, the notion of}% +\PageSep{60} +time; and the velocity with which it moves is the infinitely +small space thus described, divided by the infinitely +small time. + + +\Subsection{The Method of Fluxions.} + +The foregoing process contains the method employed +\index{Fluxions}% +by Newton, known by the name of the \emph{Method +\index{Newton}% +of Fluxions}. If we suppose $y$ to be any function of~$x$, +and that $x$~increases with a given velocity, $y$~will also +increase or decrease with a velocity depending: (1)~upon +the velocity of~$x$; (2)~upon the function which +$y$ is of~$x$. These velocities Newton called the fluxions +of $y$~and~$x$, and denoted them by $\dot{y}$~and~$\dot{x}$. Thus, if +$y = x^{2}$, and if in the interval of time~$dt$, $x$~becomes +$x + dx$, and $y$~becomes $y + dy$, we have $y + dy = (x + dx)^{2}$, +and $dy = 2x\, dx + (dx)^{2}$, or $\dfrac{dy}{dt} = 2x\, \dfrac{dx}{dt} + \dfrac{dx}{dt}\, dx$. +If we diminish~$dt$, the term $\dfrac{dx}{dt}\, dx$ will diminish +without limit, since one factor continually approaches +to a given quantity, viz., the velocity of~$x$, +and the other diminishes without limit. Hence we +obtain the velocity of $y = 2x × \text{the velocity of~$x$}$, or +$\dot{y} = 2x\, \dot{x}$, which is used in the method of fluxions instead +of $dy = 2x\, dx$ considered in the manner already +described. The processes are the same in both methods, +since the ratio of the velocities is the limiting +ratio of the corresponding increments, or, according +to Leibnitz, the ratio of the infinitely small increments. +\index{Leibnitz}% +We shall hereafter notice the common objection +to the Method of Fluxions. + + +\Subsection{Accelerated Motion.} + +When the velocity of a material point is suddenly +\index{Accelerated motion}% +\index{Motion!accelerated}% +\index{Uniformly accelerated}% +increased, an \emph{impulse} is said to be given to it, and the +\index{Impulse}% +\PageSep{61} +magnitude of the impulse or impulsive force is in proportion +\index{Force|(}% +to the velocity created by it. Thus, an impulse +which changes the velocity from $50$ to $70$~feet +per~second, is twice as great as one which changes it +from $50$ to $60$~feet. When the velocity of the point is +altered, not suddenly but continuously, so that before +the velocity can change from $50$ to $70$~feet, it goes +through all possible intermediate velocities, the point +is said to be acted on by an \emph{accelerating force}. \emph{Force} +is a name given to that which causes a change in the +velocity of a body. It is said to act uniformly, when +the velocity acquired by the point in any one interval +of time is the same as that acquired in any other interval +of equal duration. It is plain that we cannot, +by supposing any succession of impulses, however +small, and however quickly repeated, arrive at a uniformly +accelerated motion; because the length described +between any two impulses will be uniformly +described, which is inconsistent with the idea of continually +accelerated velocity. Nevertheless, by diminishing +the magnitude of the impulses, and increasing +their number, we may come as near as we please +to such a continued motion, in the same way as, by +diminishing the magnitudes of the sides of a polygon, +and increasing their number, we may approximate as +near as we please to a \Typo{continous}{continuous} curve. + +Let a point, setting out from a state of rest, increase +its velocity uniformly, so that in the time~$t$, it +may acquire the velocity~$v$---what length will have +been described during that time~$t$? Let the time~$t$ +and the velocity~$v$ be both divided into $n$~equal parts, +each of which is $t'$ and~$v'$, so that $nt' = t$, and $nv' = v$. +Let the velocity~$v'$ be communicated to the point at +rest; after an interval of~$t'$ let another velocity~$v'$ be +\PageSep{62} +communicated, so that during the second interval~$t'$ +the point has a velocity~$2v'$; during the third interval +let the point have the velocity~$3v'$, and so on; so that +in the last or $n$\th~interval the point has the velocity~$nv'$. +The space described in the first interval is, therefore,~$v't'$; +in the second,~$2v't'$; in the third~$3v't'$; and +so on, till in the $n$\th~interval it is~$nv't'$. The whole +space described is, therefore, +\[ +v't' + 2v't' + 3v't' + \dots + (n - 1)v't' + nv't'\Add{,} +\] +or +\[ +[1 + 2 + 3 \Add{+} \dots + (n - 1) + n]v't' + = n · \frac{(n + 1)}{2}\, v't' + = \frac{n^{2} v't' + nv't'}{2}. +\] +In this substitute $v$ for~$nv'$, and $t$ for~$nt'$, which gives +for the space described $\frac{1}{2}v(t + t')$. The smaller we +suppose~$t'$, the more nearly will this approach to~$\frac{1}{2}vt$. +But the smaller we suppose~$t'$, the greater must be~$n$, +the number of parts into which $t$~is divided; and the +more nearly do we render the motion of the point uniformly +accelerated. Hence the limit to which we approximate +by diminishing~$t'$ without limit, is the length +described in the time~$t$ by a uniformly accelerated +velocity, which shall increase from~$0$ to~$v$ in that time. +This is~$\frac{1}{2}vt$, or half the length which would have been +described by the velocity~$v$ continued uniformly from +the beginning of the motion. + +It is usual to measure the accelerating force by the +\index{Accelerating force}% +velocity acquired in one second. Let this be~$g$; then +since the same velocity is acquired in every other second, +the velocity acquired in $t$~seconds will be~$gt$, or +$v = gt$. Hence the space described is $\frac{1}{2}gt × t$, or~$\frac{1}{2}gt^{2}$. +If the point, instead of being at rest at the beginning +of the acceleration, had had the velocity~$a$, the lengths +\PageSep{63} +described in the successive intervals would have been +$at' + v't'$, $at' + 2v't'$, etc.; so that to the space described +by the accelerated motion would have been added~$nat'$, +or~$at$, and the whole length would have been +$at + \frac{1}{2}gt^{2}$. By similar reasoning, had the force been +a uniformly \emph{retarding} force, that is, one which diminished +\index{Force|)}% +the initial velocity~$a$ equally in equal times, the +length described in the time~$t$ would have been $at - \frac{1}{2}gt^{2}$. + +Now let the point move in such a way, that the +velocity is accelerated or retarded, but not uniformly; +that is, in different times of equal duration, let different +velocities be lost or gained. For example, let the +point, setting out from a state of rest, move in such a +\Figure{9} +way that the number of inches passed over in $t$~seconds +is always~$t^{3}$. Here $\phi t = t^{3}$, and the velocity acquired +by the body at the end of the time~$t$, is the coefficient +of~$dt$ in $(t + dt)^{3}$, or $3t^{2}$~inches per~second. +Let the point (\Fig{9}) be at~$A$ at the end of the time~$t$; +and let $AB$,~$BC$, $CD$,~etc., be lengths described in +successive equal intervals of time, each of which is~$dt$. +Then the velocities at $A$,~$B$,~$C$,~etc., are $3t^{2}$, $3(t + dt)^{2}$, +$3(t + 2\, dt)^{3}$, etc., and the lengths $AB$,~$BC$, $CD$,~etc., +are $(t + dt)^{3} - t^{3}$, $(t + 2\, dt)^{2} - (t + dt)^{3}$, $(t + 3\, dt)^{3} - (t + 2\, dt)^{3}$, +etc. +\[ +\ArrayCompress +\begin{array}{cl} +\ColHead{VELOCITY AT} & \\ +A & 3t^{2}\Add{,} \\ +B & 3t^{2} + \Z6t\, dt + \Z3(dt)^{2}\Add{,} \\ +C & 3t^{2} + 12t\, dt + 12(dt)^{2}\Add{,} \\ +\PageSep{64} +\ColHead{LENGTH OF} & \\ +AB & 3t^{2}\, dt + \Z3t(dt)^{2} + \Z\Z(dt)^{3}\Add{,} \\ +BC & 3t^{2}\, dt + \Z9t(dt)^{2} + \Z 7(dt)^{3}\Add{,} \\ +CD & 3t^{2}\, dt + 15t(dt)^{2} + 19(dt)^{3}\Add{.} +\end{array} +\] + +If we could, without error, reject the terms containing~$(dt)^{2}$ +in the velocities, and those containing~$(dt)^{3}$ +in the lengths, we should then reduce the motion +of the point to the case already considered, the +initial velocity being~$3t^{2}$, and the accelerating force~$6t$. +For we have already shown that $a$~being the initial +velocity, and $g$~the accelerating force, the space described +in the time~$t$ is $at + \frac{1}{2}gt^{2}$. Hence, $3t^{2}$~being +the initial velocity, and $6t$~the accelerating force, the +space in the time~$dt$ is $3t^{2}\, dt + 3t(dt)^{2}$, which is the +same as~$AB$ after $(dt)^{3}$~is rejected. The velocity acquired +is~$gt$, and the whole velocity is, therefore, +$a + gt$, or making the same substitutions $3t^{2} + 6t\, dt$. +This is the velocity at~$B$, after the term~$3(dt)^{2}$ is +rejected. Again, the velocity being $3t^{2} + 6t\, dt$, and +the force~$6t$, the space described in the time~$dt$ is +$(3t^{2} + 6t\, dt)\, dt + 3t(dt)^{2}$, or $3t^{2}\, dt + 9t(dt)^{2}$. This is +what the space~$BC$ becomes after $7(dt)^{3}$~is rejected. +The velocity acquired is~$6t\, dt$; and the whole velocity +is $3t^{2} + 6t\, dt + 6t\, dt$, or $3t^{2} + 12t\, dt$; which is the velocity +at~$C$ after $12(dt)^{2}$~is rejected. + +But as the terms involving $(dt)^{2}$ in the velocities, +etc., cannot be rejected without error, the above supposition +of a uniform force cannot be made. Nevertheless, +as we may take $dt$ so small that these terms +shall be as small parts as we please of those which +precede, the results of the erroneous and correct suppositions +may be brought as near to equality as we +please; hence we conclude, that though there is no +force, which, continued uniformly, would preserve +\PageSep{65} +the motion of the point~$A$, so that $OA$~should always +be~$t^{2}$ in inches, yet an interval of time may be taken +so small, that the length actually described by~$A$ in +that time, and the one which would be described if +the force~$6t$ were continued uniformly, shall have a +ratio as near to equality as we please. Hence, on a +principle similar to that by which we called~$3t^{3}$ the +velocity at~$A$, though, in truth, no space, however +small, is described with that velocity, we call~$6t$ the +accelerating force at~$A$. And it must be observed +that $6t$~is the differential coefficient of~$3t^{2}$, or the coefficient +of~$dt$, in the development of~$3(t + dt)^{2}$. + +Generally, let the point move so that the length +described in any time~$t$ is~$\phi t$. Hence the length described +at the end of the time $t + dt$ is $\phi(t + dt)$, and +that described in the interval~$dt$ is $\phi(t + dt) - \phi t$, or +\[ +\phi' t\, dt + \phi'' t\, \frac{(dt)^{2}}{2} + + \phi''' t\, \frac{(dt)^{3}}{2·3} + \etc.\Add{,} +\] +in which $dt$ may be taken so small, that either of the +first two terms shall contain the aggregate of all the +rest, as often as we please. These two first terms are +$\phi' t\, dt + \frac{1}{2}\phi'' t (dt)^{2}$, and represent the length described +during~$dt$, with a uniform velocity~$\phi' t$, and an accelerating +force~$\phi'' t$. The interval~$dt$ may then generally +be taken so small, that this supposition shall represent +the motion during that interval as nearly as we please. + + +\Subsection{Limiting Ratios of Magnitudes that Increase +Without Limit.} + +We have hitherto considered the limiting ratio of +\index{Increase without limit|EtSeq}% +\index{Limiting ratios|EtSeq}% +\index{Ratios, limiting|EtSeq}% +quantities only as to their state of \emph{decrease}: we now +proceed to some cases in which the limiting ratio of +different magnitudes which \emph{increase} without limit is +investigated. +\PageSep{66} + +It is easy to show that the increase of two magnitudes +may cause a decrease of their ratio; so that, as +the two increase without limit, their ratio may diminish +without limit. The limit of any ratio may be found +by rejecting any terms or aggregate of terms~($Q$) which +are connected with another term~($P$) by the sign of +addition or subtraction, provided that by increasing~$x$, +$Q$~may be made as small a part of~$P$ as we please. +For example, to find the limit of $\dfrac{x^{2} + 2x + 3}{2x^{2} + 5x}$, when +$x$~is increased without limit. By increasing~$x$ we can, +as will be shown immediately, cause $2x + 3$ and~$5x$ to +be contained in $x^{2}$ and~$2x^{2}$, as often as we please; rejecting +these terms, we have $\dfrac{x^{2}}{2x^{2}}$, or~$\frac{1}{2}$, for the limit. + +The demonstration is as follows: Divide both +numerator and denominator by~$x^{2}$, which gives $1 + \dfrac{2}{x} + \dfrac{3}{x^{2}}$, +and $2 + \dfrac{5}{x}$, for the numerator and denominator +of a fraction equal in value to the one proposed. +These can be brought as near as we please to $1$~and~$2$ +by making $x$ sufficiently great, or $\dfrac{1}{x}$~sufficiently small; +and, consequently, their ratio can be brought as near +as we please to~$\dfrac{1}{2}$. + +We will now prove the following: That in any +series of decreasing powers of~$x$, any one term will, if +$x$~be taken sufficiently great, contain the aggregate of +all which follow, as many times as we please. Take, +for example, +\[ +% [** TN: On two lines in the original] +ax^{m} + bx^{m-1} + cx^{m-2} + \dots + px + q + + \frac{r}{x} + \frac{s}{x^{2}} + \etc. +\] +\PageSep{67} +The ratio of the several terms will not be altered if we +divide the whole by~$x^{m}$, which gives +\[ +a + \frac{b}{x} + \frac{c}{x^{2}} + \dots + + \frac{p}{x^{m-1}} + \frac{q}{x^{m}} + \frac{r}{x^{m+1}} + + \frac{s}{x^{m+2}} + \etc. +\] +It has been shown that by taking $\dfrac{1}{x}$ sufficiently small, +that is, by taking $x$ sufficiently great, any term of this +series may be made to contain the aggregate of the +succeeding terms, as often as we please; which relation +is not altered if we multiply every term by~$x^{m}$, +and so restore the original series. + +It follows from this, that $\dfrac{(x + 1)^{m}}{x^{m}}$ has unity for its +limit when $x$~is increased without limit. For $(x + 1)^{m}$ +is $x^{m} + mx^{m-1} + \etc.$, in which $x^{m}$~can be made as +great as we please with respect to the rest of the +series. Hence $\dfrac{(x + 1)^{m}}{x^{m}} = 1 + \dfrac{mx^{m-1} + \etc.}{x^{m}}$, the numerator +of which last fraction decreases indefinitely +as compared with its denominator. + +In a similar way it may be shown that the limit of +$\dfrac{x^{m}}{(x + 1)^{m+1} - x^{m+1}}$, when $x$~is increased, is~$\dfrac{1}{m + 1}$. For +since $(x + 1)^{m+1} = x^{m+1} + (m + 1)x^{m} + \frac{1}{2}(m + 1)m x^{m-1} + \etc.$, +this fraction is +\[ +\frac{x^{m}}{(m + 1)x^{m} + \frac{1}{2}(m + 1)m x^{m-1} + \etc.} +\] +in which the first term of the denominator may be +made to contain all the rest as often as we please; +that is, if the fraction be written thus, $\dfrac{x^{m}}{(m + 1)x^{m} + A}$, +$A$~can be made as small a part of~$(m + 1)x^{m}$ as we +\PageSep{68} +please. Hence this fraction can, by a sufficient increase +of~$x$, be brought as near as we please to +$\dfrac{x^{m}}{(m + 1)x^{m}}$, or~$\dfrac{1}{m + 1}$. + +A similar proposition may be shown of the fraction +$\dfrac{(x + b)^{m}}{(x + a)^{m+1} - x^{m+1}}$, which may be immediately reduced +to the form $\dfrac{x^{m} + B}{(m + 1)ax^{m} + A}$, where $x$~may be taken +so great that $x^{m}$~shall contain $A$~and~$B$ any number of +times. + +We will now consider the sums of $x$~terms of the +following series, each of which may evidently be made +as great as we please, by taking a sufficient number +of its terms, +\begin{alignat*}{7} +&1 &&+ 2 &&+ 3 &&+ 4 &&+ \dots &&+ \; x - 1 &&+ x\Add{,} +\tag*{(1)} \\ +&1^{2} &&+ 2^{2} &&+ 3^{2} &&+ 4^{2} &&+ \dots &&+ (x - 1)^{2} &&+ x^{2}\Add{,} +\tag*{(2)} \displaybreak[0]\\ +&1^{3} &&+ 2^{3} &&+ 3^{3} &&+ 4^{3} &&+ \dots &&+ (x - 1)^{3} &&+ x^{3}\Add{,} +\tag*{(3)} \\ +\DotRow{14} \displaybreak[0]\\ +&1^{m} &&+ 2^{m} &&+ 3^{m} &&+ 4^{m} &&+ \dots &&+ (x - 1)^{m} &&+ x^{m}\Add{.} +\tag*{($m$)} +\end{alignat*} +We propose to inquire what is the limiting ratio of +any one of these series to the last term of the succeeding +one; that is, to what do the ratios of $(1 + 2 + \dots + x)$ +to~$x^{2}$, of $(1^{2} + 2^{2}\Add{+} \dots + x^{2})$ to~$x^{3}$, etc., +approach, when $x$~is increased without limit. + +To give an idea of the method of increase of these +series, we shall first show that $x$~may be taken so +great, that the last term of each series shall be as +small a part as we please of the sum of all those which +precede. To simplify the symbols, let us take the +third series $1^{3} + 2^{3} + \dots + x^{3}$, in which we are to +show that $x^{3}$~may be made less than any given part\Typo{.}{,} +\PageSep{69} +say one thousandth, of the sum of those which precede, +or of $1^{3} + 2^{3} \Add{+} \dots + (x - 1)^{3}$. + +First, $x$~may be taken so great that $x^{3}$ and $(x - 1000)^{3}$ +shall have a ratio as near to equality as we +please. For the ratio of these quantities being the +same as that of $1$ to $\left(1 - \dfrac{1000}{x}\right)^{3}$, and $\dfrac{1000}{x}$ being as +small as we please if $x$ may be as great as we please, it +follows that $1 - \dfrac{1000}{x}$, and, consequently, $\left(1 - \dfrac{1000}{x}\right)^{3}$ +may be made as near to unity as we please, or the +ratio of $1$ to $\left(1 - \dfrac{1000}{x}\right)^{3}$, may be brought as near as +we please to that of $1$ to~$1$, or a ratio of equality. But +this ratio is that of $x^{3}$ to~$(x - 1000)^{3}$. Similarly the +ratios of $x^{3}$ to~$(x - 999)^{3}$, of $x^{3}$ to~$(x - 998)^{3}$, etc., up +to the ratio of $x^{3}$ to~$(x - 1)^{3}$ may be made as near as +we please to ratios of equality; there being one thousand +in all. If, then, $(x - 1)^{3} = \alpha x^{3}$, $(x - 2)^{3} = \beta x^{3}$, +etc., up to $(x - 1000)^{3} = \omega x^{3}$, $x$~can be taken so great +that each of the fractions $\alpha$,~$\beta$,~etc., shall be as near +to unity, or $\alpha + \beta + \dots + \omega$ as near\footnote + {Observe that this conclusion depends upon the \emph{number} of quantities $\alpha$,~$\beta$,~etc., + being \emph{determinate}. If there be \emph{ten} quantities, each of which can be + brought as near to unity as we please, their sum can be brought as near to~$10$ + as we please; for, take any fraction~$A$, and make each of those quantities + differ from unity by less than the tenth part of~$A$, then will the sum differ + from~$10$ by less than~$A$. This argument fails, if the number of quantities be + unlimited.} +to~$1000$ as we +please. Hence +%[** TN: In-line in the original] +\[ +\frac{1}{\alpha + \beta + \dots + \omega}\Add{,} +\] +which is +\[ +\frac{x^{3}}{\alpha x^{3} + \beta x^{3} + \dots + \omega x^{3}}, +\] +or +\[ +\frac{x^{3}}{(x - 1)^{3} + (x - 2)^{2} + \dots + (x - 1000)^{3}}, +\] +\PageSep{70} +can be brought as near to~$\dfrac{1}{1000}$ as we please; and by +the same reasoning, the fraction +\[ +\frac{x^{3}}{(x - 1)^{3} + \dots + (x - 1001)^{3}} +\] +may be brought as near to~$\dfrac{1}{1001}$ as we please; that is, +may be made less than~$\dfrac{1}{1000}$. Still more then may +\[ +\frac{x^{3}}{(x - 1)^{3} + \dots + (x - 1001)^{3} + \dots + 2^{3} + 1^{3}} +\] +be made less than~$\dfrac{1}{1000}$, or $x^{3}$~may be less than the +thousandth part of the sum of all the preceding terms. + +In the same way it may be shown that a term may +be taken in any one of the series, which shall be less +than any given part of the sum of all the preceding +terms. It is also true that the difference of any two +succeeding terms may be made as small a part of +either as we please. For $(x + 1)^{m} - x^{m}$, when developed, +will only contain exponents less than~$m$, being +$mx^{m-1} + m\dfrac{m - 1}{2}\, x^{m-2} + \etc.$; and we have shown +(\PageRef{66}) that the sum of such a series may be made +less than any given part of~$x^{m}$. It is also evident +that, whatever number of terms we may sum, if a +sufficient number of succeeding terms be taken, the +sum of the latter shall exceed that of the former in +any ratio we please. + +Let there be a series of fractions +\[ +\frac{a}{pa + b},\quad +\frac{a'}{pa' + b'},\quad +\frac{a''}{pa'' + b''},\quad \etc., +\] +in which $a$,~$a'$,~etc., $b$,~$b'$,~etc., increase without limit; +but in which the ratio of $b$~to~$a$, $b'$~to~$a'$, etc., diminishes +without limit. If it be allowable to begin by +\PageSep{71} +supposing $b$~as small as we please with respect to~$a$, +or $\dfrac{b}{a}$~as small as we please, the first, and all the succeeding +fractions, will be as near as we please to~$\dfrac{1}{p}$, +which is evident from the equations +\[ +\frac{a}{pa + b} = \frac{1}{p + \dfrac{b}{a}},\quad +\frac{a'}{pa' + b'} = \frac{1}{p + \dfrac{b'}{a'}},\quad \etc. +\] +Form a new fraction by summing the numerators and +denominators of the preceding, such as +\[ +\frac{a + a' + a'' + \etc.} + {p(a + a' + a'' + \etc.) + b + b' + b'' + \etc.\Typo{,}{}}, +\] +the \emph{etc.}\ extending to any given number of terms. + +This may also be brought as near to~$\dfrac{1}{p}$ as we please. +For this fraction is the same as +\[ +\text{$1$~divided by } p + \frac{b + b' + \etc.}{a + a' + \etc.}; +\] +and it can be shown\footnote + {See \Title{Study of Mathematics} (Reprint Edition, Chicago: The Open Court + Publishing Co.), page~270.} +that +\[ +\frac{b + b' + \etc.}{a + a' + \etc.} +\] +must lie between the least and greatest of the fractions +$\dfrac{b}{a}$,~$\dfrac{b'}{a'}$,~etc. +If, then, each of these latter fractions +can be made as small as we please, so also can +\[ +\frac{b + b' + \etc.}{a + a' + \etc.}. +\] +No difference will be made in this result, if we use +the following fraction, +\[ +\frac{A + (a + a' + a'' + \etc.)} + {B + p(a + a' + a'' + \etc.) + b + b' + b'' + \etc.}\Add{,} +\Tag{(1)} +\] +\PageSep{72} +$A$~and~$B$ being given quantities; provided that we +can take a number of the original fractions sufficient +to make $a + a' + a'' + \etc.$, as great as we please, +compared with $A$~and~$B$. This will appear on dividing +the numerator and denominator of~\Eq{(1)} by $a + a' + a'' + \etc$. + +Let the fractions be +\begin{gather*} +\frac{(x + 1)^{3}}{(x + 1)^{4} - x^{4}},\quad +\frac{(x + 2)^{3}}{(x + 2)^{4} - (x + 1)^{4}}, \\ +\frac{(x + 3)^{3}}{(x + 3)^{4} - (x + 2)^{4}},\quad \etc. +\end{gather*} +The first of which, or $\dfrac{(x + 1)^{3}}{4x^{3} + \etc.}$ may, as we have +shown, be within any given difference of~$\dfrac{1}{4}$, and the +others still nearer, by taking a value of~$x$ sufficiently +great. Let us suppose each of these fractions to be +within $\dfrac{1}{100000}$ of~$\dfrac{1}{4}$. The fraction formed by summing +the numerators and denominators of these fractions +($n$~in number) will be within the same degree of +nearness to~$\frac{1}{4}$. But this is +\[ +\frac{(x + 1)^{3} + (x + 2)^{3} + \dots + (x + n)^{3}}{(x + 1)^{4} - x^{4}}\Add{,} +\Tag{(2)} +\] +all the terms of the denominator disappearing, except +two from the first and last. If, then, we add~$x^{4}$ to +the denominator, and $1^{3} + 2^{3} + 3^{3} \Add{+} \dots + x^{3}$ to the numerator, +we can still take $n$ so great that $(x + 1)^{3} + \dots + (x + n)^{3}$ +shall contain $1^{3} + \dots + x^{3}$ as often +as we please, and that $(x + n)^{4} - x^{4}$ shall contain~$x^{4}$ +in the same manner. To prove the latter, observe +that the ratio of $(x + n)^{4} - x^{4}$ to~$x^{4}$ being $\left(1 + \dfrac{n}{x}\right)^{4}$, +can be made as great as we please, if it be permitted +\PageSep{73} +to take for~$n$ a number containing~$x$ as often as we +please. Hence, by the preceding reasoning, the fraction, +with its numerator and denominator thus increased, +or +\[ +\frac{1^{3} + 2^{3} + 3^{3} + \dots + x^{3} + (x + 1)^{3} + \dots + (x + n)^{3}} + {(x + n)^{4}} +\Tag{(3)} +\] +may be brought to lie within the same degree of nearness +to~$\frac{1}{4}$ as~\Eq{(2)}; and since this degree of nearness +could be named at pleasure, it follows that \Eq{(3)}~can +be brought as near to~$\frac{1}{4}$ as we please. Hence the +limit of the ratio of $(1^{3} + 2^{3} + \dots + x^{3})$ to~$x^{4}$, as $x$~is +increased without limit, is~$\frac{1}{4}$; and, in a similar manner, +it may be proved that the limit of the ratio of +$(1^{m} + 2^{m} + \dots + x^{m})$ to~$x^{m+1}$ is the same as that of +$\dfrac{(x + 1)^{m}}{(x + 1)^{m+1} - x^{m+1}}$ or $\dfrac{1}{m + 1}$. + +This result will be of use when we come to the +first principles of the integral calculus. It may also +\index{Integral Calculus}% +be noticed that the limits of the ratios which $x\, \dfrac{x - 1}{2}$, +$x\, \dfrac{x - 1}{2}\, \dfrac{x - 2}{3}$, etc., bear to $x^{2}$,~$x^{3}$, etc., are severally $\dfrac{1}{2}$, +$\dfrac{1}{2·3}$, etc.; the limit being that to which the ratios approximate +as $x$~increases without limit. For $x\, \dfrac{x - 1}{2} ÷ x^{2} = \dfrac{x - 1}{2x}$, +$x\, \dfrac{x - 1}{2}\, \dfrac{x - 2}{3} ÷ x^{3} = \dfrac{x - 1}{2x}\, \dfrac{x - 2}{3x}$, etc., +and the limits of $\dfrac{x - 1}{2}$, $\dfrac{x - 2}{3}$, are severally equal to +unity. + +We now resume the elementary principles of the +Differential Calculus. +\PageSep{74} + + +\Subsection[Recapitulation of Results Reached in the Theory of Functions.] +{Recapitulation of Results.} + +The following is a recapitulation of the principal +results which have hitherto been noticed in the general +theory of functions: +\index{Functions!recapitulation of results in the theory of}% + +(1) That if in the equation $y = \phi(x)$, the variable~$x$ +receives an increment~$dx$, $y$~is increased by the series +\[ +\phi' x\, dx + \phi'' x\, \frac{(dx)^{2}}{2} + + \phi''' x\, \frac{(dx)^{3}}{2·3} + \etc. +\] + +(2) That $\phi'' x$ is derived in the same manner from~$\phi' x$, +that $\phi' x$~is from~$\phi x$; viz., that in like manner as +$\phi' x$~is the coefficient of~$dx$ in the development of +$\phi(x + dx)$, so $\phi'' x$~is the coefficient of~$dx$ in the development +of $\phi'(x + dx)$; similarly $\phi''' x$~is the coefficient +of~$dx$ in the development of~$\phi''(x + dx)$, and +so on. + +(3) That $\phi' x$ is the limit of~$\dfrac{dy}{dx}$, or the quantity to +which the latter will approach, and to which it may +be brought as near as we please, when $dx$~is diminished. +It is called the differential coefficient of~$y$. + +(4) That in every case which occurs in practice, +$dx$~may be taken so small, that any term of the series +above written may be made to contain the aggregate +of those which follow, as often as we please; whence, +though $\phi' x\, dx$~is not the actual increment produced +by changing~$x$ into~$x + dx$ in the function~$\phi x$, yet, by +taking $dx$ sufficiently small, it may be brought as near +as we please to a ratio of equality with the actual increment. + + +\Subsection[Approximations by the Differential Calculus.] +{Approximations.} + +The last of the above-mentioned principles is of +the greatest utility, since, by means of it, $\phi' x\, dx$~may +\PageSep{75} +\index{Errors, in the valuation of quantities}% +be made as nearly as we please the actual increment; +and it will generally happen in practice, that $\phi' x\, dx$ +may be used for the increment of~$\phi x$ without sensible +error; that is, if in~$\phi x$, $x$~be changed into $x + dx$, $dx$~being +very small, $\phi x$~is changed into $\phi x + \phi' x\, dx$, +very nearly. Suppose that $x$ being the correct value +of the variable, $x + h$ and $x + k$ have been successively +substituted for it, or the errors $h$~and~$k$ have +been committed in the valuation of~$x$, $h$~and~$k$ being +very small. Hence $\phi(x + h)$ and $\phi(x + k)$ will be +erroneously used for~$\phi x$. But these are nearly $\phi x + \phi' x\, h$ +and $\phi x + \phi' x\, k$, and the errors committed in +taking~$\phi x$ are $\phi' x\, h$ and $\phi' x\, k$, very nearly. These +last are in the proportion of $h$ to~$k$, and hence results +a proposition of the utmost importance in every practical +application of mathematics, viz., that if two different, +but small, errors be committed in the valuation +of any quantity, the errors arising therefrom at +the end of any process, in which both the supposed +values of~$x$ are successively adopted, are very nearly +in the proportion of the errors committed at the beginning. +For example, let there be a right-angled +triangle, whose base is~$3$, and whose other side should +be~$4$, so that the hypothenuse should be $\sqrt{3^{2} + 4^{2}}$ +or~$5$. But suppose that the other side has been twice +erroneously measured, the first measurement giving +$4.001$, and the second $4.002$, the errors being $.001$ +and~$.002$. The two values of the hypothenuse thus +obtained are +\[ +\sqrt{3^{2} + 4.001^{2}}, \quad\text{or}\quad \sqrt{25.008001}, +\] +and +\[ +\sqrt{3^{2} + 4.002^{2}}, \quad\text{or}\quad \sqrt{25.016004}, +\] +which are very nearly $5.0008$ and $5.0016$. The errors +of the hypothenuse are then $.0008$ and $.0016$ nearly; +and these last are in the proportion of $.001$ and~$.002$. +\PageSep{76} + +It also follows, that if $x$~increase by successive equal +steps, any function of~$x$ will, for a few steps, increase +so nearly in the same manner, that the supposition of +such an increase will not be materially wrong. For, +if $h$,~$2h$,~$3h$, etc., be successive small increments given +to~$x$, the successive increments of~$\phi x$ will be $\phi' x\, h$, +$\phi' x\, 2h$, $\phi' x\, 3h$,~etc.\ nearly; which being proportional +to $h$,~$2h$,~$3h$, etc., the increase of the function is nearly +doubled, trebled, etc., if the increase of~$x$ be doubled, +trebled,~etc. + +This result may be rendered conspicuous by reference +to any astronomical ephemeris, in which the +\index{Astronomical ephemeris}% +positions of a heavenly body are given from day to +day. The intervals of time at which the positions are +given differ by $24$~hours, or nearly $\frac{1}{365}$\th~part of the +whole year. And even for this interval, though it can +hardly be called \emph{small} in an astronomical point of view, +the increments or decrements will be found so nearly +the same for four or five days together, as to enable +the student to form an idea how much more near they +would be to equality, if the interval had been less, say +one hour instead of twenty-four. For example, the +sun's longitude on the following days at noon is written +\index{Sun's longitude}% +underneath, with the increments from day to day. +\[ +\ArrayCompress +\begin{array}{c*{2}{>{\ }c}c} +\ColHead[September]{1834 \\ September} & +\ColHead[Sun's longitude]{Sun's longitude \\ at noon.} & +\ColHead{Increments.} & +\ColHead[Proportion which the differences]{Proportion which the differences \\ + of the increments bear to the \\ + whole increments.} \\ +% +1\text{st} & 158\rlap{$°$}\ \ 30\rlap{$'$}\ \ 35\rlap{$''$} + & \Low{58\rlap{$'$}\ \Z9\rlap{$''$}} & \\ +2\text{nd} & 159\ \ 28\ \ 44 & \Low{58\ 12} & \frac{3}{3489} \\ +3\text{rd} & 160\ \ 26\ \ 56 & \Low{58\ 13} & \frac{1}{3492} \\ +4\text{th} & 161\ \ 25\ \ \Z9 & \Low{58\ 14} & \frac{1}{3493} \\ +5\text{th} & 162\ \ 23\ \ 23 +\end{array} +\] +The sun's longitude is a function of the time; that is, +the number of years and days from a given epoch +being given, and called~$x$, the sun's longitude can be +\PageSep{77} +found by an algebraical expression which may be +called~$\phi x$. If we date from the first of January,~1834, +$x$~is~$.666$, which is the decimal part of a year between +the first days of January and September. The increment +is one day, or nearly $.0027$~of a year. Here $x$~is +successively made equal to~$.666$, $.666 + 0027$, $.666 + 2 × .0027$, +etc.; and the intervals of the corresponding +values of~$\phi x$, if we consider only minutes, are the +same; but if we take in the seconds, they differ from +one another, though only by very small parts of themselves, +as the last column shows. + + +\Subsection[Solution of Equations by the Differential Calculus.] +{Solution of Equations.} + +This property is also used\footnote + {See \Title{Study of Mathematics} (Reprint Edition, Chicago: The Open Court + Publishing Co., 1898), page~169 et~seq.} +in finding logarithms +\index{Equations, solution of}% +intermediate to those given in the tables; and may +be applied to find a nearer solution to an equation, +than one already found. For example, suppose it required +to find the value of~$x$ in the equation $\phi x = 0$, +$a$~being a near approximation to the required value. +Let $a + h$ be the real value, in which $h$~will be a small +quantity. It follows that $\phi(a + h) = 0$, or, which is +nearly true, $\phi a + \phi' a\, h = 0$. Hence the real value of~$h$ +is nearly~$-\dfrac{\phi a}{\phi' a}$, or the value $a - \dfrac{\phi a}{\phi' a}$ is a nearer +approximation to the value of~$x$. For example, let +$x^{2} + x - 4 = 0$ be the equation. Here $\phi x = x^{2} + x - 4$, +and $\phi(x + h) = (x + h)^{2} + x + h - 4 = x^{2} + x - 4 + (2x + 1)h + h^{2}$; +so that $\phi' x = 2x + 1$. A near value +of~$x$ is~$1.57$; let this be~$a$. Then $\phi a = .0349$, and +$\phi' a = 4.14$. Hence $-\dfrac{\phi a}{\phi' a} = -.00843$. Hence +$1.57 - .00843$, or~$1.56157$, is a nearer value of~$x$. If +\PageSep{78} +we proceed in the same way with~$1.5616$, we shall +find a still nearer value of~$x$, viz., $1.561553$. We +have here chosen an equation of the second degree, +in order that the student may be able to verify the +result in the common way; it is, however, obvious +that the same method may be applied to equations +of higher degrees, and even to those which are not +to be treated by common algebraical method, such as +$\tan x = ax$. + + +\Subsection{Partial and Total Differentials.} + +We have already observed, that in a function of +\index{Differentials!partial|EtSeq}% +\index{Differentials!total|EtSeq}% +\index{Partial!differentials|EtSeq}% +\index{Total!differentials|EtSeq}% +more quantities than one, those only are mentioned +which are considered as variable; so that all which +we have said upon functions of one variable, applies +\index{Functions!of several variables|EtSeq}% +equally to functions of several variables, so far as a +\index{Variables!functions of several|EtSeq}% +change in one only is concerned. Take for example +$x^{2} y + 2xy^{3}$. If $x$~be changed into $x + dx$, $y$~remaining +the same, this function is increased by $2xy\, dx + 2y^{3}\, dx + \etc.$, +in which, as in \PageRef{29}, no terms are contained +in the~\emph{etc.}\ except those which, by diminishing~$dx$, +can be made to bear as small a proportion as we +please to the first terms. Again, if $y$~be changed into +$y + dy$, $x$~remaining the same, the function receives +the increment $x^{2}\, dy + 6xy^{2}\, dy + \etc.$; and if $x$~be changed +into $x + dx$, $y$~being at the same time changed into +$y + dy$, the increment of the function is $(2xy + 2y^{3})\, dx + (x^{2} + 6xy^{2})\, dy + \etc$. +If, then, $u = x^{2} y + 2xy^{3}$, and +$du$~denote the increment of~$u$, we have the three following +equations, answering to the various suppositions +above mentioned, \\ +(1) when $x$~only varies, +\[ +du = (2xy + 2y^{3})\, dx + \etc. +\] +\PageSep{79} +(2) when $y$~only varies, +\[ +du = (x^{2} + 6xy^{2})\, dy + \etc. +\] +(3) when both $x$~and~$y$ vary, +\[ +du = (2xy + 2y^{3})\, dx + (x^{2} + 6xy^{2})\, dy + \etc. +\] +in which, however, it must be remembered, that $du$~does +not stand for the same thing in any two of the +three equations: it is true that it always represents +an increment of~$u$, but as far as we have yet gone, we +have used it indifferently, whether the increment of~$u$ +was the result of a change in $x$~only, or $y$~only, or both +together. + +To distinguish the different increments of~$u$, we +must therefore seek an additional notation, which, +\index{Calculus, notation of|EtSeq}% +\index{Notation!of the Differential Calculus|EtSeq}% +without sacrificing the~$du$ that serves to remind us +that it was $u$ which received an increment, may also +point out from what supposition the increment arose. +For this purpose we might use $d_{x}u$~and~$d_{y}u$, and $d_{x,y}u$, +to distinguish the three; and this will appear to the +learner more simple than the one in common use, +which we shall proceed to explain. We must, however, +remind the student, that though in matters of +reasoning, he has a right to expect a solution of every +difficulty, in all that relates to notation, he must trust +entirely to his instructor; since he cannot judge between +the convenience or inconvenience of two symbols +without a degree of experience which he evidently +cannot have had. Instead of the notation above +described, the increments arising from a change in $x$ +and~$y$ are severally denoted by $\dfrac{du}{dx}\, dx$ and $\dfrac{du}{dy}\, dy$, on +the following principle: If there be a number of results +obtained by the same species of process, but on +different suppositions with regard to the quantities +\PageSep{80} +used; if, for example, $p$~be derived from some supposition +with regard to~$a$, in the same manner as are $q$ +and~$r$ with regard to $b$~and~$c$, and if it be inconvenient +and unsymmetrical to use separate letters $p$,~$q$, and~$r$, +for the three results, they may be distinguished by +using the same letter~$p$ for all, and writing the three +results thus, $\dfrac{p}{a}\, a$, $\dfrac{p}{b}\, b$, $\dfrac{p}{c}\, c$. Each of these, in common +algebra, is equal to~$p$, but the letter~$p$ does not +stand for the same thing in the three expressions. +The first is the~$p$, so to speak, which belongs to~$a$, the +second that which belongs to~$b$, the third that which +belongs to~$c$. Therefore the numerator of each of the +fractions $\dfrac{p}{a}$,~$\dfrac{p}{b}$, and~$\dfrac{p}{c}$, must never be separated +from its denominator, because the value of the former +depends, in part, upon the latter; and one~$p$ cannot +be distinguished from another without its denominator. +The numerator by itself only indicates what operation +is to be performed, and on what quantity; the +denominator shows what quantity is to be made use +of in performing it. Neither are we allowed to say +that $\dfrac{p}{a}$ divided by~$\dfrac{p}{b}$ is~$\dfrac{b}{a}$; for this supposes that $p$~means +the same thing in both quantities. + +In the expressions $\dfrac{du}{dx}\, dx$, and $\dfrac{du}{dy}\, dy$, each denotes +that $u$~has received an increment; but the first points +out that~$x$, and the second that~$y$, was supposed to increase, +in order to produce that increment; while $du$~by +itself, or sometimes $d.u$, is employed to express +the increment derived from both suppositions at once. +And since, as we have already remarked, it is not the +ratios of the increments themselves, but the limits of +those ratios, which are the objects of investigation in +\PageSep{81} +the Differential Calculus, here, as in \PageRef{28}, $\dfrac{du}{dx}\, dx$, +and $\dfrac{du}{dy}\, dy$, are generally considered as representing +those terms which are of use in obtaining the limiting +ratios, and do not include those terms, which, from +\index{Limiting ratios}% +\index{Ratios, limiting}% +their containing higher powers of $dx$~or~$dy$ than the +first, may be made as small as we please with respect +to $dx$~or~$dy$. Hence in the example just given, where +$u = x^{2} y + 2xy^{3}$, we have +\begin{align*} +&\dfrac{du}{dx}\, dx = (2xy + 2y^{3})\, dx, + &&\text{or}\quad \frac{du}{dx} = 2xy + 2y^{3}\Add{,} \\ +&\dfrac{du}{dy}\, dy = (x^{2} + 6xy^{2})\, dy, + &&\text{or}\quad \frac{du}{dy} = x^{2} + 6xy^{2}\Add{,} \\ +&du \quad\text{or}\quad d.u = \frac{du}{dx}\, dx + \frac{du}{dy}\,dy. +\end{align*} + +The last equation gives a striking illustration of +the method of notation. Treated according to the +common rules of algebra, it is $du = du + du$, which is +absurd, but which appears rational when we recollect +that the second~$du$ arises from a change in $x$~only, the +third from a change in $y$~only, and the first from a +change in both. The same equation may be proved +to be generally true for all functions of $x$~and~$y$, if we +bear in mind that no term is retained, or need be retained, +as far as the limit is concerned, which, when +$dx$~or~$dy$ is diminished, diminishes without limit as +compared with them. In using $\dfrac{du}{dx}$ and $\dfrac{du}{dy}$ as differential +coefficients of~$u$ with respect to $x$~and~$y$, the objection +(\PageRef{27}) against considering these as the +limits of the ratios, and not the ratios themselves, +does not hold, since the numerator is not to be separated +from its denominator. +\PageSep{82} + +Let $u$ be a function of $x$~and~$y$, represented\footnote + {The symbol $\phi(x, y)$ must not be confounded with~$\phi(xy)$. The former represents + any function of $x$~and~$y$; the latter a function in which $x$~and~$y$ only + enter so far as they are contained in their product. The second is therefore + a particular case of the first; but the first is not necessarily represented by + the second. For example, take the function $xy + \sin xy$, which, though it + contains both $x$~and~$y$, yet can only be altered by such a change in $x$~and~$y$ as + will alter their product, and if the product be called~$p$, will be $p + \sin p$. This + may properly be represented by~$\phi(xy)$; whereas $x + xy^{2}$ cannot be represented + in the same way, since other functions besides the product are contained + in it.} +by~$\phi(x, y)$. +It is indifferent whether $x$~and~$y$ be changed +\index{Coefficients, differential}% +\index{Differential coefficients}% +at once into $x + dx$ and $y + dy$, or whether $x$~be first +changed into $x + dx$, and $y$~be changed into $y + dy$ in +the result. Thus, $x^{2} y + y^{3}$ will become $(x + dx)^{2}(y + dy) + (y + dy)^{3}$ +in either case. If $x$~be changed +into $x + dx$, $u$~becomes $u + \ux\, dx + \etc.$, (where $\ux$~is +what we have called the differential coefficient of~$u$ +with respect to~$x$, and is itself a function of $x$~and~$y$; +and the corresponding increment of~$u$ is $\ux\, dx + \etc.$)\Add{.} +If in this result $y$~be changed into $y + dy$, $u$~will assume +the form $u + \uy\, dy + \etc.$, where $\uy$~is the differential +coefficient of~$u$ with respect to~$y$; and the increment +which $u$~receives will be $\uy + \etc$. Again, +when $y$~is changed into $y + dy$, $\ux$,~which is a function +of $x$~and~$y$, will assume the form $\ux + p\, dy + \etc.$; and +$u + \ux\, dx + \etc.$\ becomes $u + \uy\, dy + \etc. + (\ux + p\, dy + \etc.)\, dx + \etc.$, +or $u + \uy\, dy + \ux\, dx + p\, dx\, dy + \etc.$, +in which the term $p\, dx\, dy$ is useless in finding the limit. +For since $dy$~can be made as small as we please, +$p\, dx\, dy$ can be made as small a part of~$p\, dx$ as we please, +and therefore can be made as small a part of~$dx$ as +we please. Hence on the three suppositions already +made, we have the following results: +%[** TN: Re-formatted] +\begin{itemize} +\item[(1)] when $x$~only is changed into~$x + dx$, +$u$~receives the increment +$\ux\, dx + \etc$.\Add{,} + +\item[(2)] when $y$~only is changed into~$y + dy$, +$u$~receives the increment +$\uy\, dy + \etc$.\Add{,} + +\item[(3)] when $x$~becomes $x + dx$ and $y$~becomes $y + dy$ at once, +$u$~receives the increment +$\ux\, dx + \uy\, dy + \etc$.\Add{,} +\end{itemize} +\PageSep{83} +the \emph{etc.}\ in each case containing those terms only which +can be made as small as we please, with respect to +the preceding terms. In the language of Leibnitz, +\index{Leibnitz}% +we should say that if $x$~and~$y$ receive infinitely small +\index{Infinitely small, the notion of}% +increments, the sum of the infinitely small increments +of~$u$ obtained by making these changes separately, is +equal to the infinitely small increment obtained by +making them both at once. As before, we may correct +this inaccurate method of speaking. The several +increments in (1),~(2), and~(3), maybe expressed by +$\ux\, dx + P$, $\uy\, dy + Q$, and $\ux\, dx + \uy\, dy + R$; where $P$,~$Q$, +and~$R$ can be made such parts of $dx$~or~$dy$ as we +please, by taking $dx$~or~$dy$ sufficiently small. The sum +of the two first is $\ux\, dx + \uy\, dy + P + Q$, which differs +from the third by $P + Q - R$; which, since each of +its terms can be made as small a part of $dx$~or~$dy$ as +we please, can itself be made less than any given part +of $dx$~or~$dy$. + +This theorem is not confined to functions of two +variables only, but may be extended to those of any +number whatever. Thus, if $z$~be a function of $p$,~$q$,~$r$, +and~$s$, we have +\[ +d.z \quad\text{or}\quad +dz = \frac{dz}{dp}\, dp + + \frac{dz}{dq}\, dq + + \frac{dz}{dr}\, dr + + \frac{dz}{ds}\, ds + \etc. +\] +in which $\dfrac{dz}{dp}\, dp + \etc.$\ is the increment which a change +in $p$~\emph{only} gives to~$z$, and so on. The \emph{etc.}\ is the representative +of an infinite series of terms, the aggregate +of which diminishes continually with respect to $dp$,~$dq$,~etc., +as the latter are diminished, and which, therefore, +\PageSep{84} +has no effect on the \emph{limit} of the ratio of~$d.z$ to +any other quantity. + + +\Subsection[Application of the Theorem for Total Differentials to the Determination of Total Resultant Errors.] +{Practical Application of the Preceding Theorem.} + +We proceed to an important practical use of this +\index{Errors, in the valuation of quantities}% +theorem. If the increments $dp$,~$dq$,~etc., be small, +this last-mentioned equation, (the terms included in +the \emph{etc.}\ being omitted,) though not actually true, is +sufficiently near the truth for all practical purposes; +which renders the proposition, from its simplicity, of +the highest use in the applications of mathematics. +For if any result be obtained from a set of \textit{data}, no +one of which is exactly correct, the error in the result +would be a very complicated function of the errors in +the \textit{data}, if the latter were considerable. When they +are small, the error in the results is very nearly the +sum of the errors which would arise from the error in +each \textit{datum}, if all the others were correct. For if $p$,~$q$,~$r$, +and~$s$, are the \emph{presumed} values of the \textit{data}, which +give a certain value~$z$ to the function required to be +found; and if $p + dp$, $q + dq$, etc., be the \emph{correct} values +of the \textit{data}, the correction of the function~$z$ will be +very nearly made, if $z$~be increased by $\dfrac{dz}{dp}\, dp + \dfrac{dz}{dq}\, dq + \dfrac{dz}{dr}\, dr + \dfrac{dz}{ds}\, ds$, +being the sum of terms which would +arise from each separate error, if each were made in +turn by itself. + +For example: A transit instrument is a telescope +\index{Transit instrument}% +mounted on an axis, so as to move in the plane of the +meridian only, that is, the line joining the centres of +the two glasses ought, if the telescope be moved, to +pass successively through the zenith and the pole. +Hence can be determined the exact time, as shown by +a clock, at which any star passes a vertical thread, +\PageSep{85} +fixed inside the telescope so as apparently to cut the +field of view exactly in half, which thread will always +cover a part of the meridian, if the telescope be correctly +adjusted. In trying to do this, three errors +may, and generally will be committed, in some small +degree. (1)~The axis of the telescope may not be exactly +level; (2)~the ends of the same axis may not be +exactly east and west; (3)~the line which joins the +centres of the two glasses, instead of being perpendicular +to the axis of the telescope, may be inclined +to it. If each of these errors were considerable, and +the time at which a star passed the thread were observed, +the calculation of the time at which the same +star passes the real meridian would require complicated +formulæ, and be a work of much labor. But if +the errors exist in small quantities only, the calculation +is very much simplified by the preceding principle. +For, suppose only the first error to exist, and +calculate the corresponding error in the time of passing +the thread. Next suppose only the second error, +and then only the third to exist, and calculate the +effect of each separately, all which may be done by +simple formulæ. The effect of all the errors will then +be the sum of the effects of each separate error, at +least with sufficient accuracy for practical purposes. +The formulæ employed, like the equations in \PageRef{28}, +are not actually true in any case, but approach more +near to the truth as the errors are diminished. + + +\Subsection{Rules for Differentiation.} + +In order to give the student an opportunity of exercising +\index{Differentiation!of the common functions}% +himself in the principles laid down, we will +so far anticipate the treatises on the Differential Calculus +as to give the results of all the common rules +\PageSep{86} +for differentiation; that is, assuming $y$~to stand for +various functions of~$x$, we find the increment of~$y$ arising +from an increment in the value of~$x$, or rather, +that term of the increment which contains the first +power of~$dx$. This term, in theory, is the only one +on which the \emph{limit} of the ratio of the increments depends; +in practice, it is sufficiently near to the real +increment of~$y$, if the increment of~$x$ be small. + +{\Loosen (1) $y = x^{m}$, where $m$~is either whole or fractional, +\index{Differentiation!of the common functions}% +positive or negative; then $dy = mx^{m-1}\, dx$. Thus the +increment of~$x^{\efrac{2}{3}}$ or the first term of $(x + dx)^{\efrac{2}{3}} - x^{\efrac{2}{3}}$ +is $\frac{2}{3}x^{\efrac{2}{3}-1}\, dx$, or~$\dfrac{2\, dx}{3x^{\efrac{1}{3}}}$. Again, if $y = x^{8}$, $dy = 8x^{7}\, dx$. +When the exponent is negative, or when $y = \dfrac{1}{x^{m}}$, +$dy = -\dfrac{m\, dx}{x^{m+1}}$, or when $y = x^{-m}$, $dy = -mx^{-m-1}\, dx$, +which is according to the rule. The negative sign +indicates that an increase in~$x$ decreases the value +of~$y$; which, in this case, is evident.} + +(2) $y = a^{x}$. Here $dy = a^{x}\log a\, dx$ where the logarithm +(as is always the case in analysis, except +where the contrary is specially mentioned) is the Naperian +or hyperbolic logarithm. When $a$~is the base +of these logarithms, that is when $a = 2.7182818 = e$, +\index{Logarithms}% +or when $y = e^{x}$, $dy = e^{x}\, dx$. + +(3) $y = \log x$ (the Naperian logarithm). Here +$dy = \dfrac{dx}{x}$. If $y = \text{common log}~x$, $dy = -.4342944\, \dfrac{dx}{x}$. + +(4) $y = \sin x$, $dy = \cos x\, dx$; $y = \cos x$, $dy = -\sin x\, dx$; +$y = \tan x$, $dy = \dfrac{dx}{\cos^{2} x}$. + + +\Subsection[Illustration of the Rules for Differentiation.] +{Illustration of the Preceding Formulæ.} + +At the risk of being tedious to some readers, we +will proceed to illustrate these formulæ by examples +\PageSep{87} +from the tables of logarithms and sines, let $y = \text{common log}~x$. +\index{Logarithms}% +\index{Sines}% +If $x$~be changed into $x + dx$, the real increment +of~$y$ is +\[ +.4342944 \left(\frac{dx}{x} - \tfrac{1}{2}\, \frac{(dx)^{2}}{x^{2}} + + \tfrac{1}{3}\, \frac{(dx)^{3}}{x^{3}} - \etc.\right), +\] +in which the law of continuation is evident. The corresponding +series for Naperian logarithms is to be +found in \PageRef{20}. From the first term of this the +limit of the ratio of $dy$~to~$dx$ can be found; and if $dx$~be +\index{Ratio!of two increments}% +small, this will represent the increment with sufficient +accuracy. Let $x = 1000$, whence $y = \text{common log}~ 1000 =3$; +and let $dx = 1$, or let it be required to +find the common logarithm of $1000 + 1$, or~$1001$. The +first term of the series is therefore $.4342944 × \frac{1}{1000}$, or +$.0004343$, taking seven decimal places only. Hence +$\log 1001 = \log 1000 + .0004343$ or $3.0004343$ nearly. +The tables give $3.0004341$, differing from the former +only in the $7$\Chg{th}{\th}~place of decimals. + +{\Loosen Again, let $y = \sin x$; from which, by \PageRef{20}, as +before, if $x$~be increased by~$dx$, $\sin x$~is increased by +$\cos x\, dx - \frac{1}{2}\sin x\, (dx)^{2} - \etc.$, of which we take only +the first term. Let $x = 16°$, in which case $\sin x = .2756374$, +and $\cos x = .9612617$.} Let $dx = 1'$, or, as +it is represented in analysis, where the angular unit is +that angle whose arc is equal to the radius,\footnote + {See \Title{Study of Mathematics} (Chicago; The Open Court Pub. Co.), page~273 + et~seq.} +$\frac{60}{206265}$. +Hence $\sin 16°\, 1' = \sin 16° + .9612617 × \frac{60}{206265} = +.2756374 + .0002797 = .2759171$, nearly. The tables +give~$.2759170$. These examples may serve to show +how nearly the real ratio of two increments approaches +to their limit, when the increments themselves are +small. +\PageSep{88} + + +\Subsection{Differential Coefficients of Differential +Coefficients.} + +When the differential coefficient of a function of~$x$ +\index{Coefficients, differential}% +\index{Differential coefficients!of higher orders}% +\index{Finite differences|EtSeq}% +\index{Orders, differential coefficients of higher}% +\index{Successive differentiation|EtSeq}% +has been found, the result, being a function of~$x$, may +be also differentiated, which gives the differential coefficient +of the differential coefficient, or, as it is called, +the \emph{second} differential coefficient. Similarly the differential +coefficient of the second differential coefficient +is called the third differential coefficient, and so on. +We have already had occasion to notice these successive +differential coefficients in \PageRef{22}, where it appears +that $\phi' x$~being the first differential coefficient of~$\phi x$, +$\phi'' x$~is the coefficient of~$h$ in the development +$\phi'(x + h)$, and is therefore the differential coefficient +of~$\phi' x$, or what we have called the second differential +coefficient of~$\phi x$. Similarly $\phi''' x$~is the third differential +coefficient of~$\phi x$. If we were strictly to adhere +to our system of notation, we should denote the +several differential coefficients of~$\phi x$ or~$y$ by +\[ +\frac{dy}{dx}\Add{,}\quad +\frac{d.\dfrac{dy}{dx}}{dx}\Add{,}\quad +\frac{d.\dfrac{d.\frac{dy}{dx}}{dx}}{dx}\Add{,}\quad \etc. +\] +In order to avoid so cumbrous a system of notation, +the following symbols are usually preferred, +\[ +\frac{dy}{dx}\Add{,}\quad +\frac{d^{2} y}{dx^{2}}\Add{,}\quad +\frac{d^{3} y}{dx^{3}}\Add{,}\quad \etc. +\] + + +\Subsection{Calculus of Finite Differences. Successive +Differentiation.} + +We proceed to explain the manner in which this +\index{Differentiation!successive|EtSeq}% +notation is connected with our previous ideas on the +subject. +\PageSep{89} + +When in any function of~$x$, an increase is given to~$x$, +\index{Differences!calculus of}% +which is not supposed to be as small as we please, +it is usual to denote it by~$\Delta x$ instead of~$dx$, and the +corresponding increment of~$y$ or~$\phi x$, by~$\Delta y$ or~$\Delta\phi x$, +instead of~$dy$ or~$d\phi x$. The symbol~$\Delta x$ is called the +\emph{difference} of~$x$, being the difference between the value +of the variable~$x$, before and after its increase. + +Let $x$ increase at successive steps by the same difference; +that is, let a variable, whose first value is~$x$, +successively become $x + \Delta x$, $x + 2\Delta x$, $x + 3\Delta x$, etc., +and let the successive values of~$\phi x$ corresponding to +these values of~$x$ be $y$,~$y_{1}$, $y_{2}$,~$y_{3}$,~etc.; that is, $\phi x$~is +called~$y$, $\phi(x + \Delta x)$ is~$y_{1}$, $\phi(x + 2\Delta x)$ is~$y_{2}$, etc., and, +generally, $\phi(x + m\Delta x)$ is~$y_{m}$. Then, by our previous +definition $y_{1} - y$ is~$\Delta y$, $y_{2} - y_{1}$ is~$\Delta y_{1}$, $y_{3} - y_{2}$ is~$\Delta y_{2}$, +etc., the letter~$\Delta$ before a quantity always denoting +the increment it would receive if $x + \Delta x$ were substituted +for~$x$. Thus $y_{3}$ or $\phi(x + 3\Delta x)$ becomes $\phi(x + \Delta x + 3\Delta x)$, +or $\phi(x + 4\Delta x)$, when $x$~is changed into +$x + \Delta x$, and receives the increment $\phi(x + 4\Delta x) - \phi(x + 3\Delta x)$, or $y_{4} - y_{3}$. If $y$~be a function which decreases +when $x$~is increased, $y_{1} - y$, or $\Delta y$ is negative. + +It must be observed, as in \PageRef{26}, that $\Delta x$~does +not depend upon~$x$, because $x$~occurs in it; the symbol +merely signifies an increment given to~$x$, which +increment is not necessarily dependent upon the value +of~$x$. For instance, in the present case we suppose +it a given quantity; that is, when $x + \Delta x$ is changed +into $x + \Delta x + \Delta x$, or $x + 2\Delta x$, $x$~is changed, and $\Delta x$~is +not. + +In this way we get the two first of the columns underneath, +in which each term of the \emph{second} column is +formed by subtracting the term which immediately +precedes it in the first column from the one which immediately +\PageSep{90} +follows. Thus $\Delta y$ is $y_{1} - y$, $\Delta y_{1}$ is $y_{2} - y_{1}$, +etc. +\begin{gather*}%**** Tall, bad page break +\left. +\begin{alignedat}{2} +& \PadTo[l]{\phi(x + 4\Delta x)}{\phi(x)} && y \\ +& \phi(x + \Z\Delta x)\qquad && y_{1} \\ +& \phi(x + 2\Delta x) && y_{2} \\ +& \phi(x + 3\Delta x) && y_{3} \\ +& \phi(x + 4\Delta x) && y_{4} +\end{alignedat}\ +\right| +% +\left. +\begin{aligned} +& \Delta y \\ +& \Delta y_{1} \\ +& \Delta y_{2} \\ +& \Delta y_{3} +\end{aligned}\ +\right| +% +\left. +\begin{aligned} +& \Delta^{2} y \\ +& \Delta^{2} y_{1} \\ +& \Delta^{2} y_{2} +\end{aligned}\ +\right| +% +\left. +\begin{aligned} +& \Delta^{3} y \\ +& \Delta^{3} y_{1} +\end{aligned}\ +\right| +% +\begin{aligned} +& \Delta^{4} y +\end{aligned} \\ +\PadTo{\phi(x + 4\Delta x)}{\etc.}\phantom{\qquad\qquad\qquad\qquad\qquad\qquad} +\end{gather*} + +In the first column is to be found a series of successive +values of the same function~$\phi x$, that is, it contains +terms produced by substituting successively in~$\phi x$ +the quantities $x$, $x + \Delta x$, $x + 2\Delta x$, etc., instead of~$x$. +The second column contains the successive values +of another function $\phi(x + \Delta x) - \phi x$, or~$\Delta \phi x$, made by +the same substitutions; if, for example, we substitute +$x + 2\Delta x$ for~$x$, we obtain $\phi(x + 3\Delta x) - \phi(x + 2\Delta x)$, +or $y_{3} - y_{2}$, or~$\Delta y_{2}$. If, then, we form the successive +differences of the terms in the second column, we obtain +a new series, which we might call the differences +of the differences of the first column, but which are +called the \emph{second differences} of the first column. And +as we have denoted the operation which deduces the +second column from the first by~$\Delta$, so that which deduces +the third from the second may be denoted by~$\Delta\Delta$, +which is abbreviated into~$\Delta^{2}$. Hence as $y_{1} - y$ +was written~$\Delta y$, $\Delta y_{1} - \Delta y$ is written~$\Delta\Delta y$, or~$\Delta^{2} y$. And +the student must recollect, that in like manner as $\Delta$~is +not the symbol of a number, but of an operation, +so $\Delta^{2}$~does not denote a number multiplied by itself, +but an operation repeated upon its own result; just +as the logarithm of the logarithm of~$x$ might be written +$\log^{2} x$; $(\log x)^{2}$~being reserved to signify the square +of the logarithm of~$x$. We do not enlarge on this notation, +as the subject is discussed in most treatises on +\PageSep{91} +algebra.\footnote + {The reference of the original text is to ``the treatise on \Title{Algebraical Expressions},'' + Number~105 of the Library of Useful Knowledge,---the same series + in which the present work appeared. The first six pages of this treatise are + particularly recommended by De~Morgan in relation to the present point.---\Ed.} +Similarly the terms of the fourth column, +or the differences of the second differences, have the +prefix~$\Delta\Delta\Delta$ abbreviated into~$\Delta^{3}$, so that $\Delta^{2} y_{1} - \Delta^{2} y = \Delta^{3} y$, etc. + +When we have occasion to examine the results +which arise from supposing $\Delta x$~to diminish without +limit, we use~$dx$ instead of~$\Delta x$, $dy$~instead of~$\Delta y$, $d^{2} y$~instead +of~$\Delta^{2} y$, and so on. If we suppose this case, we +can show that the ratio which the term in any column +bears to its corresponding term in any preceding column, +diminishes without limit. Take for example, +$d^{2} y$~and~$dy$. The latter is $\phi(x + dx) - \phi x$, which, as +we have often noticed already, is of the form $p\, dx + q\, (dx)^{2} + \etc.$, +in which $p$,~$q$,~etc., are also functions +of~$x$. To obtain~$d^{2} y$, we must, in this series, change~$x$ +into $x + dx$, and subtract $p\, dx + q\, (dx)^{2} + \etc.$\ from +the result. But since $p$,~$q$,~etc., are functions of~$x$, +this change gives them the form +\[ +p + p'\, dx + \etc.,\quad +q + q'\, dx + \etc.; +\] +so that $d^{2} y$~is +\begin{gather*} +(p + p'\, dx + \etc.)\, dx + (q + q'\, dx + \etc.)(dx)^{2} + \etc. \\ +{} - (p\, dx + q\, (dx)^{2} + \etc.) +\end{gather*} +in which the first power of~$dx$ is destroyed. Hence +(\PageRefs{42}{44}), the ratio of $d^{2} y$ to~$dx$ diminishes without +limit, while that of $d^{2} y$ to~$(dx)^{2}$ has a finite limit, +except in those particular cases in which the second +power of~$dx$ is destroyed, in the previous subtraction, +as well as the first. In the same way it may be shown +that the ratio of $d^{3} y$ to $dx$~and $(dx)^{2}$ decreases without +limit, while that of $d^{3} y$ to~$(dx)^{3}$ remains finite; and so +\PageSep{92} +on. Hence we have a succession of ratios $\dfrac{dy}{dx}$, $\dfrac{d^{2} y}{dx^{2}}$, $\dfrac{d^{3} y}{dx^{3}}$, +etc., which tend towards finite limits when $dx$~is diminished. + +{\Loosen We now proceed to show that in the development +of $\phi(x + h)$, which has been shown to be of the form} +\[ +\phi x + \phi' x\, h + + \phi'' x\, \frac{h^{2}}{2} + + \phi''' x\, \frac{h^{3}}{2·3} + \etc., +\] +in the same manner as $\phi' x$~is the limit of~$\dfrac{dy}{dx}$ (\PageRef{23}), +so $\phi'' x$~is the limit of~$\dfrac{d^{2} y}{dx^{2}}$, $\phi''' x$~is that of~$\dfrac{d^{3} y}{dx^{3}}$, and so +forth. + +From the manner in which the preceding table +was formed, the following relations are seen immediately: +\begin{gather*} +\begin{alignedat}{4} +y_{1} &= y &&+ \Delta y & + \Delta y_{1} &= \Delta y &&+ \Delta^{2} y\Add{,} \\ +y_{2} &= y_{1} &&+ \Delta y_{1}\qquad & + \Delta y_{2} &= \Delta y_{1} &&+ \Delta^{2} y_{1}\Add{,} +\end{alignedat} \\ +\begin{alignedat}{3} +\Delta^{2} y_{1} &= \Delta^{2} y &&+ \Delta^{3} y && \etc.\Add{,} \\ +\Delta^{2} y_{2} &= \Delta^{2} y_{1} &&+ \Delta^{3} y_{1}\ && \etc. +\end{alignedat} +\end{gather*} +Hence $y_{1}$,~$y_{2}$,~etc., can be expressed in terms of $y$,~$\Delta y$, +$\Delta^{2} y$,~etc. For $y_{1} = y + \Delta y$; +%[** TN: Next two displays in-line in the original] +\begin{align*} +y_{2} = y_{1} + \Delta y_{1} + &= (y + \Delta y) + (\Delta y + \Delta^{2} y) + = y + 2\Delta y + \Delta^{2} y. \displaybreak[0] +\intertext{In the same way $\Delta y_{2} = \Delta y + 2\Delta^{2} y + \Delta^{3} y$; +hence} +y_{3} = y_{2} + \Delta y_{2} + &= (y + 2\Delta y + \Delta^{2} y) + (\Delta y + 2\Delta^{2} y + \Delta^{3} y) \\ + &= y + 3\Delta y + 3\Delta^{2} y + \Delta^{3} y. +\end{align*} + +Proceeding in this way we have +\begin{alignat*}{5} +y_{1} = y &{}+{}& \Delta y\Add{\rlap{,}} \\ +y_{2} = y &{}+{}&2\Delta y &{}+{}& \Delta^{2} y\Add{\rlap{,}} \\ +y_{3} = y &{}+{}&3\Delta y &{}+{}& 3\Delta^{2} y &{}+{}& \Delta^{3} y\Add{\rlap{,}} \\ +y_{4} = y &{}+{}&4\Delta y &{}+{}& 6\Delta^{2} y &{}+{}& 4\Delta^{3} y &{}+{}& \Delta^{5} y\Add{\rlap{,}} \\ +y_{5} = y &{}+{}&5\Delta y &{}+{}&10\Delta^{2} y &{}+{}&10\Delta^{3} y &{}+{}&5\Delta^{5} y + &+ \Delta^{6} y,\ \etc.\Add{,} +\end{alignat*} +from the whole of which it appears that $y_{n}$ or $\phi(x + n\Delta x)$ +is a series consisting of $y$,~$\Delta y$,~etc., up to~$\Delta^{n} y$, +severally multiplied by the coefficients which occur in +the expansion $(1 + a)^{n}$, or +\PageSep{93} +\begin{align*}%[** TN: Re-formatted] +y_{n} &= \phi(x + n\Delta x) \\ + &= y + n\Delta y + n\frac{n - 1}{2}\, \Delta^{2} y + + n\frac{n - 1}{2}\, \frac{n - 2}{3}\, \Delta^{3} y + \etc. +\end{align*} + +Let us now suppose that $x$~becomes $x + h$ by $n$~equal +steps; that is, $x$,~$x + \dfrac{h}{n}$, $x + \dfrac{2h}{n}$, etc.~\dots\ $x + \dfrac{nh}{h}$ +or~$x + h$, are the successive values of~$x$, so that +$n\Delta x = h$. Since the product of a number of factors is +not altered by multiplying one of them, provided we +divide another of them by the same quantity, multiply +every factor which contains~$n$ by~$\Delta x$, and divide the +accompanying difference of~$y$ by $\Delta x$ as often as there +are factors which contain~$n$, substituting~$h$ for~$n\Delta x$, +which gives +\begin{align*} +\phi(x + n\Delta x) + &= y + n\Delta x\, \frac{\Delta y}{\Delta x} + + n\Delta x\, \frac{n\Delta x - \Delta x}{2}\, \frac{\Delta^{2} y}{(\Delta x)^{2}} \\ + &\quad+ n\Delta x\, \frac{n\Delta x - \Delta x}{2}\, + \frac{n\Delta x - 2\Delta x}{3}\, \frac{\Delta^{3} y}{(\Delta x)^{3}} + \etc.\Add{,} +\end{align*} +or +\begin{align*} +\iffalse %[** TN: Commented code matches the original] +\phi(x + h) + &= \PadTo{y + n\Delta x}{y + h}\, \frac{\Delta y}{\Delta x} + + \PadTo{n\Delta x}{h}\, + \PadTo{\dfrac{n\Delta x - \Delta x}{2}}{\dfrac{h - \Delta x}{2}}\, + \frac{\Delta^{2} y}{(\Delta x)^{2}} \\ + &+ \PadTo{n\Delta x}{h}\, + \PadTo{\dfrac{n\Delta x - \Delta x}{2}}{\dfrac{h - \Delta x}{2}}\, + \PadTo{\dfrac{n\Delta x - 2\Delta x}{3}}{\dfrac{h - 2\Delta x}{3}}\, + \frac{\Delta^{3} y}{(\Delta x)^{3}} + \etc. +\fi +\phi(x + h) + = y &+ h\, \frac{\Delta y}{\Delta x} + + h\, \frac{h - \Delta x}{2}\, \frac{\Delta^{2} y}{(\Delta x)^{2}} \\ + &+ h\, \frac{h - \Delta x}{2}\, \frac{h - 2\Delta x}{3}\, + \frac{\Delta^{3} y}{(\Delta x)^{3}} + \etc. +\end{align*} +If $h$ remain the same, the more steps we make between +$x$~and~$x + h$, the smaller will each of those +steps be, and the number of steps may be increased, +until each of them is as small as we please. We can +therefore suppose $\Delta x$ to decrease without limit, without +affecting the truth of the series just deduced. +Write $dx$ for~$\Delta x$, etc., and recollect that $h - dx$, +$h - 2\, dx$, etc., continually approximate to~$h$. The series +then becomes +\[ +\phi(x + h) = y + \frac{dy}{dx}\, h + + \frac{d^{2} y}{dx^{2}}\, \frac{h^{2}}{2} + + \frac{d^{3} y}{dx^{3}}\, \frac{h^{3}}{2·3} + \etc.\Add{,} +\] +\PageSep{94} +in which, according to the view taken of the symbols +$\dfrac{dy}{dx}$~etc.\ in \PageRefs{26}{27}, $\dfrac{dy}{dx}$~stands for the \emph{limit} of the +ratio of the increments, $\dfrac{dy}{dx}$~is $\phi' x$, $\dfrac{d^{2} y}{dx^{2}}$~is $\phi'' x$,~etc. +According to the method proposed in \PageRefs{28}{29}, +the series written above is the first term of the development +of~$\phi(x + h)$, the remaining terms (which we +might include under an additional~$+$ etc.)\ being such +as to diminish without limit in comparison with the +first, when $dx$~is diminished without limit. And we +may show that the limit of~$\dfrac{d^{2} y}{dx^{2}}$ is the differential coefficient +of the limit of~$\dfrac{dy}{dx}$; or if by these fractions +themselves are understood their limits, that $\dfrac{d^{2} y}{dx^{2}}$ is the +differential coefficient of~$\dfrac{dy}{dx}$: for since $dy$, or $\phi(x + dx) - \phi x$, +becomes $dy + d^{2} y$, when $x$~is changed into +$x + dx$; and since $dx$~does not change in this process, +$\dfrac{dy}{dx}$ will become $\dfrac{dy}{dx} + \dfrac{d^{2} y}{dx}$, or its increment is~$\dfrac{d^{2} y}{dx}$. The +ratio of this to~$dx$ is~$\dfrac{d^{2} y}{(dx)^{2}}$, the limit of which, in the +definition of \PageRef{22}, is the differential coefficient of~$\dfrac{dy}{dx}$. +Similarly the limit of~$\dfrac{d^{3} y}{dx^{3}}$ is the differential coefficient +of the limit of~$\dfrac{d^{2} y}{dx^{2}}$; and so on. + + +\Subsection{Total and Partial Differential Coefficients. +Implicit Differentiation.} + +We now proceed to apply the principles laid down, +\index{Differentiation!implicit|EtSeq}% +\index{Implicit!differentiation|EtSeq}% +to some cases in which the variable enters into its +function in a less direct and more complicated manner. +\PageSep{95} + +For example, let $z$ be a given function of $x$~and~$y$, +and let $y$~be another given function of~$x$; so that $z$ +contains $x$ both directly and indirectly; the latter as +it contains~$y$, which is a function of~$x$. This will be +the case if $z = x\log y$, where $y = \sin x$. If we were to +substitute for~$y$ its value in terms of~$x$, the value of~$z$ +would then be a function of $x$~only; in the instance +just given it would be $x\log\sin x$. But if it be not convenient +to combine the two equations at the beginning +of the process, let us first consider $z$ as a function of +$x$~and~$y$, in which the two variables are independent. +In this case, if $x$~and~$y$ respectively receive the increments +$dx$~and~$dy$, the whole increment of~$z$, or~$d.z$, (or +at least that part which gives the limit of the ratios) +is represented by +\[ +\frac{dz}{dx}\, dx + \frac{dz}{dy}\, dy. +\] +If $y$ be now considered as a function of~$x$, the consequence +is that $dy$, instead of being independent of~$dx$, +is a series of the form $p\, dx + q\, (dx)^{2} + \etc.$, in which $p$~is +the differential coefficient of~$y$ with respect to~$x$. +Hence +\[ +d.z = \frac{dz}{dx}\, dx + \frac{dz}{dy}\, p\, dx \quad\text{or}\quad +\frac{d.z}{dx} = \frac{dz}{dx} + \frac{dz}{dx}\, p, +\] +in which the difference between $\dfrac{d.z}{dx}$ and $\dfrac{dz}{dx}$ is this, +that in the second, $x$~is only considered as varying +where it is directly contained in~$z$, or $z$~is considered +in the form in which it first appeared, as a function of +$x$~and~$y$, where $y$~is independent of~$x$; in the first, or +$\dfrac{d.z}{dx}$, the \emph{total variation} of~$z$ is denoted, that is, $y$~is +\index{Total!variations}% +\index{Variations, total}% +now considered as a function of~$x$, by which means if +$x$ become $x + dx$, $z$~will receive a different increment +\PageSep{96} +from that which it would have received, had $y$ been +independent of~$x$. {\Loosen In the instance above cited, where +$z = x\log y$ and $y = \sin x$, if the first equation be taken, +and $x$ becomes $x + dx$, $y$~remaining the same, $z$~becomes +$x\log y + \log y\, dx$ or $\dfrac{dz}{dx}$ is~$\log y$.} If $y$~only varies, +since (\PageRef{20}) $z$~will then become +\[ +x\log y + x\, \frac{dy}{y} - \etc., +\] +$\dfrac{dz}{dy}$ is~$\dfrac{x}{y}$ And $\dfrac{dy}{dx}$~is $\cos x$ when $y = \sin x$ (\PageRef{20}). +Hence $\dfrac{dz}{dx} + \dfrac{dz}{dy}\, p$, or $\dfrac{dz}{dx} + \dfrac{dz}{dy}\, \dfrac{dy}{dx}$ is $\log y + \dfrac{x}{y} \cos x$, +or $\log\sin x + \dfrac{x}{\sin x} \cos x$. This is~$\dfrac{d.z}{dx}$, which might +have been obtained by a more complicated process, if +$\sin x$ had been substituted for~$y$, before the operation +commenced. It is called the \emph{complete} or \emph{total} differential +\index{Coefficients, differential}% +\index{Complete Differential Coefficients}% +coefficient with respect to~$x$, the word \emph{total} indicating +that every way in which $z$ contains~$x$ has been +used; in opposition to~$\dfrac{dz}{dx}$, which is called the \emph{partial} +\index{Partial!differential coefficients}% +differential coefficient, $x$~having been considered as +varying only where it is directly contained in~$z$. + +Generally, the complete differential coefficient of~$z$ +with respect to~$x$, will contain as many terms as there +are different ways in which $z$ contains~$x$. From looking +at a complete differential coefficient, we may see +in what manner the function contained its variable. +Take, for example, the following, +\[ +\frac{d.z}{dx} + = \frac{dz}{dx} + \frac{dz}{dy}\, \frac{dy}{dx} + + \frac{dz}{da}\, \frac{da}{dy}\, \frac{dy}{dx} + + \frac{dz}{da}\, \frac{da}{dx}. +\] + +Before proceeding to demonstrate this formula, we +will collect from itself the hypothesis from which it +\PageSep{97} +\index{Functions!direct and indirect}% +must have arisen. When $x$~is contained in~$z$, we shall +say that $z$~is a \emph{direct}\footnote + {It may be right to warn the student that this phraseology is new, to the + best of our knowledge. The nomenclature of the Differential Calculus has + by no means kept pace with its wants; indeed the same may be said of algebra + generally. [Written in~1832.---\Ed.]} +function of~$x$. When $x$~is contained +in~$y$, and $y$~is contained in~$z$, we shall say that +$z$~is an indirect function of~$x$ \emph{through}~$y$. It is evident +\index{Indirect function}% +that an indirect function may be reduced to one which +is direct, by substituting for the quantities which contain~$x$, +their values in terms of~$x$. + +The first side of the equation~$\dfrac{d.z}{dx}$ is shown by the +point to be a complete differential coefficient, and indicates +that $z$~is a function of~$x$ in several ways; either +directly, and indirectly through one quantity at least, +or indirectly through several. If $z$~be a direct function +\index{Direct function}% +only, or indirectly through one quantity only, the +symbol~$\dfrac{dz}{dx}$, without the point, would represent its +total differential coefficient with respect to~$x$. + +On the second side of the equation we see: + +(1) $\dfrac{dz}{dx}$: which shows that $z$~is a direct function of~$x$, +and is that part of the differential coefficient which +we should get by changing $x$ into $x + dx$ throughout~$z$, +not supposing any other quantity which enters into~$z$ +to contain~$x$. + +(2) $\dfrac{dz}{dy}\, \dfrac{dy}{dx}$: which shows that $z$~is an indirect function +of~$x$ through~$y$. If $x$~and~$y$ had been supposed to +vary independently of each other, the increment of~$z$, +(or those terms which give the limiting ratio of this +increment to any other,) would have been $\dfrac{dz}{dx}\, dx + \dfrac{dz}{dy}\, dy$, +in which, if $dy$~had arisen from~$y$ being a function +\PageSep{98} +of~$x$, $dy$~would have been a series of the form +$p\, dx + q\, (dx)^{2} + \etc.$, of which only the differential coefficient~$p$ +would have appeared in the limit. Hence +$\dfrac{dz}{dy}\, dy$ would have given~$\dfrac{dz}{dy}\, p$, or~$\dfrac{dz}{dy}\, \dfrac{dy}{dx}$. + +(3) $\dfrac{dz}{da}\, \dfrac{da}{dy}\, \dfrac{dy}{dx}$: this arises from $z$ containing~$a$, which +contains~$y$, which contains~$x$. If $z$~had been differentiated +with respect to $a$~only, the increment would +have been represented by~$\dfrac{dz}{da}\, da$; if $da$~had arisen from +an increment of~$y$, this would have been expressed by +$\dfrac{dz}{da}\, \dfrac{da}{dy}\, dy$; if~$y$ had arisen from an increment given to~$x$, +this would have been expressed by $\dfrac{dz}{da}\, \dfrac{da}{dy}\, \dfrac{dy}{dx}\, dx$, +which, after $dx$~has been struck out, is the part of the +differential coefficient answering to that increment. + +(4) $\dfrac{dz}{da}\, \dfrac{da}{dx}$: arising from $a$~containing $x$~directly, +and $z$~therefore containing $x$ indirectly through~$a$. + +Hence $z$~is directly a function of $x$,~$y$, and~$a$, of +which $y$~is a function of~$x$, and $a$~of $y$~and~$x$. + +If we suppose $x$,~$y$ and~$a$ to vary independently, +we have +\[ +d.z = \frac{dz}{dx}\, dx + \frac{dz}{dy}\, dy + \frac{dz}{da}\, da + \etc. +\quad\text{(\PageRefs{28}{29})}. +\] +But as $a$~varies as a function of $y$~and~$x$, +\[ +da = \frac{da}{dx}\, dx + \frac{da}{dy}\, dy. +\] +If we substitute this instead of~$da$, and divide by~$dx$, +taking the limit of the ratios, we have the result first +given. + +For example, let (1) $z = x^{2} ya^{3}$, (2) $y = x^{2}$, and (3) $a = x^{3} y$. +Taking the first equation only, and substituting +\PageSep{99} +$x + dx$ for~$x$ etc., we find $\dfrac{dz}{dx} = 2xya^{3}$, $\dfrac{dz}{dy} = x^{2} a^{3}$, +and $\dfrac{dz}{da} = 3x^{2} ya^{2}$. From the second $\dfrac{dy}{dx} = 2x$, and from +the third $\dfrac{da}{dx} = 3x^{2} y$, and $\dfrac{da}{dy} = x^{3}$. Substituting these +in the value of~$\dfrac{d.z}{dx}$, we find +\begin{alignat*}{3} +%[** TN: Reformatted first line] +\frac{d.z}{dx} \text{ \ or \ } + \frac{dz}{dx} &+ \frac{dz}{dy}\, \frac{dy}{dx} + &&+ \frac{dz}{da}\, \frac{da}{dy}\, \frac{dy}{dx} + &&+ \frac{dz}{da}\, \frac{da}{dx} \\ + = 2xya^{3} &+ x^{2} a^{3} × 2x &&+ 3x^{2} ya^{2} × x^{3} × 2x &&+ 3x^{2} ya^{2} × 3x^{2} y \\ + = 2xya^{3} &+ 2x^{3} a^{3} &&+ 6x^{6} ya^{2} &&+ 9x^{4} y^{2} a^{2}\Add{.} +\end{alignat*} +If for $y$~and~$a$ in the first equation we substitute their +values $x^{2}$ and~$x^{3} y$, or~$x^{5}$, we have $z = x^{19}$, the differential +coefficient of which\Add{ is}~$19x^{18}$. This is the same as +arises from the formula just obtained, after $x^{2}$~and~$x^{5}$ +have been substituted for $y$~and~$a$; for this formula +then becomes +\[ +2x^{18} + 2x^{18} + 6x^{18} + 9x^{18} \quad\text{or}\quad 19x^{18}. +\] + +In saying that $z$~is a function of $x$~and~$y$, and that +$y$~is a function of~$x$, we have first supposed~$x$ to vary, +$y$~remaining the same. The student must not imagine +that $y$~is \emph{then} a function of~$x$; for if so, it would vary +when $x$~varied. There are two parts of the total differential +coefficient, arising from the direct and indirect +manner in which $z$ contains~$x$. That these two +parts may be obtained separately, and that their sum +constitutes the complete differential coefficient, is the +theorem we have proved. The first part~$\dfrac{dz}{dx}$ is what +\emph{would} have been obtained if $y$~had \emph{not} been a function +of~$x$; and on this supposition we therefore proceed to +find it. The other part $\dfrac{dz}{dy}\, \dfrac{dy}{dx}$ is the product (1)~of~$\dfrac{dz}{dy}$, +which would have resulted from a variation of $y$~only, +not considered as a function of~$x$; and (2)~of~$\dfrac{dy}{dx}$, +\PageSep{100} +the coefficient which arises from considering~$y$ as a +function of~$x$. These partial suppositions, however +useful in obtaining the total differential coefficient, +\index{Coefficients, differential}% +\index{Total!differential coefficient}% +cannot be separately admitted or used, except for this +purpose; since if $y$~be a function of~$x$, $x$~and~$y$ must +vary together. + +If $z$~be a function of~$x$ in various ways, the theorem +obtained may be stated as follows: + +Find the differential coefficient belonging to each +of the ways in which $z$ will contain~$x$, as if it were the +only way; the sum of these results (with their proper +signs) will be the total differential coefficient. + +Thus, if $z$~only contains $x$ indirectly through~$y$, +$\dfrac{dz}{dx}$~is $\dfrac{dz}{dy}\, \dfrac{dy}{dx}$. If $z$ contains~$a$, which contains~$b$, which +contains~$x$, $\dfrac{dz}{dx} = \dfrac{dz}{da}\, \dfrac{da}{db}\, \dfrac{db}{dx}$. + +This theorem is useful in the differentiation of complicated +\index{Differentiation!of complicated functions|EtSeq}% +functions; for example, let $z = \log(x^{2} + a^{2})$. +If we make $y = x^{2} + a^{2}$, we have $z = \log y$, and $\dfrac{dz}{dy} = \dfrac{1}{y}$; +while from the first equation $\dfrac{dy}{dx} = 2x$. Hence $\dfrac{dy}{dx}$ or +$\dfrac{dz}{dy}\, \dfrac{dy}{dx}$ is $\dfrac{2x}{y}$ or $\dfrac{2x}{x^{2} + a^{2}}$. + +%[** TN: \log\log\sin x is never real valued when x is real] +If $z = \log\log\sin x$, or the logarithm of the logarithm +of~$\sin x$, let $\sin x = y$ and $\log y = a$; whence +$z= \log a$, and contains~$x$, because $a$ contains~$y$, which +contains~$x$. Hence +\[ +\frac{dz}{dx} = \frac{dz}{da}\, \frac{da}{dy}\, \frac{dy}{dx}; +\] +but since $z = \log a$, +\[ +\frac{dz}{da} = \frac{1}{a}; +\] +\PageSep{101} +since $a = \log y$, +\[ +\frac{da}{dy} = \frac{1}{y}; +\] +and since $y = \sin x$, +\[ +\frac{dy}{dx} = \cos x. +\] +Hence +\[ +\frac{dz}{dx} = \frac{dz}{da}\, \frac{da}{dy}\, \frac{dy}{dx} + = \frac{1}{a}\, \frac{1}{y} \cos x + = \frac{\cos x}{\log\sin x \sin x}. +\] + +We now put some rules in the form of applications +of this theorem, though they may be deduced more +simply. + + +\Subsection[Applications of the Theorem for Implicit Differentiation.] +{Applications of the Preceding Theorem.} + +(1) Let $z = ab$, where $a$~and~$b$ are functions of~$x$. +The general formula, since $z$ contains~$x$ indirectly +through $a$~and~$b$, is (in this case as well as in those +which follow,) +\[ +\frac{dz}{dx} + = \frac{dz}{da}\, \frac{da}{dx} + \frac{dz}{db}\, \frac{db}{dx}. +\] + +We must leave $\dfrac{da}{dx}$ and $\dfrac{db}{dx}$ as we find them, until we +know \emph{what} functions $a$~and~$b$ are of~$x$; but as we +know what function $z$~is of $a$~and~$b$, we substitute for +$\dfrac{dz}{da}$ and~$\dfrac{dz}{db}$. Since $z = ab$, if $a$~becomes $a + da$, $z$~becomes +$ab + b\, da$, whence $\dfrac{dz}{db} = b$. In this case, and part +of the following, the limiting ratio of the increments +is the same as that of the increments themselves. +Similarly $\dfrac{dz}{db} = a$, whence from $z = ab$ follows +\[ +\frac{dz}{dx} = b\, \frac{da}{dx} + a\, \frac{db}{dx}. +\] +\PageSep{102} + +%[** TN: [sic] "become", twice] +(2) Let $z = \dfrac{a}{b}$. If $a$~become $a + da$, $z$~becomes +$\dfrac{a + da}{b}$ or $\dfrac{a}{b} + \dfrac{da}{b}$, and $\dfrac{dz}{da}$ is~$\dfrac{1}{b}$. If $b$~become $b + db$, $z$~becomes +$\dfrac{a}{b + db}$, or $\dfrac{a}{b} - \dfrac{a\, db}{b^{2}} + \etc.$, whence $\dfrac{dz}{db}$ is~$-\dfrac{a}{b^{2}}$\Add{.} +Hence from $z = \dfrac{a}{b}$ follows +\[ +\frac{dz}{dx} = \frac{1}{b}\, \frac{da}{dx} - \frac{a}{b^{2}}\, \frac{db}{dx} + = \frac{b\, \dfrac{da}{dx} - a\, \dfrac{db}{dx}}{b^{2}}. +\] + +(3) Let $z = a^{b}$. Here $(a + da)^{b} = a^{b} + ba^{b-1}\, da + \etc.$\ +(\PageRef{21}), whence $\dfrac{dz}{da} = ba^{b-1}$. Again, +$a^{b+db} = a^{b}\, a^{db} = a^{b}(1 + \log a\, db + \etc.)$ whence $\dfrac{dz}{db} = a^{b} \log a$. +Therefore from $z = a^{b}$ follows +\[ +\frac{dz}{dx} = ba^{b-1}\, \frac{da}{dx} + a^{b} \log a\, \frac{db}{dx}. +\] + + +\Subsection{Inverse Functions.} + +If $y$~be a function of~$x$, such as $y = \phi x$, we may, +\index{Functions!inverse|EtSeq}% +\index{Inverse functions|EtSeq}% +by solution of the equation, determine $x$ in terms of~$y$, +or produce another equation of the form $x = \psi y$. +For example, when $y = x^{2}$, $x = y^{\efrac{1}{2}}$. It is not necessary +that we should be able to solve the equation +$y = \phi x$ in finite terms, that is, so as to give a value +of~$x$ without infinite series; it is sufficient that $x$~can +be so expressed that the value of~$x$ corresponding to +any value of~$y$ may be found as near as we please +from $x = \psi y$, in the same manner as the value of~$y$ +corresponding to any value of~$x$ is found from $y = \phi x$. + +The equations $y = \phi x$, and $x = \psi y$, are connected, +being, in fact, the same relation in different forms; +and if the value of~$y$ from the first be substituted in +\PageSep{103} +the second, the second becomes $x = \psi(\phi x)$, or as it is +more commonly written, $\psi\phi x$. That is, the effect of +the operation or set of operations denoted by~$\psi$ is destroyed +by the effect of those denoted by~$\phi$; as in the +instances $(x^{2})^{\efrac{1}{2}}$, $(x^{3})^{\efrac{1}{3}}$, $e^{\log x}$, angle whose sine is~$(\sin x)$, +etc., each of which is equal to~$x$. + +By differentiating the first equation $y = \phi x$, we obtain +$\dfrac{dy}{dx} = \phi' x$, and from the second $\dfrac{dx}{dy} = \psi' y$. But +whatever values of $x$~and~$y$ together satisfy the first +equation, satisfy the second also; hence, if when $x$~becomes +$x + dx$ in the first, $y$~becomes $y + dy$; the same +$y + dy$ substituted for~$y$ in the second, will give the +same $x + dx$. Hence $\dfrac{dx}{dy}$ as deduced from the second, +and $\dfrac{dy}{dx}$ as deduced from the first, are reciprocals for +every value of~$dx$. The limit of one is therefore the +reciprocal of the limit of the other; the student may +easily prove that if $a$~is always equal to~$\dfrac{1}{b}$, and if $a$~continually +approaches to the limit~$\alpha$, while $b$~at the +same time approaches the limit~$\beta$, $\alpha$~is equal to~$\dfrac{1}{\beta}$. +But $\dfrac{dx}{dy}$ or $\psi' y$, deduced from $x = \psi y$, is expressed in +terms of~$y$, while $\dfrac{dy}{dx}$ or $\phi' x$, deduced from $y = \phi x$ is +expressed in terms of~$x$. Therefore $\psi' y$ and $\phi' x$ are +reciprocals for all such values of $x$~and~$y$ as satisfy +either of the two first equations. + +For example let $y = e^{x}$, from which $x = \log y$. From +the first (\PageRef{20}) $\dfrac{dy}{dx} = e^{x}$; from the second $\dfrac{dx}{dy} = \dfrac{1}{y}$; +and it is evident that $e^{x}$~and~$\dfrac{1}{y}$ are reciprocals, whenever +$y = e^{x}$. + +If we differentiate the above equations twice, we get +\PageSep{104} +$\dfrac{d^{2} y}{dx^{2}} = \phi'' x$, and $\dfrac{d^{2} x}{dy^{2}} = \psi'' x$. There is no very obvious +analogy between $\dfrac{d^{2} y}{dx^{2}}$ and $\dfrac{d^{2} x}{dy^{2}}$; indeed no such appears +from the method in which these coefficients were first +formed. Turn to the table in \PageRef{90}, and substitute +$d$ for~$\Delta$ throughout, to indicate that the increments +may be taken as small as we please. We there substitute +in~$\phi x$ what we will call a set of \emph{equidistant} values +\index{Equidistant values}% +\index{Values!equidistant}% +of~$x$, or values in arithmetical progression, viz., +$x$,~$x + dx$, $x + 2\, dx$,~etc. The resulting values of~$y$, +or $y$,~$y_{1}$, etc., are not equidistant, except in one function +only, when $y = ax + b$, where $a$~and~$b$ are constant. +Therefore $dy$,~$dy_{1}$, etc., are not equal; whence +arises the next column of second differences, or $d^{2} y$, +$d^{2} y_{1}$, etc. The limiting ratio of $d^{2} y$ to~$(dx)^{2}$, expressed +by~$\dfrac{d^{2} y}{dx^{2}}$, is the second differential coefficient of~$y$ with +respect to~$x$. If from $y = \phi x$ we deduce $x = \psi y$, and +take a set of equidistant values of~$y$, viz., $y$,~$y + dy$, +$y + 2\, dy$, etc., to which the corresponding values of~$x$ +are $x$,~$x_{1}$, $x_{2}$,~etc., a similar table may be formed, +which will give $dx$,~$dx_{1}$, etc., $d^{2} x$,~$d^{2} x_{1}$, etc., and the +limit of the ratio of~$d^{2} x$ to~$(dy)^{2}$ or $\dfrac{d^{2} x}{dy^{2}}$ is the second +differential coefficient of~$x$ with respect to~$y$. These +are entirely different suppositions, $dx$~being given in +the first table, and $dy$~varying; while in the second $dy$~is +given and $dx$~varies. We may show how to deduce +one from the other as follows: + +When, as before, $y = \phi x$ and $x = \psi y$, we have +\[ +\frac{dy}{dx} = \phi' x = \frac{1}{\psi' y} = \frac{1}{p}, +\] +if $\psi' y$ be called~$p$. Calling this~$u$, and considering it +\PageSep{105} +as a function of~$x$ from containing~$p$, which contains~$y$, +which contains~$x$, we have +\[ +\frac{du}{dp}\, \frac{dp}{dy}\, \frac{dy}{dx} +\] +for its differential coefficient with respect to~$x$. But +since +\[ +u = \frac{1}{p}, +\] +therefore +\[ +\frac{du}{dp} = -\frac{1}{p^{2}}; +\] +since $p = \psi' y$, therefore +\[ +\frac{dp}{dy} = \psi'' y; +\] +and $\psi'' y$ is the differential coefficient of~$\psi' y$, and is +$\dfrac{d^{2} x}{dy^{2}}$. Also $\dfrac{1}{p^{2}}$~is +\[ +\frac{1}{(\psi' y)^{2}} \quad\text{or}\quad +(\phi' x)^{2} \quad\text{or}\quad +\left(\frac{dy}{dx}\right)^{2}. +\] +Hence the differential coefficient of $u$ or~$\dfrac{dy}{dx}$, with respect +to~$x$, which is~$\dfrac{d^{2} y}{dx^{2}}$, is also +\[ +-\left(\frac{dy}{dx}\right)^{2} \frac{d^{2} x}{dy^{2}}\, \frac{dy}{dx} +\quad\text{or}\quad +-\left(\frac{dy}{dx}\right)^{3} \frac{d^{2} x}{dy^{2}}. +\] + +{\Loosen If $y = e^{x}$, whence $x = \log y$, we have $\dfrac{dy}{dx} = e^{x}$ and +$\dfrac{d^{2} y}{dx^{2}} = e^{x}$. But $\dfrac{dx}{dy} = \dfrac{1}{y}$ and $\dfrac{d^{2} x}{dy^{2}} = -\dfrac{1}{y^{2}}$. Therefore} +\[ +-\left(\frac{dy}{dx}\right)^{3} \frac{d^{2} x}{dy^{2}} \quad\text{is}\quad +-e^{3x} \left(-\frac{1}{y^{2}}\right) \quad\text{or}\quad +\frac{e^{3x}}{y^{2}} \quad\text{or}\quad +\frac{e^{3x}}{e^{2x}}, +\] +which is~$e^{x}$, the value just found for~$\dfrac{d^{2} y}{dx^{2}}$. +\PageSep{106} + +In the same way $\dfrac{d^{3} y}{dx^{3}}$ might be expressed in terms +of $\dfrac{dy}{dx}$, $\dfrac{d^{2} y}{dx^{2}}$, and~$\dfrac{d^{3} x}{dy^{3}}$; and so on. + + +\Subsection{Implicit Functions.} + +The variable which appears in the denominator of +the differential coefficients is called the \emph{independent} +variable. In any function, one quantity at least is +changed at pleasure; and the changes of the rest, +with the limiting ratio of the changes, follow from the +form of the function. The number of independent +variables depends upon the number of quantities +\index{Variables!independent and dependent}% +which enter into the equations, and upon the number +of equations which connect them. If there be only +one equation, all the variables except one are independent, +or may be changed at pleasure, without ceasing +to satisfy the equation; for in such a case the +common rules of algebra tell us, that as long as one +quantity is left to be determined from the rest, it can +be determined by one equation; that is, the values of +all but one are at our pleasure, it being still in our +power to satisfy one equation, by giving a proper +value to the remaining one. Similarly, if there be +two equations, all variables except two are independent, +and so on. If there be two equations with two +unknown quantities only, there are no variables; for +by algebra, a finite number of values, and a finite +number only, can satisfy these equations; whereas it +is the nature of a variable to receive any value, or at +least any value which will not give impossible values +for other variables. If then there be $m$~equations containing +$n$~variables, ($n$~must be greater than~$m$), we +have $n - m$~independent variables, to each of which +\index{Independent variables}% +\PageSep{107} +we may give what values we please, and by the equations, +deduce the values of the rest. We have thus +various sets of differential coefficients, arising out of +the various choices which we may make of independent +variables. + +If, for example, $a$,~$b$, $x$,~$y$, and~$z$, being variables, +we have +\begin{align*} +\phi(a, b, x, y, z) &= 0, \\ +\psi(a, b, x, y, z) &= 0, \\ +\chi(a, b, x, y, z) &= 0, +\end{align*} +we have two independent variables, which may be +either $x$~and~$y$, $x$~and~$z$, $a$~and~$b$, or any other combination. +If we choose $x$~and~$y$, we should determine +$a$,~$b$, and~$z$ in terms of $x$~and~$y$ from the three equations; +in which case we can obtain +\[ +\frac{da}{dx},\quad \frac{da}{dy},\quad \frac{db}{dx},\quad \etc. +\] + +When $y$~is a function of~$x$, as in $y = \phi x$, it is called +\index{Explicit functions}% +\index{Functions!implicit and explicit}% +an \emph{explicit} function of~$x$. This equation tells us not +only that $y$~is a function of~$x$, but also what function +it is. The value of~$x$ being given, nothing more is +necessary to determine the corresponding value of~$y$, +than the substitution of the value of~$x$ in the several +terms of~$\phi x$. + +But it may happen that though $y$~is a function of~$x$, +\index{Implicit!function}% +the relation between them is contained in a form +from which $y$~must be deduced by the solution of an +equation. For example, in $x^{2} - xy + y^{2} = a$, when $x$~is +known, $y$~must be determined by the solution of an +equation of the second degree. Here, though we know +that $y$~must be a function of~$x$, we do not know, without +further investigation, what function it is. In this +case $y$~is said to be \emph{implicitly} a function of~$x$, or an implicit +\PageSep{108} +function. By bringing all the terms on one side +of the equation, we may always reduce it to the form +$\phi(x, y) = 0$. Thus, in the case just cited, we have +$x^{2} - xy + y^{2} - a = 0$. + +{\Loosen We now want to deduce the differential coefficient +$\dfrac{dy}{dx}$ from an equation of the form $\phi(x, y) = 0$. If we +take the equation $u = \phi(x, y)$, in which when $x$~and~$y$ +become $x + dx$ and $y + dy$, $u$~becomes $u + du$, we have, +by our former principles,} +\[ +du = \ux\, dx + \uy\, dy + \etc., \text{(\PageRef{82})}, +\] +in which $\ux$~and~$\uy$ can be directly obtained from the +equation, as in \PageRef{82}. Here $x$~and~$y$ are independent, +as also $dx$~and~$dy$; whatever values are given to +them, it is sufficient that $u$~and~$du$ satisfy the two last +equations. But if $x$~and~$y$ must be always so taken +that $u$ may~$= 0$, (which is implied in the equation +$\phi(x, y) = 0$,) we have $u = 0$, and $du = 0$; and this, +whatever may be the values of $dx$ and~$dy$. Hence $dx$ +and~$dy$ are connected by the equation +\[ +0 = \ux\, dx + \uy\, dy + \etc., +\] +and their limiting ratio must be obtained by the equation +\[ +\ux\, dx + \uy\, dy = 0, \quad\text{or}\quad \frac{dy}{dx} = -\frac{\ux}{\uy}; +\] +{\Loosen $y$~and~$x$ are no longer independent; for, one of them +being given, the other must be so taken that the equation +$\phi(x, y) = 0$ may be satisfied. The quantities $\ux$ +\index{Functions!implicit and explicit}% +\index{Implicit!function}% +and~$\uy$ we have denoted by $\dfrac{du}{dx}$ and~$\dfrac{du}{dy}$, so that} +\[ +\frac{dy}{dx} = -\frac{\;\dfrac{du}{dx}\;}{\dfrac{du}{dy}}\Add{.} +\Tag{(1)} +\] +\PageSep{109} + +We must again call attention to the different meanings +of the same symbol~$du$ in the numerator and denominator +of the last fraction. Had $du$, $dx$, and~$dy$ +been common algebraical quantities, the first meaning +the same thing throughout, the last equation would +not have been true until the negative sign had been +removed. We will give an instance in which $du$~shall +mean the same thing in both. + +Let $u = \Chg{\phi(x)}{\phi x}$, and let $u = \psi y$, in which two equations +is implied a third $\phi x = \psi y$; and $y$~is a function +of~$x$. Here, $x$~being given, $u$~is known from the first +equation; and $u$~being known, $y$~is known from the +second. Again, $x$~and~$dx$ being given, $du$, which is +$\phi(x + dx) - \phi x$ is known, and being substituted in +the result of the second equation, we have $du = \psi(y + dy) - \psi y$, +which $dy$~must be so taken as to +satisfy. From the first equation we deduce $du = \phi'x\, dx + \etc.$\ +and from the second $du = \psi' y\, dy + \etc.$, +whence +\[ +\phi' x\, dx + \etc. = \psi' y\, dy + \etc.; +\] +the \emph{etc.}\ only containing terms which disappear in finding +the limiting ratios. Hence, +\[ +\frac{dy}{dx} = \frac{\phi' x}{\psi' y} + = \frac{\;\dfrac{du}{dx}\;}{\dfrac{du}{dy}}\Add{,} +\Tag{(2)} +\] +a result in accordance with common algebra. + +But the equation~\Eq{(1)} was obtained from $u = \phi(x, y)$, +on the supposition that $x$~and~$y$ were always so taken +that $u$ should~$= 0$, while \Eq{(2)}~was obtained from $u = \Chg{\phi(x)}{\phi x}$ +and $u = Sy$, in which no new supposition can be +made; since one more equation between $u$,~$x$, and~$y$ +would give three equations connecting these three +quantities, in which case they would cease to be variable +(\PageRef{106}). +\PageSep{110} + +As an example of~\Eq{(1)} let $xy - x = 1$, or $xy - x - 1 = 0$. +From $u = xy - x - 1$ we deduce (\PageRef{81}) +$\dfrac{du}{dx} = y - 1$, $\dfrac{du}{dy} = x$; whence, by equation~\Eq{(1)}, +\[ +\frac{dy}{dx} = -\frac{y - 1}{x}. +\Tag{(3)} +\] +By solution of $xy - x = 1$, we find $y = 1 + \dfrac{1}{x}$, and +\[ +dy = \left(1 + \frac{1}{x + dx}\right) - \left(1 + \frac{1}{x}\right) + = -\frac{dx}{x^{2}} + \etc.\footnote{See \PageRef{26}.} +\] +Hence $\dfrac{dy}{dx}$ (meaning the limit) is~$-\dfrac{1}{x^{2}}$, which will also +be the result of~\Eq{(3)} if $1 + \dfrac{1}{x}$ be substituted for~$y$. + + +\Subsection{Fluxions, and the Idea of Time.} + +To follow this subject farther would lead us beyond +\index{Time, idea of|EtSeq}% +our limits; we will therefore proceed to some +observations on the differential coefficient, which, at +this stage of his progress, may be of use to the student, +who should never take it for granted that because +he has made some progress in a science, he understands +the first principles, which are often, if not +always, the last to be learned well. If the mind were +so constituted as to receive with facility any perfectly +new idea, as soon as the same was legitimately applied +in mathematical demonstration, it would doubtless +be an advantage not to have any notion upon a +mathematical subject, previous to the time when it is +to become a subject of consideration after a strictly +mathematical method. + +This not being the case, it is a cause of embarrassment +to the student, that he is introduced at once to a +definition so refined as that of the limiting ratio which +\PageSep{111} +the increment of a function bears to the increment of +its variable. Of this he has not had that previous experience, +which is the case in regard to the words +\emph{force}, \emph{velocity}, or \emph{length}. Nevertheless, he can easily +\index{Velocity!linear}% +conceive a mathematical quantity in a state of continuous +increase or decrease, such as the distance between +two points, one of which is in motion. The +number which represents this line (reference being +made to a given linear unit) is in a corresponding +state of increase or decrease, and so is every function +of this number, or every algebraical expression in the +formation of which it is required. And the nature of +the change which takes place in the function, that is, +whether the function will increase or decrease when +the variable increases; whether that increase or decrease +corresponding to a given change in the variable +will be smaller or greater, etc., depends on the +manner in which the variable enters as a component +part of its function. + +Here we want a new word, which has not been invented +for the world at large, since none but mathematicians +consider the subject; which word, if the +change considered were change of place, depending +upon change of time, would be \emph{velocity}. Newton +adopted this word, and the corresponding idea, expressing +many numbers in succession, instead of at +once, by supposing a point to generate a straight line +by its motion, which line would at different instants +contain any different numbers of linear units. + +To this it was objected that the idea of \emph{time} is introduced, +which is foreign to the subject. We may +answer that the notion of time is only necessary, inasmuch +as we are not able to consider more than one +thing at a time. Imagine the diameter of a circle divided +\PageSep{112} +into a million of equal parts, from each of which +a perpendicular is drawn meeting the circle. A mind +which could at a view take in every one of these lines, +and compare the differences between every two contiguous +perpendiculars with one another, could, by +subdividing the diameter still further, prove those +propositions which arise from supposing a point to +move uniformly along the diameter, carrying with it +a perpendicular which lengthens or shortens itself so +as always to have one extremity on the circle. But +we, who cannot consider all these perpendiculars at +once, are obliged to take one after another. If one +perpendicular only were considered, and the differential +coefficient of that perpendicular deduced, we might +certainly appear to avoid the idea of time; but if all +the states of a function are to be considered, corresponding +to the different states of its variable, we +have no alternative, with our bounded faculties, but +to consider them in succession; and succession, disguise +it as we may, is the identical idea of time introduced +in Newton's Method of Fluxions. +\index{Fluxions}% + + +\Subsection{The Differential Coefficient Considered with Respect +to its Magnitude.} + +The differential coefficient corresponding to a particular +\index{Coefficients, differential}% +\index{Contiguous values}% +\index{Differential coefficients!as the index of the change of a function}% +\index{Logarithms|EtSeq}% +\index{Values!contiguous}% +value of the variable, is, if we may use the +phrase, the \emph{index} of the change which the function +would receive if the value of the variable were increased. +Every value of the variable, gives not only +a different value to the function, but a different quantity +of increase or decrease in passing to what we may +call \emph{contiguous} values, obtained by a given increase of +the variable. + +If, for example, we take the common logarithm of~$x$, +\PageSep{113} +and let $x$ be~$100$, we have common $\log 100 = 2$. If +$x$~be increased by~$2$, this gives common $\log 102 = 2.0086002$, +the ratio of the increment of the function +\index{Increment}% +to that of the variable being that of $.0086002$ to~$2$, or +$.0043001$. In passing from $1000$ to~$1003$, we have the +logarithms $3$ and~$3.0013009$, the above-mentioned ratio +being~$.0004336$, little more than a tenth of the +former. We do not take the increments themselves, +but the proportion they bear to the changes in the +variable which gave rise to them; so in estimating +the rate of motion of two points, we either consider +lengths described in the same time, or if that cannot +be done, we judge, not by the lengths described in +different times, but by the proportion of those lengths +to the times, or the proportions of the units which +express them. + +The above rough process, though from it some +might draw the conclusion that the logarithm of~$x$ is +increasing faster when $x = 100$ than when $x = 1000$, +is defective; for, in passing from $100$ to~$102$, the +change of the logarithm is not a sufficient index of the +change which is taking place when $x$ is~$100$; since, +for any thing we can be supposed to know to the contrary, +the logarithm might be decreasing when $x = 100$, +and might afterwards begin to increase between +$x = 100$ and $x = 102$, so as, on the whole, to cause +the increase above mentioned. The same objection +would remain good, however small the increment +might be, which we suppose $x$ to have. If, for example, +we suppose $x$ to change from $x = 100$ to $x = 100.00001$, +which increases the logarithm from~$2$ to~$2.00000004343$, +we cannot yet say but that the logarithm +may be decreasing when $x = 100$, and may begin +to increase between $x = 100$ and $x = 100.00001$. +\PageSep{114} + +In the same way, if a point is moving, so that at +the end of $1$~second it is at $3$~feet from a fixed point, +and at the end of $2$~seconds it is at $5$~feet from the +fixed point, we cannot say which way it is moving at +the end of one second. \emph{On the whole}, it increases its +distance from the fixed point in the second second; +but it is possible that at the end of the first second it +may be moving back towards the fixed point, and may +turn the contrary way during the second second. And +the same argument holds, if we attempt to ascertain +the way in which the point is moving by supposing +any finite portion to elapse after the first second. But +if on adding any interval, \emph{however small}, to the first +second, the moving point does, during that interval, +increase its distance from the fixed point, we can then +certainly say that at the end of the first second the +point is moving from the fixed point. + +On the same principle, we cannot say whether the +logarithm of~$x$ is increasing or decreasing when $x$~increases +and becomes~$100$, unless we can be sure that +any increment, however small, added to~$x$, will increase +the logarithm. Neither does the ratio of the +increment of the function to the increment of its variable +furnish any distinct idea of the change which is +taking place when the variable has attained or is passing +through a given value. For example, when $x$~passes +from $100$ to~$102$, the difference between $\log 102$ +and $\log 100$ is the united effect of all the changes +which have taken place between $x = 100$ and $x = 100\frac{1}{10}$; +$x = 100\frac{1}{10}$ and $x = 100\frac{2}{10}$, and so on. Again, +the change which takes place between $x = 100$ and +$x = 100\frac{1}{10}$ may be further compounded of those which +take place between $x = 100$ and $x = 100\frac{1}{100}$; $x = 100\frac{1}{100}$ +and $x = 100\frac{2}{100}$, and so on. The objection +\PageSep{115} +becomes of less force as the increment diminishes, +but always exists unless we take the limit of the ratio +of the increments, instead of that ratio. + +How well this answers to our previously formed +ideas on such subjects as direction, velocity, and +force, has already appeared. + + +\Subsection{The Integral Calculus.} + +We now proceed to the Integral Calculus, which +\index{Integral Calculus|EtSeq}% +is the inverse of the Differential Calculus, as will afterwards +appear. + +We have already shown, that when two functions +\emph{increase} or \emph{decrease} without limit, their \emph{ratio} may either +increase or decrease without limit, or may tend to +some finite limit. Which of these will be the case depends +upon the manner in which the functions are related +to their variable and to one another. + +This same proposition may be put in another form, +as follows: If there be two functions, the first of which +\emph{decreases} without limit, on the same supposition which +makes the second \emph{increase} without limit, the \emph{product} +of the two may either remain finite, and never exceed +a certain finite limit; or it may increase without limit, +or diminish without limit. + +For example, take $\cos\theta$ and~$\tan\theta$. As the angle~$\theta$ +\emph{approaches} a right angle, $\cos\theta$~diminishes without +limit; it is nothing when $\theta$~\emph{is} a right angle; and any +fraction being named, $\theta$~can be taken so near to a +right angle that $\cos\theta$~shall be smaller. Again, as $\theta$~approaches +to a right angle, $\tan\theta$~increases without +limit; it is called \emph{infinite} when $\theta$~is a right angle, by +which we mean that, let any number be named, however +great, $\theta$~can be taken so near a right angle that +$\tan\theta$~shall be greater. Nevertheless the product $\cos\theta × \tan\theta$, +\PageSep{116} +of which the first factor diminishes without limit, +while the second increases without limit, is always +finite, and tends towards the limit~$1$; for $\cos\theta × \tan\theta$ +is always~$\sin\theta$, which last approaches to~$1$ as $\theta$~approaches +to a right angle, and is~$1$ when $\theta$~\emph{is} a right +angle. + +Generally, if $A$~diminishes without limit at the +same time as $B$~increases without limit, the product~$AB$ +may, and often will, tend towards a finite limit. +This product~$AB$ is the representative of~$A$ divided by~$\dfrac{1}{B}$ +or the ratio of $A$ to~$\dfrac{1}{B}$. If $B$~increases without +limit, $\dfrac{1}{B}$~decreases without limit; and as $A$~also decreases +without limit, the ratio of $A$ to~$\dfrac{1}{B}$ may have a +finite limit. But it may also diminish without limit; +as in the instance of $\cos^{2}\theta × \tan\theta$, when $\theta$~approaches +to a right angle. Here $\cos^{2}\theta$~diminishes without limit, +and $\tan\theta$~increases without limit; but $\cos^{2}\theta × \tan\theta$ +being $\cos\theta × \sin\theta$, or a diminishing magnitude multiplied +by one which remains finite, diminishes without +limit. Or it may increase without limit, as in the case +of $\cos\theta × \tan^{2}\theta$, which is also $\sin\theta × \tan\theta$; which last +has one factor finite, and the other increasing without +limit. We shall soon see an instance of this. + +If we take any numbers, such as $1$~and~$2$, it is evident +that between the two we may interpose any number +of fractions, however great, either in arithmetical +progression, or according to any other law. Suppose, +for example, we wish to interpose $9$~fractions in arithmetical +progression between $1$~and~$2$. These are $1\frac{1}{10}$, +$1\frac{2}{10}$,~etc., up to~$1\frac{9}{10}$; and, generally, if $m$~fractions in +arithmetical progression be interposed between $a$~and~$a + h$, +the complete series is +\PageSep{117} +\begin{multline*} +a,\quad a + \frac{h}{m + 1},\quad + a + \frac{2h}{m + 1},\quad \etc.\Add{,} \dots\\ +\text{up to } a + \frac{mh}{m + 1},\quad a + h\Add{.} +\Tag{(1)} +\end{multline*} +The sum of these can evidently be made as great as +we please, since no one is less than the given quantity~$a$, +and the number is as great as we please. Again, +if we take~$\phi x$, any function of~$x$, and let the values +just written be successively substituted for~$x$, we shall +have the series +\begin{multline*} +\phi a,\quad \phi\left(a + \frac{h}{m + 1}\right),\quad + \phi\left(a + \frac{2h}{m + 1}\right),\quad \etc.\Add{,} \dots\\ +\text{up to } \phi(a + h); +\Tag{(2)} +\end{multline*} +the sum of which may, in many cases, also be made +as great as we please by sufficiently increasing the +number of fractions interposed, that is, by sufficiently +increasing~$m$. But though the two sums increase without +limit when $m$~increases without limit, it does not +therefore follow that their ratio increases without +limit; indeed we can show that this cannot be the +case when all the separate terms of~\Eq{(2)} remain finite. + +For let $A$~be greater than any term in~\Eq{(2)}, whence, +as there are $(m + 2)$~terms, $(m + 2)A$~is greater than +their sum. Again, every term of~\Eq{(1)}, except the first, +being greater than~$a$, and the terms being $m + 2$~in +number, $(m + 2)a$~is less than the sum of the terms in~\Eq{(1)}. +Consequently, +\[ +\frac{(m + 2)A}{(m + 2)a} + \text{ is greater than the ratio } + \frac{\text{sum of terms in~\Eq{(2)}}}{\text{sum of terms in~\Eq{(1)}}}, +\] +since its numerator is greater than the last numerator, +and its denominator less than the last denominator. +But +\PageSep{118} +\[ +\frac{(m + 2)A}{(m + 2)a} = \frac{A}{a}, +\] +which is independent of~$m$, and is a finite quantity. +Hence the ratio of the sums of the terms is always +finite, whatever may be the number of terms, at least +unless the terms in~\Eq{(2)} increase without limit. + +As the number of interposed values increases, the +interval or difference between them diminishes; if, +therefore, we multiply this difference by the sum of +the values, or form +\begin{multline*} +\frac{h}{m + 1} \Biggl[ + \phi a + \phi\left(a + \frac{h}{m + 1}\right) + \\ + \phi\left(a + \frac{2h}{m + 1}\right) \Add{+} \dots + \phi(a + h) +\Biggr]\Add{,} +\end{multline*} +we have a product, one term of which diminishes, and +the other increases, when $m$~is increased. The product +\emph{may} therefore remain finite, or never pass a certain +limit, when $m$~is increased without limit, and we +shall show that this \emph{is} the case. + +As an example, let the given function of~$x$ be~$x^{2}$, +and let the intermediate values of~$x$ be interposed between +$x = a$ and $x = a + h$. Let $v = \dfrac{h}{m + 1}$, whence +the above-mentioned product is +\begin{multline*}%[** TN: Re-formatted from the original] +v\bigl\{a^{2} + (a + v)^{2} + (a + 2v)^{2} + \dots + + \bigl(a + (m + 1)v\bigr)^{2}\bigr\} \\ + = (m + 2)va^{2} + 2av^{2} \{1 + 2 + 3 + \dots + (m + 1)\} \\ + + v^{3} \{1^{2} + 2^{2} + 3^{2} + \dots + (m + 1)^{2}\}; +\end{multline*} +{\Loosen of which, $1 + 2 + \dots + (m + 1) = \frac{1}{2}(m + 1)(m + 2)$ +and (\PageRef{73}), $1^{2} + 2^{2} + \dots + (m + 1)^{2}$ approaches +without limit to a ratio of equality with $\frac{1}{3}(m + 1)^{3}$, +when $m$~is increased without limit. Hence this last +sum may be put under the form $\frac{1}{3}(m + 1)^{3} (1 + \alpha)$, +\PageSep{119} +where $\alpha$~diminishes without limit when $m$~is increased +without limit. Making these substitutions, and putting +for~$v$ its value $\dfrac{h}{m + 1}$, the above expression becomes} +\[ +\frac{m + 2}{m + 1}\, ha^{2} + \frac{m + 2}{m + 1}\, ha^{2} + + (1 + \alpha)\, \frac{h^{3}}{3}, +\] +in which $\dfrac{m + 2}{m + 1}$ has the limit~$1$ when $m$~increases without +limit, and $1 + \alpha$~has also the limit~$1$, since, in that +case, $\alpha$~diminishes without limit. Therefore the limit +of the last expression is +\[ +ha^{2} + ha^{2} + \frac{h^{3}}{3} \quad\text{or}\quad \frac{(a + h)^{3} - a^{3}}{3}. +\] + +{\Loosen This result may be stated as follows: If the variable~$x$, +setting out from a value~$a$, becomes successively +$a + dx$, $a + 2\,dx$, etc., until the total increment +is~$h$, the smaller $dx$ is taken, the more nearly will the +sum of all the values of~$x^{2}\, dx$, or $a^{2}\, dx + (a + dx)^{2}\, dx + (a + 2\, dx)^{2}\, dx + \etc.$, +be equal to} +\[ +\frac{(a + h)^{3} - a^{3}}{3}, +\] +and to this the aforesaid sum may be brought within +any given degree of nearness, by taking $dx$ sufficiently +small. + +This result is called the \emph{integral} of~$x^{2}\, dx$, between +\index{Integral Calculus!notation of}% +\index{Integrals!definition of|EtSeq}% +\index{Notation!of the Integral Calculus}% +the limits $a$~and~$a + h$, and is written $\int x^{2}\, dx$, when it +is not necessary to specify the limits, $\int_{a}^{a+h} x^{2}\, dx$, +or\footnote + {This notation $\int x^{2}\, dx\Ibar_{a}^{a+h}$ appears to me to avoid the objections which + may be raised against $\int_{a+h}^{a} x^{2}\, dx$ as contrary to analogy, which would require + that $\int^{2} x^{2}\, dx^{2}$ should stand for the second integral of~$x^{2}\, dx$. It will be found + convenient in such integrals as $\int z\, dx\Ibar_{b}^{a}\, dy\Ibar_{0}^{\phi x}$. There is as yet no general agreement + on this point of notation.---\textit{De~Morgan}, 1832.} +$\int x^{2}\, dx\Ibar_{a}^{a+h}$, or $\int x^{2}\, dx\Ibar_{x=a}^{x=a+h}$ in the contrary case. We +\PageSep{120} +now proceed to show the connexion of this process +with the principles of the Differential Calculus. + + +\Subsection{Connexion of the Integral with the Differential +Calculus.} + +Let $x$ have the successive values $a$, $a + dx$, $a + 2\, dx$, +etc.,~\dots\ up to $a + m\, dx$, or $a + h$, $h$~being a given +quantity, and $dx$ the $m$\th~part of~$h$, so that as $m$~is increased +without limit, $dx$~is diminished without limit. +Develop the successive values $\phi x$, or $\phi a$, $\phi(a + dx)$\Add{,}~\dots\ +(\PageRef{21}),\par +{\footnotesize\begin{alignat*}{6} +& \phi a &&= \phi a\Add{,} \\ +&\phi(a + dx) &&= \phi a &&+ \phi' a\, dx + &&+ \phi'' a\, \frac{(dx)^{2}}{2} + &&+ \phi''' a\, \frac{(dx)^{3}}{2·3} &&+ \etc.\Add{,} \displaybreak[0] \\ +&\phi(a + 2\, dx) &&= \phi a &&+ \phi' a\, 2\, dx + &&+ \phi'' a\, \frac{(2\, dx)^{2}}{2} + &&+ \phi''' a\, \frac{(2\, dx)^{3}}{2·3} &&+ \etc.\Add{,} \displaybreak[0] \\ +&\phi(a + 3\, dx) &&= \phi a &&+ \phi' a\, 3\, dx + &&+ \phi'' a\, \frac{(3\, dx)^{2}}{2} + &&+ \phi''' a\, \frac{(3\, dx)^{3}}{2·3} &&+ \etc.\Add{,} \\ +\DotRow{12} \\ +&\phi(a + m\, dx) &&= \phi a &&+ \phi' a\, m\, dx + &&+ \phi'' a\, \frac{(m\, dx)^{2}}{2} + &&+ \phi''' a\, \frac{(m\, dx)^{3}}{2·3} &&+ \etc. +\end{alignat*}}% +If we multiply each development by~$dx$ and add the +results, we have a series made up of the following +terms, arising from the different columns, +\begin{alignat*}{7} +&\phi a &&×{} && && && && && \phantom{()}m\, dx\Add{,} \\ +&\phi' a &&× (1 &&+ 2 &&+ 3 &&+ \dots &&+ m &&)\, (dx)^{2}\Add{,} \\ +&\phi'' a &&× (1^{2} &&+ 2^{2} &&+ 3^{2} &&+ \dots &&+ m^{2} &&)\, \frac{(dx)^{3}}{2}\Add{,} \\ +&\phi''' a &&× (1^{3} &&+ 2^{3} &&+ 3^{3} &&+ \dots &&+ m^{3} &&)\, \frac{(dx)^{4}}{2·3} +\quad \etc.\Add{,} +\end{alignat*} +and, as in the last example, we may represent (\PageRef{73}), +\begin{alignat*}{6} +&1 &&+ 2 &&+ 3 &&+ \dots &&+ m &&\text{by}\quad + \tfrac{1}{2}m^{2}(1 + \alpha)\Add{,} \displaybreak[0] \\ +&1^{2} &&+ 2^{2} &&+ 3^{2} &&+ \dots &&+ m^{2} &&\text{by}\quad + \tfrac{1}{3}m^{3}(1 + \beta)\Add{,} \displaybreak[0] \\ +&1^{3} &&+ 2^{3} &&+ 3^{3} &&+ \dots &&+ m^{3}\quad &&\text{by}\quad + \tfrac{1}{4}m^{4}(1 + \gamma) +\quad \etc.\Add{,} +\end{alignat*} +\PageSep{121} +where $\alpha$,~$\beta$,~$\gamma$, etc., diminish without limit, when $m$~is +increased without limit. If we substitute these values, +and also put $\dfrac{h}{m}$ instead of~$dx$, we have, for the +sum of the terms, +\begin{align*} +\phi a\, h + \phi' a\, \frac{h^{2}}{2} (1 + \alpha) + &+ \phi'' a\, \frac{h^{3}}{2·3} (1 + \beta) \\ + &+ \phi''' a\, \frac{h^{4}}{2·3·4} (1 + \gamma) + \etc. +\end{align*} +which, when $m$~is increased without limit, in consequence +of which $\alpha$,~$\beta$,~etc., diminish without limit, +continually approaches to +\[ +\phi a\, h + \phi' a\, \frac{h^{2}}{2} + + \phi'' a\, \frac{h^{3}}{2·3} + + \phi''' a\, \frac{h^{4}}{2·3·4} + \etc.\Add{,} +\] +which is the limit arising from supposing $x$ to increase +from~$a$ through $a + dx$, $a + 2\, dx$, etc., up to~$a + h$, +multiplying every value of~$\phi x$ so obtained by~$dx$, summing +the results, and decreasing~$dx$ without limit. + +This is the integral of $\phi x\, dx$ from $x = a$ to $x = a + h$. +\index{Integrals!relations between differential coefficients and}% +It is evident that this series bears a great resemblance +to the development in \PageRef{21}, deprived +of its first term. Let us suppose that $\psi a$~is the function +of which $\phi a$~is the differential coefficient, that is, +that $\psi' a = \phi a$. These two functions being the same, +their differential coefficients will be the same, that is, +$\psi'' a = \phi' a$. Similarly $\psi''' a = \phi'' a$, and so on. Substituting +these, the above series becomes +\[ +\psi' a\, h + \psi'' a\, \frac{h^{2}}{2} + + \psi''' a\, \frac{h^{3}}{2·3} + + \psi^{\text{iv}} a\, \frac{h^{4}}{2·3·4} + + \etc.\Add{,} +\] +{\Loosen which is (\PageRef{21}) the same as $\psi(a + h) - \psi a$. That +is, the integral of $\phi x\, dx$ between the limits $a$~and~$a + h$, +is $\psi(a + h) - \psi a$, where $\psi x$~is the function, which, +\PageSep{122} +when differentiated, gives~$\phi x$. For $a + h$ we may +write~$b$, so that $\psi b - \psi a$ is the integral of~$\phi x\, dx$ from +$x = a$ to $x = b$. Or we may make the second limit indefinite +by writing~$x$ instead of~$b$, which gives $\psi x - \psi a$, +which is said to be the integral of~$\phi x\, dx$, beginning +when $x = a$, the summation being supposed to be continued +from $x = a$ until $x$~has the value which it may +be convenient to give it.} + + +\Subsection{Nature of Integration.} + +Hence results a new branch of the inquiry, the reverse +of the Differential Calculus, the object of which +is, not to find the differential coefficient, having given +the function, but to find the function, having given +the differential coefficient. This is called the Integral +Calculus. + +From the definition given, it is obvious that the +value of an integral is not to be determined, unless +we know the values of~$x$ corresponding to the beginning +and end of the summation, whose limit furnishes +the integral. We might, instead of defining the integral +in the manner above stated, have made the +word mean merely the converse of the differential coefficient; +thus, if $\phi x$~be the differential coefficient of~$\psi x$, +$\psi x$~might have been called the integral of~$\phi x\, dx$. +We should then have had to show that the integral, +thus defined, is equivalent to the limit of the summation +already explained. We have preferred bringing +the former method before the student first, as it is +most analogous to the manner in which he will deduce +integrals in questions of geometry or mechanics. +\index{Integrals!indefinite}% + +With the last-mentioned definition, it is also obvious +that every function has an unlimited number of +integrals. For whatever differential coefficient~$\psi x$ +\PageSep{123} +gives, $C + \psi x$ will give the same, if $C$~be a constant, +that is, not varying when $x$~varies. In this case, if $x$ +become $x + h$, $C + \psi x$ becomes $C + \psi x + \psi' x\, h + \etc.$, +from which the subtraction of the original form $C + \psi x$ +gives $\psi' x\, h + \etc.$; whence, by the process in \PageRef{23}, +$\psi' x$~is the differential coefficient of $C + \psi' x$ as well as +of~$\psi x$. As many values, therefore, positive or negative, +as can be given to~$C$, so many different integrals +\index{Integrals!indefinite}% +can be found for~$\psi' x$; and these answer to the various +limits between which the summation in our original +definition may be made. To make this problem definite, +not only $\psi' x$ the function to be integrated, must +be given, but also that value of~$x$ from which the summation +is to begin. If this be~$a$, the integral of~$\psi' x$ is, +as before determined, $\psi x - \psi a$, and $C = -\psi a$. We +may afterwards end at any value of~$x$ which we please. +If $x = a$, $\psi x - \psi a = 0$, as is evident also from the +formation of the integral. We may thus, having given +an integral in terms of~$x$, find the value at which it +began, by equating the integral to zero, and finding +the value of~$x$. Thus, since $x^{2}$, when differentiated, +gives~$2x$, $x^{2}$~is the integral of~$2x$, beginning at $x = 0$; +and $x^{2} - 4$~is the integral beginning at~$x = 2$. + +In the language of Leibnitz, an integral would be +\index{Leibnitz}% +the sum of an infinite number of infinitely small quantities, +which are the differentials or infinitely small increments +of a function. Thus, a circle being, according +to him, a rectilinear polygon of an infinite number +of infinitely small sides, the sum of these would be +the circumference of the figure. As before (\PageRefs{13}{14}, +\PageNo{38}~et~seq., \PageNo{48}~et~seq.) we proceed to interpret +this inaccuracy of language. If, in a circle, we successively +describe regular polygons of $3$,~$4$, $5$,~$6$,~etc., +sides, we may, by this means, at last attain to a polygon +\PageSep{124} +whose side shall differ from the arc of which it is +the chord, by as small a fraction, either of the chord +or arc, as we please (\PageRefs{7}{11}). That is, $A$~being +the arc, $C$~the chord, and $D$~their difference, there is +no fraction so small that $D$~cannot be made a smaller +part of~$C$. Hence, if $m$~be the number of sides of the +polygon, $mC + mD$ or $mA$ is the real circumference; +and since $mD$~is the same part of~$mC$, which $D$~is of~$C$, +$mD$~may be made as small a part of~$mC$ as we please; +so that $mC$, or the sum of all the sides of the polygon, +can be made as nearly equal to the circumference as +we please. + +As in other cases, the expressions of Leibnitz are +\index{Leibnitz}% +the most convenient and the shortest, for all who can +immediately put a rational construction upon them; +this, and the fact that, good or bad, they have been, +and are, used in the works of Lagrange, Laplace, +\index{Lagrange}% +\index{Laplace}% +Euler, and many others, which the student who really +\index{Euler}% +desires to know the present state of physical science, +cannot dispense with, must be our excuse for continually +bringing before him modes of speech, which, +taken quite literally, are absurd. + + +\Subsection{Determination of Curvilinear Areas. The Parabola.} + +We will now suppose such a part of a curve, each +\index{Curvilinear areas, determination of|EtSeq}% +\index{Parabola, the|EtSeq}% +ordinate of which is a given function of the corresponding +abscissa, as lies between two given ordinates; +for example,~$MPP'M'$. Divide the line~$MM'$ +into a number of equal parts, which we may suppose +as great as we please, and construct \Fig[Figure]{10}. Let +$O$~be the origin of co-ordinates, and let $OM$, the value +of~$x$, at which we begin, be~$a$; and $OM'$, the value +at which we end, be~$b$. Though we have only divided~$MM'$ +\PageSep{125} +into four equal parts in the figure, the reasoning +to which we proceed would apply equally, had we divided +it into four million of parts. The sum of the +parallelograms $Mr$,~$mr$,~$m'r''$, and~$m''R$, is less than +the area~$MPP'M'$, the value of which it is our object +to investigate, by the sum of the curvilinear triangles +$Prp$,~$pr'p'$,~$p'r''p''$, and~$p''RP'$. The sum of these triangles +is less than the sum of the parallelograms $Qr$,~$qr'$,~$q'r''$, +and~$q''R$; but these parallelograms are together +\Figure{10} +equal to the parallelogram~$q''w$, as appears by +inspection of the figure, since the base of each of the +above-mentioned parallelograms is equal to~$m''M$, or~$q''P'$, +and the altitude~$P'w$ is equal to the sum of the +altitudes of the same parallelograms. Hence the sum +of the parallelograms $Mr$,~$mr'$,~$m'r''$, and~$m''R$, differs +from the curvilinear area~$MPP'M'$ by less than the +parallelogram~$q''w$. But this last parallelogram may +be made as small as we please by sufficiently increasing +the number of parts into which $MM'$~is divided; +\PageSep{126} +for since one side of it,~$P'w$, is always less than~$P'M'$, +and the other side~$P'q''$, or~$m''M'$, is as small a part as +we please of~$MM'$ the number of square units in~$q''w$, +is the product of the number of linear units in $P'w$ +and~$P'q''$, the first of which numbers being finite, and +the second as small as we please, the product is +as small as we please. Hence the curvilinear area~$MPP'M'$ +is the limit towards which we continually +approach, but which we never reach, by dividing $MM'$ +into a greater and greater number of equal parts, and +adding the parallelograms $Mr$,~$mr'$,~etc., so obtained. +If each of the equal parts into which $MM'$ is divided +be called~$dx$, we have $OM = a$, $Om = a + dx$, $Om' = a + 2\,dx$, +etc. And $MP$,~$mp$, $m'p'$,~etc., are the values +of the function which expresses the ordinates, corresponding +to $a$,~$a + dx$, $a + 2\, dx$,~etc., and may therefore +be represented by $\phi a$, $\phi(a + dx)$, $\phi(a + 2\, dx)$, +etc. These are the altitudes of a set of parallelograms, +the base of each of which is~$dx$; hence the +sum of their area is +\[ +\phi a\, dx + \phi(a + dx)\, dx + \phi(a + 2\, dx)\, dx + \etc., +\] +and the limit of this, to which we approach by diminishing~$dx$, +is the area required. + +This limit is what we have defined to be the integral +of~$\phi x\, dx$ from $x = a$ to $x = b$; or if $\psi x$~be the +function, which, when differentiated, gives $\phi x$, it is +$\psi b - \psi a$. Hence, $y$~being the ordinate, the area included +between the axis of~$x$, any two values of~$y$, and +the portion of the curve they cut off, is $\int y\, dx$, beginning +at the one ordinate and ending at the other. + +Suppose that the curve is a part of a parabola +of which $O$~is the vertex, and whose equation\footnote + {If the student has not any acquaintance with the conic sections, he must + nevertheless be aware that there is some curve whose abscissa and ordinate + are connected by the equation $y^{2} = px$. This, to him, must be the definition + of \emph{parabola}: by which word he must understand, a curve whose equation is + $y^{2} = px$.} +is +\PageSep{127} +therefore $y^{2} = px$ where $p$~is the double ordinate which +passes through the focus. Here $y = p^{\efrac{1}{2}} x^{\efrac{1}{2}}$, and we +must find the integral of~$p^{\efrac{1}{2}} x^{\efrac{1}{2}}\, dx$, or the function +whose differential coefficient is~$p^{\efrac{1}{2}} x^{\efrac{1}{2}}$, $p^{\efrac{1}{2}}$~being a constant. +If we take the function~$cx^{n}$, $c$~being independent +of~$x$, and substitute $x + h$ for~$x$, we have for the +development $cx^{n} + cnx^{n-1}\, h + \etc$. Hence the differential +coefficient of~$cx^{n}$ is~$cnx^{n-1}$; and as $c$~and~$n$ may +be any numbers or fractions we please, we may take +them such that $cn$~shall $= p^{\efrac{1}{2}}$ and $n - 1 = \frac{1}{2}$, in which +case $n = \frac{3}{2}$ and $c = \frac{2}{3}p^{\efrac{1}{2}}$. Therefore the differential coefficient +of~$\frac{2}{3} p^{\efrac{1}{2}} x^{\efrac{3}{2}}$ is~$p^{\efrac{1}{2}} x^{\efrac{1}{2}}$, and conversely, the integral +of~$p^{\efrac{1}{2}} x^{\efrac{1}{2}}\, dx$ is~$\frac{2}{3} p^{\efrac{1}{2}} x^{\efrac{3}{2}}$. + +{\Loosen The area~$MPP'M'$ of the parabola is therefore +\index{Parabola, the}% +~$\frac{2}{3} p^{\efrac{1}{2}} b^{\efrac{3}{2}} - \frac{2}{3} p^{\efrac{1}{2}} a^{\efrac{3}{2}}$. If we begin the integral at the vertex~$O$, +in which case $a = 0$, we have for the area~$OM'P'$, +$\frac{2}{3} p^{\efrac{1}{2}} b^{\efrac{3}{2}}$, where $b = OM'$. This is~$\frac{2}{3} p^{\efrac{1}{2}} b^{\efrac{1}{2}} × b$, which, since +$p^{\efrac{1}{2}} b^{\efrac{1}{2}} = M'P'$ is $\frac{2}{3}P'M' × OM'$, or two-thirds of the rectangle\footnote + {This proposition is famous as having been discovered by Archimedes +\index{Archimedes}% + at a time when such a step was one of no small magnitude.} +contained by $OM'$~and~$M'P'$.} + + +\Subsection{Method of Indivisibles.} + +We may mention, in illustration of the preceding +\index{Indivisibles!method of|EtSeq}% +problem, a method of establishing the principles of +the Integral Calculus, which generally goes by the +name of the \emph{Method of Indivisibles}. A line is considered +as the sum of an infinite number of points, a +surface of an infinite number of lines, and a solid of +an infinite number of surfaces. One line twice as long +as another would be said to contain twice as many +\PageSep{128} +points, though the number of points in each is unlimited. +To this there are two objections. First, the +word infinite, in this absolute sense, really has no +\index{Infinite@\emph{Infinite}, the word}% +meaning, since it will be admitted that the mind has +no conception of a number greater than any number. +The word infinite\footnote + {See \Title{Study of Mathematics} (Chicago: The Open Court Publishing~Co\Add{.}), + page~123 et~seq.} +can only be justifiably used as an +abbreviation of a distinct and intelligible proposition; +for example, when we say that $a + \dfrac{1}{x}$ is equal to~$a$ +when $x$~is infinite, we only mean that as $x$ is increased, +$a + \dfrac{1}{x}$~becomes nearer to~$a$, and may be made as near +to it as we please, if $x$~may be as great as we please. +The second objection is, that the notion of a line +being the sum of a number of points is not true, nor +does it approach nearer the truth as we increase the +number of points. If twenty points be taken on a +straight line, the sum of the twenty-one lines which +lie between point and point is equal to the whole line; +which cannot be if the points by themselves constitute +any part of the line, however small. Nor will the sum +of the points be a part of the line, if twenty thousand +be taken instead of twenty. There is then, in this +method, neither the rigor of geometry, nor that approach +to truth, which, in the method of Leibnitz, +\index{Leibnitz}% +may be carried to any extent we please, short of absolute +correctness. We would therefore recommend to +the student not to regard any proposition derived +from this method as true on that account; for falsehoods, +as well as truths, may be deduced from it. Indeed, +the primary notion, that the number of points +in a line is proportional to its length, is manifestly incorrect. +Suppose (\Fig{6}, \PageRef{48}) that the point~$Q$ +\PageSep{129} +moves from $A$ to~$P$. It is evident that in whatever +number of points $OQ$ cuts~$AP$, it cuts~$MP$ in the same +number. But $PM$~and~$PA$ are not equal. A defender +of the system of indivisibles, if there were such a person, +\index{Indivisibles!notion of, in mechanics|EtSeq}% +would say something equivalent to supposing +that the points on the two lines are of \emph{different sizes}, +which would, in fact, be an abandonment of the +method, and an adoption of the idea of Leibnitz, using +\index{Leibnitz}% +the word \emph{point} to stand for the infinitely small +\index{Point@\emph{Point}, the word}% +line. + +This notion of indivisibles, or at least a way of +speaking which looks like it, prevails in many works +on mechanics. Though a point is not treated as a +length, or as any part of space whatever, it is considered +as having weight; and two points are spoken of +as having different weights. The same is said of a +line and a surface, neither of which can correctly be +supposed to possess weight. If a solid be of the same +density throughout, that is, if the weight of a cubic +inch of it be the same from whatever part it is cut, it +is plain that the weight may be found by finding the +number of cubic inches in the whole, and multiplying +this number by the weight of one cubic inch. But if +the weight of every two cubic inches is different, we +can only find the weight of the whole by the integral +calculus. + +Let $AB$ (\Fig{11}) be a line possessing weight, or +\index{Points, the number of, in a straight line}% +a very thin parallelepiped of matter, which is such, +that if we were to divide it into any number of equal +parts, as in the figure, the weight of the several parts +would be different. We suppose the weight to vary +continuously, that is, if two contiguous parts of equal +length be taken, as $pq$~and~$qr$, the ratio of the weights +\PageSep{130} +of these two parts may, by taking them sufficiently +small, be as near to equality as we please. + +The \emph{density} of a body is a mathematical term, which +\index{Density, continuously varying|EtSeq}% +\index{Specific gravity, continuously varying|EtSeq}% +may be explained as follows: A cubic inch of gold +weighs more than a cubic inch of water; hence gold +is \emph{denser} than water. If the first weighs $19$~times as +much as the second, gold is said to be $19$~times more +dense than water, or the density of gold is $19$~times +that of water. Hence we might define the density by +the weight of a cubic inch of the substance, but it is +usual to take, not this weight, but the proportion +which it bears to the same weight of water. Thus, +when we say the \emph{density}, or \emph{specific gravity} (these terms +are used indifferently), of cast iron is~$7.207$, we mean +\index{Iron bar continually varying in density, weight of|EtSeq}% +\index{Weight of an iron bar of which the density varies from point to point|EtSeq}% +that if any vessel of pure water were emptied and +filled with cast iron, the iron would weigh $7.207$~times +as much as the water. + +If the density of a body were uniform throughout, +we might easily determine it by dividing the weight +of any bulk of the body, by the weight of an equal +bulk of water. In the same manner (\PageRef[pages]{52} et~seq.)\ +we could, from our definition of velocity, determine +any uniform velocity by dividing the length described +by the time. But if the density vary continuously, +no such measure can be adopted. For if by the side +of~$AB$ (which we will suppose to be of iron) we placed +a similar body of water similarly divided, and if we +divided the weight of the part~$pq$ of iron by the weight +of the same part of water, we should get different +densities, according as the part~$pq$ is longer or shorter. +The water is supposed to be homogeneous, that is, +any part of it~$pr$, being twice the length of~$pq$, is twice +the weight of~$pq$, and so on. The iron, on the contrary, +being supposed to vary in density, the doubling +\PageSep{131} +the length gives either more or less than twice the +weight. But if we suppose $q$ to move towards~$p$, both +on the iron and the water, the limit of the ratio~$pq$ of +iron to $pq$~of water, may be chosen as a measure of +the density of~$p$, on the same principle as in \PageRefs{54}{55}, +the limit of the ratio of the length described to +the time of describing it, was called the velocity. If +we call $k$ this limit, and if the weight varies continuously, +though no part~$pq$, however small, of iron, +would be exactly $k$~times the same part of water in +weight, we may nevertheless take $pq$ so small that +these weights shall be as nearly as we please in the +ratio of $k$~to~$1$. + +Let us now suppose that this density, expressed +by the limiting ratio aforesaid, is always $x^{2}$ at any +\Figure{11} +point whose distance from~$A$ is $x$~feet; that is, the +density at~$q$, $2$~feet distance from~$A$, is~$4$, and so on. +Let the whole distance $AB = a$. If we divide~$a$ into +$n$~equal parts, each of which is~$dx$, so that $n\, dx = a$, +and if we call~$b$ the area of the section of the parallelepiped, +($b$~being a fraction of a square foot,) the +solid content of each of the parts will be $b\, dx$ in +cubic feet; and if $w$~be the weight of a cubic foot of +water, the weight of the same bulk of water will be~$wb\, dx$. +If the solid~$AB$ were homogeneous in the immediate +neighborhood of the point~$p$, the density being +then~$x^{2}$, would give $x^{2} × bw\, dx$ for the weight of the +same part of the substance. This is not true, but can +be brought as near to the truth as we please, by taking +$dx$ sufficiently small, or dividing~$AB$ into a sufficient +\PageSep{132} +number of parts. Hence the real weight of~$pq$ +may be represented by $bwx^{2}\, dx + \alpha$, where $\alpha$~may be +made as small a part as we please of the term which +precedes it. + +In the sum of any number of these terms, the sum +arising from the term~$\alpha$ diminishes without limit as +compared with the sum arising from the term~$bwx^{2}\, dx$; +for if $\alpha$~be less than the thousandth part of~$p$, $\alpha'$~less +than the thousandth part of~$p'$, etc., then $\alpha + \alpha' + \etc$.\ +will be less than the thousandth part of~$p + p' + \etc.$: +which is also true of any number of quantities, and of +any fraction, however small, which each term of one +set is of its corresponding term in the other. Hence +the taking of the integral of~$bwx^{2}\, dx$ dispenses with +the necessity of considering the term~$\alpha$; for in taking +the integral, we find a limit which supposes $dx$ to +have decreased without limit, and the \emph{integral} which +would arise from~$\alpha$ has therefore diminished without +limit. + +The integral of~$bwx^{2}\, dx$ is~$\frac{1}{3}bwx^{3}$, which taken from +$x = 0$ to $x = a$ is~$\frac{1}{3}bwa^{3}$. This is therefore the weight +in pounds of the bar whose length is $a$~feet, and whose +section is $b$~square feet, when the density at any point +distant by $x$~feet from the beginning is~$x^{2}$; $w$~being +the weight in pounds of a cubic foot of water. + + +\Subsection{Concluding Remarks on the Study of the Calculus.} + +We would recommend it to the student, in pursuing +\index{Advice for studying the Calculus}% +\index{Approximate solutions in the Integral Calculus}% +\index{Rough methods of solution in the Integral Calculus}% +any problem of the Integral Calculus, never for +one moment to lose sight of the manner in which he +would do it, if a rough solution for practical purposes +only were required. Thus, if he has the area of a +curve to find, instead of merely saying that~$y$, the +ordinate, being a certain function of the abscissa~$x$, +\PageSep{133} +$\int y\, dx$ within the given limits would be the area required; +and then proceeding to the mechanical solution +of the question: let him remark that if an approximate +solution only were required, it might be +obtained by dividing the curvilinear area into a number +of four-sided figures, as in \Fig[Figure]{10}, one side of +which only is curvilinear, and embracing so small an +arc that it may, without visible error, be considered +as rectilinear. The mathematical method begins with +the same principle, investigating upon this supposition, +not the sum of these rectilinear areas, but the +limit towards which this sum approaches, as the subdivision +is rendered more minute. This limit is shown +to be that of which we are in search, since it is proved +that the error diminishes without limit, as the subdivision +is indefinitely continued. + +We now leave our reader to any elementary work +which may fall in his way, having done our best to +place before him those considerations, something +equivalent to which he must turn over in his mind before +he can understand the subject. The method so +generally followed in our elementary works, of leading +the student at once into the mechanical processes +of the science, postponing entirely all other considerations, +is to many students a source of obscurity at +least, if not an absolute impediment to their progress; +since they cannot imagine what is the object of that +which they are required to do. That they shall understand +everything contained in these treatises, on +the first or second reading, we cannot promise; but +that the want of illustration and the preponderance of +\emph{technical} reasoning are the great causes of the difficulties +which students experience, is the opinion of many +\index{Advice for studying the Calculus}% +\index{Approximate solutions in the Integral Calculus}% +\index{Rough methods of solution in the Integral Calculus}% +who have had experience in teaching this subject. +\PageSep{134} +%[Blank page] +\PageSep{135} +\BackMatter + +\Section[Bibliography of Standard Text-books and Works of Reference on the Calculus] +{Brief Bibliography.\protect\footnotemark} + +\footnotetext{The information given regarding the works mentioned in this list is designed + to enable the reader to select the books which are best suited to his + needs and his purse. Where the titles do not sufficiently indicate the character + of the books, a note or extract from the Preface has been added. The + American prices have been supplied by Messrs.\ Lemcke \&~Buechner, 812~Broadway, + New~York, through whom the purchases, especially of the foreign + books, may be conveniently made.---\Ed.} + +\BibSect{Standard Text-books and Treatises on +the Calculus.} + +\BibSubsect{English.} + +\begin{Book} +Perry, John: \Title{Calculus for Engineers.} Second edition, London +and New York: Edward Arnold. 1897. Price, \Price{7s. 6d.} (\$2.50). + +\begin{Descrip} +Extract from Author's Preface: ``This book describes what has +for many years been the most important part of the regular course in +the Calculus for Mechanical and Electrical Engineering students at +the Finsbury Technical College. The students in October knew only +the most elementary mathematics, many of them did not know the +Binomial Theorem, or the definition of the sine of an angle. In July +they had not only done the work of this book, but their knowledge +was of a practical kind, ready for use in any such engineering problems +as I give here.'' + +Especially good in the character and number of practical examples +given. +\end{Descrip} +\end{Book} + +\begin{Book} +Lamb, Horace: \Title{Infinitesimal Calculus.} New York: The Macmillan +Co. 1898. Price,~\$3.00. + +\begin{Descrip} +Extract from Author's Preface: ``This book attempts to teach +those portions of the Calculus which are of primary importance in +the application to such subjects as Physics and Engineering\dots. +Stress is laid on fundamental principles\dots. Considerable attention +has been paid to the logic of the subject.'' +\end{Descrip} +\end{Book} +\PageSep{136} + +\begin{Book} +Edwards, Joseph: \Title{An Elementary Treatise on the Differential +Calculus.} Second edition, revised. 8vo,~cloth. New York +and London: The Macmillan~Co. 1892. Price, \$3.50.---% +\Title{Differential Calculus for Beginners.} 8vo,~cloth. 1893.\Chg{ }{---}\Title{The +Integral Calculus for Beginners.} 8vo,~cloth. (Same Publishers.) +Price, \$1.10~each. +\end{Book} + +\begin{Book} +Byerly, William E.: \Title{Elements of the Differential Calculus.} Boston: +Ginn \&~Co. Price, \$2.15.---\Title{Elements of the Integral +Calculus.} (Same Publishers.) Price,~\$2.15. +\end{Book} + +\begin{Book} +Rice, J.~M., and Johnson, W.~W.: \Title{An Elementary Treatise on +the Differential Calculus Founded on the Method of Rates +or Fluxions.} New~York: John Wiley \&~Sons. 8vo. 1884. +Price, \$3.50. Abridged edition, 1889. Price,~\$1.50. +\end{Book} + +\begin{Book} +Johnson, W.~W.: \Title{Elementary Treatise on the Integral Calculus +Founded on the Method of Rates or Fluxions.} 8vo,~cloth. +New~York: John Wiley \&~Sons. 1885. Price,~\$1.50. +\end{Book} + +\begin{Book} +Greenhill, A.~G.: \Title{Differential and Integral Calculus.} With applications. +8vo,~cloth. Second edition. New~York and London: +The Macmillan~Co. 1891. Price, \Price{9s.}~(\$2.60). +\end{Book} + +\begin{Book} +Price: \Title{Infinitesimal Calculus.} Four Vols. 1857--65. Out of +print and very scarce. Obtainable for about~\$27.00. +\end{Book} + +\begin{Book} +Smith, William Benjamin: \Title{Infinitesimal Analysis.} Vol.~I., Elementary: +Real Variables. New~York and London: The Macmillan~Co. +1898. Price,~\$3.25. + +\begin{Descrip} +``The aim has been, within a prescribed expense of time and +energy to penetrate as far as possible, and in as many directions, into +the subject in hand,---that the student should attain as wide knowledge +of the matter, as full comprehension of the methods, and as clear +consciousness of the spirit and power of analysis as the nature of the +case would admit.''---From Author's Preface. +\end{Descrip} +\end{Book} + +\begin{Book} +Todhunter, Isaac: \Title{A Treatise on the Differential Calculus.} London +and New~York: The Macmillan~Co. Price, \Price{10s. 6d.} +(\$2.60). \Title{A Treatise on the Integral Calculus.} (Same publishers.) +Price, \Price{10s. 6d.} (\$2\Chg{ }{.}60). + +\begin{Descrip} +Todhunter's text-books were, until recently, the most widely used +in England. His works on the Calculus still retain their standard +character, as general manuals. +\end{Descrip} +\end{Book} +\PageSep{137} + +\begin{Book} +Williamson: \Title{Differential and Integral Calculus.} London and +New~York: Longmans, Green, \&~Co. 1872--1874. Two~Vols. +Price, \$3.50~each. +\end{Book} + +\begin{Book} +De~Morgan, Augustus: \Title{Differential and Integral Calculus.} London: +Society for the Diffusion of Useful Knowledge. 1842. +Out of print. About~\$6.40. + +\begin{Descrip} +The most extensive and complete work in English. ``The object +has been to contain within the prescribed limits, the whole of the +students' course from the confines of elementary algebra and trigonometry, +to the entrance of the highest works on mathematical physics'' +(Author's Preface). Few examples. In typography, and general +arrangement of material, inferior to the best recent works. Valuable +for collateral study, and for its philosophical spirit. +\end{Descrip} +\end{Book} + + +\BibSubsect{French.} + +\begin{Book} +Sturm: \Title{Cours d'analyse de l'École Polytechnique.} 10.~édition, +revue et corrigé par E.~Prouhet, et augmentée de~la théorie +élémentaire des fonctions elliptiques, par H.~Laurent. 2~volumes +in---8. Paris: Gauthier-Villars et~fils. 1895. Bound, +16~fr.\ 50~c. \$4.95. + +\begin{Descrip} +One of the most widely used of text-books. First published in +1857. The new tenth edition has been thoroughly revised and brought +down to date. The exercises, while not numerous, are sufficient, those +which accompany the additions and complementary chapters of M.~De~Saint +Germain having been taken from the Collection of M.~Tisserand, +mentioned below. +\end{Descrip} +\end{Book} + +\begin{Book} +Duhamel: \Title{Éléments de calcul infinitésimal.} 4.~Edition, revue et +annotée par J.~Bertrand. 2~volumes in---8; avec planches. +Paris: Gauthier-Villars et~fils. 1886. 15~fr. \$4.50. + +\begin{Descrip} +The first edition was published between 1840 and 1841. ``Cordially +recommended to teachers and students'' by De~Morgan. Duhamel +paid great attention to the philosophy and logic of the mathematical +sciences, and the student may also be referred in this connexion to +his \Title{Méthodes dans les sciences de raisonnement}. 5~volumes. Paris: +Gauthier-Villars et~fils. Price, 25.50~francs. \$7.65. +\end{Descrip} +\end{Book} + +\begin{Book} +Lacroix, S.-F.: \Title{Traité élémentaire de calcul différentiel et de +calcul intégral.} 9.~Edition, revue et augmentée de notes par +Hermite et Serret. 2~vols. Paris: Gauthier-Villars et~fils. +1881. 15~fr. \$4.50. + +\begin{Descrip} +A very old work. The first edition was published in 1797. It was +the standard treatise during the early part of the century, and has +been kept revised by competent hands. +\end{Descrip} +\end{Book} +\PageSep{138} + +\begin{Book} +Appell, P.; \Title{Éléments d'analyse mathématique.} À l'usage des +ingénieurs et dés physiciens. Cours professé à l'École Centrale +des Arts et Manufactures. 1~vol.\ in---8, 720~pages, avec +figures, cartonné à l'anglaise. Paris: Georges Carré \&~C. +Naud. 1899. Price, 24~francs.\ \$7.20. +\end{Book} + +\begin{Book} +Boussinesq, J.: \Title{Cours d'analyse infinitésimal.} À l'usage des +personnes qui étudient cette science en vue de ses applications +mécaniques et physiques, 2~vols., grand in\Chg{-}{---}8, avec figures. +Tome~I\@. Calcul différentiel. Paris, 1887. 17~fr.\ (\$5.10). +Tome~II\@. Calcul intégral. Paris: Gauthier-Villars et~fils. +1890. 23~fr.\ 50~c.\ (\$7.05). +\end{Book} + +\begin{Book} +Hermite, Ch.: \Title{Cours d'analyse de l'École Polytechnique.} 2~vols. +Vol.~I\@. Paris: Gauthier-Villars et~fils. 1897. + +\begin{Descrip} +A new edition of Vol.~I. is in preparation (1899). Vol.~II. has not +yet appeared. +\end{Descrip} +\end{Book} + +\begin{Book} +Jordan, Camille: \Title{Cours d'analyse de l'École Polytechnique.} 3~volumes. +2.~édition. Paris: Gauthier-Villars et~fils. 1893--1898. +51~fr.\ \$14.70. + +\begin{Descrip} +Very comprehensive on the theoretical side. Enters deeply into +the metaphysical aspects of the subject. +\end{Descrip} +\end{Book} + +\begin{Book} +Laurent, H.: \Title{Traité d'analyse.} 7~vols in---8. Paris: Gauthier-Villars +et~fils. 1885--1891. 73~fr.\ \$21.90. + +\begin{Descrip} +The most extensive existing treatise on the Calculus. A general +handbook and work of reference for the results contained in the +more special works and memoirs. +\end{Descrip} +\end{Book} + +\begin{Book} +Picard, Émile: \Title{Traité d'analyse.} 4~volumes grand in\Chg{-}{---}8. Paris: +Gauthier-Villars et~fils. 1891. 15~fr.\ each. Vols.~I.--III., +\$14.40. Vol.~IV. has not yet appeared. + +\begin{Descrip} +An advanced treatise on the Integral Calculus and the theory of +differential equations. Presupposes a knowledge of the Differential +Calculus. +\end{Descrip} +\end{Book} + +\begin{Book} +Serret, J.-A.: \Title{Cours de calcul différentiel et intégral.} 4.~edition, +augmentée d'une note sur les fonctions elliptiques, par +Ch.~Hermite. 2~forts volumes in---8. Paris: Gauthier-Villars +et~fils. 1894. 25~fr.\ \$7.50. + +\begin{Descrip} +A good German translation of this work by Axel Harnack has +passed through its second edition (Leipsic: Teubner, 1885 and 1897). +\end{Descrip} +\end{Book} +\PageSep{139} + +\begin{Book} +Hoüel, J.: \Title{Cours de calcul infinitésimal.} 4~beaux volumes grand +in---8, avec figures. Paris: Gauthier-Villars et~fils. 1878--1879--1880--1881. +50~fr.\ \$15.00. +\end{Book} + +\begin{Book} +Bertrand, J.: \Title{Traité de calcul différentiel et de calcul intégral.} +(1)~Calcul différentiel. Paris: Gauthier-Villars et fils. 1864. +Scarce. About \$48.00. (2)~Calcul intégral (Intégrales définies +et indéfinies). Paris, 1870. Scarce. About \$24.00. +\end{Book} + +\begin{Book} +Boucharlat, J.-L.: \Title{Éléments de calcul différentiel et de calcul +intégral.} 9.~édition, revue et annotée par H.~Laurent. Paris: +Gauthier-Villars et~fils. 1891. 8~fr.\ \$2.40. +\end{Book} + +\begin{Book} +Moigno: \Title{Leçons de calcul différentiel et de calcul intégral\Typo{,}{.}} 2~vols., +Paris, 1840--1844. Scarce. About \$9.60. +\end{Book} + +\begin{Book} +Navier: \Title{Leçons d'analyse de l'École Polytechnique.} Paris, 1840. +2nd~ed. 1856. Out of print. About \$3.60. + +\begin{Descrip} +An able and practical work. Very popular in its day. The typical +course of the \textit{École Polytechnique}, and the basis of several of the treatises +that followed, including that of Sturm. Also much used in its +German translation. +\end{Descrip} +\end{Book} + +\begin{Book} +Cournot: \Title{Théorie des fonctions et du calcul infinitésimal.} 2~vols. +Paris, 1841. 2nd~ed. 1856--1858. Out of print, and +scarce. About \$3.00. + +\begin{Descrip} +The first edition (1841) was ``cordially recommended to teachers +and students'' by De~Morgan. Cournot was especially strong on the +philosophical side. He examined the foundations of many sciences +and developed original views on the theory of knowledge, which are +little known but have been largely drawn from by other philosophers. +\end{Descrip} +\end{Book} + +\begin{Book} +Cauchy, A.: \Title{\OE{}uvres complètes.} Tome~III: \Title{Cours d'analyse +de l'École Polytechnique.} Tome~IV: \Title{Résumé des leçons +données à l'École Polytechnique sur le calcul infinitésimal. +Leçons sur le calcul différentiel.} Tome~V: \Title{Leçons sur les +applications du calcul infinitésimal à la géométrie.} Paris: +Gauthier-Villars et~fils, 1885--1897. 25~fr.\ each. \$9.50~each. + +\begin{Descrip} +The works of Cauchy, as well as those of Lagrange, which follow, +are mentioned for their high historical and educational importance. +\end{Descrip} +\end{Book} + +\begin{Book} +Lagrange, J.~L.: \Title{\OE{}uvres complètes.} Tome~IX: \Title{Théorie des fonctions +analytiques.} Tome~X.: \Title{Leçons sur le calcul des fonctions.} +\PageSep{140} +Paris: Gauthier-Villars et~fils, 1881--1884. 18~fr.\ per +volume. \$5.40 per~volume. + +\begin{Descrip} +``The same power of abstraction and facility of treatment which +signalise these works are nowhere to be met with in the prior or subsequent +history of the subject. In addition, they are replete with the +profoundest aperçus into the history of the development of analytical +truths,---aperçus which could have come only from a man who combined +superior creative endowment with exact and comprehensive +knowledge of the facts. In the remarks woven into the body of the +text will be found what is virtually a detailed history of the subject, +and one which is not to be had elsewhere, least of all in diffuse histories +of mathematics. The student, thus, not only learns in these +works how to think, but also discovers how people actually have +thought, and what are the ways which human instinct and reason +have pursued in the different individuals who have participated in +the elaboration of the science.''---(E.~Dühring.) +\end{Descrip} +\end{Book} + +\begin{Book} +Euler, L.: + +\begin{Descrip} +The Latin treatises of Euler are also to be mentioned in this connexion, +for the benefit of those who wish to pursue the history of the +text-book making of this subject to its fountain-head. They are the +\Title{Differential Calculus} (St.~Petersburg, 1755), the \Title{Integral Calculus} (3~vols., +St.~Petersburg, 1768--1770), and the \Title{Introduction to the Infinitesimal +Analysis} (2~vols., Lausanne, 1748). Of the last-mentioned work +an old French translation by Labey exists (Paris: Gauthier-Villars), +and a new German translation (of Vol.~I. only) by Maser (Berlin: +Julius Springer, 1885). Of the first-mentioned treatises on the Calculus +proper there exist two old German translations, which are not +difficult to obtain. +\end{Descrip} +\end{Book} + + +\BibSubsect{German.} + +\begin{Book} +Harnack, Dr.\ Axel: \Title{Elemente der Differential- und Integralrechnung.} +Zur Einführung in das Studium dargestellt. Leipzig: +Teubner, 1881. M.~7.60. Bound, \$2.80. (English translation. +London: Williams \&~Norgate. 1891.) +\end{Book} + +\begin{Book} +Junker, Dr.\ Friedrich: \Title{Höhere Analysis.} I.~\Title{Differentialrechnung.} +Mit 63~Figuren. II.~\Title{Integralrechnung.} Leipzig: +G.~J. Göschen'sche Verlagshandlung. 1898--1899. 80~pf.\ each. +30~cents each. + +\begin{Descrip} +These books are marvellously cheap, and very concise. They +contain no examples. Pocket-size. +\end{Descrip} +\end{Book} + +\begin{Book} +Autenheimer, F.: \Title{Elementarbuch der Differential- und Integralrechnung +mit zahlreichen Anwendungen aus der Analysis, +Geometrie, Mechanik, Physik etc.} Für höhere Lehranstalten +\PageSep{141} +und den Selbstunterricht. 4te~verbesserte Auflage. Weimar: +Bernhard Friedrich Voigt. 1895. + +\begin{Descrip} +As indicated by its title, this book is specially rich in practical +applications. +\end{Descrip} +\end{Book} + +\begin{Book} +Stegemann: \Title{Grundriss der Differential- und Integralrechnung}, +8te~Auflage, herausgegeben von Kiepert. Hannover: Helwing, +1897. Two volumes, 26~marks. Two volumes, bound, +\$8.50. + +\begin{Descrip} +This work was highly recommended by Prof.\ Felix Klein at the +Evanston Colloquium in~1893. +\end{Descrip} +\end{Book} + +\begin{Book} +Schlömilch: \Title{Compendium der höheren Analysis.} Fifth edition, +1881. Two volumes, \$6.80. + +\begin{Descrip} +Schlömilch's text-books have been very successful. The present +work was long the standard manual. +\end{Descrip} +\end{Book} + +\begin{Book} +Stolz, Dr.\ Otto: \Title{Grundzüge der Differential- und Integralrechnung.} +In 2~Theilen. I.~Theil. Reelle Veränderliche und +Functionen. (460~S.) 1893. M.~8. II.~Complexe Veränderliche +und Functionen. (338~S.) Leipzig: Teubner. 1896. +M.~8. Two volumes, \$6.00. + +\begin{Descrip} +A supplementary 3rd part entitled \Title{Die Lehre von den Doppelintegralen} +has just been published (1899). Based on the works of J.~Tannery, +Peano, and Dini. +\end{Descrip} +\end{Book} + +\begin{Book} +Lipschitz, R.: \Title{Lehrbuch der Analysis.} 1877--1880. Two volumes, +bound, \$12.30. + +\begin{Descrip} +Specially good on the theoretical side. +\end{Descrip} +\end{Book} + + +\BibSubsect{Collections of Examples and Illustrations.} + +\begin{Book} +Byerly, W.~E.: \Title{Problems in Differential Calculus.} Supplementary +to a Treatise on Differential Calculus. Boston: Ginn \&~Co. +75~cents. +\end{Book} + +\begin{Book} +Gregory: \Title{Examples on the Differential and Integral Calculus.} +1841. Second edition. 1846. Out of print. About \$6.40. +\end{Book} + +\begin{Book} +Frenet: \Title{Recueil d'exercises sur le calcul infinitésimal.} 5.~édition, +augmentée d'un appendice, par H.~Laurent. Paris: +Gauthier-Villars et~fils. 1891. 8~fr.\ \$2.40. +\end{Book} +\PageSep{142} + +\begin{Book} +Tisserand, F.: \Title{Recueil complémentaire d'exercises sur le calcul +infinitésimal.} Second edition. Paris: Gauthier-Villars et~fils +1896. + +\begin{Descrip} +Complementary to Frenet. +\end{Descrip} +\end{Book} + +\begin{Book} +Laisant, C.~A.: \Title{Recueil de problèmes de mathématiques.} Tome~VII\@. +Calcul infinitésimal et calcul des fonctions. Mécanique. +Astronomie. (Announced for publication.) Paris: Gauthier-Villars +et~fils. +\end{Book} + +\begin{Book} +Schlömilch, Dr.\ Oscar: \Title{Uebungsbuch zum Studium der höheren +Analysis.} I.~Theil. Aufgaben aus der Differentialrechnung. +4te~Auflage. (336~S.) 1887. M.~6. II.~Aufgaben aus der +Integralrechnung. 3te~Auflage. (384~S.) Leipzig: Teubner, +1882. M.~7.60. Both volumes, bound, \$7.60. +\end{Book} + +\begin{Book} +Sohncke, L.~A.: \Title{Sammlung von Aufgaben aus der Differential- +und Integralrechnung.} Herausgegeben von Heis. Two volumes, +in---8. Bound, \$3.00. +\end{Book} + +\begin{Book} +Fuhrmann, Dr.\ Arwed: \Title{Anwendungen der Infinitesimalrechnung +in den Naturwissenschaften, im Hochbau und in der +Technik.} Lehrbuch und Aufgabensammlung. In sechs Theilen, +von denen jeder ein selbstständiges Ganzes bildet. Theil~I. +Naturwissenschaftliche Anwendungen der Differentialrechnung. +Theil~II. Naturwissenschaftliche Anwendungen der +Integralrechnung. Berlin: Verlag von Ernst \&~Korn. 1888--1890. +Vol.~I., Cloth, \$1.35. Vol.~II., Cloth, \$2.20. +\end{Book} +\PageSep{143} + + +\printindex + +\iffalse +%INDEX. + +Accelerated motion 57, 60 + +Accelerating force 62 + +Advice for studying the Calculus 132, 133 + +Angle, unit employed in measuring an#Angle 51 + +Approximate solutions in the Integral Calculus 132, 133 + +Arc and its chord, a continuously decreasing|EtSeq#Arc 7, 39 % et seq. + +Archimedes 127 + +Astronomical ephemeris 76 + +Calculus, notation of 25 + +Calculus, notation of|EtSeq 79 % et seq. + +Circle, equation of|EtSeq 31 % et seq. + +Circle cut by straight line, investigated|EtSeq 31 % et seq. + +Coefficients, differential|EtSeq 22 % et seq. + +Coefficients, differential 38, 55, 82, 88, 96, 100, 112 + +Complete Differential Coefficients 96 + +Constants 14 + +Contiguous values 112 + +Continuous quantities|EtSeq 7 % et seq. + +Continuous quantities 53 + +Co-ordinates 30 + +Curve, magnified 40 + +Curvilinear areas, determination of|EtSeq 124 % et seq. + +Density, continuously varying|EtSeq 130 % et seq. + +Derivatives 19, 21, 22 + +Derived Functions|EtSeq 19 % et seq. + +Derived Functions 21 + +Differences + arithmetical 4 + of increments 26 + calculus of 89 + +Differential coefficients|EtSeq 22 % et seq., + +Differential coefficients 38, 55, 82 + as the index of the change of a function 112 + of higher orders 88 + +Differentials + partial|EtSeq 78 % et seq.; + total|EtSeq 78 % et seq. + +Differentiation + of the common functions 85, 86 + successive|EtSeq 88 % et seq.; + implicit|EtSeq 94 % et seq.; + of complicated functions|EtSeq 100 % et seq. + +Direct function 97 + +Direction 36 + +Equality 4 + +Equations, solution of 77 + +Equidistant values 104 + +Euler 27, 124 + +Errors, in the valuation of quantities 75, 84 + +Explicit functions 107 + +Falling bodies 56 + +Finite differences|EtSeq 88 % et seq. + +Fluxions 11, 60, 112 + +Force 61-63 + +Functions + definition of|EtSeq 14 % et seq.; + derived|EtSeq 19 % et seq., + derived 21 + direct and indirect 97 + implicit and explicit 107, 108 + inverse|EtSeq 102 % et seq.; + of several variables|EtSeq 78 % et seq.; + recapitulation of results in the theory of 74 + +Generally@\emph{Generally}, the word 16 + +Implicit + differentiation|EtSeq 94 % et seq.; + function 107, 108 + +Impulse 60 + +Increase without limit|EtSeq 5, 65 % et seq. + +Increment 16, 113 +\PageSep{144} + +Independent variables 106 + +Indirect function 97 + +Indivisibles + method of|EtSeq 127 % et seq.; + notion of, in mechanics|EtSeq 129 % et seq. + +Infinite@\emph{Infinite}, the word#Infinite 128 + +Infinitely small, the notion of#Infinitely 12, 49, 59, 83 + +Infinitely small, the notion of|EtSeq#Infinitely 38 % et seq., + +Infinity, orders of|EtSeq 42 % et seq. + +Integral Calculus 73 + notation of 119 + +Integral Calculus|EtSeq 115 % et seq. + +Integrals + definition of|EtSeq 119 % et seq.; + relations between differential coefficients and 121 + indefinite 122, 123 + +Intersections, limit of|EtSeq 46 % et seq. + +Inverse functions|EtSeq 102 % et seq. + +Iron bar continually varying in density, weight of|EtSeq#Iron 130 % et seq. + +Ladder against wall|EtSeq 45 % et seq. + +Lagrange 124 + +Laplace 124 + +Leibnitz 11, 13, 38, 42, 48, 59, 60, 83, 123, 124, 128, 129 + +Limit of intersections|EtSeq 46 % et seq. + +Limits|EtSeq 26 % et seq. + +Limiting ratios|EtSeq 65 % et seq. + +Limiting ratios 81 + +Logarithms 20, 38, 86, 87 + +Logarithms|EtSeq 112 % et seq. + +Magnified curve 40 + +Motion + accelerated 60 + simple harmonic 57 + +Newton 11, 60 + +Notation + of the Differential Calculus 25 + of the Differential Calculus|EtSeq 79 % et seq. + of the Integral Calculus 119 + +Orders, differential coefficients of higher 88 + +Orders of infinity|EtSeq 42 % et seq. + +Parabola, the#Parabola 30, 127 + +Parabola, the|EtSeq#Parabola 124 % et seq. + +Partial + differentials|EtSeq 78 % et seq.; + differential coefficients 96 + +Point@\emph{Point}, the word#Point 129 + +Points, the number of, in a straight line 129 + +Polygon 38 + +Proportion|EtSeq 2 % et seq. + +Quantities, continuous|EtSeq 7 % et seq. + +Quantities, continuous 53 + +Ratio + defined|EtSeq 2 % et seq.; + of two increments 87 + +Ratios, limiting|EtSeq 65 % et seq. + +Ratios, limiting 81 + +Rough methods of solution in the Integral Calculus 132, 133 + +Series|EtSeq 15, 24 % et seq. + +Signs|EtSeq 31 % et seq. + +Simple harmonic motion 57 + +Sines 87 + +Singular values 16 + +Small, has no precise meaning 12 + +Specific gravity, continuously varying|EtSeq 130 % et seq. + +Successive differentiation|EtSeq 88 % et seq. + +Sun's longitude 76 + +Tangent 37, 38, 40 + +Taylor's Theorem|EtSeq 15, 19 % et seq. + +Time, idea of#Time 4 + +Time, idea of|EtSeq#Time 110 % et seq. + +Total + differential coefficient 100 + differentials|EtSeq 78 % et seq.; + variations 95 + +Transit instrument 84 + +Uniformly accelerated 57, 60 + +Values + contiguous 112 + equidistant 104 + +Variables + independent and dependent 14, 15, 106 + functions of several|EtSeq 78 % et seq. + +Variations, total#Variations 95 + +Velocity + linear|EtSeq 52 % et seq. + linear 111 + angular 59 + +Weight of an iron bar of which the density varies from point to point|EtSeq#Iron 130 % et seq. +\fi +\PageSep{145} + +\iffalse +%[** TN: Catalog text has been (lightly) proofread, but not marked up in LaTeX] + +CATALOGUE OF PUBLICATIONS +OF THE +OPEN COURT PUBLISHING CO. + +COPE, E. D. + +THE PRIMARY FACTORS OF ORGANIC EVOLUTION. + +121 cuts. Pp. xvi, 547. Cloth, \$2.00 net (10s.). + + +MULLER, F. MAX. + +THREE INTRODUCTORY LECTURES ON THE SCIENCE OF +THOUGHT. +128 pages. Cloth, 75c (3s. 6d.). + +THREE LECTURES ON THE SCIENCE OF LANGUAGE. +112 pages. 2nd Edition. Cloth, 75c (3s. 6d.). + + +ROMANES, GEORGE JOHN. + +DARWIN AND AFTER DARWIN. + +An Exposition of the Darwinian Theory and a Discussion of Post-Darwinian +Questions. Three Vols., \$4.00 net. Singly, as follows: + +1. The Darwinian Theory. 460 pages. 125 illustrations. Cloth, \$2.00. + +2. Post-Darwinian Questions. Heredity and Utility. Pp. 338. \$1.50. + +3. Post-Darwinian Questions. Isolation and Physiological Selection. +Pp. 181. \$1.00. + +AN EXAMINATION OF WEISMANNISM. +236 pages. Cloth, \$1.00 net. + +THOUGHTS ON RELIGION. +Edited by Charles Gore, M. A., Canon of Westminster. Third Edition, +Pages, 184. Cloth, gilt top, \$1.25 net. + + +SHUTE. DR. D. KERFOOT. + +FIRST BOOK IN ORGANIC EVOLUTION. + +Colored plates, and numerous diagrams. (In Preparation.) + + +MACH, ERNST. + +THE SCIENCE OF MECHANICS. + +A Critical and Historical Exposition of its Principles. Translated +by T. J. McCormack. 250 cuts. 534 pages. 1/2 in., gilt top. \$2.50 (12s. 6d.). + +POPULAR SCIENTIFIC LECTURES. +Third Edition. 415 pages. 59 cuts. Cloth, gilt top. Net, \$1.50 (7s. 6d.). + +THE ANALYSIS OF THE SENSATIONS. +Pp. 208. 37 cuts. Cloth, \$1.25 net (6s. 6d.). + + +LAGRANGE. J. L. + +LECTURES ON ELEMENTARY MATHEMATICS. +With portrait of the author. Pp. 172. Price, \$1.00 net (5s.). + + +DE MORGAN, AUGUSTUS. + +ON THE STUDY AND DIFFICULTIES OF MATHEMATICS. +New Reprint edition with notes. Pp. viii+288. Cloth, \$1.25 net (5s.). + +ELEMENTARY ILLUSTRATIONS OF THE DIFFERENTIAL AND +INTEGRAL CALCULUS. +New reprint edition. Price, \$1.00 (5s.). + + +SCHUBERT, HERMANN. + +MATHEMATICAL ESSAYS AND RECREATIONS. +Pp. 149. Cuts, 37. 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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.net % +% % +% % +% Title: Elementary Illustrations of the Differential and Integral Calculus +% % +% Author: Augustus De Morgan % +% % +% Release Date: March 3, 2012 [EBook #39041] % +% % +% Language: English % +% % +% Character set encoding: ISO-8859-1 % +% % +% *** START OF THIS PROJECT GUTENBERG EBOOK DIFFERENTIAL, INTEGRAL CALCULUS *** +% % +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % + +\def\ebook{39041} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %% +%% Packages and substitutions: %% +%% %% +%% book: Required. %% +%% inputenc: Latin-1 text encoding. Required. %% +%% %% +%% ifthen: Logical conditionals. Required. %% +%% %% +%% amsmath: AMS mathematics enhancements. Required. %% +%% amssymb: Additional mathematical symbols. 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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.net + + +Title: Elementary Illustrations of the Differential and Integral Calculus + +Author: Augustus De Morgan + +Release Date: March 3, 2012 [EBook #39041] + +Language: English + +Character set encoding: ISO-8859-1 + +*** START OF THIS PROJECT GUTENBERG EBOOK DIFFERENTIAL, INTEGRAL CALCULUS *** +\end{PGtext} +\end{minipage} +\end{center} +\newpage +%% Credits and transcriber's note %% +\begin{center} +\begin{minipage}{\textwidth} +\begin{PGtext} +Produced by Andrew D. Hwang. +\end{PGtext} +\end{minipage} +\vfill +\end{center} + +\begin{minipage}{0.85\textwidth} +\small +\BookMark{0}{Transcriber's Note.} +\subsection*{\centering\normalfont\scshape% +\normalsize\MakeLowercase{\TransNote}}% + +\raggedright +\TransNoteText +\end{minipage} +%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%% +\PageSep{ii} +\FrontMatter + + +\Section{In the Same Series.} + +\Book{ON THE STUDY AND DIFFICULTIES OF MATHEMATICS.} +By \Author{Augustus De~Morgan}. Entirely new edition, +with portrait of the author, index, and annotations, +bibliographies of modern works on algebra, the philosophy +of mathematics, pan-geometry,~etc. Pp.,~288. Cloth, \$1.25 +net~(\Price{5s.}). + +\Book{LECTURES ON ELEMENTARY MATHEMATICS.} By +\Author{Joseph Louis Lagrange}. Translated from the French by +\Translator{Thomas~J. McCormack}. With photogravure portrait of +Lagrange, notes, biography, marginal analyses,~etc. Only +separate edition in French or English, Pages,~172. Cloth, +\$1.00 net~(\Price{5s.}). + +\Book{ELEMENTARY ILLUSTRATIONS OF THE DIFFERENTIAL +AND INTEGRAL CALCULUS.} By \Author{Augustus De~Morgan}. +New reprint edition. With sub-headings, and +a brief bibliography of English, French, and German text-books +of the Calculus. Pp.,~144. Price, \$1.00 net~(\Price{5s.}). + +\Book{MATHEMATICAL ESSAYS AND RECREATIONS.} By +\Author{Hermann Schubert}, Professor of Mathematics in the +Johanneum, Hamburg, Germany. Translated from the +German by \Translator{Thomas~J. McCormack}. Containing essays on +the Notion and Definition of Number, Monism in Arithmetic, +On the Nature of Mathematical Knowledge, The +Magic Square, The Fourth Dimension, The Squaring of +the Circle. Pages,~149. Cuts,~37. Price, Cloth,~75c net~(\Price{3s. 6d.}). + +\Book{HISTORY OF ELEMENTARY MATHEMATICS.} By \Author{Dr.\ +Karl Fink}, late Professor in Tübingen. Translated from +the German by Prof.\ \Translator{Wooster Woodruff Beman} and Prof.\ +\Translator{David Eugene Smith}. (Nearly Ready.) +\vfill +\begin{center} +\small +THE OPEN COURT PUBLISHING CO. \\ +\footnotesize +324 DEARBORN ST., CHICAGO. +\end{center} +\PageSep{iii} +\cleardoublepage +%[** Title page] +\begin{center} +ELEMENTARY ILLUSTRATIONS +\vfil +\footnotesize OF THE +\vfil +\textsc{\LARGE Differential and Integral \\[4pt] +Calculus} +\vfil\vfil\vfil + +\footnotesize BY \\ +\normalsize AUGUSTUS DE MORGAN +\vfil\vfil\vfil + +\textit{\small NEW EDITION} +\vfil\vfil\vfil\vfil + +\footnotesize CHICAGO \\ +THE OPEN COURT PUBLISHING COMPANY \\ +\scriptsize FOR SALE BY \\ +\footnotesize\textsc{Kegan Paul, Trench, Trübner \&~Co., Ltd., London} \\ +1899 +\end{center} +\PageSep{iv} +% [Blank page] +\PageSep{v} + + +\Section{Editor's Preface.} + +\First{The} publication of the present reprint of De~Morgan's \Title{Elementary +Illustrations of the Differential and Integral Calculus} +forms, quite independently of its interest to professional +students of mathematics, an integral portion of the general educational +plan which the Open Court Publishing Company has been +systematically pursuing since its inception,---which is the dissemination +among the public at large of sound views of science and of +an adequate and correct appreciation of the methods by which +truth generally is reached. Of these methods, mathematics, by +its simplicity, has always formed the type and ideal, and it is +nothing less than imperative that its ways of procedure, both in +the discovery of new truth and in the demonstration of the necessity +and universality of old truth, should be laid at the foundation +of every philosophical education. The greatest achievements in +the history of thought---Plato, Descartes, Kant---are associated +with the recognition of this principle. + +But it is precisely mathematics, and the pure sciences generally, +from which the general educated public and independent +students have been debarred, and into which they have only rarely +attained more than a very meagre insight. The reason of this is +twofold. In the first place, the ascendant and consecutive character +of mathematical knowledge renders its results absolutely unsusceptible +of presentation to persons who are unacquainted with +what has gone before, and so necessitates on the part of its devotees +a thorough and patient exploration of the field from the very +beginning, as distinguished from those sciences which may, so to +speak, be begun at the end, and which are consequently cultivated +with the greatest zeal. The second reason is that, partly through +the exigencies of academic instruction, but mainly through the +martinet traditions of antiquity and the influence of mediæval +\PageSep{vi} +logic-mongers, the great bulk of the elementary text-books of +mathematics have unconsciously assumed a very repellent form,---something +similar to what is termed in the theory of protective +mimicry in biology ``the terrifying form.'' And it is mainly to +this formidableness and touch-me-not character of exterior, concealing +withal a harmless body, that the undue neglect of typical +mathematical studies is to be attributed. + +To this class of books the present work forms a notable exception. +It was originally issued as numbers 135 and 140 of the +Library of Useful Knowledge (1832), and is usually bound up with +De~Morgan's large \Title{Treatise on the Differential and Integral +Calculus} (1842). Its style is fluent and familiar; the treatment +continuous and undogmatic. The main difficulties which encompass +the early study of the Calculus are analysed and discussed in +connexion with practical and historical illustrations which in point +of simplicity and clearness leave little to be desired. No one who +will read the book through, pencil in hand, will rise from its perusal +without a clear perception of the aim and the simpler fundamental +principles of the Calculus, or without finding that the profounder +study of the science in the more advanced and more +methodical treatises has been greatly facilitated. + +The book has been reprinted substantially as it stood in its +original form; but the typography has been greatly improved, and +in order to render the subject-matter more synoptic in form and +more capable of survey, the text has been re-paragraphed and a +great number of descriptive sub-headings have been introduced, a +list of which will be found in the Contents of the book. An index +also has been added. + +Persons desirous of continuing their studies in this branch of +mathematics, will find at the end of the text a bibliography of the +principal English, French, and German works on the subject, as +well as of the main Collections of Examples. From the information +there given, they may be able to select what will suit their +special needs. + +\Signature{Thomas J. McCormack.} +{\textsc{La Salle}, Ill., August, 1899.} +\PageSep{vii} + + +\TableofContents +\iffalse +CONTENTS: + +PAGE + +On the Ratio or Proportion of Two Magnitudes 2 +On the Ratio of Magnitudes that Vanish Together.... 4 +On the Ratios of Continuously Increasing or Decreasing Quantities 7 +The Notion of Infinitely Small Quantities 11 +On Functions 14 +Infinite Series 15 +Convergent and Divergent Series 17 +Taylor's Theorem. Derived Functions 19 +Differential Coefficients 22 +The Notation of the Differential Calculus 25 +Algebraical Geometry.... 29 +On the Connexion of the Signs of Algebraical and the Directions + of Geometrical Magnitudes 31 +The Drawing of a Tangent to a Curve 36 +Rational Explanation of the Language of Leibnitz.... 38 +Orders of Infinity 42 +A Geometrical Illustration: Limit of the Intersections of Two + Coinciding Straight Lines 45 +The Same Problem Solved by the Principles of Leibnitz. . 48 +An Illustration from Dynamics; Velocity, Acceleration, etc.. 52 +Simple Harmonic Motion 57 +The Method of Fluxions 60 +Accelerated Motion 60 +Limiting Ratios of Magnitudes that Increase Without Limit. 65 +Recapitulation of Results Reached in the Theory of Functions. 74 +Approximations by the Differential Calculus 74 +Solution of Equations by the Differential Calculus.... 77 +Partial and Total Differentials 78 +\PageSep{viii} +Application of the Theorem for Total Differentials to the + Determination of Total Resultant Errors 84 +Rules for Differentiation 85 +Illustration of the Rules for Differentiation 86 +Differential Coefficients of Differential Coefficients .... 88 +Calculus of Finite Differences. Successive Differentiation . 88 +Total and Partial Differential Coefficients. Implicit Differentiation 94 +Applications of the Theorem for Implicit Differentiation .. 101 +Inverse Functions 102 +Implicit Functions 106 +Fluxions, and the Idea of Time 110 +The Differential Coefficient Considered with Respect to Its + Magnitude 112 +The Integral Calculus 115 +Connexion of the Integral with the Differential Calculus.. 120 +Nature of Integration 122 +Determination of Curvilinear Areas. The Parabola... 124 +Method of Indivisibles 125 +Concluding Remarks on the Study of the Calculus.... 132 +Bibliography of Standard Text-books and Works of Reference + on the Calculus 133 +Index 143 +\fi +\PageSep{1} +\MainMatter + + +\Section{Differential and Integral Calculus.} + +\SubSectHead{Elementary Illustrations.} + +\First{The} Differential and Integral Calculus, or, as it +was formerly called in this country [England], +the Doctrine of Fluxions, has always been supposed +to present remarkable obstacles to the beginner. It +is matter of common observation, that any one who +commences this study, even with the best elementary +works, finds himself in the dark as to the real meaning +of the processes which he learns, until, at a certain +stage of his progress, depending upon his capacity, +some accidental combination of his own ideas throws +light upon the subject. The reason of this may be, that +it is usual to introduce him at the same time to new +principles, processes, and symbols, thus preventing +his attention from being exclusively directed to one +new thing at a time. It is our belief that this should +be avoided; and we propose, therefore, to try the experiment, +whether by undertaking the solution of +some problems by common algebraical methods, without +calling for the reception of more than one new +symbol at once, or lessening the immediate evidence +of each investigation by reference to general rules, the +study of more methodical treatises may not be somewhat +\PageSep{2} +facilitated. We would not, nevertheless, that +the student should imagine we can remove all obstacles; +we must introduce notions, the consideration +of which has not hitherto occupied his mind; and +shall therefore consider our object as gained, if we +can succeed in so placing the subject before him, that +two independent difficulties shall never occupy his +mind at once. + + +\Subsection{On the Ratio or Proportion of Two Magnitudes.} + +The ratio or proportion of two magnitudes is best +\index{Proportion|EtSeq}% +\index{Ratio!defined|EtSeq}% +conceived by expressing them in numbers of some +unit when they are commensurable; or, when this is +not the case, the same may still be done as nearly as +we please by means of numbers. Thus, the ratio of +the diagonal of a square to its side is that of $\sqrt{2}$ to~$1$, +which is very nearly that of $14142$ to~$10000$, and is +certainly between this and that of $14143$ to~$10000$. +Again, any ratio, whatever numbers express it, may +be the ratio of two magnitudes, each of which is as +small as we please; by which we mean, that if we +take any given magnitude, however small, such as the +line~$A$, we may find two other lines $B$~and~$C$, each +less than~$A$, whose ratio shall be whatever we please. +Let the given ratio be that of the numbers $m$~and~$n$. +Then, $P$~being a line, $mP$~and~$nP$ are in the proportion +of $m$ to~$n$; and it is evident, that let $m$,~$n$, and~$A$ +be what they may, $P$~can be so taken that $mP$~shall be +less than~$A$. This is only saying that $P$~can be taken +less than the $m$\th~part of~$A$, which is obvious, since~$A$, +however small it may be, has its tenth, its hundredth, +its thousandth part,~etc., as certainly as if it were +larger. We are not, therefore, entitled to say that +because two magnitudes are diminished, their ratio is +\PageSep{3} +diminished; it is possible that~$B$, which we will suppose +to be at first a hundredth part of~$C$, may, after +a diminution of both, be its tenth or thousandth, or +may still remain its hundredth, as the following example +will show: +\begin{alignat*}{5} +&C && 3600 && 1800 && 36 && 90 \\ +&B && 36 && 1\tfrac{8}{10} && \tfrac{36}{100} && 9 \\ +&B &{}={}& \frac{1}{100} C\qquad + B &{}={}& \frac{1}{1000} C\qquad + B &{}={}& \frac{1}{100} C\qquad + B &{}={}& \frac{1}{10} C. +\end{alignat*} +Here the values of $B$~and~$C$ in the second, third, and +fourth column are less than those in the first; nevertheless, +the ratio of $B$ to~$C$ is less in the second column +than it was in the first, remains the same in the +third, and is greater in the fourth. + +In estimating the approach to, or departure from +equality, which two magnitudes undergo in consequence +of a change in their values, we must not look +at their differences, but at the proportions which those +differences bear to the whole magnitudes. For example, +if a geometrical figure, two of whose sides are +$3$~and $4$~inches now, be altered in dimensions, so that +the corresponding sides are $100$~and $101$~inches, they +are nearer to equality in the second case than in the +first; because, though the difference is the same in +both, namely one inch, it is one third of the least side +in the first case, and only one hundredth in the second. +This corresponds to the common usage, which +rejects quantities, not merely because they are small, +but because they are small in proportion to those of +which they are considered as parts. Thus, twenty +miles would be a material error in talking of a day's +journey, but would not be considered worth mentioning +in one of three months, and would be called totally +\PageSep{4} +insensible in stating the distance between the +earth and sun. More generally, if in the two quantities +$x$~and~$x + a$, an increase of~$m$ be given to~$x$, +the two resulting quantities $x + m$~and $x + m + a$ are +nearer to equality as to their \emph{ratio} than $x$~and~$x + a$, +\index{Equality}% +though they continue the same as to their \emph{difference}; for +\index{Differences!arithmetical}% +$\dfrac{x + a}{x} = 1 + \dfrac{a}{x}$ and $\dfrac{x + m + a}{x + m} = 1 + \dfrac{a}{x + m}$ of which +$\dfrac{a}{x + m}$~is less than~$\dfrac{a}{x}$, and therefore $1 + \dfrac{a}{x + m}$ is nearer +to unity than $1 + \dfrac{a}{x}$. In future, when we talk of an +approach towards equality, we mean that the ratio is +made more nearly equal to unity, not that the difference +is more nearly equal to nothing. The second +may follow from the first, but not necessarily; still +less does the first follow from the second. + + +\Subsection{On the Ratio of Magnitudes that Vanish Together.} + +It is conceivable that two magnitudes should decrease +\index{Time, idea of}% +simultaneously,\footnote + {In introducing the notion of time, we consult only simplicity. It would + do equally well to write any number of successive values of the two quantities, + and place them in two columns.} +so as to vanish or become +nothing, together. For example, let a point~$A$ move +on a circle towards a fixed point~$B$. The arc~$AB$ will +then diminish, as also the chord~$AB$, and by bringing +the point~$A$ sufficiently near to~$B$, we may obtain an +arc and its chord, both of which shall be smaller than +a given line, however small this last may be. But +while the magnitudes diminish, we may not assume +either that their ratio increases, diminishes, or remains +the same, for we have shown that a diminution +of two magnitudes is consistent with either of these. +\PageSep{5} +\index{Increase without limit|EtSeq}% +We must, therefore, look to each particular case for +the change, if any, which is made in the ratio by the +diminution of its terms. + +Now two suppositions are possible in every increase +or diminution of the ratio, as follows: Let $M$~and~$N$ +be two quantities which we suppose in a state +of decrease. The first possible case is that the ratio +of $M$ to~$N$ may decrease without limit, that is, $M$~may +be a smaller fraction of~$N$ after a decrease than it was +before, and a still smaller after a further decrease, +and so on; in such a way, that there is no fraction so +small, to which $\dfrac{M}{N}$~shall not be equal or inferior, if the +decrease of $M$~and~$N$ be carried sufficiently far. As +an instance, form two sets of numbers as in the adjoining +table: +\[ +\begin{array}{*{7}{c}} +\PadTxt{Ratio of }{} M \PadTxt{ to~$N$}{} + & 1 & \dfrac{1}{20} & \dfrac{1}{400} & \dfrac{1}{8000} & \dfrac{1}{160000} & \etc. \\ +\PadTxt{Ratio of }{} N \PadTxt{ to~$N$}{} + & 1 & \dfrac{1}{2} & \dfrac{1}{4} & \dfrac{1}{8} & \dfrac{1}{16} & \etc. \\ +\text{Ratio of~$M$ to~$N$} + & 1 & \dfrac{1}{10} & \dfrac{1}{100} & \dfrac{1}{1000} & \dfrac{1}{10000} & \etc. +\end{array} +\] +Here both $M$~and~$N$ decrease at every step, but $M$~loses +at each step a larger fraction of itself than~$N$, +and their ratio continually diminishes. To show that +this decrease is without limit, observe that $M$~is at +first equal to~$N$, next it is one tenth, then one hundredth, +then one thousandth of~$N$, and so on; by continuing +the values of $M$ and~$N$ according to the same +law, we should arrive at a value of~$M$ which is a +smaller part of~$N$ than any which we choose to name; +for example,~$.000003$. The second value of~$M$ beyond +our table is only one millionth of the corresponding +value of~$N$; the ratio is therefore expressed by~$.000001$ +\PageSep{6} +which is less than~$.000003$. In the same law of formation, +the ratio of $N$ to~$M$ is also \emph{increased} without limit. + +The second possible case is that in which the ratio +of $M$ to~$N$, though it increases or decreases, does not +increase or decrease without limit, that is, continually +approaches to some ratio, which it never will exactly +reach, however far the diminution of $M$ and~$N$ may +be carried. The following is an example: +\[ +\begin{array}{*{9}{c}} +\PadTxt{Ratio of }{} M \PadTxt{ to~$N$}{} + & 1 & \dfrac{1}{3} & \dfrac{1}{6} & \dfrac{1}{10} & \dfrac{1}{15} & \dfrac{1}{21} & \dfrac{1}{28} & \etc. \\ +\PadTxt{Ratio of }{} N \PadTxt{ to~$N$}{} + & 1 & \dfrac{1}{4} & \dfrac{1}{9} & \dfrac{1}{16} & \dfrac{1}{25} & \dfrac{1}{36} & \dfrac{1}{49} & \etc. \\ +\text{Ratio of~$M$ to~$N$} + & 1 & \dfrac{4}{3} & \dfrac{9}{6} & \dfrac{16}{10} & \dfrac{25}{15} & \dfrac{36}{21} & \dfrac{49}{28} & \etc. +\end{array} +\] +The ratio here increases at each step, for $\dfrac{4}{3}$~is greater +than~$1$, $\dfrac{9}{6}$~than~$\dfrac{4}{3}$, and so on. The difference between +this case and the last is, that the ratio of $M$ to~$N$, +though perpetually increasing, does not increase without +limit; it is never so great as~$2$, though it may be +brought as near to~$2$ as we please. + +To show this, observe that in the successive values +of~$M$, the denominator of the second is~$1 + 2$, that of +the third $1 + 2 + 3$, and so on; whence the denominator +of the $x$\th~value of~$M$ is +\[ +1 + 2 + 3 + \dots + x,\quad\text{or}\quad \frac{x(x + 1)}{2}\Add{.} +\] +Therefore the $x$\th~value of~$M$ is~$\dfrac{2}{x(x + 1)}$, and it is +evident that the $x$\th~value of~$N$ is~$\dfrac{1}{x^{2}}$, which gives the +$x$\th~value of the ratio $\dfrac{M}{N} = \dfrac{2x^{2}}{x(x + 1)}$, or~$\dfrac{2x}{x + 1}$, or +\PageSep{7} +$\dfrac{x}{x + 1} × 2$. If $x$~be made sufficiently great, $\dfrac{x}{x + 1}$~may +be brought as near as we please to~$1$, since, being +$1 - \dfrac{1}{x + 1}$, it differs from~$1$ by~$\dfrac{1}{x + 1}$, which may be +made as small as we please. But as $\dfrac{x}{x + 1}$, however +great $x$~may be, is always less than~$1$, $\dfrac{2x}{x + 1}$~is always +less than~$2$. Therefore (1)~$\dfrac{M}{N}$~continually increases; +(2)~may be brought as near to~$2$ as we please; (3)~can +never be greater than~$2$. This is what we mean by +saying that $\dfrac{M}{N}$~is an increasing ratio, the limit of +which is~$2$. Similarly of~$\dfrac{N}{M}$, which is the reciprocal +of~$\dfrac{M}{N}$, we may show (1)~that it continually decreases; +(2)~that it can be brought as near as we please to~$\frac{1}{2}$; +(3)~that it can never be less than~$\frac{1}{2}$. This we express +by saying that $\dfrac{N}{M}$~is a decreasing ratio, whose limit +is~$\frac{1}{2}$. + + +\Subsection{On the Ratios of Continuously Increasing or +Decreasing Quantities.} + +To the fractions here introduced, there are intermediate +\index{Continuous quantities|EtSeq}% +\index{Quantities, continuous|EtSeq}% +fractions, which we have not considered. +Thus, in the last instance, $M$~passed from $1$ to~$\frac{1}{2}$ without +any intermediate change. In geometry and mechanics, +it is necessary to consider quantities as +increasing or decreasing \emph{continuously}; that is, a magnitude +does not pass from one value to another without +passing through every intermediate value. Thus +if one point move towards another on a circle, both +the arc and its chord decrease continuously. Let $AB$ +\index{Arc and its chord, a continuously decreasing|EtSeq}% +(\Fig{1}) be an arc of a circle, the centre of which is~$O$. +\PageSep{8} +Let $A$ remain fixed, but let $B$, and with it the radius~$OB$, +move towards~$A$, the point~$B$ always remaining +on the circle. At every position of~$B$, suppose +the following figure. Draw $AT$ touching the circle at~$A$, +produce $OB$ to meet~$AT$ in~$T$, draw $BM$~and~$BN$ +perpendicular and parallel to~$OA$, and join~$BA$. Bisect +the arc~$AB$ in~$C$, and draw~$OC$ meeting the chord in~$D$ +and bisecting it. The right-angled triangles $ODA$ +and $BMA$ having a common angle, and also right +angles, are similar, as are also $BOM$ and~$TBN$. If +now we suppose $B$ to move towards~$A$, before $B$ +\Figure{1} +reaches~$A$, we shall have the following results: The +arc and chord~$BA$, the lines $BM$,~$MA$, $BT$,~$TN$, the +angles $BOA$,~$COA$,~$MBA$, and~$TBN$, will diminish +without limit; that is, assign a line and an angle, +however small, $B$~can be placed so near to~$A$ that the +lines and angles above alluded to shall be severally +less than the assigned line and angle. Again, $OT$~diminishes +and $OM$~increases, but neither without limit, +for the first is never less, nor the second greater, than +the radius. The angles $OBM$,~$MAB$, and~$BTN$, increase, +but not without limit, each being always less +than the right angle, but capable of being made as +\PageSep{9} +near to it as we please, by bringing~$B$ sufficiently near +to~$A$. + +So much for the magnitudes which compose the +figure: we proceed to consider their ratios, premising +that the arc~$AB$ is greater than the chord~$AB$, and +less than $BN + NA$. The triangle~$BMA$ being always +similar to~$ODA$, their sides change always in the same +proportion; and the sides of the first decrease without +limit, which is the case with only one side of the +second. And since $OA$~and~$OD$ differ by~$DC$, which +diminishes without limit as compared with~$OA$, the +ratio $OD ÷ OA$ is an increasing ratio whose limit is~$1$. +But $OD ÷ OA = BM ÷ BA$. We can therefore bring~$B$ +so near to~$A$ that $BM$~and~$BA$ shall differ by as +small a fraction of either of them as we please. + +To illustrate this result from the trigonometrical +tables, observe that if the radius~$OA$ be the linear +unit, and $\angle BOA = \theta$, $BM$~and~$BA$ are respectively +$\sin\theta$ and $2\sin\frac{1}{2}\theta$. Let $\theta = 1°$; then $\sin\theta = .0174524$ +and $2\sin\frac{1}{2}\theta = .0174530$; whence $2\sin\frac{1}{2}\theta ÷ \sin\theta = 1.00003$ very nearly, so that $BM$~differs from~$BA$ by +less than four of its own hundred-thousandth parts. +If $\angle BOA = 4'$, the same ratio is~$1.0000002$, differing +from unity by less than the hundredth part of the +difference in the last example. + +Again, since $DA$~diminishes continually and without +limit, which is not the case either with $OD$ or~$OA$, +the ratios $OD ÷ DA$ and $OA ÷ DA$ increase without +limit. These are respectively equal to $BM ÷ MA$ +and $BA ÷ MA$; whence it appears that, let a number +be ever so great, $B$~can be brought so near to~$A$, that +$BM$ and $BA$ shall each contain~$MA$ more times than +there are units in that number. Thus if $\angle BOA = 1°$, +$BM ÷ MA = 114.589$ and $BA ÷ MA = 114.593$ very +\PageSep{10} +nearly; that is, $BM$ and $BA$ both contain~$MA$ +more than $114$~times. If $\angle BOA = 4'$, $BM ÷ MA = 1718.8732$, +and $BA ÷ MA = 1718.8375$ very nearly; +or $BM$ and $BA$ both contain~$MA$ more than $1718$~times. + +No difficulty can arise in conceiving this result, if +the student recollect that the degree of greatness or +smallness of two magnitudes determines nothing as +to their ratio; since every quantity~$N$, however small, +can be divided into as many parts as we please, and +has therefore another small quantity which is its millionth +\Figure[nolabel]{1} +or hundred-millionth part, as certainly as if it +had been greater. There is another instance in the +line~$TN$, which, since $TBN$~is similar to~$BOM$, decreases +continually with respect to~$TB$, in the same +manner as does $BM$ with respect to~$OB$. + +The arc~$BA$ always lies between $BA$ and $BN + NA$, +or $BM + MA$; hence $\dfrac{\arc BA}{\chord BA}$ lies between $1$ and +$\dfrac{BM}{BA} + \dfrac{MA}{BA}$. But $\dfrac{BM}{BA}$~has been shown to approach +continually towards~$1$, and $\dfrac{MA}{BA}$~to decrease without +limit; hence $\dfrac{\arc BA}{\chord BA}$ continually approaches towards~$1$. +\PageSep{11} +If $\angle BOA = 1°$, $\dfrac{\arc BA}{\chord BA} = .0174533 ÷ .0174530 = 1.00002$, +very nearly. If $\angle BOA = 4'$, it is less than +$1.0000001$. + +We now proceed to illustrate the various phrases +which have been used in enunciating these and similar +propositions. + + +\Subsection{The Notion of Infinitely Small Quantities.} + +It appears that it is possible for two quantities $m$ +and $m + n$ to decrease together in such a way, that $n$~continually +decreases with respect to~$m$, that is, becomes +a less and less part of~$m$, so that $\dfrac{n}{m}$~also decreases +when $n$~and~$m$ decrease. Leibnitz,\footnote + {Leibnitz was a native of Leipsic, and died in 1716, aged~70. His dispute +\index{Leibnitz}% + with Newton, or rather with the English mathematicians in general, about +\index{Newton}% + the invention of Fluxions, and the virulence with which it was carried on, +\index{Fluxions}% + are well known. The decision of modern times appears to be that both Newton + and Leibnitz were independent inventors of this method. It has, perhaps, + not been sufficiently remarked how nearly several of their predecessors approached + the same ground; and it is a question worthy of discussion, whether + either Newton or Leibnitz might not have found broader hints in writings + accessible to both, than the latter was ever asserted to have received from + the former.} +in introducing +the Differential Calculus, presumed that in +such a case, $n$~might be taken so small as to be utterly +inconsiderable when compared with~$m$, so that $m + n$ +might be put for~$m$, or \textit{vice versa}, without any error at +all. In this case he used the phrase that $n$~is \emph{infinitely} +small with respect to~$m$. + +The following example will illustrate this term. +Since $(a + h)^{2} = a^{2} + 2ah + h^{2}$, it appears that if $a$~be +increased by~$h$, $a^{2}$~is increased by~$2ah + h^{2}$. But if $h$~be +taken very small, $h^{2}$~is very small with respect to~$h$, +for since $1:h :: h:h^{2}$, as many times as $1$~contains~$h$, +so many times does $h$~contain~$h^{2}$; so that by taking +\PageSep{12} +$h$~sufficiently small, $h$~may be made to be as many +times~$h^{2}$ as we please. Hence, in the words of Leibnitz, +if $h$~be taken \emph{infinitely} small, $h^{2}$~is \emph{infinitely} small +\index{Infinitely small, the notion of}% +with respect to~$h$, and therefore $2ah + h^{2}$ is the same +as~$2ah$; or if $a$~be increased by an infinitely small +quantity~$h$, $a^{2}$~is increased by another infinitely small +quantity~$2ah$, which is to~$h$ in the proportion of $2a$ +to~$1$. + +In this reasoning there is evidently an absolute +error; for it is impossible that $h$~can be so small, that +$2ah + h^{2}$ and $2ah$ shall be the same. The word \emph{small} +itself has no precise meaning; though the word \emph{smaller}, +\index{Small, has no precise meaning}% +or \emph{less}, as applied in comparing one of two magnitudes +with another, is perfectly intelligible. Nothing is +either small or great in itself, these terms only implying +a relation to some other magnitude of the same +kind, and even then varying their meaning with the +subject in talking of which the magnitude occurs, so +that both terms may be applied to the same magnitude: +thus a large field is a very small part of the +earth. Even in such cases there is no natural point +at which smallness or greatness commences. The +thousandth part of an inch may be called a small distance, +a mile moderate, and a thousand leagues great, +but no one can fix, even for himself, the precise mean +between any of these two, at which the one quality +ceases and the other begins. These terms are not +therefore a fit subject for mathematical discussion, +until some more precise sense can be given to them, +which shall prevent the danger of carrying away with +the words, some of the confusion attending their use +in ordinary language. It has been usual to say that +when $h$~decreases from any given value towards nothing, +$h^{2}$~will become \emph{small} as compared with~$h$, because, +\PageSep{13} +let a number be ever so great, $h$~will, before it becomes +nothing, contain $h^{2}$~more than that number of +times. Here all dispute about a standard of smallness +is avoided, because, be the standard whatever it may, +the proportion of~$h^{2}$ to~$h$ may be brought under it. It +is indifferent whether the thousandth, ten-thousandth, +or hundred-millionth part of a quantity is to be considered +small enough to be rejected by the side of the +whole, for let $h$~be $\dfrac{1}{1000}$, $\dfrac{1}{10,000}$, or $\dfrac{1}{100,000,000}$ of the +unit, and $h$~will contain~$h^{2}$, $1000$, $10,000$, or $100,000,000$ +of times. + +The proposition, therefore, that $h$~can be taken so +small that $2ah + h^{2}$ and~$2ah$ are rigorously equal, +though not true, and therefore entailing error upon +all its subsequent consequences, yet is of this character, +that, by taking $h$ sufficiently small, all errors may +be made as small as we please. The desire of combining +simplicity with the appearance of rigorous +demonstration, probably introduced the notion of infinitely +small quantities; which was further established +by observing that their careful use never led to +any error. The method of stating the above-mentioned +proposition in strict and rational terms is as follows: +If $a$~be increased by~$h$, $a^{2}$~is increased by $2ah + h^{2}$, +which, whatever may be the value of~$h$, is to~$h$ in the +proportion of $2a + h$ to~$1$. The smaller $h$~is made, +the more near does this proportion diminish towards +that of $2a$ to~$1$, to which it may be made to approach +within any quantity, if it be allowable to take $h$ as +small as we please. Hence the ratio, $\emph{increment of } a^{2} ÷ \emph{increment of } a$, is a decreasing ratio, whose limit is~$2a$. + +In further illustration of the language of Leibnitz, +\index{Leibnitz}% +we observe, that according to his phraseology, if $AB$~be +\PageSep{14} +an \emph{infinitely} small arc, the chord and arc~$AB$ are +equal, or the circle is a polygon of an \emph{infinite} number +of \emph{infinitely} small rectilinear sides. This should +be considered as an abbreviation of the proposition +proved (\PageRef{10}), and of the following: If a polygon +be inscribed in a circle, the greater the number of its +sides, and the smaller their lengths, the more nearly +will the perimeters of the polygon and circle be equal +to one another; and further, if any straight line be +given, however small, the difference between the perimeters +of the polygon and circle may be made less +than that line, by sufficient increase of the number of +sides and diminution of their lengths. Again, it would +be said (\Fig{1}) that if $AB$~be infinitely small, $MA$~is +infinitely less than~$BM$. What we have proved is, +that $MA$ may be made as small a part of~$BM$ as we +please, by sufficiently diminishing the arc~$BA$. + + +\Subsection{On Functions.} + +An algebraical expression which contains~$x$ in any +\index{Functions!definition of|EtSeq}% +way, is called a \emph{function} of~$x$. Such are $x^{2} + a^{2}$, +$\dfrac{a + x}{a - x}$, $\log(x + y)$, $\sin 2x$. An expression may be a +function of more quantities than one, but it is usual +only to name those quantities of which it is necessary +to consider a change in the value. Thus if in $x^{2} + a^{2}$ +$x$~only is considered as changing its value, this is +called a function of~$x$; if $x$~and~$a$ both change, it is +called a function of $x$~and~$a$. Quantities which change +their values during a process, are called \emph{variables}, and +\index{Variables!independent and dependent}% +those which remain the same, \emph{constants}; and variables +\index{Constants}% +which we change at pleasure are called \emph{independent}, +while those whose changes necessarily follow from +\PageSep{15} +\index{Variables!independent and dependent}% +the changes of others are called \emph{dependent}. Thus in +\Fig{1}, the length of the radius~$OB$ is a constant, the +arc~$AB$ is the independent variable, while $BM$,~$MA$, +the chord~$AB$,~etc., are dependent. And, as in algebra +we reason on numbers by means of general symbols, +each of which may afterwards be particularised +as standing for any number we please, unless specially +prevented by the conditions of the problem, so, in +treating of functions, we use general symbols, which +may, under the restrictions of the problem, stand for +any function whatever. The symbols used are the letters +$F$,~$f$, $\Phi$,~$\phi$,~$\psi$; $\phi(x)$~and~$\psi(x)$, or $\phi x$~and~$\psi x$, may +represent any functions of~$x$, just as $x$~may represent +any number. Here it must be borne in mind that $\phi$~and~$\psi$ +do not represent numbers which multiply~$x$, but +are \emph{the abbreviated directions to perform certain operations +with $x$ and constant quantities}. Thus, if $\phi x = x + x^{2}$, +$\phi$~is equivalent to a direction to add~$x$ to its +square, and the whole~$\phi x$ stands for the result of this +operation. Thus, in this case, $\phi(1) = 2$; $\phi(2) = 6$; +$\phi a = a + a^{2}$; $\phi(x + h) = x + h + (x + h)^{2}$; $\phi \sin x = \sin x + (\sin x)^{2}$. +It may be easily conceived that this +notion is useless, unless there are propositions which +are generally true of all functions, and which may be +made the foundation of general reasoning. + + +\Subsection{Infinite Series.} + +To exercise the student in this notation, we proceed +\index{Series|EtSeq}% +\index{Taylor's Theorem|EtSeq}% +to explain one of these functions which is of +most extensive application and is known by the name +of \emph{Taylor's Theorem}. If in~$\phi x$, any function of~$x$, the +value of~$x$ be increased by~$h$, or $x + h$~be substituted +instead of~$x$, the result is denoted by~$\phi(x + h)$. It +\PageSep{16} +will generally\footnote + {This word is used in making assertions which are for the most part +\index{Generally@\emph{Generally}, the word}% + true, but admit of exceptions, few in number when compared with the other + cases. Thus it generally happens that $x^{2} - 10x + 40$ is greater than~$15$, with + the exception only of the case where $x = 5$. It is generally true that a line + which meets a circle in a given point meets it again, with the exception only + of the tangent.} +happen that this is either greater or +less than~$\phi x$, and $h$~is called the \emph{increment} of~$x$, and +\index{Increment}% +$\phi(x + h) - \phi x$ is called the \emph{increment} of~$\phi x$, which is +negative when $\phi(x + h) < \phi x$. It may be proved +that $\phi(x + h)$ can generally be expanded in a series +of the form +\[ +\phi x + ph + qh^{2} + rh^{3} + \etc.,\quad \textit{ad infinitum}, +\] +which contains none but whole and positive powers +of~$h$. It will happen, however, in many functions, +that one or more values can be given to~$x$ for which +it is impossible to expand $f(x + h)$ without introducing +negative or fractional powers. These cases are +considered by themselves, and the values of~$x$ which +produce them are called \emph{singular} values. +\index{Singular values}% + +As the notion of a series which has no end of its +terms, may be new to the student, we will now proceed +to show that there may be series so constructed, +that the addition of any number of their terms, however +great, will always give a result less than some +determinate quantity. Take the series +\[ +1 + x + x^{2} + x^{3} + x^{4} + \etc., +\] +in which $x$~is supposed to be less than unity. The +first two terms of this series may be obtained by dividing +$1 - x^{2}$ by $1 - x$; the first three by dividing +$1 - x^{3}$ by $1 - x$; and the first $n$~terms by dividing +$1 - x^{n}$ by $1 - x$. If $x$~be less than unity, its successive +powers decrease without limit;\footnote + {This may be proved by means of the proposition established in \Title{Study + of Mathematics} (Chicago: The Open Court Publishing~Co., Reprint Edition), + page~247. For $\dfrac{m}{n} × \dfrac{n}{m}$ is formed (if $m$~be less than~$n$) by dividing $\dfrac{m}{n}$ into $n$~parts, + and taking away $n - m$ of them.} +that is, there is +\PageSep{17} +no quantity so small, that a power of~$x$ cannot be +found which shall be smaller. Hence by taking $n$~sufficiently +great, $\dfrac{1 - x^{n}}{1 - x}$ or $\dfrac{1}{1 - x} - \dfrac{x^{n}}{1 - x}$ may be +brought as near to~$\dfrac{1}{1 - x}$ as we please, than which, +however, it must always be less, since $\dfrac{x^{n}}{1 - x}$ can never +entirely vanish, whatever value $n$~may have, and therefore +there is always something subtracted from $\dfrac{1}{1 - x}$. +It follows, nevertheless, that $1 + x + x^{2} + \etc.$, if we +are at liberty to take as many terms as we please, can +be brought as near as we please to~$\dfrac{1}{1 - x}$, and in this +sense we say that +\[ +\frac{1}{1 - x} = 1 + x + x^{2} + x^{3} + \etc.,\quad\textit{ad infinitum}. +\] + + +\Subsection{Convergent and Divergent Series.} + +A series is said to be \emph{convergent} when the sum of +its terms tends towards some limit; that is, when, by +taking any number of terms, however great, we shall +never exceed some certain quantity. On the other +hand, a series is said to be \emph{divergent} when the sum of +a number of terms may be made to surpass any quantity, +however great. Thus of the two series, +\[ +1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \etc. +\] +and +\[ +1 + 2 + 4 + 8 + \etc.\Add{,} +\] +the first is convergent, by what has been shown, and +the second is evidently divergent. A series cannot be +convergent, unless its separate terms decrease, so as, +\PageSep{18} +at last, to become less than any given quantity. And +the terms of a series may at first increase and afterwards +decrease, being apparently divergent for a finite +number of terms, and convergent afterwards. It will +only be necessary to consider the latter part of the +series. + +Let the following series consist of terms decreasing +without limit: +\[ +a + b + c + d + \dots + k + l + m + \dots, +\] +which may be put under the form +\[ +%[** TN: Small parentheses in the original here and below, as noted] +a\left(1 + \frac{b}{a} + + \frac{c}{b}\, \frac{b}{a} + + \frac{d}{c}\, \frac{c}{b}\, \frac{b}{a} + \etc.\right); +\] +the same change of form may be made, beginning +from any term of the series, thus: +\[ +%[** TN: Small ()] +k + l + m + \etc. + = k\left(1 + \frac{l}{k} + \frac{m}{l}\, \frac{l}{k} + \etc.\right). +\] +We have introduced the new terms, $\dfrac{b}{a}$,~$\dfrac{c}{b}$,~etc., or the +ratios which the several terms of the original series +bear to those immediately preceding. It may be shown +(1)~that if the terms of the series $\dfrac{b}{a}$,~$\dfrac{c}{b}$,~$\dfrac{d}{c}$, etc., come +at last to be less than unity, and afterwards either +continue to approximate to a limit which is less than +unity, or decrease without limit, the series $a + b + c + \etc.$, +is convergent; (2)~if the limit of the terms +$\dfrac{b}{a}$,~$\dfrac{c}{b}$,~etc., is either greater than unity, or if they increase +without limit, the series is divergent. + +(1\ia). Let $\dfrac{l}{k}$~be the first which is less than unity, +and let the succeeding ratios $\dfrac{m}{l}$,~etc., decrease, either +with or without limit, so that $\dfrac{l}{k} > \dfrac{m}{l} > \dfrac{n}{m}$, etc.; +whence it follows, that of the two series, +\PageSep{19} +\begin{align*} +%[** TN: Small ()] +k\biggl(1 &+ \frac{l}{k} + + \frac{l}{k}\, \PadTo{\dfrac{m}{l}}{\dfrac{l}{k}} + + \frac{l}{k}\, + \PadTo{\dfrac{m}{l}}{\dfrac{l}{k}}\, + \PadTo{\dfrac{n}{m}}{\dfrac{l}{k}} + + \etc.\biggr), \\ +k\biggl(1 &+ \frac{l}{k} + + \frac{l}{k}\, \frac{m}{l} + + \frac{l}{k}\, \frac{m}{l}\, \frac{n}{m} + + \etc.\biggr), +\end{align*} +the first is greater than the second. But since $\dfrac{l}{k}$~is +less than unity, the first can never surpass $k × \dfrac{1}{1 - \dfrac{l}{k}}$, +or~$\dfrac{k^{2}}{k - l}$, and is convergent; the second is therefore +convergent. But the second is no other than $k + l + m + \etc.$; +therefore the series $a + b + c + \etc.$, is convergent +from the term~$k$. + +\Chg{(1\ib.)}{(1\ib).} Let $\dfrac{l}{k}$~be less than unity, and let the successive +ratios $\dfrac{l}{k}$,~$\dfrac{m}{l}$,~etc., increase, never surpassing a +limit~$A$, which is less than unity. Hence of the two +series, +\begin{align*} +%[** TN: Small ()] +k(1 &+ \PadTo{\dfrac{l}{k}}{A} + + \PadTo{\dfrac{l}{k}}{A}\,\PadTo{\dfrac{m}{l}}{A} + + \PadTo{\dfrac{l}{k}}{A}\,\PadTo{\dfrac{m}{l}}{A}\,\PadTo{\dfrac{n}{m}}{A} + + \etc.), \\ +k\biggl(1 &+ \frac{l}{k} + + \frac{l}{k}\, \frac{m}{l} + + \frac{l}{k}\, \frac{m}{l}\, \frac{n}{m} + + \etc.\biggr), +\end{align*} +the first is the greater. But since $A$~is less than unity, +the first is convergent; whence, as before, $a + b + c + \etc.$, +converges from the term~$k$. + +(2) The second theorem on the divergence of series +we leave to the student's consideration, as it is not +immediately connected with our object. + + +\Subsection{Taylor's Theorem. Derived Functions.} + +We now proceed to the series +\index{Derivatives}% +\index{Derived Functions|EtSeq}% +\index{Functions!derived|EtSeq}% +\index{Taylor's Theorem|EtSeq}% +\[ +ph + qh^{2} + rh^{3} + sh^{4} + \etc., +\] +in which we are at liberty to suppose $h$ as small as +we please. The successive ratios of the terms to those +\PageSep{20} +immediately preceding are $\dfrac{qh^{2}}{ph}$ or~$\dfrac{q}{p}h$, $\dfrac{rh^{3}}{qh^{2}}$ or~$\dfrac{r}{q}h$, +$\dfrac{sh^{4}}{rh^{3}}$ or $\dfrac{s}{r}h$,~etc. If, then, the terms $\dfrac{q}{p}$,~$\dfrac{r}{q}$,~$\dfrac{s}{r}$, etc., +are always less than a finite limit~$A$, or become so after +a definite number of terms, $\dfrac{q}{p}h$,~$\dfrac{r}{q}h$,~etc., will always +be, or will at length become, less than~$Ah$. And since $h$~may +be what we please, it may be so chosen that $Ah$~shall +be less than unity, for which $h$~must be less than~$\dfrac{1}{A}$. +In this case, by theorem~(1\ib), the series is convergent; +it follows, therefore, that a value of~$h$ can +always be found so small that $ph + qh^{2} + rh^{3} + \etc.$, +shall be convergent, at least unless the coefficients +$p$,~$q$,~$r$,~etc., be such that the ratio of any one to the +preceding increases without limit, as we take more +distant terms of the series. This never happens in +the developments which we shall be required to consider +in the Differential Calculus. + +We now return to $\phi(x +h)$, which we have asserted +(\PageRef{16}) can be expanded (with the exception +of some particular values of~$x$) in a series of the form +$\phi x + ph + qh^{2} + \etc$. The following are some instances +of this development derived from the Differential +Calculus, most of which are also to be found in +treatises on algebra: +\index{Logarithms}% + +{\scriptsize +\begin{alignat*}{4} +(x + h)^{n} &= x^{n} + &+ nx^{n-1}h + &&+ n(n - 1)x^{n-2} \frac{h^{2}}{2} + &&+ n(n - 1)(n - 2)x^{n-3} \frac{h^{3}}{2·3} &\ \etc.\Add{,} \\ +% +a^{x + h} &= a^{x} + &+ ka^{x} h\rlap{\normalsize\footnotemark[1]} + &&+ k^{2} a^{x} \frac{h^{2}}{2} + &&+ k^{3} a^{x} \frac{h^{3}}{2·3} &\ \etc.\Add{,} \\ +% +\log(x + h) &= \log x + &+ \frac{1}{x}\, h + &&- \frac{1}{x^{2}}\, \frac{h^{2}}{2} + &&+ \frac{2}{x^{3}}\, \frac{h^{3}}{2·3} &\ \etc.\Add{,} \\ +% +\sin(x + h) &= \sin x + &+ \cos x\, h + &&- \sin x\, \frac{h^{2}}{2}\rlap{\normalsize\footnotemark[2]} + &&- \cos x\, \frac{h^{3}}{2·3} &\ \etc.\Add{,} \\ +% +%[** TN: Moved up from top of page21] +\cos(x + h) &= \cos x + &- \sin x\, h + &&- \cos x\, \frac{h^{2}}{2} + &&+ \sin x\, \frac{h^{3}}{2·3} &\ \etc. +\end{alignat*}}% +\footnotetext[1]{Here $k$~is the Naperian or hyperbolic logarithm of~$a$; that is, the common + logarithm of~$a$ divided by~$.434294482$.}% +\footnotetext[2]{In the last two series the terms are positive and negative in pairs.} +\PageSep{21} + +It appears, then, that the development of~$\phi(x + h)$ +consists of certain functions of~$x$, the first of which is +$\phi x$~itself, and the remainder of which are multiplied +by $h$,~$\dfrac{h^{2}}{2}$, $\dfrac{h^{3}}{2·3}$, $\dfrac{h^{4}}{2·3·4}$, and so on. It is usual to denote +the coefficients of these divided powers of~$h$ by $\phi' x$, +$\phi'' x$, $\phi''' x$,\footnote + {Called \emph{derived functions} or \emph{derivatives}.---\Ed.} +\index{Derivatives}% +\index{Derived Functions}% +\index{Functions!derived}% +etc., where $\phi'$,~$\phi''$,~etc., are merely functional +symbols, as is $\phi$~itself; but it must be recollected +that $\phi' x$,~$\phi'' x$,~etc., are rarely, if ever, employed +to signify anything except the coefficients of~$h$, $\dfrac{h^{2}}{2}$,~etc., +in the development of~$\phi(x + h)$. Hence this development +is usually expressed as follows: +\[ +\phi(x + h) + = \phi x + \phi' x\, h + \phi''x\, \frac{h^{2}}{2} + \phi''' x\, \frac{h^{3}}{2·3} + \etc. +\] + +{\Loosen Thus, when $\phi x = x^{n}$, $\phi' x = nx^{n-1}$, $\phi'' x = n(n - 1)x^{n-2}$, etc.; +when $\phi x = \sin x$, $\phi' x = \cos x$, $\phi'' x = -\sin x$,~etc. +In the first case $\phi'(x + h) = n(x + h)^{n-1}$, +$\phi''(x + h) = n(n - 1)(x + h)^{n-2}$; and in the second +$\phi'(x + h) = \cos (x + h)$, $\phi''(x + h) = -\sin(x + h)$.} + +The following relation exists between $\phi x$,~$\phi' x$, +$\phi'' x$,~etc. In the same manner as $\phi' x$~is the coefficient +of~$h$ in the development of~$\phi(x + h)$, so $\phi'' x$~is the coefficient +of~$h$ in the development of~$\phi'(x + h)$, and +$\phi''' x$~is the coefficient of~$h$ in the development of~$\phi''(x + h)$; +$\phi^{\text{iv}} x$~is the coefficient of~$h$ in the development +of $\phi'''(x + h)$, and so on. + +The proof of this is equivalent to \emph{Taylor's Theorem} +already alluded to (\PageRef{15}); and the fact may be +verified in the examples already given. When $\phi x = a^{x}$, +$\phi' x = ka^{x}$, and $\phi'(x + h) = ka^{x+h} = k(a^{x} + ka^{x}\, h + \etc.)$. +The coefficient of~$h$ is here~$k^{2} a^{x}$, which is the +\PageSep{22} +same as~$\phi'' x$. (See the second example of the preceding +table.) Again, $\phi''(x + h) = k^{2}a^{x+h} = k^{2}(a^{x} + ka^{x}\, h + \etc.)$, +in which the coefficient of~$h$ is~$k^{3}a^{x}$, the +same as~$\phi''' x$. Again, if $\phi x = \log x$, $\phi' x = \dfrac{1}{x}$, and +$\phi'(x + h) = \dfrac{1}{x + h} = \dfrac{1}{x} - \dfrac{h}{x^{2}} + \etc.$, as appears by +common division. Here the coefficient of~$h$ is~$-\dfrac{1}{x^{2}}$, +which is the same as $\phi'' x$~in the third example. Also +$\phi''(x + h) = -\dfrac{1}{(x + h)^{2}} = -(x + h)^{-2}$, which by the +Binomial Theorem is $-(x^{-2} - 2x^{-3}\, h + \etc.)$. The +coefficient of~$h$ is~$2x^{-3}$ or~$\dfrac{2}{x^{3}}$, which is~$\phi''' x$ in the +same example. + + +\Subsection{Differential Coefficients.} + +It appears, then, that if we are able to obtain the +\index{Coefficients, differential|EtSeq}% +\index{Differential coefficients|EtSeq}% +coefficient of~$h$ in the development of \emph{any} function +whatever of~$x + h$, we can obtain all the other coefficients, +since we can thus deduce $\phi' x$ from~$\phi x$, $\phi'' x$ +from~$\phi' x$, and so on. It is usual to call~$\phi' x$ the first +differential coefficient of~$\phi x$, $\phi'' x$~the second differential +coefficient of~$\phi x$, or the first differential coefficient +of~$\phi' x$; $\phi''' x$~the third differential coefficient of~$\phi x$, +or the second of~$\phi' x$, or the first of~$\phi'' x$; and so on.\footnote + {The first, second, third, etc., differential coefficients, as thus obtained, + are also called the first, second, third, etc., \emph{derivatives}.---\Ed.} +\index{Derivatives}% +The name is derived from a method of obtaining~$\phi' x$, +etc., which we now proceed to explain. + +Let there be any function of~$x$, which we call~$\phi x$, +in which $x$~is increased by an increment~$h$; the function +then becomes +\[ +\phi x + \phi' x\, h + + \phi'' x\, \frac{h^{2}}{2} + + \phi''' x\, \frac{h^{3}}{2·3} + \etc. +\] +\PageSep{23} +The original value~$\phi x$ is increased by the increment +\[ +\phi' x\, h + \phi'' x\, \frac{h^{2}}{2} x + + \phi''' x\, \frac{h^{3}}{2·3} + \etc.; +\] +whence ($h$~being the increment of~$x$) +\[ +\frac{\emph{increment of } \phi x}{\emph{increment of } x} + = \phi' x + \phi'' x\, \frac{h}{2} x + + \phi''' x\, \frac{h^{2}}{2·3} + \etc., +\] +which is an expression for the ratio which the increment +of a function bears to the increment of its variable. +It consists of two parts. The one,~$\phi' x$, into +which $h$~does not enter, depends on $x$~only; the remainder +is a series, every term of which is multiplied +by some power of~$h$, and which therefore diminishes +as $h$~diminishes, and may be made as small as we +please by making $h$~sufficiently small. + +To make this last assertion clear, observe that all +the ratio, except its first term~$\phi' x$, may be written as +follows: +\[ +%[** TN: Small () in the original] +h\left(\phi'' x\, \frac{1}{2} + \phi''' x\, \frac{h}{2·3} + \etc.\right); +\] +the second factor of which (\PageRef{19}) is a convergent +series whenever $h$~is taken less than~$\dfrac{1}{A}$, where $A$~is +the limit towards which we approximate by taking +the coefficients $\phi'' x × \dfrac{1}{2}$, $\phi''' x × \dfrac{1}{2·3}$,~etc., and forming +the ratio of each to the one immediately preceding. +This limit, as has been observed, is finite in +every series which we have occasion to use; and +therefore a value for~$h$ can be chosen so small, that +for it the series in the last-named formula is convergent; +still more will it be so for every smaller value +of~$h$. Let the series be called~$P$. If $P$~be a finite quantity, +which decreases when $h$~decreases, $Ph$~can be +made as small as we please by sufficiently diminishing~$h$; +\PageSep{24} +whence $\phi' x + Ph$ can be brought as near as we +please to~$\phi' x$. Hence the ratio of the increments of +$\phi x$ and~$x$, produced by changing $x$ into~$x + h$, though +never equal to~$\phi' x$, approaches towards it as $h$~is diminished, +and may be brought as near as we please +to it, by sufficiently diminishing~$h$. Therefore to find +the coefficient of~$h$ in the development of~$\phi(x + h)$, +find $\phi(x + h) - \phi x$, divide it by~$h$, and find the limit +towards which it tends as $h$~is diminished. + +In any series such as +\index{Series|EtSeq}% +\[ +a + bh + ch^{2} + \dots + kh^{n} + lh^{n+1} + mh^{n+2} + \etc. +\] +which is such that some given value of~$h$ will make it +convergent, it may be shown that $h$~can be taken so +small that any one term shall contain all the succeeding +ones as often as we please. Take any one term, +as~$kh^{n}$. It is evident that, be $h$ what it may, +\[ +kh^{n} : lh^{n+1} + mh^{n+2} + \etc.,\ ::\ k : lh + mh^{2} + \etc., +\] +the last term of which is $h(l + mh + \etc.)$. By reasoning +similar to that in the last paragraph, we can +show that this may be made as small as we please, +since one factor is a series which is always finite when +$h$~is less than~$\dfrac{1}{A}$, and the other factor~$h$ can be made +as small as we please. Hence, since $k$~is a given +quantity, independent of~$h$, and which therefore remains +the same during all the changes of~$h$, the series +$h(l + mh + \etc.)$ can be made as small a part of~$k$ as +we please, since the first diminishes without limit, +and the second remains the same. By the proportion +above established, it follows then that $lh^{n+1} + mh^{n+2} + \etc.$, +can be made as small a part as we please of~$kh^{n}$. +It follows, therefore, that if, instead of the full +development of~$\phi(x + h)$, we use only its two first +\PageSep{25} +terms $\phi x + \phi' x\, h$, the error thereby introduced may, +by taking $h$ sufficiently small, be made as small a portion +as we please of the small term~$\phi' x\, h$. + + +\Subsection{The Notation of the Differential Calculus.} + +The first step usually made in the Differential Calculus +\index{Calculus, notation of}% +\index{Notation!of the Differential Calculus}% +is the determination of~$\phi' x$ for all possible values +of~$\phi x$, and the construction of general rules for +that purpose. Without entering into these we proceed +to explain the notation which is used, and to apply +the principles already established to the solution +of some of those problems which are the peculiar +province of the Differential Calculus. + +When any quantity is increased by an increment, +which, consistently with the conditions of the problem, +may be supposed as small as we please, this increment +is denoted, not by a separate letter, but by +prefixing the letter~$d$, either followed by a full stop or +not, to that already used to signify the quantity. For +example, the increment of~$x$ is denoted under these +circumstances by~$dx$; that of~$\phi x$ by~$d.\phi x$; that of~$x^{n}$ +by~$d.x^{n}$. If instead of an increment a decrement +be used, the sign of~$dx$, etc., must be changed in all +expressions which have been obtained on the supposition +of an increment; and if an increment obtained +by calculation proves to be negative, it is a sign that +a quantity which we imagined was increased by our +previous changes, was in fact diminished. Thus, if +$x$~becomes $x + dx$, $x^{2}$~becomes $x^{2} + d.x^{2}$. But this is +also $(x + dx)^{2}$ or $x^{2} + 2x\, dx + (dx)^{2}$; whence $d.x^{2} = 2x\, dx + (dx)^{2}$. +Care must be taken not to confound +$d.x^{2}$, the increment of~$x^{2}$, with~$(dx)^{2}$, or, as it is often +written,~$dx^{2}$, the square of the increment of~$x$. Again, +\PageSep{26} +if $x$~becomes $x + dx$, $\dfrac{1}{x}$ becomes $\dfrac{1}{x} + d.\dfrac{1}{x}$ and the +change of~$\dfrac{1}{x}$ is $\dfrac{1}{x + dx} - \dfrac{1}{x}$ or $-\dfrac{dx}{x^{2} + x\, dx}$; showing +that an increment of~$x$ produces a decrement in~$\dfrac{1}{x}$. + +It must not be imagined that because $x$~occurs in +the symbol~$dx$, the value of the latter in any way depends +upon that of the former: both the first value of~$x$, +and the quantity by which it is made to differ from +its first value, are at our pleasure, and the letter~$d$ must +merely be regarded as an abbreviation of the words +``\emph{difference of}.'' In the first example, if we divide +\index{Differences!of increments}% +both sides of the resulting equation by~$dx$, we have +$\dfrac{d.x^{2}}{dx} = 2x + dx$. The smaller $dx$~is supposed to be, +the more nearly will this equation assume the form +$\dfrac{d.x^{2}}{dx} = 2x$, and the ratio of $2x$ to~$1$ is the limit of the +\index{Limits|EtSeq}% +ratio of the increment of~$x^{2}$ to that of~$x$; to which +this ratio may be made to approximate as nearly as +we please, but which it can never actually reach. In +the Differential Calculus, the limit of the ratio only is +retained, to the exclusion of the rest, which may be +explained in either of the two following ways: + +(1) The fraction $\dfrac{d.x^{2}}{dx}$ may be considered as standing, +not for any value which it can actually have as +long as $dx$~has a real value, but for the limit of all +those values which it assumes while $dx$~diminishes. +In this sense the equation $\dfrac{d.x^{2}}{dx} = 2x$ is strictly true. +But here it must be observed that the algebraical +meaning of the sign of division is altered, in such a +way that it is no longer allowable to use the numerator +and denominator separately, or even at all to consider +\PageSep{27} +them as quantities. If $\dfrac{dy}{dx}$~stands, not for the +ratio of two quantities, but for the limit of that ratio, +which cannot be obtained by taking any real value of~$dx$, +however small, the whole $\dfrac{dy}{dx}$ may, by convention, +have a meaning, but the separate parts $dy$ and~$dx$ +have none, and can no more be considered as separate +quantities whose ratio is~$\dfrac{dy}{dx}$, than the two loops +of the figure~$8$ can be considered as separate numbers +whose sum is eight. This would be productive of no +great inconvenience if it were never required to separate +the two; but since all books on the Differential +Calculus and its applications are full of examples in +which deductions equivalent to assuming $dy = 2x\, dx$ +are drawn from such an equation as $\dfrac{dy}{dx} = 2x$, it becomes +necessary that the first should be explained, independently +of the meaning first given to the second. +It may be said, indeed, that if $y = x^{2}$, it follows that +$\dfrac{dy}{dx} = 2x + dx$, in which, \emph{if we make $dx = 0$}, the result +is $\dfrac{dy}{dx} = 2x$. But if $dx = 0$, $dy$~also~$= 0$, and this +equation should be written $\dfrac{0}{0} = 2x$, as is actually done +in some treatises on the Differential Calculus,\footnote + {This practice was far more common in the early part of the century + than now, and was due to the precedent of Euler (1755). For the sense in +\index{Euler}% + which Euler's view was correct, see the \Title{Encyclopedia Britannica}, art.\ \Title{Infinitesimal + Calculus}, Vol.~XII, p.~14, 2nd~column.---\Ed.} +to the +great confusion of the learner. Passing over the difficulties\footnote + {See \Title{Study of Mathematics} (Reprint Edition, Chicago: The Open Court + Publishing~Co., 1898), page~126.} +of the fraction~$\dfrac{0}{0}$, still the former objection +recurs, that the equation $dy = 2x\, dx$ cannot be used +\PageSep{28} +(and it \emph{is} used even by those who adopt this explanation) +without supposing that~$0$, which merely implies +an absence of all magnitude, can be used in different +senses, so that one~$0$ may be contained in another a +certain number of times. This, even if it can be considered +as intelligible, is a notion of much too refined +a nature for a beginner. + +(2) The presence of the letter~$d$ is an indication, +not only of an increment, but of an increment which +we are at liberty to suppose as small as we please. +The processes of the Differential Calculus are intended +to deduce relations, not between the ratios of different +increments, but between the limits to which those ratios +approximate, when the increments are decreased. +And it may be true of some parts of an equation, that +though the taking of them away would alter the relation +between $dy$ and~$dx$, it would not alter the limit +towards which their ratio approximates, when $dx$ +and~$dy$ are diminished. For example, $dy = 2x\, dx + (dx)^{2}$. +If $x = 4$ and $dx = .01$, then $dy = .0801$ and +$\dfrac{dy}{dx} = 8.01$. If $dx = .0001$, $dy = .00080001$ and $\dfrac{dy}{dx} = 8.0001$. +The limit of this ratio, to which we shall +come still nearer by making $dx$ still smaller, is~$8$. The +term~$(dx)^{2}$, though its presence affects the value of~$dy$ +and the ratio~$\dfrac{dy}{dx}$, does not affect the limit of the latter, +for in $\dfrac{dy}{dx}$ or $2x + dx$, the latter term~$dx$, which arose +from the term~$(dx)^{2}$, diminishes continually and without +limit. If, then, we throw away the term~$(dx)^{2}$, +the consequence is that, make $dx$ what we may, we +never obtain~$dy$ as it would be if correctly deduced +from the equation $y = x^{2}$, but we obtain the limit of +the ratio of~$dy$ to~$dx$. If we throw away all powers of~$dx$ +\PageSep{29} +above the first, and use the equations so obtained, +all ratios formed from these last, or their consequences, +are themselves the limiting ratios of which we are in +search. \emph{The equations which we thus use are not absolutely +true in any case, but may be brought as near as we +please to the truth}, by making $dy$~and~$dx$ sufficiently +small. If the student at first, instead of using $dy = 2x\, dx$, +were to write it thus, $dy = 2x\, dx + \etc.$, the \emph{etc.}\ +would remind him that there are other terms; \emph{necessary}, +if the value of~$dy$ corresponding to any value of~$dx$ +is to be obtained; \emph{unnecessary}, if the \emph{limit} of the +ratio of $dy$ to~$dx$ is all that is required. + +We must adopt the first of these explanations when +$dy$ and $dx$ appear in a fraction, and the second when +they are on opposite sides of an equation. + + +\Subsection{Algebraical Geometry.} + +If two straight lines be drawn at right angles to +each other, dividing the whole of their plane into four +parts, one lying in each right angle, the situation +of any point is determined when we know, (1)~in +which angle it lies, and (2)~its perpendicular distances +from the two right lines. Thus (\Fig{2}) the point~$P$ +lying in the angle~$AOB$, is known when $PM$~and~$PN$, +or when $OM$~and~$PM$ are known; for, though there +is an infinite number of points whose distance from~$OA$ +only is the same as that of~$P$, and an infinite number +of others, whose distance from~$OB$ is the same as +that of~$P$, there is no other point whose distances +from both lines are the same as those of~$P$. The line~$OA$ +is called the axis of~$x$, because it is usual to denote +any variable distance measured on or parallel to~$OA$ +by the letter~$x$. For a similar reason, $OB$~is called +\PageSep{30} +\index{Co-ordinates}% +the axis of~$y$. The \emph{co-ordinates}\footnote + {The distances $OM$ and~$MP$ are called the \emph{co-ordinates} of the point~$P$. It + is moreover usual to call the co-ordinate~$OM$, the \emph{abscissa}, and $MP$, the \emph{ordinate}, + of the point~$P$.} +or perpendicular distances +of a point~$P$ which is supposed to vary its position, +are thus denoted by $x$~and~$y$; hence $OM$ or~$PN$ +is~$x$, and $PM$ or~$ON$ is~$y$. Let a linear unit be chosen, +so that any number may be represented by a straight +line. Let the point~$M$, setting out from~$O$, move in +the direction~$OA$, always carrying with it the indefinitely +extended line~$MP$ perpendicular to~$OA$. While +this goes on, let $P$~move upon the line~$MP$ in such a +way, that $MP$ or~$y$ is always equal to a given function +of~$OM$ or~$x$; for example, let $y = x^{2}$, or let the number +\index{Parabola, the}% +\Figure{2} +of units in~$PM$ be the square of the number of +units in~$OM$. As $O$~moves towards~$A$, the point~$P$ +will, by its motion on~$MP$, compounded with the motion +of the line $MP$ itself, describe a curve~$OP$, in +which $PM$~is less than, equal to, or greater than,~$OM$, +according as $OM$~is less than, equal to, or greater +than the linear unit. It only remains to show how +the other branch of this curve is deduced from the +equation $y = x^{2}$. And to this end we shall first have +to interpolate a few remarks. +\PageSep{31} + + +\Subsection{On the Connexion of the Signs of Algebraical and +the Directions of Geometrical Magnitudes.} + +It is shown in algebra, that if, through misapprehension +\index{Signs|EtSeq}% +of a problem, we measure in one direction, a +line which ought to lie in the exactly opposite direction, +or if such a mistake be a consequence of some +previous misconstruction of the figure, any attempt +to deduce the length of that line by algebraical reasoning, +will give a negative quantity as the result. +And conversely it may be proved by any number of +examples, that when an equation in which $a$~occurs +has been deduced strictly on the supposition that $a$~is +a line measured in one direction, a change of sign in~$a$ +will turn the equation into that which would have +been deduced by the same reasoning, had we begun +by measuring the line~$a$ in the contrary direction. +Hence the change of~$+a$ into~$-a$, or of~$-a$ into~$+a$, +corresponds in geometry to a change of direction of +the line represented by~$a$, and \textit{vice versa}. + +In illustration of this general fact, the following +\index{Circle, equation of|EtSeq}% +\index{Circle cut by straight line, investigated|EtSeq}% +problem may be useful. Having a circle of given radius, +whose centre is in the intersection of the axes +of $x$~and~$y$, and also a straight line cutting the axes in +two given points, required the co-ordinates of the +points (if any) in which the straight line cuts the circle. +Let $OA$, the radius of the circle~$= r$, $OE = a$, +$OF = b$, and let the co-ordinates of~$P$, one of the +points of intersection required, be $OM = x$, $MP = y$. +(\Fig{3}.) The point~$P$ being in the circle whose radius +is~$r$, we have from the right-angled triangle~$OMP$, +$x^{2} + y^{2} = r^{2}$, which equation belongs to the co-ordinates +of every point in the circle, and is called +\PageSep{32} +the equation of the circle. Again, $EM : MP :: EO : OF$ +by similar triangles; or $a - x : y :: a : b$, whence $ay + bx = ab$, +which is true, by similar reasoning, for every +point of the line~$EF$. But for a point~$P'$ lying in~$EF$ +produced, we have $EM' : M'P' :: EO : OF$, or $x + a : y :: a : b$, +whence $ay - bx = ab$, an equation which may +be obtained from the former by changing the sign of~$x$; +and it is evident that the direction of~$x$, in the +\Figure{3} +second case, is opposite to that in the first. Again, +for a point~$P''$ in $FE$ produced, we have $EM'' : M''P'' :: EO : OF$, +or $x - a : y :: a : b$, whence $bx - ay = ab$, which +may be deduced from the first by changing the sign +of~$y$; and it is evident that $y$~is measured in different +directions in the first and third cases. Hence the +equation $ay + bx = ab$ belongs to all parts of the +straight line~$EF$, if we agree to consider $M''P''$ as +negative, when $MP$~is positive, and $OM'$~as negative +\PageSep{33} +when $OM$~is positive. Thus, if $OE = 4$, and $OF = 5$, +and $OM = 1$, we can determine~$MP$ from the equation +$ay + bx = ab$, or $4y + 5 = 20$, which gives $y$~or $MP = 3\frac{3}{4}$. +But if $OM'$~be $1$ in length, we can determine~$M'P'$ +either by calling $OM'$,~$1$, and using the equation +$ay - bx = ab$, or calling $OM'$,~$-1$, and using the equation +$ay + bx = ab$, as before. Either gives $M'P' = 6\frac{1}{4}$. +The latter method is preferable, inasmuch as it enables +us to contain, in one investigation, all the different +cases of a problem. + +We shall proceed to show that this may be done +in the present instance. We have to determine the +co-ordinates of the point~$P$, from the following equations: +\begin{align*} +ay + bx &= ab, \\ +x^{2} + y^{2} &= r^{2}. +\end{align*} +Substituting in the second the value of~$y$ derived from +the first, or $b\left(\dfrac{a - x}{b}\right)$, we have +\[ +x^{2} + b^{2}\, \frac{(a - x)^{2}}{a^{2}} = r^{2}, +\] +or +\[ +(a^{2} + b^{2}) x^{2} - 2ab^{2}x + a^{2}(b^{2} - r^{2}) = 0; +\] +and proceeding in a similar manner to find~$y$, we have +\[ +(a^{2} + b^{2}) y^{2} - 2a^{2}by + b^{2}(a^{2} - r^{2}) = 0, +\] +which \Typo{give}{gives} +\begin{align*} +x &= a\, \frac{b^{2} ± \sqrt{(a^{2} + b^{2})r^{2} - a^{2}b^{2}}}{a^{2} + b^{2}}, \\ +y &= b\, \frac{a^{2} \mp \sqrt{(a^{2} + b^{2})r^{2} - a^{2}b^{2}}}{a^{2} + b^{2}}; +\end{align*} +the upper or the lower sign to be taken in both. +Hence when $(a^{2} + b^{2})r^{2} > a^{2}b^{2}$, that is, when $r$~is greater +than the perpendicular let fall from~$O$ upon~$EF$, which +perpendicular is +\PageSep{34} +\[ +\frac{ab}{\sqrt{a^{2} + b^{2}}}, +\] +there are two points of intersection. When $(a^{2} + b^{2})r^{2} = a^{2}b^{2}$, +the two values of~$x$ become equal, and also +those of~$y$, and there is only one point in which the +straight line meets the circle; in this case $EF$~is a +tangent to the circle. And if $(a^{2} + b^{2})r^{2} < a^{2}b^{2}$, the +values of $x$~and~$y$ are impossible, and the straight line +does not meet the circle. + +Of these three cases, we confine ourselves to the +first, in which there are two points of intersection. +The product of the values of~$x$, with their proper +sign, is\footnote + {See \Title{Study of Mathematics} (Chicago: The Open Court Pub.~Co.), page~136.} +\[ +a^{2}\, \frac{b^{2} - r^{2}}{a^{2} + b^{2}}, +\] +and of~$y$, +\[ +b^{2}\, \frac{a^{2} - r^{2}}{a^{2} + b^{2}}, +\] +the signs of which are the same as those of~$b^{2} - r^{2}$, +and $a^{2} - r^{2}$. If $b$~and~$a$ be both $> r$, the two values +of~$x$ have the same sign; and it will appear from the +figure, that the lines they represent are measured in +the same direction. And this whether $b$~and~$a$ be positive +or negative, since $b^{2} - r^{2}$ and $a^{2} - r^{2}$ are both +positive when $a$~and~$b$ are numerically greater than~$r$, +whatever their signs may be. That is, if our rule, +connecting the signs of algebraical and the directions +of geometrical magnitudes, be true, let the directions +of $OE$ and $OF$ be altered in any way, so long as $OE$ +and $OF$ are both greater than~$OA$, the two values of~$OM$ +will have the same direction, and also those of~$MP$. +This result may easily be verified from the +figure. +\PageSep{35} + +Again, the values of $x$~and~$y$ having the same sign, +that sign will be (see the equations) the same as that +of $2ab^{2}$ for~$x$, and of $2a^{2}b$ for~$y$, or the same as that of +$a$~for~$x$ and of $b$~for~$y$. That is, when $OE$~and~$OF$ are +both greater than~$OA$, the direction of each set of co-ordinates +will be the same as those of $OE$ and~$OF$, +which may also be readily verified from the figure. + +Many other verifications might thus be obtained of +the same principle, viz., that any equation which corresponds +to, and is true for, all points in the angle~$AOB$, +may be used without error for all points lying +in the other three angles, by substituting the proper +numerical values, with a negative sign, for those co-ordinates +whose directions are opposite to those of +the co-ordinates in the angle~$AOB$. In this manner, +if four points be taken similarly situated in the four +angles, the numerical values of whose co-ordinates +are $x = 4$ and $y = 6$, and if the co-ordinates of that +point which lies in the angle~$AOB$, are called $+4$ and~$+6$; +those of the points lying in the angle~$BOC$ will +be $-4$~and~$+6$; in the angle~$COD$ $-4$~and~$-6$; +and in the angle~$DOE$ $+4$~and~$-6$. + +To return to \Fig{2}, if, after having completed the +branch of the curve which lies on the right of~$BC$, +and whose equation is $y = x^{2}$, we seek that which lies +on the left of~$BC$, we must, by the principles established, +substitute $-x$ instead of~$x$, when the numerical +value obtained for~$(-x)^{2}$ will be that of~$y$, and the +sign will show whether $y$~is to be measured in a similar +or contrary direction to that of~$MP$. Since $(-x)^{2} = x^{2}$, +the direction and value of~$y$, for a given value +of~$x$, remains the same as on the right of~$BC$; whence +the remaining branch of the curve is similar and equal +in all respects to~$OP$, only lying in the angle~$BOD$. +\PageSep{36} +And thus, if $y$ be any function of~$x$, we can obtain a +geometrical representation of the same, by making $y$ +the ordinate, and $x$~the abscissa of a curve, every ordinate +of which shall be the linear representation of +the numerical value of the given function corresponding +to the numerical value of the abscissa, the linear +unit being a given line. + + +\Subsection{The Drawing of a Tangent to a Curve.} + +If the point~$P$ (\Fig{2}), setting out from~$O$, move +along the branch~$OP$, it will continually change the +\Figure[nolabel]{2} +\emph{direction} of its motion, never moving, at one point, in +\index{Direction}% +the direction which it had at any previous point. Let +the moving point have reached~$P$, and let $OM = x$, +$MP = y$. Let $x$~receive the increment $MM' = dx$, in +consequence of which $y$ or $MP$ becomes~$M'P'$, and +receives the increment $QP' = dy$; so that $x + dx$ and +$y + dy$ are the co-ordinates of the moving point~$P$, +when it arrives at~$P'$. Join~$PP'$, which makes, with +$PQ$ or~$OM$, an angle, whose tangent is $\dfrac{P'Q}{PQ}$ or~$\dfrac{dy}{dx}$. +Since the relation $y = x^{2}$ is true for the co-ordinates of +every point in the curve, we have $y + dy = (x + dx)^{2}$, +\PageSep{37} +the subtraction of the former equation from which +gives $dy = 2x\, dx + (dx)^{2}$, or $\dfrac{dy}{dx} = 2x + dx$. If the +point~$P'$ be now supposed to move backwards towards~$P$, +the chord~$PP'$ will diminish without limit, and the +inclination of $PP'$ to $PQ$ will also diminish, but not +without limit, since the tangent of the angle~$P'PQ$, or~$\dfrac{dy}{dx}$, +\index{Tangent}% +is always greater than the limit~$2x$. If, therefore, +a line~$PV$ be drawn through~$P$, making with~$PQ$ an +angle whose tangent is~$2x$, the chord~$PP'$ will, as $P'$~approaches +towards~$P$, or as $dx$~is diminished, continually +approximate towards~$PV$, so that the angle~$P'PV$ +may be made smaller than any given angle, by +sufficiently diminishing~$dx$. And the line~$PV$ cannot +again meet the curve on the side of~$PP'$, nor can any +straight line be drawn between it and the curve, the +proof of which we leave to the student. + +Again, if $P'$~be placed on the other side of~$P$, so that +its co-ordinates are $x - dx$ and $y - dy$, we have $y - dy = (x - dx)^{2}$, +which, subtracted from $y = x^{2}$, gives $dy = 2x\, dx - (dx)^{2}$, +or $\dfrac{dy}{dx} = 2x - dx$. By similar reasoning, +if the straight line~$PT$ be drawn in continuation +of~$PV$, making with~$PN$ an angle, whose tangent is~$2x$, +the chord~$PP'$ will continually approach to this +line, as before. + +The line~$TPV$ indicates the direction in which the +point~$P$ is proceeding, and is called the \emph{tangent} of the +curve at the point~$P$. If the curve were the interior +of a small solid tube, in which an atom of matter were +made to move, being projected into it at~$O$, and if all +the tube above~$P$ were removed, the line~$PV$ is in the +direction which the atom would take on emerging at~$P$, +and is the line which it would describe. The angle +\PageSep{38} +which the tangent makes with the axis of~$x$ in any +\index{Tangent}% +curve, may be found by giving $x$ an increment, finding +the ratio which the corresponding increment of~$y$ +bears to that of~$x$, and determining the limit of that +ratio, or the \emph{differential coefficient}. This limit is the +\index{Coefficients, differential}% +\index{Differential coefficients}% +trigonometrical tangent\footnote + {There is some confusion between these different uses of the word tangent. + The geometrical tangent is, as already defined, the line between which + and a curve no straight line can be drawn; the trigonometrical tangent has + reference to an angle, and is the ratio which, in any right-angled triangle, + the side opposite the angle bears to that which is adjacent.} +of the angle which the geometrical +tangent makes with the axis of~$x$. If $y = \phi x$, +$\phi' x$~is this trigonometrical tangent. Thus, if the curve +be such that the ordinates are the Naperian logarithms\footnote + {It may be well to notice that in analysis the Naperian logarithms are +\index{Logarithms}% + the only ones used; while in practice the common, or Briggs's logarithms, + are always preferred.} +of the abscissæ, or $y = \log x$, and $y + dy = +\log x + \dfrac{1}{x}\, dx - \dfrac{1}{2x^{2}}\, dx^{2}$, etc., the geometrical tangent +of any point whose abscissa is~$x$, makes with the axis +an angle whose trigonometrical tangent is~$\dfrac{1}{x}$. + +This problem, of drawing a tangent to any curve, +was one, the consideration of which gave rise to the +methods of the Differential Calculus. + + +\Subsection{Rational Explanation of the Language of Leibnitz.} + +As the peculiar language of the theory of infinitely +\index{Infinitely small, the notion of|EtSeq}% +\index{Leibnitz}% +small quantities is extensively used, especially in +works of natural philosophy, it has appeared right to +us to introduce it, in order to show how the terms +which are used may be made to refer to some natural +and rational mode of explanation. In applying this +language to \Fig{2}, it would be said that the curve~$OP$ +is a polygon consisting of an infinite number of +\index{Polygon}% +\PageSep{39} +infinitely small sides, each of which produced is a +tangent to the curve; also that if $MM'$ be taken infinitely +small, the chord and arc~$PP'$ coincide with +\index{Arc and its chord, a continuously decreasing|EtSeq}% +one of these rectilinear elements; and that an infinitely +small arc coincides with its chord. All which +must be interpreted to mean that, the chord and arc +being diminished, approach more and more nearly to +a ratio of equality as to their lengths; and also that +the greatest separation between an arc and its chord +may be made as small a part as we please of the whole +chord or arc, by sufficiently diminishing the chord. + +We shall proceed to a strict proof of this; but in +the meanwhile, as a familiar illustration, imagine a +small arc to be cut off from a curve, and its extremities +joined by a chord, thus forming an arch, of which +the chord is the base. From the middle point of the +chord, erect a perpendicular to it, meeting the arc, +which will thus represent the height of the arch. +Imagine this figure to be magnified, without distortion +or alteration of its proportions, so that the larger figure +may be, as it is expressed, a true picture of the +smaller one. However the original arc may be diminished, +let the magnified base continue of a given +length. This is possible, since on any line a figure +may be constructed similar to a given figure. If the +original curve could be such that the height of the +arch could never be reduced below a certain part of +the chord, say one thousandth, the height of the magnified +arch could never be reduced below one thousandth +of the magnified chord, since the proportions +of the two figures are the same. But if, in the original +curve, an arc can be taken so small that the height +of the arch is as small a part as we please of the +chord, it will follow that in the magnified figure where +\PageSep{40} +the chord is always of one length, the height of the +arch can be made as small as we please, seeing that +it can be made as small a part as we please of a given +line. It is possible in this way to conceive a whole +curve so magnified, that a given arc, however small, +shall be represented by an arc of any given length, +however great; and the proposition amounts to this, +that let the dimensions of the magnified curve be any +\index{Curve, magnified}% +\index{Magnified curve}% +given number of times the original, however great, an +arch can be taken upon the original curve so small, +that the height of the corresponding arch in the magnified +figure shall be as small as we please. +\Figure{4} + +Let $PP'$ (\Fig{4}) be a part of a curve, whose equation +is $y = \phi(x)$, that is, $PM$~may always be found by +substituting the numerical value of~$OM$ in a given +function of~$x$. Let $OM = x$ receive the increment +$MM' = dx$, which we may afterwards suppose as small +as we please, but which, in order to render the figure +more distinct, is here considerable. The value of $PM$ +or~$y$ is~$\phi x$, and that of $P'M'$ or $y + dy$ is~$\phi(x + dx)$. + +Draw $PV$, the tangent at~$P$, which, as has been +\index{Tangent}% +shown, makes, with~$PQ$, an angle, whose trigonometrical +tangent is the limit of the ratio~$\dfrac{dy}{dx}$, when $x$~is decreased, +or~$\phi' x$. Draw the chord~$PP'$, and from any +\PageSep{41} +point in it, for example, its middle point~$p$, draw~$pv$ +parallel to~$PM$, cutting the curve in~$a$. The value of~$P'Q$, +or~$dy$, or $\phi(x + dx) - \phi x$ is %[** TN: This line displayed in the orig] +\[ +P'Q = \phi' x\, dx + + \phi'' x\, \frac{(dx)^{2}}{2} + + \phi''' x\, \frac{(dx)^{3}}{2·3} + \etc. +\] +But $\phi' x\, dx$~is $\tan VPQ·PQ = VQ$. Hence $VQ$~is the +first term of this series, and $P'V$~the aggregate of the +rest. But it has been shown that $dx$~can be taken so +small, that any one term of the above series shall contain +the rest, as often as we please. Hence $PQ$~can +be taken so small that $VQ$~shall contain~$VP'$ as often +as we please, or the ratio of $VQ$ to~$VP'$ shall be as +great as we please. And the ratio $VQ$ to~$PQ$ continues +finite, being always~$\phi' x$; hence $P'V$~also decreases +without limit as compared with~$PQ$. + +Next, the chord~$PP'$ or $\sqrt{(dx)^{2} + (dy)^{2}}$, or +\[ +dx \sqrt{1 + \left(\frac{dy}{dx}\right)^{2}} +\] +is to~$PQ$ or~$dx$ in the ratio of $\sqrt{1 + \left(\dfrac{dy}{dx}\right)^{2}} : 1$, which, +as $PQ$~is diminished, continually approximates to that +of $\sqrt{1 + (\phi' x)^{2}} : 1$, which is the ratio of~$PV : PQ$. +Hence the ratio of~$PP' : PV$ continually approaches to +unity, or $PQ$~may be taken so small that the difference +of $PP'$~and~$PV$ shall be as small a part of either +of them as we please. + +Finally, the arc~$PP'$ is greater than the chord~$PP'$ +and less than $PV + VP'$. Hence $\dfrac{\arc PP'}{\chord PP'}$ lies between +$1$~and $\dfrac{PV}{PP'} + \dfrac{VP'}{PP'}$, the former of which two +fractions can be brought as near as we please to unity, +and the latter can be made as small as we please; for +\PageSep{42} +since $P'V$~can be made as small a part of~$PQ$ as we +please, still more can it be made as small a part as we +please of~$PP'$, which is greater than~$PQ$. Therefore +the arc and chord~$PP'$ may be made to have a ratio as +nearly equal to unity as we please. And because $pa$~is +less than~$pv$, and therefore less than~$P'V$, it follows +that $pa$~may be made as small a part as we please of~$PQ$, +and still more of~$PP'$. + +In these propositions is contained the rational explanation +of the proposition of Leibnitz, that ``an infinitely +\index{Leibnitz}% +small arc is equal to, and coincides with, its +chord.'' + + +\Subsection{Orders of Infinity.} + +Let there be any number of series, arranged in +\index{Infinity, orders of|EtSeq}% +\index{Orders of infinity|EtSeq}% +powers of~$h$, so that the lowest power is first; let +them contain none but whole powers, and let them all +be such, that each will be convergent, on giving to~$h$ +a sufficiently small value: as follows, +\begin{alignat*}{4} +Ah + Bh^{2} &{}+{}& Ch^{3} &{}+{}& Dh^{4} &{}+{}& Eh^{5} &+ \etc. +\Tag{(1)} \\ + B'h^{2} &{}+{}& C'h^{3} &{}+{}& D'h^{4} &{}+{}& E'h^{5} &+ \etc. +\Tag{(2)} \\ + && C''h^{3} &{}+{}& D''h^{4} &{}+{}& E''h^{5} &+ \etc. +\Tag{(3)} \\ + &&&& D'''h^{4} &{}+{}& E'''h^{5} &+ \etc. +\Tag{(4)} \\ + &&&&&& \etc. & +\tag*{\etc.} +\end{alignat*} + +As $h$~is diminished, all these expressions decrease +without limit; but the first \emph{increases} with respect to +the second, that is, contains it more times after a decrease +of~$h$ than it did before. For the ratio of \Eq{(1)} +to~\Eq{(2)} is that of $A + Bh + Ch^{2} + \etc.$ to $B'h + C'h^{2} + \etc.$, +the ratio of the two not being changed by dividing +both by~$h$. The first term of the latter ratio +approximates continually to~$A$, as $h$~is diminished, +and the second can be made as small as we please, +and therefore can be contained in the first as often as +\PageSep{43} +we please. Hence the ratio \Eq{(1)}~to~\Eq{(2)} can be made +as great as we please. By similar reasoning, the ratio +\Eq{(2)}~to~\Eq{(3)}, of \Eq{(3)}~to~\Eq{(4)}, etc., can be made as great as +we please. We have, then, a series of quantities, +each of which, by making $h$ sufficiently small, can be +made as small as we please. Nevertheless this decrease +increases the ratio of the first to the second, of +the second to the third, and so on, and the increase is +without limit. + +Again, if we take \Eq{(1)}~and~$h$, the ratio of \Eq{(1)}~to~$h$ is +that of $A + Bh + Ch^{2} + \etc.$ to~$1$, which, by a sufficient +decrease of~$h$, may be brought as near as we +please to that of $A$~to~$1$. But if we take \Eq{(1)}~and~$h^{2}$, +the ratio of \Eq{(1)}~to~$h^{2}$ is that of $A + Bh + \etc.$ to~$h$, +which, by previous reasoning, may be increased without +limit; and the same for any higher power of~$h$. +Hence \Eq{(1)}~is said to be \emph{comparable} to the first power +of~$h$, or \emph{of the first order}, since this is the only power +of~$h$ whose ratio to~\Eq{(1)} tends towards a finite limit. +By the same reasoning, the ratio of \Eq{(2)}~to~$h^{2}$, which is +that of $B' + C'h + \etc.$ to~$1$, continually approaches +that of $B'$~to~$1$; but the ratio \Eq{(2)}~to~$h$, which is that +of $B'h + C'h^{2} + \etc.$ to~$1$, diminishes without limit, as +$h$~is decreased, while the ratio of \Eq{(2)}~to~$h^{2}$, or of $B' + C'h + \etc.$ +to~$h$, increases without limit. Hence \Eq{(2)}~is +said to be \emph{comparable} to the second power of~$h$, or \emph{of +the second order}, since this is the only power of~$h$ whose +ratio to~\Eq{(2)} tends towards a finite limit. In the language +of Leibnitz if $h$~be an infinitely small quantity, +\Eq{(1)}~is an infinitely small quantity of the first order, +\Eq{(2)}~is an infinitely small quantity of the second +order, and so on. + +We may also add that the ratio of two series of +the same order continually approximates to the ratio +\PageSep{44} +of their lowest terms. For example, the ratio of $Ah^{3} + Bh^{4} + \etc.$ +to $A'h^{3} + B'h^{4} + \etc.$ is that of $A + Bh + \etc.$ +to $A' + B'h + \etc.$, which, as $h$~is diminished, +continually approximates to the ratio of $A$ to~$A'$, which +is also that of $Ah^{3}$ to~$A'h^{3}$, or the ratio of the lowest +terms. In \Fig{4}, $PQ$~or $dx$ being put in place of~$h$, +$QP'$, or $\phi' x\, dx + \phi'' x\, \dfrac{(dx)^{2}}{2}$, etc., is of the first order, +as are~$PV$, and the chord~$PP'$; while $P'V$, or +$\phi'' x\, \dfrac{(dx)^{2}}{2} + \etc.$, is of the second order. + +The converse proposition is readily shown, that if +the ratio of two series arranged in powers of~$h$ continually +approaches to some finite limit as $h$~is diminished, +the two series are of the same order, or the exponent +of the lowest power of~$h$ is the same in both. +Let $Ah^{a}$ and $Bh^{b}$ be the lowest powers of~$h$, whose ratio, +as has just been shown, continually approximates +to the actual ratio of the two series, as $h$~is diminished. +The hypothesis is that the ratio of the two series, and +therefore that of $Ah^{a}$ to~$Bh^{b}$, has a finite limit. This +cannot be if $a > b$, for then the ratio of $Ah^{a}$ to $Bh^{b}$ is +that of $Ah^{a-b}$ to~$B$, which diminishes without limit; +neither can it be when $a < b$, for then the same ratio +is that of $A$ to~$Bh^{b-a}$, which increases without limit; +hence $a$~must be equal to~$b$. + +We leave it to the student to prove strictly a proposition +assumed in the preceding; viz., that if the +ratio of $P$~to~$Q$ has unity for its limit, when $h$~is diminished, +the limiting ratio of $P$~to~$R$ will be the same +as the limiting ratio of $Q$~to~$R$. We proceed further +to illustrate the Differential Calculus as applied to +Geometry. +\PageSep{45} + + +\Subsection[A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines.] +{A Geometrical Illustration.} + +Let $OC$ and~$OD$ (\Fig{5}) be two axes at right angles +to one another, and let a line~$AB$ of given length +be placed with one extremity in each axis. Let this +line move from its first position into that of~$A'B'$ on +one side, and afterwards into that of~$A''B''$ on the +other side, always preserving its first length. The +motion of a ladder, one end of which is against a wall, +and the other on the ground, is an instance. + +Let $A'B'$ and $A''B''$ intersect~$AB$ in $P'$~and~$P''$. If +\index{Ladder against wall|EtSeq}% +$A''B''$~were gradually moved from its present position +into that of~$A'B'$, the point~$P''$ would also move gradually +\Figure{5} +from its present position into that of~$P'$, passing, +in its course, through every point in the line~$P'P''$. +But here it is necessary to remark that $AB$~is itself +one of the positions intermediate between $A'B'$ and~$A''B''$, +and when two lines are, by the motion of one +of them, brought into one and the same straight line, +they intersect one another (if this phrase can be here +applied at all) in every point, and all idea of one distinct +point of intersection is lost. Nevertheless $P''$~describes +one part of~$P''P'$ before $A''B''$~has come into +the position~$AB$, and the rest afterwards, when it is +between $AB$ and~$A'B'$. +\PageSep{46} + +Let $P$~be the point of separation; then every point +of~$P'P''$, except~$P$, is a real point of intersection of~$AB$, +with one of the positions of~$A''B''$, and when +$A''B''$~has moved very near to~$AB$, the point~$P''$ will +be very near to~$P$; and there is no point so near to~$P$, +that it may not be made the intersection of $A''B''$ and~$AB$, +by bringing the former sufficiently near to the +latter. This point~$P$ is, therefore, the \emph{limit} of the intersections +\index{Intersections, limit of|EtSeq}% +\index{Limit of intersections|EtSeq}% +of $A''B''$~and~$AB$, and cannot be found by +the ordinary application of algebra to geometry, but +may be made the subject of an inquiry similar to those +\Figure[nolabel]{5} +which have hitherto occupied us, in the following +manner: + +Let $OA = a$, $OB = b$, $AB = A'B' = A''B'' = l$. Let +$AA' = da$, $BB' = db$, whence $OA' = a + da$, $OB' = b - db$. +We have then $a^{2} + b^{2} = l^{2}$, and $(a + da)^{2} + (b - db)^{2} = l^{2}$; +subtracting the former of which from +the development of the latter, we have +\[ +2a\, da + (da)^{2} - 2b\, db + (db)^{2} = 0\Add{,} +\] +or +\[ +\frac{db}{da} = \frac{2a + da}{2b - db}\Add{.} +\Tag{(1)} +\] +As $A'B'$ moves towards~$AB$, $da$~and~$db$ are diminished +without limit, $a$~and~$b$ remaining the same; hence the +limit of the ratio~$\dfrac{db}{da}$ is $\dfrac{2a}{2b}$ or~$\dfrac{a}{b}$. +\PageSep{47} + +Let the co-ordinates\footnote + {The lines $OM'$ and $M'P'$ are omitted, to avoid crowding the figure.} +of~$P'$ be $OM' = x$ and $M'P = y$. +Then (\PageRef{32}) the co-ordinates of any point in~$AB$ +have the equation +\[ +ay + bx = ab\Add{.} +\Tag{(2)} +\] +The point~$P'$ is in this line, and also in the one which +cuts off $a + da$ and $b - db$ from the axes, whence +\[ +(a + da)y + (b - db)x = (a + da)(b - db)\Add{;} +\Tag{(3)} +\] +subtract \Eq{(2)} from~\Eq{(3)} after developing the latter, which +gives +\[ +y\, da - x\, db = b\, da - a\, db - da\, db\Add{.} +\Tag{(4)} +\] +If we now suppose $A'B'$~to move towards~$AB$, equation~\Eq{(4)} +gives no result, since each of its terms diminishes +without limit. If, however, we divide~\Eq{(4)} by~$da$, +and substitute in the result the value of~$\dfrac{db}{da}$ obtained +from~\Eq{(1)} we have +\[ +y - x\, \frac{2a + da}{2b - db} + = b - a\, \frac{2a + da}{2b - db} - db\Add{.} +\Tag{(5)} +\] +From this and~\Eq{(2)} we might deduce the values of $y$ +and~$x$, for the point~$P'$, as the figure actually stands. +Then by diminishing $db$~and $da$ without limit, and +observing the limit towards which $x$~and~$y$ tend, we +might deduce the co-ordinates of~$P$, the limit of the +intersections. + +The same result may be more simply obtained, by +diminishing $da$~and~$db$ in equation~\Eq{(5)}, before obtaining +the values of $y$~and~$x$. This gives +\[ +y - \frac{a}{b}\, x = b - \frac{a^{2}}{b} \quad\text{or}\quad +by - ax = b^{2} - a^{2}\Add{.} +\Tag{(6)} +\] +From \Eq{(6)}~and~\Eq{(2)} we find (\Fig{6}) +\[ +x = OM = \frac{a^{3}}{a^{2} + b^{2}} = \frac{a^{3}}{l^{2}} \quad\text{and}\quad +y = MP = \frac{b^{3}}{a^{2} + b^{2}} = \frac{b^{3}}{l^{2}}. +\] +\PageSep{48} + +This limit of the intersections is different for every +different position of the line~$AB$, but may be determined, +in every case, by the following simple construction. + +Since (\Fig{6}) $BP: PN$, or $OM :: BA : AO$, we +have $BP = OM\, \dfrac{BA}{AO} = \dfrac{a^{3}}{l^{2}}\, \dfrac{l}{a} = \dfrac{a^{2}}{l}$; and, similarly, +$PA = \dfrac{b^{2}}{l}$. Let $OQ$~be drawn perpendicular to~$BA$; +then since $OA$~is a mean proportional between $AQ$ +and~$AB$, we have $AQ = \dfrac{a^{2}}{l}$, and similarly $BQ = \dfrac{b^{2}}{l}$. +Hence $BP = AQ$ and $AP = BQ$, or the point~$P$ is +as far from either extremity of~$AB$ as $Q$~is from the +other. +\Figure{6} + + +\Subsection{The Same Problem Solved by the Principles of +Leibnitz.} + +We proceed to solve the same problem, using the +\index{Leibnitz}% +principles of Leibnitz, that is, supposing magnitudes +can be taken so small, that those proportions may be +regarded as absolutely correct, which are not so in +reality, but which only approach more nearly to the +truth, the smaller the magnitudes are taken. The inaccuracy +of this supposition has been already pointed +out; yet it must be confessed that this once got over, +\PageSep{49} +the results are deduced with a degree of simplicity +and consequent clearness, not to be found in any other +method. The following cannot be regarded as a demonstration, +except by a mind so accustomed to the +subject that it can readily convert the various inaccuracies +into their corresponding truths, and see, at one +glance, how far any proposition will affect the final +result. The beginner will be struck with the extraordinary +assertions which follow, given in their most +naked form, without any attempt at a less startling +mode of expression. +\Figure{7} + +Let $A'B'$ (\Fig{7}) be a position of~$AB$ infinitely +\index{Infinitely small, the notion of}% +near to it; that is, let $A'PA$~be an infinitely small +angle. With the centre~$P$, and the radii $PA'$ and~$PB$, +describe the infinitely small arcs $A'a$,~$Bb$. An infinitely +small arc of a circle is a straight line perpendicular +to its radius; hence $A'aA$~and~$BbB'$ are right-angled +triangles, the first similar to~$BOA$, the two +having the angle~$A$ in common, and the second similar +to~$B'OA'$. Again, since the angles of~$BOA$, which +are finite, only differ from those of~$B'OA'$ by the infinitely +small angle~$A'PA$, they may be regarded as +\PageSep{50} +equal; whence $A'aA$~and~$B'bB$ are similar to~$BOA$, +and to one another. Also $P$~is the point of which we +are in search, or infinitely near to it; and since $BA = B'A'$, +of which $BP = bP$ and $aP = A'P$, the remainders +$B'b$~and~$Aa$ are equal. Moreover, $Bb$~and~$A'a$ +being arcs of circles subtending equal angles, are in +the proportion of the radii $BP$~and~$PA'$. + +Hence we have the following proportions: +\begin{gather*} +Aa : A'a :: OA : OB :: a : b \\ +Bb : B'b :: OA : OB :: a : b\rlap{.} +\end{gather*} +The composition of which gives, since $Aa = B'b$: +\[ +\PadTo[r]{BP + Pa}{Bb} : A'a :: \PadTo[l]{a^{2} + b^{2}}{a^{2}} : b^{2}. +\] +Also +\[ +\PadTo[r]{BP + Pa}{Bb} : A'a :: \PadTo[l]{a^{2} + b^{2}}{BP} : Pa, +\] +whence +\[ +\PadTo[r]{BP + Pa}{BP} : Pa :: \PadTo[l]{a^{2} + b^{2}}{a^{2}} : b^{2}, +\] +and +\[ +BP + Pa : Pa :: a^{2} + b^{2} : b^{2}. +\] +But $Pa$~only differs from~$PA$ by the infinitely small +quantity~$Aa$, and $BP + PA = l$, and $a^{2} + b^{2} = l^{2}$; +whence +\[ +l : PA :: l^{2} : b^{2},\quad\text{or}\quad PA = \frac{b^{2}}{l}, +\] +which is the result already obtained. + +In this reasoning we observe four independent +errors, from which others follow: (1)~that $Bb$~and~$A'a$ +are straight lines at right-angles to~$Pa$; (2)~that $BOA$\Add{,}~$B'OA'$ +are similar triangles; (3)~that $P$~is really the +point of which we are in search; (4)~that $PA$~and~$Pa$ +are equal. But at the same time we observe that +every one of these assumptions approaches the truth, +as we diminish the angle~$A'PA$, so that there is no +magnitude, line or angle, so small that the linear or +angular errors, arising from the above-mentioned suppositions, +may not be made smaller. + +We now proceed to put the same demonstration +\PageSep{51} +in a stricter form, so as to neglect no quantity during +the process. This should always be done by the beginner, +until he is so far master of the subject as to be +able to annex to the inaccurate terms the ideas necessary +for their rational explanation. To the former figure +add $B\beta$ and~$A\alpha$, the real perpendiculars, with +which the arcs have been confounded. Let $\angle A'PA = d\theta$, +\index{Angle, unit employed in measuring an}% +$PA = p$, $Aa = dp$, $BP = q$, $B'b = dq$; and $OA = a$, +$OB = b$, and $AB = l$. Then\footnote + {For the unit employed in measuring an angle, see \Title{Study of Mathematics} + (Chicago, 1898), pages 273--277.} +$A'a = (p - dp)\, d\theta$, $Bb = q\, d\theta$, +and the triangles $A'A\alpha$ and $B'B\beta$ are similar to +\Figure[nolabel]{7} +$BOA$~and~$B'OA'$. The perpendiculars $A'\alpha$ and~$B\beta$ +are equal to $PA' \sin d\theta$ and $PB \sin d\theta$, or $(p - dp) \sin d\theta$ +and $q \sin d\theta$. Let $a\alpha = \mu$ and $b\beta = \nu$. These +(\PageRef[p.]{9}) will diminish without limit as compared with +$A'\alpha$ and~$B\beta$; and since the ratios of $A'\alpha$ to~$\alpha A$ and $B\beta$ +to~$\beta B'$ continue finite (these being sides of triangles +similar to $AOB$ and~$A'OB'$), $a\alpha$~and~$b\beta$ will diminish +indefinitely with respect to $\alpha A$~and~$\beta B'$. Hence the +ratio $A\alpha$ to~$\beta B'$ or $dp + \mu$ to $dq + \nu$ will continually +approximate to that of $dp$ to~$dq$, or a ratio of equality. +\PageSep{52} + +The exact proportions, to which those in the last +page are approximations, are as follows: +\begin{alignat*}{3} +dp + \mu &: (p - dp) \sin d\theta &&:: a &&: b, \\ +q \sin d\theta &: \PadTo{(p - dp) \sin d\theta}{dq + \nu} &&:: a - da &&: b + db; +\end{alignat*} +by composition of which, recollecting that $dp = dq$ +(which is rigorously true) and dividing the two first +terms of the resulting proportion by~$dp$, we have +\[ +q\left(1 + \frac{\mu}{dp}\right) : (p - dp)\left(1 + \frac{\nu}{dp}\right) + :: a(a - da) : b(b + db). +\] + +If $d\theta$ be diminished without limit, the quantities +$da$,~$db$, and~$dp$, and also the ratios $\dfrac{\mu}{dp}$ and~$\dfrac{\nu}{dp}$, as +above-mentioned, are diminished without limit, so +that the limit of the proportion just obtained, or the +proportion which gives the limits of the lines into +which $P$~divides~$AB$, is +\begin{alignat*}{3} +q &: p &&:: a^{2} &&: b^{2}, \\ +\intertext{hence} +q + p = l &: p &&:: a^{2} + b^{2} = l^{2} &&: b^{2}, +\end{alignat*} +the same as before. + + +\Subsection[An Illustration from Dynamics: Velocity, Acceleration, etc.] +{An Illustration from Dynamics.} + +We proceed to apply the preceding principles to +dynamics, or the theory of motion. + +Suppose a point moving along a straight line uniformly; +that is, if the whole length described be divided +into any number of equal parts, however great, +each of those parts is described in the same time. +Thus, whatever length is described in the first second +of time, or in any part of the first second, the same +is described in any other second, or in the same part +of any other second. The number of units of length +described in a unit of time is called the \emph{velocity}; thus +\index{Velocity!linear|EtSeq}% +a velocity of $3.01$~feet in a second means that the +\PageSep{53} +point describes three feet and one hundredth in each +second, and a proportional part of the same in any +part of a second. Hence, if $v$~be the velocity, and +$t$~the units of time elapsed from the beginning of the +motion, $vt$~is the length described; and if any length +described be known, the velocity can be determined +by dividing that length by the time of describing it. +Thus, a point which moves uniformly through $3$~feet +in $1\frac{1}{2}$~second, moves with a velocity of $3 ÷ 1\frac{1}{2}$, or $2$~feet +per~second. + +Let the point not move uniformly; that is, let different +\index{Continuous quantities}% +\index{Quantities, continuous}% +parts of the line, having the same length, be +described in different times; at the same time let the +motion be \emph{continuous}, that is, not suddenly increased +or decreased, as it would be if the point were composed +of some hard matter, and received a blow while +it was moving. This will be the case if its motion be +represented by some algebraical function of the time, +or if, $t$~being the number of units of time during which +the point has moved, the number of units of length +described can be represented by~$\phi t$. This, for example, +we will suppose to be~$t + t^{2}$, the unit of time +being one second, and the unit of length one inch; +so that $\frac{1}{2} + \frac{1}{4}$, or $\frac{3}{4}$~of an inch, is described in the first +half second; $1 + 1$, or two inches, in the first second; +$2 + 4$, or six inches, in the first two seconds, and so on. + +Here we have no longer an evident measure of the +velocity of the point; we can only say that it obviously +increases from the beginning of the motion to +the end, and is different at every two different points. +Let the time~$t$ elapse, during which the point will describe +the distance $t + t^{2}$; let a further time~$dt$ elapse, +during which the point will increase its distance to +$t + dt + (t + dt)^{2}$, which, diminished by~$t + t^{2}$, gives +\PageSep{54} +$dt + 2t\, dt + (dt)^{2}$ for the length described during the +increment of time~$dt$. This varies with the value of~$t$; +thus, in the interval~$dt$ after the first second, the +length described is $3\, dt + dt^{2}$; after the second second, +it is $5\, dt + (dt)^{2}$, and so on. Nor can we, as in the +case of uniform motion, divide the length described, +by the time, and call the result the velocity with which +that length is described; for no length, however small, +is here uniformly described. If we were to divide a +length by the time in which it is described, and also +its first and second halves by the times in which they +are respectively described, the three results would be +all different from one another. + +Here a difficulty arises, similar to that already noticed, +when a point moves along a curve; in which, +as we have seen, it is improper to say that it is moving +in any one direction through an arc, however +small. Nevertheless a straight line was found at every +point, which did, more nearly than any other straight +line, represent the direction of the motion. So, in +this case, though it is incorrect to say that there is +any uniform velocity with which the point continues +to move for any portion of time, however small, we +can, at the end of every time, assign a uniform velocity, +which shall represent, more nearly than any +other, the rate at which the point is moving. If we +say that, at the end of the time~$t$, the point is moving +with a velocity~$v$, we must not now say that the length~$v\, dt$ +is described in the succeeding interval of time~$dt$; +but we mean that $dt$~may be taken so small, that $v\, dt$~shall +bear to the distance actually described a ratio as +near to equality as we please. + +Let the point have moved during the time~$t$, after +which let successive intervals of time elapse, each +\PageSep{55} +\index{Coefficients, differential}% +\index{Differential coefficients}% +equal to~$dt$. At the end of the times, $t$,~$t + dt$, $t + 2\, dt$, +$t + 3\, dt$,~etc., the whole lengths described will be $t + t^{2}$, +$t + dt + (t + dt)^{2}$, $t + 2\, dt + (t + 2\, dt)^{2}$, $t + 3\, dt + (t + 3\, dt)^{2}$, +etc.; the differences of which, or $dt + 2t\, dt + (dt)^{2}$, +$dt + 2t\, dt + 3(dt)^{2}$, $dt + 2t\, dt + 5(dt)^{2}$, etc., +are the lengths described in the first, second, third, +etc., intervals~$dt$. These are not equal to one another, +as would be the case if the velocity were uniform; but +by making $dt$ sufficiently small, their ratio may be +brought as near to equality as we please, since the +terms $(dt)^{2}$,~$3(dt)^{2}$,~etc., by which they all differ from +the common part $(1 + 2t)\, dt$, may be made as small as +we please, in comparison of this common part. If we +divide the above-mentioned lengths by~$dt$, which does +not alter their ratio, they become $1 + 2t + dt$, $1 + 2t + 3\, dt$, +$1 + 2t + 5\, dt$, etc., which may be brought as +near as we please to equality, by sufficient diminution +of~$dt$. Hence $1 + 2t$ is said to be the velocity of the +point after the time~$t$; and if we take a succession of +equal intervals of time, each equal to~$dt$, and sufficiently +small, the lengths described in those intervals +will bear to $(1 + 2t)\, dt$, the length which would be described +in the same interval with the uniform velocity +$1 + 2t$, a ratio as near to equality as we please. And +observe, that if $\phi t$ is $t + t^{2}$, $\phi' t$~is $1 + 2t$, or the coefficient +of~$h$ in $(t + h) + (t + h)^{2}$. + +In the same way it may be shown, that if the point +moves so that $\phi t$~always represents the length described +in the time~$t$, the differential coefficient of~$\phi t$\Add{,} +or~$\phi' t$, is the velocity with which the point is moving +at the end of the time~$t$. For the time~$t$ having elapsed, +the whole lengths described at the end of the times $t$ +and $t + dt$ are $\phi t$ and $\phi(t + dt)$; whence the length +described during the time~$dt$ is +\PageSep{56} +\[ +\phi(t + dt) - \phi t, \quad\text{or}\quad +\phi't\, dt + \phi'' t\, \frac{(dt)^{2}}{2} + \etc. +\] +Similarly, the length described in the next interval +$dt$ is +\begin{gather*} +\phi(t + 2\, dt) - \phi(t + dt); \quad\text{or}, \displaybreak[0] \\ + \phi t + \phi' t\, 2\, dt + \phi'' t\, \frac{(2\, dt)^{2}}{2} + \etc. \displaybreak[0] \\ +-(\phi t + \phi' t\, dt + \phi'' t\, \frac{(dt)^{2}}{2} + \etc.), \displaybreak[0] \\ +\intertext{which is} +\phi' t\, dt + 3\phi'' t\, \frac{(dt)^{2}}{2} + \etc.; +\end{gather*} +the length described in the third interval~$dt$ is +$\phi' t\, dt + 5\phi'' t\, \dfrac{(dt)^{2}}{2} + \etc.$,~etc. + +Now, it has been shown for each of these, that the +first term can be made to contain the aggregate of all +the rest as often as we please, by making $dt$ sufficiently +small; this first term is $\phi' t\, dt$ in all, or the length +which would be described in the time~$dt$ by the velocity +$\phi' t$ continued uniformly: it is possible, therefore, +to take $dt$ so small, that the lengths actually described +in a succession of intervals equal to~$dt$, shall be as +nearly as we please in a ratio of equality with those +described in the same intervals of time by the velocity~$\phi' t$. +For example, it is observed in bodies which fall +\index{Falling bodies}% +to the earth from a height above it, when the resistance +of the air is removed, that if the time be taken +in seconds, and the distance in feet, the number of +feet fallen through in $t$~seconds is always~$at^{2}$, where +$a = 16\frac{1}{12}$ very nearly; what is the velocity of a body +which has fallen \textit{in~vacuo} for four seconds? Here $\phi t$ +being~$at^{2}$, we find, by substituting $t + h$, or $t + dt$, instead +of~$t$, that $\phi' t$~is~$2at$, or $2 × 16\frac{1}{12} × t$; which, at +\PageSep{57} +the end of four seconds, is $32\frac{1}{6} × 4$, or $128\frac{2}{3}$~feet. That +is, at the end of four seconds a falling body moves at +the rate of $128\frac{2}{3}$~feet per~second. By which we do +not mean that it continues to move with this velocity +for any appreciable time, since the rate is always +varying; but that the length described in the interval~$dt$ +after the fourth second, may be made as nearly as +we please in a ratio of equality with $128\frac{2}{3} × dt$, by +taking $dt$ sufficiently small. This velocity~$2at$ is said +to be \emph{uniformly} accelerated; since in each second the +\index{Accelerated motion}% +\index{Uniformly accelerated}% +same velocity~$2a$ is gained. And since, when $x$~is the +space described, $\phi' t$~is the limit of~$\dfrac{dx}{dt}$, the velocity is +also this limit; that is, when a point does not move +uniformly, the velocity is not represented by any increment +of length divided by its increment of time, +but by the limit to which that ratio continually tends, +as the increment of time is diminished. + + +\Subsection{Simple Harmonic Motion.} + +We now propose the following problem: A point +\index{Motion!simple harmonic}% +\index{Simple harmonic motion}% +moves uniformly round a circle; with what velocities +do the abscissa and ordinate increase or decrease, at +any given point? (\Fig{8}.) + +Let the point~$P$, setting out from~$A$, describe the +arc~$AP$, etc., with the uniform velocity of $a$~inches +per~second. Let $OA = r$, $\angle A0P = \theta$, $\angle POP' = d\theta$, +$0M = x$, $MP = y$, $MM' = dx$, $QP' = dy$. + +From the first principles of trigonometry +\begin{alignat*}{4}%[** TN: Re-aligned from the original] +&x &&= r \cos\theta\Add{,} \\ +&x - dx &&= r \cos(\theta + d\theta) + &&= r \cos\theta \cos d\theta - r \sin\theta \sin d\theta\Add{,} \displaybreak[0] \\ +&y &&= r \sin\theta\Add{,} \\ +&y + dy &&= r \sin(\theta + d\theta) + &&= r \sin\theta \cos d\theta + r \cos\theta \sin d\theta. +\end{alignat*} +\PageSep{58} +Subtracting the second from the first, and the third +from the fourth, we have +\begin{alignat*}{2} +dx &= r \sin\theta \sin d\theta + r \cos\theta(1 - \cos d\theta)\Add{,} +\Tag{(1)} \\ +dy &= r \cos\theta \sin d\theta + r \sin\theta(1 - \cos d\theta)\Add{.} +\Tag{(2)} \\ +\end{alignat*} +But if $d\theta$ be taken sufficiently small, $\sin d\theta$, and~$d\theta$, +may be made as nearly in a ratio of equality as we +please, and $1 - \cos d\theta$ may be made as small a part +as we please, either of $d\theta$ or $\sin d\theta$. These follow from +\Fig{1}, in which it was shown that $BM$ and the arc~$BA$, +or (if $OA = r$ and $AOB = d\theta$), $r \sin d\theta$ and~$r\, d\theta$, +may be brought as near to a ratio of equality as we +\Figure{8} +please, which is therefore true of $\sin d\theta$ and~$d\theta$. Again, +it was shown that~$AM$, or $r - r \cos d\theta$, can be made +as small a part as we please, either of~$BM$ or the arc~$BA$, +that is, either of $r \sin d\theta$, or~$r\, d\theta$; the same is +therefore true of $1 - \cos d\theta$, and either $\sin d\theta$ or~$d\theta$. +Hence, if we write equations \Eq{(1)}~and~\Eq{(2)} thus, +\[ +dx = r \sin\theta\, d\theta\quad (1)\qquad\qquad +dy = r \cos\theta\, d\theta\quad (2), +\] +we have equations, which, though never exactly true, +are such that by making $d\theta$ sufficiently small, the +errors may be made as small parts of~$d\theta$ as we please. +Again, since the arc~$AP$ is uniformly described, so +also is the angle~$POA$; and since an arc~$a$ is described +\PageSep{59} +in one second, the angle~$\dfrac{a}{r}$ is described in the same +\index{Velocity!angular}% +time; this is, therefore, the \emph{angular velocity}.\footnote + {The same considerations of velocity which have been applied to the + motion of a point along a line may also be applied to the motion of a line + round a point. If the angle so described be always increased by equal angles + in equal portions of time, the angular velocity is said to be uniform, and is + measured by the number of angular units described in a unit of time. By + similar reasoning to that already described, if the velocity with which the + angle increases be not uniform, so that at the end of the time~$t$ the angle described + is $\theta = \phi t$, the angular velocity is~$\phi' t$, or the limit of the ratio~$\dfrac{d\theta}{dt}$.} +If we +divide equations \Eq{(1)}~and~\Eq{(2)} by~$dt$, we have +\[ +%[** TN: Signs OK; De Morgan absorbs the - in dx/dt at the bottom of p. 57] +\frac{dx}{dt} = r \sin\theta\, \frac{d\theta}{dt}\qquad +\frac{dy}{dt} = r \cos\theta\, \frac{d\theta}{dt}; +\] +these become more nearly true as $dt$~and~$d\theta$ are diminished, +so that if for $\dfrac{dx}{dt}$,~etc., the limits of these ratios +be substituted, the equations will become rigorously +true. But these limits are the velocities of $x$,~$y$, and~$\theta$, +the last of which is also~$\dfrac{a}{r}$; hence +\begin{alignat*}{2} +\text{velocity of~$x$} &= r \sin\theta × \frac{a}{r} &&= a \sin\theta, \\ +\text{velocity of~$y$} &= r \cos\theta × \frac{a}{r} &&= a \cos\theta; +\end{alignat*} +that is, the point~$M$ moves towards~$O$ with a variable +velocity, which is always such a part of the velocity +of~$P$, as $\sin\theta$~is of unity, or as $PM$~is of~$OB$; and the +distance~$PM$ increases, or the point~$N$ moves from~$O$, +with a velocity which is such a part of the velocity of~$P$ +as $\cos\theta$~is of unity, or as $OM$~is of~$OA$. [The motion +of the point~$M$ or the point~$N$ is called in physics +a \emph{simple harmonic motion}.] + +In the language of Leibnitz, the results of the two +\index{Leibnitz}% +foregoing sections would be expressed thus: If a +point move, but not uniformly, it may still be considered +as moving uniformly for any infinitely small +\index{Infinitely small, the notion of}% +\PageSep{60} +time; and the velocity with which it moves is the infinitely +small space thus described, divided by the infinitely +small time. + + +\Subsection{The Method of Fluxions.} + +The foregoing process contains the method employed +\index{Fluxions}% +by Newton, known by the name of the \emph{Method +\index{Newton}% +of Fluxions}. If we suppose $y$ to be any function of~$x$, +and that $x$~increases with a given velocity, $y$~will also +increase or decrease with a velocity depending: (1)~upon +the velocity of~$x$; (2)~upon the function which +$y$ is of~$x$. These velocities Newton called the fluxions +of $y$~and~$x$, and denoted them by $\dot{y}$~and~$\dot{x}$. Thus, if +$y = x^{2}$, and if in the interval of time~$dt$, $x$~becomes +$x + dx$, and $y$~becomes $y + dy$, we have $y + dy = (x + dx)^{2}$, +and $dy = 2x\, dx + (dx)^{2}$, or $\dfrac{dy}{dt} = 2x\, \dfrac{dx}{dt} + \dfrac{dx}{dt}\, dx$. +If we diminish~$dt$, the term $\dfrac{dx}{dt}\, dx$ will diminish +without limit, since one factor continually approaches +to a given quantity, viz., the velocity of~$x$, +and the other diminishes without limit. Hence we +obtain the velocity of $y = 2x × \text{the velocity of~$x$}$, or +$\dot{y} = 2x\, \dot{x}$, which is used in the method of fluxions instead +of $dy = 2x\, dx$ considered in the manner already +described. The processes are the same in both methods, +since the ratio of the velocities is the limiting +ratio of the corresponding increments, or, according +to Leibnitz, the ratio of the infinitely small increments. +\index{Leibnitz}% +We shall hereafter notice the common objection +to the Method of Fluxions. + + +\Subsection{Accelerated Motion.} + +When the velocity of a material point is suddenly +\index{Accelerated motion}% +\index{Motion!accelerated}% +\index{Uniformly accelerated}% +increased, an \emph{impulse} is said to be given to it, and the +\index{Impulse}% +\PageSep{61} +magnitude of the impulse or impulsive force is in proportion +\index{Force|(}% +to the velocity created by it. Thus, an impulse +which changes the velocity from $50$ to $70$~feet +per~second, is twice as great as one which changes it +from $50$ to $60$~feet. When the velocity of the point is +altered, not suddenly but continuously, so that before +the velocity can change from $50$ to $70$~feet, it goes +through all possible intermediate velocities, the point +is said to be acted on by an \emph{accelerating force}. \emph{Force} +is a name given to that which causes a change in the +velocity of a body. It is said to act uniformly, when +the velocity acquired by the point in any one interval +of time is the same as that acquired in any other interval +of equal duration. It is plain that we cannot, +by supposing any succession of impulses, however +small, and however quickly repeated, arrive at a uniformly +accelerated motion; because the length described +between any two impulses will be uniformly +described, which is inconsistent with the idea of continually +accelerated velocity. Nevertheless, by diminishing +the magnitude of the impulses, and increasing +their number, we may come as near as we please +to such a continued motion, in the same way as, by +diminishing the magnitudes of the sides of a polygon, +and increasing their number, we may approximate as +near as we please to a \Typo{continous}{continuous} curve. + +Let a point, setting out from a state of rest, increase +its velocity uniformly, so that in the time~$t$, it +may acquire the velocity~$v$---what length will have +been described during that time~$t$? Let the time~$t$ +and the velocity~$v$ be both divided into $n$~equal parts, +each of which is $t'$ and~$v'$, so that $nt' = t$, and $nv' = v$. +Let the velocity~$v'$ be communicated to the point at +rest; after an interval of~$t'$ let another velocity~$v'$ be +\PageSep{62} +communicated, so that during the second interval~$t'$ +the point has a velocity~$2v'$; during the third interval +let the point have the velocity~$3v'$, and so on; so that +in the last or $n$\th~interval the point has the velocity~$nv'$. +The space described in the first interval is, therefore,~$v't'$; +in the second,~$2v't'$; in the third~$3v't'$; and +so on, till in the $n$\th~interval it is~$nv't'$. The whole +space described is, therefore, +\[ +v't' + 2v't' + 3v't' + \dots + (n - 1)v't' + nv't'\Add{,} +\] +or +\[ +[1 + 2 + 3 \Add{+} \dots + (n - 1) + n]v't' + = n · \frac{(n + 1)}{2}\, v't' + = \frac{n^{2} v't' + nv't'}{2}. +\] +In this substitute $v$ for~$nv'$, and $t$ for~$nt'$, which gives +for the space described $\frac{1}{2}v(t + t')$. The smaller we +suppose~$t'$, the more nearly will this approach to~$\frac{1}{2}vt$. +But the smaller we suppose~$t'$, the greater must be~$n$, +the number of parts into which $t$~is divided; and the +more nearly do we render the motion of the point uniformly +accelerated. Hence the limit to which we approximate +by diminishing~$t'$ without limit, is the length +described in the time~$t$ by a uniformly accelerated +velocity, which shall increase from~$0$ to~$v$ in that time. +This is~$\frac{1}{2}vt$, or half the length which would have been +described by the velocity~$v$ continued uniformly from +the beginning of the motion. + +It is usual to measure the accelerating force by the +\index{Accelerating force}% +velocity acquired in one second. Let this be~$g$; then +since the same velocity is acquired in every other second, +the velocity acquired in $t$~seconds will be~$gt$, or +$v = gt$. Hence the space described is $\frac{1}{2}gt × t$, or~$\frac{1}{2}gt^{2}$. +If the point, instead of being at rest at the beginning +of the acceleration, had had the velocity~$a$, the lengths +\PageSep{63} +described in the successive intervals would have been +$at' + v't'$, $at' + 2v't'$, etc.; so that to the space described +by the accelerated motion would have been added~$nat'$, +or~$at$, and the whole length would have been +$at + \frac{1}{2}gt^{2}$. By similar reasoning, had the force been +a uniformly \emph{retarding} force, that is, one which diminished +\index{Force|)}% +the initial velocity~$a$ equally in equal times, the +length described in the time~$t$ would have been $at - \frac{1}{2}gt^{2}$. + +Now let the point move in such a way, that the +velocity is accelerated or retarded, but not uniformly; +that is, in different times of equal duration, let different +velocities be lost or gained. For example, let the +point, setting out from a state of rest, move in such a +\Figure{9} +way that the number of inches passed over in $t$~seconds +is always~$t^{3}$. Here $\phi t = t^{3}$, and the velocity acquired +by the body at the end of the time~$t$, is the coefficient +of~$dt$ in $(t + dt)^{3}$, or $3t^{2}$~inches per~second. +Let the point (\Fig{9}) be at~$A$ at the end of the time~$t$; +and let $AB$,~$BC$, $CD$,~etc., be lengths described in +successive equal intervals of time, each of which is~$dt$. +Then the velocities at $A$,~$B$,~$C$,~etc., are $3t^{2}$, $3(t + dt)^{2}$, +$3(t + 2\, dt)^{3}$, etc., and the lengths $AB$,~$BC$, $CD$,~etc., +are $(t + dt)^{3} - t^{3}$, $(t + 2\, dt)^{2} - (t + dt)^{3}$, $(t + 3\, dt)^{3} - (t + 2\, dt)^{3}$, +etc. +\[ +\ArrayCompress +\begin{array}{cl} +\ColHead{VELOCITY AT} & \\ +A & 3t^{2}\Add{,} \\ +B & 3t^{2} + \Z6t\, dt + \Z3(dt)^{2}\Add{,} \\ +C & 3t^{2} + 12t\, dt + 12(dt)^{2}\Add{,} \\ +\PageSep{64} +\ColHead{LENGTH OF} & \\ +AB & 3t^{2}\, dt + \Z3t(dt)^{2} + \Z\Z(dt)^{3}\Add{,} \\ +BC & 3t^{2}\, dt + \Z9t(dt)^{2} + \Z 7(dt)^{3}\Add{,} \\ +CD & 3t^{2}\, dt + 15t(dt)^{2} + 19(dt)^{3}\Add{.} +\end{array} +\] + +If we could, without error, reject the terms containing~$(dt)^{2}$ +in the velocities, and those containing~$(dt)^{3}$ +in the lengths, we should then reduce the motion +of the point to the case already considered, the +initial velocity being~$3t^{2}$, and the accelerating force~$6t$. +For we have already shown that $a$~being the initial +velocity, and $g$~the accelerating force, the space described +in the time~$t$ is $at + \frac{1}{2}gt^{2}$. Hence, $3t^{2}$~being +the initial velocity, and $6t$~the accelerating force, the +space in the time~$dt$ is $3t^{2}\, dt + 3t(dt)^{2}$, which is the +same as~$AB$ after $(dt)^{3}$~is rejected. The velocity acquired +is~$gt$, and the whole velocity is, therefore, +$a + gt$, or making the same substitutions $3t^{2} + 6t\, dt$. +This is the velocity at~$B$, after the term~$3(dt)^{2}$ is +rejected. Again, the velocity being $3t^{2} + 6t\, dt$, and +the force~$6t$, the space described in the time~$dt$ is +$(3t^{2} + 6t\, dt)\, dt + 3t(dt)^{2}$, or $3t^{2}\, dt + 9t(dt)^{2}$. This is +what the space~$BC$ becomes after $7(dt)^{3}$~is rejected. +The velocity acquired is~$6t\, dt$; and the whole velocity +is $3t^{2} + 6t\, dt + 6t\, dt$, or $3t^{2} + 12t\, dt$; which is the velocity +at~$C$ after $12(dt)^{2}$~is rejected. + +But as the terms involving $(dt)^{2}$ in the velocities, +etc., cannot be rejected without error, the above supposition +of a uniform force cannot be made. Nevertheless, +as we may take $dt$ so small that these terms +shall be as small parts as we please of those which +precede, the results of the erroneous and correct suppositions +may be brought as near to equality as we +please; hence we conclude, that though there is no +force, which, continued uniformly, would preserve +\PageSep{65} +the motion of the point~$A$, so that $OA$~should always +be~$t^{2}$ in inches, yet an interval of time may be taken +so small, that the length actually described by~$A$ in +that time, and the one which would be described if +the force~$6t$ were continued uniformly, shall have a +ratio as near to equality as we please. Hence, on a +principle similar to that by which we called~$3t^{3}$ the +velocity at~$A$, though, in truth, no space, however +small, is described with that velocity, we call~$6t$ the +accelerating force at~$A$. And it must be observed +that $6t$~is the differential coefficient of~$3t^{2}$, or the coefficient +of~$dt$, in the development of~$3(t + dt)^{2}$. + +Generally, let the point move so that the length +described in any time~$t$ is~$\phi t$. Hence the length described +at the end of the time $t + dt$ is $\phi(t + dt)$, and +that described in the interval~$dt$ is $\phi(t + dt) - \phi t$, or +\[ +\phi' t\, dt + \phi'' t\, \frac{(dt)^{2}}{2} + + \phi''' t\, \frac{(dt)^{3}}{2·3} + \etc.\Add{,} +\] +in which $dt$ may be taken so small, that either of the +first two terms shall contain the aggregate of all the +rest, as often as we please. These two first terms are +$\phi' t\, dt + \frac{1}{2}\phi'' t (dt)^{2}$, and represent the length described +during~$dt$, with a uniform velocity~$\phi' t$, and an accelerating +force~$\phi'' t$. The interval~$dt$ may then generally +be taken so small, that this supposition shall represent +the motion during that interval as nearly as we please. + + +\Subsection{Limiting Ratios of Magnitudes that Increase +Without Limit.} + +We have hitherto considered the limiting ratio of +\index{Increase without limit|EtSeq}% +\index{Limiting ratios|EtSeq}% +\index{Ratios, limiting|EtSeq}% +quantities only as to their state of \emph{decrease}: we now +proceed to some cases in which the limiting ratio of +different magnitudes which \emph{increase} without limit is +investigated. +\PageSep{66} + +It is easy to show that the increase of two magnitudes +may cause a decrease of their ratio; so that, as +the two increase without limit, their ratio may diminish +without limit. The limit of any ratio may be found +by rejecting any terms or aggregate of terms~($Q$) which +are connected with another term~($P$) by the sign of +addition or subtraction, provided that by increasing~$x$, +$Q$~may be made as small a part of~$P$ as we please. +For example, to find the limit of $\dfrac{x^{2} + 2x + 3}{2x^{2} + 5x}$, when +$x$~is increased without limit. By increasing~$x$ we can, +as will be shown immediately, cause $2x + 3$ and~$5x$ to +be contained in $x^{2}$ and~$2x^{2}$, as often as we please; rejecting +these terms, we have $\dfrac{x^{2}}{2x^{2}}$, or~$\frac{1}{2}$, for the limit. + +The demonstration is as follows: Divide both +numerator and denominator by~$x^{2}$, which gives $1 + \dfrac{2}{x} + \dfrac{3}{x^{2}}$, +and $2 + \dfrac{5}{x}$, for the numerator and denominator +of a fraction equal in value to the one proposed. +These can be brought as near as we please to $1$~and~$2$ +by making $x$ sufficiently great, or $\dfrac{1}{x}$~sufficiently small; +and, consequently, their ratio can be brought as near +as we please to~$\dfrac{1}{2}$. + +We will now prove the following: That in any +series of decreasing powers of~$x$, any one term will, if +$x$~be taken sufficiently great, contain the aggregate of +all which follow, as many times as we please. Take, +for example, +\[ +% [** TN: On two lines in the original] +ax^{m} + bx^{m-1} + cx^{m-2} + \dots + px + q + + \frac{r}{x} + \frac{s}{x^{2}} + \etc. +\] +\PageSep{67} +The ratio of the several terms will not be altered if we +divide the whole by~$x^{m}$, which gives +\[ +a + \frac{b}{x} + \frac{c}{x^{2}} + \dots + + \frac{p}{x^{m-1}} + \frac{q}{x^{m}} + \frac{r}{x^{m+1}} + + \frac{s}{x^{m+2}} + \etc. +\] +It has been shown that by taking $\dfrac{1}{x}$ sufficiently small, +that is, by taking $x$ sufficiently great, any term of this +series may be made to contain the aggregate of the +succeeding terms, as often as we please; which relation +is not altered if we multiply every term by~$x^{m}$, +and so restore the original series. + +It follows from this, that $\dfrac{(x + 1)^{m}}{x^{m}}$ has unity for its +limit when $x$~is increased without limit. For $(x + 1)^{m}$ +is $x^{m} + mx^{m-1} + \etc.$, in which $x^{m}$~can be made as +great as we please with respect to the rest of the +series. Hence $\dfrac{(x + 1)^{m}}{x^{m}} = 1 + \dfrac{mx^{m-1} + \etc.}{x^{m}}$, the numerator +of which last fraction decreases indefinitely +as compared with its denominator. + +In a similar way it may be shown that the limit of +$\dfrac{x^{m}}{(x + 1)^{m+1} - x^{m+1}}$, when $x$~is increased, is~$\dfrac{1}{m + 1}$. For +since $(x + 1)^{m+1} = x^{m+1} + (m + 1)x^{m} + \frac{1}{2}(m + 1)m x^{m-1} + \etc.$, +this fraction is +\[ +\frac{x^{m}}{(m + 1)x^{m} + \frac{1}{2}(m + 1)m x^{m-1} + \etc.} +\] +in which the first term of the denominator may be +made to contain all the rest as often as we please; +that is, if the fraction be written thus, $\dfrac{x^{m}}{(m + 1)x^{m} + A}$, +$A$~can be made as small a part of~$(m + 1)x^{m}$ as we +\PageSep{68} +please. Hence this fraction can, by a sufficient increase +of~$x$, be brought as near as we please to +$\dfrac{x^{m}}{(m + 1)x^{m}}$, or~$\dfrac{1}{m + 1}$. + +A similar proposition may be shown of the fraction +$\dfrac{(x + b)^{m}}{(x + a)^{m+1} - x^{m+1}}$, which may be immediately reduced +to the form $\dfrac{x^{m} + B}{(m + 1)ax^{m} + A}$, where $x$~may be taken +so great that $x^{m}$~shall contain $A$~and~$B$ any number of +times. + +We will now consider the sums of $x$~terms of the +following series, each of which may evidently be made +as great as we please, by taking a sufficient number +of its terms, +\begin{alignat*}{7} +&1 &&+ 2 &&+ 3 &&+ 4 &&+ \dots &&+ \; x - 1 &&+ x\Add{,} +\tag*{(1)} \\ +&1^{2} &&+ 2^{2} &&+ 3^{2} &&+ 4^{2} &&+ \dots &&+ (x - 1)^{2} &&+ x^{2}\Add{,} +\tag*{(2)} \displaybreak[0]\\ +&1^{3} &&+ 2^{3} &&+ 3^{3} &&+ 4^{3} &&+ \dots &&+ (x - 1)^{3} &&+ x^{3}\Add{,} +\tag*{(3)} \\ +\DotRow{14} \displaybreak[0]\\ +&1^{m} &&+ 2^{m} &&+ 3^{m} &&+ 4^{m} &&+ \dots &&+ (x - 1)^{m} &&+ x^{m}\Add{.} +\tag*{($m$)} +\end{alignat*} +We propose to inquire what is the limiting ratio of +any one of these series to the last term of the succeeding +one; that is, to what do the ratios of $(1 + 2 + \dots + x)$ +to~$x^{2}$, of $(1^{2} + 2^{2}\Add{+} \dots + x^{2})$ to~$x^{3}$, etc., +approach, when $x$~is increased without limit. + +To give an idea of the method of increase of these +series, we shall first show that $x$~may be taken so +great, that the last term of each series shall be as +small a part as we please of the sum of all those which +precede. To simplify the symbols, let us take the +third series $1^{3} + 2^{3} + \dots + x^{3}$, in which we are to +show that $x^{3}$~may be made less than any given part\Typo{.}{,} +\PageSep{69} +say one thousandth, of the sum of those which precede, +or of $1^{3} + 2^{3} \Add{+} \dots + (x - 1)^{3}$. + +First, $x$~may be taken so great that $x^{3}$ and $(x - 1000)^{3}$ +shall have a ratio as near to equality as we +please. For the ratio of these quantities being the +same as that of $1$ to $\left(1 - \dfrac{1000}{x}\right)^{3}$, and $\dfrac{1000}{x}$ being as +small as we please if $x$ may be as great as we please, it +follows that $1 - \dfrac{1000}{x}$, and, consequently, $\left(1 - \dfrac{1000}{x}\right)^{3}$ +may be made as near to unity as we please, or the +ratio of $1$ to $\left(1 - \dfrac{1000}{x}\right)^{3}$, may be brought as near as +we please to that of $1$ to~$1$, or a ratio of equality. But +this ratio is that of $x^{3}$ to~$(x - 1000)^{3}$. Similarly the +ratios of $x^{3}$ to~$(x - 999)^{3}$, of $x^{3}$ to~$(x - 998)^{3}$, etc., up +to the ratio of $x^{3}$ to~$(x - 1)^{3}$ may be made as near as +we please to ratios of equality; there being one thousand +in all. If, then, $(x - 1)^{3} = \alpha x^{3}$, $(x - 2)^{3} = \beta x^{3}$, +etc., up to $(x - 1000)^{3} = \omega x^{3}$, $x$~can be taken so great +that each of the fractions $\alpha$,~$\beta$,~etc., shall be as near +to unity, or $\alpha + \beta + \dots + \omega$ as near\footnote + {Observe that this conclusion depends upon the \emph{number} of quantities $\alpha$,~$\beta$,~etc., + being \emph{determinate}. If there be \emph{ten} quantities, each of which can be + brought as near to unity as we please, their sum can be brought as near to~$10$ + as we please; for, take any fraction~$A$, and make each of those quantities + differ from unity by less than the tenth part of~$A$, then will the sum differ + from~$10$ by less than~$A$. This argument fails, if the number of quantities be + unlimited.} +to~$1000$ as we +please. Hence +%[** TN: In-line in the original] +\[ +\frac{1}{\alpha + \beta + \dots + \omega}\Add{,} +\] +which is +\[ +\frac{x^{3}}{\alpha x^{3} + \beta x^{3} + \dots + \omega x^{3}}, +\] +or +\[ +\frac{x^{3}}{(x - 1)^{3} + (x - 2)^{2} + \dots + (x - 1000)^{3}}, +\] +\PageSep{70} +can be brought as near to~$\dfrac{1}{1000}$ as we please; and by +the same reasoning, the fraction +\[ +\frac{x^{3}}{(x - 1)^{3} + \dots + (x - 1001)^{3}} +\] +may be brought as near to~$\dfrac{1}{1001}$ as we please; that is, +may be made less than~$\dfrac{1}{1000}$. Still more then may +\[ +\frac{x^{3}}{(x - 1)^{3} + \dots + (x - 1001)^{3} + \dots + 2^{3} + 1^{3}} +\] +be made less than~$\dfrac{1}{1000}$, or $x^{3}$~may be less than the +thousandth part of the sum of all the preceding terms. + +In the same way it may be shown that a term may +be taken in any one of the series, which shall be less +than any given part of the sum of all the preceding +terms. It is also true that the difference of any two +succeeding terms may be made as small a part of +either as we please. For $(x + 1)^{m} - x^{m}$, when developed, +will only contain exponents less than~$m$, being +$mx^{m-1} + m\dfrac{m - 1}{2}\, x^{m-2} + \etc.$; and we have shown +(\PageRef{66}) that the sum of such a series may be made +less than any given part of~$x^{m}$. It is also evident +that, whatever number of terms we may sum, if a +sufficient number of succeeding terms be taken, the +sum of the latter shall exceed that of the former in +any ratio we please. + +Let there be a series of fractions +\[ +\frac{a}{pa + b},\quad +\frac{a'}{pa' + b'},\quad +\frac{a''}{pa'' + b''},\quad \etc., +\] +in which $a$,~$a'$,~etc., $b$,~$b'$,~etc., increase without limit; +but in which the ratio of $b$~to~$a$, $b'$~to~$a'$, etc., diminishes +without limit. If it be allowable to begin by +\PageSep{71} +supposing $b$~as small as we please with respect to~$a$, +or $\dfrac{b}{a}$~as small as we please, the first, and all the succeeding +fractions, will be as near as we please to~$\dfrac{1}{p}$, +which is evident from the equations +\[ +\frac{a}{pa + b} = \frac{1}{p + \dfrac{b}{a}},\quad +\frac{a'}{pa' + b'} = \frac{1}{p + \dfrac{b'}{a'}},\quad \etc. +\] +Form a new fraction by summing the numerators and +denominators of the preceding, such as +\[ +\frac{a + a' + a'' + \etc.} + {p(a + a' + a'' + \etc.) + b + b' + b'' + \etc.\Typo{,}{}}, +\] +the \emph{etc.}\ extending to any given number of terms. + +This may also be brought as near to~$\dfrac{1}{p}$ as we please. +For this fraction is the same as +\[ +\text{$1$~divided by } p + \frac{b + b' + \etc.}{a + a' + \etc.}; +\] +and it can be shown\footnote + {See \Title{Study of Mathematics} (Reprint Edition, Chicago: The Open Court + Publishing Co.), page~270.} +that +\[ +\frac{b + b' + \etc.}{a + a' + \etc.} +\] +must lie between the least and greatest of the fractions +$\dfrac{b}{a}$,~$\dfrac{b'}{a'}$,~etc. +If, then, each of these latter fractions +can be made as small as we please, so also can +\[ +\frac{b + b' + \etc.}{a + a' + \etc.}. +\] +No difference will be made in this result, if we use +the following fraction, +\[ +\frac{A + (a + a' + a'' + \etc.)} + {B + p(a + a' + a'' + \etc.) + b + b' + b'' + \etc.}\Add{,} +\Tag{(1)} +\] +\PageSep{72} +$A$~and~$B$ being given quantities; provided that we +can take a number of the original fractions sufficient +to make $a + a' + a'' + \etc.$, as great as we please, +compared with $A$~and~$B$. This will appear on dividing +the numerator and denominator of~\Eq{(1)} by $a + a' + a'' + \etc$. + +Let the fractions be +\begin{gather*} +\frac{(x + 1)^{3}}{(x + 1)^{4} - x^{4}},\quad +\frac{(x + 2)^{3}}{(x + 2)^{4} - (x + 1)^{4}}, \\ +\frac{(x + 3)^{3}}{(x + 3)^{4} - (x + 2)^{4}},\quad \etc. +\end{gather*} +The first of which, or $\dfrac{(x + 1)^{3}}{4x^{3} + \etc.}$ may, as we have +shown, be within any given difference of~$\dfrac{1}{4}$, and the +others still nearer, by taking a value of~$x$ sufficiently +great. Let us suppose each of these fractions to be +within $\dfrac{1}{100000}$ of~$\dfrac{1}{4}$. The fraction formed by summing +the numerators and denominators of these fractions +($n$~in number) will be within the same degree of +nearness to~$\frac{1}{4}$. But this is +\[ +\frac{(x + 1)^{3} + (x + 2)^{3} + \dots + (x + n)^{3}}{(x + 1)^{4} - x^{4}}\Add{,} +\Tag{(2)} +\] +all the terms of the denominator disappearing, except +two from the first and last. If, then, we add~$x^{4}$ to +the denominator, and $1^{3} + 2^{3} + 3^{3} \Add{+} \dots + x^{3}$ to the numerator, +we can still take $n$ so great that $(x + 1)^{3} + \dots + (x + n)^{3}$ +shall contain $1^{3} + \dots + x^{3}$ as often +as we please, and that $(x + n)^{4} - x^{4}$ shall contain~$x^{4}$ +in the same manner. To prove the latter, observe +that the ratio of $(x + n)^{4} - x^{4}$ to~$x^{4}$ being $\left(1 + \dfrac{n}{x}\right)^{4}$, +can be made as great as we please, if it be permitted +\PageSep{73} +to take for~$n$ a number containing~$x$ as often as we +please. Hence, by the preceding reasoning, the fraction, +with its numerator and denominator thus increased, +or +\[ +\frac{1^{3} + 2^{3} + 3^{3} + \dots + x^{3} + (x + 1)^{3} + \dots + (x + n)^{3}} + {(x + n)^{4}} +\Tag{(3)} +\] +may be brought to lie within the same degree of nearness +to~$\frac{1}{4}$ as~\Eq{(2)}; and since this degree of nearness +could be named at pleasure, it follows that \Eq{(3)}~can +be brought as near to~$\frac{1}{4}$ as we please. Hence the +limit of the ratio of $(1^{3} + 2^{3} + \dots + x^{3})$ to~$x^{4}$, as $x$~is +increased without limit, is~$\frac{1}{4}$; and, in a similar manner, +it may be proved that the limit of the ratio of +$(1^{m} + 2^{m} + \dots + x^{m})$ to~$x^{m+1}$ is the same as that of +$\dfrac{(x + 1)^{m}}{(x + 1)^{m+1} - x^{m+1}}$ or $\dfrac{1}{m + 1}$. + +This result will be of use when we come to the +first principles of the integral calculus. It may also +\index{Integral Calculus}% +be noticed that the limits of the ratios which $x\, \dfrac{x - 1}{2}$, +$x\, \dfrac{x - 1}{2}\, \dfrac{x - 2}{3}$, etc., bear to $x^{2}$,~$x^{3}$, etc., are severally $\dfrac{1}{2}$, +$\dfrac{1}{2·3}$, etc.; the limit being that to which the ratios approximate +as $x$~increases without limit. For $x\, \dfrac{x - 1}{2} ÷ x^{2} = \dfrac{x - 1}{2x}$, +$x\, \dfrac{x - 1}{2}\, \dfrac{x - 2}{3} ÷ x^{3} = \dfrac{x - 1}{2x}\, \dfrac{x - 2}{3x}$, etc., +and the limits of $\dfrac{x - 1}{2}$, $\dfrac{x - 2}{3}$, are severally equal to +unity. + +We now resume the elementary principles of the +Differential Calculus. +\PageSep{74} + + +\Subsection[Recapitulation of Results Reached in the Theory of Functions.] +{Recapitulation of Results.} + +The following is a recapitulation of the principal +results which have hitherto been noticed in the general +theory of functions: +\index{Functions!recapitulation of results in the theory of}% + +(1) That if in the equation $y = \phi(x)$, the variable~$x$ +receives an increment~$dx$, $y$~is increased by the series +\[ +\phi' x\, dx + \phi'' x\, \frac{(dx)^{2}}{2} + + \phi''' x\, \frac{(dx)^{3}}{2·3} + \etc. +\] + +(2) That $\phi'' x$ is derived in the same manner from~$\phi' x$, +that $\phi' x$~is from~$\phi x$; viz., that in like manner as +$\phi' x$~is the coefficient of~$dx$ in the development of +$\phi(x + dx)$, so $\phi'' x$~is the coefficient of~$dx$ in the development +of $\phi'(x + dx)$; similarly $\phi''' x$~is the coefficient +of~$dx$ in the development of~$\phi''(x + dx)$, and +so on. + +(3) That $\phi' x$ is the limit of~$\dfrac{dy}{dx}$, or the quantity to +which the latter will approach, and to which it may +be brought as near as we please, when $dx$~is diminished. +It is called the differential coefficient of~$y$. + +(4) That in every case which occurs in practice, +$dx$~may be taken so small, that any term of the series +above written may be made to contain the aggregate +of those which follow, as often as we please; whence, +though $\phi' x\, dx$~is not the actual increment produced +by changing~$x$ into~$x + dx$ in the function~$\phi x$, yet, by +taking $dx$ sufficiently small, it may be brought as near +as we please to a ratio of equality with the actual increment. + + +\Subsection[Approximations by the Differential Calculus.] +{Approximations.} + +The last of the above-mentioned principles is of +the greatest utility, since, by means of it, $\phi' x\, dx$~may +\PageSep{75} +\index{Errors, in the valuation of quantities}% +be made as nearly as we please the actual increment; +and it will generally happen in practice, that $\phi' x\, dx$ +may be used for the increment of~$\phi x$ without sensible +error; that is, if in~$\phi x$, $x$~be changed into $x + dx$, $dx$~being +very small, $\phi x$~is changed into $\phi x + \phi' x\, dx$, +very nearly. Suppose that $x$ being the correct value +of the variable, $x + h$ and $x + k$ have been successively +substituted for it, or the errors $h$~and~$k$ have +been committed in the valuation of~$x$, $h$~and~$k$ being +very small. Hence $\phi(x + h)$ and $\phi(x + k)$ will be +erroneously used for~$\phi x$. But these are nearly $\phi x + \phi' x\, h$ +and $\phi x + \phi' x\, k$, and the errors committed in +taking~$\phi x$ are $\phi' x\, h$ and $\phi' x\, k$, very nearly. These +last are in the proportion of $h$ to~$k$, and hence results +a proposition of the utmost importance in every practical +application of mathematics, viz., that if two different, +but small, errors be committed in the valuation +of any quantity, the errors arising therefrom at +the end of any process, in which both the supposed +values of~$x$ are successively adopted, are very nearly +in the proportion of the errors committed at the beginning. +For example, let there be a right-angled +triangle, whose base is~$3$, and whose other side should +be~$4$, so that the hypothenuse should be $\sqrt{3^{2} + 4^{2}}$ +or~$5$. But suppose that the other side has been twice +erroneously measured, the first measurement giving +$4.001$, and the second $4.002$, the errors being $.001$ +and~$.002$. The two values of the hypothenuse thus +obtained are +\[ +\sqrt{3^{2} + 4.001^{2}}, \quad\text{or}\quad \sqrt{25.008001}, +\] +and +\[ +\sqrt{3^{2} + 4.002^{2}}, \quad\text{or}\quad \sqrt{25.016004}, +\] +which are very nearly $5.0008$ and $5.0016$. The errors +of the hypothenuse are then $.0008$ and $.0016$ nearly; +and these last are in the proportion of $.001$ and~$.002$. +\PageSep{76} + +It also follows, that if $x$~increase by successive equal +steps, any function of~$x$ will, for a few steps, increase +so nearly in the same manner, that the supposition of +such an increase will not be materially wrong. For, +if $h$,~$2h$,~$3h$, etc., be successive small increments given +to~$x$, the successive increments of~$\phi x$ will be $\phi' x\, h$, +$\phi' x\, 2h$, $\phi' x\, 3h$,~etc.\ nearly; which being proportional +to $h$,~$2h$,~$3h$, etc., the increase of the function is nearly +doubled, trebled, etc., if the increase of~$x$ be doubled, +trebled,~etc. + +This result may be rendered conspicuous by reference +to any astronomical ephemeris, in which the +\index{Astronomical ephemeris}% +positions of a heavenly body are given from day to +day. The intervals of time at which the positions are +given differ by $24$~hours, or nearly $\frac{1}{365}$\th~part of the +whole year. And even for this interval, though it can +hardly be called \emph{small} in an astronomical point of view, +the increments or decrements will be found so nearly +the same for four or five days together, as to enable +the student to form an idea how much more near they +would be to equality, if the interval had been less, say +one hour instead of twenty-four. For example, the +sun's longitude on the following days at noon is written +\index{Sun's longitude}% +underneath, with the increments from day to day. +\[ +\ArrayCompress +\begin{array}{c*{2}{>{\ }c}c} +\ColHead[September]{1834 \\ September} & +\ColHead[Sun's longitude]{Sun's longitude \\ at noon.} & +\ColHead{Increments.} & +\ColHead[Proportion which the differences]{Proportion which the differences \\ + of the increments bear to the \\ + whole increments.} \\ +% +1\text{st} & 158\rlap{$°$}\ \ 30\rlap{$'$}\ \ 35\rlap{$''$} + & \Low{58\rlap{$'$}\ \Z9\rlap{$''$}} & \\ +2\text{nd} & 159\ \ 28\ \ 44 & \Low{58\ 12} & \frac{3}{3489} \\ +3\text{rd} & 160\ \ 26\ \ 56 & \Low{58\ 13} & \frac{1}{3492} \\ +4\text{th} & 161\ \ 25\ \ \Z9 & \Low{58\ 14} & \frac{1}{3493} \\ +5\text{th} & 162\ \ 23\ \ 23 +\end{array} +\] +The sun's longitude is a function of the time; that is, +the number of years and days from a given epoch +being given, and called~$x$, the sun's longitude can be +\PageSep{77} +found by an algebraical expression which may be +called~$\phi x$. If we date from the first of January,~1834, +$x$~is~$.666$, which is the decimal part of a year between +the first days of January and September. The increment +is one day, or nearly $.0027$~of a year. Here $x$~is +successively made equal to~$.666$, $.666 + 0027$, $.666 + 2 × .0027$, +etc.; and the intervals of the corresponding +values of~$\phi x$, if we consider only minutes, are the +same; but if we take in the seconds, they differ from +one another, though only by very small parts of themselves, +as the last column shows. + + +\Subsection[Solution of Equations by the Differential Calculus.] +{Solution of Equations.} + +This property is also used\footnote + {See \Title{Study of Mathematics} (Reprint Edition, Chicago: The Open Court + Publishing Co., 1898), page~169 et~seq.} +in finding logarithms +\index{Equations, solution of}% +intermediate to those given in the tables; and may +be applied to find a nearer solution to an equation, +than one already found. For example, suppose it required +to find the value of~$x$ in the equation $\phi x = 0$, +$a$~being a near approximation to the required value. +Let $a + h$ be the real value, in which $h$~will be a small +quantity. It follows that $\phi(a + h) = 0$, or, which is +nearly true, $\phi a + \phi' a\, h = 0$. Hence the real value of~$h$ +is nearly~$-\dfrac{\phi a}{\phi' a}$, or the value $a - \dfrac{\phi a}{\phi' a}$ is a nearer +approximation to the value of~$x$. For example, let +$x^{2} + x - 4 = 0$ be the equation. Here $\phi x = x^{2} + x - 4$, +and $\phi(x + h) = (x + h)^{2} + x + h - 4 = x^{2} + x - 4 + (2x + 1)h + h^{2}$; +so that $\phi' x = 2x + 1$. A near value +of~$x$ is~$1.57$; let this be~$a$. Then $\phi a = .0349$, and +$\phi' a = 4.14$. Hence $-\dfrac{\phi a}{\phi' a} = -.00843$. Hence +$1.57 - .00843$, or~$1.56157$, is a nearer value of~$x$. If +\PageSep{78} +we proceed in the same way with~$1.5616$, we shall +find a still nearer value of~$x$, viz., $1.561553$. We +have here chosen an equation of the second degree, +in order that the student may be able to verify the +result in the common way; it is, however, obvious +that the same method may be applied to equations +of higher degrees, and even to those which are not +to be treated by common algebraical method, such as +$\tan x = ax$. + + +\Subsection{Partial and Total Differentials.} + +We have already observed, that in a function of +\index{Differentials!partial|EtSeq}% +\index{Differentials!total|EtSeq}% +\index{Partial!differentials|EtSeq}% +\index{Total!differentials|EtSeq}% +more quantities than one, those only are mentioned +which are considered as variable; so that all which +we have said upon functions of one variable, applies +\index{Functions!of several variables|EtSeq}% +equally to functions of several variables, so far as a +\index{Variables!functions of several|EtSeq}% +change in one only is concerned. Take for example +$x^{2} y + 2xy^{3}$. If $x$~be changed into $x + dx$, $y$~remaining +the same, this function is increased by $2xy\, dx + 2y^{3}\, dx + \etc.$, +in which, as in \PageRef{29}, no terms are contained +in the~\emph{etc.}\ except those which, by diminishing~$dx$, +can be made to bear as small a proportion as we +please to the first terms. Again, if $y$~be changed into +$y + dy$, $x$~remaining the same, the function receives +the increment $x^{2}\, dy + 6xy^{2}\, dy + \etc.$; and if $x$~be changed +into $x + dx$, $y$~being at the same time changed into +$y + dy$, the increment of the function is $(2xy + 2y^{3})\, dx + (x^{2} + 6xy^{2})\, dy + \etc$. +If, then, $u = x^{2} y + 2xy^{3}$, and +$du$~denote the increment of~$u$, we have the three following +equations, answering to the various suppositions +above mentioned, \\ +(1) when $x$~only varies, +\[ +du = (2xy + 2y^{3})\, dx + \etc. +\] +\PageSep{79} +(2) when $y$~only varies, +\[ +du = (x^{2} + 6xy^{2})\, dy + \etc. +\] +(3) when both $x$~and~$y$ vary, +\[ +du = (2xy + 2y^{3})\, dx + (x^{2} + 6xy^{2})\, dy + \etc. +\] +in which, however, it must be remembered, that $du$~does +not stand for the same thing in any two of the +three equations: it is true that it always represents +an increment of~$u$, but as far as we have yet gone, we +have used it indifferently, whether the increment of~$u$ +was the result of a change in $x$~only, or $y$~only, or both +together. + +To distinguish the different increments of~$u$, we +must therefore seek an additional notation, which, +\index{Calculus, notation of|EtSeq}% +\index{Notation!of the Differential Calculus|EtSeq}% +without sacrificing the~$du$ that serves to remind us +that it was $u$ which received an increment, may also +point out from what supposition the increment arose. +For this purpose we might use $d_{x}u$~and~$d_{y}u$, and $d_{x,y}u$, +to distinguish the three; and this will appear to the +learner more simple than the one in common use, +which we shall proceed to explain. We must, however, +remind the student, that though in matters of +reasoning, he has a right to expect a solution of every +difficulty, in all that relates to notation, he must trust +entirely to his instructor; since he cannot judge between +the convenience or inconvenience of two symbols +without a degree of experience which he evidently +cannot have had. Instead of the notation above +described, the increments arising from a change in $x$ +and~$y$ are severally denoted by $\dfrac{du}{dx}\, dx$ and $\dfrac{du}{dy}\, dy$, on +the following principle: If there be a number of results +obtained by the same species of process, but on +different suppositions with regard to the quantities +\PageSep{80} +used; if, for example, $p$~be derived from some supposition +with regard to~$a$, in the same manner as are $q$ +and~$r$ with regard to $b$~and~$c$, and if it be inconvenient +and unsymmetrical to use separate letters $p$,~$q$, and~$r$, +for the three results, they may be distinguished by +using the same letter~$p$ for all, and writing the three +results thus, $\dfrac{p}{a}\, a$, $\dfrac{p}{b}\, b$, $\dfrac{p}{c}\, c$. Each of these, in common +algebra, is equal to~$p$, but the letter~$p$ does not +stand for the same thing in the three expressions. +The first is the~$p$, so to speak, which belongs to~$a$, the +second that which belongs to~$b$, the third that which +belongs to~$c$. Therefore the numerator of each of the +fractions $\dfrac{p}{a}$,~$\dfrac{p}{b}$, and~$\dfrac{p}{c}$, must never be separated +from its denominator, because the value of the former +depends, in part, upon the latter; and one~$p$ cannot +be distinguished from another without its denominator. +The numerator by itself only indicates what operation +is to be performed, and on what quantity; the +denominator shows what quantity is to be made use +of in performing it. Neither are we allowed to say +that $\dfrac{p}{a}$ divided by~$\dfrac{p}{b}$ is~$\dfrac{b}{a}$; for this supposes that $p$~means +the same thing in both quantities. + +In the expressions $\dfrac{du}{dx}\, dx$, and $\dfrac{du}{dy}\, dy$, each denotes +that $u$~has received an increment; but the first points +out that~$x$, and the second that~$y$, was supposed to increase, +in order to produce that increment; while $du$~by +itself, or sometimes $d.u$, is employed to express +the increment derived from both suppositions at once. +And since, as we have already remarked, it is not the +ratios of the increments themselves, but the limits of +those ratios, which are the objects of investigation in +\PageSep{81} +the Differential Calculus, here, as in \PageRef{28}, $\dfrac{du}{dx}\, dx$, +and $\dfrac{du}{dy}\, dy$, are generally considered as representing +those terms which are of use in obtaining the limiting +ratios, and do not include those terms, which, from +\index{Limiting ratios}% +\index{Ratios, limiting}% +their containing higher powers of $dx$~or~$dy$ than the +first, may be made as small as we please with respect +to $dx$~or~$dy$. Hence in the example just given, where +$u = x^{2} y + 2xy^{3}$, we have +\begin{align*} +&\dfrac{du}{dx}\, dx = (2xy + 2y^{3})\, dx, + &&\text{or}\quad \frac{du}{dx} = 2xy + 2y^{3}\Add{,} \\ +&\dfrac{du}{dy}\, dy = (x^{2} + 6xy^{2})\, dy, + &&\text{or}\quad \frac{du}{dy} = x^{2} + 6xy^{2}\Add{,} \\ +&du \quad\text{or}\quad d.u = \frac{du}{dx}\, dx + \frac{du}{dy}\,dy. +\end{align*} + +The last equation gives a striking illustration of +the method of notation. Treated according to the +common rules of algebra, it is $du = du + du$, which is +absurd, but which appears rational when we recollect +that the second~$du$ arises from a change in $x$~only, the +third from a change in $y$~only, and the first from a +change in both. The same equation may be proved +to be generally true for all functions of $x$~and~$y$, if we +bear in mind that no term is retained, or need be retained, +as far as the limit is concerned, which, when +$dx$~or~$dy$ is diminished, diminishes without limit as +compared with them. In using $\dfrac{du}{dx}$ and $\dfrac{du}{dy}$ as differential +coefficients of~$u$ with respect to $x$~and~$y$, the objection +(\PageRef{27}) against considering these as the +limits of the ratios, and not the ratios themselves, +does not hold, since the numerator is not to be separated +from its denominator. +\PageSep{82} + +Let $u$ be a function of $x$~and~$y$, represented\footnote + {The symbol $\phi(x, y)$ must not be confounded with~$\phi(xy)$. The former represents + any function of $x$~and~$y$; the latter a function in which $x$~and~$y$ only + enter so far as they are contained in their product. The second is therefore + a particular case of the first; but the first is not necessarily represented by + the second. For example, take the function $xy + \sin xy$, which, though it + contains both $x$~and~$y$, yet can only be altered by such a change in $x$~and~$y$ as + will alter their product, and if the product be called~$p$, will be $p + \sin p$. This + may properly be represented by~$\phi(xy)$; whereas $x + xy^{2}$ cannot be represented + in the same way, since other functions besides the product are contained + in it.} +by~$\phi(x, y)$. +It is indifferent whether $x$~and~$y$ be changed +\index{Coefficients, differential}% +\index{Differential coefficients}% +at once into $x + dx$ and $y + dy$, or whether $x$~be first +changed into $x + dx$, and $y$~be changed into $y + dy$ in +the result. Thus, $x^{2} y + y^{3}$ will become $(x + dx)^{2}(y + dy) + (y + dy)^{3}$ +in either case. If $x$~be changed +into $x + dx$, $u$~becomes $u + \ux\, dx + \etc.$, (where $\ux$~is +what we have called the differential coefficient of~$u$ +with respect to~$x$, and is itself a function of $x$~and~$y$; +and the corresponding increment of~$u$ is $\ux\, dx + \etc.$)\Add{.} +If in this result $y$~be changed into $y + dy$, $u$~will assume +the form $u + \uy\, dy + \etc.$, where $\uy$~is the differential +coefficient of~$u$ with respect to~$y$; and the increment +which $u$~receives will be $\uy + \etc$. Again, +when $y$~is changed into $y + dy$, $\ux$,~which is a function +of $x$~and~$y$, will assume the form $\ux + p\, dy + \etc.$; and +$u + \ux\, dx + \etc.$\ becomes $u + \uy\, dy + \etc. + (\ux + p\, dy + \etc.)\, dx + \etc.$, +or $u + \uy\, dy + \ux\, dx + p\, dx\, dy + \etc.$, +in which the term $p\, dx\, dy$ is useless in finding the limit. +For since $dy$~can be made as small as we please, +$p\, dx\, dy$ can be made as small a part of~$p\, dx$ as we please, +and therefore can be made as small a part of~$dx$ as +we please. Hence on the three suppositions already +made, we have the following results: +%[** TN: Re-formatted] +\begin{itemize} +\item[(1)] when $x$~only is changed into~$x + dx$, +$u$~receives the increment +$\ux\, dx + \etc$.\Add{,} + +\item[(2)] when $y$~only is changed into~$y + dy$, +$u$~receives the increment +$\uy\, dy + \etc$.\Add{,} + +\item[(3)] when $x$~becomes $x + dx$ and $y$~becomes $y + dy$ at once, +$u$~receives the increment +$\ux\, dx + \uy\, dy + \etc$.\Add{,} +\end{itemize} +\PageSep{83} +the \emph{etc.}\ in each case containing those terms only which +can be made as small as we please, with respect to +the preceding terms. In the language of Leibnitz, +\index{Leibnitz}% +we should say that if $x$~and~$y$ receive infinitely small +\index{Infinitely small, the notion of}% +increments, the sum of the infinitely small increments +of~$u$ obtained by making these changes separately, is +equal to the infinitely small increment obtained by +making them both at once. As before, we may correct +this inaccurate method of speaking. The several +increments in (1),~(2), and~(3), maybe expressed by +$\ux\, dx + P$, $\uy\, dy + Q$, and $\ux\, dx + \uy\, dy + R$; where $P$,~$Q$, +and~$R$ can be made such parts of $dx$~or~$dy$ as we +please, by taking $dx$~or~$dy$ sufficiently small. The sum +of the two first is $\ux\, dx + \uy\, dy + P + Q$, which differs +from the third by $P + Q - R$; which, since each of +its terms can be made as small a part of $dx$~or~$dy$ as +we please, can itself be made less than any given part +of $dx$~or~$dy$. + +This theorem is not confined to functions of two +variables only, but may be extended to those of any +number whatever. Thus, if $z$~be a function of $p$,~$q$,~$r$, +and~$s$, we have +\[ +d.z \quad\text{or}\quad +dz = \frac{dz}{dp}\, dp + + \frac{dz}{dq}\, dq + + \frac{dz}{dr}\, dr + + \frac{dz}{ds}\, ds + \etc. +\] +in which $\dfrac{dz}{dp}\, dp + \etc.$\ is the increment which a change +in $p$~\emph{only} gives to~$z$, and so on. The \emph{etc.}\ is the representative +of an infinite series of terms, the aggregate +of which diminishes continually with respect to $dp$,~$dq$,~etc., +as the latter are diminished, and which, therefore, +\PageSep{84} +has no effect on the \emph{limit} of the ratio of~$d.z$ to +any other quantity. + + +\Subsection[Application of the Theorem for Total Differentials to the Determination of Total Resultant Errors.] +{Practical Application of the Preceding Theorem.} + +We proceed to an important practical use of this +\index{Errors, in the valuation of quantities}% +theorem. If the increments $dp$,~$dq$,~etc., be small, +this last-mentioned equation, (the terms included in +the \emph{etc.}\ being omitted,) though not actually true, is +sufficiently near the truth for all practical purposes; +which renders the proposition, from its simplicity, of +the highest use in the applications of mathematics. +For if any result be obtained from a set of \textit{data}, no +one of which is exactly correct, the error in the result +would be a very complicated function of the errors in +the \textit{data}, if the latter were considerable. When they +are small, the error in the results is very nearly the +sum of the errors which would arise from the error in +each \textit{datum}, if all the others were correct. For if $p$,~$q$,~$r$, +and~$s$, are the \emph{presumed} values of the \textit{data}, which +give a certain value~$z$ to the function required to be +found; and if $p + dp$, $q + dq$, etc., be the \emph{correct} values +of the \textit{data}, the correction of the function~$z$ will be +very nearly made, if $z$~be increased by $\dfrac{dz}{dp}\, dp + \dfrac{dz}{dq}\, dq + \dfrac{dz}{dr}\, dr + \dfrac{dz}{ds}\, ds$, +being the sum of terms which would +arise from each separate error, if each were made in +turn by itself. + +For example: A transit instrument is a telescope +\index{Transit instrument}% +mounted on an axis, so as to move in the plane of the +meridian only, that is, the line joining the centres of +the two glasses ought, if the telescope be moved, to +pass successively through the zenith and the pole. +Hence can be determined the exact time, as shown by +a clock, at which any star passes a vertical thread, +\PageSep{85} +fixed inside the telescope so as apparently to cut the +field of view exactly in half, which thread will always +cover a part of the meridian, if the telescope be correctly +adjusted. In trying to do this, three errors +may, and generally will be committed, in some small +degree. (1)~The axis of the telescope may not be exactly +level; (2)~the ends of the same axis may not be +exactly east and west; (3)~the line which joins the +centres of the two glasses, instead of being perpendicular +to the axis of the telescope, may be inclined +to it. If each of these errors were considerable, and +the time at which a star passed the thread were observed, +the calculation of the time at which the same +star passes the real meridian would require complicated +formulæ, and be a work of much labor. But if +the errors exist in small quantities only, the calculation +is very much simplified by the preceding principle. +For, suppose only the first error to exist, and +calculate the corresponding error in the time of passing +the thread. Next suppose only the second error, +and then only the third to exist, and calculate the +effect of each separately, all which may be done by +simple formulæ. The effect of all the errors will then +be the sum of the effects of each separate error, at +least with sufficient accuracy for practical purposes. +The formulæ employed, like the equations in \PageRef{28}, +are not actually true in any case, but approach more +near to the truth as the errors are diminished. + + +\Subsection{Rules for Differentiation.} + +In order to give the student an opportunity of exercising +\index{Differentiation!of the common functions}% +himself in the principles laid down, we will +so far anticipate the treatises on the Differential Calculus +as to give the results of all the common rules +\PageSep{86} +for differentiation; that is, assuming $y$~to stand for +various functions of~$x$, we find the increment of~$y$ arising +from an increment in the value of~$x$, or rather, +that term of the increment which contains the first +power of~$dx$. This term, in theory, is the only one +on which the \emph{limit} of the ratio of the increments depends; +in practice, it is sufficiently near to the real +increment of~$y$, if the increment of~$x$ be small. + +{\Loosen (1) $y = x^{m}$, where $m$~is either whole or fractional, +\index{Differentiation!of the common functions}% +positive or negative; then $dy = mx^{m-1}\, dx$. Thus the +increment of~$x^{\efrac{2}{3}}$ or the first term of $(x + dx)^{\efrac{2}{3}} - x^{\efrac{2}{3}}$ +is $\frac{2}{3}x^{\efrac{2}{3}-1}\, dx$, or~$\dfrac{2\, dx}{3x^{\efrac{1}{3}}}$. Again, if $y = x^{8}$, $dy = 8x^{7}\, dx$. +When the exponent is negative, or when $y = \dfrac{1}{x^{m}}$, +$dy = -\dfrac{m\, dx}{x^{m+1}}$, or when $y = x^{-m}$, $dy = -mx^{-m-1}\, dx$, +which is according to the rule. The negative sign +indicates that an increase in~$x$ decreases the value +of~$y$; which, in this case, is evident.} + +(2) $y = a^{x}$. Here $dy = a^{x}\log a\, dx$ where the logarithm +(as is always the case in analysis, except +where the contrary is specially mentioned) is the Naperian +or hyperbolic logarithm. When $a$~is the base +of these logarithms, that is when $a = 2.7182818 = e$, +\index{Logarithms}% +or when $y = e^{x}$, $dy = e^{x}\, dx$. + +(3) $y = \log x$ (the Naperian logarithm). Here +$dy = \dfrac{dx}{x}$. If $y = \text{common log}~x$, $dy = -.4342944\, \dfrac{dx}{x}$. + +(4) $y = \sin x$, $dy = \cos x\, dx$; $y = \cos x$, $dy = -\sin x\, dx$; +$y = \tan x$, $dy = \dfrac{dx}{\cos^{2} x}$. + + +\Subsection[Illustration of the Rules for Differentiation.] +{Illustration of the Preceding Formulæ.} + +At the risk of being tedious to some readers, we +will proceed to illustrate these formulæ by examples +\PageSep{87} +from the tables of logarithms and sines, let $y = \text{common log}~x$. +\index{Logarithms}% +\index{Sines}% +If $x$~be changed into $x + dx$, the real increment +of~$y$ is +\[ +.4342944 \left(\frac{dx}{x} - \tfrac{1}{2}\, \frac{(dx)^{2}}{x^{2}} + + \tfrac{1}{3}\, \frac{(dx)^{3}}{x^{3}} - \etc.\right), +\] +in which the law of continuation is evident. The corresponding +series for Naperian logarithms is to be +found in \PageRef{20}. From the first term of this the +limit of the ratio of $dy$~to~$dx$ can be found; and if $dx$~be +\index{Ratio!of two increments}% +small, this will represent the increment with sufficient +accuracy. Let $x = 1000$, whence $y = \text{common log}~ 1000 =3$; +and let $dx = 1$, or let it be required to +find the common logarithm of $1000 + 1$, or~$1001$. The +first term of the series is therefore $.4342944 × \frac{1}{1000}$, or +$.0004343$, taking seven decimal places only. Hence +$\log 1001 = \log 1000 + .0004343$ or $3.0004343$ nearly. +The tables give $3.0004341$, differing from the former +only in the $7$\Chg{th}{\th}~place of decimals. + +{\Loosen Again, let $y = \sin x$; from which, by \PageRef{20}, as +before, if $x$~be increased by~$dx$, $\sin x$~is increased by +$\cos x\, dx - \frac{1}{2}\sin x\, (dx)^{2} - \etc.$, of which we take only +the first term. Let $x = 16°$, in which case $\sin x = .2756374$, +and $\cos x = .9612617$.} Let $dx = 1'$, or, as +it is represented in analysis, where the angular unit is +that angle whose arc is equal to the radius,\footnote + {See \Title{Study of Mathematics} (Chicago; The Open Court Pub. Co.), page~273 + et~seq.} +$\frac{60}{206265}$. +Hence $\sin 16°\, 1' = \sin 16° + .9612617 × \frac{60}{206265} = +.2756374 + .0002797 = .2759171$, nearly. The tables +give~$.2759170$. These examples may serve to show +how nearly the real ratio of two increments approaches +to their limit, when the increments themselves are +small. +\PageSep{88} + + +\Subsection{Differential Coefficients of Differential +Coefficients.} + +When the differential coefficient of a function of~$x$ +\index{Coefficients, differential}% +\index{Differential coefficients!of higher orders}% +\index{Finite differences|EtSeq}% +\index{Orders, differential coefficients of higher}% +\index{Successive differentiation|EtSeq}% +has been found, the result, being a function of~$x$, may +be also differentiated, which gives the differential coefficient +of the differential coefficient, or, as it is called, +the \emph{second} differential coefficient. Similarly the differential +coefficient of the second differential coefficient +is called the third differential coefficient, and so on. +We have already had occasion to notice these successive +differential coefficients in \PageRef{22}, where it appears +that $\phi' x$~being the first differential coefficient of~$\phi x$, +$\phi'' x$~is the coefficient of~$h$ in the development +$\phi'(x + h)$, and is therefore the differential coefficient +of~$\phi' x$, or what we have called the second differential +coefficient of~$\phi x$. Similarly $\phi''' x$~is the third differential +coefficient of~$\phi x$. If we were strictly to adhere +to our system of notation, we should denote the +several differential coefficients of~$\phi x$ or~$y$ by +\[ +\frac{dy}{dx}\Add{,}\quad +\frac{d.\dfrac{dy}{dx}}{dx}\Add{,}\quad +\frac{d.\dfrac{d.\frac{dy}{dx}}{dx}}{dx}\Add{,}\quad \etc. +\] +In order to avoid so cumbrous a system of notation, +the following symbols are usually preferred, +\[ +\frac{dy}{dx}\Add{,}\quad +\frac{d^{2} y}{dx^{2}}\Add{,}\quad +\frac{d^{3} y}{dx^{3}}\Add{,}\quad \etc. +\] + + +\Subsection{Calculus of Finite Differences. Successive +Differentiation.} + +We proceed to explain the manner in which this +\index{Differentiation!successive|EtSeq}% +notation is connected with our previous ideas on the +subject. +\PageSep{89} + +When in any function of~$x$, an increase is given to~$x$, +\index{Differences!calculus of}% +which is not supposed to be as small as we please, +it is usual to denote it by~$\Delta x$ instead of~$dx$, and the +corresponding increment of~$y$ or~$\phi x$, by~$\Delta y$ or~$\Delta\phi x$, +instead of~$dy$ or~$d\phi x$. The symbol~$\Delta x$ is called the +\emph{difference} of~$x$, being the difference between the value +of the variable~$x$, before and after its increase. + +Let $x$ increase at successive steps by the same difference; +that is, let a variable, whose first value is~$x$, +successively become $x + \Delta x$, $x + 2\Delta x$, $x + 3\Delta x$, etc., +and let the successive values of~$\phi x$ corresponding to +these values of~$x$ be $y$,~$y_{1}$, $y_{2}$,~$y_{3}$,~etc.; that is, $\phi x$~is +called~$y$, $\phi(x + \Delta x)$ is~$y_{1}$, $\phi(x + 2\Delta x)$ is~$y_{2}$, etc., and, +generally, $\phi(x + m\Delta x)$ is~$y_{m}$. Then, by our previous +definition $y_{1} - y$ is~$\Delta y$, $y_{2} - y_{1}$ is~$\Delta y_{1}$, $y_{3} - y_{2}$ is~$\Delta y_{2}$, +etc., the letter~$\Delta$ before a quantity always denoting +the increment it would receive if $x + \Delta x$ were substituted +for~$x$. Thus $y_{3}$ or $\phi(x + 3\Delta x)$ becomes $\phi(x + \Delta x + 3\Delta x)$, +or $\phi(x + 4\Delta x)$, when $x$~is changed into +$x + \Delta x$, and receives the increment $\phi(x + 4\Delta x) - \phi(x + 3\Delta x)$, or $y_{4} - y_{3}$. If $y$~be a function which decreases +when $x$~is increased, $y_{1} - y$, or $\Delta y$ is negative. + +It must be observed, as in \PageRef{26}, that $\Delta x$~does +not depend upon~$x$, because $x$~occurs in it; the symbol +merely signifies an increment given to~$x$, which +increment is not necessarily dependent upon the value +of~$x$. For instance, in the present case we suppose +it a given quantity; that is, when $x + \Delta x$ is changed +into $x + \Delta x + \Delta x$, or $x + 2\Delta x$, $x$~is changed, and $\Delta x$~is +not. + +In this way we get the two first of the columns underneath, +in which each term of the \emph{second} column is +formed by subtracting the term which immediately +precedes it in the first column from the one which immediately +\PageSep{90} +follows. Thus $\Delta y$ is $y_{1} - y$, $\Delta y_{1}$ is $y_{2} - y_{1}$, +etc. +\begin{gather*}%**** Tall, bad page break +\left. +\begin{alignedat}{2} +& \PadTo[l]{\phi(x + 4\Delta x)}{\phi(x)} && y \\ +& \phi(x + \Z\Delta x)\qquad && y_{1} \\ +& \phi(x + 2\Delta x) && y_{2} \\ +& \phi(x + 3\Delta x) && y_{3} \\ +& \phi(x + 4\Delta x) && y_{4} +\end{alignedat}\ +\right| +% +\left. +\begin{aligned} +& \Delta y \\ +& \Delta y_{1} \\ +& \Delta y_{2} \\ +& \Delta y_{3} +\end{aligned}\ +\right| +% +\left. +\begin{aligned} +& \Delta^{2} y \\ +& \Delta^{2} y_{1} \\ +& \Delta^{2} y_{2} +\end{aligned}\ +\right| +% +\left. +\begin{aligned} +& \Delta^{3} y \\ +& \Delta^{3} y_{1} +\end{aligned}\ +\right| +% +\begin{aligned} +& \Delta^{4} y +\end{aligned} \\ +\PadTo{\phi(x + 4\Delta x)}{\etc.}\phantom{\qquad\qquad\qquad\qquad\qquad\qquad} +\end{gather*} + +In the first column is to be found a series of successive +values of the same function~$\phi x$, that is, it contains +terms produced by substituting successively in~$\phi x$ +the quantities $x$, $x + \Delta x$, $x + 2\Delta x$, etc., instead of~$x$. +The second column contains the successive values +of another function $\phi(x + \Delta x) - \phi x$, or~$\Delta \phi x$, made by +the same substitutions; if, for example, we substitute +$x + 2\Delta x$ for~$x$, we obtain $\phi(x + 3\Delta x) - \phi(x + 2\Delta x)$, +or $y_{3} - y_{2}$, or~$\Delta y_{2}$. If, then, we form the successive +differences of the terms in the second column, we obtain +a new series, which we might call the differences +of the differences of the first column, but which are +called the \emph{second differences} of the first column. And +as we have denoted the operation which deduces the +second column from the first by~$\Delta$, so that which deduces +the third from the second may be denoted by~$\Delta\Delta$, +which is abbreviated into~$\Delta^{2}$. Hence as $y_{1} - y$ +was written~$\Delta y$, $\Delta y_{1} - \Delta y$ is written~$\Delta\Delta y$, or~$\Delta^{2} y$. And +the student must recollect, that in like manner as $\Delta$~is +not the symbol of a number, but of an operation, +so $\Delta^{2}$~does not denote a number multiplied by itself, +but an operation repeated upon its own result; just +as the logarithm of the logarithm of~$x$ might be written +$\log^{2} x$; $(\log x)^{2}$~being reserved to signify the square +of the logarithm of~$x$. We do not enlarge on this notation, +as the subject is discussed in most treatises on +\PageSep{91} +algebra.\footnote + {The reference of the original text is to ``the treatise on \Title{Algebraical Expressions},'' + Number~105 of the Library of Useful Knowledge,---the same series + in which the present work appeared. The first six pages of this treatise are + particularly recommended by De~Morgan in relation to the present point.---\Ed.} +Similarly the terms of the fourth column, +or the differences of the second differences, have the +prefix~$\Delta\Delta\Delta$ abbreviated into~$\Delta^{3}$, so that $\Delta^{2} y_{1} - \Delta^{2} y = \Delta^{3} y$, etc. + +When we have occasion to examine the results +which arise from supposing $\Delta x$~to diminish without +limit, we use~$dx$ instead of~$\Delta x$, $dy$~instead of~$\Delta y$, $d^{2} y$~instead +of~$\Delta^{2} y$, and so on. If we suppose this case, we +can show that the ratio which the term in any column +bears to its corresponding term in any preceding column, +diminishes without limit. Take for example, +$d^{2} y$~and~$dy$. The latter is $\phi(x + dx) - \phi x$, which, as +we have often noticed already, is of the form $p\, dx + q\, (dx)^{2} + \etc.$, +in which $p$,~$q$,~etc., are also functions +of~$x$. To obtain~$d^{2} y$, we must, in this series, change~$x$ +into $x + dx$, and subtract $p\, dx + q\, (dx)^{2} + \etc.$\ from +the result. But since $p$,~$q$,~etc., are functions of~$x$, +this change gives them the form +\[ +p + p'\, dx + \etc.,\quad +q + q'\, dx + \etc.; +\] +so that $d^{2} y$~is +\begin{gather*} +(p + p'\, dx + \etc.)\, dx + (q + q'\, dx + \etc.)(dx)^{2} + \etc. \\ +{} - (p\, dx + q\, (dx)^{2} + \etc.) +\end{gather*} +in which the first power of~$dx$ is destroyed. Hence +(\PageRefs{42}{44}), the ratio of $d^{2} y$ to~$dx$ diminishes without +limit, while that of $d^{2} y$ to~$(dx)^{2}$ has a finite limit, +except in those particular cases in which the second +power of~$dx$ is destroyed, in the previous subtraction, +as well as the first. In the same way it may be shown +that the ratio of $d^{3} y$ to $dx$~and $(dx)^{2}$ decreases without +limit, while that of $d^{3} y$ to~$(dx)^{3}$ remains finite; and so +\PageSep{92} +on. Hence we have a succession of ratios $\dfrac{dy}{dx}$, $\dfrac{d^{2} y}{dx^{2}}$, $\dfrac{d^{3} y}{dx^{3}}$, +etc., which tend towards finite limits when $dx$~is diminished. + +{\Loosen We now proceed to show that in the development +of $\phi(x + h)$, which has been shown to be of the form} +\[ +\phi x + \phi' x\, h + + \phi'' x\, \frac{h^{2}}{2} + + \phi''' x\, \frac{h^{3}}{2·3} + \etc., +\] +in the same manner as $\phi' x$~is the limit of~$\dfrac{dy}{dx}$ (\PageRef{23}), +so $\phi'' x$~is the limit of~$\dfrac{d^{2} y}{dx^{2}}$, $\phi''' x$~is that of~$\dfrac{d^{3} y}{dx^{3}}$, and so +forth. + +From the manner in which the preceding table +was formed, the following relations are seen immediately: +\begin{gather*} +\begin{alignedat}{4} +y_{1} &= y &&+ \Delta y & + \Delta y_{1} &= \Delta y &&+ \Delta^{2} y\Add{,} \\ +y_{2} &= y_{1} &&+ \Delta y_{1}\qquad & + \Delta y_{2} &= \Delta y_{1} &&+ \Delta^{2} y_{1}\Add{,} +\end{alignedat} \\ +\begin{alignedat}{3} +\Delta^{2} y_{1} &= \Delta^{2} y &&+ \Delta^{3} y && \etc.\Add{,} \\ +\Delta^{2} y_{2} &= \Delta^{2} y_{1} &&+ \Delta^{3} y_{1}\ && \etc. +\end{alignedat} +\end{gather*} +Hence $y_{1}$,~$y_{2}$,~etc., can be expressed in terms of $y$,~$\Delta y$, +$\Delta^{2} y$,~etc. For $y_{1} = y + \Delta y$; +%[** TN: Next two displays in-line in the original] +\begin{align*} +y_{2} = y_{1} + \Delta y_{1} + &= (y + \Delta y) + (\Delta y + \Delta^{2} y) + = y + 2\Delta y + \Delta^{2} y. \displaybreak[0] +\intertext{In the same way $\Delta y_{2} = \Delta y + 2\Delta^{2} y + \Delta^{3} y$; +hence} +y_{3} = y_{2} + \Delta y_{2} + &= (y + 2\Delta y + \Delta^{2} y) + (\Delta y + 2\Delta^{2} y + \Delta^{3} y) \\ + &= y + 3\Delta y + 3\Delta^{2} y + \Delta^{3} y. +\end{align*} + +Proceeding in this way we have +\begin{alignat*}{5} +y_{1} = y &{}+{}& \Delta y\Add{\rlap{,}} \\ +y_{2} = y &{}+{}&2\Delta y &{}+{}& \Delta^{2} y\Add{\rlap{,}} \\ +y_{3} = y &{}+{}&3\Delta y &{}+{}& 3\Delta^{2} y &{}+{}& \Delta^{3} y\Add{\rlap{,}} \\ +y_{4} = y &{}+{}&4\Delta y &{}+{}& 6\Delta^{2} y &{}+{}& 4\Delta^{3} y &{}+{}& \Delta^{5} y\Add{\rlap{,}} \\ +y_{5} = y &{}+{}&5\Delta y &{}+{}&10\Delta^{2} y &{}+{}&10\Delta^{3} y &{}+{}&5\Delta^{5} y + &+ \Delta^{6} y,\ \etc.\Add{,} +\end{alignat*} +from the whole of which it appears that $y_{n}$ or $\phi(x + n\Delta x)$ +is a series consisting of $y$,~$\Delta y$,~etc., up to~$\Delta^{n} y$, +severally multiplied by the coefficients which occur in +the expansion $(1 + a)^{n}$, or +\PageSep{93} +\begin{align*}%[** TN: Re-formatted] +y_{n} &= \phi(x + n\Delta x) \\ + &= y + n\Delta y + n\frac{n - 1}{2}\, \Delta^{2} y + + n\frac{n - 1}{2}\, \frac{n - 2}{3}\, \Delta^{3} y + \etc. +\end{align*} + +Let us now suppose that $x$~becomes $x + h$ by $n$~equal +steps; that is, $x$,~$x + \dfrac{h}{n}$, $x + \dfrac{2h}{n}$, etc.~\dots\ $x + \dfrac{nh}{h}$ +or~$x + h$, are the successive values of~$x$, so that +$n\Delta x = h$. Since the product of a number of factors is +not altered by multiplying one of them, provided we +divide another of them by the same quantity, multiply +every factor which contains~$n$ by~$\Delta x$, and divide the +accompanying difference of~$y$ by $\Delta x$ as often as there +are factors which contain~$n$, substituting~$h$ for~$n\Delta x$, +which gives +\begin{align*} +\phi(x + n\Delta x) + &= y + n\Delta x\, \frac{\Delta y}{\Delta x} + + n\Delta x\, \frac{n\Delta x - \Delta x}{2}\, \frac{\Delta^{2} y}{(\Delta x)^{2}} \\ + &\quad+ n\Delta x\, \frac{n\Delta x - \Delta x}{2}\, + \frac{n\Delta x - 2\Delta x}{3}\, \frac{\Delta^{3} y}{(\Delta x)^{3}} + \etc.\Add{,} +\end{align*} +or +\begin{align*} +\iffalse %[** TN: Commented code matches the original] +\phi(x + h) + &= \PadTo{y + n\Delta x}{y + h}\, \frac{\Delta y}{\Delta x} + + \PadTo{n\Delta x}{h}\, + \PadTo{\dfrac{n\Delta x - \Delta x}{2}}{\dfrac{h - \Delta x}{2}}\, + \frac{\Delta^{2} y}{(\Delta x)^{2}} \\ + &+ \PadTo{n\Delta x}{h}\, + \PadTo{\dfrac{n\Delta x - \Delta x}{2}}{\dfrac{h - \Delta x}{2}}\, + \PadTo{\dfrac{n\Delta x - 2\Delta x}{3}}{\dfrac{h - 2\Delta x}{3}}\, + \frac{\Delta^{3} y}{(\Delta x)^{3}} + \etc. +\fi +\phi(x + h) + = y &+ h\, \frac{\Delta y}{\Delta x} + + h\, \frac{h - \Delta x}{2}\, \frac{\Delta^{2} y}{(\Delta x)^{2}} \\ + &+ h\, \frac{h - \Delta x}{2}\, \frac{h - 2\Delta x}{3}\, + \frac{\Delta^{3} y}{(\Delta x)^{3}} + \etc. +\end{align*} +If $h$ remain the same, the more steps we make between +$x$~and~$x + h$, the smaller will each of those +steps be, and the number of steps may be increased, +until each of them is as small as we please. We can +therefore suppose $\Delta x$ to decrease without limit, without +affecting the truth of the series just deduced. +Write $dx$ for~$\Delta x$, etc., and recollect that $h - dx$, +$h - 2\, dx$, etc., continually approximate to~$h$. The series +then becomes +\[ +\phi(x + h) = y + \frac{dy}{dx}\, h + + \frac{d^{2} y}{dx^{2}}\, \frac{h^{2}}{2} + + \frac{d^{3} y}{dx^{3}}\, \frac{h^{3}}{2·3} + \etc.\Add{,} +\] +\PageSep{94} +in which, according to the view taken of the symbols +$\dfrac{dy}{dx}$~etc.\ in \PageRefs{26}{27}, $\dfrac{dy}{dx}$~stands for the \emph{limit} of the +ratio of the increments, $\dfrac{dy}{dx}$~is $\phi' x$, $\dfrac{d^{2} y}{dx^{2}}$~is $\phi'' x$,~etc. +According to the method proposed in \PageRefs{28}{29}, +the series written above is the first term of the development +of~$\phi(x + h)$, the remaining terms (which we +might include under an additional~$+$ etc.)\ being such +as to diminish without limit in comparison with the +first, when $dx$~is diminished without limit. And we +may show that the limit of~$\dfrac{d^{2} y}{dx^{2}}$ is the differential coefficient +of the limit of~$\dfrac{dy}{dx}$; or if by these fractions +themselves are understood their limits, that $\dfrac{d^{2} y}{dx^{2}}$ is the +differential coefficient of~$\dfrac{dy}{dx}$: for since $dy$, or $\phi(x + dx) - \phi x$, +becomes $dy + d^{2} y$, when $x$~is changed into +$x + dx$; and since $dx$~does not change in this process, +$\dfrac{dy}{dx}$ will become $\dfrac{dy}{dx} + \dfrac{d^{2} y}{dx}$, or its increment is~$\dfrac{d^{2} y}{dx}$. The +ratio of this to~$dx$ is~$\dfrac{d^{2} y}{(dx)^{2}}$, the limit of which, in the +definition of \PageRef{22}, is the differential coefficient of~$\dfrac{dy}{dx}$. +Similarly the limit of~$\dfrac{d^{3} y}{dx^{3}}$ is the differential coefficient +of the limit of~$\dfrac{d^{2} y}{dx^{2}}$; and so on. + + +\Subsection{Total and Partial Differential Coefficients. +Implicit Differentiation.} + +We now proceed to apply the principles laid down, +\index{Differentiation!implicit|EtSeq}% +\index{Implicit!differentiation|EtSeq}% +to some cases in which the variable enters into its +function in a less direct and more complicated manner. +\PageSep{95} + +For example, let $z$ be a given function of $x$~and~$y$, +and let $y$~be another given function of~$x$; so that $z$ +contains $x$ both directly and indirectly; the latter as +it contains~$y$, which is a function of~$x$. This will be +the case if $z = x\log y$, where $y = \sin x$. If we were to +substitute for~$y$ its value in terms of~$x$, the value of~$z$ +would then be a function of $x$~only; in the instance +just given it would be $x\log\sin x$. But if it be not convenient +to combine the two equations at the beginning +of the process, let us first consider $z$ as a function of +$x$~and~$y$, in which the two variables are independent. +In this case, if $x$~and~$y$ respectively receive the increments +$dx$~and~$dy$, the whole increment of~$z$, or~$d.z$, (or +at least that part which gives the limit of the ratios) +is represented by +\[ +\frac{dz}{dx}\, dx + \frac{dz}{dy}\, dy. +\] +If $y$ be now considered as a function of~$x$, the consequence +is that $dy$, instead of being independent of~$dx$, +is a series of the form $p\, dx + q\, (dx)^{2} + \etc.$, in which $p$~is +the differential coefficient of~$y$ with respect to~$x$. +Hence +\[ +d.z = \frac{dz}{dx}\, dx + \frac{dz}{dy}\, p\, dx \quad\text{or}\quad +\frac{d.z}{dx} = \frac{dz}{dx} + \frac{dz}{dx}\, p, +\] +in which the difference between $\dfrac{d.z}{dx}$ and $\dfrac{dz}{dx}$ is this, +that in the second, $x$~is only considered as varying +where it is directly contained in~$z$, or $z$~is considered +in the form in which it first appeared, as a function of +$x$~and~$y$, where $y$~is independent of~$x$; in the first, or +$\dfrac{d.z}{dx}$, the \emph{total variation} of~$z$ is denoted, that is, $y$~is +\index{Total!variations}% +\index{Variations, total}% +now considered as a function of~$x$, by which means if +$x$ become $x + dx$, $z$~will receive a different increment +\PageSep{96} +from that which it would have received, had $y$ been +independent of~$x$. {\Loosen In the instance above cited, where +$z = x\log y$ and $y = \sin x$, if the first equation be taken, +and $x$ becomes $x + dx$, $y$~remaining the same, $z$~becomes +$x\log y + \log y\, dx$ or $\dfrac{dz}{dx}$ is~$\log y$.} If $y$~only varies, +since (\PageRef{20}) $z$~will then become +\[ +x\log y + x\, \frac{dy}{y} - \etc., +\] +$\dfrac{dz}{dy}$ is~$\dfrac{x}{y}$ And $\dfrac{dy}{dx}$~is $\cos x$ when $y = \sin x$ (\PageRef{20}). +Hence $\dfrac{dz}{dx} + \dfrac{dz}{dy}\, p$, or $\dfrac{dz}{dx} + \dfrac{dz}{dy}\, \dfrac{dy}{dx}$ is $\log y + \dfrac{x}{y} \cos x$, +or $\log\sin x + \dfrac{x}{\sin x} \cos x$. This is~$\dfrac{d.z}{dx}$, which might +have been obtained by a more complicated process, if +$\sin x$ had been substituted for~$y$, before the operation +commenced. It is called the \emph{complete} or \emph{total} differential +\index{Coefficients, differential}% +\index{Complete Differential Coefficients}% +coefficient with respect to~$x$, the word \emph{total} indicating +that every way in which $z$ contains~$x$ has been +used; in opposition to~$\dfrac{dz}{dx}$, which is called the \emph{partial} +\index{Partial!differential coefficients}% +differential coefficient, $x$~having been considered as +varying only where it is directly contained in~$z$. + +Generally, the complete differential coefficient of~$z$ +with respect to~$x$, will contain as many terms as there +are different ways in which $z$ contains~$x$. From looking +at a complete differential coefficient, we may see +in what manner the function contained its variable. +Take, for example, the following, +\[ +\frac{d.z}{dx} + = \frac{dz}{dx} + \frac{dz}{dy}\, \frac{dy}{dx} + + \frac{dz}{da}\, \frac{da}{dy}\, \frac{dy}{dx} + + \frac{dz}{da}\, \frac{da}{dx}. +\] + +Before proceeding to demonstrate this formula, we +will collect from itself the hypothesis from which it +\PageSep{97} +\index{Functions!direct and indirect}% +must have arisen. When $x$~is contained in~$z$, we shall +say that $z$~is a \emph{direct}\footnote + {It may be right to warn the student that this phraseology is new, to the + best of our knowledge. The nomenclature of the Differential Calculus has + by no means kept pace with its wants; indeed the same may be said of algebra + generally. [Written in~1832.---\Ed.]} +function of~$x$. When $x$~is contained +in~$y$, and $y$~is contained in~$z$, we shall say that +$z$~is an indirect function of~$x$ \emph{through}~$y$. It is evident +\index{Indirect function}% +that an indirect function may be reduced to one which +is direct, by substituting for the quantities which contain~$x$, +their values in terms of~$x$. + +The first side of the equation~$\dfrac{d.z}{dx}$ is shown by the +point to be a complete differential coefficient, and indicates +that $z$~is a function of~$x$ in several ways; either +directly, and indirectly through one quantity at least, +or indirectly through several. If $z$~be a direct function +\index{Direct function}% +only, or indirectly through one quantity only, the +symbol~$\dfrac{dz}{dx}$, without the point, would represent its +total differential coefficient with respect to~$x$. + +On the second side of the equation we see: + +(1) $\dfrac{dz}{dx}$: which shows that $z$~is a direct function of~$x$, +and is that part of the differential coefficient which +we should get by changing $x$ into $x + dx$ throughout~$z$, +not supposing any other quantity which enters into~$z$ +to contain~$x$. + +(2) $\dfrac{dz}{dy}\, \dfrac{dy}{dx}$: which shows that $z$~is an indirect function +of~$x$ through~$y$. If $x$~and~$y$ had been supposed to +vary independently of each other, the increment of~$z$, +(or those terms which give the limiting ratio of this +increment to any other,) would have been $\dfrac{dz}{dx}\, dx + \dfrac{dz}{dy}\, dy$, +in which, if $dy$~had arisen from~$y$ being a function +\PageSep{98} +of~$x$, $dy$~would have been a series of the form +$p\, dx + q\, (dx)^{2} + \etc.$, of which only the differential coefficient~$p$ +would have appeared in the limit. Hence +$\dfrac{dz}{dy}\, dy$ would have given~$\dfrac{dz}{dy}\, p$, or~$\dfrac{dz}{dy}\, \dfrac{dy}{dx}$. + +(3) $\dfrac{dz}{da}\, \dfrac{da}{dy}\, \dfrac{dy}{dx}$: this arises from $z$ containing~$a$, which +contains~$y$, which contains~$x$. If $z$~had been differentiated +with respect to $a$~only, the increment would +have been represented by~$\dfrac{dz}{da}\, da$; if $da$~had arisen from +an increment of~$y$, this would have been expressed by +$\dfrac{dz}{da}\, \dfrac{da}{dy}\, dy$; if~$y$ had arisen from an increment given to~$x$, +this would have been expressed by $\dfrac{dz}{da}\, \dfrac{da}{dy}\, \dfrac{dy}{dx}\, dx$, +which, after $dx$~has been struck out, is the part of the +differential coefficient answering to that increment. + +(4) $\dfrac{dz}{da}\, \dfrac{da}{dx}$: arising from $a$~containing $x$~directly, +and $z$~therefore containing $x$ indirectly through~$a$. + +Hence $z$~is directly a function of $x$,~$y$, and~$a$, of +which $y$~is a function of~$x$, and $a$~of $y$~and~$x$. + +If we suppose $x$,~$y$ and~$a$ to vary independently, +we have +\[ +d.z = \frac{dz}{dx}\, dx + \frac{dz}{dy}\, dy + \frac{dz}{da}\, da + \etc. +\quad\text{(\PageRefs{28}{29})}. +\] +But as $a$~varies as a function of $y$~and~$x$, +\[ +da = \frac{da}{dx}\, dx + \frac{da}{dy}\, dy. +\] +If we substitute this instead of~$da$, and divide by~$dx$, +taking the limit of the ratios, we have the result first +given. + +For example, let (1) $z = x^{2} ya^{3}$, (2) $y = x^{2}$, and (3) $a = x^{3} y$. +Taking the first equation only, and substituting +\PageSep{99} +$x + dx$ for~$x$ etc., we find $\dfrac{dz}{dx} = 2xya^{3}$, $\dfrac{dz}{dy} = x^{2} a^{3}$, +and $\dfrac{dz}{da} = 3x^{2} ya^{2}$. From the second $\dfrac{dy}{dx} = 2x$, and from +the third $\dfrac{da}{dx} = 3x^{2} y$, and $\dfrac{da}{dy} = x^{3}$. Substituting these +in the value of~$\dfrac{d.z}{dx}$, we find +\begin{alignat*}{3} +%[** TN: Reformatted first line] +\frac{d.z}{dx} \text{ \ or \ } + \frac{dz}{dx} &+ \frac{dz}{dy}\, \frac{dy}{dx} + &&+ \frac{dz}{da}\, \frac{da}{dy}\, \frac{dy}{dx} + &&+ \frac{dz}{da}\, \frac{da}{dx} \\ + = 2xya^{3} &+ x^{2} a^{3} × 2x &&+ 3x^{2} ya^{2} × x^{3} × 2x &&+ 3x^{2} ya^{2} × 3x^{2} y \\ + = 2xya^{3} &+ 2x^{3} a^{3} &&+ 6x^{6} ya^{2} &&+ 9x^{4} y^{2} a^{2}\Add{.} +\end{alignat*} +If for $y$~and~$a$ in the first equation we substitute their +values $x^{2}$ and~$x^{3} y$, or~$x^{5}$, we have $z = x^{19}$, the differential +coefficient of which\Add{ is}~$19x^{18}$. This is the same as +arises from the formula just obtained, after $x^{2}$~and~$x^{5}$ +have been substituted for $y$~and~$a$; for this formula +then becomes +\[ +2x^{18} + 2x^{18} + 6x^{18} + 9x^{18} \quad\text{or}\quad 19x^{18}. +\] + +In saying that $z$~is a function of $x$~and~$y$, and that +$y$~is a function of~$x$, we have first supposed~$x$ to vary, +$y$~remaining the same. The student must not imagine +that $y$~is \emph{then} a function of~$x$; for if so, it would vary +when $x$~varied. There are two parts of the total differential +coefficient, arising from the direct and indirect +manner in which $z$ contains~$x$. That these two +parts may be obtained separately, and that their sum +constitutes the complete differential coefficient, is the +theorem we have proved. The first part~$\dfrac{dz}{dx}$ is what +\emph{would} have been obtained if $y$~had \emph{not} been a function +of~$x$; and on this supposition we therefore proceed to +find it. The other part $\dfrac{dz}{dy}\, \dfrac{dy}{dx}$ is the product (1)~of~$\dfrac{dz}{dy}$, +which would have resulted from a variation of $y$~only, +not considered as a function of~$x$; and (2)~of~$\dfrac{dy}{dx}$, +\PageSep{100} +the coefficient which arises from considering~$y$ as a +function of~$x$. These partial suppositions, however +useful in obtaining the total differential coefficient, +\index{Coefficients, differential}% +\index{Total!differential coefficient}% +cannot be separately admitted or used, except for this +purpose; since if $y$~be a function of~$x$, $x$~and~$y$ must +vary together. + +If $z$~be a function of~$x$ in various ways, the theorem +obtained may be stated as follows: + +Find the differential coefficient belonging to each +of the ways in which $z$ will contain~$x$, as if it were the +only way; the sum of these results (with their proper +signs) will be the total differential coefficient. + +Thus, if $z$~only contains $x$ indirectly through~$y$, +$\dfrac{dz}{dx}$~is $\dfrac{dz}{dy}\, \dfrac{dy}{dx}$. If $z$ contains~$a$, which contains~$b$, which +contains~$x$, $\dfrac{dz}{dx} = \dfrac{dz}{da}\, \dfrac{da}{db}\, \dfrac{db}{dx}$. + +This theorem is useful in the differentiation of complicated +\index{Differentiation!of complicated functions|EtSeq}% +functions; for example, let $z = \log(x^{2} + a^{2})$. +If we make $y = x^{2} + a^{2}$, we have $z = \log y$, and $\dfrac{dz}{dy} = \dfrac{1}{y}$; +while from the first equation $\dfrac{dy}{dx} = 2x$. Hence $\dfrac{dy}{dx}$ or +$\dfrac{dz}{dy}\, \dfrac{dy}{dx}$ is $\dfrac{2x}{y}$ or $\dfrac{2x}{x^{2} + a^{2}}$. + +%[** TN: \log\log\sin x is never real valued when x is real] +If $z = \log\log\sin x$, or the logarithm of the logarithm +of~$\sin x$, let $\sin x = y$ and $\log y = a$; whence +$z= \log a$, and contains~$x$, because $a$ contains~$y$, which +contains~$x$. Hence +\[ +\frac{dz}{dx} = \frac{dz}{da}\, \frac{da}{dy}\, \frac{dy}{dx}; +\] +but since $z = \log a$, +\[ +\frac{dz}{da} = \frac{1}{a}; +\] +\PageSep{101} +since $a = \log y$, +\[ +\frac{da}{dy} = \frac{1}{y}; +\] +and since $y = \sin x$, +\[ +\frac{dy}{dx} = \cos x. +\] +Hence +\[ +\frac{dz}{dx} = \frac{dz}{da}\, \frac{da}{dy}\, \frac{dy}{dx} + = \frac{1}{a}\, \frac{1}{y} \cos x + = \frac{\cos x}{\log\sin x \sin x}. +\] + +We now put some rules in the form of applications +of this theorem, though they may be deduced more +simply. + + +\Subsection[Applications of the Theorem for Implicit Differentiation.] +{Applications of the Preceding Theorem.} + +(1) Let $z = ab$, where $a$~and~$b$ are functions of~$x$. +The general formula, since $z$ contains~$x$ indirectly +through $a$~and~$b$, is (in this case as well as in those +which follow,) +\[ +\frac{dz}{dx} + = \frac{dz}{da}\, \frac{da}{dx} + \frac{dz}{db}\, \frac{db}{dx}. +\] + +We must leave $\dfrac{da}{dx}$ and $\dfrac{db}{dx}$ as we find them, until we +know \emph{what} functions $a$~and~$b$ are of~$x$; but as we +know what function $z$~is of $a$~and~$b$, we substitute for +$\dfrac{dz}{da}$ and~$\dfrac{dz}{db}$. Since $z = ab$, if $a$~becomes $a + da$, $z$~becomes +$ab + b\, da$, whence $\dfrac{dz}{db} = b$. In this case, and part +of the following, the limiting ratio of the increments +is the same as that of the increments themselves. +Similarly $\dfrac{dz}{db} = a$, whence from $z = ab$ follows +\[ +\frac{dz}{dx} = b\, \frac{da}{dx} + a\, \frac{db}{dx}. +\] +\PageSep{102} + +%[** TN: [sic] "become", twice] +(2) Let $z = \dfrac{a}{b}$. If $a$~become $a + da$, $z$~becomes +$\dfrac{a + da}{b}$ or $\dfrac{a}{b} + \dfrac{da}{b}$, and $\dfrac{dz}{da}$ is~$\dfrac{1}{b}$. If $b$~become $b + db$, $z$~becomes +$\dfrac{a}{b + db}$, or $\dfrac{a}{b} - \dfrac{a\, db}{b^{2}} + \etc.$, whence $\dfrac{dz}{db}$ is~$-\dfrac{a}{b^{2}}$\Add{.} +Hence from $z = \dfrac{a}{b}$ follows +\[ +\frac{dz}{dx} = \frac{1}{b}\, \frac{da}{dx} - \frac{a}{b^{2}}\, \frac{db}{dx} + = \frac{b\, \dfrac{da}{dx} - a\, \dfrac{db}{dx}}{b^{2}}. +\] + +(3) Let $z = a^{b}$. Here $(a + da)^{b} = a^{b} + ba^{b-1}\, da + \etc.$\ +(\PageRef{21}), whence $\dfrac{dz}{da} = ba^{b-1}$. Again, +$a^{b+db} = a^{b}\, a^{db} = a^{b}(1 + \log a\, db + \etc.)$ whence $\dfrac{dz}{db} = a^{b} \log a$. +Therefore from $z = a^{b}$ follows +\[ +\frac{dz}{dx} = ba^{b-1}\, \frac{da}{dx} + a^{b} \log a\, \frac{db}{dx}. +\] + + +\Subsection{Inverse Functions.} + +If $y$~be a function of~$x$, such as $y = \phi x$, we may, +\index{Functions!inverse|EtSeq}% +\index{Inverse functions|EtSeq}% +by solution of the equation, determine $x$ in terms of~$y$, +or produce another equation of the form $x = \psi y$. +For example, when $y = x^{2}$, $x = y^{\efrac{1}{2}}$. It is not necessary +that we should be able to solve the equation +$y = \phi x$ in finite terms, that is, so as to give a value +of~$x$ without infinite series; it is sufficient that $x$~can +be so expressed that the value of~$x$ corresponding to +any value of~$y$ may be found as near as we please +from $x = \psi y$, in the same manner as the value of~$y$ +corresponding to any value of~$x$ is found from $y = \phi x$. + +The equations $y = \phi x$, and $x = \psi y$, are connected, +being, in fact, the same relation in different forms; +and if the value of~$y$ from the first be substituted in +\PageSep{103} +the second, the second becomes $x = \psi(\phi x)$, or as it is +more commonly written, $\psi\phi x$. That is, the effect of +the operation or set of operations denoted by~$\psi$ is destroyed +by the effect of those denoted by~$\phi$; as in the +instances $(x^{2})^{\efrac{1}{2}}$, $(x^{3})^{\efrac{1}{3}}$, $e^{\log x}$, angle whose sine is~$(\sin x)$, +etc., each of which is equal to~$x$. + +By differentiating the first equation $y = \phi x$, we obtain +$\dfrac{dy}{dx} = \phi' x$, and from the second $\dfrac{dx}{dy} = \psi' y$. But +whatever values of $x$~and~$y$ together satisfy the first +equation, satisfy the second also; hence, if when $x$~becomes +$x + dx$ in the first, $y$~becomes $y + dy$; the same +$y + dy$ substituted for~$y$ in the second, will give the +same $x + dx$. Hence $\dfrac{dx}{dy}$ as deduced from the second, +and $\dfrac{dy}{dx}$ as deduced from the first, are reciprocals for +every value of~$dx$. The limit of one is therefore the +reciprocal of the limit of the other; the student may +easily prove that if $a$~is always equal to~$\dfrac{1}{b}$, and if $a$~continually +approaches to the limit~$\alpha$, while $b$~at the +same time approaches the limit~$\beta$, $\alpha$~is equal to~$\dfrac{1}{\beta}$. +But $\dfrac{dx}{dy}$ or $\psi' y$, deduced from $x = \psi y$, is expressed in +terms of~$y$, while $\dfrac{dy}{dx}$ or $\phi' x$, deduced from $y = \phi x$ is +expressed in terms of~$x$. Therefore $\psi' y$ and $\phi' x$ are +reciprocals for all such values of $x$~and~$y$ as satisfy +either of the two first equations. + +For example let $y = e^{x}$, from which $x = \log y$. From +the first (\PageRef{20}) $\dfrac{dy}{dx} = e^{x}$; from the second $\dfrac{dx}{dy} = \dfrac{1}{y}$; +and it is evident that $e^{x}$~and~$\dfrac{1}{y}$ are reciprocals, whenever +$y = e^{x}$. + +If we differentiate the above equations twice, we get +\PageSep{104} +$\dfrac{d^{2} y}{dx^{2}} = \phi'' x$, and $\dfrac{d^{2} x}{dy^{2}} = \psi'' x$. There is no very obvious +analogy between $\dfrac{d^{2} y}{dx^{2}}$ and $\dfrac{d^{2} x}{dy^{2}}$; indeed no such appears +from the method in which these coefficients were first +formed. Turn to the table in \PageRef{90}, and substitute +$d$ for~$\Delta$ throughout, to indicate that the increments +may be taken as small as we please. We there substitute +in~$\phi x$ what we will call a set of \emph{equidistant} values +\index{Equidistant values}% +\index{Values!equidistant}% +of~$x$, or values in arithmetical progression, viz., +$x$,~$x + dx$, $x + 2\, dx$,~etc. The resulting values of~$y$, +or $y$,~$y_{1}$, etc., are not equidistant, except in one function +only, when $y = ax + b$, where $a$~and~$b$ are constant. +Therefore $dy$,~$dy_{1}$, etc., are not equal; whence +arises the next column of second differences, or $d^{2} y$, +$d^{2} y_{1}$, etc. The limiting ratio of $d^{2} y$ to~$(dx)^{2}$, expressed +by~$\dfrac{d^{2} y}{dx^{2}}$, is the second differential coefficient of~$y$ with +respect to~$x$. If from $y = \phi x$ we deduce $x = \psi y$, and +take a set of equidistant values of~$y$, viz., $y$,~$y + dy$, +$y + 2\, dy$, etc., to which the corresponding values of~$x$ +are $x$,~$x_{1}$, $x_{2}$,~etc., a similar table may be formed, +which will give $dx$,~$dx_{1}$, etc., $d^{2} x$,~$d^{2} x_{1}$, etc., and the +limit of the ratio of~$d^{2} x$ to~$(dy)^{2}$ or $\dfrac{d^{2} x}{dy^{2}}$ is the second +differential coefficient of~$x$ with respect to~$y$. These +are entirely different suppositions, $dx$~being given in +the first table, and $dy$~varying; while in the second $dy$~is +given and $dx$~varies. We may show how to deduce +one from the other as follows: + +When, as before, $y = \phi x$ and $x = \psi y$, we have +\[ +\frac{dy}{dx} = \phi' x = \frac{1}{\psi' y} = \frac{1}{p}, +\] +if $\psi' y$ be called~$p$. Calling this~$u$, and considering it +\PageSep{105} +as a function of~$x$ from containing~$p$, which contains~$y$, +which contains~$x$, we have +\[ +\frac{du}{dp}\, \frac{dp}{dy}\, \frac{dy}{dx} +\] +for its differential coefficient with respect to~$x$. But +since +\[ +u = \frac{1}{p}, +\] +therefore +\[ +\frac{du}{dp} = -\frac{1}{p^{2}}; +\] +since $p = \psi' y$, therefore +\[ +\frac{dp}{dy} = \psi'' y; +\] +and $\psi'' y$ is the differential coefficient of~$\psi' y$, and is +$\dfrac{d^{2} x}{dy^{2}}$. Also $\dfrac{1}{p^{2}}$~is +\[ +\frac{1}{(\psi' y)^{2}} \quad\text{or}\quad +(\phi' x)^{2} \quad\text{or}\quad +\left(\frac{dy}{dx}\right)^{2}. +\] +Hence the differential coefficient of $u$ or~$\dfrac{dy}{dx}$, with respect +to~$x$, which is~$\dfrac{d^{2} y}{dx^{2}}$, is also +\[ +-\left(\frac{dy}{dx}\right)^{2} \frac{d^{2} x}{dy^{2}}\, \frac{dy}{dx} +\quad\text{or}\quad +-\left(\frac{dy}{dx}\right)^{3} \frac{d^{2} x}{dy^{2}}. +\] + +{\Loosen If $y = e^{x}$, whence $x = \log y$, we have $\dfrac{dy}{dx} = e^{x}$ and +$\dfrac{d^{2} y}{dx^{2}} = e^{x}$. But $\dfrac{dx}{dy} = \dfrac{1}{y}$ and $\dfrac{d^{2} x}{dy^{2}} = -\dfrac{1}{y^{2}}$. Therefore} +\[ +-\left(\frac{dy}{dx}\right)^{3} \frac{d^{2} x}{dy^{2}} \quad\text{is}\quad +-e^{3x} \left(-\frac{1}{y^{2}}\right) \quad\text{or}\quad +\frac{e^{3x}}{y^{2}} \quad\text{or}\quad +\frac{e^{3x}}{e^{2x}}, +\] +which is~$e^{x}$, the value just found for~$\dfrac{d^{2} y}{dx^{2}}$. +\PageSep{106} + +In the same way $\dfrac{d^{3} y}{dx^{3}}$ might be expressed in terms +of $\dfrac{dy}{dx}$, $\dfrac{d^{2} y}{dx^{2}}$, and~$\dfrac{d^{3} x}{dy^{3}}$; and so on. + + +\Subsection{Implicit Functions.} + +The variable which appears in the denominator of +the differential coefficients is called the \emph{independent} +variable. In any function, one quantity at least is +changed at pleasure; and the changes of the rest, +with the limiting ratio of the changes, follow from the +form of the function. The number of independent +variables depends upon the number of quantities +\index{Variables!independent and dependent}% +which enter into the equations, and upon the number +of equations which connect them. If there be only +one equation, all the variables except one are independent, +or may be changed at pleasure, without ceasing +to satisfy the equation; for in such a case the +common rules of algebra tell us, that as long as one +quantity is left to be determined from the rest, it can +be determined by one equation; that is, the values of +all but one are at our pleasure, it being still in our +power to satisfy one equation, by giving a proper +value to the remaining one. Similarly, if there be +two equations, all variables except two are independent, +and so on. If there be two equations with two +unknown quantities only, there are no variables; for +by algebra, a finite number of values, and a finite +number only, can satisfy these equations; whereas it +is the nature of a variable to receive any value, or at +least any value which will not give impossible values +for other variables. If then there be $m$~equations containing +$n$~variables, ($n$~must be greater than~$m$), we +have $n - m$~independent variables, to each of which +\index{Independent variables}% +\PageSep{107} +we may give what values we please, and by the equations, +deduce the values of the rest. We have thus +various sets of differential coefficients, arising out of +the various choices which we may make of independent +variables. + +If, for example, $a$,~$b$, $x$,~$y$, and~$z$, being variables, +we have +\begin{align*} +\phi(a, b, x, y, z) &= 0, \\ +\psi(a, b, x, y, z) &= 0, \\ +\chi(a, b, x, y, z) &= 0, +\end{align*} +we have two independent variables, which may be +either $x$~and~$y$, $x$~and~$z$, $a$~and~$b$, or any other combination. +If we choose $x$~and~$y$, we should determine +$a$,~$b$, and~$z$ in terms of $x$~and~$y$ from the three equations; +in which case we can obtain +\[ +\frac{da}{dx},\quad \frac{da}{dy},\quad \frac{db}{dx},\quad \etc. +\] + +When $y$~is a function of~$x$, as in $y = \phi x$, it is called +\index{Explicit functions}% +\index{Functions!implicit and explicit}% +an \emph{explicit} function of~$x$. This equation tells us not +only that $y$~is a function of~$x$, but also what function +it is. The value of~$x$ being given, nothing more is +necessary to determine the corresponding value of~$y$, +than the substitution of the value of~$x$ in the several +terms of~$\phi x$. + +But it may happen that though $y$~is a function of~$x$, +\index{Implicit!function}% +the relation between them is contained in a form +from which $y$~must be deduced by the solution of an +equation. For example, in $x^{2} - xy + y^{2} = a$, when $x$~is +known, $y$~must be determined by the solution of an +equation of the second degree. Here, though we know +that $y$~must be a function of~$x$, we do not know, without +further investigation, what function it is. In this +case $y$~is said to be \emph{implicitly} a function of~$x$, or an implicit +\PageSep{108} +function. By bringing all the terms on one side +of the equation, we may always reduce it to the form +$\phi(x, y) = 0$. Thus, in the case just cited, we have +$x^{2} - xy + y^{2} - a = 0$. + +{\Loosen We now want to deduce the differential coefficient +$\dfrac{dy}{dx}$ from an equation of the form $\phi(x, y) = 0$. If we +take the equation $u = \phi(x, y)$, in which when $x$~and~$y$ +become $x + dx$ and $y + dy$, $u$~becomes $u + du$, we have, +by our former principles,} +\[ +du = \ux\, dx + \uy\, dy + \etc., \text{(\PageRef{82})}, +\] +in which $\ux$~and~$\uy$ can be directly obtained from the +equation, as in \PageRef{82}. Here $x$~and~$y$ are independent, +as also $dx$~and~$dy$; whatever values are given to +them, it is sufficient that $u$~and~$du$ satisfy the two last +equations. But if $x$~and~$y$ must be always so taken +that $u$ may~$= 0$, (which is implied in the equation +$\phi(x, y) = 0$,) we have $u = 0$, and $du = 0$; and this, +whatever may be the values of $dx$ and~$dy$. Hence $dx$ +and~$dy$ are connected by the equation +\[ +0 = \ux\, dx + \uy\, dy + \etc., +\] +and their limiting ratio must be obtained by the equation +\[ +\ux\, dx + \uy\, dy = 0, \quad\text{or}\quad \frac{dy}{dx} = -\frac{\ux}{\uy}; +\] +{\Loosen $y$~and~$x$ are no longer independent; for, one of them +being given, the other must be so taken that the equation +$\phi(x, y) = 0$ may be satisfied. The quantities $\ux$ +\index{Functions!implicit and explicit}% +\index{Implicit!function}% +and~$\uy$ we have denoted by $\dfrac{du}{dx}$ and~$\dfrac{du}{dy}$, so that} +\[ +\frac{dy}{dx} = -\frac{\;\dfrac{du}{dx}\;}{\dfrac{du}{dy}}\Add{.} +\Tag{(1)} +\] +\PageSep{109} + +We must again call attention to the different meanings +of the same symbol~$du$ in the numerator and denominator +of the last fraction. Had $du$, $dx$, and~$dy$ +been common algebraical quantities, the first meaning +the same thing throughout, the last equation would +not have been true until the negative sign had been +removed. We will give an instance in which $du$~shall +mean the same thing in both. + +Let $u = \Chg{\phi(x)}{\phi x}$, and let $u = \psi y$, in which two equations +is implied a third $\phi x = \psi y$; and $y$~is a function +of~$x$. Here, $x$~being given, $u$~is known from the first +equation; and $u$~being known, $y$~is known from the +second. Again, $x$~and~$dx$ being given, $du$, which is +$\phi(x + dx) - \phi x$ is known, and being substituted in +the result of the second equation, we have $du = \psi(y + dy) - \psi y$, +which $dy$~must be so taken as to +satisfy. From the first equation we deduce $du = \phi'x\, dx + \etc.$\ +and from the second $du = \psi' y\, dy + \etc.$, +whence +\[ +\phi' x\, dx + \etc. = \psi' y\, dy + \etc.; +\] +the \emph{etc.}\ only containing terms which disappear in finding +the limiting ratios. Hence, +\[ +\frac{dy}{dx} = \frac{\phi' x}{\psi' y} + = \frac{\;\dfrac{du}{dx}\;}{\dfrac{du}{dy}}\Add{,} +\Tag{(2)} +\] +a result in accordance with common algebra. + +But the equation~\Eq{(1)} was obtained from $u = \phi(x, y)$, +on the supposition that $x$~and~$y$ were always so taken +that $u$ should~$= 0$, while \Eq{(2)}~was obtained from $u = \Chg{\phi(x)}{\phi x}$ +and $u = Sy$, in which no new supposition can be +made; since one more equation between $u$,~$x$, and~$y$ +would give three equations connecting these three +quantities, in which case they would cease to be variable +(\PageRef{106}). +\PageSep{110} + +As an example of~\Eq{(1)} let $xy - x = 1$, or $xy - x - 1 = 0$. +From $u = xy - x - 1$ we deduce (\PageRef{81}) +$\dfrac{du}{dx} = y - 1$, $\dfrac{du}{dy} = x$; whence, by equation~\Eq{(1)}, +\[ +\frac{dy}{dx} = -\frac{y - 1}{x}. +\Tag{(3)} +\] +By solution of $xy - x = 1$, we find $y = 1 + \dfrac{1}{x}$, and +\[ +dy = \left(1 + \frac{1}{x + dx}\right) - \left(1 + \frac{1}{x}\right) + = -\frac{dx}{x^{2}} + \etc.\footnote{See \PageRef{26}.} +\] +Hence $\dfrac{dy}{dx}$ (meaning the limit) is~$-\dfrac{1}{x^{2}}$, which will also +be the result of~\Eq{(3)} if $1 + \dfrac{1}{x}$ be substituted for~$y$. + + +\Subsection{Fluxions, and the Idea of Time.} + +To follow this subject farther would lead us beyond +\index{Time, idea of|EtSeq}% +our limits; we will therefore proceed to some +observations on the differential coefficient, which, at +this stage of his progress, may be of use to the student, +who should never take it for granted that because +he has made some progress in a science, he understands +the first principles, which are often, if not +always, the last to be learned well. If the mind were +so constituted as to receive with facility any perfectly +new idea, as soon as the same was legitimately applied +in mathematical demonstration, it would doubtless +be an advantage not to have any notion upon a +mathematical subject, previous to the time when it is +to become a subject of consideration after a strictly +mathematical method. + +This not being the case, it is a cause of embarrassment +to the student, that he is introduced at once to a +definition so refined as that of the limiting ratio which +\PageSep{111} +the increment of a function bears to the increment of +its variable. Of this he has not had that previous experience, +which is the case in regard to the words +\emph{force}, \emph{velocity}, or \emph{length}. Nevertheless, he can easily +\index{Velocity!linear}% +conceive a mathematical quantity in a state of continuous +increase or decrease, such as the distance between +two points, one of which is in motion. The +number which represents this line (reference being +made to a given linear unit) is in a corresponding +state of increase or decrease, and so is every function +of this number, or every algebraical expression in the +formation of which it is required. And the nature of +the change which takes place in the function, that is, +whether the function will increase or decrease when +the variable increases; whether that increase or decrease +corresponding to a given change in the variable +will be smaller or greater, etc., depends on the +manner in which the variable enters as a component +part of its function. + +Here we want a new word, which has not been invented +for the world at large, since none but mathematicians +consider the subject; which word, if the +change considered were change of place, depending +upon change of time, would be \emph{velocity}. Newton +adopted this word, and the corresponding idea, expressing +many numbers in succession, instead of at +once, by supposing a point to generate a straight line +by its motion, which line would at different instants +contain any different numbers of linear units. + +To this it was objected that the idea of \emph{time} is introduced, +which is foreign to the subject. We may +answer that the notion of time is only necessary, inasmuch +as we are not able to consider more than one +thing at a time. Imagine the diameter of a circle divided +\PageSep{112} +into a million of equal parts, from each of which +a perpendicular is drawn meeting the circle. A mind +which could at a view take in every one of these lines, +and compare the differences between every two contiguous +perpendiculars with one another, could, by +subdividing the diameter still further, prove those +propositions which arise from supposing a point to +move uniformly along the diameter, carrying with it +a perpendicular which lengthens or shortens itself so +as always to have one extremity on the circle. But +we, who cannot consider all these perpendiculars at +once, are obliged to take one after another. If one +perpendicular only were considered, and the differential +coefficient of that perpendicular deduced, we might +certainly appear to avoid the idea of time; but if all +the states of a function are to be considered, corresponding +to the different states of its variable, we +have no alternative, with our bounded faculties, but +to consider them in succession; and succession, disguise +it as we may, is the identical idea of time introduced +in Newton's Method of Fluxions. +\index{Fluxions}% + + +\Subsection{The Differential Coefficient Considered with Respect +to its Magnitude.} + +The differential coefficient corresponding to a particular +\index{Coefficients, differential}% +\index{Contiguous values}% +\index{Differential coefficients!as the index of the change of a function}% +\index{Logarithms|EtSeq}% +\index{Values!contiguous}% +value of the variable, is, if we may use the +phrase, the \emph{index} of the change which the function +would receive if the value of the variable were increased. +Every value of the variable, gives not only +a different value to the function, but a different quantity +of increase or decrease in passing to what we may +call \emph{contiguous} values, obtained by a given increase of +the variable. + +If, for example, we take the common logarithm of~$x$, +\PageSep{113} +and let $x$ be~$100$, we have common $\log 100 = 2$. If +$x$~be increased by~$2$, this gives common $\log 102 = 2.0086002$, +the ratio of the increment of the function +\index{Increment}% +to that of the variable being that of $.0086002$ to~$2$, or +$.0043001$. In passing from $1000$ to~$1003$, we have the +logarithms $3$ and~$3.0013009$, the above-mentioned ratio +being~$.0004336$, little more than a tenth of the +former. We do not take the increments themselves, +but the proportion they bear to the changes in the +variable which gave rise to them; so in estimating +the rate of motion of two points, we either consider +lengths described in the same time, or if that cannot +be done, we judge, not by the lengths described in +different times, but by the proportion of those lengths +to the times, or the proportions of the units which +express them. + +The above rough process, though from it some +might draw the conclusion that the logarithm of~$x$ is +increasing faster when $x = 100$ than when $x = 1000$, +is defective; for, in passing from $100$ to~$102$, the +change of the logarithm is not a sufficient index of the +change which is taking place when $x$ is~$100$; since, +for any thing we can be supposed to know to the contrary, +the logarithm might be decreasing when $x = 100$, +and might afterwards begin to increase between +$x = 100$ and $x = 102$, so as, on the whole, to cause +the increase above mentioned. The same objection +would remain good, however small the increment +might be, which we suppose $x$ to have. If, for example, +we suppose $x$ to change from $x = 100$ to $x = 100.00001$, +which increases the logarithm from~$2$ to~$2.00000004343$, +we cannot yet say but that the logarithm +may be decreasing when $x = 100$, and may begin +to increase between $x = 100$ and $x = 100.00001$. +\PageSep{114} + +In the same way, if a point is moving, so that at +the end of $1$~second it is at $3$~feet from a fixed point, +and at the end of $2$~seconds it is at $5$~feet from the +fixed point, we cannot say which way it is moving at +the end of one second. \emph{On the whole}, it increases its +distance from the fixed point in the second second; +but it is possible that at the end of the first second it +may be moving back towards the fixed point, and may +turn the contrary way during the second second. And +the same argument holds, if we attempt to ascertain +the way in which the point is moving by supposing +any finite portion to elapse after the first second. But +if on adding any interval, \emph{however small}, to the first +second, the moving point does, during that interval, +increase its distance from the fixed point, we can then +certainly say that at the end of the first second the +point is moving from the fixed point. + +On the same principle, we cannot say whether the +logarithm of~$x$ is increasing or decreasing when $x$~increases +and becomes~$100$, unless we can be sure that +any increment, however small, added to~$x$, will increase +the logarithm. Neither does the ratio of the +increment of the function to the increment of its variable +furnish any distinct idea of the change which is +taking place when the variable has attained or is passing +through a given value. For example, when $x$~passes +from $100$ to~$102$, the difference between $\log 102$ +and $\log 100$ is the united effect of all the changes +which have taken place between $x = 100$ and $x = 100\frac{1}{10}$; +$x = 100\frac{1}{10}$ and $x = 100\frac{2}{10}$, and so on. Again, +the change which takes place between $x = 100$ and +$x = 100\frac{1}{10}$ may be further compounded of those which +take place between $x = 100$ and $x = 100\frac{1}{100}$; $x = 100\frac{1}{100}$ +and $x = 100\frac{2}{100}$, and so on. The objection +\PageSep{115} +becomes of less force as the increment diminishes, +but always exists unless we take the limit of the ratio +of the increments, instead of that ratio. + +How well this answers to our previously formed +ideas on such subjects as direction, velocity, and +force, has already appeared. + + +\Subsection{The Integral Calculus.} + +We now proceed to the Integral Calculus, which +\index{Integral Calculus|EtSeq}% +is the inverse of the Differential Calculus, as will afterwards +appear. + +We have already shown, that when two functions +\emph{increase} or \emph{decrease} without limit, their \emph{ratio} may either +increase or decrease without limit, or may tend to +some finite limit. Which of these will be the case depends +upon the manner in which the functions are related +to their variable and to one another. + +This same proposition may be put in another form, +as follows: If there be two functions, the first of which +\emph{decreases} without limit, on the same supposition which +makes the second \emph{increase} without limit, the \emph{product} +of the two may either remain finite, and never exceed +a certain finite limit; or it may increase without limit, +or diminish without limit. + +For example, take $\cos\theta$ and~$\tan\theta$. As the angle~$\theta$ +\emph{approaches} a right angle, $\cos\theta$~diminishes without +limit; it is nothing when $\theta$~\emph{is} a right angle; and any +fraction being named, $\theta$~can be taken so near to a +right angle that $\cos\theta$~shall be smaller. Again, as $\theta$~approaches +to a right angle, $\tan\theta$~increases without +limit; it is called \emph{infinite} when $\theta$~is a right angle, by +which we mean that, let any number be named, however +great, $\theta$~can be taken so near a right angle that +$\tan\theta$~shall be greater. Nevertheless the product $\cos\theta × \tan\theta$, +\PageSep{116} +of which the first factor diminishes without limit, +while the second increases without limit, is always +finite, and tends towards the limit~$1$; for $\cos\theta × \tan\theta$ +is always~$\sin\theta$, which last approaches to~$1$ as $\theta$~approaches +to a right angle, and is~$1$ when $\theta$~\emph{is} a right +angle. + +Generally, if $A$~diminishes without limit at the +same time as $B$~increases without limit, the product~$AB$ +may, and often will, tend towards a finite limit. +This product~$AB$ is the representative of~$A$ divided by~$\dfrac{1}{B}$ +or the ratio of $A$ to~$\dfrac{1}{B}$. If $B$~increases without +limit, $\dfrac{1}{B}$~decreases without limit; and as $A$~also decreases +without limit, the ratio of $A$ to~$\dfrac{1}{B}$ may have a +finite limit. But it may also diminish without limit; +as in the instance of $\cos^{2}\theta × \tan\theta$, when $\theta$~approaches +to a right angle. Here $\cos^{2}\theta$~diminishes without limit, +and $\tan\theta$~increases without limit; but $\cos^{2}\theta × \tan\theta$ +being $\cos\theta × \sin\theta$, or a diminishing magnitude multiplied +by one which remains finite, diminishes without +limit. Or it may increase without limit, as in the case +of $\cos\theta × \tan^{2}\theta$, which is also $\sin\theta × \tan\theta$; which last +has one factor finite, and the other increasing without +limit. We shall soon see an instance of this. + +If we take any numbers, such as $1$~and~$2$, it is evident +that between the two we may interpose any number +of fractions, however great, either in arithmetical +progression, or according to any other law. Suppose, +for example, we wish to interpose $9$~fractions in arithmetical +progression between $1$~and~$2$. These are $1\frac{1}{10}$, +$1\frac{2}{10}$,~etc., up to~$1\frac{9}{10}$; and, generally, if $m$~fractions in +arithmetical progression be interposed between $a$~and~$a + h$, +the complete series is +\PageSep{117} +\begin{multline*} +a,\quad a + \frac{h}{m + 1},\quad + a + \frac{2h}{m + 1},\quad \etc.\Add{,} \dots\\ +\text{up to } a + \frac{mh}{m + 1},\quad a + h\Add{.} +\Tag{(1)} +\end{multline*} +The sum of these can evidently be made as great as +we please, since no one is less than the given quantity~$a$, +and the number is as great as we please. Again, +if we take~$\phi x$, any function of~$x$, and let the values +just written be successively substituted for~$x$, we shall +have the series +\begin{multline*} +\phi a,\quad \phi\left(a + \frac{h}{m + 1}\right),\quad + \phi\left(a + \frac{2h}{m + 1}\right),\quad \etc.\Add{,} \dots\\ +\text{up to } \phi(a + h); +\Tag{(2)} +\end{multline*} +the sum of which may, in many cases, also be made +as great as we please by sufficiently increasing the +number of fractions interposed, that is, by sufficiently +increasing~$m$. But though the two sums increase without +limit when $m$~increases without limit, it does not +therefore follow that their ratio increases without +limit; indeed we can show that this cannot be the +case when all the separate terms of~\Eq{(2)} remain finite. + +For let $A$~be greater than any term in~\Eq{(2)}, whence, +as there are $(m + 2)$~terms, $(m + 2)A$~is greater than +their sum. Again, every term of~\Eq{(1)}, except the first, +being greater than~$a$, and the terms being $m + 2$~in +number, $(m + 2)a$~is less than the sum of the terms in~\Eq{(1)}. +Consequently, +\[ +\frac{(m + 2)A}{(m + 2)a} + \text{ is greater than the ratio } + \frac{\text{sum of terms in~\Eq{(2)}}}{\text{sum of terms in~\Eq{(1)}}}, +\] +since its numerator is greater than the last numerator, +and its denominator less than the last denominator. +But +\PageSep{118} +\[ +\frac{(m + 2)A}{(m + 2)a} = \frac{A}{a}, +\] +which is independent of~$m$, and is a finite quantity. +Hence the ratio of the sums of the terms is always +finite, whatever may be the number of terms, at least +unless the terms in~\Eq{(2)} increase without limit. + +As the number of interposed values increases, the +interval or difference between them diminishes; if, +therefore, we multiply this difference by the sum of +the values, or form +\begin{multline*} +\frac{h}{m + 1} \Biggl[ + \phi a + \phi\left(a + \frac{h}{m + 1}\right) + \\ + \phi\left(a + \frac{2h}{m + 1}\right) \Add{+} \dots + \phi(a + h) +\Biggr]\Add{,} +\end{multline*} +we have a product, one term of which diminishes, and +the other increases, when $m$~is increased. The product +\emph{may} therefore remain finite, or never pass a certain +limit, when $m$~is increased without limit, and we +shall show that this \emph{is} the case. + +As an example, let the given function of~$x$ be~$x^{2}$, +and let the intermediate values of~$x$ be interposed between +$x = a$ and $x = a + h$. Let $v = \dfrac{h}{m + 1}$, whence +the above-mentioned product is +\begin{multline*}%[** TN: Re-formatted from the original] +v\bigl\{a^{2} + (a + v)^{2} + (a + 2v)^{2} + \dots + + \bigl(a + (m + 1)v\bigr)^{2}\bigr\} \\ + = (m + 2)va^{2} + 2av^{2} \{1 + 2 + 3 + \dots + (m + 1)\} \\ + + v^{3} \{1^{2} + 2^{2} + 3^{2} + \dots + (m + 1)^{2}\}; +\end{multline*} +{\Loosen of which, $1 + 2 + \dots + (m + 1) = \frac{1}{2}(m + 1)(m + 2)$ +and (\PageRef{73}), $1^{2} + 2^{2} + \dots + (m + 1)^{2}$ approaches +without limit to a ratio of equality with $\frac{1}{3}(m + 1)^{3}$, +when $m$~is increased without limit. Hence this last +sum may be put under the form $\frac{1}{3}(m + 1)^{3} (1 + \alpha)$, +\PageSep{119} +where $\alpha$~diminishes without limit when $m$~is increased +without limit. Making these substitutions, and putting +for~$v$ its value $\dfrac{h}{m + 1}$, the above expression becomes} +\[ +\frac{m + 2}{m + 1}\, ha^{2} + \frac{m + 2}{m + 1}\, ha^{2} + + (1 + \alpha)\, \frac{h^{3}}{3}, +\] +in which $\dfrac{m + 2}{m + 1}$ has the limit~$1$ when $m$~increases without +limit, and $1 + \alpha$~has also the limit~$1$, since, in that +case, $\alpha$~diminishes without limit. Therefore the limit +of the last expression is +\[ +ha^{2} + ha^{2} + \frac{h^{3}}{3} \quad\text{or}\quad \frac{(a + h)^{3} - a^{3}}{3}. +\] + +{\Loosen This result may be stated as follows: If the variable~$x$, +setting out from a value~$a$, becomes successively +$a + dx$, $a + 2\,dx$, etc., until the total increment +is~$h$, the smaller $dx$ is taken, the more nearly will the +sum of all the values of~$x^{2}\, dx$, or $a^{2}\, dx + (a + dx)^{2}\, dx + (a + 2\, dx)^{2}\, dx + \etc.$, +be equal to} +\[ +\frac{(a + h)^{3} - a^{3}}{3}, +\] +and to this the aforesaid sum may be brought within +any given degree of nearness, by taking $dx$ sufficiently +small. + +This result is called the \emph{integral} of~$x^{2}\, dx$, between +\index{Integral Calculus!notation of}% +\index{Integrals!definition of|EtSeq}% +\index{Notation!of the Integral Calculus}% +the limits $a$~and~$a + h$, and is written $\int x^{2}\, dx$, when it +is not necessary to specify the limits, $\int_{a}^{a+h} x^{2}\, dx$, +or\footnote + {This notation $\int x^{2}\, dx\Ibar_{a}^{a+h}$ appears to me to avoid the objections which + may be raised against $\int_{a+h}^{a} x^{2}\, dx$ as contrary to analogy, which would require + that $\int^{2} x^{2}\, dx^{2}$ should stand for the second integral of~$x^{2}\, dx$. It will be found + convenient in such integrals as $\int z\, dx\Ibar_{b}^{a}\, dy\Ibar_{0}^{\phi x}$. There is as yet no general agreement + on this point of notation.---\textit{De~Morgan}, 1832.} +$\int x^{2}\, dx\Ibar_{a}^{a+h}$, or $\int x^{2}\, dx\Ibar_{x=a}^{x=a+h}$ in the contrary case. We +\PageSep{120} +now proceed to show the connexion of this process +with the principles of the Differential Calculus. + + +\Subsection{Connexion of the Integral with the Differential +Calculus.} + +Let $x$ have the successive values $a$, $a + dx$, $a + 2\, dx$, +etc.,~\dots\ up to $a + m\, dx$, or $a + h$, $h$~being a given +quantity, and $dx$ the $m$\th~part of~$h$, so that as $m$~is increased +without limit, $dx$~is diminished without limit. +Develop the successive values $\phi x$, or $\phi a$, $\phi(a + dx)$\Add{,}~\dots\ +(\PageRef{21}),\par +{\footnotesize\begin{alignat*}{6} +& \phi a &&= \phi a\Add{,} \\ +&\phi(a + dx) &&= \phi a &&+ \phi' a\, dx + &&+ \phi'' a\, \frac{(dx)^{2}}{2} + &&+ \phi''' a\, \frac{(dx)^{3}}{2·3} &&+ \etc.\Add{,} \displaybreak[0] \\ +&\phi(a + 2\, dx) &&= \phi a &&+ \phi' a\, 2\, dx + &&+ \phi'' a\, \frac{(2\, dx)^{2}}{2} + &&+ \phi''' a\, \frac{(2\, dx)^{3}}{2·3} &&+ \etc.\Add{,} \displaybreak[0] \\ +&\phi(a + 3\, dx) &&= \phi a &&+ \phi' a\, 3\, dx + &&+ \phi'' a\, \frac{(3\, dx)^{2}}{2} + &&+ \phi''' a\, \frac{(3\, dx)^{3}}{2·3} &&+ \etc.\Add{,} \\ +\DotRow{12} \\ +&\phi(a + m\, dx) &&= \phi a &&+ \phi' a\, m\, dx + &&+ \phi'' a\, \frac{(m\, dx)^{2}}{2} + &&+ \phi''' a\, \frac{(m\, dx)^{3}}{2·3} &&+ \etc. +\end{alignat*}}% +If we multiply each development by~$dx$ and add the +results, we have a series made up of the following +terms, arising from the different columns, +\begin{alignat*}{7} +&\phi a &&×{} && && && && && \phantom{()}m\, dx\Add{,} \\ +&\phi' a &&× (1 &&+ 2 &&+ 3 &&+ \dots &&+ m &&)\, (dx)^{2}\Add{,} \\ +&\phi'' a &&× (1^{2} &&+ 2^{2} &&+ 3^{2} &&+ \dots &&+ m^{2} &&)\, \frac{(dx)^{3}}{2}\Add{,} \\ +&\phi''' a &&× (1^{3} &&+ 2^{3} &&+ 3^{3} &&+ \dots &&+ m^{3} &&)\, \frac{(dx)^{4}}{2·3} +\quad \etc.\Add{,} +\end{alignat*} +and, as in the last example, we may represent (\PageRef{73}), +\begin{alignat*}{6} +&1 &&+ 2 &&+ 3 &&+ \dots &&+ m &&\text{by}\quad + \tfrac{1}{2}m^{2}(1 + \alpha)\Add{,} \displaybreak[0] \\ +&1^{2} &&+ 2^{2} &&+ 3^{2} &&+ \dots &&+ m^{2} &&\text{by}\quad + \tfrac{1}{3}m^{3}(1 + \beta)\Add{,} \displaybreak[0] \\ +&1^{3} &&+ 2^{3} &&+ 3^{3} &&+ \dots &&+ m^{3}\quad &&\text{by}\quad + \tfrac{1}{4}m^{4}(1 + \gamma) +\quad \etc.\Add{,} +\end{alignat*} +\PageSep{121} +where $\alpha$,~$\beta$,~$\gamma$, etc., diminish without limit, when $m$~is +increased without limit. If we substitute these values, +and also put $\dfrac{h}{m}$ instead of~$dx$, we have, for the +sum of the terms, +\begin{align*} +\phi a\, h + \phi' a\, \frac{h^{2}}{2} (1 + \alpha) + &+ \phi'' a\, \frac{h^{3}}{2·3} (1 + \beta) \\ + &+ \phi''' a\, \frac{h^{4}}{2·3·4} (1 + \gamma) + \etc. +\end{align*} +which, when $m$~is increased without limit, in consequence +of which $\alpha$,~$\beta$,~etc., diminish without limit, +continually approaches to +\[ +\phi a\, h + \phi' a\, \frac{h^{2}}{2} + + \phi'' a\, \frac{h^{3}}{2·3} + + \phi''' a\, \frac{h^{4}}{2·3·4} + \etc.\Add{,} +\] +which is the limit arising from supposing $x$ to increase +from~$a$ through $a + dx$, $a + 2\, dx$, etc., up to~$a + h$, +multiplying every value of~$\phi x$ so obtained by~$dx$, summing +the results, and decreasing~$dx$ without limit. + +This is the integral of $\phi x\, dx$ from $x = a$ to $x = a + h$. +\index{Integrals!relations between differential coefficients and}% +It is evident that this series bears a great resemblance +to the development in \PageRef{21}, deprived +of its first term. Let us suppose that $\psi a$~is the function +of which $\phi a$~is the differential coefficient, that is, +that $\psi' a = \phi a$. These two functions being the same, +their differential coefficients will be the same, that is, +$\psi'' a = \phi' a$. Similarly $\psi''' a = \phi'' a$, and so on. Substituting +these, the above series becomes +\[ +\psi' a\, h + \psi'' a\, \frac{h^{2}}{2} + + \psi''' a\, \frac{h^{3}}{2·3} + + \psi^{\text{iv}} a\, \frac{h^{4}}{2·3·4} + + \etc.\Add{,} +\] +{\Loosen which is (\PageRef{21}) the same as $\psi(a + h) - \psi a$. That +is, the integral of $\phi x\, dx$ between the limits $a$~and~$a + h$, +is $\psi(a + h) - \psi a$, where $\psi x$~is the function, which, +\PageSep{122} +when differentiated, gives~$\phi x$. For $a + h$ we may +write~$b$, so that $\psi b - \psi a$ is the integral of~$\phi x\, dx$ from +$x = a$ to $x = b$. Or we may make the second limit indefinite +by writing~$x$ instead of~$b$, which gives $\psi x - \psi a$, +which is said to be the integral of~$\phi x\, dx$, beginning +when $x = a$, the summation being supposed to be continued +from $x = a$ until $x$~has the value which it may +be convenient to give it.} + + +\Subsection{Nature of Integration.} + +Hence results a new branch of the inquiry, the reverse +of the Differential Calculus, the object of which +is, not to find the differential coefficient, having given +the function, but to find the function, having given +the differential coefficient. This is called the Integral +Calculus. + +From the definition given, it is obvious that the +value of an integral is not to be determined, unless +we know the values of~$x$ corresponding to the beginning +and end of the summation, whose limit furnishes +the integral. We might, instead of defining the integral +in the manner above stated, have made the +word mean merely the converse of the differential coefficient; +thus, if $\phi x$~be the differential coefficient of~$\psi x$, +$\psi x$~might have been called the integral of~$\phi x\, dx$. +We should then have had to show that the integral, +thus defined, is equivalent to the limit of the summation +already explained. We have preferred bringing +the former method before the student first, as it is +most analogous to the manner in which he will deduce +integrals in questions of geometry or mechanics. +\index{Integrals!indefinite}% + +With the last-mentioned definition, it is also obvious +that every function has an unlimited number of +integrals. For whatever differential coefficient~$\psi x$ +\PageSep{123} +gives, $C + \psi x$ will give the same, if $C$~be a constant, +that is, not varying when $x$~varies. In this case, if $x$ +become $x + h$, $C + \psi x$ becomes $C + \psi x + \psi' x\, h + \etc.$, +from which the subtraction of the original form $C + \psi x$ +gives $\psi' x\, h + \etc.$; whence, by the process in \PageRef{23}, +$\psi' x$~is the differential coefficient of $C + \psi' x$ as well as +of~$\psi x$. As many values, therefore, positive or negative, +as can be given to~$C$, so many different integrals +\index{Integrals!indefinite}% +can be found for~$\psi' x$; and these answer to the various +limits between which the summation in our original +definition may be made. To make this problem definite, +not only $\psi' x$ the function to be integrated, must +be given, but also that value of~$x$ from which the summation +is to begin. If this be~$a$, the integral of~$\psi' x$ is, +as before determined, $\psi x - \psi a$, and $C = -\psi a$. We +may afterwards end at any value of~$x$ which we please. +If $x = a$, $\psi x - \psi a = 0$, as is evident also from the +formation of the integral. We may thus, having given +an integral in terms of~$x$, find the value at which it +began, by equating the integral to zero, and finding +the value of~$x$. Thus, since $x^{2}$, when differentiated, +gives~$2x$, $x^{2}$~is the integral of~$2x$, beginning at $x = 0$; +and $x^{2} - 4$~is the integral beginning at~$x = 2$. + +In the language of Leibnitz, an integral would be +\index{Leibnitz}% +the sum of an infinite number of infinitely small quantities, +which are the differentials or infinitely small increments +of a function. Thus, a circle being, according +to him, a rectilinear polygon of an infinite number +of infinitely small sides, the sum of these would be +the circumference of the figure. As before (\PageRefs{13}{14}, +\PageNo{38}~et~seq., \PageNo{48}~et~seq.) we proceed to interpret +this inaccuracy of language. If, in a circle, we successively +describe regular polygons of $3$,~$4$, $5$,~$6$,~etc., +sides, we may, by this means, at last attain to a polygon +\PageSep{124} +whose side shall differ from the arc of which it is +the chord, by as small a fraction, either of the chord +or arc, as we please (\PageRefs{7}{11}). That is, $A$~being +the arc, $C$~the chord, and $D$~their difference, there is +no fraction so small that $D$~cannot be made a smaller +part of~$C$. Hence, if $m$~be the number of sides of the +polygon, $mC + mD$ or $mA$ is the real circumference; +and since $mD$~is the same part of~$mC$, which $D$~is of~$C$, +$mD$~may be made as small a part of~$mC$ as we please; +so that $mC$, or the sum of all the sides of the polygon, +can be made as nearly equal to the circumference as +we please. + +As in other cases, the expressions of Leibnitz are +\index{Leibnitz}% +the most convenient and the shortest, for all who can +immediately put a rational construction upon them; +this, and the fact that, good or bad, they have been, +and are, used in the works of Lagrange, Laplace, +\index{Lagrange}% +\index{Laplace}% +Euler, and many others, which the student who really +\index{Euler}% +desires to know the present state of physical science, +cannot dispense with, must be our excuse for continually +bringing before him modes of speech, which, +taken quite literally, are absurd. + + +\Subsection{Determination of Curvilinear Areas. The Parabola.} + +We will now suppose such a part of a curve, each +\index{Curvilinear areas, determination of|EtSeq}% +\index{Parabola, the|EtSeq}% +ordinate of which is a given function of the corresponding +abscissa, as lies between two given ordinates; +for example,~$MPP'M'$. Divide the line~$MM'$ +into a number of equal parts, which we may suppose +as great as we please, and construct \Fig[Figure]{10}. Let +$O$~be the origin of co-ordinates, and let $OM$, the value +of~$x$, at which we begin, be~$a$; and $OM'$, the value +at which we end, be~$b$. Though we have only divided~$MM'$ +\PageSep{125} +into four equal parts in the figure, the reasoning +to which we proceed would apply equally, had we divided +it into four million of parts. The sum of the +parallelograms $Mr$,~$mr$,~$m'r''$, and~$m''R$, is less than +the area~$MPP'M'$, the value of which it is our object +to investigate, by the sum of the curvilinear triangles +$Prp$,~$pr'p'$,~$p'r''p''$, and~$p''RP'$. The sum of these triangles +is less than the sum of the parallelograms $Qr$,~$qr'$,~$q'r''$, +and~$q''R$; but these parallelograms are together +\Figure{10} +equal to the parallelogram~$q''w$, as appears by +inspection of the figure, since the base of each of the +above-mentioned parallelograms is equal to~$m''M$, or~$q''P'$, +and the altitude~$P'w$ is equal to the sum of the +altitudes of the same parallelograms. Hence the sum +of the parallelograms $Mr$,~$mr'$,~$m'r''$, and~$m''R$, differs +from the curvilinear area~$MPP'M'$ by less than the +parallelogram~$q''w$. But this last parallelogram may +be made as small as we please by sufficiently increasing +the number of parts into which $MM'$~is divided; +\PageSep{126} +for since one side of it,~$P'w$, is always less than~$P'M'$, +and the other side~$P'q''$, or~$m''M'$, is as small a part as +we please of~$MM'$ the number of square units in~$q''w$, +is the product of the number of linear units in $P'w$ +and~$P'q''$, the first of which numbers being finite, and +the second as small as we please, the product is +as small as we please. Hence the curvilinear area~$MPP'M'$ +is the limit towards which we continually +approach, but which we never reach, by dividing $MM'$ +into a greater and greater number of equal parts, and +adding the parallelograms $Mr$,~$mr'$,~etc., so obtained. +If each of the equal parts into which $MM'$ is divided +be called~$dx$, we have $OM = a$, $Om = a + dx$, $Om' = a + 2\,dx$, +etc. And $MP$,~$mp$, $m'p'$,~etc., are the values +of the function which expresses the ordinates, corresponding +to $a$,~$a + dx$, $a + 2\, dx$,~etc., and may therefore +be represented by $\phi a$, $\phi(a + dx)$, $\phi(a + 2\, dx)$, +etc. These are the altitudes of a set of parallelograms, +the base of each of which is~$dx$; hence the +sum of their area is +\[ +\phi a\, dx + \phi(a + dx)\, dx + \phi(a + 2\, dx)\, dx + \etc., +\] +and the limit of this, to which we approach by diminishing~$dx$, +is the area required. + +This limit is what we have defined to be the integral +of~$\phi x\, dx$ from $x = a$ to $x = b$; or if $\psi x$~be the +function, which, when differentiated, gives $\phi x$, it is +$\psi b - \psi a$. Hence, $y$~being the ordinate, the area included +between the axis of~$x$, any two values of~$y$, and +the portion of the curve they cut off, is $\int y\, dx$, beginning +at the one ordinate and ending at the other. + +Suppose that the curve is a part of a parabola +of which $O$~is the vertex, and whose equation\footnote + {If the student has not any acquaintance with the conic sections, he must + nevertheless be aware that there is some curve whose abscissa and ordinate + are connected by the equation $y^{2} = px$. This, to him, must be the definition + of \emph{parabola}: by which word he must understand, a curve whose equation is + $y^{2} = px$.} +is +\PageSep{127} +therefore $y^{2} = px$ where $p$~is the double ordinate which +passes through the focus. Here $y = p^{\efrac{1}{2}} x^{\efrac{1}{2}}$, and we +must find the integral of~$p^{\efrac{1}{2}} x^{\efrac{1}{2}}\, dx$, or the function +whose differential coefficient is~$p^{\efrac{1}{2}} x^{\efrac{1}{2}}$, $p^{\efrac{1}{2}}$~being a constant. +If we take the function~$cx^{n}$, $c$~being independent +of~$x$, and substitute $x + h$ for~$x$, we have for the +development $cx^{n} + cnx^{n-1}\, h + \etc$. Hence the differential +coefficient of~$cx^{n}$ is~$cnx^{n-1}$; and as $c$~and~$n$ may +be any numbers or fractions we please, we may take +them such that $cn$~shall $= p^{\efrac{1}{2}}$ and $n - 1 = \frac{1}{2}$, in which +case $n = \frac{3}{2}$ and $c = \frac{2}{3}p^{\efrac{1}{2}}$. Therefore the differential coefficient +of~$\frac{2}{3} p^{\efrac{1}{2}} x^{\efrac{3}{2}}$ is~$p^{\efrac{1}{2}} x^{\efrac{1}{2}}$, and conversely, the integral +of~$p^{\efrac{1}{2}} x^{\efrac{1}{2}}\, dx$ is~$\frac{2}{3} p^{\efrac{1}{2}} x^{\efrac{3}{2}}$. + +{\Loosen The area~$MPP'M'$ of the parabola is therefore +\index{Parabola, the}% +~$\frac{2}{3} p^{\efrac{1}{2}} b^{\efrac{3}{2}} - \frac{2}{3} p^{\efrac{1}{2}} a^{\efrac{3}{2}}$. If we begin the integral at the vertex~$O$, +in which case $a = 0$, we have for the area~$OM'P'$, +$\frac{2}{3} p^{\efrac{1}{2}} b^{\efrac{3}{2}}$, where $b = OM'$. This is~$\frac{2}{3} p^{\efrac{1}{2}} b^{\efrac{1}{2}} × b$, which, since +$p^{\efrac{1}{2}} b^{\efrac{1}{2}} = M'P'$ is $\frac{2}{3}P'M' × OM'$, or two-thirds of the rectangle\footnote + {This proposition is famous as having been discovered by Archimedes +\index{Archimedes}% + at a time when such a step was one of no small magnitude.} +contained by $OM'$~and~$M'P'$.} + + +\Subsection{Method of Indivisibles.} + +We may mention, in illustration of the preceding +\index{Indivisibles!method of|EtSeq}% +problem, a method of establishing the principles of +the Integral Calculus, which generally goes by the +name of the \emph{Method of Indivisibles}. A line is considered +as the sum of an infinite number of points, a +surface of an infinite number of lines, and a solid of +an infinite number of surfaces. One line twice as long +as another would be said to contain twice as many +\PageSep{128} +points, though the number of points in each is unlimited. +To this there are two objections. First, the +word infinite, in this absolute sense, really has no +\index{Infinite@\emph{Infinite}, the word}% +meaning, since it will be admitted that the mind has +no conception of a number greater than any number. +The word infinite\footnote + {See \Title{Study of Mathematics} (Chicago: The Open Court Publishing~Co\Add{.}), + page~123 et~seq.} +can only be justifiably used as an +abbreviation of a distinct and intelligible proposition; +for example, when we say that $a + \dfrac{1}{x}$ is equal to~$a$ +when $x$~is infinite, we only mean that as $x$ is increased, +$a + \dfrac{1}{x}$~becomes nearer to~$a$, and may be made as near +to it as we please, if $x$~may be as great as we please. +The second objection is, that the notion of a line +being the sum of a number of points is not true, nor +does it approach nearer the truth as we increase the +number of points. If twenty points be taken on a +straight line, the sum of the twenty-one lines which +lie between point and point is equal to the whole line; +which cannot be if the points by themselves constitute +any part of the line, however small. Nor will the sum +of the points be a part of the line, if twenty thousand +be taken instead of twenty. There is then, in this +method, neither the rigor of geometry, nor that approach +to truth, which, in the method of Leibnitz, +\index{Leibnitz}% +may be carried to any extent we please, short of absolute +correctness. We would therefore recommend to +the student not to regard any proposition derived +from this method as true on that account; for falsehoods, +as well as truths, may be deduced from it. Indeed, +the primary notion, that the number of points +in a line is proportional to its length, is manifestly incorrect. +Suppose (\Fig{6}, \PageRef{48}) that the point~$Q$ +\PageSep{129} +moves from $A$ to~$P$. It is evident that in whatever +number of points $OQ$ cuts~$AP$, it cuts~$MP$ in the same +number. But $PM$~and~$PA$ are not equal. A defender +of the system of indivisibles, if there were such a person, +\index{Indivisibles!notion of, in mechanics|EtSeq}% +would say something equivalent to supposing +that the points on the two lines are of \emph{different sizes}, +which would, in fact, be an abandonment of the +method, and an adoption of the idea of Leibnitz, using +\index{Leibnitz}% +the word \emph{point} to stand for the infinitely small +\index{Point@\emph{Point}, the word}% +line. + +This notion of indivisibles, or at least a way of +speaking which looks like it, prevails in many works +on mechanics. Though a point is not treated as a +length, or as any part of space whatever, it is considered +as having weight; and two points are spoken of +as having different weights. The same is said of a +line and a surface, neither of which can correctly be +supposed to possess weight. If a solid be of the same +density throughout, that is, if the weight of a cubic +inch of it be the same from whatever part it is cut, it +is plain that the weight may be found by finding the +number of cubic inches in the whole, and multiplying +this number by the weight of one cubic inch. But if +the weight of every two cubic inches is different, we +can only find the weight of the whole by the integral +calculus. + +Let $AB$ (\Fig{11}) be a line possessing weight, or +\index{Points, the number of, in a straight line}% +a very thin parallelepiped of matter, which is such, +that if we were to divide it into any number of equal +parts, as in the figure, the weight of the several parts +would be different. We suppose the weight to vary +continuously, that is, if two contiguous parts of equal +length be taken, as $pq$~and~$qr$, the ratio of the weights +\PageSep{130} +of these two parts may, by taking them sufficiently +small, be as near to equality as we please. + +The \emph{density} of a body is a mathematical term, which +\index{Density, continuously varying|EtSeq}% +\index{Specific gravity, continuously varying|EtSeq}% +may be explained as follows: A cubic inch of gold +weighs more than a cubic inch of water; hence gold +is \emph{denser} than water. If the first weighs $19$~times as +much as the second, gold is said to be $19$~times more +dense than water, or the density of gold is $19$~times +that of water. Hence we might define the density by +the weight of a cubic inch of the substance, but it is +usual to take, not this weight, but the proportion +which it bears to the same weight of water. Thus, +when we say the \emph{density}, or \emph{specific gravity} (these terms +are used indifferently), of cast iron is~$7.207$, we mean +\index{Iron bar continually varying in density, weight of|EtSeq}% +\index{Weight of an iron bar of which the density varies from point to point|EtSeq}% +that if any vessel of pure water were emptied and +filled with cast iron, the iron would weigh $7.207$~times +as much as the water. + +If the density of a body were uniform throughout, +we might easily determine it by dividing the weight +of any bulk of the body, by the weight of an equal +bulk of water. In the same manner (\PageRef[pages]{52} et~seq.)\ +we could, from our definition of velocity, determine +any uniform velocity by dividing the length described +by the time. But if the density vary continuously, +no such measure can be adopted. For if by the side +of~$AB$ (which we will suppose to be of iron) we placed +a similar body of water similarly divided, and if we +divided the weight of the part~$pq$ of iron by the weight +of the same part of water, we should get different +densities, according as the part~$pq$ is longer or shorter. +The water is supposed to be homogeneous, that is, +any part of it~$pr$, being twice the length of~$pq$, is twice +the weight of~$pq$, and so on. The iron, on the contrary, +being supposed to vary in density, the doubling +\PageSep{131} +the length gives either more or less than twice the +weight. But if we suppose $q$ to move towards~$p$, both +on the iron and the water, the limit of the ratio~$pq$ of +iron to $pq$~of water, may be chosen as a measure of +the density of~$p$, on the same principle as in \PageRefs{54}{55}, +the limit of the ratio of the length described to +the time of describing it, was called the velocity. If +we call $k$ this limit, and if the weight varies continuously, +though no part~$pq$, however small, of iron, +would be exactly $k$~times the same part of water in +weight, we may nevertheless take $pq$ so small that +these weights shall be as nearly as we please in the +ratio of $k$~to~$1$. + +Let us now suppose that this density, expressed +by the limiting ratio aforesaid, is always $x^{2}$ at any +\Figure{11} +point whose distance from~$A$ is $x$~feet; that is, the +density at~$q$, $2$~feet distance from~$A$, is~$4$, and so on. +Let the whole distance $AB = a$. If we divide~$a$ into +$n$~equal parts, each of which is~$dx$, so that $n\, dx = a$, +and if we call~$b$ the area of the section of the parallelepiped, +($b$~being a fraction of a square foot,) the +solid content of each of the parts will be $b\, dx$ in +cubic feet; and if $w$~be the weight of a cubic foot of +water, the weight of the same bulk of water will be~$wb\, dx$. +If the solid~$AB$ were homogeneous in the immediate +neighborhood of the point~$p$, the density being +then~$x^{2}$, would give $x^{2} × bw\, dx$ for the weight of the +same part of the substance. This is not true, but can +be brought as near to the truth as we please, by taking +$dx$ sufficiently small, or dividing~$AB$ into a sufficient +\PageSep{132} +number of parts. Hence the real weight of~$pq$ +may be represented by $bwx^{2}\, dx + \alpha$, where $\alpha$~may be +made as small a part as we please of the term which +precedes it. + +In the sum of any number of these terms, the sum +arising from the term~$\alpha$ diminishes without limit as +compared with the sum arising from the term~$bwx^{2}\, dx$; +for if $\alpha$~be less than the thousandth part of~$p$, $\alpha'$~less +than the thousandth part of~$p'$, etc., then $\alpha + \alpha' + \etc$.\ +will be less than the thousandth part of~$p + p' + \etc.$: +which is also true of any number of quantities, and of +any fraction, however small, which each term of one +set is of its corresponding term in the other. Hence +the taking of the integral of~$bwx^{2}\, dx$ dispenses with +the necessity of considering the term~$\alpha$; for in taking +the integral, we find a limit which supposes $dx$ to +have decreased without limit, and the \emph{integral} which +would arise from~$\alpha$ has therefore diminished without +limit. + +The integral of~$bwx^{2}\, dx$ is~$\frac{1}{3}bwx^{3}$, which taken from +$x = 0$ to $x = a$ is~$\frac{1}{3}bwa^{3}$. This is therefore the weight +in pounds of the bar whose length is $a$~feet, and whose +section is $b$~square feet, when the density at any point +distant by $x$~feet from the beginning is~$x^{2}$; $w$~being +the weight in pounds of a cubic foot of water. + + +\Subsection{Concluding Remarks on the Study of the Calculus.} + +We would recommend it to the student, in pursuing +\index{Advice for studying the Calculus}% +\index{Approximate solutions in the Integral Calculus}% +\index{Rough methods of solution in the Integral Calculus}% +any problem of the Integral Calculus, never for +one moment to lose sight of the manner in which he +would do it, if a rough solution for practical purposes +only were required. Thus, if he has the area of a +curve to find, instead of merely saying that~$y$, the +ordinate, being a certain function of the abscissa~$x$, +\PageSep{133} +$\int y\, dx$ within the given limits would be the area required; +and then proceeding to the mechanical solution +of the question: let him remark that if an approximate +solution only were required, it might be +obtained by dividing the curvilinear area into a number +of four-sided figures, as in \Fig[Figure]{10}, one side of +which only is curvilinear, and embracing so small an +arc that it may, without visible error, be considered +as rectilinear. The mathematical method begins with +the same principle, investigating upon this supposition, +not the sum of these rectilinear areas, but the +limit towards which this sum approaches, as the subdivision +is rendered more minute. This limit is shown +to be that of which we are in search, since it is proved +that the error diminishes without limit, as the subdivision +is indefinitely continued. + +We now leave our reader to any elementary work +which may fall in his way, having done our best to +place before him those considerations, something +equivalent to which he must turn over in his mind before +he can understand the subject. The method so +generally followed in our elementary works, of leading +the student at once into the mechanical processes +of the science, postponing entirely all other considerations, +is to many students a source of obscurity at +least, if not an absolute impediment to their progress; +since they cannot imagine what is the object of that +which they are required to do. That they shall understand +everything contained in these treatises, on +the first or second reading, we cannot promise; but +that the want of illustration and the preponderance of +\emph{technical} reasoning are the great causes of the difficulties +which students experience, is the opinion of many +\index{Advice for studying the Calculus}% +\index{Approximate solutions in the Integral Calculus}% +\index{Rough methods of solution in the Integral Calculus}% +who have had experience in teaching this subject. +\PageSep{134} +%[Blank page] +\PageSep{135} +\BackMatter + +\Section[Bibliography of Standard Text-books and Works of Reference on the Calculus] +{Brief Bibliography.\protect\footnotemark} + +\footnotetext{The information given regarding the works mentioned in this list is designed + to enable the reader to select the books which are best suited to his + needs and his purse. Where the titles do not sufficiently indicate the character + of the books, a note or extract from the Preface has been added. The + American prices have been supplied by Messrs.\ Lemcke \&~Buechner, 812~Broadway, + New~York, through whom the purchases, especially of the foreign + books, may be conveniently made.---\Ed.} + +\BibSect{Standard Text-books and Treatises on +the Calculus.} + +\BibSubsect{English.} + +\begin{Book} +Perry, John: \Title{Calculus for Engineers.} Second edition, London +and New York: Edward Arnold. 1897. Price, \Price{7s. 6d.} (\$2.50). + +\begin{Descrip} +Extract from Author's Preface: ``This book describes what has +for many years been the most important part of the regular course in +the Calculus for Mechanical and Electrical Engineering students at +the Finsbury Technical College. The students in October knew only +the most elementary mathematics, many of them did not know the +Binomial Theorem, or the definition of the sine of an angle. In July +they had not only done the work of this book, but their knowledge +was of a practical kind, ready for use in any such engineering problems +as I give here.'' + +Especially good in the character and number of practical examples +given. +\end{Descrip} +\end{Book} + +\begin{Book} +Lamb, Horace: \Title{Infinitesimal Calculus.} New York: The Macmillan +Co. 1898. Price,~\$3.00. + +\begin{Descrip} +Extract from Author's Preface: ``This book attempts to teach +those portions of the Calculus which are of primary importance in +the application to such subjects as Physics and Engineering\dots. +Stress is laid on fundamental principles\dots. Considerable attention +has been paid to the logic of the subject.'' +\end{Descrip} +\end{Book} +\PageSep{136} + +\begin{Book} +Edwards, Joseph: \Title{An Elementary Treatise on the Differential +Calculus.} Second edition, revised. 8vo,~cloth. New York +and London: The Macmillan~Co. 1892. Price, \$3.50.---% +\Title{Differential Calculus for Beginners.} 8vo,~cloth. 1893.\Chg{ }{---}\Title{The +Integral Calculus for Beginners.} 8vo,~cloth. (Same Publishers.) +Price, \$1.10~each. +\end{Book} + +\begin{Book} +Byerly, William E.: \Title{Elements of the Differential Calculus.} Boston: +Ginn \&~Co. Price, \$2.15.---\Title{Elements of the Integral +Calculus.} (Same Publishers.) Price,~\$2.15. +\end{Book} + +\begin{Book} +Rice, J.~M., and Johnson, W.~W.: \Title{An Elementary Treatise on +the Differential Calculus Founded on the Method of Rates +or Fluxions.} New~York: John Wiley \&~Sons. 8vo. 1884. +Price, \$3.50. Abridged edition, 1889. Price,~\$1.50. +\end{Book} + +\begin{Book} +Johnson, W.~W.: \Title{Elementary Treatise on the Integral Calculus +Founded on the Method of Rates or Fluxions.} 8vo,~cloth. +New~York: John Wiley \&~Sons. 1885. Price,~\$1.50. +\end{Book} + +\begin{Book} +Greenhill, A.~G.: \Title{Differential and Integral Calculus.} With applications. +8vo,~cloth. Second edition. New~York and London: +The Macmillan~Co. 1891. Price, \Price{9s.}~(\$2.60). +\end{Book} + +\begin{Book} +Price: \Title{Infinitesimal Calculus.} Four Vols. 1857--65. Out of +print and very scarce. Obtainable for about~\$27.00. +\end{Book} + +\begin{Book} +Smith, William Benjamin: \Title{Infinitesimal Analysis.} Vol.~I., Elementary: +Real Variables. New~York and London: The Macmillan~Co. +1898. Price,~\$3.25. + +\begin{Descrip} +``The aim has been, within a prescribed expense of time and +energy to penetrate as far as possible, and in as many directions, into +the subject in hand,---that the student should attain as wide knowledge +of the matter, as full comprehension of the methods, and as clear +consciousness of the spirit and power of analysis as the nature of the +case would admit.''---From Author's Preface. +\end{Descrip} +\end{Book} + +\begin{Book} +Todhunter, Isaac: \Title{A Treatise on the Differential Calculus.} London +and New~York: The Macmillan~Co. Price, \Price{10s. 6d.} +(\$2.60). \Title{A Treatise on the Integral Calculus.} (Same publishers.) +Price, \Price{10s. 6d.} (\$2\Chg{ }{.}60). + +\begin{Descrip} +Todhunter's text-books were, until recently, the most widely used +in England. His works on the Calculus still retain their standard +character, as general manuals. +\end{Descrip} +\end{Book} +\PageSep{137} + +\begin{Book} +Williamson: \Title{Differential and Integral Calculus.} London and +New~York: Longmans, Green, \&~Co. 1872--1874. Two~Vols. +Price, \$3.50~each. +\end{Book} + +\begin{Book} +De~Morgan, Augustus: \Title{Differential and Integral Calculus.} London: +Society for the Diffusion of Useful Knowledge. 1842. +Out of print. About~\$6.40. + +\begin{Descrip} +The most extensive and complete work in English. ``The object +has been to contain within the prescribed limits, the whole of the +students' course from the confines of elementary algebra and trigonometry, +to the entrance of the highest works on mathematical physics'' +(Author's Preface). Few examples. In typography, and general +arrangement of material, inferior to the best recent works. Valuable +for collateral study, and for its philosophical spirit. +\end{Descrip} +\end{Book} + + +\BibSubsect{French.} + +\begin{Book} +Sturm: \Title{Cours d'analyse de l'École Polytechnique.} 10.~édition, +revue et corrigé par E.~Prouhet, et augmentée de~la théorie +élémentaire des fonctions elliptiques, par H.~Laurent. 2~volumes +in---8. Paris: Gauthier-Villars et~fils. 1895. Bound, +16~fr.\ 50~c. \$4.95. + +\begin{Descrip} +One of the most widely used of text-books. First published in +1857. The new tenth edition has been thoroughly revised and brought +down to date. The exercises, while not numerous, are sufficient, those +which accompany the additions and complementary chapters of M.~De~Saint +Germain having been taken from the Collection of M.~Tisserand, +mentioned below. +\end{Descrip} +\end{Book} + +\begin{Book} +Duhamel: \Title{Éléments de calcul infinitésimal.} 4.~Edition, revue et +annotée par J.~Bertrand. 2~volumes in---8; avec planches. +Paris: Gauthier-Villars et~fils. 1886. 15~fr. \$4.50. + +\begin{Descrip} +The first edition was published between 1840 and 1841. ``Cordially +recommended to teachers and students'' by De~Morgan. Duhamel +paid great attention to the philosophy and logic of the mathematical +sciences, and the student may also be referred in this connexion to +his \Title{Méthodes dans les sciences de raisonnement}. 5~volumes. Paris: +Gauthier-Villars et~fils. Price, 25.50~francs. \$7.65. +\end{Descrip} +\end{Book} + +\begin{Book} +Lacroix, S.-F.: \Title{Traité élémentaire de calcul différentiel et de +calcul intégral.} 9.~Edition, revue et augmentée de notes par +Hermite et Serret. 2~vols. Paris: Gauthier-Villars et~fils. +1881. 15~fr. \$4.50. + +\begin{Descrip} +A very old work. The first edition was published in 1797. It was +the standard treatise during the early part of the century, and has +been kept revised by competent hands. +\end{Descrip} +\end{Book} +\PageSep{138} + +\begin{Book} +Appell, P.; \Title{Éléments d'analyse mathématique.} À l'usage des +ingénieurs et dés physiciens. Cours professé à l'École Centrale +des Arts et Manufactures. 1~vol.\ in---8, 720~pages, avec +figures, cartonné à l'anglaise. Paris: Georges Carré \&~C. +Naud. 1899. Price, 24~francs.\ \$7.20. +\end{Book} + +\begin{Book} +Boussinesq, J.: \Title{Cours d'analyse infinitésimal.} À l'usage des +personnes qui étudient cette science en vue de ses applications +mécaniques et physiques, 2~vols., grand in\Chg{-}{---}8, avec figures. +Tome~I\@. Calcul différentiel. Paris, 1887. 17~fr.\ (\$5.10). +Tome~II\@. Calcul intégral. Paris: Gauthier-Villars et~fils. +1890. 23~fr.\ 50~c.\ (\$7.05). +\end{Book} + +\begin{Book} +Hermite, Ch.: \Title{Cours d'analyse de l'École Polytechnique.} 2~vols. +Vol.~I\@. Paris: Gauthier-Villars et~fils. 1897. + +\begin{Descrip} +A new edition of Vol.~I. is in preparation (1899). Vol.~II. has not +yet appeared. +\end{Descrip} +\end{Book} + +\begin{Book} +Jordan, Camille: \Title{Cours d'analyse de l'École Polytechnique.} 3~volumes. +2.~édition. Paris: Gauthier-Villars et~fils. 1893--1898. +51~fr.\ \$14.70. + +\begin{Descrip} +Very comprehensive on the theoretical side. Enters deeply into +the metaphysical aspects of the subject. +\end{Descrip} +\end{Book} + +\begin{Book} +Laurent, H.: \Title{Traité d'analyse.} 7~vols in---8. Paris: Gauthier-Villars +et~fils. 1885--1891. 73~fr.\ \$21.90. + +\begin{Descrip} +The most extensive existing treatise on the Calculus. A general +handbook and work of reference for the results contained in the +more special works and memoirs. +\end{Descrip} +\end{Book} + +\begin{Book} +Picard, Émile: \Title{Traité d'analyse.} 4~volumes grand in\Chg{-}{---}8. Paris: +Gauthier-Villars et~fils. 1891. 15~fr.\ each. Vols.~I.--III., +\$14.40. Vol.~IV. has not yet appeared. + +\begin{Descrip} +An advanced treatise on the Integral Calculus and the theory of +differential equations. Presupposes a knowledge of the Differential +Calculus. +\end{Descrip} +\end{Book} + +\begin{Book} +Serret, J.-A.: \Title{Cours de calcul différentiel et intégral.} 4.~edition, +augmentée d'une note sur les fonctions elliptiques, par +Ch.~Hermite. 2~forts volumes in---8. Paris: Gauthier-Villars +et~fils. 1894. 25~fr.\ \$7.50. + +\begin{Descrip} +A good German translation of this work by Axel Harnack has +passed through its second edition (Leipsic: Teubner, 1885 and 1897). +\end{Descrip} +\end{Book} +\PageSep{139} + +\begin{Book} +Hoüel, J.: \Title{Cours de calcul infinitésimal.} 4~beaux volumes grand +in---8, avec figures. Paris: Gauthier-Villars et~fils. 1878--1879--1880--1881. +50~fr.\ \$15.00. +\end{Book} + +\begin{Book} +Bertrand, J.: \Title{Traité de calcul différentiel et de calcul intégral.} +(1)~Calcul différentiel. Paris: Gauthier-Villars et fils. 1864. +Scarce. About \$48.00. (2)~Calcul intégral (Intégrales définies +et indéfinies). Paris, 1870. Scarce. About \$24.00. +\end{Book} + +\begin{Book} +Boucharlat, J.-L.: \Title{Éléments de calcul différentiel et de calcul +intégral.} 9.~édition, revue et annotée par H.~Laurent. Paris: +Gauthier-Villars et~fils. 1891. 8~fr.\ \$2.40. +\end{Book} + +\begin{Book} +Moigno: \Title{Leçons de calcul différentiel et de calcul intégral\Typo{,}{.}} 2~vols., +Paris, 1840--1844. Scarce. About \$9.60. +\end{Book} + +\begin{Book} +Navier: \Title{Leçons d'analyse de l'École Polytechnique.} Paris, 1840. +2nd~ed. 1856. Out of print. About \$3.60. + +\begin{Descrip} +An able and practical work. Very popular in its day. The typical +course of the \textit{École Polytechnique}, and the basis of several of the treatises +that followed, including that of Sturm. Also much used in its +German translation. +\end{Descrip} +\end{Book} + +\begin{Book} +Cournot: \Title{Théorie des fonctions et du calcul infinitésimal.} 2~vols. +Paris, 1841. 2nd~ed. 1856--1858. Out of print, and +scarce. About \$3.00. + +\begin{Descrip} +The first edition (1841) was ``cordially recommended to teachers +and students'' by De~Morgan. Cournot was especially strong on the +philosophical side. He examined the foundations of many sciences +and developed original views on the theory of knowledge, which are +little known but have been largely drawn from by other philosophers. +\end{Descrip} +\end{Book} + +\begin{Book} +Cauchy, A.: \Title{\OE{}uvres complètes.} Tome~III: \Title{Cours d'analyse +de l'École Polytechnique.} Tome~IV: \Title{Résumé des leçons +données à l'École Polytechnique sur le calcul infinitésimal. +Leçons sur le calcul différentiel.} Tome~V: \Title{Leçons sur les +applications du calcul infinitésimal à la géométrie.} Paris: +Gauthier-Villars et~fils, 1885--1897. 25~fr.\ each. \$9.50~each. + +\begin{Descrip} +The works of Cauchy, as well as those of Lagrange, which follow, +are mentioned for their high historical and educational importance. +\end{Descrip} +\end{Book} + +\begin{Book} +Lagrange, J.~L.: \Title{\OE{}uvres complètes.} Tome~IX: \Title{Théorie des fonctions +analytiques.} Tome~X.: \Title{Leçons sur le calcul des fonctions.} +\PageSep{140} +Paris: Gauthier-Villars et~fils, 1881--1884. 18~fr.\ per +volume. \$5.40 per~volume. + +\begin{Descrip} +``The same power of abstraction and facility of treatment which +signalise these works are nowhere to be met with in the prior or subsequent +history of the subject. In addition, they are replete with the +profoundest aperçus into the history of the development of analytical +truths,---aperçus which could have come only from a man who combined +superior creative endowment with exact and comprehensive +knowledge of the facts. In the remarks woven into the body of the +text will be found what is virtually a detailed history of the subject, +and one which is not to be had elsewhere, least of all in diffuse histories +of mathematics. The student, thus, not only learns in these +works how to think, but also discovers how people actually have +thought, and what are the ways which human instinct and reason +have pursued in the different individuals who have participated in +the elaboration of the science.''---(E.~Dühring.) +\end{Descrip} +\end{Book} + +\begin{Book} +Euler, L.: + +\begin{Descrip} +The Latin treatises of Euler are also to be mentioned in this connexion, +for the benefit of those who wish to pursue the history of the +text-book making of this subject to its fountain-head. They are the +\Title{Differential Calculus} (St.~Petersburg, 1755), the \Title{Integral Calculus} (3~vols., +St.~Petersburg, 1768--1770), and the \Title{Introduction to the Infinitesimal +Analysis} (2~vols., Lausanne, 1748). Of the last-mentioned work +an old French translation by Labey exists (Paris: Gauthier-Villars), +and a new German translation (of Vol.~I. only) by Maser (Berlin: +Julius Springer, 1885). Of the first-mentioned treatises on the Calculus +proper there exist two old German translations, which are not +difficult to obtain. +\end{Descrip} +\end{Book} + + +\BibSubsect{German.} + +\begin{Book} +Harnack, Dr.\ Axel: \Title{Elemente der Differential- und Integralrechnung.} +Zur Einführung in das Studium dargestellt. Leipzig: +Teubner, 1881. M.~7.60. Bound, \$2.80. (English translation. +London: Williams \&~Norgate. 1891.) +\end{Book} + +\begin{Book} +Junker, Dr.\ Friedrich: \Title{Höhere Analysis.} I.~\Title{Differentialrechnung.} +Mit 63~Figuren. II.~\Title{Integralrechnung.} Leipzig: +G.~J. Göschen'sche Verlagshandlung. 1898--1899. 80~pf.\ each. +30~cents each. + +\begin{Descrip} +These books are marvellously cheap, and very concise. They +contain no examples. Pocket-size. +\end{Descrip} +\end{Book} + +\begin{Book} +Autenheimer, F.: \Title{Elementarbuch der Differential- und Integralrechnung +mit zahlreichen Anwendungen aus der Analysis, +Geometrie, Mechanik, Physik etc.} Für höhere Lehranstalten +\PageSep{141} +und den Selbstunterricht. 4te~verbesserte Auflage. Weimar: +Bernhard Friedrich Voigt. 1895. + +\begin{Descrip} +As indicated by its title, this book is specially rich in practical +applications. +\end{Descrip} +\end{Book} + +\begin{Book} +Stegemann: \Title{Grundriss der Differential- und Integralrechnung}, +8te~Auflage, herausgegeben von Kiepert. Hannover: Helwing, +1897. Two volumes, 26~marks. Two volumes, bound, +\$8.50. + +\begin{Descrip} +This work was highly recommended by Prof.\ Felix Klein at the +Evanston Colloquium in~1893. +\end{Descrip} +\end{Book} + +\begin{Book} +Schlömilch: \Title{Compendium der höheren Analysis.} Fifth edition, +1881. Two volumes, \$6.80. + +\begin{Descrip} +Schlömilch's text-books have been very successful. The present +work was long the standard manual. +\end{Descrip} +\end{Book} + +\begin{Book} +Stolz, Dr.\ Otto: \Title{Grundzüge der Differential- und Integralrechnung.} +In 2~Theilen. I.~Theil. Reelle Veränderliche und +Functionen. (460~S.) 1893. M.~8. II.~Complexe Veränderliche +und Functionen. (338~S.) Leipzig: Teubner. 1896. +M.~8. Two volumes, \$6.00. + +\begin{Descrip} +A supplementary 3rd part entitled \Title{Die Lehre von den Doppelintegralen} +has just been published (1899). Based on the works of J.~Tannery, +Peano, and Dini. +\end{Descrip} +\end{Book} + +\begin{Book} +Lipschitz, R.: \Title{Lehrbuch der Analysis.} 1877--1880. Two volumes, +bound, \$12.30. + +\begin{Descrip} +Specially good on the theoretical side. +\end{Descrip} +\end{Book} + + +\BibSubsect{Collections of Examples and Illustrations.} + +\begin{Book} +Byerly, W.~E.: \Title{Problems in Differential Calculus.} Supplementary +to a Treatise on Differential Calculus. Boston: Ginn \&~Co. +75~cents. +\end{Book} + +\begin{Book} +Gregory: \Title{Examples on the Differential and Integral Calculus.} +1841. Second edition. 1846. Out of print. About \$6.40. +\end{Book} + +\begin{Book} +Frenet: \Title{Recueil d'exercises sur le calcul infinitésimal.} 5.~édition, +augmentée d'un appendice, par H.~Laurent. Paris: +Gauthier-Villars et~fils. 1891. 8~fr.\ \$2.40. +\end{Book} +\PageSep{142} + +\begin{Book} +Tisserand, F.: \Title{Recueil complémentaire d'exercises sur le calcul +infinitésimal.} Second edition. Paris: Gauthier-Villars et~fils +1896. + +\begin{Descrip} +Complementary to Frenet. +\end{Descrip} +\end{Book} + +\begin{Book} +Laisant, C.~A.: \Title{Recueil de problèmes de mathématiques.} Tome~VII\@. +Calcul infinitésimal et calcul des fonctions. Mécanique. +Astronomie. (Announced for publication.) Paris: Gauthier-Villars +et~fils. +\end{Book} + +\begin{Book} +Schlömilch, Dr.\ Oscar: \Title{Uebungsbuch zum Studium der höheren +Analysis.} I.~Theil. Aufgaben aus der Differentialrechnung. +4te~Auflage. (336~S.) 1887. M.~6. II.~Aufgaben aus der +Integralrechnung. 3te~Auflage. (384~S.) Leipzig: Teubner, +1882. M.~7.60. Both volumes, bound, \$7.60. +\end{Book} + +\begin{Book} +Sohncke, L.~A.: \Title{Sammlung von Aufgaben aus der Differential- +und Integralrechnung.} Herausgegeben von Heis. Two volumes, +in---8. Bound, \$3.00. +\end{Book} + +\begin{Book} +Fuhrmann, Dr.\ Arwed: \Title{Anwendungen der Infinitesimalrechnung +in den Naturwissenschaften, im Hochbau und in der +Technik.} Lehrbuch und Aufgabensammlung. In sechs Theilen, +von denen jeder ein selbstständiges Ganzes bildet. Theil~I. +Naturwissenschaftliche Anwendungen der Differentialrechnung. +Theil~II. Naturwissenschaftliche Anwendungen der +Integralrechnung. Berlin: Verlag von Ernst \&~Korn. 1888--1890. +Vol.~I., Cloth, \$1.35. Vol.~II., Cloth, \$2.20. +\end{Book} +\PageSep{143} + + +\printindex + +\iffalse +%INDEX. + +Accelerated motion 57, 60 + +Accelerating force 62 + +Advice for studying the Calculus 132, 133 + +Angle, unit employed in measuring an#Angle 51 + +Approximate solutions in the Integral Calculus 132, 133 + +Arc and its chord, a continuously decreasing|EtSeq#Arc 7, 39 % et seq. + +Archimedes 127 + +Astronomical ephemeris 76 + +Calculus, notation of 25 + +Calculus, notation of|EtSeq 79 % et seq. + +Circle, equation of|EtSeq 31 % et seq. + +Circle cut by straight line, investigated|EtSeq 31 % et seq. + +Coefficients, differential|EtSeq 22 % et seq. + +Coefficients, differential 38, 55, 82, 88, 96, 100, 112 + +Complete Differential Coefficients 96 + +Constants 14 + +Contiguous values 112 + +Continuous quantities|EtSeq 7 % et seq. + +Continuous quantities 53 + +Co-ordinates 30 + +Curve, magnified 40 + +Curvilinear areas, determination of|EtSeq 124 % et seq. + +Density, continuously varying|EtSeq 130 % et seq. + +Derivatives 19, 21, 22 + +Derived Functions|EtSeq 19 % et seq. + +Derived Functions 21 + +Differences + arithmetical 4 + of increments 26 + calculus of 89 + +Differential coefficients|EtSeq 22 % et seq., + +Differential coefficients 38, 55, 82 + as the index of the change of a function 112 + of higher orders 88 + +Differentials + partial|EtSeq 78 % et seq.; + total|EtSeq 78 % et seq. + +Differentiation + of the common functions 85, 86 + successive|EtSeq 88 % et seq.; + implicit|EtSeq 94 % et seq.; + of complicated functions|EtSeq 100 % et seq. + +Direct function 97 + +Direction 36 + +Equality 4 + +Equations, solution of 77 + +Equidistant values 104 + +Euler 27, 124 + +Errors, in the valuation of quantities 75, 84 + +Explicit functions 107 + +Falling bodies 56 + +Finite differences|EtSeq 88 % et seq. + +Fluxions 11, 60, 112 + +Force 61-63 + +Functions + definition of|EtSeq 14 % et seq.; + derived|EtSeq 19 % et seq., + derived 21 + direct and indirect 97 + implicit and explicit 107, 108 + inverse|EtSeq 102 % et seq.; + of several variables|EtSeq 78 % et seq.; + recapitulation of results in the theory of 74 + +Generally@\emph{Generally}, the word 16 + +Implicit + differentiation|EtSeq 94 % et seq.; + function 107, 108 + +Impulse 60 + +Increase without limit|EtSeq 5, 65 % et seq. + +Increment 16, 113 +\PageSep{144} + +Independent variables 106 + +Indirect function 97 + +Indivisibles + method of|EtSeq 127 % et seq.; + notion of, in mechanics|EtSeq 129 % et seq. + +Infinite@\emph{Infinite}, the word#Infinite 128 + +Infinitely small, the notion of#Infinitely 12, 49, 59, 83 + +Infinitely small, the notion of|EtSeq#Infinitely 38 % et seq., + +Infinity, orders of|EtSeq 42 % et seq. + +Integral Calculus 73 + notation of 119 + +Integral Calculus|EtSeq 115 % et seq. + +Integrals + definition of|EtSeq 119 % et seq.; + relations between differential coefficients and 121 + indefinite 122, 123 + +Intersections, limit of|EtSeq 46 % et seq. + +Inverse functions|EtSeq 102 % et seq. + +Iron bar continually varying in density, weight of|EtSeq#Iron 130 % et seq. + +Ladder against wall|EtSeq 45 % et seq. + +Lagrange 124 + +Laplace 124 + +Leibnitz 11, 13, 38, 42, 48, 59, 60, 83, 123, 124, 128, 129 + +Limit of intersections|EtSeq 46 % et seq. + +Limits|EtSeq 26 % et seq. + +Limiting ratios|EtSeq 65 % et seq. + +Limiting ratios 81 + +Logarithms 20, 38, 86, 87 + +Logarithms|EtSeq 112 % et seq. + +Magnified curve 40 + +Motion + accelerated 60 + simple harmonic 57 + +Newton 11, 60 + +Notation + of the Differential Calculus 25 + of the Differential Calculus|EtSeq 79 % et seq. + of the Integral Calculus 119 + +Orders, differential coefficients of higher 88 + +Orders of infinity|EtSeq 42 % et seq. + +Parabola, the#Parabola 30, 127 + +Parabola, the|EtSeq#Parabola 124 % et seq. + +Partial + differentials|EtSeq 78 % et seq.; + differential coefficients 96 + +Point@\emph{Point}, the word#Point 129 + +Points, the number of, in a straight line 129 + +Polygon 38 + +Proportion|EtSeq 2 % et seq. + +Quantities, continuous|EtSeq 7 % et seq. + +Quantities, continuous 53 + +Ratio + defined|EtSeq 2 % et seq.; + of two increments 87 + +Ratios, limiting|EtSeq 65 % et seq. + +Ratios, limiting 81 + +Rough methods of solution in the Integral Calculus 132, 133 + +Series|EtSeq 15, 24 % et seq. + +Signs|EtSeq 31 % et seq. + +Simple harmonic motion 57 + +Sines 87 + +Singular values 16 + +Small, has no precise meaning 12 + +Specific gravity, continuously varying|EtSeq 130 % et seq. + +Successive differentiation|EtSeq 88 % et seq. + +Sun's longitude 76 + +Tangent 37, 38, 40 + +Taylor's Theorem|EtSeq 15, 19 % et seq. + +Time, idea of#Time 4 + +Time, idea of|EtSeq#Time 110 % et seq. + +Total + differential coefficient 100 + differentials|EtSeq 78 % et seq.; + variations 95 + +Transit instrument 84 + +Uniformly accelerated 57, 60 + +Values + contiguous 112 + equidistant 104 + +Variables + independent and dependent 14, 15, 106 + functions of several|EtSeq 78 % et seq. + +Variations, total#Variations 95 + +Velocity + linear|EtSeq 52 % et seq. + linear 111 + angular 59 + +Weight of an iron bar of which the density varies from point to point|EtSeq#Iron 130 % et seq. +\fi +\PageSep{145} + +\iffalse +%[** TN: Catalog text has been (lightly) proofread, but not marked up in LaTeX] + +CATALOGUE OF PUBLICATIONS +OF THE +OPEN COURT PUBLISHING CO. + +COPE, E. 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