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| author | Roger Frank <rfrank@pglaf.org> | 2025-10-14 20:06:13 -0700 |
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| committer | Roger Frank <rfrank@pglaf.org> | 2025-10-14 20:06:13 -0700 |
| commit | 76248a42fb61ecb72a001f5328e53658067305d8 (patch) | |
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diff --git a/.gitattributes b/.gitattributes new file mode 100644 index 0000000..6833f05 --- /dev/null +++ b/.gitattributes @@ -0,0 +1,3 @@ +* text=auto +*.txt text +*.md text diff --git a/36640-pdf.pdf b/36640-pdf.pdf Binary files differnew file mode 100644 index 0000000..eae58a1 --- /dev/null +++ b/36640-pdf.pdf diff --git a/36640-pdf.zip b/36640-pdf.zip Binary files differnew file mode 100644 index 0000000..c881b5d --- /dev/null +++ b/36640-pdf.zip diff --git a/36640-t.zip b/36640-t.zip Binary files differnew file mode 100644 index 0000000..70fa9ff --- /dev/null +++ b/36640-t.zip diff --git a/36640-t/36640-t.tex b/36640-t/36640-t.tex new file mode 100644 index 0000000..7c7a601 --- /dev/null +++ b/36640-t/36640-t.tex @@ -0,0 +1,8810 @@ +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % +% % +% The Project Gutenberg EBook of Lectures on Elementary Mathematics, by % +% Joseph Louis Lagrange % +% % +% This eBook is for the use of anyone anywhere at no cost and with % +% almost no restrictions whatsoever. You may copy it, give it away or % +% re-use it under the terms of the Project Gutenberg License included % +% with this eBook or online at www.gutenberg.org % +% % +% % +% Title: Lectures on Elementary Mathematics % +% % +% Author: Joseph Louis Lagrange % +% % +% Translator: Thomas Joseph McCormack % +% % +% Release Date: July 6, 2011 [EBook #36640] % +% Most recently updated: June 11, 2021 % +% % +% Language: English % +% % +% Character set encoding: UTF-8 % +% % +% *** START OF THIS PROJECT GUTENBERG EBOOK LECTURES ON ELEMENTARY MATHEMATICS *** +% % +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % + +\def\ebook{36640} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %% +%% Packages and substitutions: %% +%% %% +%% book: Required. %% +%% inputenc: Latin-1 text encoding. Required. %% +%% %% +%% babel: Greek snippets. Required. %% +%% %% +%% ifthen: Logical conditionals. 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+\begin{document} +%% PG BOILERPLATE %% +\PGBoilerPlate +\begin{center} +\begin{minipage}{\textwidth} +\small +\begin{PGtext} +The Project Gutenberg EBook of Lectures on Elementary Mathematics, by +Joseph Louis Lagrange + +This eBook is for the use of anyone anywhere at no cost and with +almost no restrictions whatsoever. You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.org + + +Title: Lectures on Elementary Mathematics + +Author: Joseph Louis Lagrange + +Translator: Thomas Joseph McCormack + +Release Date: July 6, 2011 [EBook #36640] +Most recently updated: June 11, 2021 + +Language: English + +Character set encoding: UTF-8 + +*** START OF THIS PROJECT GUTENBERG EBOOK LECTURES ON ELEMENTARY MATHEMATICS *** +\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} +\end{center} +\vfill + +\begin{minipage}{0.85\textwidth} +\small +\BookMark{0}{Transcriber's Note.} +\subsection*{\centering\normalfont\scshape% +\normalsize\MakeLowercase{\TransNote}}% + +\raggedright +\TransNoteText +\end{minipage} +%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%% +\FrontMatter +\null\vfill +\noindent {\Large ON ELEMENTARY MATHEMATICS} +\vfill +\PageSep{} +\FrontCatalog{IN THE SAME SERIES.} + +\tb + +\Book{ON CONTINUITY AND IRRATIONAL NUMBERS, and +ON THE NATURE AND MEANING OF NUMBERS\@. +By R.~\textsc{Dedekind}. From the German by \textit{W.~W. Beman}. +Pages,~115. Cloth, 75~cents net (3s.~6d.~net).} + +\Book{GEOMETRIC EXERCISES IN PAPER-FOLDING\@. By \textsc{T.~Sundara Row}. +Edited and revised by \textit{W.~W. Beman} and +\textit{D.~E. Smith}. With many half-tone engravings from photographs +of actual exercises, and a package of papers for +folding. Pages, circa~200. Cloth, \$1.00\Typo{.}{} net (4s.~6d.~net). +(In Preparation.)} + +\Book{ON THE STUDY AND DIFFICULTIES OF MATHEMATICS\@. +By \textsc{Augustus De~Morgan}. Reprint edition\Typo{`}{} +with portrait and bibliographies. Pp.,~288. Cloth, \$1.25 +net (4s.~6d.~net).} + +\Book{LECTURES ON ELEMENTARY MATHEMATICS\@. By +\textsc{Joseph Louis Lagrange}. From the French by \textit{Thomas~J. +McCormack}, With portrait and biography. Pages,~172. +Cloth, \$1.00 net (4s.~6d.~net).} + +\Book{ELEMENTARY ILLUSTRATIONS OF THE DIFFERENTIAL +AND INTEGRAL CALCULUS\@. By \textsc{Augustus De~Morgan}. +Reprint edition. With a bibliography of text-books +of the Calculus. Pp.,~144. Price, \$1.00 net (4s.~6d.~net).} + +\Book{MATHEMATICAL ESSAYS AND RECREATIONS\@. By +\textsc{Prof.\ Hermann Schubert}, of Hamburg, Germany. From +the German by \textit{T.~J. McCormack}. Essays on Number\Typo{.}{,} +The Magic Square, The Fourth Dimension, The Squaring +of the Circle. Pages,~149. Price, Cloth, 75c.~net (3s.~net).} + +\Book{A BRIEF HISTORY OF ELEMENTARY MATHEMATICS\@. +By \textsc{Dr.\ Karl Fink}, of Tübingen. From the German by \textit{W.~W. +Beman} and \textit{D.~E. Smith}. Pp.~333. Cloth, \$1.50 net +(5s.~6d.~net).} + +\tb +\vfill + +\noindent\makebox[\textwidth][s]{\large THE OPEN COURT PUBLISHING COMPANY} +\begin{center} +\footnotesize 324 DEARBORN ST., CHICAGO. \\ +\normalsize LONDON: Kegan Paul, Trench, Trübner \& Co. +\end{center} +\PageSep{i} +%[Blank page] +\PageSep{ii} +\Frontispiece +\PageSep{iii} +\begin{center} +\Large LECTURES\\[24pt] +\footnotesize ON\\[24pt] +\LARGE ELEMENTARY MATHEMATICS +\vfill + +\footnotesize BY\\[18pt] +\large JOSEPH LOUIS LAGRANGE +\vfill + +\footnotesize FROM THE FRENCH BY\\[18pt] +\normalsize THOMAS J. McCORMACK +\vfill\vfill + +\small SECOND EDITION +\vfill\vfill + +\large CHICAGO \\ +\normalsize THE OPEN COURT PUBLISHING COMPANY \\[12pt] +\footnotesize LONDON AGENTS \\ +\textsc{Kegan Paul, Trench, Trübner \& Co., Ltd.} \\ +1901 +\end{center} +\newpage +\PageSep{iv} +\null\vfill +\begin{center} +\footnotesize TRANSLATION COPYRIGHTED \\ +BY \\ +\small\textsc{The Open Court Publishing Co.} \\ +1898. +\end{center} +\vfill +\PageSep{v} + + +\Preface + +\First{The} present work, which is a translation of the \textit{Leçons élémentaires +sur les \Typo{mathematiques}{mathématiques}} of Joseph Louis Lagrange, +\index{Lagrange, J. L.}% +the greatest of modern analysts, and which is to be found in Volume~VII. +of the new edition of his collected works, consists of a +series of lectures delivered in the year 1795 at the \textit{\Typo{Ecole}{École} Normale},---an +institution which was the direct outcome of the French Revolution +and which gave the first impulse to modern practical +ideals of education. With Lagrange, at this institution, were associated, +as professors of mathematics. Monge and Laplace, and we +\index{Laplace}% +\index{Monge}% +owe to the same historical event the final form of the famous \textit{Géométrie +descriptive}, as well as a second course of lectures on arithmetic +and algebra, introductory to these of Lagrange, by Laplace. + +With the exception of a German translation by Niedermüller +\index{Ecole@{\Typo{Ecole}{École} Normale}}% +(Leipsic, 1880), the lectures of Lagrange have never been published +in separate form; originally they appeared in a fragmentary +shape in the \textit{Séances des \Typo{Ecoles}{Écoles} Normales}, as they had been reported +by the stenographers, and were subsequently reprinted in +the journal of the Polytechnic School. From references in them +\index{Polytechnic School}% +to subjects afterwards to be treated it is to be inferred that a fuller +development of higher algebra was intended,---an intention which +the brief existence of the \textit{\Typo{Ecole}{École} Normale} defeated. With very few +exceptions, we have left the expositions in their historical form, +having only referred in an Appendix to a point in the early history +of algebra. + +The originality, elegance, and symmetrical character of these +lectures have been pointed out by \Typo{DeMorgan}{De~Morgan}, and notably by Dühring, +\index{DeMorgan@{\Typo{DeMorgan}{De Morgan}}}% +\index{Duhring@{Dühring, E.}}% +who places them in the front rank of elementary expositions, +as an exemplar of their kind. Coming, as they do, from one of +the greatest mathematicians of modern times, and with all the excellencies +which such a source implies, unique in their character +\PageSep{vi} +as a \emph{reading-book} in mathematics, and interwoven with historical +and philosophical remarks of great helpfulness, they cannot fail +to have a beneficent and stimulating influence\Typo{,}{.} + +The thanks of the translator of the present volume are due to +Professor Henry~B. Fine, of Princeton, N.~J., for having read the +proofs. + +\Signature{Thomas J. McCormack.} +{\textsc{La Salle, Illinois}, August~1, 1898.} +\PageSep{vii} + + +\BioSketch{Joseph Louis Lagrange.} +{Biographical Sketch.} +\index{Economy of thought}% +\index{Lagrange, J. L.|EtSeq}% +\index{Short-mind symbols|EtSeq}% +\index{Stenophrenic symbols|EtSeq}% +\index{Symbols|EtSeq}% + +\First{A great} part of the progress of formal thought, where it is +not hampered by outward causes, has been due to the invention +of what we may call \emph{stenophrenic}, or \emph{short-mind}, symbols. +These, of which all written language and scientific notations are +examples, disengage the mind from the consideration of ponderous +and circuitous mechanical operations and economise its energies +for the performance of new and unaccomplished tasks of thought. +And the advancement of those sciences has been most notable +which have made the most extensive use of these short-mind symbols. +Here mathematics and chemistry stand pre-eminent. The +\index{Greeks, mathematics of the}% +\index{Mathematics!evolution of}% +ancient Greeks, with all their mathematical endowment as a race, +and even admitting that their powers were more visualistic than +analytic, were yet so impeded by their lack of short-mind symbols +as to have made scarcely any progress whatever in analysis. Their +arithmetic was a species of geometry. They did not possess the +sign for zero, and also did not make use of position as an indicator +of value. Even later, when the germs of the indeterminate analysis +were disseminated in Europe by Diophantus, progress ceased +here in the science, doubtless from this very cause. The historical +\index{Science!development of|EtSeq}% +calculations of Archimedes, his approximation to the value of~$\pi$,~etc, +owing to this lack of appropriate arithmetical and algebraical +symbols, entailed enormous and incredible labors, which, if +they had been avoided, would, with his genius, indubitably have +led to great discoveries. +\PageSep{viii} + +Subsequently, at the close of the Middle Ages, when the so-called +Arabic figures became established throughout Europe with +the symbol~$0$ and the principle of local value, immediate progress +was made in the art of reckoning. The problems which arose +gave rise to questions of increasing complexity and led up to the +general solutions of equations of the third and fourth degree by +the Italian mathematicians of the sixteenth century. Yet even +these discoveries were made in somewhat the same manner as +problems in mental arithmetic are now solved in common schools; +for the present signs of plus, minus, and equality, the radical and +exponential signs, and especially the systematic use of letters for +denoting general quantities in algebra, had not yet become universal. +The last step was definitively due to the French mathematician +Vieta (1540--1603), and the mighty advancement of analysis +\index{Vieta}% +resulting therefrom can hardly be measured or imagined. The +trammels were here removed from algebraic thought, and it ever +afterwards pursued its way unincumbered in development as if impelled +by some intrinsic and irresistible potency. Then followed +the introduction of exponents by Descartes, the representation of +\index{Descartes}% +geometrical magnitudes by algebraical symbols, the extension of +the theory of exponents to fractional and negative numbers by +Wallis (1616--1703), and other symbolic artifices, which rendered +\index{Wallis}% +the language of analysis as economic, unequivocal, and appropriate +as the needs of the science appeared to demand. In the famous +dispute regarding the invention of the infinitesimal calculus, while +not denying and even granting for the nonce the priority of Newton +\index{Newton, his problem}% +in the matter, some writers have gone so far as to regard Leibnitz's +\index{Leibnitz}% +introduction of the integral symbol~$\int$ as alone a sufficient substantiation +of his claims to originality and independence, so far as the +power of the new science was concerned. + +For the \emph{development} of science all such short-mind symbols +are of paramount importance, and seem to carry within themselves +the germ of a perpetual mental motion which needs no outward +power for its unfoldment. Euler's well-known saying that his +\index{Euler}% +\PageSep{ix} +pencil seemed to surpass him in intelligence finds its explanation +here, and will be understood by all who have experienced the uncanny +feeling attending the rapid development of algebraical formulæ, +where the urned thought of centuries, so to speak, rolls from +one's finger's ends. + +But it should never be forgotten that the mighty stenophrenic +engine of which we here speak, like all machinery, affords us rather +a mastery over nature than an insight into it; and for some, unfortunately, +the higher symbols of mathematics are merely brambles +that hide the living springs of reality. Many of the greatest +discoveries of science,---for example, those of Galileo, Huygens, +\index{Galileo}% +\index{Huygens}% +and Newton,---were made without the mechanism which afterwards +becomes so indispensable for their development and application. +Galileo's reasoning anent the summation of the impulses imparted +to a falling stone is virtual integration; and Newton's mechanical +discoveries were made by the man who invented, but evidently did +not use to that end, the doctrine of fluxions. +\stars + +We have been following here, briefly and roughly, a line of +progressive abstraction and generalisation which even in its beginning +was, psychologically speaking, at an exalted height, but in the +course of centuries had been carried to points of literally ethereal +refinement and altitude. In that long succession of inquirers by +whom this result was effected, the process reached, we may say, +its culmination and purest expression in Joseph Louis Lagrange, +born in Turin, Italy, the 30th~of January,~1736, died in Paris, April~10, +1813. Lagrange's power over symbols has, perhaps, never been +paralleled either before his day or since. It is amusing to hear his +biographers relate that in early life he evinced no aptitude for +mathematics, but seemed to have been given over entirely to the +pursuits of pure literature; for at fifteen we find him teaching +mathematics in an artillery school in Turin, and at nineteen he +had made the greatest discovery in mathematical science since that +of the infinitesimal calculus, namely, the creation of the algorism +\PageSep{x} +\index{Variations, calculus of}% +and method of the Calculus of Variations. ``Your analytical solution +of the isoperimetrical problem,'' writes Euler, then the prince +\index{Euler}% +of European mathematicians, to him, ``leaves nothing to be desired +in this department of inquiry, and I am delighted beyond measure +that it has been your lot to carry to the highest pitch of perfection, +a theory, which since its inception I have been almost the only one +to cultivate.'' But the exact nature of a ``variation'' even Euler +did not grasp, and even as late as~1810 in the English treatise of +Woodhouse on this subject we read regarding a certain new sign +\index{Woodhouse}% +introduced, that M.~Lagrange's ``power over symbols is so unbounded +that the possession of it seems to have made him capricious.'' + +Lagrange himself was conscious of his wonderful capacities in +this direction. His was a time when geometry, as he himself +phrased it, had become a dead language, the abstractions of analysis +were being pushed to their highest pitch, and he felt that with +his achievements its possibilities within certain limits were being +rapidly exhausted. The saying is attributed to him that chairs of +mathematics, so far as creation was concerned, and unless new +fields were opened up, would soon be as rare at universities as +chairs of Arabic. In both research and exposition, he totally reversed +the methods of his predecessors. They had proceeded in +their exposition from special cases by a species of induction; his +eye was always directed to the highest and most general points of +view; and it was by his suppression of details and neglect of minor, +unimportant considerations that he swept the whole field of analysis +with a generality of insight and power never excelled, adding +to his originality and profundity a conciseness, elegance, and lucidity +which have made him the model of mathematical writers. +\stars + +Lagrange came of an old French family of Touraine, France, +said to have been allied to that of Descartes. At the age of twenty-six +he found himself at the zenith of European fame. But his +reputation had been purchased at a great cost. Although of ordinary +\PageSep{xi} +height and well proportioned, he had by his ecstatic devotion +to study,---periods always accompanied by an irregular pulse and +high febrile \Typo{excitatian}{excitation},---almost ruined his health. At this age, +accordingly, he was seized with a hypochondriacal affection and +with bilious disorders, which accompanied him \Typo{thronghout}{throughout} his life, +and which were only allayed by his great abstemiousness and careful +regimen. He was bled twenty-nine times, an infliction which +alone would have affected the most robust constitution. Through +his great care for his health he gave much attention to medicine. +He was, in fact, conversant with all the sciences, although knowing +his \textit{forte} he rarely expressed an opinion on anything unconnected +with mathematics. + +When Euler left Berlin for St.~Petersburg in~1766 he and +D'Alembert induced Frederick the Great to make Lagrange president +of the Academy of Sciences at Berlin. Lagrange accepted +the position and lived in Berlin twenty years, where he wrote some +of his greatest works. He was a great favorite of the Berlin people, +and enjoyed the profoundest respect of Frederick the Great, +although the latter seems to have preferred the noisy reputation of +Maupertuis, Lamettrie, and Voltaire to the unobtrusive fame and +personality of the man whose achievements were destined to shed +more lasting light on his reign than those of any of his more strident +literary predecessors: Lagrange was, as he himself said, \textit{philosophe +sans crier}. + +The climate of Prussia agreed with the mathematician. He +refused the most seductive offers of foreign courts and princes, and +it was not until the death of Frederick and the intellectual reaction +of the Prussian court that he returned to Paris, where his career +broke forth in renewed splendor. He published in~1788 his great +\textit{Mécanique analytique}, that ``scientific poem'' of Sir William +Rowan Hamilton, which gave the quietus to mechanics as then +formulated, and having been made during the Revolution Professor +of Mathematics at the new \textit{\Typo{Ecole}{École} Normale} and the \textit{\Typo{Ecole}{École} Polytechnique}, +\index{Ecole@{\Typo{Ecole}{École} Normale}}% +\index{Polytechnic School}% +he entered with Laplace and Monge upon the activity +\index{Laplace}% +\index{Monge}% +\PageSep{xii} +which made these schools for generations to come exemplars of +practical scientific education, systematising by his lectures there, +and putting into definitive form, the science of mathematical analysis +of which he had developed the extremest capacities. Lagrange's +activity at Paris was interrupted only once by a brief period +of melancholy aversion for mathematics, a lull which he +devoted to the adolescent science of chemistry and to philosophical +studies; but he afterwards resumed his old love with increased ardor +and assiduity. His significance for thought generally is far +beyond what we have space to insist upon. Not least of all, theology, +which had invariably mingled itself with the researches of his +predecessors, was with him forever divorced from a legitimate influence +of science. + +The honors of the world sat ill upon Lagrange: \textit{la magnificence +le gênait}, he said; but he lived at a time when proffered +things were usually accepted, not refused. He was loaded with +personal favors and official distinctions by Napoleon who called +\index{Napoleon}% +him \textit{la haute pyramide des sciences mathématiques}, was made a +Senator, a Count of the Empire, a Grand Officer of the Legion of +Honor, and, just before his death, received the grand cross of the +Order of Reunion. He never feared death, which he termed \textit{une +dernière fonction, ni pénible ni désagréable}, much less the disapproval +of the great. He remained in Paris during the Revolution +when \textit{savants} were decidedly in disfavor, but was suspected +of aspiring to no throne but that of mathematics. When Lavoisier +\index{Lavoisier}% +was executed he said: ``It took them but a moment to lay low that +head; yet a hundred years will not suffice perhaps to produce its +like again.'' + +Lagrange would never allow his portrait to be painted, maintaining +that a man's works and not his personality deserved preservation. +The frontispiece to the present work is from a steel +engraving based on a sketch obtained by stealth at a meeting of +the Institute. His genius was excelled only by the purity and +nobleness of his character, in which the world never even sought +\PageSep{xiii} +to find a blot, and by the exalted Pythagorean simplicity of his +life. He was twice married, and by his wonderful care of his person +lived to the advanced age of seventy-seven years, not one of +which had been misspent. His life was the veriest incarnation of +the scientific spirit; he lived for nothing else. He left his weak +body, which retained its intellectual powers to the very last, as an +offering upon the altar of science, happily made when his work +had been done; but to the world he bequeathed his ``ever-living'' +thoughts now recently resurgent in a new and monumental edition +of his works (published by Gauthier-Villars, Paris). \textit{Ma vie est +là !} he said, pointing to his brain the day before his death. + +\Signature{Thomas J. McCormack.}{} +\PageSep{xiv} +%[Blank page] +\PageSep{xv} +\TableofContents +\iffalse +%[** TN: Used marginal notes to generate entries; entries in original ToC +% don't obviously match the book's units.] +CONTENTS. + +PAGES + +Preface + +Biographical Sketch of Joseph Louis LaGrange. + +Lecture I. On Arithmetic, and in Particular Fractions +and Logarithms. 1-23 + +Systems of Numeration. Fractions. Greatest Common +Divisor. Continued Fractions. Theory of +Powers, Proportions, and Progressions. Involution +and Evolution. Rule of Three. Interest. Annuities. +Logarithms. + +Lecture II. On the Operations of Arithmetic . . . 24-53 + +Arithmetic and Geometry. New Method of Subtraction. +Abridged and Approximate Multiplication. +Decimals. Property of the Number 9. +Tests of Divisibility. Theory of Remainders. +Checks on Multiplication and Division. Evolution. +Rule of Three. Theory and Practice. Probability +of Life. Alligation or the Rule of Mixtures. + +Lecture III. On Algebra, Particularly the Resolution +of Equations of the Third and Fourth Degree 54-95 + +Origin of Greek Algebra. Diophantus. Indeterminate +Analysis. Equations of the Second Degree. +Translations of Diophantus. Algebra Among the +Arabs. History of Algebra in Italy, France, and +Germany. History of Equations of the Third and +Fourth Degree and of the Irreducible Case. Theory +of Equations. Discussion of Cubic Equations. +Discussion of the Irreducible Case. The Theory +\PageSep{xvi} +of Roots. Extraction of the Square and Cube Roots +of Two Imaginary Binomials. Theory of Imaginary +Expressions. Trisection of an Angle. Method +of Indeterminates. Discussion of Biquadratic Equations. + +Lecture IV. On the Resolution of Numerical Equations ... 96-126 + +Algebraical Resolution of Equations. Numerical +Resolution of Equations. Position of the Roots. +Representation of Equations by Curves. Graphic +Resolution of Equations. Character of the Roots of +Equations. Limits of the Roots of Numerical Equations. +Separation of the Roots. Method of Substitutions. +The Equation of Differences. Method of +Elimination. Constructions and Instruments for +Solving Equations. + +Lecture V. On the Employment of Curves in the Solution +of Problems 127-149 + +Application of Geometry to Algebra. Resolution of +Problems by Curves. The Problem of Two Lights. +Variable Quantities Minimal Values. Analysis +of Biquadratic Equations Conformably to the Problem +of the Two Lights. Advantages of the Method +of Curves The Curve of Errors. \textit{Regula falsi.} +Solution of Problems by the Curve of Errors. +Problem of the Circle and Inscribed Polygon. +Problem of the Observer and Three Objects. Parabolic +Curves. Newton's Problem. Interpolation +of Intermediate Terms in Series of Observations, +Experiments, etc. + +Appendix . 151 + +Note on the Origin of Algebra. +\fi +\PageSep{1} +\MainMatter +\index{Numerical equations|See{Equations}}% + + +\Lecture[On Arithmetic.]{I.}{On Arithmetic, and in Particular Fractions +and Logarithms.} + +\First{Arithmetic} is divided into two parts. The first +is based on the decimal system of notation and +\MNote{Systems of numeration\Add{.}} +on the manner of arranging numeral characters to express +numbers. This first part comprises the four +common operations of addition, subtraction, multiplication, +and division,---operations which, as you +know, would be different if a different system were +adopted, but, which it would not be difficult to transform +from one system to another, if a change of systems +were desirable. + +The second part is independent of the system of +\index{Numeration, systems of}% +numeration. It is based on the consideration of quantities +and on the general properties of numbers. The +theory of fractions, the theory of powers and of roots, +the theory of arithmetical and geometrical progressions, +and, lastly, the theory of logarithms, fall under +this head. I purpose to advance, here, some remarks +on the different branches of this part of arithmetic. +\PageSep{2} + +It may be regarded as \emph{universal arithmetic}, having +\index{Arithmetic!universal|EtSeq}% +an intimate affinity to algebra. For, if instead of +\index{Algebra!definition of}% +particularising the quantities considered, if instead of +assigning them numerically, we treat them in quite a +general way, designating them by letters, we have +algebra. + +You know what a fraction is. The notion of a +\index{Fractions|EtSeq}% +\index{Ratios, constant|EtSeq}% +\MNote{Fractions.} +fraction is slightly more composite than that of whole +numbers. In whole numbers we consider simply a +quantity repeated. To reach the notion of a fraction +it is necessary to consider the quantity divided into a +certain number of parts. Fractions represent in general +ratios, and serve to express one quantity by means +of another. In general, nothing measurable can be +measured except by fractions expressing the result of +the measurement, unless the measure be contained an +exact number of times in the thing to be measured. + +You also know how a fraction can be reduced to +\index{Divisor, greatest common|EtSeq}% +its lowest terms. When the numerator and the denominator +are both divisible by the same number, +their greatest common divisor can be found by a very +ingenious method which we owe to Euclid. This +\index{Euclid}% +method is exceedingly simple and lucid, but it may +be rendered even more palpable to the eye by the following +consideration. Suppose, for example, that you +have a given length, and that you wish to measure it. +The unit of measure is given, and you wish to know +how many times it is contained in the length. You +first lay off your measure as many times as you can on +\PageSep{3} +the given length, and that gives you a certain whole +number of measures. If there is no remainder your +operation is finished. But if there be a remainder, +\MNote{Greatest common divisor.} +that remainder is still to be evaluated. If the measure +is divided into equal parts, for example, into ten, +twelve, or more equal parts, the natural procedure is +to use one of these parts as a new measure and to see +how many times it is contained in the remainder. +You will then have for the value of your remainder, +a fraction of which the numerator is the number of +parts contained in the remainder and the denominator +the total number of parts into which the given measure +is divided. + +I will suppose, now, that your measure is not so +divided but that you still wish to determine the ratio +of the proposed length to the length which you have +adopted as your measure. The following is the procedure +which most naturally suggests itself. + +If you have a remainder, since that is less than the +\index{Fractions!continued|EtSeq}% +measure, naturally you will seek to find how many +times your remainder is contained in this measure. +Let us say two times, and that a remainder is still +left. Lay this remainder on the preceding remainder. +Since it is necessarily smaller, it will still be contained +a certain number of times in the preceding remainder, +say three times, and there will be another remainder +or there will not; and so on. In these different remainders +you will have what is called a \emph{continued fraction}. +For example, you have found that the measure +\PageSep{4} +is contained three times in the proposed length. You +have, to start with, the number \emph{three}. Then you have +\MNote{Continued fractions.} +found that your first remainder is contained twice in +your measure. You will have the fraction \emph{one} divided +by \emph{two}. But this last denominator is not complete, +for it was supposed there was still a remainder. That +remainder will give another and similar fraction, which +is to be added to the last denominator, and which by +our supposition is \emph{one} divided by \emph{three}. And so with +the rest. You will then have the fraction +\[ +3 + \cfrac{1}{2 + \cfrac{1}{3 + \ddots}} +\] +as the expression of your ratio between the proposed +length and the adopted measure. + +Fractions of this form are called \emph{continued fractions}, +and can be reduced to ordinary fractions by the common +rules. Thus, if we stop at the first fraction, i.e., +if we consider only the first remainder and neglect the +second, we shall have $3 + \frac{1}{2}$, which is equal to~$\frac{7}{2}$. Considering +only the first and the second remainders, we +stop at the second fraction, and shall have $3 + \dfrac{1}{2 + \frac{1}{3}}$. +Now $2 + \frac{1}{3} = \frac{7}{3}$. We shall have therefore $3 + \frac{3}{7}$, which +is equal to~$\frac{24}{7}$. And so on with the rest. If we arrive +in the course of the operation at a remainder which is +contained exactly in the preceding remainder, the +operation is terminated, and we shall have in the continued +\PageSep{5} +fraction a common fraction that is the exact +value of the length to be measured, in terms of the +length which served as our measure. If the operation +\MNote{Terminating continued fractions.} +is not thus terminated, it can be continued to infinity, +and we shall have only fractions which approach more +and more nearly to the true value. + +If we now compare this procedure with that employed +for finding the greatest common divisor of two +numbers, we shall see that it is virtually the same +thing; the difference being that in finding the greatest +common divisor we devote our attention solely to +the different remainders, of which the last is the divisor +sought, whereas by employing the successive +quotients, as we have done above, we obtain fractions +which constantly approach nearer and nearer to the +fraction formed by the two numbers given, and of +which the last is that fraction itself reduced to its +lowest terms. + +As the theory of continued fractions is little known, +but is yet of great utility in the solution of important +numerical questions, I shall enter here somewhat +more fully into the formation and properties of these +fractions. And, first, let us suppose that the quotients +found, whether by the mechanical operation, or by +the method for finding the greatest common divisor, +are, as above, $3$,~$2$, $3$, $5$, $7$,~$3$. The following is a rule +by which we can write down at once the convergent +fractions which result from these quotients, without +developing the continued fraction. +\PageSep{6} + +The first quotient, supposed divided by unity, +will give the first fraction, which will be too small, +\MNote{Converging fractions.} +\index{Fractions!converging}% +namely,~$\frac{3}{1}$. Then, multiplying the numerator and denominator +of this fraction by the second quotient and +adding unity to the numerator, we shall have the second +fraction,~$\frac{7}{2}$, which will be too large. Multiplying +in like manner the numerator and denominator of this +fraction by the third quotient, and adding to the numerator +the numerator of the preceding fraction, and +to the denominator the denominator of the preceding +fraction, we shall have the third fraction, which will +be too small. Thus, the third quotient being~$3$, we +have for our numerator $(7 × 3 = 21) + 3 = 24$, and for +our denominator $(2 × 3 = 6) + 1 = 7$. The third convergent, +therefore, is~$\frac{24}{7}$. We proceed in the same +manner for the fourth convergent. The fourth quotient +being~$5$, we say $24$~times~$5$ is~$120$, and this plus~$7$, +the numerator of the fraction preceding, is~$127$; +similarly, $7$~times~$5$ is~$35$, and this plus~$2$ is~$37$. The +new fraction, therefore, is~$\frac{127}{37}$. And so with the rest. + +In this manner, by employing the six quotients $3$,~$2$, +$3$, $5$, $7$,~$3$ we obtain the six fractions +\[ +\frac{3}{1},\quad +\frac{7}{2},\quad +\frac{24}{7},\quad +\frac{127}{37},\quad +\frac{913}{266},\quad +\frac{2866}{835}, +\] +of which the last, supposing the operation to be completed +at the sixth quotient~$3$, will be the required +value of the length measured, or the fraction itself +reduced to its lowest terms. + +The fractions which precede the last are alternately +\PageSep{7} +smaller and larger than the last, and have the advantage +of approaching more and more nearly to its value +in such wise that no other fraction can approach it +\MNote{Convergents.} +\index{Convergents}% +more nearly except its denominator be larger than the +product of the denominator of the fraction in question +and the denominator of the fraction following. For +example, the fraction~$\frac{24}{7}$ is less than the true value +which is that of the fraction~$\frac{2866}{835}$, but it approaches +to it more nearly than any other fraction does whose +denominator is not greater than the product of~$7$ by~$37$, +that is,~$259$. Thus, any fraction expressed in large +numbers may be reduced to a series of fractions expressed +in smaller numbers and which approach as +near to it as possible in value. + +The demonstration of the foregoing properties is +deduced from the nature of continued fractions, and +from the fact that if we seek the difference between +one of the convergent fractions and that next adjacent +to it we shall obtain a fraction of which the numerator +is always unity and the denominator the product of +the two denominators; a consequence which follows +\textit{\Typo{a}{à }~priori} from the very law of formation of these fractions. +Thus the difference between $\frac{7}{2}$~and~$\frac{3}{1}$ is~$\frac{1}{2}$, in +excess; between $\frac{24}{7}$~and~$\frac{7}{2}$, $\frac{1}{14}$,~in defect; between $\frac{127}{37}$ +and~$\frac{24}{7}$, $\frac{1}{259}$,~in excess; and so on. The result being, +that by employing this series of differences we can +express in another and very simple manner the fractions +with which we are here concerned, by means of +a second series of fractions of which the numerators +\PageSep{8} +are all unity and the denominators successively the +products of every two adjacent denominators. Instead +\MNote{A second method of expression.} +of the fractions written above, we have thus the +series: +\[ +\frac{3}{1} + \frac{1}{1 × 2} + - \frac{1}{2 × 7} + + \frac{1}{7 × 37} + - \frac{1}{37 × 266} + + \frac{1}{266 × 835}. +\] + +The first term, as we see, is the first fraction, the +first and second together give the second fraction~$\frac{7}{2}$, +the first, the second, and the third give the third fraction~$\frac{24}{7}$, +and so on with the rest; the result being that +the series entire is equivalent to the last fraction. + +There is still another way, less known but in some +respects more simple, of treating the same question---which +leads directly to a series similar to the preceding. +Reverting to the previous example, after having +found that the measure goes three times into the length +to be measured and that after the first remainder has +been applied to the measure there is left a new remainder, +instead of comparing this second remainder +with the preceding, as we did above, we may compare +it with the measure itself. Thus, supposing it goes +into the latter seven times with a remainder, we again +compare this last remainder with the measure, and so +on, until we arrive, if possible, at a remainder which +is an aliquot part of the measure,---which will terminate +the operation. In the contrary event, if the +measure and the length to be measured are incommensurable, +the process may be continued to infinity. +\PageSep{9} +We shall have then, as the expression of the length +measured, the series +\MNote{A third method of expression.} +\[ +3 + \frac{1}{2} - \frac{1}{2 × 7} + \ldots. +\] + +It is clear that this method is also applicable to +ordinary fractions. We constantly retain the denominator +of the fraction as the dividend, and take the different +remainders successively as divisors. Thus, the +fraction~$\frac{2866}{835}$ gives the quotients $3$,~$2$, $7$, $18$, $19$, $46$, +$119$, $417$\Typo{}{,}~$835$; from which we obtain the series +\[ +3 + \frac{1}{2} - \frac{1}{2 × 7} + + \frac{1}{2 × 7 × 18} + - \frac{1}{2 × 7 × 18 × 19} + \ldots; +\] +and as these partial fractions rapidly diminish, we +shall have, by combining them successively, the simple +fractions, +\[ +\frac{7}{2},\quad +\frac{48}{2 × 7},\quad +\frac{865}{2 × 7 × 18}, \ldots, +\] +which will constantly approach nearer and nearer to +the true value sought, and the error will be less than +the first of the partial fractions neglected. + +Our remarks on the foregoing methods of evaluating +fractions should not be construed as signifying +that the employment of decimal fractions is not nearly +\index{Decimal!fractions}% +\index{Fractions!decimal}% +always preferable for expressing the values of fractions +to whatever degree of exactness we wish. But cases +occur where it is necessary that these values should +be expressed by as few figures as possible. For example, +if it were required to construct a planetarium, +\index{Planetarium}% +\PageSep{10} +since the ratios of the revolutions of the planets to one +another are expressed by very large numbers, it would +\MNote{Origin of continued fractions.} +\index{Fractions!origin of continued}% +be necessary, in order not to multiply unduly the +number of the teeth on the wheels, to avail ourselves +of smaller numbers, but at the same time so to select +them that their ratios should approach as nearly as +possible to the actual ratios. It was, in fact, this very +question that prompted Huygens, in his search for its +\index{Huygens}% +solution, to resort to continued fractions and that so +gave birth to the theory of these fractions. Afterwards, +in the elaboration of this theory, it was found +adapted to the solution of other important questions, +and this is the reason, since it is not found in elementary +works, that I have deemed it necessary to go +somewhat into detail in expounding its principles. + +We will now pass to the theory of powers, proportions, +and progressions. + +As you already know, a number multiplied by itself +\index{Powers|EtSeq}% +gives its square, and multiplied again by itself +gives its cube, and so on. In geometry we do not go +beyond the cube, because no body can have more than +three dimensions. But in algebra and arithmetic we +may go as far as we please. And here the theory of +the extraction of roots takes its origin. For, although +every number can be raised to its square and to its +cube and so forth, it is not true reciprocally that every +number is an exact square or an exact cube. The +number~$2$, for example, is not a square; for the square +of~$1$ is~$1$, and the square of~$2$ is four; and there being +\PageSep{11} +no other whole numbers between these two, it is impossible +to find a whole number which multiplied by +itself will give~$2$. It cannot be found in fractions, for +\MNote{Involution and evolution.} +\index{Evolution}% +\index{Involution and evolution}% +if you take a fraction reduced to its lowest terms, the +square of that fraction will again be a fraction reduced +to its lowest terms, and consequently cannot be equal +to the whole number~$2$. But though we cannot obtain +the square root of~$2$ exactly, we can yet approach to it +as nearly as we please, particularly by decimal fractions. +By following the common rules for the extraction +of square roots, cube roots, and so forth, the process +may be extended to infinity, and the true values +of the roots may be approximated to any degree of +exactitude we wish. + +But I shall not enter into details here. The theory +of powers has given rise to that of progressions, before +entering on which a word is necessary on proportions. + +Every fraction expresses a ratio. Having two equal +\index{Proportion|EtSeq}% +\index{Ratios, constant|EtSeq}% +fractions, therefore, we have two equal ratios; and +the numbers constituting the fractions or the ratios +form what is called a \emph{proportion}. Thus the equality +of the ratios $2$~to~$4$ and $3$~to~$6$ gives the proportion +$2 : 4 :: 3 : 6$, because $4$~is the double of~$2$ as $6$~is the +double of~$3$. Many of the rules of arithmetic depend +on the theory of proportions. First, it is the foundation +of the famous \emph{rule of three}, which is so extensively +\index{Rule!three@of three|EtSeq}% +used. You know that when the first three terms of a +proportion are given, to obtain the fourth you have +\PageSep{12} +only to multiply the last two together and divide the +product by the first. Various special rules have also +\MNote{Proportions\Add{.}} +been conceived and have found a place in the books +on arithmetic; but they are all reducible to the rule +of three and may be neglected if we once thoroughly +grasp the conditions of the problem. There are direct, +inverse, simple, and compound rules of three, rules of +partnership, of mixtures, and so forth. In all cases +it is only necessary to consider carefully the conditions +of the problem and to arrange the terms of the +proportion correspondingly. + +I shall not enter into further details here. There +\index{Progressions, theory of}% +is, however, another theory which is useful on numerous +occasions,---namely, the \emph{theory of progressions}. +When you have several numbers that bear the same +proportion to one another, and which follow one another +in such a manner that the second is to the first +as the third is to the second, as the fourth is to the +third, and so forth, these numbers form a progression. +I shall begin with an observation. + +The books of arithmetic and algebra ordinarily distinguish +between two kinds of progression, arithmetical +and geometrical, corresponding to the proportions +called arithmetical and geometrical. But the appellation +proportion appears to me extremely inappropriate +as applied to \emph{arithmetical proportion}. And as it +\index{Arithmetical proportion}% +is one of the objects of the \textit{École Normale} to rectify +\index{Ecole@{\Typo{Ecole}{École} Normale}}% +the language of science, the present slight digression +will not be considered irrelevant. +\PageSep{13} + +I take it, then, that the idea of proportion is already +well established by usage and that it corresponds solely +to what is called \emph{geometrical proportion}. When we +\index{Geometrical!proportion}% +\MNote{Arithmetical and geometrical proportions.} +speak of the proportion of the parts of a man's body, +of the proportion of the parts of an edifice,~etc.; when +we say that a plan should be reduced proportionately +in size,~etc.; in fact, when we say generally that one +thing is proportional to another, we understand by +proportion equality of ratios only, as in geometrical +proportion, and never equality of differences as in +arithmetical proportion. Therefore, instead of saying +\index{Equi-different numbers}% +that the numbers, $3$,~$5$, $7$,~$9$, are in arithmetical proportion, +because the difference between $5$~and~$3$ is the +same as that between $9$~and~$7$, I deem it desirable that +some other term should be employed, so as to avoid +all ambiguity. We might, for instance, call such numbers +\emph{equi-different}, reserving the name of \emph{proportionals} +for numbers that are in geometrical proportion, as $2$,~$4$, +$6$,~$8$,~etc. + +As for the rest, I cannot see why the proportion +called \emph{arithmetical} is any more arithmetical than that +which is called geometrical, nor why the latter is more +geometrical than the former. On the contrary, the +primitive idea of geometrical proportion is based on +arithmetic, for the notion of ratios springs essentially +from the consideration of numbers. + +Still, in waiting for these inappropriate designations +to be changed, I shall continue to make use of +them, as a matter of simplicity and convenience. +\PageSep{14} + +The theory of arithmetical progressions presents +few difficulties. Arithmetical progressions consist of +\MNote{Progressions.} +\index{Progressions, theory of}% +quantities which increase or diminish constantly by +the same amount. But the theory of geometrical progressions +is more difficult and more important, as a +large number of interesting questions depend upon it---for +example, all problems of compound interest, all +problems that relate to discount, and many others of +like nature. + +In general, quantities in geometrical proportion +are produced, when a quantity increases and the force +generating the increase, so to speak, is proportional +to that quantity. It has been observed that in countries +where the means of subsistence are easy of acquisition, +as in the first American colonies, the population +is doubled at the expiration of twenty years; if +it is doubled at the end of twenty years it will be quadrupled +at the end of forty, octupled at the end of sixty, +and so on; the result being, as we see, a geometrical +progression, corresponding to intervals of time in +arithmetical progression. It is the same with compound +interest. If a given sum of money produces, +at the expiration of a certain time, a certain sum, at +the end of double that time, the original sum will have +produced an equivalent additional sum, and in addition +the sum produced in the first space of time will, +in its proportion, likewise have produced during the +second space of time a certain sum; and so with the +rest. The original sum is commonly called the \emph{principal}, +\PageSep{15} +the sum produced the \emph{interest}, and the constant +\index{Interest}% +ratio of the principal to the interest per annum, the +\emph{rate}. Thus, the rate \emph{twenty} signifies that the interest +\MNote{Compound interest.} +is the twentieth part of the principal,---a rate which +is commonly called $5$~\emph{per cent.}, $5$~being the twentieth +part of~$100$. On this basis, the principal, at the end +of one year, will have increased by its one-twentieth +part; consequently, it will have been augmented in +the ratio of $21$~to~$20$. At the end of two years, it will +have been increased again in the same ratio, that is in +the ratio of $\frac{21}{20}$~multiplied by~$\frac{21}{20}$; at the end of three +years, in the ratio of $\frac{21}{20}$~multiplied twice by itself; and +so on. In this manner we shall find that at the end of +fifteen years it will almost have doubled itself, and that +at the end of fifty-three years it will have increased +tenfold. Conversely, then, since a sum paid now will +be doubled at the end of fifteen years, it is clear that +a sum not payable till after the expiration of fifteen +years is now worth only one-half its amount. This +is what is termed the \emph{present value} of a sum payable +\index{Present value}% +at the end of a certain time; and it is plain, that to +find that value, it is only necessary to divide the sum +promised by the fraction~$\frac{21}{20}$, or to multiply it by the +fraction~$\frac{20}{21}$, as many times as there are years for the +sum to run. In this way we shall find that a sum +payable at the end of fifty-three years, is worth at +present only one-tenth. From this it is evident what +little advantage is to be derived from surrendering the +absolute ownership of a sum of money in order to obtain +\PageSep{16} +the enjoyment of it for a period of only fifty +years, say; seeing that we gain by such a transaction +\MNote{Present values and annuities.} +\index{Annuities}% +only one-tenth in actual use, whilst we lose the ownership +of the property forever. + +In \emph{annuities}, the consideration of interest is combined +with that of the probability of life; and as +every one is prone to believe that he will live very +long, and as, on the other hand, one is apt to under-*estimate +the value of property which must be abandoned +on death, a peculiar temptation arises, when +one is without children, to invest one's fortune, wholly +or in part, in annuities. Nevertheless, when put to +the test of rigorous calculation, annuities are not +found to offer sufficient advantages to induce people +to sacrifice for them the ownership of the original +capital. Accordingly, whenever it has been attempted +to create annuities sufficiently attractive to induce individuals +to invest in them, it has been necessary to +offer them on terms which are onerous to the company. + +But we shall have more to say on this subject when +we expound the theory of annuities, which is a branch +of the calculus of probabilities. + +I shall conclude the present lecture with a word +\index{Logarithms|EtSeq}% +on \emph{logarithms}. The simplest idea which we can form +of the theory of logarithms, as they are found in the +ordinary tables, is that of conceiving all numbers +as powers of~$10$; the exponents of these powers, +then, will be the logarithms of the numbers. From +\PageSep{17} +this it is evident that the multiplication and division +of two numbers is reducible to the addition and subtraction +of their respective exponents, that is, of their +\MNote{Logarithms\Add{.}} +logarithms. And, consequently, involution and the +extraction of roots are reducible to multiplication and +division, which is of immense advantage in arithmetic +and renders logarithms of priceless value in that science. + +But in the period when logarithms were invented, +mathematicians were not in possession of the theory +of powers. They did not know that the root of a number +could be represented by a fractional power. The +following was the way in which they approached the +problem. + +The primitive idea was that of two corresponding +progressions, one arithmetical, and the other geometrical. +In this way the general notion of a logarithm +was reached. But the means for finding the logarithms +of all numbers were still lacking. As the numbers +follow one another in arithmetical progression, it +was requisite, in order that they might all be found +among the terms of a geometrical progression, so to +establish that progression that its successive terms +should differ by extremely small quantities from one +another; and, to prove the possibility of expressing +all numbers in this way, Napier, the inventor, first +\index{Napier|EtSeq}% +considered them as expressed by lines and parts of +lines, and these lines he considered as generated by +\PageSep{18} +the continuous motion of a point, which was quite +natural. + +\MNote{Napier (1550--1617).} +He considered, accordingly, two lines, the first of +which was generated by the motion of a point describing +in equal times spaces in geometrical progression, +and the other generated by a point which described +spaces that increased as the times and consequently +formed an arithmetical progression corresponding to +the geometrical progression. And he supposed, for +the sake of simplicity, that the initial velocities of +these two points were equal. This gave him the logarithms, +at first called \emph{natural}, and afterwards \emph{hyperbolical}, +when it was discovered that they could be expressed +as parts of the area included between a +hyperbola and its asymptotes. By this method it is +clear that to find the logarithm of any given number, +it is only necessary to take a part on the first line +equal to the given number, and to seek the part on +the second line which shall have been described in +the same interval of time as the part on the first. + +Conformably to this idea, if we take as the two +first terms of our geometrical progression the numbers +with very small differences $1$~and~$1.0000001$, and as +those of our arithmetical progression $0$~and $0.0000001$, +and if we seek successively, by the known rules, all +the following terms of the two progressions, we shall +find that the number~$2$ expressed approximately to the +eighth place of decimals is the $6931472$th~term of the +geometrical progression, that is, that the logarithm of~$2$ +\PageSep{19} +is~$0.6931472$. The number~$10$ will be found to be the +$23025851$th~term of the same progression; therefore, +the logarithm of~$10$ is~$2.3025851$, and so with the rest. +\MNote{Origin of logarithms\Add{.}} +\index{Logarithms!origin of}% +But Napier, having to determine only the logarithms +of numbers less than unity for the purposes of trigonometry, +where the sines and cosines of angles are +expressed as fractions of the radius, considered a decreasing +geometrical progression of which the first +two terms were $1$~and~$0.9999999$; and of this progression +he determined the succeeding terms by enormous +computations. On this last hypothesis, the logarithm +which we have just found for~$2$ becomes that of the +number~$\frac{1}{5}$ or~$0.5$, and that of the number~$10$ becomes +that of the number~$\frac{1}{10}$ or~$0.1$; as is readily apparent +from the nature of the two progressions. + +Napier's work appeared in~1614. Its utility was +felt at once. But it was also immediately seen that it +would conform better to the decimal system of our +arithmetic, and would be simpler, if the logarithm of~$10$ +were made unity, conformably to which that of~$100$ +would be~$2$, and so with the rest. To that end, instead +of taking as the first two terms of our geometrical +progression the numbers $1$~and~$\Typo{0.0000001}{1.0000001}$, we should +have to take the numbers $1$~and~$1.0000002302$, retaining +$0$~and~$0.0000001$ as the corresponding terms of the +arithmetical progression. Whence it will be seen, +that, while the point which is supposed to generate by +its motion the geometrical line, or the numbers, is +describing the very small portion~$0.0000002302\dots$, +\PageSep{20} +the other point, the office of which is to generate +simultaneously the arithmetical line, will have described +\MNote{Briggs (1556--1631). Vlacq.} +\index{Briggs}% +\index{Vlacq}% +the portion~$0.0000001$; and that therefore the +spaces described in the same time by the two points +at the beginning of their motion, that is to say, their +initial velocities, instead of being equal, as in the +preceding system, will be in the proportion of the +numbers $2.302\dots$~to~$1$, where it will be remarked +that the number~$2.302\dots$ is exactly the number +which in the original system of natural logarithms +stood for the logarithm of~$10$,---a result demonstrable +\textit{à ~priori}, as we shall see when we come to apply +the formulæ of algebra to the theory of logarithms. +Briggs, a contemporary of Napier, is the author of this +change in the system of logarithms, as he is also of +the tables of logarithms now in common use. A portion +\index{Logarithms!tables of}% +of these was calculated by Briggs himself, and +the remainder by Vlacq, a Dutchman. + +These tables appeared at Gouda, in~1628. They +contain the logarithms of all numbers from~$1$ to~$100000$ +to ten decimal places, and are now extremely rare. +But it was afterwards discovered that for ordinary purposes +seven decimals were sufficient, and the logarithms +are found in this form in the tables which are +used to-day. Briggs and Vlacq employed a number +of highly ingenious artifices for facilitating their work. +The device which offered itself most naturally and +which is still one of the simplest, consists in taking +the numbers $1$,~$10$, $100$,~$\dots$, of which the logarithms +\PageSep{21} +are $0$,~$1$,~$2$,~$\dots$, and in interpolating between the successive +terms of these two series as many corresponding +terms as we desire, in the first series by geometrical +\MNote{Computation of logarithms.} +mean proportionals and in the second by +arithmetical means. In this manner, when we have +arrived at a term of the first series approaching, to the +eighth decimal place, the number whose logarithm +we seek, the corresponding term of the other series +will be, to the eighth decimal place approximately, +the logarithm of that number. Thus, to obtain the +logarithm of~$2$, since $2$~lies between $1$~and~$10$, we seek +first by the extraction of the square root of~$10$, the +geometrical mean between $1$~and~$10$, which we find to +be~$3.16227766$, while the corresponding arithmetical +mean between $0$~and~$1$ is~$\frac{1}{2}$ or~$0.50000000$; we are +assured thus that this last number is the logarithm of +the first. Again, as $2$~lies between $1$~and~$3.16227766$, +the number just found, we seek in the same manner +the geometrical mean between these two numbers, +and find the number~$1.77827941$. As before, taking +the arithmetical mean between $0$~and~$5.0000000$, we +shall have for the logarithm of~$1.77827941$ the number~$0.25000000$. +Again, $2$~lying between $1.77827941$ +and~$3.16227766$, it will be necessary, for still further +approximation, to find the geometrical mean between +these two, and likewise the arithmetical mean between +their logarithms. And so on. In this manner, +by a large number of similar operations, we find that +the logarithm of~$2$ is~$0.3010300$, that of~$3$ is~$0.4771213$, +\PageSep{22} +and so on, not carrying the degree of exactness beyond +the seventh decimal place. But the preceding +\MNote{Value of the history of science.} +\index{Science!history of}% +calculation is necessary only for prime numbers; because +the logarithms of numbers which are the product +of two or several others, are found by simply +taking the sum of the logarithms of their factors. + +As for the rest, since the calculation of logarithms +is now a thing of the past, except in isolated instances, +it may be thought that the details into which we have +here entered are devoid of value. We may, however, +justly be curious to know the trying and tortuous +paths which the great inventors have trodden, the different +\index{Inventors, great}% +steps which they have taken to attain their goal, +and the extent to which we are indebted to these veritable +benefactors of the human race. Such knowledge, +moreover, is not matter of idle curiosity. It can +afford us guidance in similar inquiries and sheds an +increased light on the subjects with which we are +employed. + +Logarithms are an instrument universally employed +in the sciences, and in the arts depending on calculation. +The following, for example, is a very evident +application of their use. + +Persons not entirely unacquainted with music know +\index{Music}% +that the different notes of the octave are expressed by +numbers which give the divisions of a stretched cord +producing those notes. Thus, the principal note being +denoted by~$1$, its octave will be denoted by~$\frac{1}{2}$, +its fifth by~$\frac{2}{3}$, its third by~$\frac{4}{5}$, its fourth by~$\frac{3}{4}$, its second +\PageSep{23} +by~$\frac{8}{9}$, and so on. The distance of one of these notes +from that next adjacent to it is called an \emph{interval}, and +is measured, not by the difference, but by the ratio of +the numbers expressing the two sounds. Thus, the +interval between the fourth and fifth, which is called +the \emph{major tone}, is regarded as sensibly double of that +between the third and fourth, which is called the \emph{semi-major}. +In fact, the first being expressed by~$\frac{8}{9}$, the +second by~$\frac{15}{16}$, it can be easily proved that the first +does not differ by much from the square of the second. +Now, it is clear that this conception of intervals, on +\MNote{Musical temperament.} +\index{Temperament, theory of}% +which the whole theory of temperament is founded, +conducts us naturally to logarithms. For if we express +the value of the different notes by the logarithms +of the lengths of the cords answering to them, +then the interval of one note from another will be +expressed by the simple difference of values of the +two notes; and if it were required to divide the octave +into twelve equal semi-tones, which would give the +temperament that is simplest and most exact, we +should simply have to divide the logarithm of one +half, the value of the octave, into twelve equal parts. +\PageSep{24} + + +\Lecture{II.}{On the Operations of Arithmetic.} +\index{Arithmetic!operations of|EtSeq}% + +\First{An ancient} writer once remarked that arithmetic +and geometry were \emph{the wings of mathematics}. +\index{Geometry}% +\index{Mathematics!wings of}% +\MNote{Arithmetic and geometry.} +I believe we can say, without metaphor, that +these two sciences are the foundation and essence of +all the sciences that treat of magnitude. But not +only are they the foundation, they are also, so to +speak, the capstone of these sciences. For, whenever +we have reached a result, in order to make use of it, +it is requisite that it be translated into numbers or +into lines; to translate it into numbers, arithmetic is +necessary; to translate it into lines, we must have +recourse to geometry. + +The importance of arithmetic, accordingly, leads +me to the further discussion of that subject to-day, +although we have begun algebra. I shall take up its +several parts, and shall offer new observations, which +will serve to supplement what I have already expounded +to you. I shall employ, moreover, the geometrical +\index{Geometrical!calculus}% +calculus, wherever that is necessary for giving +\PageSep{25} +greater generality to the demonstrations and +methods. + +First, then, as regards addition, there is nothing +to be added to what has already been said. Addition +is an operation so simple in character that its conception +is a matter of course. But with regard to subtraction, +\MNote{New method of subtraction\Add{.}} +\index{Subtraction, new method of|EtSeq}% +there is another manner of performing that +operation which is frequently more advantageous than +the common method, particularly for those familiar +with it. It consists in converting the subtraction into +addition by taking the complement of every figure of +the number which is to be subtracted, first with respect +to~$10$ and afterwards with respect to~$9$. Suppose, +for example, that the number~$2635$ is to be subtracted +from the number~$7853$. Instead of saying $5$~from~$13$ +\begin{figure}[hbt!] +\centering +$\begin{array}{r} +7853 \\ +2635 \\ +\hline +5218 +\end{array}$ +\end{figure} +leaves~$8$; $3$~from~$4$ leaves~$1$; $6$~from~$8$ leaves~$2$; +and $2$~from~$7$ leaves~$5$, giving a total remainder of~$5218$,---I +say: $5$~the complement of~$5$ with respect to~$10$ +added to~$3$ gives~$8$,---I write down~$8$; $6$~the complement +of~$3$ with respect to~$9$ added to~$5$ gives~$11$,---I +write down~$1$ and carry~$1$; $3$~the complement of~$6$ +with respect to~$9$, plus~$9$, by reason of the $1$~carried, +gives~$12$,---I put down~$2$ and carry~$1$; lastly, $7$~the +complement of~$2$ with respect to~$9$ plus~$8$, on account +of the $1$~carried, gives~$15$,---I put down~$5$ and this time +carry nothing, for the operation is completed, and the +\PageSep{26} +last~$10$ which was borrowed in the course of the operation +must be rejected. In this manner we obtain the +same remainder as above,~$5218$. + +The foregoing method is extremely convenient +\MNote{Subtraction by complements.} +\index{Complements, subtraction by}% +when the numbers are large; for in the common +method of subtraction, where borrowing is necessary +in subtracting single numbers from one another, mistakes +are frequently made, whereas in the method +with which we are here concerned we never borrow +but simply carry, the subtraction being converted into +addition. With regard to the complements they are +discoverable at the merest glance, for every one knows +that $3$~is the complement of~$7$ with respect to~$10$, $4$~the +complement of~$5$ with respect to~$9$,~etc. And as +to the reason of the method, it too is quite palpable. +The different complements taken together form the +total complement of the number to be subtracted +either with respect to~$10$ or~$100$ or~$1000$, etc., according +as the number has $1$,~$2$,~$3$~$\dots$ figures; so that the +operation performed is virtually equivalent to first +adding $10$,~$100$, $1000$~$\dots$ to the minuend and then +taking the subtrahend from the minuend as so augmented. +Whence it is likewise apparent why the~$10$ +of the sum found by the last partial addition must be +rejected. + +As to multiplication, there are various abridged +\index{Multiplication!abridged methods of|EtSeq}% +methods possible, based on the decimal system of +numbers. In multiplying by~$10$, for example, we have, +as we know, simply to add a cipher; in multiplying +\PageSep{27} +by~$100$ we add two ciphers; by~$1000$, three ciphers,~etc. +Consequently, to multiply by any aliquot part of~$10$, +for example~$5$, we have simply to multiply by~$10$ +\MNote{Abridged multiplication.} +and then divide by~$2$; to multiply by~$25$ we multiply +by~$100$ and divide by~$4$, and so on for all the products +of~$5$. + +When decimal numbers are to be multiplied by +\index{Decimal!numbers|EtSeq}% +decimal numbers, the general rule is to consider the +two numbers as integers and when the operation is +finished to mark off from the right to the left as many +places in the product as there are decimal places in +the multiplier and the multiplicand together. But in +practice this rule is frequently attended with the inconvenience +of unnecessarily lengthening the operation, +for when we have numbers containing decimals +these numbers are ordinarily exact only to a certain +number of places, so that it is necessary to retain in +the product only the decimal places of an equivalent +order. For example, if the multiplicand and the multiplier +each contain two places of decimals and are exact +only to two decimal places, we should have in the +product by the ordinary method four decimal places, +the two last of which we should have to reject as useless +and inexact. I shall give you now a method for +obtaining in the product only just so many decimal +places as you desire. + +I observe first that in the ordinary method of multiplying +we begin with the units of the multiplier which +we multiply with the units of the multiplicand, and so +\PageSep{28} +continue from the right to the left. But there is nothing +compelling us to begin at the right of the multiplier. +\MNote{Inverted multiplication.} +\index{Multiplication!inverted}% +We may equally well begin at the left. And +I cannot in truth understand why the latter method +should not be preferred, since it possesses the advantage +of giving at once the figures having the greatest +value, and since, in the majority of cases where large +numbers are multiplied together, it is just these last +and highest places that concern us most; we frequently, +in fact, perform multiplications only to find +what these last figures are. And herein, be it parenthetically +remarked, consists one of the great advantages +in calculating by logarithms, which always +\index{Logarithms!advantages in calculating by}% +give, be it in multiplication or division, in involution +or evolution, the figures in the descending order of +their value, beginning with the highest and proceeding +from the left to the right. + +By performing multiplication in this manner, no +difference is caused in the total product. The sole +distinction is, that by the new method the first line, +the first partial product, is that which in the ordinary +method is last, and the second partial product is that +which in the ordinary method is next to the last, and +so with the rest. + +Where whole numbers are concerned and the exact +product is required, it is indifferent which method we +employ. But when decimal places are involved the +prime essential is to have the figures of the whole +numbers first in the product and to descend afterwards +\PageSep{29} +successively to the figures of the decimal parts, +instead of, as in the ordinary method, beginning with +the last decimal places and successively ascending to +the figures forming the whole numbers. + +In applying this method practically, we write the +multiplier underneath the multiplicand so that the +units' figure of the multiplier falls beneath the last +\MNote{Approximate multiplication.} +\index{Multiplication!approximate}% +figure of the multiplicand. We then begin with the +last left-hand figure of the multiplier which we multiply +as in the ordinary method by all the figures of the +multiplicand, beginning with the last to the right and +proceeding successively to the left; observing that the +first figure of the product is to be placed underneath +the figure with which we are multiplying, while the +others follow in their successive order to the left. We +proceed in the same manner with the second figure of +the multiplier, likewise placing beneath this figure the +first figure of the product, and so on with the rest. +The place of the decimal point in these different products +will be the same as in the multiplicand, that is +to say, the units of the products will all fall in the +same vertical line with those of the multiplicand and +consequently those of the sum of all the products or +of the total product will also fall in that line. In this +manner it is an easy matter to calculate only as many +decimal places as we wish. I give below an example +of this method in which the multiplicand is~$437.25$ +and the multiplier~$27.34$: +\PageSep{30} +\MNote{The new method exemplified.} +\[ +\begin{array}{r@{\,}l} +437\PadTo[l]{\,}{.} & 25 \\ + & 27.34 \\ +\hline +\MultRow{8745}{0} \\ +\MultRow{3060}{75} \\ +\MultRow{131}{17\phantom{.}5} \\ +\MultRow{17}{49\phantom{.}00} \\ +\hline +\MultRow{11954}{41\phantom{.}50} +\end{array} +\] + +I have written all the decimals in the product, but +\index{Decimals!multiplication of}% +it is easy to see how we may omit calculating the decimals +which we wish to neglect. The vertical line is +used to mark more distinctly the place of the decimal +point. + +The preceding rule appears to me simpler and +more natural than that which is attributed to Oughtred +\index{Oughtred}% +and which consists in writing the multiplier underneath +the multiplicand in the reverse order. + +There is one more point, finally, to be remarked +in connexion with the multiplication of numbers containing +\index{Multiplication!decimals@of decimals}% +decimals, and that is that we may alter the +place of the decimal point of either number at will. +For seeing that moving the decimal point from the +right to the left in one of the numbers is equivalent to +dividing the number by~$10$, by~$100$, or by~$1000\dots$, and +that moving the decimal point back in the other number +the same number of places from the left to the +right is tantamount to multiplying that number by~$10$, +$100$, or~$1000$,~$\dots$, it follows that we may push the +decimal point forward in one of the numbers as many +places as we please provided we move it back in the +other number the same number of places, without in +\PageSep{31} +any wise altering the product. In this way we can +always so arrange it that one of the two numbers shall +contain no decimals---which simplifies the question. + +Division is susceptible of a like simplification, for +\index{Decimals!division of}% +\index{Division!decimals@of decimals}% +since the quotient is not altered by multiplying or dividing +\MNote{Division of decimals.} +the dividend and the divisor by the same number, +it follows that in division we may move the decimal +point of both numbers forwards or backwards as +many places as we please, provided we move it the +same distance in each case. Consequently, we can +always reduce the divisor to a whole number---which +facilitates infinitely the operation for the reason that +when there are decimal places in the dividend only, +we may proceed with the division by the common +method and neglect all places giving decimals of a +lower order than those we desire to take account of. + +You know the remarkable property of the number~$9$, +\index{Nine!property of the number|EtSeq}% +whereby if a number be divisible by~$9$ the sum of +its digits is also divisible by~$9$. This property enables +us to tell at once, not only whether a number is divisible +by~$9$ but also what is its remainder from such division. +For we have only to take the sum of its digits +and to divide that sum by~$9$, when the remainder will +be the same as that of the original number divided +by~$9$. + +The demonstration of the foregoing proposition is +not difficult. It reposes upon the fact that the numbers +$10$~less~$1$, $100$~less~$1$, $1000$~less~$1$,~$\dots$ are all divisible +\PageSep{32} +by~$9$,---which seeing that the resulting numbers +are $9$,~$99$, $999$,~$\dots$ is quite obvious. + +If, now, you subtract from a given number the +sum of all its digits, you will have as your remainder +\MNote{Property of the number~$9$.} +the tens' digit multiplied by~$9$, the hundreds' digit +multiplied by~$99$, the thousands' digit multiplied by~$999$, +and so on,---a remainder which is plainly divisible +by~$9$. Consequently, if the sum of the digits is +divisible by~$9$, the original number itself will be so +divisible, and if it is not divisible by~$9$ the original +number likewise will not be divisible thereby. But +the remainder in the one case will be the same as in +the other. + +In the case of the number~$9$, it is evident immediately +that $10$~less~$1$, $100$~less~$1$,~$\dots$ are divisible by~$9$; +but algebra demonstrates that the property in +question holds good for every number~$a$. For it can +be shown that +\[ +a - 1,\quad a^{2} - 1,\quad a^{3} - 1,\quad a^{4} - 1, \dots +\] +are all quantities divisible by~$a - 1$, actual division +giving the quotients +\[ +1,\quad a + 1,\quad a^{2} + a + 1,\quad a^{3} + a^{2} + a + 1, \dots. +\] + +The conclusion is therefore obvious that the aforesaid +property of the number~$9$ holds good in our decimal +system of arithmetic because $9$~is $10$~less~$1$, and +that in any other system founded upon the progression +$a$,~$a^{2}$,~$a^{3}$,~$\dots$ the number~$a - 1$ would enjoy the +same property. Thus in the duodecimal system it +\index{Duodecimal system}% +\PageSep{33} +would be the number~$11$; and in this system every +number, the sum of whose digits was divisible by~$11$, +would also itself be divisible by that number. + +The foregoing property of the number~$9$, now, admits +\index{Nine!property of the number generalised}% +of generalisation, as the following consideration +\MNote{Property of the number~$9$ generalised.} +will show. Since every number in our system is represented +by the sum of certain terms of the progression +$1$,~$10$, $100$, $1000$,~$\dots$, each multiplied by one of +the nine digits $1$,~$2$, $3$, $4$,~$\dots$\Add{,}~$9$, it is easy to see that +the remainder resulting from the division of any number +by a given divisor will be equal to the sum of the +remainders resulting from the division of the terms $1$, +$10$, $100$, $1000$,~$\dots$ by that divisor, each multiplied by +the digit showing how many times the corresponding +term has been taken. Hence, generally, if the given +divisor be called~$D$, and if $m$,~$n$,~$p$,~$\dots$ be the remainders +of the division of the numbers $1$, $10$, $100$, $1000$ +by~$D$, the remainder from the division of any number +whatever~$N$, of which the characters proceeding from +the right to the left are $a$,~$b$,~$c$,~$\dots$, by~$D$ will obviously +be equal to +\[ +ma + nb + pc + \dots. +\] +Accordingly, if for a given divisor~$D$ we know the remainders +$m$,~$n$,~$p$,~$\dots$, which depend solely upon that +divisor and which are always the same for the same +divisor, we have only to write the remainders underneath +the original number, proceeding from the right +to the left, and then to find the different products of +\PageSep{34} +each digit of the number by the digit which is underneath +it. The sum of all these products will be the +\MNote{Theory of remainders\Add{.}} +\index{Remainders!theory of|EtSeq}% +total remainder resulting from the division of the proposed +number by the same divisor~$D$. And if the sum +found is greater than~$D$, we can proceed in the same +manner to seek its remainder from division by~$D$, and +so on until we arrive finally at a remainder which is +less than~$D$, which will be the true remainder sought. +It follows from this that the proposed number cannot +be exactly divisible by the given divisor unless the +last remainder found by this method is zero. + +The remainders resulting from the division of the +terms $1$, $10$, $100$,~$\dots$\Add{,} $1000$, by~$9$ are always unity. +\index{Division!nine@by \textit{nine}}% +Hence, the sum of the digits of any number whatever +is the remainder resulting from the division of that +number by~$9$. The remainders resulting from the division +of the same terms by~$8$ are $1$,~$2$, $4$, $0$, $0$, $0$,~$\dots$. +\index{Division!eight@by \textit{eight}}% +We shall obtain, accordingly, the remainder resulting +from dividing any number by~$8$, by taking the sum +of the first digit to the right, the second digit next +thereto to the left multiplied by~$2$, and the third digit +multiplied by~$4$. + +The remainders resulting from the divisions of the +\index{Division!seven@by \textit{seven}|EtSeq}% +terms $1$, $10$, $100$, $1000$,~$\dots$ by~$7$ are $1$, $3$, $2$, $6$, $4$, $5$, +$1$, $3$,~$\dots$, where the same remainders continually recur +in the same order. If I have, now, the number +$13527541$ to be divided by~$7$, I write it thus with the +above remainders underneath it: +\PageSep{35} +\index{Seven, tests of divisibility by}% +\MNote{Test of divisibility by~$7$.} +\[ +\begin{array}{@{\,}*{2}{r@{}}r@{\,}} +13527&5&41 \\ +31546&2&31 \\ +\hline +&& 1 \\ +&& 12 \\ +&& 10 \\ +&& 42 \\ +&& 8 \\ +&& 25 \\ +&& 3 \\ +&& 3 \\ +\cline{2-3} +& 1&04 \\ +& 2&31 \\ +\cline{2-3} +&& 4 \\ +&& 0 \\ +&& 2 \\ +\cline{3-3} +&& 6 +\end{array} +\] + +Taking the partial products and adding them, I +obtain~$104$, which would be the remainder from the +division of the given number by~$7$, were it not greater +than the divisor. I accordingly repeat the operation +with this remainder, and find for my second remainder~$6$, +which is the real remainder in question. + +I have still to remark with regard to the preceding +remainders and the multiplications which result from +them, that they may be simplified by introducing negative +remainders in the place of remainders which are +greater than half the divisor, and to accomplish this +we have simply to subtract the divisor from each of +such remainders. We obtain thus, instead of the remainders +$6$,~$5$,~$4$, the following: +\[ +-1,\quad -2,\quad -3. +\] +\PageSep{36} +The remainders for the divisor~$7$, accordingly, are +\[ +1,\quad 3,\quad 2,\quad -1,\quad -3,\quad -2,\quad 1,\quad 3, \dots +\] +and so on to infinity. + +\MNote{Negative remainders\Add{.}} +\index{Remainders!negative|EtSeq}% +The preceding example, then, takes the following +form: +\[ +\begin{array}{@{\,}*{3}{r@{}}r@{\,}} +135&27&5&41 \\ +31\underline{2}&\underline{31}&2&31 \\ +\hline + & 7& & 1 \\ + & 6& &12 \\ + &10& &10 \\ +\cline{2-2} + &23& & 3 \\ + & & & 3 \\ +\cline{4-4} + & & &29 \\ +\multicolumn{2}{r}{\llap{\text{subtract}}} & &23 \\ +\cline{4-4} + & & & 6 +\end{array} +\] + +I have placed a bar beneath the digits which are +to be taken negatively, and I have subtracted the sum +of the products of these numbers by those above them +from the sum of the other products. + +The whole question, therefore, resolves itself into +finding for every divisor the remainders resulting from +dividing $1$, $10$, $100$, $1000$\Add{,~$\dots$} by that divisor. This can be +readily done by actual division; but it can be accomplished +more simply by the following consideration. +If $r$~be the remainder from the division of~$10$ by a +given divisor, $r^{2}$~will be the remainder from the division +of~$100$, the square of~$10$, by that divisor; and +consequently it will be necessary merely to subtract +the given divisor from~$r^{2}$ as many times as is requisite +to obtain a positive or negative remainder less than +\PageSep{37} +half of that divisor. Let $s$ be that remainder; we shall +then only have to multiply $s$~by~$r$, the remainder from +the division of~$10$, to obtain the remainder from the +division of~$1000$ by the given divisor, because $1000$~is +$100 × 10$, and so~on. + +For example, dividing $10$ by~$7$ we have a remainder +of~$3$; hence, the remainder from dividing $100$ by~$7$ +will be~$9$, or, subtracting from~$9$ the given divisor~$7$,~$2$. +The remainder from dividing $1000$ by~$7$, then, will +be the product of~$2$ by $3$~or~$6$, or, subtracting the divisor,~$7$,~$-1$. +Again, the remainder from dividing +%[** TN: Removed comma in 10,000 for consistency] +$\Typo{10,000}{10000}$ by~$7$ will be the product of $-1$~and~$3$, or~$-3$, +and so~on. + +Let us now take the divisor~$11$. The remainder +\index{Eleven, the number, test of divisibility by}% +from dividing~$1$ by~$11$ is~$1$, from dividing~$10$ by~$11$ is~$10$, +\MNote{Test of divisibility by~$11$.} +or, subtracting the divisor,~$-1$. The remainder +from dividing~$100$ by~$11$, then, will be the square of~$-1$, +or~$1$; from dividing $1000$ by~$11$ it will be $1$~multiplied +by~$-1$ or\Add{ }$-1$~again, and so on forever, the remainders +forming the infinite series +\[ +1,\quad -1,\quad 1,\quad -1,\quad 1,\quad -1,\dots\Add{.} +\] + +Hence results the remarkable property of the number~$11$, +that if the digits of any number be alternately +added and subtracted, that is to say, if we take the +sum of the first, the third, and the fifth, etc., and subtract +from it the sum of the second, the fourth, the +sixth, etc., we shall obtain the remainder which results +from dividing that number by the number~$11$. +\PageSep{38} + +The preceding theory of remainders is fraught +\index{Remainders!theory of}% +with remarkable consequences, and has given rise to +\MNote{Theory of remainders\Add{.}} +many ingenious and difficult investigations. We can +demonstrate, for example, that if the divisor is a prime +number, the remainders of any progression $1$, $a$, $a^{2}$, +$a^{3}$, $a^{4}$,~$\dots$ form periods which will recur continually +to infinity, and all of which, like the first, begin with +unity; in such wise that when unity reappears among +the remainders we may continue them to infinity by +simply repeating the remainders which precede. It +has also been demonstrated that these periods can +only contain a number of terms which is equal to the +divisor less~$1$ or to an aliquot part of the divisor less~$1$. +But we have not yet been able to determine \textit{à ~priori} +this number for any divisor whatever. + +As to the utility of this method for finding the remainder +\index{Theory of remainders, utility of the}% +resulting from dividing a given number by a +given divisor, it is frequently very useful when one +has several numbers to divide by the same number, +and it is required to prepare a table of the remainders. +While as to division by $9$~and~$11$, since that is very +simple, it can be employed as a check upon multiplication +and division. Having found the remainders +from dividing the multiplicand and the multiplier by +either of these numbers it is simply necessary to take +the product of the two remainders so resulting, from +which, after subtracting the divisor as many times as +is requisite, we shall obtain the remainder from dividing +their product by the given divisor,---a remainder +\PageSep{39} +which should agree with the remainder obtained +from treating the actual product in this manner. And +since in division the dividend less the remainder should +\MNote{checks on multiplication and division.} +\index{Checks on multiplication and division}% +be equal to the product of the divisor and the quotient, +the same check may also be applied here to advantage. + +The supposition which I have just made that the +product of the remainders from dividing two numbers +by the same divisor is equal to the remainder from +dividing the product of these numbers by the same +divisor is easily proved, and I here give a general +demonstration of it. + +Let $M$~and~$N$ be two numbers, $D$~the divisor, $p$~and~$q$ +the quotients, and $r$,~$s$ the two remainders. We +shall plainly have +\[ +M = pD + r,\quad +N = qD + s, +\] +from which by multiplying we obtain +\[ +MN = pqD^{2} + spD + rqD + rs; +\] +where it will be seen that all the terms are divisible +by~$D$ with the exception of the last,~$rs$, whence it follows +that $rs$~will be the remainder from dividing~$MN$ +by~$D$. It is further evident that if any multiple whatever +of~$D$, as~$mD$, be subtracted from~$rs$, the result +$rs - mD$ will also be the remainder from dividing~$MN$ +by~$D$. For, putting the value of~$MN$ in the following +form: +\[ +pqD^{2} + spD + rqD + mD + rs - mD, +\] +it is obvious that the remaining terms are all divisible +\PageSep{40} +by~$D$. And this remainder $rs - mD$ can always be +made less than~$D$, or, by employing negative remainders, +less even than~$\dfrac{D}{2}$. + +This is all that I have to say upon multiplication +\MNote{Evolution.} +\index{Evolution}% +and division. I shall not speak of the \emph{extraction of +roots}. The rule is quite simple for square roots; it +leads directly to its goal; trials are unnecessary. As +to cube and higher roots, the occasion rarely arises +for extracting them, and when it does arise the extraction +can be performed with great facility by means +of logarithms, where the degree of exactitude can be +\index{Logarithms}% +carried to as many decimal places as the logarithms +themselves have decimal places. Thus, with seven-place +logarithms we can extract roots having seven +figures, and with the large tables where the logarithms +have been calculated to ten decimal places we +can obtain even ten figures of the result. + +One of the most important operations in arithmetic +\index{Rule!three@of three|EtSeq}% +is the so-called \emph{rule of three}, which consists in +finding the fourth term of a proportion of which the +first three terms are given. + +In the ordinary text-books of arithmetic this rule +has been unnecessarily complicated, having been divided +into simple, direct, inverse, and compound rules +of three. In general it is sufficient to comprehend the +conditions of the problem thoroughly, for the common +rule of three is always applicable where a quantity increases +or diminishes in the same proportion as another. +\PageSep{41} +For example, the price of things augments in +proportion to the quantity of the things, so that the +quantity of the thing being doubled, the price also +\MNote{Rule of three.} +will be doubled, and so on. Similarly, the amount of +work done increases proportionally to the number of +persons employed. Again, things may increase simultaneously +in two different proportions. For example, +the quantity of work done increases with the +number of the persons employed, and also with the +time during which they are employed. Further, there +are things that decrease as others increase. + +Now all this may be embraced in a single, simple +proposition. If a quantity increases both in the ratio +in which one or several other quantities increase and +in that in which one or several other quantities decrease, +it is the same thing as saying that the proposed +quantity increases proportionally to the product of the +quantities which increase with it, divided by the product +of the quantities which simultaneously decrease. +For example, since the quantity of work done increases +proportionally with the number of laborers +\index{Laborers, work of}% +and with the time during which they work and since +it diminishes in proportion as the work becomes more +difficult, we may say that the result is proportional to +the number of laborers multiplied by the number +measuring the time during which they labor, divided +by the number which measures or expresses the difficulty +of the work. + +The further fact should not be lost sight of that +\PageSep{42} +the rule of three is properly applicable only to things +which increase in a constant ratio. For example, it is +\index{Ratios, constant}% +\MNote{Applicability of the rule of three.} +assumed that if a man does a certain amount of work +in one day, two men will do twice that amount in one +day, three men three times that amount, four men +four times that amount,~etc. In reality this is not the +case, but in the rule of proportion it is assumed to be +such, since otherwise we should not be able to employ +it. + +When the law of augmentation or diminution varies, +the rule of three is not applicable, and the ordinary +methods of arithmetic are found wanting. We +must then have recourse to algebra. + +A cask of a certain capacity empties itself in a certain +\index{Efflux, law of}% +time. If we were to conclude from this that a +cask of double that capacity would empty itself in +double the time, we should be mistaken, for it will +empty itself in a much shorter time. The law of efflux +does not follow a constant ratio but a variable +ratio which diminishes with the quantity of liquid remaining +in the cask. + +We know from mechanics that the spaces traversed +\index{Falling stone, spaces traversed by a}% +by a body in uniform motion bear a constant ratio to +the times elapsed. If we travel one mile in one hour, +in two hours we shall travel two miles. But the spaces +traversed by a falling stone are not in a fixed ratio to +the time. If it falls sixteen feet in the first second, it +will fall forty-eight feet in the second second. + +The rule of three is applicable when the ratios are +\PageSep{43} +constant only. And in the majority of affairs of ordinary +life constant ratios are the rule. In general, the +price is always proportional to the quantity, so that if +\MNote{Theory and practice.} +\index{Practice, theory and}% +\index{Theory and practice}% +a given thing has a certain value, two such things will +have twice that value, three three times that value, +four four times that value,~etc. It is the same with +the product of labor relatively to the number of laborers +and to the duration of the labor. Nevertheless, +cases occur in which we may be easily led into error. +If two horses, for example, can pull a load of a certain +\index{Horses}% +weight, it is natural to suppose that four horses +could pull a load of double that weight, six horses a +load of three times that weight. Yet, strictly speaking, +such is not the case. For the inference is based +upon the assumption that the four horses pull alike in +amount and direction, which in practice can scarcely +ever be the case. It so happens that we are frequently +led in our reckonings to results which diverge widely +from reality. But the fault is not the fault of mathematics; +\index{Mathematics!exactness of}% +for mathematics always gives back to us exactly +what we have put into it. The ratio was constant +according to the supposition. The result is founded +upon that supposition. If the supposition is false the +result is necessarily false. Whenever it has been attempted +to charge mathematics with inexactitude, the +accusers have simply attributed to mathematics the +error of the calculator. False or inexact data having +been employed by him, the result also has been necessarily +false or inexact. +\PageSep{44} + +Among the other rules of arithmetic there is one +called \emph{alligation} which deserves special consideration +\index{Alligation!generally|EtSeq}% +\MNote{Alligation.} +from the numerous applications which it has. Although +alligation is mainly used with reference to the +mingling of metals by fusion, it is yet applied generally +\index{Metals, mingling of, by fusion}% +to mixtures of any number of articles of different +values which are to be compounded into a whole of a +like number of parts having a mean value. The rule +\index{Mixtures, rule of|EtSeq}% +\index{Rule!mixtures@of mixtures|EtSeq}% +of alligation, or mixtures, accordingly, has two parts. + +In one we seek the mean and common value of +each part of the mixture, having given the number +of the parts and the particular value of each. In the +second, having given the total number of the parts +and their mean value, we seek the composition of the +mixture itself, or the proportional number of parts of +each ingredient which must be mixed or alligated together. + +Let us suppose, for example, that we have several +\index{Grain, of different prices}% +bushels of grain of different prices, and that we are +desirous of knowing the mean price. The mean price +must be such that if each bushel were of that price the +total price of all the bushels together would still be +the same. Whence it is easy to see that to find the +mean price in the present case we have first simply to +find the total price and to divide it by the number of +bushels. + +In general if we multiply the number of things of +each kind by the value of the unit of that kind and +then divide the sum of all these products by the total +\PageSep{45} +number of things, we shall have the mean value, because +that value multiplied by the number of the +things will again give the total value of all the things +taken together. + +This mean or average value as it is called, is of +\index{Mean values|EtSeq}% +\index{Values!mean|EtSeq}% +great utility in almost all the affairs of life. Whenever +\MNote{Mean values.} +we arrive at a number of different results, we +always like to reduce them to a mean or average expression +which will yield the same total result. + +You will see when you come to the calculus of +\index{Probabilities, calculus of|EtSeq}% +probabilities that this science is almost entirely based +upon the principle we are discussing. + +The registration of births and deaths has rendered +\index{Average life|EtSeq}% +\index{Life insurance|EtSeq}% +\index{Mortality, tables of}% +possible the construction of so-called \emph{tables of mortality} +which show what proportion of a given number of +children born at the same time or in the same year +survive at the end of one year, two years, three years,~etc. +So that we may ask upon this basis what is the +mean or average value of the life of a person at any +given age. If we look up in the tables the number of +people living at a certain age, and then add to this +the number of persons living at all subsequent ages, +it is clear that this sum will give the total number of +years which all living persons of the age in question +have still to live. Consequently, it is only necessary +to divide this sum by the number of living persons of +a certain age in order to obtain the average duration +of life of such persons, or better, the number of years +which each person must live that the total number of +\PageSep{46} +years lived by all shall be the same and that each +person shall have lived an equal number. It has been +\MNote{Probability of life.} +\index{Life, probability of}% +found in this manner by taking the mean of the results +of different tables of mortality, that for an infant +one year old the average duration of life is about +$40$~years; for a child ten years old it is still $40$~years; +for~$20$ it is~$34$; for~$30$ it is~$26$; for~$40$ it is~$23$; for~$50$ +it is~$17$; for~$60$ it is~$12$; for~$70$,~$8$; and for~$80$,~$5$. + +To take another example, a number of different +experiments are made. Three experiments have given~$4$ +\index{Experiments!average of}% +as a result; two experiments have given~$5$; and one +has given~$6$. To find the mean we multiply~$4$ by~$3$, $5$~by~$2$, +and $1$~by~$6$, add the products which gives~$28$, +and divide~$28$ by the number of experiments or~$6$, +which gives~$4\frac{2}{3}$ as the mean result of all the experiments. + +But it will be apparent that this result can be regarded +as exact only upon the condition of our having +supposed that the experiments were all conducted with +equal precision. But it is impossible that such could +have been the case, and it is consequently imperative +to take account of these inequalities, a requirement +which would demand a far more complicated calculus +than that which we have employed, and one which is +now engaging the attention of mathematicians. + +The foregoing is the substance of the first part of +the rule of alligation; the second part is the opposite +of the first. Given the mean value, to find how much +\PageSep{47} +must be taken of each ingredient to produce the required +mean value. + +The problems of the first class are always determinate, +because, as we have just seen, the number of +\MNote{Alternate alligation.} +\index{Alligation!alternate}% +units of each ingredient has simply to be multiplied +by the value of each ingredient and the sum of all +these products divided by the number of the ingredients. + +The problems of the second class, on the other +\index{Analysis!indeterminate|EtSeq}% +\index{Indeterminate analysis|EtSeq}% +hand, are always indeterminate. But the condition +that only positive whole numbers shall be admitted +in the result serves to limit the number of the solutions. + +Suppose that we have two kinds of things, that +the value of the unit of one kind is~$a$, and that of the +unit of the second is~$b$, and that it is required to find +how many units of the first kind and how many units +of the second must be taken to form a mixture or +whole of which the mean value shall be~$m$. + +Call $x$~the number of units of the first kind that +must enter into the mixture, and $y$~the number of units +of the second kind. It is clear that $ax$~will be the +value of the $x$~units of the first kind, and $by$~the value +of the $y$~units of the second. Hence $ax + by$ will be +the total value of the mixture. But the mean value +of the mixture being by supposition~$m$, the sum~$x + y$ +of the units of the mixture multiplied by~$m$, the mean +value of each unit, must give the same total value. +We shall have, therefore, the equation +\PageSep{48} +\[ +ax + by = mx + my. +\] +Transposing to one side the terms multiplied by~$x$ +and to the other the terms multiplied by~$y$, we obtain: +\MNote{Two ingredients.} +\index{Ingredients}% +\[ +(a - m)x = (m - b)y, +\] +and dividing by~$a - m$ we get +\[ +x = \frac{(m - b)y}{a - m}, +\] +whence it appears that the number~$y$ may be taken at +pleasure, for whatever be the value given to~$y$, there +will always be a corresponding value of~$x$ which will +satisfy the problem. + +Such is the general solution which algebra gives. +But if the condition be added that the two numbers $x$~and~$y$ +shall be integers, then $y$~may not be taken at +pleasure. In order to see how we can satisfy this last +condition in the simplest manner, let us divide the +last equation by~$y$, and we shall have +\[ +\frac{x}{y} = \frac{m - b}{a - m}. +\] +For $x$~and~$y$ both to be positive, it is necessary that +the quantities +\[ +m - b \quad\text{and}\quad a - m +\] +should both have the same sign; that is to say, if $a$~is +greater or less than~$m$, then conversely $b$~must be less +or greater than~$m$; or again, $m$~must lie between $a$~and~$b$, +which is evident from the condition of the +problem. Suppose $a$, then, to be the greater and $b$~the +\PageSep{49} +smaller of the two prices. It remains to find the +value of the fraction +\MNote{Rule of mixtures.} +\index{Mixtures, rule of}% +\[ +\frac{m - b}{a - m}, +\] +which if necessary is to be reduced to its lowest terms. +Let~$\dfrac{B}{A}$ be that fraction reduced to its lowest terms. It +is clear that the simplest solution will be that in which +\[ +x = B \quad\text{and}\quad y = A. +\] +But since a fraction is not altered by multiplying its +numerator and denominator by the same number, it +is clear that we may also take $x = nB$ and $y = nA$, $n$~being +any number whatever, provided it is an integer, +for by supposition $x$~and~$y$ must be integers. And it +is easy to prove that these expressions of $x$~and~$y$ are +the only ones which will resolve the proposed problem. +According to the ordinary rule of mixtures, $x$, +the quantity of the dearer ingredient, is made equal +to~$m - b$, the excess of the average price above the +lower price, and $y$~the quantity of the cheaper ingredient +is made equal to~$a - m$, the excess of the higher +price above the average price,---a rule which is contained +directly in the general solution above given. + +Suppose, now, that instead of two kinds of things, +we have three kinds, the values of which beginning +with the highest are $a$,~$b$, and~$c$. Let $x$,~$y$,~$z$ be the +quantities which must be taken of each to form a mixture +or compound having the mean value~$m$. The +sum of the values of the three quantities $x$,~$y$,~$z$ will +then be +\[ +ax + by + cz. +\] +\PageSep{50} +But this total value must be the same as that produced +if all the individual values were~$m$, in which +\MNote{Three ingredients.} +case the total value is obviously +\[ +mx + my + mz. +\] +The following equation, therefore, must be satisfied: +\[ +ax + by + cz = mx + my + mz, +\] +or, more simply, +\[ +(a - m)x + (b - m)y + (c - m)z = 0. +\] +Since there are three unknown quantities in this equation, +two of them may be taken at pleasure. But if +the condition is that they shall be expressed by positive +integers, it is to be observed first that the numbers +\[ +a - m \quad\text{and}\quad m - c +\] +are necessarily positive; so that putting the equation +in the form +\[ +(a - m)x - (m - c)z = (m - b)y, +\] +the question resolves itself into finding two multiples +of the given numbers +\[ +a - m \quad\text{and}\quad m - c +\] +whose difference shall be equal to~$(m - b)y$. + +This question is always resolvable in whole numbers +whatever the given numbers be of which we seek +the multiples, and whatever be the difference between +these multiples. As it is sufficiently remarkable in itself +and may be of utility in many emergencies, we +shall give here a general solution of it, derived from +the properties of continued fractions. +\index{Continued fractions, solution of alligation by|EtSeq}% +\PageSep{51} + +Let $M$~and~$N$ be two whole numbers. Of these +numbers two multiples $xM$,~$zN$ are sought whose difference +is given and equal to~$D$. The following equation +\MNote{General solution.} +will then have to be satisfied +\[ +xM - zN = D, +\] +where $x$~and~$z$ by supposition are whole numbers. In +the first place, it is plain that if $M$~and~$N$ are not +prime to each other, the number~$D$ is divisible by the +greatest common divisor of $M$~and~$N$; and the division +having been performed, we should have a similar +equation in which the numbers $M$~and~$N$ are prime +to each other, so that we are at liberty always to suppose +them reduced to that condition. I now observe +that if we know the solution of the equation for the +case in which the number~$D$ is equal to $+1$~or~$-1$, +we can deduce the solution of it for any value whatever +of~$D$. For example, suppose that we know two +multiples of $M$~and~$N$, say $pM$~and~$qN$, the difference +of which $pM - qN$ is equal to~$±1$. Then obviously +we shall merely have to multiply both these multiples +by the number~$D$ to obtain a difference equal to~$±D$. +For, multiplying the preceding equation by~$D$, we +have +\[ +pDM - qDN = ±D; +\] +and subtracting the latter equation from the original +equation +\[ +xM - zN = D, +\] +or adding it, according as the term~$D$ has the sign +$+$~or~$-$ before it, we obtain +\PageSep{52} +\[ +(x \mp pD)M - (z \mp qD)N = 0, +\] +which gives at once, as we saw above in the rule for +the mixture of two different ingredients, +\MNote{Development.} +\[ +(x \mp pD) = nN,\quad +(z \mp qD) = nM, +\] +$n$~being any number whatever. So that we have generally +\[ +x = nN ± pD \quad\text{and}\quad z = nM ± qD +\] +where $n$~is any whole number, positive or negative. +It remains merely to find two numbers $p$~and~$q$ such +that +\[ +pM - qN = ±1. +\] +Now this question is easily resolvable by continued +fractions. For we have seen in treating of these fractions +that if the fraction~$\dfrac{M}{N}$ be reduced to a continued +fraction, and all the successive fractions approximating +to its value be calculated, the last of these successive +fractions being the fraction~$\dfrac{M}{N}$ itself, then the series +of fractions so reached is such that the difference +between any two consecutive fractions is always equal +to a fraction of which the numerator is unity and the +denominator the product of the two denominators. +For example, designating by~$\dfrac{K}{L}$ the fraction which +immediately precedes the last fraction~$\dfrac{M}{N}$ we obtain +necessarily +\[ +LM - KN = 1 \quad\text{or}\quad -1, +\] +according as $\dfrac{M}{N}$~is greater or less than~$\dfrac{K}{L}$, in other +\PageSep{53} +words, according as the place occupied by the last +fraction~$\dfrac{M}{N}$ in the series of fractions successively approximating +to its value is even or odd; for, the first +\MNote{Resolution by continued fractions.} +fraction of the approximating series is always smaller, +the second larger, the third smaller,~etc., than the +original fraction which is identical with the last fraction +of the series. Making, therefore, +\[ +p = L \quad\text{and}\quad q = K, +\] +the problem of the two multiples will be resolved in +all its generality. + +It is now clear that in order to apply the foregoing +solution to the initial question regarding alligation we +have simply to put +\[ +M = a - m,\quad N = m - c, \quad\text{and}\quad D = (m - b)y; +\] +so that the number~$y$ remains undetermined and may +be taken at pleasure, as may also the number~$N$ which +appears in the expressions for $x$~and~$z$. +\PageSep{54} + + +\Lecture[On Algebra.]{III.}{On Algebra, Particularly the Resolution of +Equations of the Third and +Fourth Degree.} +\index{Algebra!history of|EtSeq}% +\index{Diophantus|EtSeq}% +\index{Geometers, ancient|EtSeq}% +\index{Greeks, mathematics of the|EtSeq}% +\index{Romans, mathematics of the}% +\PgLabel{54} + +\First{Algebra} is a science almost entirely due to the +moderns. I say almost entirely, for we have +\MNote{Algebra among the ancients.} +one treatise from the Greeks, that of Diophantus, who +flourished in the third\footnote + {The period is uncertain. Some say in the fourth century. See Cantor, + \index{Cantor|FN}% + \textit{Geschichte der Mathematik}, 2nd.~ed., Vol.~I., p.~434.---\textit{Trans.}} +century of the Christian era. +This work is the only one which we owe to the ancients +in this branch of mathematics. When I speak +of the ancients I speak of the Greeks only, for the +Romans have left nothing in the sciences, and to all +appearances did nothing. + +Diophantus may be regarded as the inventor of +algebra.\footnote + {On this point, see \textit{Appendix}, \PgRef{151}.---\textit{Trans.}} +From a word in his preface, or rather in his +letter of dedication, (for the ancient geometers were +wont to address their productions to certain of their +friends, a practice exemplified in the prefaces of Apollonius +\index{Apollonius}% +and Archimedes), from a word in his preface, I +\index{Archimedes}% +say, we learn that he was the first to occupy himself +\PageSep{55} +with that branch of arithmetic which has since been +called algebra. + +His work contains the first elements of this science. +He employed to express the unknown quantity a Greek +\index{Unknown quantity}% +\MNote{Diophantus\Add{.}} +letter which corresponds to our~$st$\footnote + {According to a recent conjecture, the character in question is an abbreviation + of~\textgreek{ar} the first letters of \textgreek{>arijm'os}, \textit{number}, the appellation technically + applied by Diophantus to the unknown quantity.---\textit{Trans.}} +and which has +been replaced in the translations by~$N$. To express +the known quantities he employed numbers solely, for +algebra was long destined to be restricted entirely to +the solution of numerical problems. We find, however, +that in setting up his equations consonantly with +the conditions of the problem he uses the known and +the unknown quantities alike. And herein consists +\index{Algebra!essence of}% +virtually the essence of algebra, which is to employ +unknown quantities, to calculate with them as we do +with known quantities, and to form from them one +or several equations from which the value of the unknown +quantities can be determined. Although the +work of Diophantus contains indeterminate problems +\index{Analysis!indeterminate}% +\index{Indeterminate analysis}% +almost exclusively, the solution of which he seeks in +rational numbers,---problems which have been designated +after him \emph{Diophantine problems},---we nevertheless +\index{Diophantine problems}% +find in his work the solution of a number of determinate +problems of the first degree, and even of such +as involve several unknown quantities. In the latter +case, however, the author invariably has recourse to +particular artifices for reducing the problem to a single +unknown quantity,---which is not difficult. He gives, +\PageSep{56} +also, the solution of \emph{equations of the second degree}, but +\index{Equations!second@of the second degree}% +is careful so to arrange them that they never assume +the affected form containing the square and the first +power of the unknown quantity. + +He proposed, for example, the following question +\MNote{Equations of the second degree.} +which involves the general theory of equations of the +second degree: + +\textit{To find two numbers the sum and the product of which +are given.} +\index{Sum and difference, of two numbers}% + +If we call the sum~$a$ and the product~$b$ we have at +once, by the theory of equations, the equation +\[ +x^{2} - ax + b = 0. +\] + +Diophantus resolves this problem in the following +manner. The sum of the two numbers being given, +he seeks their difference, and takes the latter as the +unknown quantity. He then expresses the two numbers +in terms of their sum and difference,---the one +by half the sum plus half the difference, the other by +half the sum less half the difference,---and he has +then simply to satisfy the other condition by equating +their product to the given number. Calling the given +sum~$a$, the unknown difference~$x$, one of the numbers +will be~$\dfrac{a + x}{2}$ and the other will be~$\dfrac{a - x}{2}$. Multiplying +these together we have~$\dfrac{a^{2} - x^{2}}{4}$. The term containing~$x$ +is here eliminated, and equating the quantity +last obtained to the given product, we have the +simple equation +\[ +\frac{a^{2} - x^{2}}{4} = b, +\] +\PageSep{57} +from which we obtain +\[ +x^{2} = a^{2} - 4b, +\] +and from the latter +\[ +x = \sqrt{a^{2} - 4b}. +\] + +Diophantus resolves several other problems of this +class. By appropriately treating the sum or difference +\MNote{Other problems solved by Diophantus.} +as the unknown quantity he always arrives at an +equation in which he has only to extract a square root +to reach the solution of his problem. + +But in the books which have come down to us +(for the entire work of Diophantus has not been preserved) +this author does not proceed beyond equations +of the second degree, and we do not know if he +or any of his successors (for no other work on this +subject has been handed down from antiquity) ever +pushed their researches beyond this point. + +I have still to remark in connexion with the work +\index{Signs $+$ and $-$}% +of Diophantus that he enunciated the principle that +$+$~and~$-$ give~$-$ in multiplication, and $-$~and~$-$,~$+$, +in the form of a definition. But I am of opinion that +this is an error of the copyists, since he is more likely +to have considered it as an axiom, as did Euclid some +\index{Euclid}% +of the principles of geometry. However that may be, +it will be seen that Diophantus regarded the rule of +the signs as a self-evident principle not in need of demonstration. + +The work of Diophantus is of incalculable value +from its containing the first germs of a science which +because of the enormous progress which it has since +\PageSep{58} +made constitutes one of the chiefest glories of the human +intellect. Diophantus was not known in Europe +\MNote{Translations of Diophantus\Add{.}} +until the end of the sixteenth century, the first translation +having been a wretched one by Xylander made +\index{Xylander}% +in~1575 and based upon a manuscript found about the +middle of the sixteenth century in the Vatican library, +\index{Vatican library}% +where it had probably been carried from Greece when +the Turks took possession of Constantinople. +\index{Constantinople}% +\index{Turks}% + +Bachet de Méziriac, one of the earliest members +\index{Bachet de Méziriac}% +\index{Meziriac@Méziriac, Bachet de}% +of the French Academy, and a tolerably good mathematician +for his time, subsequently published~(1621) +a new translation of the work of Diophantus accompanied +by lengthy commentaries, now superfluous. +Bachet's translation was afterwards reprinted with observations +and notes by Fermat, one of the most celebrated +\index{Fermat}% +mathematicians of France, who flourished +\index{France}% +about the middle of the seventeenth century, and of +whom we shall have occasion to speak in the sequel +for the important discoveries which he has made in +analysis. Fermat's edition bears the date of~1670.\footnote + {There have since been published a new critical edition of the text by + M.~Paul Tannery (Leipsic, 1893), and two German translations, one by O.~Schulz + \index{Tannery, M. Paul|FN}% + \index{Wertheim, G.|FN}% + (Berlin, 1822) and one by G.~Wertheim (Leipsic, 1890). Fermat's notes + on Diophantus have been republished in Vol.~I. of the new edition of Fermat's + works (Paris, Gauthier-Villars et Fils, 1891).---\textit{Trans.}} + +It is much to be desired that good translations +\index{Geometers, ancient}% +should be made, not only of the work of Diophantus, +but also of the small number of other mathematical +works which the Greeks have left us.\footnote + {Since Lagrange's time this want has been partly supplied. Not to mention + Euclid, we have, for example, of Archimedes the German translation of + \index{Archimedes|FN}% + Nizze (Stralsund, 1824) and the French translation of Peyrard (Paris, 1807); of + \index{Nizze|FN}% + \index{Peyrard}% + Apollonius, several translations; also modern translations of Hero, Ptolemy, + \index{Apollonius}% + \index{Geometers, ancient}% + \index{Hero}% + \index{Pappus}% + \index{Proclus}% + \index{Ptolemy}% + \index{Theon}% + Pappus, Theon, Proclus, and several others.} +\PageSep{59} + +Prior to the discovery and publication of Diophantus, +however, algebra had already found its way into +\index{Algebra!name@the name of}% +\index{Algebra!among the Arabs|EtSeq}% +Europe. Towards the end of the fifteenth century +there appeared in Venice a work by an Italian Franciscan +monk named Lucas Paciolus on arithmetic and +\index{Paciolus, Lucas}% +geometry in which the elementary rules of algebra +were stated. This book was published (1494) in the +\MNote{Algebra among the Arabs.} +\index{Arabs!Algebra among the|EtSeq}% +early days of the invention of printing, and the fact +\index{Printing, invention of}% +that the name of \emph{algebra} was given to the new science +shows clearly that it came from the Arabs. It is true +that the signification of this Arabic word is still disputed, +but we shall not stop to discuss such matters, +for they are foreign to our purpose. Let it suffice +that the word has become the name for a science that +is universally known, and that there is not the slightest +ambiguity concerning its meaning, since up to the +present time it has never been employed to designate +anything else. + +We do not know whether the Arabs invented algebra +\PgLabel{59} +themselves or whether they took it from the +Greeks.\footnote + {See Appendix, \PgRef{152}.} +There is reason to believe that they possessed +the work of Diophantus, for when the ages of +barbarism and ignorance which followed their first +conquests had passed by, they began to devote themselves +to the sciences and to translate into Arabic all +the Greek works which treated of scientific subjects. +It is reasonable to suppose, therefore, that they also +\PageSep{60} +translated the work of Diophantus and that the same +work stimulated them to push their inquiries farther +in this science. + +Be that as it may, the Europeans, having received +\MNote{Algebra in Europe.} +\index{Algebra!Europe@in Europe}% +\index{Europe, algebra in}% +algebra from the Arabs, were in possession of it one +hundred years before the work of Diophantus was +known to them. They made, however, no progress +beyond equations of the first and second degree. In +\index{Equations!third@of the third degree}% +the work of Paciolus, which we mentioned above, the +\index{Paciolus, Lucas}% +general resolution of equations of the second degree, +such as we now have it, was not given. We find in +this work simply rules, expressed in bad Latin verses, +for resolving each particular case according to the +different combinations of the signs of the terms of +equation, and even these rules applied only to the +case where the roots were real and positive. Negative +\index{Negative roots}% +\index{Roots!negative}% +roots were still regarded as meaningless and superfluous. +It was geometry really that suggested to us the +\index{Geometry}% +use of negative quantities, and herein consists one of +the greatest advantages that have resulted from the +application of algebra to geometry,---a step which we +owe to Descartes. +\index{Descartes}% +\PgLabel{60} + +In the subsequent period the resolution of \emph{equations +of the third degree} was investigated and the discovery +for a particular case ultimately made by a mathematician +\index{Ferrous, Scipio|EtSeq}% +of Bologna named Scipio Ferreus (1515).\footnote + {The date is uncertain. Tartaglia gives 1506, Cardan 1515. Cantor prefers + \index{Cantor|FN}% + \index{Cardan}% + \index{Tartaglia}% + the latter.---\textit{Trans.}} +Two +other Italian mathematicians, Tartaglia and Cardan, +\PageSep{61} +subsequently perfected the solution of Ferreus and +rendered it general for all equations of the third degree. +At this period, Italy, which was the cradle of +\index{Italy, cradle of algebra in Europe}% +\MNote{Tartaglia (1500--1559). Cardan (1501--1576).} +\index{Cardan}% +\index{Tartaglia}% +algebra in Europe, was still almost the sole cultivator +of the science, and it was not until about the middle +of the sixteenth century that treatises on algebra began +to appear in France, Germany, and other countries. +\index{France}% +\index{Germany}% +The works of Peletier and Buteo were the first +\index{Buteo}% +\index{Peletier}% +which France produced in this science, the treatise of +the former having been printed in~1554 and that of +the latter in~1559. + +Tartaglia expounded his solution in bad Italian +verses in a work treating of divers questions and inventions +printed in~1546, a work which enjoys the +distinction of being one of the first to treat of modern +fortifications by bastions. + +About the same time (1545) Cardan published his +treatise \textit{Ars Magna}, or \textit{Algebra}, in which he left +scarcely anything to be desired in the resolution of +equations of the third degree. Cardan was the first to +perceive that equations had several roots and to distinguish +them into positive and negative. But he is +particularly known for having first remarked the so-called +\emph{irreducible case} in which the expression of the +\index{Irreducible case}% +real roots appears in an imaginary form. Cardan convinced +himself from several special cases in which the +equation had rational divisors that the imaginary form +did not prevent the roots from having a real value. +But it remained to be proved that not only were the +\PageSep{62} +roots real in the irreducible case, but that it was impossible +for all three together to be real except in that +case. This proof was afterwards supplied by Vieta, +\index{Vieta}% +and particularly by Albert Girard, from considerations +\index{Girard, Albert}% +touching the trisection of an angle. +\index{Angle, trisection of an}% +\index{Trisection of an angle}% + +We shall revert later on to the \emph{irreducible case of +equations of the third degree}, not solely because it presents +\MNote{The irreducible case.} +a new form of algebraical expressions which +have found extensive application in analysis, but because +it is constantly giving rise to unprofitable inquiries +with a view to reducing the imaginary form to +a real form and because it thus presents in algebra a +problem which may be placed upon the same footing +with the famous problems of the duplication of the +\index{Problems!solution@for solution}% +cube and the squaring of the circle in geometry. +\index{Circle!squaring of the}% +\index{Cube, duplication of the}% +\index{Squaring of the circle}% + +The mathematicians of the period under discussion +\index{Academies, rise of}% +were wont to propound to one another problems +for solution. These problems were in the nature of +public challenges and served to excite and to maintain +in the minds of thinkers that fermentation which +is necessary for the pursuit of science. The challenges +in question were continued down to the beginning of +the eighteenth century by the foremost mathematicians +of Europe, and really did not cease until the rise +of the Academies which fulfilled the same end in a +manner even more conducive to the progress of science, +partly by the union of the knowledge of their +various members, partly by the intercourse which they +maintained between them, and not least by the publication +\PageSep{63} +of their memoirs, which served to disseminate +the new discoveries and observations among all persons +interested in science. + +The challenges of which we speak supplied in a +\index{Academies, rise of}% +measure the lack of Academies, which were not yet +\MNote{Biquadratic equations.} +\index{Biquadratic equations}% +\index{Equations!fourth@of the fourth degree}% +in existence, and we owe to these passages at arms +many important discoveries in analysis. Such was +the resolution of \emph{equations of the fourth degree}, which +was propounded in the following problem. + +%[** TN: Next paragraph centered in the original] +\textit{To find three numbers in continued proportion of which +the sum is~$10$, and the product of the first two~$6$.} + +Generalising and calling the sum of the three numbers~$a$, +the product of the first two~$b$, and the first two +numbers themselves $x$,~$y$, we shall have, first, $xy = b$. +Owing to the continued proportion, the third number +will then be expressed by~$\dfrac{y^{2}}{x}$, so that the remaining +condition will give +\[ +x + y + \frac{y^{2}}{x} = a. +\] +From the first equation we obtain $x = \dfrac{b}{y}$, which substituted +in the second gives +\[ +\frac{b}{y} + y + \frac{y^{2}}{b} = a\Typo{,}{.} +\] +Removing the fractions and arranging the terms, we +get finally +\[ +y^{4} + by^{2} - aby + b^{2} = 0, +\] +an equation of the fourth degree with the second term +missing. + +According to Bombelli, of whom we shall speak +\index{Bombelli}% +\PageSep{64} +again, Louis Ferrari of Bologna resolved the problem +\index{Ferrari, Louis}% +by a highly ingenious method, which consists in +\MNote{Ferrari (1522-1565). Bombelli.} +\index{Bombelli}% +dividing the equation into two parts both of which +permit of the extraction of the square root. To do +this it is necessary to add to the two numbers quantities +whose determination depends on an equation of +the third degree, so that the resolution of equations +\index{Equations!fifth@of the fifth degree}% +of the fourth degree depends upon the resolution of +equations of the third and is therefore subject to the +same drawbacks of the irreducible case. + +The \textit{Algebra} of Bombelli was printed in Bologna +\index{Algebra!Italy@in Italy}% +in~1579\footnote + {This was the second edition. The first edition appeared in Venice in~1572.---\textit{Trans.}} +in the Italian language. It contains not only +the discovery of Ferrari but also divers other important +remarks on equations of the second and third +degree and particularly on the theory of radicals by +means of which the author succeeded in several cases +in extracting the imaginary cube roots of the two +binomials of the formula of the third degree in the irreducible +case, so finding a perfectly real result and +furnishing thus the most direct proof possible of the +reality of this species of expressions. + +Such is a succinct history of the first progress of +algebra in Italy. The solution of equations of the +\index{Italy, cradle of algebra in Europe}% +third and fourth degree was quickly accomplished. +But the successive efforts of mathematicians for over +two centuries have not succeeded in surmounting the +difficulties of the equation of the fifth degree. +\PageSep{65} + +Yet these efforts are far from having been in vain. +They have given rise to the many beautiful theorems +which we possess on the formation of equations, on +\MNote{Theory of equations.} +\index{Equations!theory of}% +the character and signs of the roots, on the transformation +of a given equation into others of which the +roots may be formed at pleasure from the roots of the +given equation, and finally, to the beautiful considerations +concerning the metaphysics of the resolution +of equations from which the most direct method of +arriving at their solution, when possible, has resulted. +All this has been presented to you in previous lectures +and would leave nothing to be desired if it were +but applicable to the resolution of equations of higher +degree. + +Vieta and Descartes in France, Harriot in England, +\index{Descartes}% +\index{Harriot}% +\index{Vieta}% +and Hudde in Holland, were the first after the +\index{Hudde}% +Italians whom we have just mentioned to perfect the +theory of equations, and since their time there is +scarcely a mathematician of note that has not applied +himself to its investigation, so that in its present state +this theory is the result of so many different inquiries +that it is difficult in the extreme to assign the author +of each of the numerous discoveries which constitute it. + +I promised to revert to the irreducible case. To +\index{Irreducible case}% +this end it will be necessary to recall the method +which seems to have led to the original resolution of +equations of the third degree and which is still employed +in the majority of the treatises on algebra. +\PageSep{66} +Let us consider the general equation of the third degree +deprived of its second term, which can always be +removed; in a word, let us consider the equation +\MNote{Equations of the third degree.} +\index{Equations!third@of the third degree}% +\[ +x^{3} + px + q = 0. +\] +Suppose +\[ +x = y + z, +\] +where $y$~and~$z$ are two new unknown quantities, of +which one consequently may be taken at pleasure and +determined as we think most convenient. Substituting +this value for~$x$, we obtain \emph{the transformed equation} +\[ +y^{3} + 3y^{2}z + 3yz^{2} + z^{3} + p(y + z) + q = 0. +\] +Factoring the two terms $3y^{2}z + 3yz^{2}$ we get +\[ +3yz(y + z), +\] +and the transformed equation may be written as follows: +\[ +y^{3} + z^{3} + (3yz + p)(y + z) + q = 0. +\] +Putting the factor multiplying $y + z$ equal to zero,---which +is permissible owing to the two undetermined +quantities involved,---we shall have the two equations +\[ +3yz + p = 0\Typo{.}{} +\] +and +\[ +y^{3} + z^{3} + q = 0\Typo{.}{,} +\] +from which $y$~and~$z$ can be determined. The means +which most naturally suggests itself to this end is to +take from the first equation the value of~$z$, +\[ +z = -\frac{p}{3y}, +\] +and to substitute it in the second equation, removing +the fractions by multiplication. So proceeding, we +\PageSep{67} +obtain the following equation of the sixth degree in~$y$, +called \emph{the reduced equation}, +\MNote{The reduced equation.} +\[ +y^{6} + qy^{3} - \frac{p^{3}}{27} = 0, +\] +which, since it contains two powers only of the unknown +quantity, of which one is the square of the +other, is resolvable after the manner of equations of +the second degree and gives immediately +\[ +y^{3} = -\frac{q}{2} + \sqrt{\frac{q^{2}}{4} + \frac{p^{3}}{27}}, +\] +from which, by extracting the cube root, we get +\[ +y = \sqrt[3]{-\frac{q}{2} + \sqrt{\frac{q^{2}}{4} + \frac{p^{3}}{27}}}, +\] +and finally, +\[ +x = y + z = y - \frac{p}{3y}\Add{.} +\] +This expression for~$x$ may be simplified by remarking +that the product of~$y$ by the radical +\[ + \sqrt[3]{-\frac{q}{2} - \sqrt{\frac{q^{2}}{4} + \frac{p^{3}}{27}}}\Add{,} +\] +supposing all the quantities under the sign to be multiplied +together, is +\[ +\sqrt[3]{-\frac{p^{3}}{27}} = -\frac{p}{3}. +\] +The term $\dfrac{p}{3y}$, accordingly, takes the form +\[ +-\sqrt[3]{-\frac{q}{2} - \sqrt{\frac{q^{2}}{4} + \frac{p^{3}}{27}}}, +\] +and we have +\[ +x = \sqrt[3]{-\frac{q}{2} + \sqrt{\frac{q^{2}}{4} + \frac{p^{3}}{27}}} + + \sqrt[3]{-\frac{q}{2} - \sqrt{\frac{q^{2}}{4} + \frac{p^{3}}{27}}}, +\] +\PageSep{68} +an expression in which the square root underneath the +cubic radical occurs in both its plus and minus forms +and where consequently there can, on this score, be +no occasion for ambiguity. + +This last expression is known as the \emph{Rule of Cardan}, +\index{Cardan}% +\index{Rule!Cardan's}% +\MNote{Cardan's rule.} +and there has hitherto been no method devised +for the resolution of equations of the third degree +which does not lead to it. Since cubic radicals naturally +present but a single value, it was long thought +that Cardan's rule could give but one of the roots of +the equation, and that in order to find the two others +we must have recourse to the original equation and divide +it by~$x - a$, $a$~being the first root found. The +resulting quotient being an equation of the second degree +may be resolved in the usual manner. The division +in question is not only always possible, but it is +also very easy to perform. For in the case we are +considering the equation being +\[ +x^{3} + px + q = 0, +\] +if $a$~is one of the roots we shall have +\[ +a^{3} + pa + q = 0, +\] +which subtracted from the preceding will give +\[ +x^{3} - a^{3} + p(x - a) = 0, +\] +a quantity divisible by~$x - a$ and having as its resulting +quotient +\[ +x^{2} + ax + a^{2} + p = 0; +\] +so that the new equation which is to be resolved for +finding the two other roots will be +\PageSep{69} +\[ +x^{2} + ax + a^{2} + p = 0, +\] +from which we have at once +\[ +x = -\frac{a}{2} ± \sqrt{-p - \frac{3a^{2}}{4}}. +\] + +I see by the \textit{Algebra} of Clairaut, printed in~1746, +\index{Clairaut}% +and by D'Alem\-bert's article on the \emph{Irreducible Case} in +\index{Irreducible case}% +\MNote{The generality of algebra.} +the first \textit{Encyclopædia} that the idea referred to prevailed +even in that period. But it would be the height +of injustice to algebra to accuse it of not yielding results +\index{Algebra!generality@the generality of}% +which were possessed of all the generality of +which the question was susceptible. The sole requisite +is to be able to read the peculiar hand-writing +\index{Algebra!hand-writing of}% +\index{Hand-writing of algebra}% +of algebra, and we shall then be able to see in it everything +which by its nature it can be made to contain. +In the case which we are considering it was forgotten +that every cube root may have three values, as every +square root has two. For the extraction of the cube +root of~$a$ for example is merely equivalent to the resolution +of the equation of the third degree $x^{3} - a = 0$. +Making $x = y\sqrt[3]{a}$, this last equation passes into the +simpler form $y^{3} - 1 = 0$, which has the root $y = 1$. +Then dividing by~$y - 1$ we have +\[ +y^{2} + y + 1 = 0, +\] +from which we deduce directly the two other roots +\[ +y = \frac{-1 ± \sqrt{-3}}{2}. +\] +These three roots, accordingly, are the three cube +roots of unity, and they may be made to give the three +cube roots of any other quantity~$a$ by multiplying +\PageSep{70} +them by the ordinary cube root of that quantity. It +is the same with roots of the fourth, the fifth, and all +the following degrees. For brevity, let us designate +the two roots +\MNote{The three cube roots of a quantity.} +\index{Cube roots of a quantity, the three}% +\[ +\frac{-1 + \sqrt{-3}}{2} \quad\text{and}\quad \frac{-1 - \sqrt{-3}}{2}\Typo{,}{} +\] +by $m$~and~$n$. It will be seen that they are imaginary, +although their cube is real and equal to~$1$, as we may +readily convince ourselves by raising them to the +third power. We have, therefore, for the three cube +roots of~$a$, +\[ +\sqrt[3]{a},\quad m\sqrt[3]{a},\quad n\sqrt[3]{a}. +\] + +Now, in the resolution of the equation of the third +degree above considered, on coming to the reduced +expression $y^{3} = A$, where for brevity we suppose +\[ +A = -\frac{q}{2} + \sqrt{\frac{q^{2}}{4} + \frac{p^{3}}{27}}, +\] +we deduced the following result only: +\[ +y = \sqrt[3]{A}. +\] +But from what we have just seen, it is clear that we +shall have not only +\[ +y = \sqrt[3]{A}, +\] +but also +\[ +y = m\sqrt[3]{A} \quad\text{and}\quad y = n\sqrt[3]{A}. +\] +The root~$x$ of the equation of the third degree which +we found equal to +\[ +y - \frac{p}{3y}, +\] +will therefore have the three following values +\PageSep{71} +\[ +\sqrt[3]{A} - \frac{p}{3\sqrt[3]{A}},\quad +m\sqrt[3]{A} - \frac{p}{3m\sqrt[3]{A}},\quad +n\sqrt[3]{A} - \frac{p}{3n\sqrt[3]{A}}, +\] +\MNote{The roots of equations of the third degree.} +\index{Roots!equations@of equations of the third degree}% +\index{Third degree, equations of the}% +which will be the three roots of the equation proposed. +But making +\[ +B = -\frac{q}{2} - \sqrt{\frac{q^{2}}{4} + \frac{p^{3}}{27}}, +\] +it is clear that +\[ +AB = -\frac{p^{3}}{27}, +\] +whence +\[ +\sqrt[3]{A} × \sqrt[3]{B} = -\frac{p}{3}. +\] +Substituting $\sqrt[3]{B}$ for $-\dfrac{p}{3\sqrt[3]{A}}$, and remarking that +$mn = 1$, and that consequently +\[ +\frac{1}{m} = n,\quad \frac{1}{n} = m, +\] +the three roots which we are considering will be expressed +as follows: +%[** TN: Set on two lines in the original] +\[ +x = \sqrt[3]{A} + \sqrt[3]{B},\quad +x = m\sqrt[3]{A} + n\sqrt[3]{B},\quad +x = n\sqrt[3]{A} + m\sqrt[3]{B}. +\] + +We see, accordingly, that when properly understood +the ordinary method gives the three roots directly, +and gives three only. I have deemed it necessary +to enter upon these slight details for the reason +that if on the one hand the method was long taxed +with being able to give but one root, on the other +hand when it was seen that it really gave three it was +thought that it should have given six, owing to the +\PageSep{72} +false employment of all the possible combinations of +the three cubic roots of unity, viz., $1$,~$m$,~$n$, with the +\index{Unity, three cubic roots of}% +two cubic radicals $\sqrt[3]{A}$~and~$\sqrt[3]{B}$. + +We could have arrived directly at the results which +\MNote{A direct method of reaching the roots.} +we have just found by remarking that the two equations +\[ +y^{3} + z^{3} + q = 0 \quad\text{and}\quad 3yz + p = 0 +\] +give +\[ +y^{3} + z^{3} = -q \quad\text{and}\quad y^{3}z^{3} = -\frac{p^{3}}{27}; +\] +where it will be seen at once that $y^{3}$~and~$z^{3}$ are the +roots of an equation of the second degree of which +the second term is~$q$ and the third~$-\dfrac{p^{3}}{27}$. This equation, +which is called \emph{the reduced equation}, will accordingly +have the form +\[ +u^{2} + qu - \frac{p^{3}}{27} = 0; +\] +and calling $A$~and~$B$ its two roots we shall have immediately +\[ +y = \sqrt[3]{A},\quad z = \sqrt[3]{B}, +\] +where it will be observed that $A$~and~$B$ have the same +values that they had in the previous discussion. Now, +from what has gone before, we shall likewise have +\[ +y = m\sqrt[3]{A} \quad\text{or}\quad y = n\sqrt[3]{A}, +\] +and the same will also hold good for~$z$. But the equation +\[ +zy = -\frac{p}{3}, +\] +of which we have employed the cube only, limits these +\PageSep{73} +values and it is easy to see that the restriction requires +the three corresponding values of~$z$ to be +\[ +\sqrt[3]{B},\quad m\sqrt[3]{B},\quad n\sqrt[3]{B}; +\] +whence follow for the value of~$x$, which is equal to~$y + z$, +the same three values which we found above. + +As to the form of these values it is apparent, first, +that so long as $A$~and~$B$ are real quantities, one only +\MNote{The form of the roots\Add{.}} +of them can be real, for $m$~and~$n$ are imaginary. They +can consequently all three be real only in the case +where the roots $A$~and~$B$ of the reduced equation are +imaginary, that is, when the quantity +\[ +\frac{q^{2}}{4} + \frac{p^{3}}{27} +\] +beneath the radical sign is negative, which happens +only when $p$~is negative and greater than +\[ +3\sqrt[3]{\frac{q^{2}}{4}}. +\] +And this is the so-called \emph{irreducible case}. +\index{Irreducible case}% + +Since in this event +\[ +\frac{q^{2}}{4} + \frac{p^{3}}{27} +\] +is a negative quantity, let us suppose it equal to~$-g^{2}$, +$g$~being any real quantity whatever. Then making, +for the sake of simplicity, +\[ +-\frac{q}{2} = f, +\] +the two roots $A$~and~$B$ of the reduced equation assume +the form +\[ +A = f + g\sqrt{-1},\quad B = f - g\sqrt{-1}. +\] +\PageSep{74} + +Now I say that if $\sqrt[3]{A} + \sqrt[3]{B}$, which is one of the +\MNote{The reality of the roots\Add{.}} +\index{Roots!reality@the reality of the}% +roots of the equation of the third degree, is real, then +the two other roots, expressed by +\[ +m\sqrt[3]{A} + n\sqrt[3]{B} \quad\text{and}\quad n\sqrt[3]{A} + m\sqrt[3]{B}, +\] +will also be real. Put +\[ +\sqrt[3]{A} = t,\quad \sqrt[3]{B} = u; +\] +we shall have +\[ +t + u = h, +\] +where $h$~by hypothesis is a real quantity. Now, +\[ +tu = \sqrt[3]{AB} \quad\text{and}\quad AB = f^{2} + g^{2}, +\] +therefore +\[ +tu = \sqrt[3]{f^{2} + g^{2}}; +\] +squaring the equation $t + u = h$ we have +\[ +t^{2} + 2tu + u^{2} = h^{2}; +\] +from which subtracting~$4tu$ we obtain +\[ +(t - u)^{2} = h^{2} - 4\sqrt[3]{f^{2} + g^{2}}. +\] +I observe that this quantity must necessarily be negative, +for if it were positive and equal to~$k^{2}$ we should +have +\[ +(t - u)^{2} = k^{2}, +\] +whence +\[ +t - u = k. +\] +Then since +\[ +t + u = h, +\] +it would follow that +\[ +t = \frac{h + k}{2} \quad\text{and}\quad u = \frac{h - k}{2}, +\] +\PageSep{75} +both of which are real quantities. But then $t^{3}$~and~$u^{3}$ +would also be real quantities, which is contrary to +our hypothesis, since these quantities are equal to $A$~and~$B$, +both of which are imaginary. + +The quantity +\[ +h^{2} - 4\sqrt[3]{f^{2} + g^{2}} +\] +therefore, is necessarily negative. Let us suppose it +equal to~$-k^{2}$; we shall have then +\[ +(t - u)^{2} = -k^{2}, +\] +and extracting the square root +\[ +t - u = k\sqrt{-1}; +\] +\MNote{The form of the two cubic radicals.} +\index{Cubic radicals}% +\index{Radicals, cubic}% +whence +\[ +t = \frac{h + k\sqrt{-1}}{2} = \sqrt[3]{A},\quad +u = \frac{h - k\sqrt{-1}}{2} = \sqrt[3]{B}. +\] + +Such necessarily will be the form of the two cubic +radicals +\[ +\sqrt[3]{f + g\sqrt{-1}} \quad\text{and}\quad \sqrt[3]{f - g\sqrt{-1}}, +\] +a form at which we can arrive directly by expanding +these roots according to the Newtonian theorem into +series. But since proofs by series are apt to leave +some doubt in the mind, I have sought to render the +preceding discussion entirely independent of them. + +If, therefore, +\[ +\sqrt[3]{A} + \sqrt[3]{B} = h, +\] +we shall have +\[ +\sqrt[3]{A} = \frac{h + k\sqrt{-1}}{2} \quad\text{and}\quad +\sqrt[3]{B} = \frac{h - k\sqrt{-1}}{2}. +\] +Now we have found above that +\[ +m = \frac{-1 + \sqrt{-3}}{2},\quad n = \frac{-1 - \sqrt{-3}}{2}; +\] +\PageSep{76} +wherefore, multiplying these quantities together, we +have +\begin{align*} +m\sqrt[3]{A} + n\sqrt[3]{B} &= \frac{-h + k\sqrt{-3}}{2} \\ +\intertext{and} +n\sqrt[3]{A} + m\sqrt[3]{B} &= \frac{-h - k\sqrt{-3}}{2}, +\end{align*} +which are real quantities. Consequently, if the root~$h$ +\MNote{Condition of the reality of the roots.} +\index{Reality of roots}% +\index{Roots!reality@the reality of the}% +is real, the two other roots also will be real in the +irreducible case and they will be real in that case only. + +But the invariable difficulty is, to demonstrate directly +that +\[ +\sqrt[3]{f + g\sqrt{-1}} + \sqrt[3]{f - g\sqrt{-1}}, +\] +which we have supposed equal to~$h$, is always a real +quantity whatever be the values of $f$~and~$g$. In particular +cases the demonstration can be effected by the +extraction of the cube root, when that is possible. For +example, if $f = 2$, $g = 11$, we shall find that the cube +root of~$2 + 11\sqrt{-1}$ will be~$2 + \sqrt{-1}$, and similarly +that the cube root of~$2 - 11\sqrt{-1}$ will be~$2 - \sqrt{-1}$, +and the sum of the radicals will be~$4$. An infinite +number of examples of this class may be constructed +and it was through the consideration of such instances +that Bombelli became convinced of the reality of the +imaginary expression in the formula for the irreducible +case. But forasmuch as the extraction of cube roots +is in general possible only by means of series, we cannot +arrive in this way at a general and direct demonstration +of the proposition under consideration. +\PageSep{77} + +It is otherwise with square roots and with all roots +of which the exponents are powers of~$2$. For example, +\MNote{Extraction of the square roots of two imaginary binomials.} +\index{Binomials, extraction of the square roots of two imaginary}% +\index{Imaginary binomials, square roots of}% +if we have the expression +\[ +\sqrt{f + g\sqrt{-1}} + \sqrt{f - g\sqrt{-1}}, +\] +composed of two imaginary radicals, its square will be +\[ +2f + 2\sqrt{f^{2} + g^{2}}, +\] +a quantity which is necessarily positive. Extracting +the square root, so as to obtain the equivalent expression, +we have +\[ +\sqrt{2f + 2\sqrt{f^{2} + g^{2}}}, +\] +for the real value of the imaginary quantity we started +with. But if instead of the sum we had had the difference +between the two proposed imaginary radicals +we should then have obtained for its square the following +expression +\[ +2f - 2\sqrt{f^{2} + g^{2}}, +\] +a quantity which is necessarily negative; and, taking +the square root of the latter, we should have obtained +the simple imaginary expression +\[ +\sqrt{2f - 2\sqrt{f^{2} + g^{2}}}. +\] + +Further, if the quantity +\[ +\sqrt[4]{f + g\sqrt{-1}} + \sqrt[4]{f - g\sqrt{-1}} +\] +were given, we should have, by squaring, the form +\begin{multline*} +%[** TN: Moved equality sign to second line] +\sqrt{f + g\sqrt{-1}} + \sqrt{f - g\sqrt{-1}} + 2\sqrt[4]{f^{2} + g^{2}} \\ += \sqrt{2f + 2\sqrt{f^{2} + g^{2}}} + 2\sqrt[4]{f^{2} + g^{2}}, +\end{multline*} +a real and positive quantity. Extracting the square +\PageSep{78} +root of this expression we should obtain a real value +for the original quantity; and so on for all the other +remaining even roots. But if we should attempt to +apply the preceding method to cubic radicals we +should be led again to equations of the third degree +in the irreducible case. + +For example, let +\MNote{Extraction of the cube roots of two imaginary binomials.} +\[ +\sqrt[3]{f + g\sqrt{-1}} + \sqrt[3]{f - g\sqrt{-1}} = x. +\] +Cubing, we get +\[ +2f + 3\sqrt[3]{f^{2} + g^{2}}\left( +\sqrt[3]{f + g\sqrt{-1}} + \sqrt[3]{f - g\sqrt{-1}} +\right) = x^{3}; +\] +that is +\[ +2f + 3x\sqrt[3]{f^{2} + g^{2}} = x^{3}, +\] +or, with the terms properly arranged, +\[ +x^{3} - 3x\sqrt[3]{f^{2} + g^{2}} - 2f = 0, +\] +the general formula of the irreducible case, for +\[ +\frac{1}{4}(2f)^{2} + \frac{1}{27}\bigl(-3\sqrt[3]{f^{2} + g^{2}}\bigr)^{3} + = -g^{2}. +\] +If $g = 0$ we shall have $x = 2\sqrt[3]{f}$. The sole \textit{desideratum}, +therefore, is to demonstrate that if $g$~have any value +whatever, $x$~has a corresponding real value. Now the +second last equation gives +\[ +\sqrt[3]{f^{2} + g^{2}} = \frac{x^{3} - 2f}{3x}\Add{,} +\] +and cubing we get +\[ +f^{2} + g^{2} = \frac{x^{9} - 6x^{6}f + 12x^{3}f^{2} - 8f^{3}}{27x^{3}}, +\] +whence +\[ +g^{2} = \frac{x^{9} - 6x^{6}f - 15x^{3}f^{2} - 8f^{3}}{27x^{3}}, +\] +\PageSep{79} +an equation which may be written as follows +\[ +g^{2} = \frac{(x^{3} - 8f)(x^{3} + f)^{2}}{27x^{3}}, +\] +or, better, thus: +\[ +g^{2} = \frac{1}{27}\left(1 - \frac{8f}{x^{3}}\right)(x^{3} + f)^{2}. +\] + +It is plain from the last expression that $g$~is zero +when $x^{3} = 8f$; further, that $g$~constantly and uninterruptedly +\MNote{General theory of the reality of the roots\Add{.}} +\index{Roots!reality@the reality of the}% +increases as $x$~increases; for the factor +$(x^{3} + f)^{2}$ augments constantly, and the other factor +$1 - \dfrac{8f}{x^{3}}$ also keeps increasing, seeing that as the denominator~$x^{3}$ +increases the negative part~$\dfrac{8f}{x^{3}}$, which is +originally equal to~$1$, keeps constantly growing less +than~$1$. Therefore, if the value of~$x^{3}$ be increased by +insensible degrees from~$8f$ to infinity, the value of~$g^{2}$ +will also augment by insensible and corresponding +degrees from zero to infinity. And therefore, reciprocally, +to every value of~$g^{2}$ from zero to infinity there +must correspond some value of~$x^{3}$ lying between the +limits of~$8f$ and infinity, and since this is so whatever +be the value of~$f$ we may legitimately conclude that, +be the values of $f$~and~$g$ what they may, the corresponding +value of~$x^{3}$ and consequently also of~$x$ is +always real. + +But how is this value of~$x$ to be assigned? It would +\index{Imaginary expressions|EtSeq}% +seem that it can be represented only by an imaginary +expression or by a series which is the development of +an imaginary expression. Are we to regard this class +of imaginary expressions, which correspond to real +\PageSep{80} +values, as constituting a new species of algebraical expressions +which although they are not, like other expressions, +\MNote{Imaginary expressions\Add{.}} +susceptible of being numerically evaluated +in the form in which they exist, yet possess the indisputable +advantage---and this is the chief requisite---that +they can be employed in the operations of algebra +exactly as if they did not contain imaginary expressions. +They further enjoy the advantage of having a +wide range of usefulness in geometrical constructions, +as we shall see in the theory of angular sections, so +\index{Angular sections, theory of}% +that they can always be exactly represented by lines; +while as to their numerical value, we can always find +it approximately and to any degree of exactness that +we desire, by the approximate resolution of the equation +on which they depend, or by the use of the common +trigonometrical tables. + +It is demonstrated in geometry that if in a circle +having the radius~$r$ an arc be taken of which the chord +is~$c$, and that if the chord of the third part of that arc +be called~$x$, we shall have for the determination of~$x$ +the following equation of the third degree +\[ +x^{3} - 3r^{2}x + r^{2}c = 0, +\] +an equation which leads to the irreducible case since +$c$~is always necessarily less than~$2r$, and which, owing +to the two undetermined quantities $r$~and~$c$, may be +taken as the type of all equations of this class. For, +if we compare it with the general equation +\[ +x^{3} + px + q = 0, +\] +we shall have +\PageSep{81} +\[ +r = \sqrt{-\frac{p}{3}} \quad\text{and}\quad c = -\frac{3q}{p} +\] +so that by trisecting the arc corresponding to the +chord~$c$ in a circle of the radius~$r$ we shall obtain at +\MNote{Trisection of an angle.} +\index{Angle, trisection of an}% +\index{Trisection of an angle}% +once the value of a root~$x$, which will be the chord of +the third part of that arc. Now, from the nature of a +circle the same chord~$c$ corresponds not only to the +arc~$s$ but (calling the entire circumference~$u$) also to +the arcs +\[ +u - s,\quad 2u + s,\quad 3u - s, \dots\Add{.} +\] +Also the arcs +\[ +u + s,\quad 2u - s,\quad 3u + s, \dots +\] +have the same chord, but taken negatively, for on +completing a full circumference the chords become +zero and then negative, and they do not become positive +again until the completion of the second circumference, +as you may readily see. Therefore, the values +of~$x$ are not only the chord of the arc~$\dfrac{s}{3}$ but also +the chords of the arcs +\[ +\frac{u - s}{3},\quad \frac{2u + s}{3}, +\] +and these chords will be the three roots of the equation +proposed. If we were to take the succeeding arcs +which have the same chord~$c$ we should be led simply +to the same roots, for the arc~$3u - s$ would give the +chord of~$\dfrac{3u - s}{3}$, that is, of~$u - \dfrac{s}{3}$, which we have already +seen is the same as that of~$\dfrac{s}{3}$, and so with the +rest. +\PageSep{82} + +Since in the irreducible case the coefficient~$p$ is +\index{Irreducible case}% +necessarily negative, the value of the given chord~$c$ +\MNote{Trigonometrical solution.} +will be positive or negative according as $q$~is positive +or negative. In the first case, we take for~$s$ the arc +subtended by the positive chord $c = -\dfrac{3q}{p}$. The second +case is reducible to the first by making $x$~negative, +whereby the sign of the last term is changed; so +that if again we take for~$s$ an arc subtended by the +positive chord~$\dfrac{3q}{p}$, we shall have simply to change +the sign of the three roots. + +Although the preceding discussion may be deemed +sufficient to dispel all doubts concerning the nature +of the roots of equations of the third degree, we propose +\index{Equations!third@of the third degree}% +\index{Third degree, equations of the}% +adding to it a few reflexions concerning the +method by which the roots are found. The method +which we have propounded in the foregoing and which +is commonly called \emph{Cardan's method}, although it seems +\index{Cardan}% +to me that we owe it to Hudde, has been frequently +\index{Hudde}% +criticised, and will doubtless always be criticised, for +giving the roots in the irreducible case in an imaginary +form, solely because a supposition is here made which +is contradictory to the nature of the equation. For +the very gist of the method consists in its supposing +\index{Undetermined quantities}% +the unknown quantity equal to two undetermined +quantities $y + z$, in order to enable us afterwards to +separate the resulting equation +\[ +y^{3} + z^{3} + (3yz + p)(y + z) + q = 0 +\] +into the two following: +\PageSep{83} +\[ +3yz + p = 0 \quad\text{and}\quad y^{3} + z^{3} + q = 0. +\] +Now, throwing the first of these into the form +\MNote{The method of indeterminates.} +\index{Indeterminates, the method of}% +\[ +y^{3}z^{3} = -\frac{p^{3}}{27} +\] +it is plain that the question reduces itself to finding +two numbers $y^{3}$~and~$z^{3}$ of which the sum is~$-q$ and +the product~$-\dfrac{p^{3}}{27}$, which is impossible unless the +square of half the sum exceed the product, for the +difference between these two quantities is equal to the +square of half the difference of the numbers sought. + +The natural conclusion was that it was not at all +astonishing that we should reach imaginary expressions +\index{Imaginary expressions}% +when proceeding from a supposition which it +was impossible to express in numbers, and so some +writers have been induced to believe that by adopting +a different course the expression in question could be +avoided and the roots all obtained in their real form. +\index{Reality of roots}% +\index{Roots!reality@the reality of the}% + +Since pretty much the same objection can be advanced +against the other methods which have since +been found and which are all more or less based upon +the method of indeterminates, that is, the introduction +of certain arbitrary quantities to be determined +so as to satisfy the conditions of the problem,---we +propose to consider the question of the reality of the +roots by itself and independently of any supposition +whatever. Let us take again the equation +\[ +x^{3} + px + q = 0; +\] +and let us suppose that its three roots are $a$,~$b$,~$c$. +\PageSep{84} + +By the theory of equations the left-hand side of +\index{Equations!theory of}% +the preceding expression is the product of three quantities +\MNote{An independent consideration.} +\[ +x - a,\quad x - b,\quad x - c, +\] +which, multiplied together, give +\[ +x^{3} - (a + b + c)x^{2} + (ab + ac + bc)x - abc; +\] +and comparing the corresponding terms, we have +\[ +a + b + c = 0,\quad +ab + ac + bc = p,\quad +abc = -q. +\] +As the degree of the equation is odd we may be certain, +as you doubtless already know and in any event +will clearly see from the lecture which is to follow, +that it has necessarily one real root. Let that root +be~$c$. The first of the three equations which we have +just found will then give +\[ +c = -a - b, +\] +whence it is plain that $a + b$ is also necessarily a real +quantity. Substituting the last value of~$c$ in the second +and third equations, we have +\[ +ab - a^{2} - ab - ab - b^{2} = p,\quad -ab(a + b) = -q, +\] +or +\[ +a^{2} + ab + b^{2} = -p,\quad ab(a + b) = q, +\] +from which are to be found $a$~and~$b$. The last equation +gives $ab = \dfrac{q}{a + b}$ from which I conclude that $ab$ +also is necessarily a real quantity. Let us consider +now the quantity $\dfrac{q^{2}}{4} + \dfrac{p^{3}}{27}$ or, clearing of fractions, the +quantity $27q^{2} + 4p^{3}$, upon the sign of which the irreducible +case depends. Substituting in this for $p$~and~$q$ +their value as given above in terms of $a$~and~$b$, +\PageSep{85} +we shall find that when the necessary reductions are +made the quantity in question is equal to the square of +\MNote{New view of the reality of the roots.} +\index{Reality of roots}% +\index{Roots!reality@the reality of the}% +\[ +2a^{3} - 2b^{3} + 3a^{2}b - 3ab^{2} +\] +taken negatively; so that by changing the signs and +extracting the square root we shall have +\[ +2a^{3} - 2b^{3} + 3a^{2}b - 3ab^{2} = \sqrt{-27q^{2} - 4p^{3}}, +\] +whence it is easy to infer that the two roots $a$~and~$b$ +cannot be real unless the quantity $27q^{2} + 4p^{3}$ be negative. +But I shall show that in that case, which is as +we know the irreducible case, the two roots $a$~and~$b$ +are necessarily real. The quantity +\[ +2a^{3} - 2b^{3} + 3a^{2}b - 3ab^{2} +\] +may be reduced to the form +\[ +(a - b)(2a^{2} + 2b^{2} + 5ab), +\] +as multiplication will show. Now, we have already +seen that the two quantities $a + b$ and $ab$ are necessarily +real, whence it follows that +\[ +2a^{2} + 2b^{2} + 5ab = 2(a + b)^2 + ab +\] +is also necessarily real. Hence the other factor~$a - b$ +is also real when the radical $\sqrt{-27q^{2} - 4p^{3}}$ is real. +Therefore $a + b$ and $a - b$ being real quantities, it follows +that $a$~and~$b$ are real. + +We have already derived the preceding theorems +from the form of the roots themselves. But the present +demonstration is in some respects more general +and more direct, being deduced from the fundamental +principles of the problem itself. We have made no +\PageSep{86} +suppositions, and the particular nature of the irreducible +case has introduced no imaginary quantities. + +\MNote{Final solution on the new view.} +But the values of $a$~and~$b$ still remain to be found +from the preceding equations. And to this end I observe +that the left-hand side of the equation +\[ +a^{3} - b^{3} + \frac{3}{2}(a^{2}b - ab^{2}) + = \frac{1}{2}\sqrt{-27q^{2} - 4p^{3}} +\] +can be made a perfect cube by adding the left-hand +side of the equation +\[ +ab(a + b) = q, +\] +multiplied by $\dfrac{3\sqrt{-3}}{2}$, and that the root of this cube is +\[ +\frac{1 - \sqrt{-3}}{2}b - \frac{1 + \sqrt{-3}}{2}a +\] +so that, extracting the cube root of both sides, we +shall have the expression +\[ +\frac{1 - \sqrt{-3}}{2}b - \frac{1 + \sqrt{-3}}{2}a +\] +expressed in known quantities. And since the radical +$\sqrt{-3}$ may also be taken negatively, we shall also +have the expression +\[ +\frac{1 + \sqrt{-3}}{2}b - \frac{1 - \sqrt{-3}}{2}a +\] +expressed in known quantities, from which the values +of $a$~and~$b$ can be deduced. And these values will +contain the imaginary quantity~$\sqrt{-3}$, which was introduced +by multiplication, and will be reducible to +the same form with the two roots +\PageSep{87} +\[ +m\sqrt[3]{A} + n\sqrt[3]{B} \quad\text{and}\quad n\sqrt[3]{A} + m\sqrt[3]{B}, +\] +which we found above. The third root +\MNote{Office of imaginary quantities.} +\[ +c = -a - b +\] +will then be expressed by $\sqrt[3]{A} + \sqrt[3]{B}$. + +By this method we see that the imaginary quantities +\index{Imaginary quantities, office of the}% +employed have simply served to facilitate the extraction +of the cube root without which we could not +determine separately the values of $a$~and~$b$. And since +it is apparently impossible to attain this object by a +different method, we may regard it as a demonstrated +truth that the general expression of the roots of an +equation of the third degree in the irreducible case +cannot be rendered independent of imaginary quantities. + +Let us now pass to \emph{equations of the fourth degree}. +\index{Equations!fourth@of the fourth degree}% +We have already said that the artifice which was originally +employed for resolving these equations consisted +in so arranging them that the square root of +the two sides could be extracted, by which they were +reduced to equations of the second degree. The following +is the procedure employed. Let +\[ +x^{4} + px^{2} + qx + r = 0 +\] +be the general equation of the fourth degree deprived +of its second term, which can always be eliminated, +as you know, by increasing or diminishing the roots +by a suitable quantity. Let the equation be put in +the form +\[ +x^{4} = -px^{2} - qx - r, +\] +\PageSep{88} +and to each side let there be added the terms $2x^{2}y + y^{2}$, +which contain a new undetermined quantity~$y$ but +\MNote{Biquadratic equations.} +\index{Biquadratic equations}% +\index{Equations!biquadratic}% +which still leave the left-hand side of the equation a +square. We shall then have +\[ +(x^{2} + y)^{2} = (2y - p)x^{2} - qx + y^{2} - r. +\] +We must now make the right-hand side also a square. +To this end it is necessary that +\[ +4(2y - p)(y^{2} - r) = q^{2}, +\] +in which case the square root of the right-hand side +will have the form +\[ +x\sqrt{2y - p} - \frac{q}{2\sqrt{2y - p}}. +\] +Supposing then that the quantity~$y$ satisfies the equation +\[ +4(2y - p)(y^{2} - r) = q^{2}, +\] +which developed becomes +\[ +y^{3} - \frac{py^{2}}{2} - ry + \frac{pr}{2} - \frac{q^{2}}{8} = 0, +\] +and which, as we see, is an equation of the third degree, +the equation originally given may be reduced to +the following by extracting the square root of its two +members,~viz.: +\[ +x^{2} + y = x\sqrt{2y - p} - \frac{q}{2\sqrt{2y - p}}, +\] +where we may take either the plus or the positive +value for the radical $\sqrt{2y - p}$, and shall consequently +have two equations of the second degree to which the +given equation has been reduced and the roots of +which will give the four roots of the original equation. +\PageSep{89} +All of which furnishes us with our first instance of the +decomposition of equations into others of lower degree. + +The method of Descartes which is commonly followed +\index{Descartes}% +in the elements of algebra is based upon the +\MNote{The method of Descartes.} +same principle and consists in assuming at the outset +that the proposed equation is produced by the multiplication +of two equations of the second degree, as +\[ +x^{2} - ux + s = 0 \quad\text{and}\quad x^{2} + ux + t = 0, +\] +where $u$,~$s$, and~$t$ are indeterminate coefficients. Multiplying +\index{Coefficients!indeterminate}% +\index{Indeterminate coefficients}% +them together we have +\[ +x^{4} + (s + t - u^{2})x + (s - t)ux + st = 0, +\] +comparison of which with the original equation gives +\[ +s + t - u^{2} = p,\quad (s - t)u = q \quad\text{and}\quad st = r. +\] +The first two equations give +\[ +2s = p + u^{2} + \frac{q}{u},\quad 2t = p + u^{2} - \frac{q}{u}. +\] +And if these values be substituted in the third equation +of condition $st = r$, we shall have an equation of +the sixth degree in~$u$, which owing to its containing +only even powers of~$u$ is resolvable by the rules for +cubic equations. And if we substitute in this equation +$2y - p$ for~$u^{2}$, we shall obtain in~$y$ the same reduced +equation that we found above by the old method. + +Having the value of~$u^{2}$ we have also the values of +$s$~and~$t$, and our equation of the fourth degree will be +decomposed into two equations of the second degree +which will give the four roots sought. This method, +as well as the preceding, has been the occasion of some +\PageSep{90} +hesitancy as to which of the three roots of the reduced +cubic equation in $u^{2}$ or~$y$ should be employed. +\MNote{The determined character of the roots\Add{.}} +The difficulty has been well resolved in Clairaut's +\index{Clairaut}% +\textit{Algebra}, where we are led to see directly that we always +obtain the same four roots or values of~$x$ whatever +root of the reduced equation we employ. But +this generality is needless and prejudicial to the simplicity +which is to be desired in the expression of +the roots of the proposed equation, and we should +prefer the formulæ which you have learned in the +principal course and in which the three roots of the +reduced equation are contained in exactly the same +manner. + +The following is another method of reaching the +same formulæ, less direct than that which has already +been expounded to you, but which, on the other hand +has the advantage of being analogous to the method +of Cardan for equations of the third degree. +\index{Cardan}% + +I take up again the equation +\[ +x^{4} + px^{2} + qx + r = 0, +\] +and I suppose +\[ +x = y + z + t. +\] +Squaring I obtain +\[ +x^{2} = y^{2} + z^{2} + t^{2} + 2(yz + yt + zt). +\] +Squaring again I have +\[ +%[** TN: Set on two lines in original] +x^{4} = (y^{2} + z^{2} + t^{2})^{2} + 4(y^{2} + z^{2} + t^{2})(yz + yt + zt) ++ 4(yz + yt + zt)^{2}; +\] +but +\begin{align*} +%[** TN: Re-broken] +(yz + yt + zt)^{2} + &= y^{2}z^{2} + y^{2}t^{2} + z^{2}t^{2} + + 2y^{2}zt + 2yz^{2}t + 2yzt^{2} \\ + &= y^{2}z^{2} + y^{2}t^{2} + z^{2}t^{2} + 2yzt(y + z + t). +\end{align*} +\PageSep{91} +Substituting these three values of $x$,~$x^{2}$, and~$x^{4}$ in the +original equation, and bringing together the terms +multiplied by~$y + z + t$ and the terms multiplied by~$yz + yt + zt$, +\MNote{A third method.} +I have the transformed equation +\begin{gather*} +%[** TN: Re-broken] +(y^{2} + z^{2} + t^{2})^{2} + p(y^{2} + z^{2} + t^{2}) \\ + + \bigl[4(y^{2} + z^{2} + t^{2}) + 2p\bigr](yz + yt + zt) \\ + + 4(y^{2}z^{2} + y^{2}t^{2} + z^{2}t^{2}) + + (8yzt + q)(y + z + t) + r = 0. +\end{gather*} +We now proceed as we did with equations of the third +degree, where we caused the terms containing $y + z$ +to vanish, and in the same manner cause here the +terms containing $y + z + t$ and $yz + yt + zt$ to disappear, +which will give us the two equations of condition +\[ +8yzt + q = 0 \quad\text{and}\quad 4(y^{2} + z^{2} + t^{2}) + 2p = 0. +\] + +There remains the equation +\[ +(y^{2} + z^{2} + t^{2})^{2} + p(y^{2} + z^{2} + t^{2}) + + 4(y^{2}z^{2} + y^{2}t^{2} + z^{2}t^{2}) + r = 0; +\] +and the three together will determine the quantities +$y$,~$z$, and~$t$. The second gives immediately +\[ +y^{2} + z^{2} + t^{2} = -\frac{p}{2}, +\] +which substituted in the third gives +\[ +y^{2}z^{2} + y^{2}t^{2} + z^{2}t^{2} = \frac{p^{2}}{16} - \frac{r}{4}\Add{.} +\] +The first, raised to its square, gives +\[ +y^{2}z^{2}t^{2} = \frac{q^{2}}{64}. +\] +Hence, by the general theory of equations the three +\PageSep{92} +quantities $y^{2}$,~$z^{2}$,~$t^{2}$ will be the roots of an equation of +the third degree having the form +\MNote{The reduced equation.} +\[ +u^{3} + \frac{p}{2} u^{2} + + \left(\frac{p^{2}}{16} - \frac{r}{4}\right)u + - \frac{q^{2}}{64} = 0; +\] +so that if the three roots of this equation, which we +will call \emph{the reduced equation}, be designated by $a$,~$b$,~$c$, +we shall have +\[ +y = \sqrta,\quad z = \sqrt{b},\quad t = \sqrtc, +\] +and the value of~$x$ will be expressed by +\[ +\sqrta + \sqrt{b} + \sqrtc. +\] +Since the three radicals may each be taken with the +plus sign or the minus sign, we should have, if all +possible combinations were taken, eight different values +for~$x$. It is to be observed, however, that in the +preceding analysis we employed the equation $y^{2}z^{2}t^{2} = \dfrac{q^{2}}{64}$, +whereas the equation immediately given is $yzt = -\dfrac{q}{8}$. +Hence the product of the three quantities $y$,~$z$,~$t$, +that is to say of the three radicals +\[ +\sqrta,\quad \sqrt{b}, \quad \sqrtc, +\] +must have the contrary sign to that of the quantity~$q$. +Therefore, if $q$~be a negative quantity, either three +positive radicals or one positive and two negative radicals +must be contained in the expression for~$x$. And +in this case we shall have the following four combinations +only: +\begin{alignat*}{2} + &\sqrta + \sqrt{b} + \sqrtc,\qquad && \sqrta - \sqrt{b} - \sqrtc,\\ +-&\sqrta + \sqrt{b} - \sqrtc, &\Typo{}{-}&\sqrta - \sqrt{b} + \sqrtc, +\end{alignat*} +\PageSep{93} +which will be the four roots of the proposed equation +of the fourth degree. But if $q$~be a positive quantity, +either three negative radicals or one negative and two +\MNote{Euler's formulæ.} +positive radicals must be contained in the expression +for~$x$, which will give the following four other combinations +as the roots of the proposed equation:\footnote + {These simple and elegant formulæ are due to Euler. But M.~Bret, Professor + \index{Bret, M.|FN}% + \index{Euler}% + of Mathematics at Grenoble, has made the important observation (see + the \textit{Correspondance sur l'\Typo{Ecole}{École} Polytechnique}, t.~II., 3\ieme~Cahier, p.~217) that + they can give false values when imaginary quantities occur among the four + roots. + + In order to remove all difficulty and ambiguity we have only to substitute + for one of these radicals its value as derived from the equation $\sqrta\sqrt{b}\sqrtc = -\dfrac{q}{8}$. + Then the formula + \[ + \sqrta + \sqrt{b} - \frac{q}{8\sqrta\sqrt{b}} + \] + will give the four roots of the original equation by taking for $a$~and~$b$ any two + of the three roots of the reduced equation, and by taking the two radicals + successively positive and negative. + + The preceding remark should be added to article~777 of Euler's \textit{Algebra} + and to article~37 of the author's Note~XIII of the \textit{Traité de la résolution des + équations numériques}.} +\begin{alignat*}{2} +-&\sqrta - \sqrt{b} - \sqrtc,\qquad & -&\sqrta + \sqrt{b} + \sqrtc, \displaybreak[1] \\ + &\sqrta - \sqrt{b} + \sqrtc, &&\sqrta + \sqrt{b} - \sqrtc. +\end{alignat*} + +Now if the three roots $a$,~$b$,~$c$ of the reduced equation +\index{Reality of roots}% +\index{Roots!reality@the reality of the}% +\index{Three roots, reality of the}% +of the third degree are all real and positive, it is +evident that the four preceding roots will also all be +real. But if among the three real roots $a$,~$b$,~$c$, any +are negative, obviously the four roots of the given +biquadratic equation will be imaginary. Hence, besides +the condition for the reality of the three roots of +the reduced equation it is also requisite in the first +case, agreeably to the well-known rule of Descartes, +\index{Descartes}% +\PageSep{94} +that the coefficients of the terms of the reduced equation +should be alternatively positive and negative, and +\MNote{Roots of a biquadratic equation.} +\index{Biquadratic equations}% +\index{Roots!biquadratic@of a biquadratic equation}% +consequently that $p$~should be negative and $\dfrac{p^{2}}{16} - \dfrac{r}{4}$ +positive, that is, $p^{2} > 4r$. If one of these conditions +is not realised the proposed biquadratic equation cannot +have four real roots. If the reduced equation have +but one real root, it will be observed, first, that by +reason of its last term being negative the one real root +of the equation must necessarily be positive. It is +then easy to see from the general expressions which +we gave for the roots of cubic equations deprived of +their second term,---a form to which the reduced equation +in~$u$ can easily be brought by simply increasing +all the roots by the quantity~$\dfrac{p}{6}$,---it is easy to see, I +say, that the two imaginary roots of this equation will +be of the form +\[ +f + g\sqrt{-1} \quad\text{and}\quad f - g\sqrt{-1}. +\] +Therefore, supposing $a$~to be the real root and $b$,~$c$ the +two imaginary roots, $\sqrta$~will be a real quantity and +$\sqrt{b} + \sqrtc$ will also be real for reasons which we have +given above; while $\sqrt{b} - \sqrtc$ on the other hand will +be imaginary. Whence it follows that of the four +roots of the proposed biquadratic equation, the two +first will be real and the two others will be imaginary. + +As for the rest, if we make $u = s - \dfrac{p}{6}$ in the reduced +equation in~$u$, so as to eliminate the second +term and to reduce it to the form which we have above +\PageSep{95} +examined, we shall have the following transformed +equation in~$s$: +\[ +s^{3} - \left(\frac{p^{2}}{48} + \frac{r}{4}\right)s + - \frac{p^{3}}{864} + \frac{pr}{24} - \frac{q^{2}}{64} = 0; +\] +and the condition for the reality of the three roots of +the reduced equation will be +\[ +4\left(\frac{p^{2}}{48} + \frac{r}{4}\right)^{3} + > 27\left(\frac{p^{3}}{864} - \frac{pr}{24} + \frac{q^{2}}{64}\right)^{2}. +\] +\PageSep{96}%XXXX + + +\Lecture{IV.}{On the Resolution of Numerical Equations.} +\index{Numerical equations!resolution of|(}% + +\First{We} have seen how equations of the second, the +third, and the fourth degree can be resolved. +\MNote{Limits of the algebraical resolution of equations.} +\index{Algebraical resolution of equations!limits of the}% +\index{Equations!limits of the algebraical resolution of}% +The fifth degree constitutes a sort of barrier to analysts, +\index{Equations!fifth@of the fifth degree}% +\index{Fifth degree, equations of the}% +which by their greatest efforts they have never +yet been able to surmount, and the general resolution +of equations is one of the things that are still to be +desired in algebra. I say in algebra, for if with the +third degree the analytical expression of the roots is +insufficient for determining in all cases their numerical +value, \textit{a~fortiori} must it be so with equations of a +higher degree; and so we find ourselves constantly +under the necessity of having recourse to other means +for determining numerically the roots of a given equation,---for +to determine these roots is in the last resort +the object of the solution of all problems which +necessity or curiosity may offer. + +I propose here to set forth the principal artifices +which have been devised for accomplishing this important +object. Let us consider any equation of the +\index{Equations!mth@of the $m$th degree}% +$m$th~degree, represented by the formula +\PageSep{97} +\[ +x^{m} + px^{m-1} + qx^{m-2} + rx^{m-3} + \dots + u = 0, +\] +in which $x$~is the unknown quantity, $p$,~$q$,~$r$,~$\dots$ the +known positive or negative coefficients, and $u$~the +\MNote{Conditions of the resolution of numerical equations.} +\index{Numerical equations!conditions of the resolution of}% +last term, not containing~$x$ and consequently also a +known quantity. It is assumed that the values of +these coefficients are given either in numbers or in +lines; (it is indifferent which, seeing that by taking a +given line as the unit or common measure of the rest +we can assign to all the lines numerical values;) and it +is clear that this assumption is always permissible +when the equation is the result of a real and determinate +problem. The problem set us is to find the value, +or, if there be several, the values, of~$x$ which satisfy the +equation, i.e.\Add{,} which render the sum of all its terms +zero. Now any other value which may be given to~$x$ +will render that sum equal to some positive or negative +quantity, for since only integral powers of~$x$ enter +the equation, it is plain that every real value of~$x$ +will also give a real value for the quantity in question. +The more that value approaches to zero, the more +will the value of~$x$ which has produced it approach to +a root of the equation. And if we find two values of~$x$, +of which one renders the sum of the terms equal to +a positive quantity and the other to a negative quantity, +we may be assured in advance that between these +two values there will of necessity be at least one value +which will render the expression zero and will consequently +be a root of the equation. + +Let $P$~stand for the sum of all the terms of the +\PageSep{98} +equation having the sign~$+$ and $Q$~for the sum of all +the terms having the sign~$-$; then the equation will +be represented by +\[ +P - Q = 0. +\] +Let us suppose, for further simplicity, that the two +\MNote{Position of the roots of numerical equations.} +\index{Numerical equations!position of the roots of}% +values of~$x$ in question are positive, that $A$~is the +smaller, $B$~the greater, and that the substitution of~$A$ +for~$x$ gives a negative result and the substitution of~$B$ +for~$x$ a positive result; i.e., that the value of~$P - Q$ +is negative when $x = A$, and positive when $x = B$. + +Consequently, when $x = A$, $P$~will be less than~$Q$, +and when $x = B$, $P$~will be greater than~$Q$. Now, +from the very form of the quantities $P$~and~$Q$, which +contain only positive terms and whole positive powers +of~$x$, it is clear that these quantities augment continuously +as $x$~augments, and that by making $x$ augment by +insensible degrees through all values from $A$~to~$B$, they +also will augment by insensible degrees but in such +wise that $P$~will increase more than~$Q$, seeing that +from having been smaller than~$Q$ it will have become +greater. Therefore, there must of necessity be some +expression for the value of~$x$ between $A$~and~$B$ which +will make $P = Q$; just as two moving bodies which +\index{Moving bodies, two}% +we suppose to be travelling along the same straight +line and which having started simultaneously from +two different points arrive simultaneously at two other +points but in such wise that the body which was at first +in the rear is now in advance of the other,---just as +two such bodies, I say, must necessarily meet at some +\PageSep{99} +point in their path. That value of~$x$, therefore, which +will make $P = Q$ will be one of the roots of the equation, +and such a value will lie of necessity between $A$~and~$B$. + +The same reasoning may be employed for the +\MNote{Position of the roots of numerical equations.} +other cases, and always with the same result. + +The proposition in question is also demonstrable +by a direct consideration of the equation itself, which +may be regarded as made up of the product of the +factors, +\[ +x - a,\quad x - b,\quad x - c,\dots, +\] +where $a$,~$b$,~$c$,~$\dots$ are the roots. For it is obvious +that this product cannot, by the substitution of two +different values for~$x$, be made to change its sign, unless +at least one of the factors changes its sign. And +it is likewise easy to see that if more than one of the +factors changes its sign, their number must be odd. +Thus, if $A$~and~$B$ are two values of~$x$ for which the +factor $x - b$, for example, has opposite signs, then if +$A$~be larger than~$b$, necessarily $B$~must be smaller +than~$b$, or \textit{vice versa}. Perforce, then, the root~$b$ will +fall between the two quantities $A$~and~$B$. + +As for imaginary roots, if there be any in the equation, +\index{Imaginary roots, occur in pairs}% +since it has been demonstrated that they always +occur in pairs and are of the form +\[ +f + g\sqrt{-1},\quad f - g\sqrt{-1}, +\] +therefore if $a$~and~$b$ are imaginary, the product of the +factors $x - a$ and $x - b$ will be +\PageSep{100} +\[ +(x - f - g\sqrt{-1})(x - f + g\sqrt{-1}) = (x - f)^{2} + g^{2}, +\] +a quantity which is always positive whatever value be +given to~$x$. From this it follows that alterations in +the sign can be due only to real roots. But since the +theorem respecting the form of imaginary roots cannot +be rigorously demonstrated without employing the +other theorem that every equation of an odd degree +has necessarily one real root, a theorem of which the +general demonstration itself depends on the proposition +which we are concerned in proving, it follows +that that demonstration must be regarded as a sort of +vicious circle, and that it must be replaced by another +which is unassailable. + +But there is a more general and simpler method +\MNote{Application of geometry to algebra.} +\index{Algebra!application of geometry to|EtSeq}% +\index{Geometry!application of to algebra|EtSeq}% +of considering equations, which enjoys the advantage +\index{Equations!constructions for solving|EtSeq}% +of affording direct demonstration to the eye of the +principal properties of equations. It is founded upon +a species of application of geometry to algebra which +is the more deserving of exposition as it finds extended +employment in all branches of mathematics. + +Let us take up again the general equation proposed +above and let us represent by straight lines all +the successive values which are given to the unknown +quantity~$x$ and let us do the same for the corresponding +values which the left-hand side of the equation +assumes in this manner. To this end, instead of supposing +the right-hand side of the equation equal to +zero, we suppose it equal to an undetermined quantity~$y$. +We lay off the values of~$x$ upon an indefinite +\PageSep{101} +straight line~$AB$ (Fig.~1), starting from a fixed point~$O$ +at which $x$~is zero and taking the positive values of~$x$ +in the direction~$OB$ to the right of~$O$ and the negative +values of~$x$ in the opposite direction to the left of~$O$. +Then let~$OP$ be any value of~$x$. To represent +the corresponding value of~$y$ we erect at~$P$ a perpendicular +to the line~$OB$ and lay off on it the value of~$y$ +in the direction~$PQ$ above the straight line~$OB$ if it is +positive, and on the same perpendicular below~$OB$ if +it is negative. We do the same for all the values of~$x$, +\MNote{Representation of equations by curves.} +\index{Curves!representation of equations by|EtSeq}% +\Figure{1}{0.8\textwidth} +positive as well as negative; that is, we lay off +corresponding values of~$y$ upon perpendiculars to the +straight line through all the points whose distance +from the point~$O$ is equal to~$x$. The extremities of all +these perpendiculars will together form a straight line +or a curve, which will furnish, so to speak, a picture +of the equation +\[ +x^{m} + px^{m-1} + qx^{m-2} + \dots + u = y. +\] +The line~$AB$ is called the axis of the curve, $O$~the origin +of the abscissæ, $OP = x$ an abscissa, $PQ = y$ the corresponding +\PageSep{102} +ordinate, and the equations in $x$~and~$y$ the +\index{Equations!general remarks upon the roots of|EtSeq}% +equations of the curve. A curve such as that of Fig.~1 +having been described in the manner indicated, it is +clear that its intersections with the axis~$AB$ will give +the roots of the proposed equation +\MNote{Graphic resolution of equations.} +\index{Equations!graphic resolution of}% +\index{Intersections, with the axis give roots|EtSeq}% +\[ +x^{m} + px^{m-1} + qx^{m-2} + \dots + u = 0. +\] +For seeing that this equation is realised only when in +the equation of the curve $y$~becomes zero, therefore +those values of~$x$ which satisfy the equation in question +and which are its roots can only be the abscissæ +\ifthenelse{\not\boolean{ForPrinting}}{% +\Figure{1}{0.8\textwidth} %[** TN: [sic], figure repeated] +}{}% [Discard second copy if formatting for printing] +that correspond to the points at which the ordinates +are zero, that is, to the points at which the curve cuts +the axis~$AB$. Thus, supposing the curve of the equation +in $x$~and~$y$ is that represented in Fig.~1, the roots +of the proposed equation will be +\[ +OM,\quad ON,\quad OR,\dots \quad\text{and}\quad -OI,\quad -OG,\dots. +\] +I give the sign~$-$ to the latter because the intersections +$I$,~$G$,~$\dots$ fall on the other side of the point~$O$. +The consideration of the curve in question gives rise +to the following general remarks upon equations: +\PageSep{103} + +(1) Since the equation of the curve contains only +whole and positive powers of the unknown quantity~$x$ +it is clear that to every value of~$x$ there must correspond +\MNote{The consequences of the graphic resolution.} +a determinate value of~$y$, and that the value in +question will be unique and finite so long as $x$~is finite. +But since there is nothing to limit the values of~$x$ they +may be supposed infinitely great, positive as well as +negative, and to them will correspond also values of~$y$ +which are infinitely great. Whence it follows that +the curve will have a continuous and single course, +and that it may be extended to infinity on both sides +of the origin~$O$. + +(2) It also follows that the curve cannot pass from +one side of the axis to the other without cutting it, +and that it cannot return to the same side without +having cut it twice. Consequently, between any two +points of the curve on the same side of the axis there +will necessarily be either no intersections or an even +number of intersections; for example, between the +points $H$~and~$Q$ we find two intersections $I$~and~$M$, +and between the points $H$~and~$S$ we find four, $I$, $M$ +$N$, $R$, and so on. Contrariwise, between a point on +one side of the axis and a point on the other side, the +curve will have an odd number of intersections; for +example, between the points $L$~and~$Q$ there is one intersection~$M$, +and between the points $H$~and~$K$ there +are three intersections, $I$, $M$, $N$, and so on. + +For the same reason there can be no simple intersection +unless on both sides of the point of intersection, +\PageSep{104} +above and below the axis, points of the curve are +situated as are the points $L$,~$Q$ with respect to the intersection~$M$. +\MNote{Intersections indicate the roots.} +But two intersections, such as $N$~and~$R$, +may approach each other so as ultimately to coincide +at~$T$. Then the branch~$QKS$ will take the form +of the dotted line~$QTS$ and touch the axis at~$T$, and +will consequently lie in its whole extent above the +axis; this is the case in which the two roots $ON$,~$OR$ +are equal. If three intersections coincide at a point,---a +coincidence which occurs when there are three +equal roots,---then the curve will cut the axis in one +additional point only, as in the case of a single point +of intersection, and so on. + +Consequently, if we have found for~$y$ two values +having the same sign, we may be assured that between +the two corresponding values of~$x$ there can fall only +an even number of roots of the proposed equation; +that is, that there will be none or there will be two, or +there will be four, etc. On the other hand, if we have +found for~$y$ two values having contrary signs, we may +be assured that between the corresponding values of~$x$ +there will necessarily fall an odd number of roots of +the proposed equation; that is, there will be one, or +there will be three, or there will be five, etc.; so that, +in the case last mentioned, we may infer immediately +that there will be at least one root of the proposed +equation between the two values of~$x$. + +Conversely, every value of~$x$ which is a root of the +equation will be found between some larger and some +\PageSep{105} +smaller value of~$x$ which on being substituted for~$x$ in +the equation will yield values of~$y$ with contrary signs. + +This will not be the case, however, if the value of~$x$ +is a double root; that is, if the equation contains +\MNote{Case of multiple roots.} +\index{Multiple roots}% +\index{Roots!multiple}% +two roots of the same value. On the other hand, if +the value of~$x$ is a triple root, there will again exist +a larger and a smaller value for~$x$ which will give to +the corresponding values of~$y$ contrary signs, and so +on with the rest. + +If, now, we consider the equation of the curve, it +is plain in the first place, that by making $x = 0$ we +shall have $y = u$; and consequently that the sign of +the ordinate~$y$ will be the same as that of the quantity~$u$, +the last term of the proposed equation. It is also +easy to see that there can be given to~$x$ a positive or +negative value sufficiently great to make the first term~$x^{m}$ +of the equation exceed the sum of all the other +terms which have the opposite sign to~$x^{m}$; with the +result that the corresponding value of~$y$ will have the +same sign as the first term~$x^{m}$. Now, if $m$~is odd $x^{m}$~will +be positive or negative according as $x$~is positive +or negative, and if $m$~is even, $x^{m}$~will always be positive +whether $x$~be positive or not. + +Whence we may conclude: + +(1) That every equation of an odd degree of which +\index{Equations!odd@of an odd degree, roots of}% +the last term is negative has an odd number of roots +between $x = 0$ and some very large positive value of~$x$, +and an even number of roots between $x = 0$ and +some very large negative value of~$x$, and consequently +\PageSep{106} +that it has at least one real positive root. That, contrariwise, +if the last term of the equation is positive it +\MNote{General conclusions as to the character of the roots.} +will have an odd number of roots between $x = 0$ and +some very large negative value of~$x$, and an even +number of roots between $x = 0$ and some very large +positive value of~$x$, and consequently that it will have +at least one real negative root. + +(2) That every equation of an even degree, of +\index{Equations!even@of an even degree, roots of}% +which the last term is negative, has an odd number of +roots between $x = 0$ and some very large positive value +of~$x$, as well as an odd number of roots between $x = 0$ +and some very large negative value of~$x$, and consequently +that it has at least one real positive root and +one real negative root. That, on the other hand, if +the last term is positive there will be an even number +of roots between $x = 0$ and some very large positive +value of~$x$, and also an even number of roots between +$x = 0$ and some very large negative value of~$x$; with +the result that in this case the equation may have no +real root, whether positive or negative. + +We have said that there could always be given to~$x$ +a value sufficiently great to make the first term~$x^{m}$ of +the equation exceed the sum of all the terms of contrary +sign. Although this proposition is not in need +of demonstration, seeing that, since the power~$x^{m}$ is +higher than any of the other powers of~$x$ which enter +the equation, it is bound, as $x$~increases, to increase +much more rapidly than these other powers; nevertheless, +in order to leave no doubts in the mind, we +\PageSep{107} +shall offer a very simple demonstration of it,---a demonstration +which will enjoy the collateral advantage +of furnishing a limit beyond which we may be certain +no root of the equation can be found. + +To this end, let us first suppose that $x$~is positive, +\index{Limits of roots|(}% +and that $k$~is the greatest of the coefficients of the +\index{Coefficients!greatest negative|EtSeq}% +\MNote{Limits of the real roots of equations.} +\index{Equations!real roots of, limits of the|EtSeq}% +negative terms. If we make $x = k + 1$ we shall have +\[ +x^{m} = (k + 1)^{m} = k(k + 1)^{m-1} + (k + 1)^{m-1}. +\] +Similarly, +\begin{align*} +(k + 1)^{m-1} &= k(k + 1)^{m-2} + (k + 1)^{m-2}, \\ +(k + 1)^{m-2} &= k(k + 1)^{m-3} + (k + 1)^{m-3} +\end{align*} +and so on; so that we shall finally have +\[ +(k + 1)^{m} + = k(k + 1)^{m-1} + + k(k + 1)^{m-2} + + k(k + 1)^{m-3} + \dots + k + 1. +\] +Now this quantity is evidently greater than the sum +of all the negative terms of the equation taken positively, +on the supposition that $x = k + 1$. Therefore, +the supposition $x = k + 1$ necessarily renders the first +term~$x^{m}$ greater than the sum of all the negative terms. +Consequently, the value of~$y$ will have the same sign +as~$x$. + +The same reasoning and the same result hold good +when $x$~is negative. We have here merely to change~$x$ +into~$-x$ in the proposed equation, in order to change +the positive roots into negative roots, and \textit{vice versa}. + +In the same way it may be proved that if any value +be given to~$x$ greater than~$k + 1$, the value of~$y$ will +still have the same sign. From this and from what +has been developed above, it follows immediately that +\PageSep{108} +the equation can have no root equal to or greater than~$k + 1$. + +Therefore, in general, if $k$~is the greatest of the +\MNote{Limits of the positive and negative roots.} +coefficients of the negative terms of an equation, and +changing the unknown quantity~$x$ into~$-x$, $h$~is +the greatest of the coefficients of the negative terms +of the new equation,---the first term always being supposed +positive,---then all the real roots of the equation +will necessarily be comprised between the limits +\[ +k + 1 \quad\text{and}\quad -h - 1. +\] + +But if there are several positive terms in the equation +preceding the first negative term, we may take +for~$k$ a quantity less than the greatest negative coefficient. +In fact it is easy to see that the formula given +above can be put into the form +\[ +(k + 1)^{m} + = k(k + 1)(k + 1)^{m-2} + + k(k + 1)(k + 1)^{m-3} + \dots + (k + 1)^{2} +\] +and similarly into the following +\[ +(k + 1)^{m} + = k(k + 1)^{2}(k + 1)^{m-3} + + k(k + 1)^{2}(k + 1)^{m-4} + \dots + (k + 1)^{3} +\] +and so on. + +Whence it is easy to infer that if $m - n$ is the exponent +of the first negative term of the proposed equation +of the $m$th~degree, and if $l$~is the largest coefficient +of the negative terms, it will be sufficient if $k$~is +so determined that +\[ +k(k + 1)^{n-1} = l. +\] +And since we may take for~$k$ any larger value that we +please, it will be sufficient to take +\PageSep{109} +\[ +k^{n} = l,\quad\text{or}\quad k = \sqrt[n]{l}. +\] +And the same will hold good for the quantity~$h$ as the +limit of the negative roots. +\index{Positive roots, superior and inferior limits of the}% +\index{Roots!superior and inferior limits of the positive}% + +If, now, the unknown quantity~$x$ be changed into~$\dfrac{1}{z}$, +the largest roots of the equation in~$x$ will be converted +\MNote{Superior and inferior limits of the positive roots.} +into the smallest in the new equation in~$z$, and +conversely. Having effected this transformation, and +having so arranged the terms according to the powers +of~$z$ that the first term of the equation is~$z^{m}$, we may +then in the same manner seek for the limits $K + 1$ and +$-H - 1$ of the positive and negative roots of the +equation in~$z$. + +Thus $K + 1$ being larger than the largest value of~$z$ +or of~$\dfrac{1}{x}$, therefore, by the nature of fractions, $\dfrac{1}{K + 1}$ +will be smaller than the smallest value of~$x$ and similarly +$\dfrac{1}{H + 1}$ will be smaller than the smallest negative +value of~$x$. + +Whence it may be inferred that all the positive +real roots will necessarily be comprised between the +limits +\[ +\frac{1}{K + 1} \quad\text{and}\quad k + 1, +\] +and that the negative real roots will fall between the +limits +\[ +-\frac{1}{H + 1} \quad\text{and}\quad -h - 1. +\] + +There are methods for finding still closer limits; +but since they require considerable labor, the preceding +\PageSep{110} +method is, in the majority of cases, preferable, as +being more simple and convenient. + +For example, if in the proposed equation $l + z$ be +\MNote{A further method for finding the limits.} +\index{Roots!method for finding the limits of}% +substituted for~$x$, and if after having arranged the +terms according to the powers of~$z$, there be given to~$l$ +a value such that the coefficients of all the terms +become positive, it is plain that there will then be no +positive value of~$z$ that can satisfy the equation. The +equation will have negative roots only, and consequently +$l$~will be a quantity greater than the greatest +value of~$x$. Now it is easy to see that these coefficients +will be expressed as follows: +\begin{gather*} +%[** TN: Re-broken] +p + ml, \\ +q + (m - 1)pl + \frac{m(m - 1)}{2}\, l^{2}, \\ +r + (m - 2)ql + \frac{(m - 1)(m - 2)}{2}\, pl^{2} + + \frac{m(m - 1)(m - 2)}{2·3}\, l^{3}, +\end{gather*} +and so on. Accordingly, it is only necessary to seek +by trial the smallest value of~$l$ which will render them +all positive. + +But in the majority of cases it is not sufficient to +\index{Problems}% +know the limits of the roots of an equation; the thing +necessary is to know the values of those roots, at +least as approximately as the conditions of the problem +require. For every problem leads in its last analysis +to an equation which contains its solution; and +if it is not in our power to resolve this equation, all +\PageSep{111} +the pains expended upon its formulation are a sheer +loss. We may regard this point, therefore, as the +most important in all analysis, and for this reason I +\MNote{The real problem, the finding of the roots.} +have felt constrained to make it the principal subject +of the present lecture. + +From the principles established above regarding +\index{Substitutions|EtSeq}% +the nature of the curve of which the ordinates~$y$ represent +all the values which the left-hand side of an +equation assumes, it follows that if we possessed +some means of describing this curve we should obtain +at once, by its intersections with the axis, all the roots +of the proposed equation. But for this purpose it is +not necessary to have all of the curve; it is sufficient +to know the parts which lie immediately above and +below each point of intersection. Now it is possible +to find as many points of a curve as we please, and as +near to one another as we please by successively substituting +for~$x$ numbers which are very little different +from one another, but which are still near enough for +our purpose, and by taking for~$y$ the results of these +substitutions in the left-hand side of the equation. If +among the results of these substitutions two be found +having contrary signs, we may be certain, by the principles +established above, that there will be between +these two values of~$x$ at least one real root. We can +then by new substitutions bring these two limits still +closer together and approach as nearly as we wish to +the roots sought. + +Calling the smaller of the two values of~$x$ which +\PageSep{112} +have given results with contrary signs,~$A$, and the +larger~$B$, and supposing that we wish to find the +\MNote{Separation of the roots.} +\index{Roots!separation of the}% +\index{Roots!arithmetical@the arithmetical progression revealing the|EtSeq}% +value of the root within a degree of exactness denoted +by~$n$, where $n$~is a fraction of any degree of smallness +we please, we proceed to substitute successively for~$x$ +the following numbers in arithmetical progression: +\index{Arithmetical progression revealing the roots|EtSeq}% +\[ +A + n,\quad A + 2n,\quad A + 3n, \dots, +\] +or +\[ +B - n,\quad B - 2n,\quad B - 3n, \dots, +\] +until a result is reached having the contrary sign to +that obtained by the substitution of~$A$ or of~$B$. Then +one of the two successive values of~$x$ which have given +results with contrary signs will necessarily be larger +than the root sought, and the other smaller; and since +by hypothesis these values differ from one another +only by the quantity~$n$, it follows that each of them +approaches to within less than~$n$ of the root sought, +and that the error is therefore less than~$n$. + +But how are the initial values substituted for~$x$ to +be determined, so as on the one hand to avoid as +many useless trials as possible, and on the other to +make us confident that we have discovered by this +method all the real roots of this equation. If we examine +the curve of the equation it will be readily seen +that the question resolves itself into so selecting the +values of~$x$ that at least one of them shall fall between +two adjacent intersections, which will be necessarily +the case if the difference between two consecutive values +\PageSep{113} +is less than the smallest distance between two +adjacent intersections. + +Thus, supposing that $D$~is a quantity smaller than +the smallest distance between two intersections immediately +\MNote{To find a quantity less than the difference between any two roots.} +\index{Roots!quantity less than the difference between any two}% +following each other, we form the arithmetical +progression +\[ +0,\quad D,\quad 2D,\quad 3D,\quad 4D,\dots, +\] +and we select from this progression only the terms +which fall between the limits +\[ +\frac{1}{K + 1} \quad\text{and}\quad k + 1, +\] +as determined by the method already given. We obtain, +in this manner, values which on being substituted +for~$x$ ultimately give us all the positive roots of +the equation, and at the same time give the initial +limits of each root. In the same manner, for obtaining +the negative roots we form the progression +\[ +0,\quad -D,\quad -2D,\quad -3D,\quad -4D,\dots, +\] +from which we also take only the terms comprised +between the limits +\[ +-\frac{1}{H + 1} \quad\text{and}\quad -h - 1. +\] + +Thus this difficulty is resolved. But it still remains +to find the quantity~$D$,---that is, a quantity +smaller than the smallest interval between any two adjacent +intersections of the curve with the axis. Since +the abscissæ which correspond to the intersections are +\index{Intersections, with the axis give roots}% +the roots of the proposed equation, it is clear that the +question reduces itself to finding a quantity smaller +\PageSep{114} +than the smallest difference between two roots, neglecting +the signs. We have, therefore, to seek, by the +methods which were discussed in the lectures of the +principal course, the equation whose roots are the differences +between the roots of the proposed equation. +And we must then seek, by the methods expounded +above, a quantity smaller than the smallest root of +this last equation, and take that quantity for the value +of~$D$. + +This method, as we see, leaves nothing to be desired +\MNote{The equation of differences.} +\index{Differences, the equation of|EtSeq}% +as regards the rigorous solution of the problem, +but it labors under great disadvantage in requiring +extremely long calculations, especially if the proposed +equation is at all high in degree. For example, if $m$~is +the degree of the original equation, that of the equation +of differences will be~$m(m - 1)$, because each root +can be subtracted from all the remaining roots, the +number of which is~$m - 1$,---which gives $m(m - 1)$ +differences. But since each difference can be positive +or negative, it follows that the equation of differences +must have the same roots both in a positive and in a +negative form; that consequently the equation must +be wanting in all terms in which the unknown quantity +is raised to an odd power; so that by taking the +square of the differences as the unknown quantity, this +unknown quantity can occur only in the $\dfrac{m(m - 1)}{2}$th +degree. For an equation of the $m$th~degree, accordingly, +there is requisite at the start a transformed +\PageSep{115} +equation of the $\dfrac{m(m - 1)}{2}$th degree, which necessitates +an enormous amount of tedious labor, if $m$~is at all +large. For example, for an equation of the $10$th~degree, +\MNote{Impracticability of the method.} +the transformed equation would be of the~$45$th. +And since in the majority of cases this disadvantage +renders the method almost impracticable, it is of great +importance to find a means of remedying it. + +To this end let us resume the proposed equation of +the $m$th~degree, +\[ +x^{m} + px^{m-1} + qx^{m-2} + \dots + u = 0, +\] +of which the roots are $a$,~$b$,~$c$,~$\dots$. We shall have +then +\[ +a^{m} + pa^{m-1} + qa^{m-2} + \dots + u = 0 +\] +and also +\[ +b^{m} + pb^{m-1} + qb^{m-2} + \dots + u = 0. +\] +Let $b - a = i$. Substitute this value of~$b$ in the second +equation, and after developing the different powers of~$a + i$ +according to the well known binomial theorem, +\index{Binomial theorem}% +arrange the resulting equation according to the powers +of~$i$, beginning with the lowest. We shall have the +transformed equation +\[ +P + Qi + Ri^{2} + \dots + i^{m} = 0, +\] +in which the coefficients $P$,~$Q$,~$R$,~$\dots$ have the following +values +\begin{align*} +P &= a^{m} + pa^{m-1} + qa^{m-2} + \dots + u, \displaybreak[1] \\ +Q &= ma^{m-1} + (m - 1)pa^{m-2} + (m - 2)qa^{m-3} + \dots\Add{,} \displaybreak[1] \\ +\PageSep{116} +R &= \begin{aligned}[t] + \frac{m(m - 1)}{2}\, a^{m-2} + &+ \frac{(m - 1)(m - 2)}{2}\, pa^{m-3} \\ + &+ \frac{(m - 2)(m - 3)}{2}\, qa^{m-4} + \dots\Add{,} +\end{aligned} +\end{align*} +\MNote{Attempt to remedy the method.} +and so on. The law of formation of these expressions +is evident. + +Now, by the first equation in~$a$ we have~$P = 0$. +Rejecting, therefore, the term~$P$ of the equation in~$i$ +and dividing all the remaining terms by~$i$, the equation +in question will be reduced to the $(m - 1)$th~degree, +and will have the form +\[ +Q + Ri + Si^{2} + \dots + i^{m-1} = 0. +\] + +This equation will have for its roots the $m - 1$~differences +between the root~$a$ and the remaining roots +$b$,~$c$,~$\dots$\Add{.} Similarly, if $b$~be substituted for~$a$ in the expressions +for the coefficients $Q$,~$R$,~$\dots$, we shall obtain +an equation of which the roots are the difference +between the root~$b$ and the remaining roots $a$,~$c$,~$\dots$, +and so on. + +Accordingly, if a quantity can be found smaller +\index{Roots!smallest|EtSeq}% +than the smallest root of all these equations, it will +possess the property required and may be taken for +the quantity~$D$, the value of which we are seeking. + +If, by means of the equation $P = 0$, $a$~be eliminated +from the equation in~$i$, we shall get a new equation in~$i$ +which will contain all the other equations of which +we have just spoken, and of which it would only be +necessary to seek the smallest root. But this new +\PageSep{117} +equation in~$i$ is nothing else than the equation of differences +which we sought to dispense with. + +\MNote{Further improvement.} +In the above equation in~$i$ let us put it $i = \dfrac{1}{z}$. We +shall have then the transformed equation in~$z$, +\[ +z^{m-1} + + \frac{R}{Q}\, z^{m-2} + + \frac{S}{Q}\, z^{m-3} + \dots + \frac{1}{Q} = 0, +\] +and the greatest negative coefficient of this equation +will, from what has been demonstrated above, give a +value greater than its greatest root; so that calling~$L$ +this greatest coefficient, $L + 1$~will be a quantity +greater than the greatest value of~$z$. Consequently, +$\dfrac{1}{L + 1}$ will be a quantity smaller than the smallest +positive value of~$i$; and in like manner we shall find +a quantity smaller than the smallest negative value +of~$i$. Accordingly, we may take for~$D$ the smallest of +these two quantities, or some quantity smaller than +either of them. + +For a simpler result, and one which is independent +of signs, we may reduce the question to finding a +quantity~$L$ numerically greater than any of the coefficients +\index{Coefficients!greatest negative}% +of the equation in~$z$, and it is clear that if we +find a quantity~$N$ numerically smaller than the smallest +value of~$Q$ and a quantity~$M$ numerically greater +than the greatest value of any of the quantities $R$, +$S$,~$\dots$, we may put $L = \dfrac{M}{N}$. + +Let us begin with finding the values of~$M$. It is +not difficult to demonstrate, by the principles established +above, that if $k + 1$~is the limit of the positive +\PageSep{118} +roots and $-h - 1$~the limit of the negative roots of +the proposed equation, and if for~$a$, $k + 1$~and~$-h - 1$ +\MNote{Final resolution.} +be successively substituted in the expressions for $R$, +$S$,~$\dots$, considering only the terms which have the +same sign as the first,---it is easy to demonstrate that +we shall obtain in this manner quantities which are +greater than the greatest positive and negative values +of $R$, $S$,~$\dots$ corresponding to the roots $a$,~$b$, $c$\Add{,}~$\dots$ of +the proposed equation; so that we may take for~$M$ +the quantity which is numerically the greatest of +these. + +It accordingly only remains to find a value smaller +than the smallest value of~$Q$. Now it would seem +that we could arrive at this in no other way than by +employing the equation of which the different values +of~$Q$ are the roots,---an equation which can only be +reached by eliminating~$a$ from the following equations: +\begin{gather*} +a^{m} + pa^{m-1} + qa^{m-2} + \dots + u = 0, \\ +ma^{m-1} + (m - 1)pa^{m-2} + (m - 2)qa^{m-3} + \dots = Q. +\end{gather*} + +It can be easily demonstrated by the theory of +elimination that the resulting equation in~$Q$ will be of +the $m$th~degree, that is to say, of the same degree with +the proposed equation; and it can also be demonstrated +from the form of the roots of this equation +that its next to the last term will be missing. If, accordingly, +we seek by the method given above a quantity +numerically smaller than the smallest root of this +equation, the quantity found can be taken for~$N$. The +\PageSep{119} +problem is therefore resolved by means of an equation +of the same degree as the proposed equation. + +The upshot of the whole is \Typo{a}{as} follows,---where for +\MNote{Recapitulation.} +the sake of simplicity I retain the letter~$x$ instead of +the letter~$a$. + +Let the following be the proposed equation of the +$m$th~degree: +\[ +x^{m} + px^{m-1} + qx^{m-2} + rx^{m-3} + \dots = 0; +\] +let $k$~be the largest coefficient of the negative terms, +and $m - n$~the exponent of~$x$ in the first negative term. +Similarly, let $h$ be the greatest coefficient of the terms +having a contrary sign to the first term after $x$~has +been changed into~$-x$; and let $m - n'$ be the exponent +of~$x$ in the first term having a contrary sign to +the first term of the equation as thus altered. Putting, then, +\[ +f = \sqrt[n]{k} + 1 \quad\text{and}\quad g = \sqrt[n]{h} + 1, +\] +we shall have $f$~and~$-g$ for the limits of the positive +and negative roots. These limits are then substituted +\index{Roots!limits of the positive and negative}% +successively for~$x$ in the following formulæ, neglecting +the terms which have the same sign as the first +term: +\begin{gather*} +%[** TN: Re-broken] +\begin{aligned} +\frac{m(m - 1)}{2}\, x^{m-2} + &+ \frac{(m - 1)(m - 2)}{2}\, px^{m-3} \\ + &+ \frac{(m - 2)(m - 3)}{2}\, qx^{m-4} + \dots, +\end{aligned} \\ +\frac{m(m - 1)(m - 2)}{2·3}\, x^{m-3} + + \frac{(m - 1)(m - 2)(m - 3)}{2·3}\, px^{m-4} + \dots, +\end{gather*} +\PageSep{120} +and so on. Of these formulæ there will be~$m - 2$. Let +the greatest of the numerical quantities obtained in +this manner be called M. We then take the equation +\MNote{The arithmetical progression revealing the roots.} +\index{Arithmetical progression revealing the roots}% +\index{Roots!arithmetical@the arithmetical progression revealing the}% +\[ +mx^{m-1} + (m - 1)px^{m-2} + (m - 2)qx^{m-3} + (m - 3)rx^{m-4} + \dots = y +\] +and eliminate~$x$ from it by means of the proposed +equation,---which gives an equation in~$y$ of the $m$th~degree +with its next to the last term wanting. Let $V$~be +the last term of this equation in~$y$, and $T$~the largest +coefficient of the terms having the contrary sign +to~$V$, supposing $y$~positive as well as negative. Then +taking these two quantities $T$~and~$V$ positive, $N$~will +be determined by the equation +\[ +\frac{N}{1 - N} = \sqrt[n]{\frac{V}{T}} +\] +where $n$~is equal to the exponent of the last term having +the contrary sign to~$V$. We then take $D$ equal to +or smaller than the quantity~$\dfrac{N}{M + N}$, and interpolate +the arithmetical progression: +\[ +0,\quad D,\quad 2D,\quad 3D,\dots,\quad +-D,\quad -2D,\quad -3D, \dots +\] +between the limits $f$~and~$-g$. The terms of these +progressions being successively substituted for~$x$ in +the proposed equation will reveal all the real roots, +positive as well as negative, by the changes of sign +in the series of results produced by these substitutions, +and they will at the same time give the first +limits of these roots,---limits which can be narrowed +as much as we please, as we already know. +\index{Limits of roots|)}% +\PageSep{121} + +If the last term~$V$ of the equation in~$y$ resulting +from the elimination of~$x$ is zero, then $N$~will be zero, +and consequently $D$~will be equal to zero. But in +\MNote{Method of elimination\Add{.}} +\index{Elimination!method of}% +this case it is clear that the equation in~$y$ will have +one root equal to zero and even two, because its next +to the last term is wanting. Consequently the equation +\[ +mx^{m-1} + (m - 1)px^{m-2} + (m - 2)qx^{m-3} + \dots = 0\Typo{.}{} +\] +will hold good at the same time with the proposed +equation. These two equations will, accordingly, have +\index{Common divisor of two equations}% +\index{Equations!common divisor of two}% +a common divisor which can be found by the ordinary +method, and this divisor, put equal to zero, will give +one or several roots of the proposed equation, which +roots will be double or multiple, as is easily apparent +from the preceding theory; for if the last term~$Q$ of +the equation in~$i$ is zero, it follows that +\[ +i = 0 \quad\text{and}\quad a = b. +\] +The equation in~$y$ is reduced, by the vanishing of its +last term, to the $(m - 2)$th~degree,---being divisible +by~$y^{2}$. If after this division its last term should still +be zero, this would be an indication that it had more +than two roots equal to zero, and so on. In such a +contingency we should divide it by~$y$ as many times +as possible, and then take its last term for~$V$, and the +greatest coefficient of the terms of contrary sign to~$V$ +for~$T$, in order to obtain the value of~$D$, which will +enable us to find all the remaining roots of the proposed +equation. If the proposed equation is of the +third degree, as +\PageSep{122} +\[ +x^{3} + qx + r = 0, +\] +we shall get for the equation in~$y$, +\[ +y^{3} + 3qy^{2} - 4q^{3} - 27r^{2} = 0. +\] + +If the proposed equation is +\[ +x^{4} + qx^{2} + rx + s = 0 +\] +we shall obtain for the equation in~$y$ the following +\begin{multline*} +%[** TN: Re-broken] +y^{4} + 8ry^{3} + (4q^{3} - 16qs + 18r^{2})y^{2} \\ + + 256s^{3} - 128s^{2}q^{2} + 16sq^{4} + 144r^{2}sq - 4r^{2}q^{3} - 27r^{4} + = 0 +\end{multline*} +and so on. + +Since, however, the finding of the equation in~$y$ by +\MNote{General formulæ for elimination.} +\index{Elimination!general formulæ for}% +the ordinary methods of elimination may be fraught +with considerable difficulty, I here give the general +formulæ for the purpose, derived from the known +properties of equations. We form, first, from the coefficients +$p$,~$q$,~$r$ of the proposed equation, the quantities +$x_{1}$,~$x_{2}$,~$x_{3}$,~$\dots$, in the following manner: +\[ +\begin{array}{r@{\,}l} +x_{1} &= -p, \\ +x_{2} &= -px_{1} - 2q, \\ +x_{3} &= -px_{2} - qx_{1} - 3r, \\ +\hdotsfor{2}. +\end{array} +\] +We then substitute in the expressions for $y$,~$y^{2}$,~$y^{3}$,~$\dots$ +up to~$y^{m}$, after the terms in~$x$ have been developed +the quantities $x_{1}$~for~$x$, $x_{2}$~for~$x^{}$, $x_{3}$~for~$x^{3}$, and so forth, +and designate by $y_{1}$,~$y_{2}$, $y_{3}$,~$\dots$ the values of $y$,~$y^{2}$, $y^{3}$,~$\dots$ +resulting from these substitutions. We have then +simply to form the quantities $A$,~$B$,~$C$ from the formulæ +\PageSep{123} +\index{Differences, the equation of}% +\[ +\begin{array}{r@{\,}l} +A &= y_{1}, \\ +B &= \dfrac{Ay_{1} - y_{2}}{2}, \\ +C &= \dfrac{By_{1} - Ay_{2} + y_{3}}{3}, \\ +\hdotsfor{2}, +\end{array} +\] +and we shall have the following equation in~$y$: +\[ +y^{m} - Ay^{m-1} + By^{m-2} - Cy^{m-3} + \dots = 0. +\] + +The value, or rather the limit of~$D$, which we find +by the method just expounded may often be much +\MNote{General result.} +smaller than is necessary for finding all the roots, but +there would be no further inconvenience in this than +to increase the number of successive substitutions for~$x$ +\index{Substitutions}% +in the proposed equation. Furthermore, when there +are as many results found as there are units in the +highest exponent of the equation, we can continue +these results as far as we wish by the simple addition +of the first, second, third differences, etc., because +the differences of the order corresponding to the degree +of the equation are always constant. + +We have seen above how the curve of the proposed +equation can be constructed by successively giving +different values to the abscissæ~$x$ and taking for the +ordinates~$y$ the values resulting from these substitutions +in the left-hand side of the equation. But these +values for~$y$ can also be found by another very simple +construction, which deserves to be brought to your +notice. Let us represent the proposed equation by +\[ +a + bx + cx^{2} + dx^{3} + \dots = 0 +\] +\PageSep{124} +where the terms are taken in the inverse order. The +equation of the curve will then be +\[ +y = a + bx + cx^{2} + dx^{3} + \dots\Add{.} +\] +Drawing (Fig.~2) the straight line~$OX$, which we take +\MNote{A second construction for solving equations.} +\index{Equations!constructions for solving}% +\index{Machine for solving equations|(}% +as the axis of abscissæ with $O$~as origin, we lay off on +this line the segment~$OI$ equal to the unit in terms of +which we may suppose the quantities $a$,~$b$,~$c$\Add{,}~$\dots$, to +be expressed; and we erect at the points~$OI$ the perpendiculars +\Figure{2}{0.5\textwidth} +$OD$,~$IM$. We then lay off upon the line~$OD$ +the segments +\[ +OA = a,\quad AB = b,\quad BC = c,\quad CD = d, \dots, +\] +and so on. Let $OP = x$, and at the point~$P$ let the +perpendicular~$PT$\Typo{}{ }be erected. Suppose, for example, +that $d$~is the last of the coefficients $a$,~$b$,~$c$,~$\dots$, so that +the proposed equation is only of the third degree, and +that the problem is to find the value of +\[ +y = a + bx + cx^{2} + dx^{3}. +\] +The point~$D$ being the last of the points determined +upon the perpendicular~$OD$, and the point~$C$ the next +\PageSep{125} +to the last, we draw through~$D$ the line~$DM$ parallel +to the axis~$OI$, and through the point~$M$ where this +line cuts the perpendicular~$IM$ we draw the straight +\MNote{The development and solution.} +line~$CM$ connecting $M$ with~$C$. Then through the +point~$S$ where this last straight line cuts the perpendicular~$PT$, +we draw $HSL$ parallel to~$OI$, and through +the point~$L$ where this parallel cuts the perpendicular~$IM$ +we draw to the point~$B$ the straight line~$BL$. +Similarly, through the point~$R$, where this last line +cuts the perpendicular~$PT$, we draw $GRK$ parallel to~$OI$, +and through the point~$K$, where this parallel cuts +the perpendicular~$IM$ we draw to the first division +point~$A$ of the perpendicular~$DO$ the straight line~$AK$. +The point~$Q$ where this straight line cuts the perpendicular~$PT$ +will give the segment $PQ = y$. + +Through $Q$ draw the line $FQ$ parallel to the axis~$OP$. +The two similar triangles $CDM$~and~$CHS$ give +\[ +DM(1) : DC(d) = HS(x) : CH(= dx). +\] +Adding $CB(c)$ we have +\[ +BH = c + dx. +\] +Also the two similar triangles $BHL$~and~$BGR$ give +\[ +HL(1) : HB(c + dx)= GR(x) : BG(= cx + dx^{2}). +\] +Adding $AB(b)$ we have +\[ +AG = b + cx + dx^{2}. +\] +Finally the similar triangles $AGK$~and~$AFQ$ give +\[ +%[** TN: Set on two lines in original] +GK(1) : GA(b + cx + dx^{2}) = FQ(x) : FA(= bx + cx^{2} + dx^{3}), +\] +and we obtain by adding $OA(a)$ +\[ +OF = PQ = a + bx + cx^{2} + dx^{3} = y. +\] +\PageSep{126} + +The same construction and the same demonstration +hold, whatever be the number of terms in the +proposed equation. When negative coefficients occur +among $a$,~$b$, $c$,~$\dots$, it is simply necessary to take +them in the opposite direction to that of the positive +coefficients. For example, if $a$~were negative we +should have to lay off the segment~$OA$ below the axis~$OI$. +Then we should start from the point~$A$ and add +to it the segment $AB = b$. If $b$~were positive, $AB$~would +be taken in the direction of~$OD$; but if $b$~were +negative, $AB$~would be taken in the opposite direction, +and so on with the rest. + +With regard to~$x$, $OP$~is taken in the direction of~$OI$, +which is supposed to be equal to positive unity, +when $x$~is positive; but in the opposite direction when +$x$~is negative. + +It would not be difficult to construct, on the foregoing +\MNote{A machine for solving equations.} +\index{Equations!machine@a machine for solving}% +model, an instrument which would be applicable +to all values of the coefficients $a$,~$b$, $c$,~$\dots$, and which +by means of a number of movable and properly jointed +rulers would give for every point~$P$ of the straight +line~$OP$ the corresponding point~$Q$, and which could +be even made by a continuous movement to describe +the curve. Such an instrument might be used for +solving equations of all degrees; at least it could be +used for finding the first approximate values of the +roots, by means of which afterwards more exact values +could be reached. +\index{Machine for solving equations|)}% +\index{Numerical equations!resolution of|)}% +\PageSep{127} + + +\Lecture[The Employment of Curves.] +{V.}{On the Employment of Curves in the Solution +of Problems.} +\index{Curves!employment of in the solution of problems|(}% +\index{Problems!employment of curves in the solution of|(}% + +\First{As long} as algebra and geometry travelled separate +\index{Algebra!application of geometry to|EtSeq}% +\index{Geometry!application of to algebra|EtSeq}% +paths their advance was slow and their +\MNote{Geometry applied to algebra.} +applications limited. But when these two sciences +joined company, they drew from each other fresh vitality +and thenceforward marched on at a rapid pace +towards perfection. It is to Descartes that we owe +\index{Descartes}% +the application of algebra to geometry,---an application +which has furnished the key to the greatest discoveries +in all branches of mathematics. The method +which I last expounded to you for finding and demonstrating +divers general properties of equations by considering +the curves which represent them, is, properly +speaking, a species of application of geometry to algebra, +and since this method has extended \Typo{applicacations}{applications}, +and is capable of readily solving problems +whose direct solution would be extremely difficult or +even impossible, I deem it proper to engage your attention +in this lecture with a further view of this subject,---especially +\PageSep{128} +since it is not ordinarily found in +elementary works on algebra. + +You have seen how an equation of any degree +\MNote{Method of resolution by curves.} +whatsoever can be resolved by means of a curve, of +which the abscissæ represent the unknown quantity +of the equation, and the ordinates the values which +the left-hand member assumes for every value of the +unknown quantity. It is clear that this method can be +applied generally to all equations, whatever their form, +and that it only requires them to be developed and +arranged according to the different powers of the unknown +quantity. It is simply necessary to bring all +the terms of the equation to one side, so that the other +side shall be equal to zero. Then taking the unknown +quantity for the abscissa~$x$, and the function of the +unknown quantity, or the quantity compounded of +that quantity and the known quantities, which forms +one side of the equation, for the ordinate~$y$, the curve +described by these co-ordinates $x$~and~$y$ will give by +its intersections with the axis those values of~$x$ which +are the required roots of the equation. And since +most frequently it is not necessary to know all possible +values of the unknown quantity but only such as +solve the problem in hand, it will be sufficient to describe +that portion of the curve which corresponds to +these roots, thus saving much unnecessary calculation. +We can even determine in this manner, from the shape +of the curve itself, whether the problem has possible +solutions satisfying the proposed conditions. +\PageSep{129} + +Suppose, for instance, that it is required to find on +\index{Light, law of the intensity of}% +\index{Lights, problem of the two|EtSeq}% +the line joining two luminous points of given intensity, +the point which receives a given quantity of light,---the +\MNote{Problem of the two lights.} +law of physics being that the intensity of light decreases +with the square of the distance. + +Let $a$~be the distance between the two lights and +$x$~the distance between the point sought and one of +the lights, the intensity' of which at unit distance is~$M$, +the intensity of the other at that distance being~$N$. +The expressions $\dfrac{M}{x^{2}}$ and $\dfrac{N}{(a - x)^{2}}$, accordingly, +give the intensity of the two lights at the point in +question, so that, designating the total given effect by~$A$, +we have the equation +\[ +\frac{M}{x^{2}} + \frac{N}{(a - x)^{2}} = A\Add{,} +\] +or +\[ +\frac{M}{x^{2}} + \frac{N}{(a - x)^{2}} - A = 0. +\] + +We will now consider the curve having the equation +\[ +\frac{M}{x^{2}} + \frac{N}{(a - x)^{2}} - A = y +\] +in which it will be seen at once that by giving to~$x$ a +very small value, positive or negative, the term~$\dfrac{M}{x^{2}}$, +while continuing positive, will grow very large, because +a fraction increases in proportion as its denominator +decreases, and it will be infinite when $x = 0$. +Further, if $x$~be made to increase, the expression~$\dfrac{M}{x^{2}}$ +will constantly diminish; but the other expression~$\dfrac{N}{(a - x)^{2}}$, +\PageSep{130} +which was $\dfrac{N}{a^{2}}$ when $x = 0$, will constantly increase +until it becomes very large or infinite when $x$ +has a value very near to or equal to~$a$. + +Accordingly, if, by giving to~$x$ values from zero to~$a$, +\MNote{Various solutions.} +the sum of these two expressions can be made to +become less than the given quantity~$A$, then the value +of~$y$, which at first was very large and positive, will +become negative, and afterwards again become very +large and positive. Consequently, the curve will cut +the axis twice between the two lights, and the problem +will have two solutions. These two solutions will +be reduced to a single solution if the smallest value of +\[ +\frac{M}{x^{2}} + \frac{N}{(a - x)^{2}} +\] +is exactly equal to~$A$, and they will become imaginary +if that value is greater than~$A$, because then the value +of~$y$ will always be positive from $x = 0$ to $x = a$. +Whence it is plain that if one of the conditions of the +problem be that the required point shall fall between +the two lights it is possible that the problem has no +solution. But if the point be allowed to fall on the +prolongation of the line joining the two lights, we +shall see that the problem is always resolvable in two +ways. In fact, supposing $x$~negative, it is plain that +the term~$\dfrac{M}{x^{2}}$ will always remain positive and from being +very large when $x$~is near to zero, it will commence +and keep decreasing as $x$~increases until it grows very +small or becomes zero when $x$~is very great or infinite. +\PageSep{131} +The other term~$\dfrac{N}{(a - x)^{2}}$, which at first was equal to~$\dfrac{N}{a^{2}}$, +also goes on diminishing until it becomes zero +when $x$~is negative infinity. It will be the same if $x$~is +positive and greater than~$a$; for when $x = a$, the +expression $\dfrac{N}{(a - x)^{2}}$ will be infinitely great; afterwards +it will keep on decreasing until it becomes zero when $x$~is +infinite, while the other expression $\dfrac{M}{x^{2}}$ will first be +equal to $\dfrac{M}{a^{2}}$ and will also go on diminishing towards +zero as $x$~increases. + +Hence, whatever be the value of the quantity~$A$, +it is plain that the values of~$y$ will necessarily pass +\MNote{General solution.} +from positive to negative, both for $x$~negative and for +$x$~positive and greater than~$a$. Accordingly, there +will be a negative value of~$x$ and a positive value of~$x$ +greater than~$a$ which will resolve the problem in all +cases. These values may be found by the general +method by successively causing the values of~$x$ which +give values of~$y$ with contrary signs, to approach +nearer and nearer to each other. + +With regard to the values of~$x$ which are less than~$a$ +we have seen that the reality of these values depends +on the smallest value of the quantity +\[ +\frac{M}{x^{2}} + \frac{N}{(a - x)^{2}}. +\] +Directions for finding the smallest and greatest values +of variable quantities are given in the Differential Calculus. +\index{Differential Calculus}% +We shall here content ourselves with remarking +\PageSep{132} +that the quantity in question will be a minimum +when +\MNote{Minimal values.} +\index{Minimal values}% +\index{Values!minimal}% +\[ +\frac{x}{a - x} = \sqrt[3]{\frac{M}{N}}; +\] +so that we shall have +\[ +x = \frac{a\sqrt[3]{M}}{\sqrt[3]{M} + \sqrt[3]{N}}, +\] +from which we get, as the smallest value of the expression +\[ +\frac{M}{x^{2}} + \frac{N}{(a - x)^{2}}, +\] +the quantity +\[ +\frac{(\sqrt[3]{M} + \sqrt[3]{N})^{3}}{a^{2}}. +\] +Hence there will be two real values for~$x$ if this quantity +is less than~$A$; but these values will be imaginary +if it is greater. The case of equality will give two +equal values for~$x$. + +I have dwelt at considerable length on the analysis +of this problem, (though in itself it is of slight importance,) +for the reason that it can be made to serve +as a type for all analogous cases. + +The equation of the foregoing problem, having +been freed from fractions, will assume the following +form: +\[ +Ax^{2}(a - x)^{2} - M(a - x)^{2} - Nx^{2} = 0. +\] +With its terms developed and properly arranged it +will be found to be of the fourth degree, and will consequently +have four roots. Now by the analysis which +we have just given, we can recognise at once the character +\PageSep{133} +of these roots. And since a method may spring +from this consideration applicable to all equations of +\index{Equations!fourth@of the fourth degree}% +\index{Fourth degree, equations of the}% +the fourth degree, we shall make a few brief remarks +\MNote{Preceding analysis applied to bi-quadratic equations.} +\index{Biquadratic equations}% +upon it in passing. Let the general equation be +\[ +x^{4} + px^{2} + qx + r = 0. +\] +We have already seen that if the last term of this +equation be negative it will necessarily have two real +roots, one positive and one negative; but that if the +last term be positive we can in general infer nothing +as to the character of its roots. If we give to this +equation the following form +\[ +(x^{2} - a^{2})^{2} + b(x + a)^{2} + c(x - a)^{2} = 0, +\] +a form which developed becomes +\[ +x^{4} + (b + c - 2a^{2})x^{2} + 2a(b - c)x + a^{4} + a^{2}(b + c) = 0, +\] +and from this by comparison derive the following +equations of condition +\[ +b + c - 2a^{2} = p,\quad 2a(b - c) = q,\quad a^{4} + a^{2}(b + c) = r, +\] +and from these, again, the following, +\[ +b + c = p + 2a^{2},\quad b - c = \frac{q}{2a},\quad 3a^{4} + pa^{2} = r, +\] +we shall obtain, by resolving the last equation, +\[ +a^{2} = -\frac{p}{6} + \sqrt{\frac{r}{3} + \frac{p^{2}}{36}}. +\] +If $r$~be supposed positive, $a^{2}$~will be positive and real, +and consequently $a$~will be real, and therefore, also, +$b$~and~$c$ will be real. + +Having determined in this manner the three quantities +$a$,~$b$,~$c$, we obtain the transformed equation +\[ +(x^{2} - a^{2})^{2} + b(x + a)^{2} + c(x - a)^{2} = 0. +\] +\PageSep{134} + +Putting the right-hand side of this equation equal +to~$y$, and considering the curve having for abscissæ +\MNote{Consideration of equations of the fourth degree.} +the different values of~$y$, it is plain, that when $b$~and~$c$ +are positive quantities this curve will lie wholly +above the axis and that consequently the equation +will have no real root. Secondly, suppose that $b$~is a +negative quantity and $c$~a positive quantity; then $x = a$ +will give $y = 4ba^{2}$,---a negative quantity. A very +large positive or negative~$x$ will then give a very large +positive~$y$,---whence it is easy to conclude that the +equation will have two real roots, one larger than~$a$ +and one less than~$a$. We shall likewise find that if +$b$~is positive and $c$~is negative, the equation will have +two real roots, one greater and one less than~$-a$. +Finally, if $b$~and~$c$ are both negative, then $y$~will become +negative by making +\[ +x = a \quad\text{and}\quad x = -a +\] +and it will be positive and very large for a very large +positive or negative value of~$x$,---whence it follows +that the equation will have two real roots, one greater +than~$a$ and one less than~$-a$. The preceding considerations +might be greatly extended, but at present we +must forego their pursuit. + +It will be seen from the preceding example that +the consideration of the curve does not require the +equation to be freed from fractional expressions. The +\index{Fractional expressions in equations}% +\index{Radical expressions in equations}% +same may be said of radical expressions. There is +an advantage even in retaining these expressions in +\PageSep{135} +the form given by the analysis of the problem; the +advantage being that we may in this way restrict our +attention to those signs of the radicals which answer +\MNote{Advantages of the method of curves.} +\index{Curves!advantages of the method of}% +to the special exigencies of each problem, instead of +causing the fractions and the radicals to disappear +and obtaining an equation arranged according to the +different whole powers of the unknown quantity in +which frequently roots are introduced which are entirely +foreign to the question proposed. It is true that +these roots are always part of the question viewed in +its entire extent; but this wealth of algebraical analysis, +although in itself and from a general point of view +extremely valuable, may be inconvenient and burdensome +in particular cases where the solution of which +we are in need cannot by direct methods be found independently +of all other possible solutions. When +the equation which immediately flows from the conditions +of the problem contains radicals which are essentially +ambiguous in sign, the curve of that equation +(constructed by making the side which is equal to +zero, equal to the ordinate~$y$) will necessarily have as +many branches as there are possible different combinations +of these signs, and for the complete solution it +would be necessary to consider each of these branches. +But this generality may be restricted by the particular +conditions of the problem which determine the branch +on which the solution is to be sought; the result being +that we are spared much needless calculation,---an +advantage which is not the least of those offered by +\PageSep{136} +the method of solving equations from the consideration +of curves. + +But this method can be still further generalised +\MNote{The curve of errors.} +\index{Errors, curve of|EtSeq}% +and even rendered independent of the equation of the +problem. It is sufficient in applying it to consider +the conditions of the problem in and for themselves, +to give to the unknown quantity different arbitrary +values, and to determine by calculation or construction +the errors which result from such suppositions +according to the original conditions. Taking these +errors as the ordinates~$y$ of a curve having for abscissæ +the corresponding values of the unknown quantity, +we obtain a continuous curve called \emph{the curve of errors}, +which by its intersections with the axis also gives all +solutions of the problem. Thus, if two successive errors +be found, one of which is an excess, and another +a defect, that is, one positive and one negative, we +may conclude at once that between these two corresponding +values of the unknown quantity there will +be one for which the error is zero, and to which we +can approach as near as we please by successive substitutions, +or by the mechanical description of the +curve. + +This mode of resolving questions by curves of errors +\index{Astronomy, mechanics, and physics, curves of errors in}% +\index{Mechanics, astronomy, and physics, curves of errors in}% +\index{Physics, astronomy, and mechanics, curves of errors in}% +is one of the most useful that have been devised. +It is constantly employed in astronomy when direct +solutions are difficult or impossible. It can be employed +for resolving important problems of geometry +and mechanics and even of physics. It is properly +\PageSep{137} +speaking the \textit{regula falsi}, taken in its most general +\index{False, rule of}% +\index{Regula@\textit{Regula falsi}}% +\index{Rule!false@of false}% +sense and rendered applicable to all questions where +there is an unknown quantity to be determined. It +\MNote{Solution of a problem by the curve of errors.} +can also be applied to problems that depend on two +or several unknown quantities by successively giving +to these unknown quantities different arbitrary values +and calculating the errors which result therefrom, afterwards +linking them together by different curves, or +reducing them to tables; the result being that we may +\index{Tables}% +by this method obtain directly the solution sought +\Figure{3}{0.4\textwidth} +without preliminary elimination of the unknown quantities. + +We shall illustrate its use by a few examples. + +\textit{Required a circle in which a polygon of given sides can +be inscribed.} + +This problem gives an equation which is proportionate +in degree to the number of sides of the polygon. +To solve it by the method just expounded we +describe any circle~$ABCD$ (Fig.~3) and lay off in this +circle the given sides $AB$,~$BC$, $CD$, $DE$,~$EF$ of the +\PageSep{138} +polygon, which for the sake of simplicity I here suppose +to be pentagonal. If the extremity of the last +\MNote{Problem of the circle and inscribed polygon.} +\index{Circle!and inscribed polygon, problem of the}% +\index{Polygon, problem of the circle and inscribed}% +side falls on~$A$, the problem is solved. But since it +is very improbable that this should happen at the first +trial we lay off on the straight line~$PR$ (Fig.~4) the +radius~$PA$ of the circle, and erect on it at the point~$A$ +the perpendicular~$AF$ equal to the chord~$AF$ of the +arc~$AF$ which represents the error in the supposition +\index{Supposition, rule of}% +\index{Trial and error, rule of}% +made regarding the length of the radius~$PA$. Since +this error is an excess, it will be necessary to describe +\Figure{4}{0.3\textwidth} +a circle having a larger radius and to perform the +same operation as before, and so on, trying circles of +various sizes. Thus, the circle having the radius~$PA$ +gives the error~$F'A'$ which, since it falls on the hither +side of the point~$A'$, should be accounted negative. It +will consequently be necessary in Fig.~4 in applying +the ordinate~$A'F'$ to the abscissa~$PA'$ to draw that +ordinate below the axis. In this manner we shall obtain +several points $F$,~$F'$,~$\dots$, which will lie on a +curve of which the intersection~$R$ with the axis~$PA$ +\PageSep{139} +will give the true radius~$PR$ of the circle satisfying +the problem, and we shall find this intersection by +successively causing the points of the curve lying on +\MNote{Solution of a second problem by the curve of errors.} +the two sides of the axis as $F$,~$F'$,~$\dots$ to approach +nearer and nearer to one another. + +\textit{From a point, the position of which is unknown, three +\index{Point in space, position of a}% +objects are observed, the distances of which from one another +are known. The three angles formed by the rays of +light from these three objects to the eye of the observer are +also known. Required the position of the observer with +respect to the three objects.} + +If the three objects be joined by three straight +lines, it is plain that these three lines will form with +the visual rays from the eye of the observer a triangular +pyramid of which the base and the three face angles +forming the solid angle at the vertex are given. +And since the observer is supposed to be stationed at +the vertex, the question is accordingly reduced to determining +the dimensions of this pyramid. + +Since the position of a point in space is completely +determined by its three distances from three given +points, it is clear that the problem will be resolved, if +the distances of the point at which the observer is +stationed from each of the three objects can be determined. +Taking these three distances as the unknown +quantities we shall have three equations of the second +degree, which after elimination will give a resultant +equation of the eighth degree; but taking only one of +these distances and the relations of the two others to it +\PageSep{140} +for the unknown quantities, the final equation will be +only of the fourth degree. We can accordingly rigorously +\MNote{Problem of the observer and three objects.} +solve this problem by the known methods; but +the direct solution, which is complicated and inconvenient +in practice, may be replaced by the following +which is reached by the curve of errors. + +Let the three successive angles $APB$, $BPC$, $CPD$ +\index{Observer, problem of the, and three objects}% +(Fig.~5) be constructed, having the vertex~$P$ and +respectively equal to the angles observed between the +first object and the second, the second and the third, +\Figure{5}{0.4\textwidth} +the third and the first; and let the straight line~$PA$ +be taken at random to represent the distance from the +observer to the first object. Since the distance of +that object to the second is supposed to be known, +let it be denoted by~$AB$, and let it be laid off on the +line~$AB$. We shall in this way obtain the distance~$BP$ +of the second object to the observer. In like manner, +let $BC$, the distance of the second object to the +third, be laid off on~$BC$, and we shall have the distance~$PC$ +of that object to the observer. If, now, the +\PageSep{141} +distance of the third object to the first be laid off on +the line~$CD$, we shall obtain~$PD$ as the distance of +the first object to the observer. Consequently, if the +\MNote{Employment of the curve of errors.} +distance first assumed is exact, the two lines $PA$~and~$PD$ +will necessarily coincide. Making, therefore, on +the line~$PA$, prolonged if necessary, the segment +$PE = PD$, if the point~$E$ does not fall upon the point~$A$, +the difference will be the error of the first assumption~$PA$. +Having drawn the straight line~$PR$ (Fig.~6) +we lay off upon it from the fixed point~$P$, the abscissa~$PA$, +and apply to it at right angles the ordinate~$EA$; +we shall have the point~$E$ of the curve of errors~$ERS$. +\Figure{6}{0.4\textwidth} +Taking other distances for~$PA$, and making the same +construction, we shall obtain other errors which can be +similarly applied to the line~$PR$, and which will give +other points in the same curve. + +We can thus trace this curve through several +points, and the point~$R$ where it cuts the axis~$PR$ will +give the distance~$PR$, of which the error is zero, and +which will consequently represent the exact distance +of the observer from the first object. This distance +being known, the others may be obtained by the same +construction. + +It is well to remark that the construction we have +been considering gives for each point~$A$ of the line~$PA$, +\PageSep{142} +two points $B$~and~$B'$ of the line~$PB$; for, since +the distance~$AB$ is given, to find the point~$B$ it is only +\MNote{Eight possible solutions of the preceding problem.} +necessary to describe from the point~$A$ as centre and +with radius~$AB$ an arc of a circle cutting the straight +line~$PB$ at the two points $B$~and~$B'$,---both of which +points satisfy the conditions of the problem. In the +same manner, each of these last-mentioned points will +give two more upon the straight line~$PC$, and each of +the last will give two more on the straight line~$PD$. +Whence it follows that every point~$A$ taken upon the +straight line~$PA$ will in general give eight upon the +straight line~$PD$, all of which must be separately and +successively considered to obtain all the possible solutions. +I have said, \emph{in general}, because it is possible +(1)~for the two points $B$~and~$B'$ to coincide at a single +point, which will happen when the circle described +with the centre~$A$ and radius~$AB$ touches the straight +line~$PB$; and (2)~that the circle may not cut the +straight line~$PB$ at all, in which case the rest of the +construction is impossible, and the same is also to be +said regarding the points $C$,~$D$. Accordingly, drawing +the line~$GF$ parallel to~$BP$ and at a distance from it +equal to the given line~$AB$, the point~$F$ at which this +line cuts the line~$PE$, prolonged if necessary, will be +the limit beyond which the points~$A$ must not be taken +if we desire to obtain possible solutions. There exist +also limits for the points $B$~and~$C$, which may be employed +in restricting the primitive suppositions made +with respect to the distance~$PA$. +\PageSep{143} + +The eight points~$D$, which depend in general on +each point~$A$, answer to the eight solutions of which +the problem is susceptible, and when one has no special +\MNote{Reduction of the possible solutions in practice.} +datum by means of which it can be determined +which of these solutions answer best to the case proposed, +it is indispensable to ascertain them all by employing +for each one of the eight combinations a special +curve of errors. But if it be known, for example, +that the distance of the observer to the second object +is greater or less than his distance to the first, it will +then be necessary to take on the line~$PB$ only the +point~$B$ in the first case and the point~$B'$ in the second,---a +course which will reduce the eight combinations +one-half. If we had the same datum with regard +to the third object relatively to the second, and with +regard to the first object relatively to the third, then +the points $C$~and~$D$ would be determined, and we +should have but a single solution. + +These two examples may suffice to illustrate the +uses to which the method of curves can be put in solving +\index{Curves!method of, submitted to analysis|EtSeq}% +problems. But this method, which we have presented, +so to speak, in a mechanical manner, can also +be submitted to analysis. + +The entire question in fact is reducible to the description +of a curve which shall pass through a certain +number of points, whether these points be given by +calculation or construction, or whether they be given +by observation or single experiences entirely independent +of one another. The problem is in truth indeterminate, +\PageSep{144} +for strictly speaking there can be made +to pass through a given number of points an infinite +\MNote{General conclusion on the method of curves.} +\index{Curves!advantages of the method of}% +number of different curves, regular or irregular, that +is, subject to equations or arbitrarily drawn by the +hand. But the question is not to find any solutions +whatever but the simplest and easiest in practice. + +Thus if there are only two points given, the simplest +solution is a straight line between the two points. +\index{Straight line}% +If there are three points given, the arc of a circle is +\index{Circle}% +drawn through these points, for the arc of a circle +after the straight line is the simplest line that can be +described. + +But if the circle is the simplest curve with respect +to description, it is not so with respect to the equation +between its abscissæ and rectangular ordinates. +In this latter point of view, those curves may be regarded +as the simplest of which the ordinates are expressed +by an integral rational function of the abscissæ, +as in the following equation +\[ +y = a + bx + cx^{2} + dx^{3} + \dots, +\] +where $y$~is the ordinate and $x$~the abscissa. Curves +of this class are called in general \emph{parabolic}, because +\index{Parabolic@\textit{Parabolic} curves|EtSeq}% +they may be regarded as a generalisation of the parabola,---a +curve represented by the foregoing equation +when it has only the first three terms. We have already +illustrated their employment in resolving equations, +and their consideration is always useful in the +approximate description of curves, for the reason that +a curve of this kind can always be made to pass +\PageSep{145} +through as many points of a given curve as we please,---it +being only necessary to take as many undetermined +coefficients $a$,~$b$,~$c$,~$\dots$ as there are points given, +\MNote{Parabolic curves.} +and to determine these coefficients so as to obtain the +abscissæ and ordinates for these points. Now it is +clear that whatever be the curve proposed, the parabolic +curve so described will always differ from it by +less and less according as the number of the different +points is larger and larger and their distance from +one another smaller and smaller. + +Newton was the first to propose this problem. The +\index{Newton, his problem}% +following is the solution which he gave of it: + +Let $P$,~$Q$, $R$,~$S$,~$\dots$ be the values of the ordinates~$y$ +corresponding to the values $p$,~$q$, $r$,~$s$,~$\dots$ of +the abscissæ~$x$; we shall have the following equations +\[ +\begin{array}{r@{\,}*{3}{l@{\,}}l} +P &= a + bp &+ cp^{2} &+ dp^{3} &+ \dots, \\ +Q &= a + bq &+ cq^{2} &+ dq^{3} &+ \dots, \\ +R &= a + br &+ cr^{2} &+ dr^{3} &+ \dots, \\ +\hdotsfor{5}\Add{.} +\end{array} +\] +The number of these equations must be equal to the +number of the undetermined coefficients $a$,~$b$,~$c$,~$\dots$. +Subtracting these equations from one another, the remainders +will be divisible by $q - p$, $r - q$,~$\dots$, and +we shall have after such division +\[ +\begin{array}{r@{\,}*{2}{l@{\,}}l} +\dfrac{Q - P}{q - p} &= b + c(q + p) &= d(q^{2} + qp + p^{2}) &+ \dots, \\[8pt] +\dfrac{R - Q}{r - q} &= b + c(r + q) &= d(r^{2} + rq + q^{2}) &+ \dots, \\ +\hdotsfor{4}\Add{.} +\end{array} +\] +\PageSep{146} + +Let +\[ +\frac{Q - P}{q - p} = Q_{1},\quad +\frac{R - Q}{r - q} = R_{1},\quad +\frac{S - R}{s - r} = S_{1},\dots\Add{.} +\] +\MNote{Newton's problem.} +We shall find in like manner, by subtraction and division, +the following: +\[ +\begin{array}{r@{\,}l@{\,}l} +\dfrac{R_{1} - Q_{1}}{r - p} &= c + d(r + q + p) &+ \dots, \\[8pt] +\dfrac{S_{1} - R_{1}}{s - q} &= c + d(s + r + q) &+ \dots, \\ +\hdotsfor{3}\Add{.} +\end{array} +\] + +Further let +\[ +\frac{R_{1} - Q_{1}}{r - p} = R_{2},\quad +\frac{S_{1} - R_{1}}{s - q} = S_{2},\dots. +\] +We shall have +\[ +\frac{S_{2} - R_{2}}{s - p} = d + \dots, +\] +and so on. + +In this manner we shall find the value of the coefficients +$a$,~$b$,~$c$,~$\dots$ commencing with the last; and, +substituting them in the general equation +\[ +y = a + bx + cx^{2} + dx^{3} + \dots, +\] +we shall obtain, after the appropriate reductions have +been made, the formula +\[ +y = P + + Q_{1}(x - p) + + R_{2}(x - p)(x - q) + + S_{3}(x - p)(x - q)(x - r) + \dots, +\Tag{(1)} +\] +which can be carried as far as we please. + +But this solution may be simplified by the following +consideration. + +Since $y$~necessarily becomes $P$,~$Q$,~$R$\Add{,}~$\dots$, when $x$~becomes +\PageSep{147} +$p$,~$q$,~$r$, it is easy to see that the expression +for~$y$ will be of the form +\MNote{Simplification of Newton's solution.} +\[ +y = AP + BQ + CR + DS + \dots +\Tag{(2)} +\] +where the quantities $A$,~$B$, $C$,~$\dots$ are so expressed in +terms of~$x$ that by making $x = p$ we shall have +\[ +A = 1,\quad B = 0,\quad C = 0,\dots, +\] +and by making $x = q$ we shall have +\[ +A = 0,\quad B = 1,\quad C = 0,\quad D = 0,\dots, +\] +and by making $x = r$ we shall similarly have +\[ +A = 0,\quad B = 0,\quad C = 1,\quad D = 0,\dots\ \text{etc.} +\] +Whence it is easy to conclude that the values of $A$, +$B$, $C$,~$\dots$ must be of the form +\begin{align*} +A &= \frac{(x - q)(x - r)(x - s)\dots}{(p - q)(p - r)(p - s)\dots}, \\ +B &= \frac{(x - p)(x - r)(x - s)\dots}{(q - p)(q - r)(q - s)\dots}, \\ +C &= \frac{(x - p)(x - q)(x - s)\dots}{(r - p)(r - q)(r - s)\dots}, +\end{align*} +where there are as many factors in the numerators +and denominators as there are points given of the +curve less one. + +The last expression for~$y$ (see equation~2), although +different in form, is the same as equation~1. To show +this, the values of the quantities $Q_{1}$,~$R_{2}$, $S_{3}$,~$\dots$ need +only be developed and substituted in equation~1 and +the terms arranged with respect to the quantities $P$, +$Q$, $R$,~$\dots$\Add{.} But the last expression for~$y$ (equation~2) +is preferable, partly because of the simplicity of the +\PageSep{148} +analysis from which it is derived, and also because of +its form, which is more convenient for computation. + +\MNote{Possible uses of Newton's problem.} +Now, by means of this formula, which it is not +difficult to reduce to a geometrical construction, we +are able to find the value of the ordinate~$y$ for any abscissa~$x$, +because the ordinates $P$,~$Q$, $R$,~$\dots$ for the +given abscissæ $p$,~$q$, $r$,~$\dots$ are known. Thus, if we +have several of the terms of any series, we can find +any intermediate term that we wish,---an expedient +which is extremely valuable for supplying lacunæ +which may arise in a series of observations or experiments, +\index{Experiments!expedient@an expedient for supplying lacunæ in a series of}% +\index{Observations, expedient for supplying lacunæ in series of}% +or in tables calculated by formulæ or in given +\index{Tables!expedient for supplying lacunæ in}% +constructions. + +If this theory now be applied to the two examples +\index{Regula@\textit{Regula falsi}}% +\index{Supposition, rule of}% +\index{Trial and error, rule of}% +discussed above and to similar examples in which we +have errors corresponding to different suppositions, we +can directly find the error~$y$ which corresponds to any +intermediate supposition~$x$ by taking the quantities +$P$,~$Q$, $R$,~$\dots$, for the errors found, and $p$,~$q$, $r$,~$\dots$ for +the suppositions from which they result. But since +in these examples the question is to find not the error +which corresponds to a given supposition, but the +supposition for which the error is zero, it is clear that +the present question is the opposite of the preceding +and that it can also be resolved by the same formula +by reciprocally taking the quantities $p$,~$q$, $r$,~$\dots$ for +the errors, and the quantities $P$,~$Q$, $R$,~$\dots$ for the +corresponding suppositions. Then $x$~will be the error +for the supposition~$y$; and consequently, by making +\PageSep{149} +$x = 0$, the value of~$y$ will be that of the supposition +for which the error is zero. + +Let $P$,~$Q$, $R$,~$\dots$ be the values of the unknown +quantity in the different suppositions, and $p$,~$q$, $r$\Add{,}~$\dots$ +\MNote{Application of Newton's problem to the preceding examples.} +the errors resulting from these suppositions, to which +the appropriate signs are given. We shall then have +for the value of the unknown quantity of which the +error is zero, the expression +\[ +AP + BQ + CR + \dots, +\] +in which the values of $A$,~$B$,~$C$\Add{,}~$\dots$ are +\begin{align*} +A &= \frac{q}{q - r} × \frac{r}{r - p} × \dots, \displaybreak[1] \\ +B &= \frac{P}{p - q} × \frac{r}{r - q} × \dots, \displaybreak[1] \\ +C &= \frac{p}{p - r} × \frac{q}{q - r} × \dots, +\end{align*} +where as many factors are taken as there are suppositions +less one. +\index{Curves!employment of in the solution of problems|)}% +\index{Problems!employment of curves in the solution of|)}% +\PageSep{150} +%[Blank page] +\PageSep{151} + + +\Appendix{Note on the Origin of Algebra.} +\PgLabel{151} +\index{Algebra!history of}% + +\First{The} impression (\PgRef{54}) that Diophantus was the +\index{Diophantus}% +``inventor'' of algebra, which sprang, in its Diophantine +form, full-fledged from his brain, was a widespread +one in the eighteenth and in the beginning of +the nineteenth century. But, apart from the intrinsic +improbability of this view which is at variance with +the truth that science is nearly always gradual and +organic in growth, modern historical researches have +traced the germs and beginnings of algebra to a much +remoter date, even in the line of European historical +continuity. The Egyptian book of Ahmes contains +\index{Ahmes}% +examples of equations of the first degree. The early +Greek mathematicians performed the partial resolution +\index{Greeks, mathematics of the}% +of equations of the second and third degree +by geometrical methods. According to Tannery, an +\index{Tannery, M. Paul}% +embryonic indeterminate analysis existed in Pre-Christian +times (Archimedes, Hero, Hypsicles). But +\index{Archimedes}% +\index{Hero}% +\index{Hypsicles}% +the merit of Diophantus as organiser and inaugurator +of a more systematic short-hand notation, at +least in the European line, remains; he enriched +whatever was handed down to him with the most +manifold extensions and applications, betokening his +\PageSep{152} +originality and genius, and carried the science of algebra +\index{Algebra!among the Arabs}% +\index{Algebra!India@in India}% +to its highest pitch of perfection among the +\PgLabel{152} +Greeks. (See Cantor, \textit{Geschichte der Mathematik}, second +\index{Cantor}% +edition, Vol.~I., p.~438, et~seq.; Ball, \textit{Short Account +\index{Ball}% +of the History of Mathematics}, second edition, p.~104 +et~seq.; Fink, \textit{A Brief History of Mathematics}, pp.~63 +\index{Fink}% +et~seq., 77~et~seq. (Chicago: The Open Court +Publishing~Co.) + +The development of Hindu algebra is also to be +noted in connexion with the text of \PgRange{59}{60}. The +Arabs, who had considerable commerce with India, +\index{Arabs!Algebra among the}% +drew not a little of their early knowledge from the +works of the Hindus. Their algebra rested on both +that of the Hindus and the Greeks. (See Ball, \textit{op.~cit.}, +p.~150 et~seq.; Cantor, \textit{op.~cit.}, Vol.~I., p.~651 et~seq.).---\textit{Trans.} +\PageSep{153} +\BackMatter +\printindex +\iffalse +INDEX. + +Academies, rise of 62, 63 + +Ahmes 151 + +Algebra + definition of 2 + history of|EtSeq#history 54 % et seq., + history of 151 + essence of 55 + name@the name of 59 + among the Arabs|EtSeq 59 % et seq, + among the Arabs 152 + Europe@in Europe 60 + Italy@in Italy 64 + India@in India 152 + generality@the generality of 69 + hand-writing of 69 + application of geometry to|EtSeq 100, 127 % et seq. + +Algebraical resolution of equations + limits of the 96 + +Alligation + generally|EtSeq 44 % et seq.; + alternate 47 + +Analysis + indeterminate|EtSeq 47 % et seq., + indeterminate 55 + +Angle, trisection of an 62, 81 + +Angular sections, theory of 80 + +Annuities 16 + +Apollonius 54, 59 + +Arabs + Algebra among the|EtSeq 59 % et seq., + Algebra among the 152 + +Archimedes 54, 151 + +Archimedes|FN 58 % footnote + +Arithmetic + universal|EtSeq 2 % et seq.; + operations of|EtSeq 24 % et seq. + +Arithmetical progression revealing the roots 120 + +Arithmetical progression revealing the roots|EtSeq 112 % et seq. + +Arithmetical proportion 12 + +Astronomy, mechanics, and physics, curves of errors in 136 + +Average life|EtSeq 45 % et seq. + +Bachet de Méziriac 58 + +Ball 152 + +Binomial theorem 115 + +Binomials, extraction of the square roots of two imaginary 77 + +Biquadratic equations 63, 88, 94, 133 + +Bombelli 63, 64 + +Bret, M.|FN 93 % footnote. + +Briggs 20 + +Buteo 61 + +Cantor|FN 54, 60 % footnote, + +Cantor 152 + +Cardan 60, 61, 68, 82, 90 + +Checks on multiplication and division 39 + +Circle 144 + squaring of the 62 + and inscribed polygon, problem of the 138 + +Clairaut 69, 90 + +Coefficients + indeterminate 89 + greatest negative|EtSeq 107 % et seq., + greatest negative 117 + +Common divisor of two equations 121 + +Complements, subtraction by 26 + +Constantinople 58 + +Continued fractions, solution of alligation by|EtSeq 50 % et seq. + +Convergents 7 + +Cube, duplication of the 62 + +Cube roots of a quantity, the three 70 + +Cubic radicals 75 + +Curves + representation of equations by|EtSeq 101 % et seq; + employment of in the solution of problems 127-149 + method of, submitted to analysis|EtSeq 143 % et seq.; + advantages of the method of 135, 144 + +Decimal + fractions 9 + numbers|EtSeq 27 % et seq. + +Decimals + multiplication of 30 + division of 31 +\PageSep{154} + +DeMorgan@{\Typo{DeMorgan}{De Morgan}} v + +Descartes viii, 60, 65, 89, 93, 127 + +Differences, the equation of|EtSeq 114 % et seq., + +Differences, the equation of 123 + +Differential Calculus 131 + +Diophantine problems 55 + +Diophantus|EtSeq 54 % et seq + +Diophantus 151 + +Division + nine@by \textit{nine} 34 + eight@by \textit{eight} 34 + seven@by \textit{seven}|EtSeq 34 % et seq.; + decimals@of decimals 31 + +Divisor, greatest common|EtSeq 2 % et seq. + +Duhring@{Dühring, E.} v + +Duodecimal system 32 + +Ecole@{\Typo{Ecole}{École} Normale} v, xi, 12 + +Economy of thought vii + +Efflux, law of 42 + +Eleven, the number, test of divisibility by 37 + +Elimination + method of 121 + general formulæ for 122 + +Equations + second@of the second degree 56 + third@of the third degree 60, 66, 82 + fourth@of the fourth degree 63, 87, 133 + fifth@of the fifth degree 64 + theory of 65, 84 + biquadratic 88 + limits of the algebraical resolution of 96 + fifth@of the fifth degree 96 + mth@of the $m$th degree 96 + general remarks upon the roots of|EtSeq 102 % et seq.; + graphic resolution of 102 + odd@of an odd degree, roots of 105 + even@of an even degree, roots of 106 + real roots of, limits of the|EtSeq 107 % et seq.; + common divisor of two 121 + constructions for solving|EtSeq 100 % et seq. + constructions for solving 124 + machine@a machine for solving 126 + +Equi-different numbers 13 + +Errors, curve of|EtSeq 136 % et seq. + +Euclid 2, 57 + +Euler viii, x, 93 + +Europe, algebra in 60 + +Evolution 11, 40 + +Experiments + average of 46 + expedient@an expedient for supplying lacunæ in a series of 148 + +Falling stone, spaces traversed by a 42 + +False, rule of 137 + +Fermat 58 + +Ferrari, Louis 64 + +Ferrous, Scipio|EtSeq 60 % et seq. + +Fifth degree, equations of the 96 + +Fink 152 + +Fourth degree, equations of the 133 + +Fractional expressions in equations 134 + +Fractions|EtSeq 2 % et seq.; + +Fractions + continued|EtSeq 3 % et seq.; + converging 6 + decimal 9 + origin of continued 10 + +France 58, 61 + +Galileo ix + +Geometers, ancient|EtSeq 54 % et seq. + +Geometers, ancient 58, 59 + +Geometrical + proportion 13 + calculus 24 + +Geometry 24, 60 + application of to algebra|EtSeq 100, 127 % et seq. + +Germany 61 + +Girard, Albert 62 + +Grain, of different prices 44 + +Greeks, mathematics of the vii, 151 + +Greeks, mathematics of the|EtSeq 54 % et seq. + +Hand-writing of algebra 69 + +Harriot 65 + +Hero 59, 151 + +Horses 43 + +Hudde 65, 82 + +Huygens ix, 10 + +Hypsicles 151 + +Imaginary binomials, square roots of 77 + +Imaginary expressions|EtSeq 79 % et seq. + +Imaginary expressions 83 + +Imaginary quantities, office of the 87 + +Imaginary roots, occur in pairs 99 + +Indeterminate analysis|EtSeq 47 % et seq. + +Indeterminate analysis 55 + +Indeterminate coefficients 89 + +Indeterminates, the method of 83 + +Ingredients 48 + +Interest 15 + +Intersections, with the axis give roots|EtSeq 102 % et seq , + +Intersections, with the axis give roots 113 + +Inventors, great 22 + +Involution and evolution 11 + +Irreducible case 61, 65, 69, 73, 82 + +Italy, cradle of algebra in Europe 61, 64 + +Laborers, work of 41 + +Lagrange, J. L.#Lagrange v + +Lagrange, J. L.|EtSeq#Lagrange vii % et seq. +\PageSep{155} + +Laplace v, xi + +Lavoisier xii + +Leibnitz viii + +Life insurance|EtSeq 45 % et seq. + +Life, probability of 46 + +Light, law of the intensity of 129 + +Lights, problem of the two|EtSeq 129 % et seq. + +Limits of roots 107-120 + +Logarithms|EtSeq 16 % et seq. + +Logarithms 40 + advantages in calculating by 28 + origin of 19 + tables of 20 + +Machine for solving equations 124-126 + +Mathematics + wings of 24 + exactness of 43 + evolution of vii + +Mean values|EtSeq 45 % et seq. + +Mechanics, astronomy, and physics, curves of errors in 136 + +Metals, mingling of, by fusion 44 + +Meziriac@Méziriac, Bachet de 58 + +Minimal values 132 + +Mixtures, rule of|EtSeq 44 % et seq. + +Mixtures, rule of 49 + +Monge v, xi + +Mortality, tables of 45 + +Moving bodies, two 98 + +Multiple roots 105 + +Multiplication + abridged methods of|EtSeq 26 % et seq.; + inverted 28 + approximate 29 + decimals@of decimals 30 + +Music 22 + +Napier|EtSeq 17 % et seq. + +Napoleon xii + +Negative roots 60 + +Newton, his problem 145, viii + +Nine + property of the number|EtSeq 31 % et seq.; + property of the number generalised 33 + +Nizze|FN 58 % footnote. + +Numeration, systems of 1 + +Numerical equations |See Equations 0 + +Numerical equations + resolution of 96-126 + conditions of the resolution of 97 + position of the roots of 98 + +Observations, expedient for supplying lacunæ in series of 148 + +Observer, problem of the, and three objects 140 + +Oughtred 30 + +Paciolus, Lucas 59, 60 + +Pappus 59 + +Parabolic@\textit{Parabolic} curves|EtSeq 144 % et seq. + +Peletier 61 + +Peyrard 58 + +Physics, astronomy, and mechanics, curves of errors in 136 + +Planetarium 9 + +Point in space, position of a 139 + +Polygon, problem of the circle and inscribed 138 + +Polytechnic School v, xi + +Positive roots, superior and inferior limits of the 109 + +Powers|EtSeq 10 % et seq. + +Practice, theory and 43 + +Present value 15 + +Printing, invention of 59 + +Probabilities, calculus of|EtSeq 45 % et seq. + +Problems 110 + solution@for solution 62 + employment of curves in the solution of 127-149 + +Proclus 59 + +Progressions, theory of 12, 14 + +Proportion|EtSeq 11 % et seq. + +Ptolemy 59 + +Radical expressions in equations 134 + +Radicals, cubic 75 + +Ratios, constant 42 + +Ratios, constant|EtSeq 2, 11 % et seq. + +Reality of roots 76, 83, 85, 93 + +Regula@\textit{Regula falsi} 137, 148 + +Remainders + theory of|EtSeq 34 % et seq. + theory of 38 + negative|EtSeq 35 % et seq. + +Romans, mathematics of the 54 + +Roots + negative 60 + equations@of equations of the third degree 71 + reality@the reality of the 74, 76, 79, 83, 85, 93 + biquadratic@of a biquadratic equation 94 + multiple 105 + superior and inferior limits of the positive 109 + method for finding the limits of 110 + separation of the 112 + arithmetical@the arithmetical progression revealing the|EtSeq 112 % et seq. + arithmetical@the arithmetical progression revealing the 120 + quantity less than the difference between any two 113 + smallest|EtSeq 116 % et seq.; + limits of the positive and negative 119 + +Rule + Cardan's 68 + false@of false 137 + mixtures@of mixtures|EtSeq 44 % et seq.; + three@of three|EtSeq 11, 40 % et seq. +\PageSep{156} + +Science + history of 22 + development of|EtSeq vii % et seq. + +Seven, tests of divisibility by 35 + +Short-mind symbols|EtSeq vii % et seq. + +Signs $+$ and $-$ 57 + +Squaring of the circle 62 + +Stenophrenic symbols|EtSeq vii % et seq. + +Straight line 144 + +Substitutions|EtSeq 111 % et seq. + +Substitutions 123 + +Subtraction, new method of|EtSeq 25 % et seq. + +Sum and difference, of two numbers 56 + +Supposition, rule of 137, 148 + +Symbols|EtSeq vii % et seq. + +Tables 137 + expedient for supplying lacunæ in 148 + +Tannery, M. Paul|FN 58 % footnote + +Tannery, M. Paul 151 + +Tartaglia 60, 61 + +Temperament, theory of 23 + +Theon 59 + +Theory and practice 43 + +Theory of remainders, utility of the 38 + +Third degree, equations of the 71, 82 + +Three roots, reality of the 93 + +Trial and error, rule of 137, 148 + +Trisection of an angle 62, 81 + +Turks 58 + +Undetermined quantities 82 + +Unity, three cubic roots of 72 + +Unknown quantity 55 + +Values + mean|EtSeq 45 % et seq.; + minimal 132 + +Variations, calculus of x + +Vatican library 58 + +Vieta viii, 62, 65 + +Vlacq 20 + +Wallis viii + +Wertheim, G.|FN 58 % footnote. + +Woodhouse x + +Xylander 58 +\fi +\PageSep{157} + +\Catalog +%[** TN: Macro prints the following text] +% Catalogue of Publications +% of the +% Open Court Publishing Co. + +\begin{Author}{COPE, E. D.} +\Title{THE PRIMARY FACTORS OF ORGANIC EVOLUTION.} +{121~cuts. Pp.~xvi,~547. Cloth,~\$2.00 (10s.).} +\end{Author} + +\begin{Author}{MÃœLLER, F. 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+\begin{document} +%% PG BOILERPLATE %% +\PGBoilerPlate +\begin{center} +\begin{minipage}{\textwidth} +\small +\begin{PGtext} +The Project Gutenberg EBook of Lectures on Elementary Mathematics, by +Joseph Louis Lagrange + +This eBook is for the use of anyone anywhere at no cost and with +almost no restrictions whatsoever. You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.org + + +Title: Lectures on Elementary Mathematics + +Author: Joseph Louis Lagrange + +Translator: Thomas Joseph McCormack + +Release Date: July 6, 2011 [EBook #36640] + +Language: English + +Character set encoding: ISO-8859-1 + +*** START OF THIS PROJECT GUTENBERG EBOOK LECTURES ON ELEMENTARY MATHEMATICS *** +\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} +\end{center} +\vfill + +\begin{minipage}{0.85\textwidth} +\small +\BookMark{0}{Transcriber's Note.} +\subsection*{\centering\normalfont\scshape% +\normalsize\MakeLowercase{\TransNote}}% + +\raggedright +\TransNoteText +\end{minipage} +%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%% +\FrontMatter +\null\vfill +\noindent {\Large ON ELEMENTARY MATHEMATICS} +\vfill +\PageSep{} +\FrontCatalog{IN THE SAME SERIES.} + +\tb + +\Book{ON CONTINUITY AND IRRATIONAL NUMBERS, and +ON THE NATURE AND MEANING OF NUMBERS\@. +By R.~\textsc{Dedekind}. From the German by \textit{W.~W. Beman}. +Pages,~115. Cloth, 75~cents net (3s.~6d.~net).} + +\Book{GEOMETRIC EXERCISES IN PAPER-FOLDING\@. By \textsc{T.~Sundara Row}. +Edited and revised by \textit{W.~W. Beman} and +\textit{D.~E. Smith}. With many half-tone engravings from photographs +of actual exercises, and a package of papers for +folding. Pages, circa~200. Cloth, \$1.00\Typo{.}{} net (4s.~6d.~net). +(In Preparation.)} + +\Book{ON THE STUDY AND DIFFICULTIES OF MATHEMATICS\@. +By \textsc{Augustus De~Morgan}. Reprint edition\Typo{`}{} +with portrait and bibliographies. Pp.,~288. Cloth, \$1.25 +net (4s.~6d.~net).} + +\Book{LECTURES ON ELEMENTARY MATHEMATICS\@. By +\textsc{Joseph Louis Lagrange}. From the French by \textit{Thomas~J. +McCormack}, With portrait and biography. Pages,~172. +Cloth, \$1.00 net (4s.~6d.~net).} + +\Book{ELEMENTARY ILLUSTRATIONS OF THE DIFFERENTIAL +AND INTEGRAL CALCULUS\@. By \textsc{Augustus De~Morgan}. +Reprint edition. With a bibliography of text-books +of the Calculus. Pp.,~144. Price, \$1.00 net (4s.~6d.~net).} + +\Book{MATHEMATICAL ESSAYS AND RECREATIONS\@. By +\textsc{Prof.\ Hermann Schubert}, of Hamburg, Germany. From +the German by \textit{T.~J. McCormack}. Essays on Number\Typo{.}{,} +The Magic Square, The Fourth Dimension, The Squaring +of the Circle. Pages,~149. Price, Cloth, 75c.~net (3s.~net).} + +\Book{A BRIEF HISTORY OF ELEMENTARY MATHEMATICS\@. +By \textsc{Dr.\ Karl Fink}, of Tübingen. From the German by \textit{W.~W. +Beman} and \textit{D.~E. Smith}. Pp.~333. Cloth, \$1.50 net +(5s.~6d.~net).} + +\tb +\vfill + +\noindent\makebox[\textwidth][s]{\large THE OPEN COURT PUBLISHING COMPANY} +\begin{center} +\footnotesize 324 DEARBORN ST., CHICAGO. \\ +\normalsize LONDON: Kegan Paul, Trench, Trübner \& Co. +\end{center} +\PageSep{i} +%[Blank page] +\PageSep{ii} +\Frontispiece +\PageSep{iii} +\begin{center} +\Large LECTURES\\[24pt] +\footnotesize ON\\[24pt] +\LARGE ELEMENTARY MATHEMATICS +\vfill + +\footnotesize BY\\[18pt] +\large JOSEPH LOUIS LAGRANGE +\vfill + +\footnotesize FROM THE FRENCH BY\\[18pt] +\normalsize THOMAS J. McCORMACK +\vfill\vfill + +\small SECOND EDITION +\vfill\vfill + +\large CHICAGO \\ +\normalsize THE OPEN COURT PUBLISHING COMPANY \\[12pt] +\footnotesize LONDON AGENTS \\ +\textsc{Kegan Paul, Trench, Trübner \& Co., Ltd.} \\ +1901 +\end{center} +\newpage +\PageSep{iv} +\null\vfill +\begin{center} +\footnotesize TRANSLATION COPYRIGHTED \\ +BY \\ +\small\textsc{The Open Court Publishing Co.} \\ +1898. +\end{center} +\vfill +\PageSep{v} + + +\Preface + +\First{The} present work, which is a translation of the \textit{Leçons élémentaires +sur les \Typo{mathematiques}{mathématiques}} of Joseph Louis Lagrange, +\index{Lagrange, J. L.}% +the greatest of modern analysts, and which is to be found in Volume~VII. +of the new edition of his collected works, consists of a +series of lectures delivered in the year 1795 at the \textit{\Typo{Ecole}{École} Normale},---an +institution which was the direct outcome of the French Revolution +and which gave the first impulse to modern practical +ideals of education. With Lagrange, at this institution, were associated, +as professors of mathematics. Monge and Laplace, and we +\index{Laplace}% +\index{Monge}% +owe to the same historical event the final form of the famous \textit{Géométrie +descriptive}, as well as a second course of lectures on arithmetic +and algebra, introductory to these of Lagrange, by Laplace. + +With the exception of a German translation by Niedermüller +\index{Ecole@{\Typo{Ecole}{École} Normale}}% +(Leipsic, 1880), the lectures of Lagrange have never been published +in separate form; originally they appeared in a fragmentary +shape in the \textit{Séances des \Typo{Ecoles}{Écoles} Normales}, as they had been reported +by the stenographers, and were subsequently reprinted in +the journal of the Polytechnic School. From references in them +\index{Polytechnic School}% +to subjects afterwards to be treated it is to be inferred that a fuller +development of higher algebra was intended,---an intention which +the brief existence of the \textit{\Typo{Ecole}{École} Normale} defeated. With very few +exceptions, we have left the expositions in their historical form, +having only referred in an Appendix to a point in the early history +of algebra. + +The originality, elegance, and symmetrical character of these +lectures have been pointed out by \Typo{DeMorgan}{De~Morgan}, and notably by Dühring, +\index{DeMorgan@{\Typo{DeMorgan}{De Morgan}}}% +\index{Duhring@{Dühring, E.}}% +who places them in the front rank of elementary expositions, +as an exemplar of their kind. Coming, as they do, from one of +the greatest mathematicians of modern times, and with all the excellencies +which such a source implies, unique in their character +\PageSep{vi} +as a \emph{reading-book} in mathematics, and interwoven with historical +and philosophical remarks of great helpfulness, they cannot fail +to have a beneficent and stimulating influence\Typo{,}{.} + +The thanks of the translator of the present volume are due to +Professor Henry~B. Fine, of Princeton, N.~J., for having read the +proofs. + +\Signature{Thomas J. McCormack.} +{\textsc{La Salle, Illinois}, August~1, 1898.} +\PageSep{vii} + + +\BioSketch{Joseph Louis Lagrange.} +{Biographical Sketch.} +\index{Economy of thought}% +\index{Lagrange, J. L.|EtSeq}% +\index{Short-mind symbols|EtSeq}% +\index{Stenophrenic symbols|EtSeq}% +\index{Symbols|EtSeq}% + +\First{A great} part of the progress of formal thought, where it is +not hampered by outward causes, has been due to the invention +of what we may call \emph{stenophrenic}, or \emph{short-mind}, symbols. +These, of which all written language and scientific notations are +examples, disengage the mind from the consideration of ponderous +and circuitous mechanical operations and economise its energies +for the performance of new and unaccomplished tasks of thought. +And the advancement of those sciences has been most notable +which have made the most extensive use of these short-mind symbols. +Here mathematics and chemistry stand pre-eminent. The +\index{Greeks, mathematics of the}% +\index{Mathematics!evolution of}% +ancient Greeks, with all their mathematical endowment as a race, +and even admitting that their powers were more visualistic than +analytic, were yet so impeded by their lack of short-mind symbols +as to have made scarcely any progress whatever in analysis. Their +arithmetic was a species of geometry. They did not possess the +sign for zero, and also did not make use of position as an indicator +of value. Even later, when the germs of the indeterminate analysis +were disseminated in Europe by Diophantus, progress ceased +here in the science, doubtless from this very cause. The historical +\index{Science!development of|EtSeq}% +calculations of Archimedes, his approximation to the value of~$\pi$,~etc, +owing to this lack of appropriate arithmetical and algebraical +symbols, entailed enormous and incredible labors, which, if +they had been avoided, would, with his genius, indubitably have +led to great discoveries. +\PageSep{viii} + +Subsequently, at the close of the Middle Ages, when the so-called +Arabic figures became established throughout Europe with +the symbol~$0$ and the principle of local value, immediate progress +was made in the art of reckoning. The problems which arose +gave rise to questions of increasing complexity and led up to the +general solutions of equations of the third and fourth degree by +the Italian mathematicians of the sixteenth century. Yet even +these discoveries were made in somewhat the same manner as +problems in mental arithmetic are now solved in common schools; +for the present signs of plus, minus, and equality, the radical and +exponential signs, and especially the systematic use of letters for +denoting general quantities in algebra, had not yet become universal. +The last step was definitively due to the French mathematician +Vieta (1540--1603), and the mighty advancement of analysis +\index{Vieta}% +resulting therefrom can hardly be measured or imagined. The +trammels were here removed from algebraic thought, and it ever +afterwards pursued its way unincumbered in development as if impelled +by some intrinsic and irresistible potency. Then followed +the introduction of exponents by Descartes, the representation of +\index{Descartes}% +geometrical magnitudes by algebraical symbols, the extension of +the theory of exponents to fractional and negative numbers by +Wallis (1616--1703), and other symbolic artifices, which rendered +\index{Wallis}% +the language of analysis as economic, unequivocal, and appropriate +as the needs of the science appeared to demand. In the famous +dispute regarding the invention of the infinitesimal calculus, while +not denying and even granting for the nonce the priority of Newton +\index{Newton, his problem}% +in the matter, some writers have gone so far as to regard Leibnitz's +\index{Leibnitz}% +introduction of the integral symbol~$\int$ as alone a sufficient substantiation +of his claims to originality and independence, so far as the +power of the new science was concerned. + +For the \emph{development} of science all such short-mind symbols +are of paramount importance, and seem to carry within themselves +the germ of a perpetual mental motion which needs no outward +power for its unfoldment. Euler's well-known saying that his +\index{Euler}% +\PageSep{ix} +pencil seemed to surpass him in intelligence finds its explanation +here, and will be understood by all who have experienced the uncanny +feeling attending the rapid development of algebraical formulæ, +where the urned thought of centuries, so to speak, rolls from +one's finger's ends. + +But it should never be forgotten that the mighty stenophrenic +engine of which we here speak, like all machinery, affords us rather +a mastery over nature than an insight into it; and for some, unfortunately, +the higher symbols of mathematics are merely brambles +that hide the living springs of reality. Many of the greatest +discoveries of science,---for example, those of Galileo, Huygens, +\index{Galileo}% +\index{Huygens}% +and Newton,---were made without the mechanism which afterwards +becomes so indispensable for their development and application. +Galileo's reasoning anent the summation of the impulses imparted +to a falling stone is virtual integration; and Newton's mechanical +discoveries were made by the man who invented, but evidently did +not use to that end, the doctrine of fluxions. +\stars + +We have been following here, briefly and roughly, a line of +progressive abstraction and generalisation which even in its beginning +was, psychologically speaking, at an exalted height, but in the +course of centuries had been carried to points of literally ethereal +refinement and altitude. In that long succession of inquirers by +whom this result was effected, the process reached, we may say, +its culmination and purest expression in Joseph Louis Lagrange, +born in Turin, Italy, the 30th~of January,~1736, died in Paris, April~10, +1813. Lagrange's power over symbols has, perhaps, never been +paralleled either before his day or since. It is amusing to hear his +biographers relate that in early life he evinced no aptitude for +mathematics, but seemed to have been given over entirely to the +pursuits of pure literature; for at fifteen we find him teaching +mathematics in an artillery school in Turin, and at nineteen he +had made the greatest discovery in mathematical science since that +of the infinitesimal calculus, namely, the creation of the algorism +\PageSep{x} +\index{Variations, calculus of}% +and method of the Calculus of Variations. ``Your analytical solution +of the isoperimetrical problem,'' writes Euler, then the prince +\index{Euler}% +of European mathematicians, to him, ``leaves nothing to be desired +in this department of inquiry, and I am delighted beyond measure +that it has been your lot to carry to the highest pitch of perfection, +a theory, which since its inception I have been almost the only one +to cultivate.'' But the exact nature of a ``variation'' even Euler +did not grasp, and even as late as~1810 in the English treatise of +Woodhouse on this subject we read regarding a certain new sign +\index{Woodhouse}% +introduced, that M.~Lagrange's ``power over symbols is so unbounded +that the possession of it seems to have made him capricious.'' + +Lagrange himself was conscious of his wonderful capacities in +this direction. His was a time when geometry, as he himself +phrased it, had become a dead language, the abstractions of analysis +were being pushed to their highest pitch, and he felt that with +his achievements its possibilities within certain limits were being +rapidly exhausted. The saying is attributed to him that chairs of +mathematics, so far as creation was concerned, and unless new +fields were opened up, would soon be as rare at universities as +chairs of Arabic. In both research and exposition, he totally reversed +the methods of his predecessors. They had proceeded in +their exposition from special cases by a species of induction; his +eye was always directed to the highest and most general points of +view; and it was by his suppression of details and neglect of minor, +unimportant considerations that he swept the whole field of analysis +with a generality of insight and power never excelled, adding +to his originality and profundity a conciseness, elegance, and lucidity +which have made him the model of mathematical writers. +\stars + +Lagrange came of an old French family of Touraine, France, +said to have been allied to that of Descartes. At the age of twenty-six +he found himself at the zenith of European fame. But his +reputation had been purchased at a great cost. Although of ordinary +\PageSep{xi} +height and well proportioned, he had by his ecstatic devotion +to study,---periods always accompanied by an irregular pulse and +high febrile \Typo{excitatian}{excitation},---almost ruined his health. At this age, +accordingly, he was seized with a hypochondriacal affection and +with bilious disorders, which accompanied him \Typo{thronghout}{throughout} his life, +and which were only allayed by his great abstemiousness and careful +regimen. He was bled twenty-nine times, an infliction which +alone would have affected the most robust constitution. Through +his great care for his health he gave much attention to medicine. +He was, in fact, conversant with all the sciences, although knowing +his \textit{forte} he rarely expressed an opinion on anything unconnected +with mathematics. + +When Euler left Berlin for St.~Petersburg in~1766 he and +D'Alembert induced Frederick the Great to make Lagrange president +of the Academy of Sciences at Berlin. Lagrange accepted +the position and lived in Berlin twenty years, where he wrote some +of his greatest works. He was a great favorite of the Berlin people, +and enjoyed the profoundest respect of Frederick the Great, +although the latter seems to have preferred the noisy reputation of +Maupertuis, Lamettrie, and Voltaire to the unobtrusive fame and +personality of the man whose achievements were destined to shed +more lasting light on his reign than those of any of his more strident +literary predecessors: Lagrange was, as he himself said, \textit{philosophe +sans crier}. + +The climate of Prussia agreed with the mathematician. He +refused the most seductive offers of foreign courts and princes, and +it was not until the death of Frederick and the intellectual reaction +of the Prussian court that he returned to Paris, where his career +broke forth in renewed splendor. He published in~1788 his great +\textit{Mécanique analytique}, that ``scientific poem'' of Sir William +Rowan Hamilton, which gave the quietus to mechanics as then +formulated, and having been made during the Revolution Professor +of Mathematics at the new \textit{\Typo{Ecole}{École} Normale} and the \textit{\Typo{Ecole}{École} Polytechnique}, +\index{Ecole@{\Typo{Ecole}{École} Normale}}% +\index{Polytechnic School}% +he entered with Laplace and Monge upon the activity +\index{Laplace}% +\index{Monge}% +\PageSep{xii} +which made these schools for generations to come exemplars of +practical scientific education, systematising by his lectures there, +and putting into definitive form, the science of mathematical analysis +of which he had developed the extremest capacities. Lagrange's +activity at Paris was interrupted only once by a brief period +of melancholy aversion for mathematics, a lull which he +devoted to the adolescent science of chemistry and to philosophical +studies; but he afterwards resumed his old love with increased ardor +and assiduity. His significance for thought generally is far +beyond what we have space to insist upon. Not least of all, theology, +which had invariably mingled itself with the researches of his +predecessors, was with him forever divorced from a legitimate influence +of science. + +The honors of the world sat ill upon Lagrange: \textit{la magnificence +le gênait}, he said; but he lived at a time when proffered +things were usually accepted, not refused. He was loaded with +personal favors and official distinctions by Napoleon who called +\index{Napoleon}% +him \textit{la haute pyramide des sciences mathématiques}, was made a +Senator, a Count of the Empire, a Grand Officer of the Legion of +Honor, and, just before his death, received the grand cross of the +Order of Reunion. He never feared death, which he termed \textit{une +dernière fonction, ni pénible ni désagréable}, much less the disapproval +of the great. He remained in Paris during the Revolution +when \textit{savants} were decidedly in disfavor, but was suspected +of aspiring to no throne but that of mathematics. When Lavoisier +\index{Lavoisier}% +was executed he said: ``It took them but a moment to lay low that +head; yet a hundred years will not suffice perhaps to produce its +like again.'' + +Lagrange would never allow his portrait to be painted, maintaining +that a man's works and not his personality deserved preservation. +The frontispiece to the present work is from a steel +engraving based on a sketch obtained by stealth at a meeting of +the Institute. His genius was excelled only by the purity and +nobleness of his character, in which the world never even sought +\PageSep{xiii} +to find a blot, and by the exalted Pythagorean simplicity of his +life. He was twice married, and by his wonderful care of his person +lived to the advanced age of seventy-seven years, not one of +which had been misspent. His life was the veriest incarnation of +the scientific spirit; he lived for nothing else. He left his weak +body, which retained its intellectual powers to the very last, as an +offering upon the altar of science, happily made when his work +had been done; but to the world he bequeathed his ``ever-living'' +thoughts now recently resurgent in a new and monumental edition +of his works (published by Gauthier-Villars, Paris). \textit{Ma vie est +là!} he said, pointing to his brain the day before his death. + +\Signature{Thomas J. McCormack.}{} +\PageSep{xiv} +%[Blank page] +\PageSep{xv} +\TableofContents +\iffalse +%[** TN: Used marginal notes to generate entries; entries in original ToC +% don't obviously match the book's units.] +CONTENTS. + +PAGES + +Preface + +Biographical Sketch of Joseph Louis LaGrange. + +Lecture I. On Arithmetic, and in Particular Fractions +and Logarithms. 1-23 + +Systems of Numeration. Fractions. Greatest Common +Divisor. Continued Fractions. Theory of +Powers, Proportions, and Progressions. Involution +and Evolution. Rule of Three. Interest. Annuities. +Logarithms. + +Lecture II. On the Operations of Arithmetic . . . 24-53 + +Arithmetic and Geometry. New Method of Subtraction. +Abridged and Approximate Multiplication. +Decimals. Property of the Number 9. +Tests of Divisibility. Theory of Remainders. +Checks on Multiplication and Division. Evolution. +Rule of Three. Theory and Practice. Probability +of Life. Alligation or the Rule of Mixtures. + +Lecture III. On Algebra, Particularly the Resolution +of Equations of the Third and Fourth Degree 54-95 + +Origin of Greek Algebra. Diophantus. Indeterminate +Analysis. Equations of the Second Degree. +Translations of Diophantus. Algebra Among the +Arabs. History of Algebra in Italy, France, and +Germany. History of Equations of the Third and +Fourth Degree and of the Irreducible Case. Theory +of Equations. Discussion of Cubic Equations. +Discussion of the Irreducible Case. The Theory +\PageSep{xvi} +of Roots. Extraction of the Square and Cube Roots +of Two Imaginary Binomials. Theory of Imaginary +Expressions. Trisection of an Angle. Method +of Indeterminates. Discussion of Biquadratic Equations. + +Lecture IV. On the Resolution of Numerical Equations ... 96-126 + +Algebraical Resolution of Equations. Numerical +Resolution of Equations. Position of the Roots. +Representation of Equations by Curves. Graphic +Resolution of Equations. Character of the Roots of +Equations. Limits of the Roots of Numerical Equations. +Separation of the Roots. Method of Substitutions. +The Equation of Differences. Method of +Elimination. Constructions and Instruments for +Solving Equations. + +Lecture V. On the Employment of Curves in the Solution +of Problems 127-149 + +Application of Geometry to Algebra. Resolution of +Problems by Curves. The Problem of Two Lights. +Variable Quantities Minimal Values. Analysis +of Biquadratic Equations Conformably to the Problem +of the Two Lights. Advantages of the Method +of Curves The Curve of Errors. \textit{Regula falsi.} +Solution of Problems by the Curve of Errors. +Problem of the Circle and Inscribed Polygon. +Problem of the Observer and Three Objects. Parabolic +Curves. Newton's Problem. Interpolation +of Intermediate Terms in Series of Observations, +Experiments, etc. + +Appendix . 151 + +Note on the Origin of Algebra. +\fi +\PageSep{1} +\MainMatter +\index{Numerical equations|See{Equations}}% + + +\Lecture[On Arithmetic.]{I.}{On Arithmetic, and in Particular Fractions +and Logarithms.} + +\First{Arithmetic} is divided into two parts. The first +is based on the decimal system of notation and +\MNote{Systems of numeration\Add{.}} +on the manner of arranging numeral characters to express +numbers. This first part comprises the four +common operations of addition, subtraction, multiplication, +and division,---operations which, as you +know, would be different if a different system were +adopted, but, which it would not be difficult to transform +from one system to another, if a change of systems +were desirable. + +The second part is independent of the system of +\index{Numeration, systems of}% +numeration. It is based on the consideration of quantities +and on the general properties of numbers. The +theory of fractions, the theory of powers and of roots, +the theory of arithmetical and geometrical progressions, +and, lastly, the theory of logarithms, fall under +this head. I purpose to advance, here, some remarks +on the different branches of this part of arithmetic. +\PageSep{2} + +It may be regarded as \emph{universal arithmetic}, having +\index{Arithmetic!universal|EtSeq}% +an intimate affinity to algebra. For, if instead of +\index{Algebra!definition of}% +particularising the quantities considered, if instead of +assigning them numerically, we treat them in quite a +general way, designating them by letters, we have +algebra. + +You know what a fraction is. The notion of a +\index{Fractions|EtSeq}% +\index{Ratios, constant|EtSeq}% +\MNote{Fractions.} +fraction is slightly more composite than that of whole +numbers. In whole numbers we consider simply a +quantity repeated. To reach the notion of a fraction +it is necessary to consider the quantity divided into a +certain number of parts. Fractions represent in general +ratios, and serve to express one quantity by means +of another. In general, nothing measurable can be +measured except by fractions expressing the result of +the measurement, unless the measure be contained an +exact number of times in the thing to be measured. + +You also know how a fraction can be reduced to +\index{Divisor, greatest common|EtSeq}% +its lowest terms. When the numerator and the denominator +are both divisible by the same number, +their greatest common divisor can be found by a very +ingenious method which we owe to Euclid. This +\index{Euclid}% +method is exceedingly simple and lucid, but it may +be rendered even more palpable to the eye by the following +consideration. Suppose, for example, that you +have a given length, and that you wish to measure it. +The unit of measure is given, and you wish to know +how many times it is contained in the length. You +first lay off your measure as many times as you can on +\PageSep{3} +the given length, and that gives you a certain whole +number of measures. If there is no remainder your +operation is finished. But if there be a remainder, +\MNote{Greatest common divisor.} +that remainder is still to be evaluated. If the measure +is divided into equal parts, for example, into ten, +twelve, or more equal parts, the natural procedure is +to use one of these parts as a new measure and to see +how many times it is contained in the remainder. +You will then have for the value of your remainder, +a fraction of which the numerator is the number of +parts contained in the remainder and the denominator +the total number of parts into which the given measure +is divided. + +I will suppose, now, that your measure is not so +divided but that you still wish to determine the ratio +of the proposed length to the length which you have +adopted as your measure. The following is the procedure +which most naturally suggests itself. + +If you have a remainder, since that is less than the +\index{Fractions!continued|EtSeq}% +measure, naturally you will seek to find how many +times your remainder is contained in this measure. +Let us say two times, and that a remainder is still +left. Lay this remainder on the preceding remainder. +Since it is necessarily smaller, it will still be contained +a certain number of times in the preceding remainder, +say three times, and there will be another remainder +or there will not; and so on. In these different remainders +you will have what is called a \emph{continued fraction}. +For example, you have found that the measure +\PageSep{4} +is contained three times in the proposed length. You +have, to start with, the number \emph{three}. Then you have +\MNote{Continued fractions.} +found that your first remainder is contained twice in +your measure. You will have the fraction \emph{one} divided +by \emph{two}. But this last denominator is not complete, +for it was supposed there was still a remainder. That +remainder will give another and similar fraction, which +is to be added to the last denominator, and which by +our supposition is \emph{one} divided by \emph{three}. And so with +the rest. You will then have the fraction +\[ +3 + \cfrac{1}{2 + \cfrac{1}{3 + \ddots}} +\] +as the expression of your ratio between the proposed +length and the adopted measure. + +Fractions of this form are called \emph{continued fractions}, +and can be reduced to ordinary fractions by the common +rules. Thus, if we stop at the first fraction, i.e., +if we consider only the first remainder and neglect the +second, we shall have $3 + \frac{1}{2}$, which is equal to~$\frac{7}{2}$. Considering +only the first and the second remainders, we +stop at the second fraction, and shall have $3 + \dfrac{1}{2 + \frac{1}{3}}$. +Now $2 + \frac{1}{3} = \frac{7}{3}$. We shall have therefore $3 + \frac{3}{7}$, which +is equal to~$\frac{24}{7}$. And so on with the rest. If we arrive +in the course of the operation at a remainder which is +contained exactly in the preceding remainder, the +operation is terminated, and we shall have in the continued +\PageSep{5} +fraction a common fraction that is the exact +value of the length to be measured, in terms of the +length which served as our measure. If the operation +\MNote{Terminating continued fractions.} +is not thus terminated, it can be continued to infinity, +and we shall have only fractions which approach more +and more nearly to the true value. + +If we now compare this procedure with that employed +for finding the greatest common divisor of two +numbers, we shall see that it is virtually the same +thing; the difference being that in finding the greatest +common divisor we devote our attention solely to +the different remainders, of which the last is the divisor +sought, whereas by employing the successive +quotients, as we have done above, we obtain fractions +which constantly approach nearer and nearer to the +fraction formed by the two numbers given, and of +which the last is that fraction itself reduced to its +lowest terms. + +As the theory of continued fractions is little known, +but is yet of great utility in the solution of important +numerical questions, I shall enter here somewhat +more fully into the formation and properties of these +fractions. And, first, let us suppose that the quotients +found, whether by the mechanical operation, or by +the method for finding the greatest common divisor, +are, as above, $3$,~$2$, $3$, $5$, $7$,~$3$. The following is a rule +by which we can write down at once the convergent +fractions which result from these quotients, without +developing the continued fraction. +\PageSep{6} + +The first quotient, supposed divided by unity, +will give the first fraction, which will be too small, +\MNote{Converging fractions.} +\index{Fractions!converging}% +namely,~$\frac{3}{1}$. Then, multiplying the numerator and denominator +of this fraction by the second quotient and +adding unity to the numerator, we shall have the second +fraction,~$\frac{7}{2}$, which will be too large. Multiplying +in like manner the numerator and denominator of this +fraction by the third quotient, and adding to the numerator +the numerator of the preceding fraction, and +to the denominator the denominator of the preceding +fraction, we shall have the third fraction, which will +be too small. Thus, the third quotient being~$3$, we +have for our numerator $(7 × 3 = 21) + 3 = 24$, and for +our denominator $(2 × 3 = 6) + 1 = 7$. The third convergent, +therefore, is~$\frac{24}{7}$. We proceed in the same +manner for the fourth convergent. The fourth quotient +being~$5$, we say $24$~times~$5$ is~$120$, and this plus~$7$, +the numerator of the fraction preceding, is~$127$; +similarly, $7$~times~$5$ is~$35$, and this plus~$2$ is~$37$. The +new fraction, therefore, is~$\frac{127}{37}$. And so with the rest. + +In this manner, by employing the six quotients $3$,~$2$, +$3$, $5$, $7$,~$3$ we obtain the six fractions +\[ +\frac{3}{1},\quad +\frac{7}{2},\quad +\frac{24}{7},\quad +\frac{127}{37},\quad +\frac{913}{266},\quad +\frac{2866}{835}, +\] +of which the last, supposing the operation to be completed +at the sixth quotient~$3$, will be the required +value of the length measured, or the fraction itself +reduced to its lowest terms. + +The fractions which precede the last are alternately +\PageSep{7} +smaller and larger than the last, and have the advantage +of approaching more and more nearly to its value +in such wise that no other fraction can approach it +\MNote{Convergents.} +\index{Convergents}% +more nearly except its denominator be larger than the +product of the denominator of the fraction in question +and the denominator of the fraction following. For +example, the fraction~$\frac{24}{7}$ is less than the true value +which is that of the fraction~$\frac{2866}{835}$, but it approaches +to it more nearly than any other fraction does whose +denominator is not greater than the product of~$7$ by~$37$, +that is,~$259$. Thus, any fraction expressed in large +numbers may be reduced to a series of fractions expressed +in smaller numbers and which approach as +near to it as possible in value. + +The demonstration of the foregoing properties is +deduced from the nature of continued fractions, and +from the fact that if we seek the difference between +one of the convergent fractions and that next adjacent +to it we shall obtain a fraction of which the numerator +is always unity and the denominator the product of +the two denominators; a consequence which follows +\textit{\Typo{a}{à}~priori} from the very law of formation of these fractions. +Thus the difference between $\frac{7}{2}$~and~$\frac{3}{1}$ is~$\frac{1}{2}$, in +excess; between $\frac{24}{7}$~and~$\frac{7}{2}$, $\frac{1}{14}$,~in defect; between $\frac{127}{37}$ +and~$\frac{24}{7}$, $\frac{1}{259}$,~in excess; and so on. The result being, +that by employing this series of differences we can +express in another and very simple manner the fractions +with which we are here concerned, by means of +a second series of fractions of which the numerators +\PageSep{8} +are all unity and the denominators successively the +products of every two adjacent denominators. Instead +\MNote{A second method of expression.} +of the fractions written above, we have thus the +series: +\[ +\frac{3}{1} + \frac{1}{1 × 2} + - \frac{1}{2 × 7} + + \frac{1}{7 × 37} + - \frac{1}{37 × 266} + + \frac{1}{266 × 835}. +\] + +The first term, as we see, is the first fraction, the +first and second together give the second fraction~$\frac{7}{2}$, +the first, the second, and the third give the third fraction~$\frac{24}{7}$, +and so on with the rest; the result being that +the series entire is equivalent to the last fraction. + +There is still another way, less known but in some +respects more simple, of treating the same question---which +leads directly to a series similar to the preceding. +Reverting to the previous example, after having +found that the measure goes three times into the length +to be measured and that after the first remainder has +been applied to the measure there is left a new remainder, +instead of comparing this second remainder +with the preceding, as we did above, we may compare +it with the measure itself. Thus, supposing it goes +into the latter seven times with a remainder, we again +compare this last remainder with the measure, and so +on, until we arrive, if possible, at a remainder which +is an aliquot part of the measure,---which will terminate +the operation. In the contrary event, if the +measure and the length to be measured are incommensurable, +the process may be continued to infinity. +\PageSep{9} +We shall have then, as the expression of the length +measured, the series +\MNote{A third method of expression.} +\[ +3 + \frac{1}{2} - \frac{1}{2 × 7} + \ldots. +\] + +It is clear that this method is also applicable to +ordinary fractions. We constantly retain the denominator +of the fraction as the dividend, and take the different +remainders successively as divisors. Thus, the +fraction~$\frac{2866}{835}$ gives the quotients $3$,~$2$, $7$, $18$, $19$, $46$, +$119$, $417$\Typo{}{,}~$835$; from which we obtain the series +\[ +3 + \frac{1}{2} - \frac{1}{2 × 7} + + \frac{1}{2 × 7 × 18} + - \frac{1}{2 × 7 × 18 × 19} + \ldots; +\] +and as these partial fractions rapidly diminish, we +shall have, by combining them successively, the simple +fractions, +\[ +\frac{7}{2},\quad +\frac{48}{2 × 7},\quad +\frac{865}{2 × 7 × 18}, \ldots, +\] +which will constantly approach nearer and nearer to +the true value sought, and the error will be less than +the first of the partial fractions neglected. + +Our remarks on the foregoing methods of evaluating +fractions should not be construed as signifying +that the employment of decimal fractions is not nearly +\index{Decimal!fractions}% +\index{Fractions!decimal}% +always preferable for expressing the values of fractions +to whatever degree of exactness we wish. But cases +occur where it is necessary that these values should +be expressed by as few figures as possible. For example, +if it were required to construct a planetarium, +\index{Planetarium}% +\PageSep{10} +since the ratios of the revolutions of the planets to one +another are expressed by very large numbers, it would +\MNote{Origin of continued fractions.} +\index{Fractions!origin of continued}% +be necessary, in order not to multiply unduly the +number of the teeth on the wheels, to avail ourselves +of smaller numbers, but at the same time so to select +them that their ratios should approach as nearly as +possible to the actual ratios. It was, in fact, this very +question that prompted Huygens, in his search for its +\index{Huygens}% +solution, to resort to continued fractions and that so +gave birth to the theory of these fractions. Afterwards, +in the elaboration of this theory, it was found +adapted to the solution of other important questions, +and this is the reason, since it is not found in elementary +works, that I have deemed it necessary to go +somewhat into detail in expounding its principles. + +We will now pass to the theory of powers, proportions, +and progressions. + +As you already know, a number multiplied by itself +\index{Powers|EtSeq}% +gives its square, and multiplied again by itself +gives its cube, and so on. In geometry we do not go +beyond the cube, because no body can have more than +three dimensions. But in algebra and arithmetic we +may go as far as we please. And here the theory of +the extraction of roots takes its origin. For, although +every number can be raised to its square and to its +cube and so forth, it is not true reciprocally that every +number is an exact square or an exact cube. The +number~$2$, for example, is not a square; for the square +of~$1$ is~$1$, and the square of~$2$ is four; and there being +\PageSep{11} +no other whole numbers between these two, it is impossible +to find a whole number which multiplied by +itself will give~$2$. It cannot be found in fractions, for +\MNote{Involution and evolution.} +\index{Evolution}% +\index{Involution and evolution}% +if you take a fraction reduced to its lowest terms, the +square of that fraction will again be a fraction reduced +to its lowest terms, and consequently cannot be equal +to the whole number~$2$. But though we cannot obtain +the square root of~$2$ exactly, we can yet approach to it +as nearly as we please, particularly by decimal fractions. +By following the common rules for the extraction +of square roots, cube roots, and so forth, the process +may be extended to infinity, and the true values +of the roots may be approximated to any degree of +exactitude we wish. + +But I shall not enter into details here. The theory +of powers has given rise to that of progressions, before +entering on which a word is necessary on proportions. + +Every fraction expresses a ratio. Having two equal +\index{Proportion|EtSeq}% +\index{Ratios, constant|EtSeq}% +fractions, therefore, we have two equal ratios; and +the numbers constituting the fractions or the ratios +form what is called a \emph{proportion}. Thus the equality +of the ratios $2$~to~$4$ and $3$~to~$6$ gives the proportion +$2 : 4 :: 3 : 6$, because $4$~is the double of~$2$ as $6$~is the +double of~$3$. Many of the rules of arithmetic depend +on the theory of proportions. First, it is the foundation +of the famous \emph{rule of three}, which is so extensively +\index{Rule!three@of three|EtSeq}% +used. You know that when the first three terms of a +proportion are given, to obtain the fourth you have +\PageSep{12} +only to multiply the last two together and divide the +product by the first. Various special rules have also +\MNote{Proportions\Add{.}} +been conceived and have found a place in the books +on arithmetic; but they are all reducible to the rule +of three and may be neglected if we once thoroughly +grasp the conditions of the problem. There are direct, +inverse, simple, and compound rules of three, rules of +partnership, of mixtures, and so forth. In all cases +it is only necessary to consider carefully the conditions +of the problem and to arrange the terms of the +proportion correspondingly. + +I shall not enter into further details here. There +\index{Progressions, theory of}% +is, however, another theory which is useful on numerous +occasions,---namely, the \emph{theory of progressions}. +When you have several numbers that bear the same +proportion to one another, and which follow one another +in such a manner that the second is to the first +as the third is to the second, as the fourth is to the +third, and so forth, these numbers form a progression. +I shall begin with an observation. + +The books of arithmetic and algebra ordinarily distinguish +between two kinds of progression, arithmetical +and geometrical, corresponding to the proportions +called arithmetical and geometrical. But the appellation +proportion appears to me extremely inappropriate +as applied to \emph{arithmetical proportion}. And as it +\index{Arithmetical proportion}% +is one of the objects of the \textit{École Normale} to rectify +\index{Ecole@{\Typo{Ecole}{École} Normale}}% +the language of science, the present slight digression +will not be considered irrelevant. +\PageSep{13} + +I take it, then, that the idea of proportion is already +well established by usage and that it corresponds solely +to what is called \emph{geometrical proportion}. When we +\index{Geometrical!proportion}% +\MNote{Arithmetical and geometrical proportions.} +speak of the proportion of the parts of a man's body, +of the proportion of the parts of an edifice,~etc.; when +we say that a plan should be reduced proportionately +in size,~etc.; in fact, when we say generally that one +thing is proportional to another, we understand by +proportion equality of ratios only, as in geometrical +proportion, and never equality of differences as in +arithmetical proportion. Therefore, instead of saying +\index{Equi-different numbers}% +that the numbers, $3$,~$5$, $7$,~$9$, are in arithmetical proportion, +because the difference between $5$~and~$3$ is the +same as that between $9$~and~$7$, I deem it desirable that +some other term should be employed, so as to avoid +all ambiguity. We might, for instance, call such numbers +\emph{equi-different}, reserving the name of \emph{proportionals} +for numbers that are in geometrical proportion, as $2$,~$4$, +$6$,~$8$,~etc. + +As for the rest, I cannot see why the proportion +called \emph{arithmetical} is any more arithmetical than that +which is called geometrical, nor why the latter is more +geometrical than the former. On the contrary, the +primitive idea of geometrical proportion is based on +arithmetic, for the notion of ratios springs essentially +from the consideration of numbers. + +Still, in waiting for these inappropriate designations +to be changed, I shall continue to make use of +them, as a matter of simplicity and convenience. +\PageSep{14} + +The theory of arithmetical progressions presents +few difficulties. Arithmetical progressions consist of +\MNote{Progressions.} +\index{Progressions, theory of}% +quantities which increase or diminish constantly by +the same amount. But the theory of geometrical progressions +is more difficult and more important, as a +large number of interesting questions depend upon it---for +example, all problems of compound interest, all +problems that relate to discount, and many others of +like nature. + +In general, quantities in geometrical proportion +are produced, when a quantity increases and the force +generating the increase, so to speak, is proportional +to that quantity. It has been observed that in countries +where the means of subsistence are easy of acquisition, +as in the first American colonies, the population +is doubled at the expiration of twenty years; if +it is doubled at the end of twenty years it will be quadrupled +at the end of forty, octupled at the end of sixty, +and so on; the result being, as we see, a geometrical +progression, corresponding to intervals of time in +arithmetical progression. It is the same with compound +interest. If a given sum of money produces, +at the expiration of a certain time, a certain sum, at +the end of double that time, the original sum will have +produced an equivalent additional sum, and in addition +the sum produced in the first space of time will, +in its proportion, likewise have produced during the +second space of time a certain sum; and so with the +rest. The original sum is commonly called the \emph{principal}, +\PageSep{15} +the sum produced the \emph{interest}, and the constant +\index{Interest}% +ratio of the principal to the interest per annum, the +\emph{rate}. Thus, the rate \emph{twenty} signifies that the interest +\MNote{Compound interest.} +is the twentieth part of the principal,---a rate which +is commonly called $5$~\emph{per cent.}, $5$~being the twentieth +part of~$100$. On this basis, the principal, at the end +of one year, will have increased by its one-twentieth +part; consequently, it will have been augmented in +the ratio of $21$~to~$20$. At the end of two years, it will +have been increased again in the same ratio, that is in +the ratio of $\frac{21}{20}$~multiplied by~$\frac{21}{20}$; at the end of three +years, in the ratio of $\frac{21}{20}$~multiplied twice by itself; and +so on. In this manner we shall find that at the end of +fifteen years it will almost have doubled itself, and that +at the end of fifty-three years it will have increased +tenfold. Conversely, then, since a sum paid now will +be doubled at the end of fifteen years, it is clear that +a sum not payable till after the expiration of fifteen +years is now worth only one-half its amount. This +is what is termed the \emph{present value} of a sum payable +\index{Present value}% +at the end of a certain time; and it is plain, that to +find that value, it is only necessary to divide the sum +promised by the fraction~$\frac{21}{20}$, or to multiply it by the +fraction~$\frac{20}{21}$, as many times as there are years for the +sum to run. In this way we shall find that a sum +payable at the end of fifty-three years, is worth at +present only one-tenth. From this it is evident what +little advantage is to be derived from surrendering the +absolute ownership of a sum of money in order to obtain +\PageSep{16} +the enjoyment of it for a period of only fifty +years, say; seeing that we gain by such a transaction +\MNote{Present values and annuities.} +\index{Annuities}% +only one-tenth in actual use, whilst we lose the ownership +of the property forever. + +In \emph{annuities}, the consideration of interest is combined +with that of the probability of life; and as +every one is prone to believe that he will live very +long, and as, on the other hand, one is apt to under-*estimate +the value of property which must be abandoned +on death, a peculiar temptation arises, when +one is without children, to invest one's fortune, wholly +or in part, in annuities. Nevertheless, when put to +the test of rigorous calculation, annuities are not +found to offer sufficient advantages to induce people +to sacrifice for them the ownership of the original +capital. Accordingly, whenever it has been attempted +to create annuities sufficiently attractive to induce individuals +to invest in them, it has been necessary to +offer them on terms which are onerous to the company. + +But we shall have more to say on this subject when +we expound the theory of annuities, which is a branch +of the calculus of probabilities. + +I shall conclude the present lecture with a word +\index{Logarithms|EtSeq}% +on \emph{logarithms}. The simplest idea which we can form +of the theory of logarithms, as they are found in the +ordinary tables, is that of conceiving all numbers +as powers of~$10$; the exponents of these powers, +then, will be the logarithms of the numbers. From +\PageSep{17} +this it is evident that the multiplication and division +of two numbers is reducible to the addition and subtraction +of their respective exponents, that is, of their +\MNote{Logarithms\Add{.}} +logarithms. And, consequently, involution and the +extraction of roots are reducible to multiplication and +division, which is of immense advantage in arithmetic +and renders logarithms of priceless value in that science. + +But in the period when logarithms were invented, +mathematicians were not in possession of the theory +of powers. They did not know that the root of a number +could be represented by a fractional power. The +following was the way in which they approached the +problem. + +The primitive idea was that of two corresponding +progressions, one arithmetical, and the other geometrical. +In this way the general notion of a logarithm +was reached. But the means for finding the logarithms +of all numbers were still lacking. As the numbers +follow one another in arithmetical progression, it +was requisite, in order that they might all be found +among the terms of a geometrical progression, so to +establish that progression that its successive terms +should differ by extremely small quantities from one +another; and, to prove the possibility of expressing +all numbers in this way, Napier, the inventor, first +\index{Napier|EtSeq}% +considered them as expressed by lines and parts of +lines, and these lines he considered as generated by +\PageSep{18} +the continuous motion of a point, which was quite +natural. + +\MNote{Napier (1550--1617).} +He considered, accordingly, two lines, the first of +which was generated by the motion of a point describing +in equal times spaces in geometrical progression, +and the other generated by a point which described +spaces that increased as the times and consequently +formed an arithmetical progression corresponding to +the geometrical progression. And he supposed, for +the sake of simplicity, that the initial velocities of +these two points were equal. This gave him the logarithms, +at first called \emph{natural}, and afterwards \emph{hyperbolical}, +when it was discovered that they could be expressed +as parts of the area included between a +hyperbola and its asymptotes. By this method it is +clear that to find the logarithm of any given number, +it is only necessary to take a part on the first line +equal to the given number, and to seek the part on +the second line which shall have been described in +the same interval of time as the part on the first. + +Conformably to this idea, if we take as the two +first terms of our geometrical progression the numbers +with very small differences $1$~and~$1.0000001$, and as +those of our arithmetical progression $0$~and $0.0000001$, +and if we seek successively, by the known rules, all +the following terms of the two progressions, we shall +find that the number~$2$ expressed approximately to the +eighth place of decimals is the $6931472$th~term of the +geometrical progression, that is, that the logarithm of~$2$ +\PageSep{19} +is~$0.6931472$. The number~$10$ will be found to be the +$23025851$th~term of the same progression; therefore, +the logarithm of~$10$ is~$2.3025851$, and so with the rest. +\MNote{Origin of logarithms\Add{.}} +\index{Logarithms!origin of}% +But Napier, having to determine only the logarithms +of numbers less than unity for the purposes of trigonometry, +where the sines and cosines of angles are +expressed as fractions of the radius, considered a decreasing +geometrical progression of which the first +two terms were $1$~and~$0.9999999$; and of this progression +he determined the succeeding terms by enormous +computations. On this last hypothesis, the logarithm +which we have just found for~$2$ becomes that of the +number~$\frac{1}{5}$ or~$0.5$, and that of the number~$10$ becomes +that of the number~$\frac{1}{10}$ or~$0.1$; as is readily apparent +from the nature of the two progressions. + +Napier's work appeared in~1614. Its utility was +felt at once. But it was also immediately seen that it +would conform better to the decimal system of our +arithmetic, and would be simpler, if the logarithm of~$10$ +were made unity, conformably to which that of~$100$ +would be~$2$, and so with the rest. To that end, instead +of taking as the first two terms of our geometrical +progression the numbers $1$~and~$\Typo{0.0000001}{1.0000001}$, we should +have to take the numbers $1$~and~$1.0000002302$, retaining +$0$~and~$0.0000001$ as the corresponding terms of the +arithmetical progression. Whence it will be seen, +that, while the point which is supposed to generate by +its motion the geometrical line, or the numbers, is +describing the very small portion~$0.0000002302\dots$, +\PageSep{20} +the other point, the office of which is to generate +simultaneously the arithmetical line, will have described +\MNote{Briggs (1556--1631). Vlacq.} +\index{Briggs}% +\index{Vlacq}% +the portion~$0.0000001$; and that therefore the +spaces described in the same time by the two points +at the beginning of their motion, that is to say, their +initial velocities, instead of being equal, as in the +preceding system, will be in the proportion of the +numbers $2.302\dots$~to~$1$, where it will be remarked +that the number~$2.302\dots$ is exactly the number +which in the original system of natural logarithms +stood for the logarithm of~$10$,---a result demonstrable +\textit{à~priori}, as we shall see when we come to apply +the formulæ of algebra to the theory of logarithms. +Briggs, a contemporary of Napier, is the author of this +change in the system of logarithms, as he is also of +the tables of logarithms now in common use. A portion +\index{Logarithms!tables of}% +of these was calculated by Briggs himself, and +the remainder by Vlacq, a Dutchman. + +These tables appeared at Gouda, in~1628. They +contain the logarithms of all numbers from~$1$ to~$100000$ +to ten decimal places, and are now extremely rare. +But it was afterwards discovered that for ordinary purposes +seven decimals were sufficient, and the logarithms +are found in this form in the tables which are +used to-day. Briggs and Vlacq employed a number +of highly ingenious artifices for facilitating their work. +The device which offered itself most naturally and +which is still one of the simplest, consists in taking +the numbers $1$,~$10$, $100$,~$\dots$, of which the logarithms +\PageSep{21} +are $0$,~$1$,~$2$,~$\dots$, and in interpolating between the successive +terms of these two series as many corresponding +terms as we desire, in the first series by geometrical +\MNote{Computation of logarithms.} +mean proportionals and in the second by +arithmetical means. In this manner, when we have +arrived at a term of the first series approaching, to the +eighth decimal place, the number whose logarithm +we seek, the corresponding term of the other series +will be, to the eighth decimal place approximately, +the logarithm of that number. Thus, to obtain the +logarithm of~$2$, since $2$~lies between $1$~and~$10$, we seek +first by the extraction of the square root of~$10$, the +geometrical mean between $1$~and~$10$, which we find to +be~$3.16227766$, while the corresponding arithmetical +mean between $0$~and~$1$ is~$\frac{1}{2}$ or~$0.50000000$; we are +assured thus that this last number is the logarithm of +the first. Again, as $2$~lies between $1$~and~$3.16227766$, +the number just found, we seek in the same manner +the geometrical mean between these two numbers, +and find the number~$1.77827941$. As before, taking +the arithmetical mean between $0$~and~$5.0000000$, we +shall have for the logarithm of~$1.77827941$ the number~$0.25000000$. +Again, $2$~lying between $1.77827941$ +and~$3.16227766$, it will be necessary, for still further +approximation, to find the geometrical mean between +these two, and likewise the arithmetical mean between +their logarithms. And so on. In this manner, +by a large number of similar operations, we find that +the logarithm of~$2$ is~$0.3010300$, that of~$3$ is~$0.4771213$, +\PageSep{22} +and so on, not carrying the degree of exactness beyond +the seventh decimal place. But the preceding +\MNote{Value of the history of science.} +\index{Science!history of}% +calculation is necessary only for prime numbers; because +the logarithms of numbers which are the product +of two or several others, are found by simply +taking the sum of the logarithms of their factors. + +As for the rest, since the calculation of logarithms +is now a thing of the past, except in isolated instances, +it may be thought that the details into which we have +here entered are devoid of value. We may, however, +justly be curious to know the trying and tortuous +paths which the great inventors have trodden, the different +\index{Inventors, great}% +steps which they have taken to attain their goal, +and the extent to which we are indebted to these veritable +benefactors of the human race. Such knowledge, +moreover, is not matter of idle curiosity. It can +afford us guidance in similar inquiries and sheds an +increased light on the subjects with which we are +employed. + +Logarithms are an instrument universally employed +in the sciences, and in the arts depending on calculation. +The following, for example, is a very evident +application of their use. + +Persons not entirely unacquainted with music know +\index{Music}% +that the different notes of the octave are expressed by +numbers which give the divisions of a stretched cord +producing those notes. Thus, the principal note being +denoted by~$1$, its octave will be denoted by~$\frac{1}{2}$, +its fifth by~$\frac{2}{3}$, its third by~$\frac{4}{5}$, its fourth by~$\frac{3}{4}$, its second +\PageSep{23} +by~$\frac{8}{9}$, and so on. The distance of one of these notes +from that next adjacent to it is called an \emph{interval}, and +is measured, not by the difference, but by the ratio of +the numbers expressing the two sounds. Thus, the +interval between the fourth and fifth, which is called +the \emph{major tone}, is regarded as sensibly double of that +between the third and fourth, which is called the \emph{semi-major}. +In fact, the first being expressed by~$\frac{8}{9}$, the +second by~$\frac{15}{16}$, it can be easily proved that the first +does not differ by much from the square of the second. +Now, it is clear that this conception of intervals, on +\MNote{Musical temperament.} +\index{Temperament, theory of}% +which the whole theory of temperament is founded, +conducts us naturally to logarithms. For if we express +the value of the different notes by the logarithms +of the lengths of the cords answering to them, +then the interval of one note from another will be +expressed by the simple difference of values of the +two notes; and if it were required to divide the octave +into twelve equal semi-tones, which would give the +temperament that is simplest and most exact, we +should simply have to divide the logarithm of one +half, the value of the octave, into twelve equal parts. +\PageSep{24} + + +\Lecture{II.}{On the Operations of Arithmetic.} +\index{Arithmetic!operations of|EtSeq}% + +\First{An ancient} writer once remarked that arithmetic +and geometry were \emph{the wings of mathematics}. +\index{Geometry}% +\index{Mathematics!wings of}% +\MNote{Arithmetic and geometry.} +I believe we can say, without metaphor, that +these two sciences are the foundation and essence of +all the sciences that treat of magnitude. But not +only are they the foundation, they are also, so to +speak, the capstone of these sciences. For, whenever +we have reached a result, in order to make use of it, +it is requisite that it be translated into numbers or +into lines; to translate it into numbers, arithmetic is +necessary; to translate it into lines, we must have +recourse to geometry. + +The importance of arithmetic, accordingly, leads +me to the further discussion of that subject to-day, +although we have begun algebra. I shall take up its +several parts, and shall offer new observations, which +will serve to supplement what I have already expounded +to you. I shall employ, moreover, the geometrical +\index{Geometrical!calculus}% +calculus, wherever that is necessary for giving +\PageSep{25} +greater generality to the demonstrations and +methods. + +First, then, as regards addition, there is nothing +to be added to what has already been said. Addition +is an operation so simple in character that its conception +is a matter of course. But with regard to subtraction, +\MNote{New method of subtraction\Add{.}} +\index{Subtraction, new method of|EtSeq}% +there is another manner of performing that +operation which is frequently more advantageous than +the common method, particularly for those familiar +with it. It consists in converting the subtraction into +addition by taking the complement of every figure of +the number which is to be subtracted, first with respect +to~$10$ and afterwards with respect to~$9$. Suppose, +for example, that the number~$2635$ is to be subtracted +from the number~$7853$. Instead of saying $5$~from~$13$ +\begin{figure}[hbt!] +\centering +$\begin{array}{r} +7853 \\ +2635 \\ +\hline +5218 +\end{array}$ +\end{figure} +leaves~$8$; $3$~from~$4$ leaves~$1$; $6$~from~$8$ leaves~$2$; +and $2$~from~$7$ leaves~$5$, giving a total remainder of~$5218$,---I +say: $5$~the complement of~$5$ with respect to~$10$ +added to~$3$ gives~$8$,---I write down~$8$; $6$~the complement +of~$3$ with respect to~$9$ added to~$5$ gives~$11$,---I +write down~$1$ and carry~$1$; $3$~the complement of~$6$ +with respect to~$9$, plus~$9$, by reason of the $1$~carried, +gives~$12$,---I put down~$2$ and carry~$1$; lastly, $7$~the +complement of~$2$ with respect to~$9$ plus~$8$, on account +of the $1$~carried, gives~$15$,---I put down~$5$ and this time +carry nothing, for the operation is completed, and the +\PageSep{26} +last~$10$ which was borrowed in the course of the operation +must be rejected. In this manner we obtain the +same remainder as above,~$5218$. + +The foregoing method is extremely convenient +\MNote{Subtraction by complements.} +\index{Complements, subtraction by}% +when the numbers are large; for in the common +method of subtraction, where borrowing is necessary +in subtracting single numbers from one another, mistakes +are frequently made, whereas in the method +with which we are here concerned we never borrow +but simply carry, the subtraction being converted into +addition. With regard to the complements they are +discoverable at the merest glance, for every one knows +that $3$~is the complement of~$7$ with respect to~$10$, $4$~the +complement of~$5$ with respect to~$9$,~etc. And as +to the reason of the method, it too is quite palpable. +The different complements taken together form the +total complement of the number to be subtracted +either with respect to~$10$ or~$100$ or~$1000$, etc., according +as the number has $1$,~$2$,~$3$~$\dots$ figures; so that the +operation performed is virtually equivalent to first +adding $10$,~$100$, $1000$~$\dots$ to the minuend and then +taking the subtrahend from the minuend as so augmented. +Whence it is likewise apparent why the~$10$ +of the sum found by the last partial addition must be +rejected. + +As to multiplication, there are various abridged +\index{Multiplication!abridged methods of|EtSeq}% +methods possible, based on the decimal system of +numbers. In multiplying by~$10$, for example, we have, +as we know, simply to add a cipher; in multiplying +\PageSep{27} +by~$100$ we add two ciphers; by~$1000$, three ciphers,~etc. +Consequently, to multiply by any aliquot part of~$10$, +for example~$5$, we have simply to multiply by~$10$ +\MNote{Abridged multiplication.} +and then divide by~$2$; to multiply by~$25$ we multiply +by~$100$ and divide by~$4$, and so on for all the products +of~$5$. + +When decimal numbers are to be multiplied by +\index{Decimal!numbers|EtSeq}% +decimal numbers, the general rule is to consider the +two numbers as integers and when the operation is +finished to mark off from the right to the left as many +places in the product as there are decimal places in +the multiplier and the multiplicand together. But in +practice this rule is frequently attended with the inconvenience +of unnecessarily lengthening the operation, +for when we have numbers containing decimals +these numbers are ordinarily exact only to a certain +number of places, so that it is necessary to retain in +the product only the decimal places of an equivalent +order. For example, if the multiplicand and the multiplier +each contain two places of decimals and are exact +only to two decimal places, we should have in the +product by the ordinary method four decimal places, +the two last of which we should have to reject as useless +and inexact. I shall give you now a method for +obtaining in the product only just so many decimal +places as you desire. + +I observe first that in the ordinary method of multiplying +we begin with the units of the multiplier which +we multiply with the units of the multiplicand, and so +\PageSep{28} +continue from the right to the left. But there is nothing +compelling us to begin at the right of the multiplier. +\MNote{Inverted multiplication.} +\index{Multiplication!inverted}% +We may equally well begin at the left. And +I cannot in truth understand why the latter method +should not be preferred, since it possesses the advantage +of giving at once the figures having the greatest +value, and since, in the majority of cases where large +numbers are multiplied together, it is just these last +and highest places that concern us most; we frequently, +in fact, perform multiplications only to find +what these last figures are. And herein, be it parenthetically +remarked, consists one of the great advantages +in calculating by logarithms, which always +\index{Logarithms!advantages in calculating by}% +give, be it in multiplication or division, in involution +or evolution, the figures in the descending order of +their value, beginning with the highest and proceeding +from the left to the right. + +By performing multiplication in this manner, no +difference is caused in the total product. The sole +distinction is, that by the new method the first line, +the first partial product, is that which in the ordinary +method is last, and the second partial product is that +which in the ordinary method is next to the last, and +so with the rest. + +Where whole numbers are concerned and the exact +product is required, it is indifferent which method we +employ. But when decimal places are involved the +prime essential is to have the figures of the whole +numbers first in the product and to descend afterwards +\PageSep{29} +successively to the figures of the decimal parts, +instead of, as in the ordinary method, beginning with +the last decimal places and successively ascending to +the figures forming the whole numbers. + +In applying this method practically, we write the +multiplier underneath the multiplicand so that the +units' figure of the multiplier falls beneath the last +\MNote{Approximate multiplication.} +\index{Multiplication!approximate}% +figure of the multiplicand. We then begin with the +last left-hand figure of the multiplier which we multiply +as in the ordinary method by all the figures of the +multiplicand, beginning with the last to the right and +proceeding successively to the left; observing that the +first figure of the product is to be placed underneath +the figure with which we are multiplying, while the +others follow in their successive order to the left. We +proceed in the same manner with the second figure of +the multiplier, likewise placing beneath this figure the +first figure of the product, and so on with the rest. +The place of the decimal point in these different products +will be the same as in the multiplicand, that is +to say, the units of the products will all fall in the +same vertical line with those of the multiplicand and +consequently those of the sum of all the products or +of the total product will also fall in that line. In this +manner it is an easy matter to calculate only as many +decimal places as we wish. I give below an example +of this method in which the multiplicand is~$437.25$ +and the multiplier~$27.34$: +\PageSep{30} +\MNote{The new method exemplified.} +\[ +\begin{array}{r@{\,}l} +437\PadTo[l]{\,}{.} & 25 \\ + & 27.34 \\ +\hline +\MultRow{8745}{0} \\ +\MultRow{3060}{75} \\ +\MultRow{131}{17\phantom{.}5} \\ +\MultRow{17}{49\phantom{.}00} \\ +\hline +\MultRow{11954}{41\phantom{.}50} +\end{array} +\] + +I have written all the decimals in the product, but +\index{Decimals!multiplication of}% +it is easy to see how we may omit calculating the decimals +which we wish to neglect. The vertical line is +used to mark more distinctly the place of the decimal +point. + +The preceding rule appears to me simpler and +more natural than that which is attributed to Oughtred +\index{Oughtred}% +and which consists in writing the multiplier underneath +the multiplicand in the reverse order. + +There is one more point, finally, to be remarked +in connexion with the multiplication of numbers containing +\index{Multiplication!decimals@of decimals}% +decimals, and that is that we may alter the +place of the decimal point of either number at will. +For seeing that moving the decimal point from the +right to the left in one of the numbers is equivalent to +dividing the number by~$10$, by~$100$, or by~$1000\dots$, and +that moving the decimal point back in the other number +the same number of places from the left to the +right is tantamount to multiplying that number by~$10$, +$100$, or~$1000$,~$\dots$, it follows that we may push the +decimal point forward in one of the numbers as many +places as we please provided we move it back in the +other number the same number of places, without in +\PageSep{31} +any wise altering the product. In this way we can +always so arrange it that one of the two numbers shall +contain no decimals---which simplifies the question. + +Division is susceptible of a like simplification, for +\index{Decimals!division of}% +\index{Division!decimals@of decimals}% +since the quotient is not altered by multiplying or dividing +\MNote{Division of decimals.} +the dividend and the divisor by the same number, +it follows that in division we may move the decimal +point of both numbers forwards or backwards as +many places as we please, provided we move it the +same distance in each case. Consequently, we can +always reduce the divisor to a whole number---which +facilitates infinitely the operation for the reason that +when there are decimal places in the dividend only, +we may proceed with the division by the common +method and neglect all places giving decimals of a +lower order than those we desire to take account of. + +You know the remarkable property of the number~$9$, +\index{Nine!property of the number|EtSeq}% +whereby if a number be divisible by~$9$ the sum of +its digits is also divisible by~$9$. This property enables +us to tell at once, not only whether a number is divisible +by~$9$ but also what is its remainder from such division. +For we have only to take the sum of its digits +and to divide that sum by~$9$, when the remainder will +be the same as that of the original number divided +by~$9$. + +The demonstration of the foregoing proposition is +not difficult. It reposes upon the fact that the numbers +$10$~less~$1$, $100$~less~$1$, $1000$~less~$1$,~$\dots$ are all divisible +\PageSep{32} +by~$9$,---which seeing that the resulting numbers +are $9$,~$99$, $999$,~$\dots$ is quite obvious. + +If, now, you subtract from a given number the +sum of all its digits, you will have as your remainder +\MNote{Property of the number~$9$.} +the tens' digit multiplied by~$9$, the hundreds' digit +multiplied by~$99$, the thousands' digit multiplied by~$999$, +and so on,---a remainder which is plainly divisible +by~$9$. Consequently, if the sum of the digits is +divisible by~$9$, the original number itself will be so +divisible, and if it is not divisible by~$9$ the original +number likewise will not be divisible thereby. But +the remainder in the one case will be the same as in +the other. + +In the case of the number~$9$, it is evident immediately +that $10$~less~$1$, $100$~less~$1$,~$\dots$ are divisible by~$9$; +but algebra demonstrates that the property in +question holds good for every number~$a$. For it can +be shown that +\[ +a - 1,\quad a^{2} - 1,\quad a^{3} - 1,\quad a^{4} - 1, \dots +\] +are all quantities divisible by~$a - 1$, actual division +giving the quotients +\[ +1,\quad a + 1,\quad a^{2} + a + 1,\quad a^{3} + a^{2} + a + 1, \dots. +\] + +The conclusion is therefore obvious that the aforesaid +property of the number~$9$ holds good in our decimal +system of arithmetic because $9$~is $10$~less~$1$, and +that in any other system founded upon the progression +$a$,~$a^{2}$,~$a^{3}$,~$\dots$ the number~$a - 1$ would enjoy the +same property. Thus in the duodecimal system it +\index{Duodecimal system}% +\PageSep{33} +would be the number~$11$; and in this system every +number, the sum of whose digits was divisible by~$11$, +would also itself be divisible by that number. + +The foregoing property of the number~$9$, now, admits +\index{Nine!property of the number generalised}% +of generalisation, as the following consideration +\MNote{Property of the number~$9$ generalised.} +will show. Since every number in our system is represented +by the sum of certain terms of the progression +$1$,~$10$, $100$, $1000$,~$\dots$, each multiplied by one of +the nine digits $1$,~$2$, $3$, $4$,~$\dots$\Add{,}~$9$, it is easy to see that +the remainder resulting from the division of any number +by a given divisor will be equal to the sum of the +remainders resulting from the division of the terms $1$, +$10$, $100$, $1000$,~$\dots$ by that divisor, each multiplied by +the digit showing how many times the corresponding +term has been taken. Hence, generally, if the given +divisor be called~$D$, and if $m$,~$n$,~$p$,~$\dots$ be the remainders +of the division of the numbers $1$, $10$, $100$, $1000$ +by~$D$, the remainder from the division of any number +whatever~$N$, of which the characters proceeding from +the right to the left are $a$,~$b$,~$c$,~$\dots$, by~$D$ will obviously +be equal to +\[ +ma + nb + pc + \dots. +\] +Accordingly, if for a given divisor~$D$ we know the remainders +$m$,~$n$,~$p$,~$\dots$, which depend solely upon that +divisor and which are always the same for the same +divisor, we have only to write the remainders underneath +the original number, proceeding from the right +to the left, and then to find the different products of +\PageSep{34} +each digit of the number by the digit which is underneath +it. The sum of all these products will be the +\MNote{Theory of remainders\Add{.}} +\index{Remainders!theory of|EtSeq}% +total remainder resulting from the division of the proposed +number by the same divisor~$D$. And if the sum +found is greater than~$D$, we can proceed in the same +manner to seek its remainder from division by~$D$, and +so on until we arrive finally at a remainder which is +less than~$D$, which will be the true remainder sought. +It follows from this that the proposed number cannot +be exactly divisible by the given divisor unless the +last remainder found by this method is zero. + +The remainders resulting from the division of the +terms $1$, $10$, $100$,~$\dots$\Add{,} $1000$, by~$9$ are always unity. +\index{Division!nine@by \textit{nine}}% +Hence, the sum of the digits of any number whatever +is the remainder resulting from the division of that +number by~$9$. The remainders resulting from the division +of the same terms by~$8$ are $1$,~$2$, $4$, $0$, $0$, $0$,~$\dots$. +\index{Division!eight@by \textit{eight}}% +We shall obtain, accordingly, the remainder resulting +from dividing any number by~$8$, by taking the sum +of the first digit to the right, the second digit next +thereto to the left multiplied by~$2$, and the third digit +multiplied by~$4$. + +The remainders resulting from the divisions of the +\index{Division!seven@by \textit{seven}|EtSeq}% +terms $1$, $10$, $100$, $1000$,~$\dots$ by~$7$ are $1$, $3$, $2$, $6$, $4$, $5$, +$1$, $3$,~$\dots$, where the same remainders continually recur +in the same order. If I have, now, the number +$13527541$ to be divided by~$7$, I write it thus with the +above remainders underneath it: +\PageSep{35} +\index{Seven, tests of divisibility by}% +\MNote{Test of divisibility by~$7$.} +\[ +\begin{array}{@{\,}*{2}{r@{}}r@{\,}} +13527&5&41 \\ +31546&2&31 \\ +\hline +&& 1 \\ +&& 12 \\ +&& 10 \\ +&& 42 \\ +&& 8 \\ +&& 25 \\ +&& 3 \\ +&& 3 \\ +\cline{2-3} +& 1&04 \\ +& 2&31 \\ +\cline{2-3} +&& 4 \\ +&& 0 \\ +&& 2 \\ +\cline{3-3} +&& 6 +\end{array} +\] + +Taking the partial products and adding them, I +obtain~$104$, which would be the remainder from the +division of the given number by~$7$, were it not greater +than the divisor. I accordingly repeat the operation +with this remainder, and find for my second remainder~$6$, +which is the real remainder in question. + +I have still to remark with regard to the preceding +remainders and the multiplications which result from +them, that they may be simplified by introducing negative +remainders in the place of remainders which are +greater than half the divisor, and to accomplish this +we have simply to subtract the divisor from each of +such remainders. We obtain thus, instead of the remainders +$6$,~$5$,~$4$, the following: +\[ +-1,\quad -2,\quad -3. +\] +\PageSep{36} +The remainders for the divisor~$7$, accordingly, are +\[ +1,\quad 3,\quad 2,\quad -1,\quad -3,\quad -2,\quad 1,\quad 3, \dots +\] +and so on to infinity. + +\MNote{Negative remainders\Add{.}} +\index{Remainders!negative|EtSeq}% +The preceding example, then, takes the following +form: +\[ +\begin{array}{@{\,}*{3}{r@{}}r@{\,}} +135&27&5&41 \\ +31\underline{2}&\underline{31}&2&31 \\ +\hline + & 7& & 1 \\ + & 6& &12 \\ + &10& &10 \\ +\cline{2-2} + &23& & 3 \\ + & & & 3 \\ +\cline{4-4} + & & &29 \\ +\multicolumn{2}{r}{\llap{\text{subtract}}} & &23 \\ +\cline{4-4} + & & & 6 +\end{array} +\] + +I have placed a bar beneath the digits which are +to be taken negatively, and I have subtracted the sum +of the products of these numbers by those above them +from the sum of the other products. + +The whole question, therefore, resolves itself into +finding for every divisor the remainders resulting from +dividing $1$, $10$, $100$, $1000$\Add{,~$\dots$} by that divisor. This can be +readily done by actual division; but it can be accomplished +more simply by the following consideration. +If $r$~be the remainder from the division of~$10$ by a +given divisor, $r^{2}$~will be the remainder from the division +of~$100$, the square of~$10$, by that divisor; and +consequently it will be necessary merely to subtract +the given divisor from~$r^{2}$ as many times as is requisite +to obtain a positive or negative remainder less than +\PageSep{37} +half of that divisor. Let $s$ be that remainder; we shall +then only have to multiply $s$~by~$r$, the remainder from +the division of~$10$, to obtain the remainder from the +division of~$1000$ by the given divisor, because $1000$~is +$100 × 10$, and so~on. + +For example, dividing $10$ by~$7$ we have a remainder +of~$3$; hence, the remainder from dividing $100$ by~$7$ +will be~$9$, or, subtracting from~$9$ the given divisor~$7$,~$2$. +The remainder from dividing $1000$ by~$7$, then, will +be the product of~$2$ by $3$~or~$6$, or, subtracting the divisor,~$7$,~$-1$. +Again, the remainder from dividing +%[** TN: Removed comma in 10,000 for consistency] +$\Typo{10,000}{10000}$ by~$7$ will be the product of $-1$~and~$3$, or~$-3$, +and so~on. + +Let us now take the divisor~$11$. The remainder +\index{Eleven, the number, test of divisibility by}% +from dividing~$1$ by~$11$ is~$1$, from dividing~$10$ by~$11$ is~$10$, +\MNote{Test of divisibility by~$11$.} +or, subtracting the divisor,~$-1$. The remainder +from dividing~$100$ by~$11$, then, will be the square of~$-1$, +or~$1$; from dividing $1000$ by~$11$ it will be $1$~multiplied +by~$-1$ or\Add{ }$-1$~again, and so on forever, the remainders +forming the infinite series +\[ +1,\quad -1,\quad 1,\quad -1,\quad 1,\quad -1,\dots\Add{.} +\] + +Hence results the remarkable property of the number~$11$, +that if the digits of any number be alternately +added and subtracted, that is to say, if we take the +sum of the first, the third, and the fifth, etc., and subtract +from it the sum of the second, the fourth, the +sixth, etc., we shall obtain the remainder which results +from dividing that number by the number~$11$. +\PageSep{38} + +The preceding theory of remainders is fraught +\index{Remainders!theory of}% +with remarkable consequences, and has given rise to +\MNote{Theory of remainders\Add{.}} +many ingenious and difficult investigations. We can +demonstrate, for example, that if the divisor is a prime +number, the remainders of any progression $1$, $a$, $a^{2}$, +$a^{3}$, $a^{4}$,~$\dots$ form periods which will recur continually +to infinity, and all of which, like the first, begin with +unity; in such wise that when unity reappears among +the remainders we may continue them to infinity by +simply repeating the remainders which precede. It +has also been demonstrated that these periods can +only contain a number of terms which is equal to the +divisor less~$1$ or to an aliquot part of the divisor less~$1$. +But we have not yet been able to determine \textit{à~priori} +this number for any divisor whatever. + +As to the utility of this method for finding the remainder +\index{Theory of remainders, utility of the}% +resulting from dividing a given number by a +given divisor, it is frequently very useful when one +has several numbers to divide by the same number, +and it is required to prepare a table of the remainders. +While as to division by $9$~and~$11$, since that is very +simple, it can be employed as a check upon multiplication +and division. Having found the remainders +from dividing the multiplicand and the multiplier by +either of these numbers it is simply necessary to take +the product of the two remainders so resulting, from +which, after subtracting the divisor as many times as +is requisite, we shall obtain the remainder from dividing +their product by the given divisor,---a remainder +\PageSep{39} +which should agree with the remainder obtained +from treating the actual product in this manner. And +since in division the dividend less the remainder should +\MNote{checks on multiplication and division.} +\index{Checks on multiplication and division}% +be equal to the product of the divisor and the quotient, +the same check may also be applied here to advantage. + +The supposition which I have just made that the +product of the remainders from dividing two numbers +by the same divisor is equal to the remainder from +dividing the product of these numbers by the same +divisor is easily proved, and I here give a general +demonstration of it. + +Let $M$~and~$N$ be two numbers, $D$~the divisor, $p$~and~$q$ +the quotients, and $r$,~$s$ the two remainders. We +shall plainly have +\[ +M = pD + r,\quad +N = qD + s, +\] +from which by multiplying we obtain +\[ +MN = pqD^{2} + spD + rqD + rs; +\] +where it will be seen that all the terms are divisible +by~$D$ with the exception of the last,~$rs$, whence it follows +that $rs$~will be the remainder from dividing~$MN$ +by~$D$. It is further evident that if any multiple whatever +of~$D$, as~$mD$, be subtracted from~$rs$, the result +$rs - mD$ will also be the remainder from dividing~$MN$ +by~$D$. For, putting the value of~$MN$ in the following +form: +\[ +pqD^{2} + spD + rqD + mD + rs - mD, +\] +it is obvious that the remaining terms are all divisible +\PageSep{40} +by~$D$. And this remainder $rs - mD$ can always be +made less than~$D$, or, by employing negative remainders, +less even than~$\dfrac{D}{2}$. + +This is all that I have to say upon multiplication +\MNote{Evolution.} +\index{Evolution}% +and division. I shall not speak of the \emph{extraction of +roots}. The rule is quite simple for square roots; it +leads directly to its goal; trials are unnecessary. As +to cube and higher roots, the occasion rarely arises +for extracting them, and when it does arise the extraction +can be performed with great facility by means +of logarithms, where the degree of exactitude can be +\index{Logarithms}% +carried to as many decimal places as the logarithms +themselves have decimal places. Thus, with seven-place +logarithms we can extract roots having seven +figures, and with the large tables where the logarithms +have been calculated to ten decimal places we +can obtain even ten figures of the result. + +One of the most important operations in arithmetic +\index{Rule!three@of three|EtSeq}% +is the so-called \emph{rule of three}, which consists in +finding the fourth term of a proportion of which the +first three terms are given. + +In the ordinary text-books of arithmetic this rule +has been unnecessarily complicated, having been divided +into simple, direct, inverse, and compound rules +of three. In general it is sufficient to comprehend the +conditions of the problem thoroughly, for the common +rule of three is always applicable where a quantity increases +or diminishes in the same proportion as another. +\PageSep{41} +For example, the price of things augments in +proportion to the quantity of the things, so that the +quantity of the thing being doubled, the price also +\MNote{Rule of three.} +will be doubled, and so on. Similarly, the amount of +work done increases proportionally to the number of +persons employed. Again, things may increase simultaneously +in two different proportions. For example, +the quantity of work done increases with the +number of the persons employed, and also with the +time during which they are employed. Further, there +are things that decrease as others increase. + +Now all this may be embraced in a single, simple +proposition. If a quantity increases both in the ratio +in which one or several other quantities increase and +in that in which one or several other quantities decrease, +it is the same thing as saying that the proposed +quantity increases proportionally to the product of the +quantities which increase with it, divided by the product +of the quantities which simultaneously decrease. +For example, since the quantity of work done increases +proportionally with the number of laborers +\index{Laborers, work of}% +and with the time during which they work and since +it diminishes in proportion as the work becomes more +difficult, we may say that the result is proportional to +the number of laborers multiplied by the number +measuring the time during which they labor, divided +by the number which measures or expresses the difficulty +of the work. + +The further fact should not be lost sight of that +\PageSep{42} +the rule of three is properly applicable only to things +which increase in a constant ratio. For example, it is +\index{Ratios, constant}% +\MNote{Applicability of the rule of three.} +assumed that if a man does a certain amount of work +in one day, two men will do twice that amount in one +day, three men three times that amount, four men +four times that amount,~etc. In reality this is not the +case, but in the rule of proportion it is assumed to be +such, since otherwise we should not be able to employ +it. + +When the law of augmentation or diminution varies, +the rule of three is not applicable, and the ordinary +methods of arithmetic are found wanting. We +must then have recourse to algebra. + +A cask of a certain capacity empties itself in a certain +\index{Efflux, law of}% +time. If we were to conclude from this that a +cask of double that capacity would empty itself in +double the time, we should be mistaken, for it will +empty itself in a much shorter time. The law of efflux +does not follow a constant ratio but a variable +ratio which diminishes with the quantity of liquid remaining +in the cask. + +We know from mechanics that the spaces traversed +\index{Falling stone, spaces traversed by a}% +by a body in uniform motion bear a constant ratio to +the times elapsed. If we travel one mile in one hour, +in two hours we shall travel two miles. But the spaces +traversed by a falling stone are not in a fixed ratio to +the time. If it falls sixteen feet in the first second, it +will fall forty-eight feet in the second second. + +The rule of three is applicable when the ratios are +\PageSep{43} +constant only. And in the majority of affairs of ordinary +life constant ratios are the rule. In general, the +price is always proportional to the quantity, so that if +\MNote{Theory and practice.} +\index{Practice, theory and}% +\index{Theory and practice}% +a given thing has a certain value, two such things will +have twice that value, three three times that value, +four four times that value,~etc. It is the same with +the product of labor relatively to the number of laborers +and to the duration of the labor. Nevertheless, +cases occur in which we may be easily led into error. +If two horses, for example, can pull a load of a certain +\index{Horses}% +weight, it is natural to suppose that four horses +could pull a load of double that weight, six horses a +load of three times that weight. Yet, strictly speaking, +such is not the case. For the inference is based +upon the assumption that the four horses pull alike in +amount and direction, which in practice can scarcely +ever be the case. It so happens that we are frequently +led in our reckonings to results which diverge widely +from reality. But the fault is not the fault of mathematics; +\index{Mathematics!exactness of}% +for mathematics always gives back to us exactly +what we have put into it. The ratio was constant +according to the supposition. The result is founded +upon that supposition. If the supposition is false the +result is necessarily false. Whenever it has been attempted +to charge mathematics with inexactitude, the +accusers have simply attributed to mathematics the +error of the calculator. False or inexact data having +been employed by him, the result also has been necessarily +false or inexact. +\PageSep{44} + +Among the other rules of arithmetic there is one +called \emph{alligation} which deserves special consideration +\index{Alligation!generally|EtSeq}% +\MNote{Alligation.} +from the numerous applications which it has. Although +alligation is mainly used with reference to the +mingling of metals by fusion, it is yet applied generally +\index{Metals, mingling of, by fusion}% +to mixtures of any number of articles of different +values which are to be compounded into a whole of a +like number of parts having a mean value. The rule +\index{Mixtures, rule of|EtSeq}% +\index{Rule!mixtures@of mixtures|EtSeq}% +of alligation, or mixtures, accordingly, has two parts. + +In one we seek the mean and common value of +each part of the mixture, having given the number +of the parts and the particular value of each. In the +second, having given the total number of the parts +and their mean value, we seek the composition of the +mixture itself, or the proportional number of parts of +each ingredient which must be mixed or alligated together. + +Let us suppose, for example, that we have several +\index{Grain, of different prices}% +bushels of grain of different prices, and that we are +desirous of knowing the mean price. The mean price +must be such that if each bushel were of that price the +total price of all the bushels together would still be +the same. Whence it is easy to see that to find the +mean price in the present case we have first simply to +find the total price and to divide it by the number of +bushels. + +In general if we multiply the number of things of +each kind by the value of the unit of that kind and +then divide the sum of all these products by the total +\PageSep{45} +number of things, we shall have the mean value, because +that value multiplied by the number of the +things will again give the total value of all the things +taken together. + +This mean or average value as it is called, is of +\index{Mean values|EtSeq}% +\index{Values!mean|EtSeq}% +great utility in almost all the affairs of life. Whenever +\MNote{Mean values.} +we arrive at a number of different results, we +always like to reduce them to a mean or average expression +which will yield the same total result. + +You will see when you come to the calculus of +\index{Probabilities, calculus of|EtSeq}% +probabilities that this science is almost entirely based +upon the principle we are discussing. + +The registration of births and deaths has rendered +\index{Average life|EtSeq}% +\index{Life insurance|EtSeq}% +\index{Mortality, tables of}% +possible the construction of so-called \emph{tables of mortality} +which show what proportion of a given number of +children born at the same time or in the same year +survive at the end of one year, two years, three years,~etc. +So that we may ask upon this basis what is the +mean or average value of the life of a person at any +given age. If we look up in the tables the number of +people living at a certain age, and then add to this +the number of persons living at all subsequent ages, +it is clear that this sum will give the total number of +years which all living persons of the age in question +have still to live. Consequently, it is only necessary +to divide this sum by the number of living persons of +a certain age in order to obtain the average duration +of life of such persons, or better, the number of years +which each person must live that the total number of +\PageSep{46} +years lived by all shall be the same and that each +person shall have lived an equal number. It has been +\MNote{Probability of life.} +\index{Life, probability of}% +found in this manner by taking the mean of the results +of different tables of mortality, that for an infant +one year old the average duration of life is about +$40$~years; for a child ten years old it is still $40$~years; +for~$20$ it is~$34$; for~$30$ it is~$26$; for~$40$ it is~$23$; for~$50$ +it is~$17$; for~$60$ it is~$12$; for~$70$,~$8$; and for~$80$,~$5$. + +To take another example, a number of different +experiments are made. Three experiments have given~$4$ +\index{Experiments!average of}% +as a result; two experiments have given~$5$; and one +has given~$6$. To find the mean we multiply~$4$ by~$3$, $5$~by~$2$, +and $1$~by~$6$, add the products which gives~$28$, +and divide~$28$ by the number of experiments or~$6$, +which gives~$4\frac{2}{3}$ as the mean result of all the experiments. + +But it will be apparent that this result can be regarded +as exact only upon the condition of our having +supposed that the experiments were all conducted with +equal precision. But it is impossible that such could +have been the case, and it is consequently imperative +to take account of these inequalities, a requirement +which would demand a far more complicated calculus +than that which we have employed, and one which is +now engaging the attention of mathematicians. + +The foregoing is the substance of the first part of +the rule of alligation; the second part is the opposite +of the first. Given the mean value, to find how much +\PageSep{47} +must be taken of each ingredient to produce the required +mean value. + +The problems of the first class are always determinate, +because, as we have just seen, the number of +\MNote{Alternate alligation.} +\index{Alligation!alternate}% +units of each ingredient has simply to be multiplied +by the value of each ingredient and the sum of all +these products divided by the number of the ingredients. + +The problems of the second class, on the other +\index{Analysis!indeterminate|EtSeq}% +\index{Indeterminate analysis|EtSeq}% +hand, are always indeterminate. But the condition +that only positive whole numbers shall be admitted +in the result serves to limit the number of the solutions. + +Suppose that we have two kinds of things, that +the value of the unit of one kind is~$a$, and that of the +unit of the second is~$b$, and that it is required to find +how many units of the first kind and how many units +of the second must be taken to form a mixture or +whole of which the mean value shall be~$m$. + +Call $x$~the number of units of the first kind that +must enter into the mixture, and $y$~the number of units +of the second kind. It is clear that $ax$~will be the +value of the $x$~units of the first kind, and $by$~the value +of the $y$~units of the second. Hence $ax + by$ will be +the total value of the mixture. But the mean value +of the mixture being by supposition~$m$, the sum~$x + y$ +of the units of the mixture multiplied by~$m$, the mean +value of each unit, must give the same total value. +We shall have, therefore, the equation +\PageSep{48} +\[ +ax + by = mx + my. +\] +Transposing to one side the terms multiplied by~$x$ +and to the other the terms multiplied by~$y$, we obtain: +\MNote{Two ingredients.} +\index{Ingredients}% +\[ +(a - m)x = (m - b)y, +\] +and dividing by~$a - m$ we get +\[ +x = \frac{(m - b)y}{a - m}, +\] +whence it appears that the number~$y$ may be taken at +pleasure, for whatever be the value given to~$y$, there +will always be a corresponding value of~$x$ which will +satisfy the problem. + +Such is the general solution which algebra gives. +But if the condition be added that the two numbers $x$~and~$y$ +shall be integers, then $y$~may not be taken at +pleasure. In order to see how we can satisfy this last +condition in the simplest manner, let us divide the +last equation by~$y$, and we shall have +\[ +\frac{x}{y} = \frac{m - b}{a - m}. +\] +For $x$~and~$y$ both to be positive, it is necessary that +the quantities +\[ +m - b \quad\text{and}\quad a - m +\] +should both have the same sign; that is to say, if $a$~is +greater or less than~$m$, then conversely $b$~must be less +or greater than~$m$; or again, $m$~must lie between $a$~and~$b$, +which is evident from the condition of the +problem. Suppose $a$, then, to be the greater and $b$~the +\PageSep{49} +smaller of the two prices. It remains to find the +value of the fraction +\MNote{Rule of mixtures.} +\index{Mixtures, rule of}% +\[ +\frac{m - b}{a - m}, +\] +which if necessary is to be reduced to its lowest terms. +Let~$\dfrac{B}{A}$ be that fraction reduced to its lowest terms. It +is clear that the simplest solution will be that in which +\[ +x = B \quad\text{and}\quad y = A. +\] +But since a fraction is not altered by multiplying its +numerator and denominator by the same number, it +is clear that we may also take $x = nB$ and $y = nA$, $n$~being +any number whatever, provided it is an integer, +for by supposition $x$~and~$y$ must be integers. And it +is easy to prove that these expressions of $x$~and~$y$ are +the only ones which will resolve the proposed problem. +According to the ordinary rule of mixtures, $x$, +the quantity of the dearer ingredient, is made equal +to~$m - b$, the excess of the average price above the +lower price, and $y$~the quantity of the cheaper ingredient +is made equal to~$a - m$, the excess of the higher +price above the average price,---a rule which is contained +directly in the general solution above given. + +Suppose, now, that instead of two kinds of things, +we have three kinds, the values of which beginning +with the highest are $a$,~$b$, and~$c$. Let $x$,~$y$,~$z$ be the +quantities which must be taken of each to form a mixture +or compound having the mean value~$m$. The +sum of the values of the three quantities $x$,~$y$,~$z$ will +then be +\[ +ax + by + cz. +\] +\PageSep{50} +But this total value must be the same as that produced +if all the individual values were~$m$, in which +\MNote{Three ingredients.} +case the total value is obviously +\[ +mx + my + mz. +\] +The following equation, therefore, must be satisfied: +\[ +ax + by + cz = mx + my + mz, +\] +or, more simply, +\[ +(a - m)x + (b - m)y + (c - m)z = 0. +\] +Since there are three unknown quantities in this equation, +two of them may be taken at pleasure. But if +the condition is that they shall be expressed by positive +integers, it is to be observed first that the numbers +\[ +a - m \quad\text{and}\quad m - c +\] +are necessarily positive; so that putting the equation +in the form +\[ +(a - m)x - (m - c)z = (m - b)y, +\] +the question resolves itself into finding two multiples +of the given numbers +\[ +a - m \quad\text{and}\quad m - c +\] +whose difference shall be equal to~$(m - b)y$. + +This question is always resolvable in whole numbers +whatever the given numbers be of which we seek +the multiples, and whatever be the difference between +these multiples. As it is sufficiently remarkable in itself +and may be of utility in many emergencies, we +shall give here a general solution of it, derived from +the properties of continued fractions. +\index{Continued fractions, solution of alligation by|EtSeq}% +\PageSep{51} + +Let $M$~and~$N$ be two whole numbers. Of these +numbers two multiples $xM$,~$zN$ are sought whose difference +is given and equal to~$D$. The following equation +\MNote{General solution.} +will then have to be satisfied +\[ +xM - zN = D, +\] +where $x$~and~$z$ by supposition are whole numbers. In +the first place, it is plain that if $M$~and~$N$ are not +prime to each other, the number~$D$ is divisible by the +greatest common divisor of $M$~and~$N$; and the division +having been performed, we should have a similar +equation in which the numbers $M$~and~$N$ are prime +to each other, so that we are at liberty always to suppose +them reduced to that condition. I now observe +that if we know the solution of the equation for the +case in which the number~$D$ is equal to $+1$~or~$-1$, +we can deduce the solution of it for any value whatever +of~$D$. For example, suppose that we know two +multiples of $M$~and~$N$, say $pM$~and~$qN$, the difference +of which $pM - qN$ is equal to~$±1$. Then obviously +we shall merely have to multiply both these multiples +by the number~$D$ to obtain a difference equal to~$±D$. +For, multiplying the preceding equation by~$D$, we +have +\[ +pDM - qDN = ±D; +\] +and subtracting the latter equation from the original +equation +\[ +xM - zN = D, +\] +or adding it, according as the term~$D$ has the sign +$+$~or~$-$ before it, we obtain +\PageSep{52} +\[ +(x \mp pD)M - (z \mp qD)N = 0, +\] +which gives at once, as we saw above in the rule for +the mixture of two different ingredients, +\MNote{Development.} +\[ +(x \mp pD) = nN,\quad +(z \mp qD) = nM, +\] +$n$~being any number whatever. So that we have generally +\[ +x = nN ± pD \quad\text{and}\quad z = nM ± qD +\] +where $n$~is any whole number, positive or negative. +It remains merely to find two numbers $p$~and~$q$ such +that +\[ +pM - qN = ±1. +\] +Now this question is easily resolvable by continued +fractions. For we have seen in treating of these fractions +that if the fraction~$\dfrac{M}{N}$ be reduced to a continued +fraction, and all the successive fractions approximating +to its value be calculated, the last of these successive +fractions being the fraction~$\dfrac{M}{N}$ itself, then the series +of fractions so reached is such that the difference +between any two consecutive fractions is always equal +to a fraction of which the numerator is unity and the +denominator the product of the two denominators. +For example, designating by~$\dfrac{K}{L}$ the fraction which +immediately precedes the last fraction~$\dfrac{M}{N}$ we obtain +necessarily +\[ +LM - KN = 1 \quad\text{or}\quad -1, +\] +according as $\dfrac{M}{N}$~is greater or less than~$\dfrac{K}{L}$, in other +\PageSep{53} +words, according as the place occupied by the last +fraction~$\dfrac{M}{N}$ in the series of fractions successively approximating +to its value is even or odd; for, the first +\MNote{Resolution by continued fractions.} +fraction of the approximating series is always smaller, +the second larger, the third smaller,~etc., than the +original fraction which is identical with the last fraction +of the series. Making, therefore, +\[ +p = L \quad\text{and}\quad q = K, +\] +the problem of the two multiples will be resolved in +all its generality. + +It is now clear that in order to apply the foregoing +solution to the initial question regarding alligation we +have simply to put +\[ +M = a - m,\quad N = m - c, \quad\text{and}\quad D = (m - b)y; +\] +so that the number~$y$ remains undetermined and may +be taken at pleasure, as may also the number~$N$ which +appears in the expressions for $x$~and~$z$. +\PageSep{54} + + +\Lecture[On Algebra.]{III.}{On Algebra, Particularly the Resolution of +Equations of the Third and +Fourth Degree.} +\index{Algebra!history of|EtSeq}% +\index{Diophantus|EtSeq}% +\index{Geometers, ancient|EtSeq}% +\index{Greeks, mathematics of the|EtSeq}% +\index{Romans, mathematics of the}% +\PgLabel{54} + +\First{Algebra} is a science almost entirely due to the +moderns. I say almost entirely, for we have +\MNote{Algebra among the ancients.} +one treatise from the Greeks, that of Diophantus, who +flourished in the third\footnote + {The period is uncertain. Some say in the fourth century. See Cantor, + \index{Cantor|FN}% + \textit{Geschichte der Mathematik}, 2nd.~ed., Vol.~I., p.~434.---\textit{Trans.}} +century of the Christian era. +This work is the only one which we owe to the ancients +in this branch of mathematics. When I speak +of the ancients I speak of the Greeks only, for the +Romans have left nothing in the sciences, and to all +appearances did nothing. + +Diophantus may be regarded as the inventor of +algebra.\footnote + {On this point, see \textit{Appendix}, \PgRef{151}.---\textit{Trans.}} +From a word in his preface, or rather in his +letter of dedication, (for the ancient geometers were +wont to address their productions to certain of their +friends, a practice exemplified in the prefaces of Apollonius +\index{Apollonius}% +and Archimedes), from a word in his preface, I +\index{Archimedes}% +say, we learn that he was the first to occupy himself +\PageSep{55} +with that branch of arithmetic which has since been +called algebra. + +His work contains the first elements of this science. +He employed to express the unknown quantity a Greek +\index{Unknown quantity}% +\MNote{Diophantus\Add{.}} +letter which corresponds to our~$st$\footnote + {According to a recent conjecture, the character in question is an abbreviation + of~\textgreek{ar} the first letters of \textgreek{>arijm'os}, \textit{number}, the appellation technically + applied by Diophantus to the unknown quantity.---\textit{Trans.}} +and which has +been replaced in the translations by~$N$. To express +the known quantities he employed numbers solely, for +algebra was long destined to be restricted entirely to +the solution of numerical problems. We find, however, +that in setting up his equations consonantly with +the conditions of the problem he uses the known and +the unknown quantities alike. And herein consists +\index{Algebra!essence of}% +virtually the essence of algebra, which is to employ +unknown quantities, to calculate with them as we do +with known quantities, and to form from them one +or several equations from which the value of the unknown +quantities can be determined. Although the +work of Diophantus contains indeterminate problems +\index{Analysis!indeterminate}% +\index{Indeterminate analysis}% +almost exclusively, the solution of which he seeks in +rational numbers,---problems which have been designated +after him \emph{Diophantine problems},---we nevertheless +\index{Diophantine problems}% +find in his work the solution of a number of determinate +problems of the first degree, and even of such +as involve several unknown quantities. In the latter +case, however, the author invariably has recourse to +particular artifices for reducing the problem to a single +unknown quantity,---which is not difficult. He gives, +\PageSep{56} +also, the solution of \emph{equations of the second degree}, but +\index{Equations!second@of the second degree}% +is careful so to arrange them that they never assume +the affected form containing the square and the first +power of the unknown quantity. + +He proposed, for example, the following question +\MNote{Equations of the second degree.} +which involves the general theory of equations of the +second degree: + +\textit{To find two numbers the sum and the product of which +are given.} +\index{Sum and difference, of two numbers}% + +If we call the sum~$a$ and the product~$b$ we have at +once, by the theory of equations, the equation +\[ +x^{2} - ax + b = 0. +\] + +Diophantus resolves this problem in the following +manner. The sum of the two numbers being given, +he seeks their difference, and takes the latter as the +unknown quantity. He then expresses the two numbers +in terms of their sum and difference,---the one +by half the sum plus half the difference, the other by +half the sum less half the difference,---and he has +then simply to satisfy the other condition by equating +their product to the given number. Calling the given +sum~$a$, the unknown difference~$x$, one of the numbers +will be~$\dfrac{a + x}{2}$ and the other will be~$\dfrac{a - x}{2}$. Multiplying +these together we have~$\dfrac{a^{2} - x^{2}}{4}$. The term containing~$x$ +is here eliminated, and equating the quantity +last obtained to the given product, we have the +simple equation +\[ +\frac{a^{2} - x^{2}}{4} = b, +\] +\PageSep{57} +from which we obtain +\[ +x^{2} = a^{2} - 4b, +\] +and from the latter +\[ +x = \sqrt{a^{2} - 4b}. +\] + +Diophantus resolves several other problems of this +class. By appropriately treating the sum or difference +\MNote{Other problems solved by Diophantus.} +as the unknown quantity he always arrives at an +equation in which he has only to extract a square root +to reach the solution of his problem. + +But in the books which have come down to us +(for the entire work of Diophantus has not been preserved) +this author does not proceed beyond equations +of the second degree, and we do not know if he +or any of his successors (for no other work on this +subject has been handed down from antiquity) ever +pushed their researches beyond this point. + +I have still to remark in connexion with the work +\index{Signs $+$ and $-$}% +of Diophantus that he enunciated the principle that +$+$~and~$-$ give~$-$ in multiplication, and $-$~and~$-$,~$+$, +in the form of a definition. But I am of opinion that +this is an error of the copyists, since he is more likely +to have considered it as an axiom, as did Euclid some +\index{Euclid}% +of the principles of geometry. However that may be, +it will be seen that Diophantus regarded the rule of +the signs as a self-evident principle not in need of demonstration. + +The work of Diophantus is of incalculable value +from its containing the first germs of a science which +because of the enormous progress which it has since +\PageSep{58} +made constitutes one of the chiefest glories of the human +intellect. Diophantus was not known in Europe +\MNote{Translations of Diophantus\Add{.}} +until the end of the sixteenth century, the first translation +having been a wretched one by Xylander made +\index{Xylander}% +in~1575 and based upon a manuscript found about the +middle of the sixteenth century in the Vatican library, +\index{Vatican library}% +where it had probably been carried from Greece when +the Turks took possession of Constantinople. +\index{Constantinople}% +\index{Turks}% + +Bachet de Méziriac, one of the earliest members +\index{Bachet de Méziriac}% +\index{Meziriac@Méziriac, Bachet de}% +of the French Academy, and a tolerably good mathematician +for his time, subsequently published~(1621) +a new translation of the work of Diophantus accompanied +by lengthy commentaries, now superfluous. +Bachet's translation was afterwards reprinted with observations +and notes by Fermat, one of the most celebrated +\index{Fermat}% +mathematicians of France, who flourished +\index{France}% +about the middle of the seventeenth century, and of +whom we shall have occasion to speak in the sequel +for the important discoveries which he has made in +analysis. Fermat's edition bears the date of~1670.\footnote + {There have since been published a new critical edition of the text by + M.~Paul Tannery (Leipsic, 1893), and two German translations, one by O.~Schulz + \index{Tannery, M. Paul|FN}% + \index{Wertheim, G.|FN}% + (Berlin, 1822) and one by G.~Wertheim (Leipsic, 1890). Fermat's notes + on Diophantus have been republished in Vol.~I. of the new edition of Fermat's + works (Paris, Gauthier-Villars et Fils, 1891).---\textit{Trans.}} + +It is much to be desired that good translations +\index{Geometers, ancient}% +should be made, not only of the work of Diophantus, +but also of the small number of other mathematical +works which the Greeks have left us.\footnote + {Since Lagrange's time this want has been partly supplied. Not to mention + Euclid, we have, for example, of Archimedes the German translation of + \index{Archimedes|FN}% + Nizze (Stralsund, 1824) and the French translation of Peyrard (Paris, 1807); of + \index{Nizze|FN}% + \index{Peyrard}% + Apollonius, several translations; also modern translations of Hero, Ptolemy, + \index{Apollonius}% + \index{Geometers, ancient}% + \index{Hero}% + \index{Pappus}% + \index{Proclus}% + \index{Ptolemy}% + \index{Theon}% + Pappus, Theon, Proclus, and several others.} +\PageSep{59} + +Prior to the discovery and publication of Diophantus, +however, algebra had already found its way into +\index{Algebra!name@the name of}% +\index{Algebra!among the Arabs|EtSeq}% +Europe. Towards the end of the fifteenth century +there appeared in Venice a work by an Italian Franciscan +monk named Lucas Paciolus on arithmetic and +\index{Paciolus, Lucas}% +geometry in which the elementary rules of algebra +were stated. This book was published (1494) in the +\MNote{Algebra among the Arabs.} +\index{Arabs!Algebra among the|EtSeq}% +early days of the invention of printing, and the fact +\index{Printing, invention of}% +that the name of \emph{algebra} was given to the new science +shows clearly that it came from the Arabs. It is true +that the signification of this Arabic word is still disputed, +but we shall not stop to discuss such matters, +for they are foreign to our purpose. Let it suffice +that the word has become the name for a science that +is universally known, and that there is not the slightest +ambiguity concerning its meaning, since up to the +present time it has never been employed to designate +anything else. + +We do not know whether the Arabs invented algebra +\PgLabel{59} +themselves or whether they took it from the +Greeks.\footnote + {See Appendix, \PgRef{152}.} +There is reason to believe that they possessed +the work of Diophantus, for when the ages of +barbarism and ignorance which followed their first +conquests had passed by, they began to devote themselves +to the sciences and to translate into Arabic all +the Greek works which treated of scientific subjects. +It is reasonable to suppose, therefore, that they also +\PageSep{60} +translated the work of Diophantus and that the same +work stimulated them to push their inquiries farther +in this science. + +Be that as it may, the Europeans, having received +\MNote{Algebra in Europe.} +\index{Algebra!Europe@in Europe}% +\index{Europe, algebra in}% +algebra from the Arabs, were in possession of it one +hundred years before the work of Diophantus was +known to them. They made, however, no progress +beyond equations of the first and second degree. In +\index{Equations!third@of the third degree}% +the work of Paciolus, which we mentioned above, the +\index{Paciolus, Lucas}% +general resolution of equations of the second degree, +such as we now have it, was not given. We find in +this work simply rules, expressed in bad Latin verses, +for resolving each particular case according to the +different combinations of the signs of the terms of +equation, and even these rules applied only to the +case where the roots were real and positive. Negative +\index{Negative roots}% +\index{Roots!negative}% +roots were still regarded as meaningless and superfluous. +It was geometry really that suggested to us the +\index{Geometry}% +use of negative quantities, and herein consists one of +the greatest advantages that have resulted from the +application of algebra to geometry,---a step which we +owe to Descartes. +\index{Descartes}% +\PgLabel{60} + +In the subsequent period the resolution of \emph{equations +of the third degree} was investigated and the discovery +for a particular case ultimately made by a mathematician +\index{Ferrous, Scipio|EtSeq}% +of Bologna named Scipio Ferreus (1515).\footnote + {The date is uncertain. Tartaglia gives 1506, Cardan 1515. Cantor prefers + \index{Cantor|FN}% + \index{Cardan}% + \index{Tartaglia}% + the latter.---\textit{Trans.}} +Two +other Italian mathematicians, Tartaglia and Cardan, +\PageSep{61} +subsequently perfected the solution of Ferreus and +rendered it general for all equations of the third degree. +At this period, Italy, which was the cradle of +\index{Italy, cradle of algebra in Europe}% +\MNote{Tartaglia (1500--1559). Cardan (1501--1576).} +\index{Cardan}% +\index{Tartaglia}% +algebra in Europe, was still almost the sole cultivator +of the science, and it was not until about the middle +of the sixteenth century that treatises on algebra began +to appear in France, Germany, and other countries. +\index{France}% +\index{Germany}% +The works of Peletier and Buteo were the first +\index{Buteo}% +\index{Peletier}% +which France produced in this science, the treatise of +the former having been printed in~1554 and that of +the latter in~1559. + +Tartaglia expounded his solution in bad Italian +verses in a work treating of divers questions and inventions +printed in~1546, a work which enjoys the +distinction of being one of the first to treat of modern +fortifications by bastions. + +About the same time (1545) Cardan published his +treatise \textit{Ars Magna}, or \textit{Algebra}, in which he left +scarcely anything to be desired in the resolution of +equations of the third degree. Cardan was the first to +perceive that equations had several roots and to distinguish +them into positive and negative. But he is +particularly known for having first remarked the so-called +\emph{irreducible case} in which the expression of the +\index{Irreducible case}% +real roots appears in an imaginary form. Cardan convinced +himself from several special cases in which the +equation had rational divisors that the imaginary form +did not prevent the roots from having a real value. +But it remained to be proved that not only were the +\PageSep{62} +roots real in the irreducible case, but that it was impossible +for all three together to be real except in that +case. This proof was afterwards supplied by Vieta, +\index{Vieta}% +and particularly by Albert Girard, from considerations +\index{Girard, Albert}% +touching the trisection of an angle. +\index{Angle, trisection of an}% +\index{Trisection of an angle}% + +We shall revert later on to the \emph{irreducible case of +equations of the third degree}, not solely because it presents +\MNote{The irreducible case.} +a new form of algebraical expressions which +have found extensive application in analysis, but because +it is constantly giving rise to unprofitable inquiries +with a view to reducing the imaginary form to +a real form and because it thus presents in algebra a +problem which may be placed upon the same footing +with the famous problems of the duplication of the +\index{Problems!solution@for solution}% +cube and the squaring of the circle in geometry. +\index{Circle!squaring of the}% +\index{Cube, duplication of the}% +\index{Squaring of the circle}% + +The mathematicians of the period under discussion +\index{Academies, rise of}% +were wont to propound to one another problems +for solution. These problems were in the nature of +public challenges and served to excite and to maintain +in the minds of thinkers that fermentation which +is necessary for the pursuit of science. The challenges +in question were continued down to the beginning of +the eighteenth century by the foremost mathematicians +of Europe, and really did not cease until the rise +of the Academies which fulfilled the same end in a +manner even more conducive to the progress of science, +partly by the union of the knowledge of their +various members, partly by the intercourse which they +maintained between them, and not least by the publication +\PageSep{63} +of their memoirs, which served to disseminate +the new discoveries and observations among all persons +interested in science. + +The challenges of which we speak supplied in a +\index{Academies, rise of}% +measure the lack of Academies, which were not yet +\MNote{Biquadratic equations.} +\index{Biquadratic equations}% +\index{Equations!fourth@of the fourth degree}% +in existence, and we owe to these passages at arms +many important discoveries in analysis. Such was +the resolution of \emph{equations of the fourth degree}, which +was propounded in the following problem. + +%[** TN: Next paragraph centered in the original] +\textit{To find three numbers in continued proportion of which +the sum is~$10$, and the product of the first two~$6$.} + +Generalising and calling the sum of the three numbers~$a$, +the product of the first two~$b$, and the first two +numbers themselves $x$,~$y$, we shall have, first, $xy = b$. +Owing to the continued proportion, the third number +will then be expressed by~$\dfrac{y^{2}}{x}$, so that the remaining +condition will give +\[ +x + y + \frac{y^{2}}{x} = a. +\] +From the first equation we obtain $x = \dfrac{b}{y}$, which substituted +in the second gives +\[ +\frac{b}{y} + y + \frac{y^{2}}{b} = a\Typo{,}{.} +\] +Removing the fractions and arranging the terms, we +get finally +\[ +y^{4} + by^{2} - aby + b^{2} = 0, +\] +an equation of the fourth degree with the second term +missing. + +According to Bombelli, of whom we shall speak +\index{Bombelli}% +\PageSep{64} +again, Louis Ferrari of Bologna resolved the problem +\index{Ferrari, Louis}% +by a highly ingenious method, which consists in +\MNote{Ferrari (1522-1565). Bombelli.} +\index{Bombelli}% +dividing the equation into two parts both of which +permit of the extraction of the square root. To do +this it is necessary to add to the two numbers quantities +whose determination depends on an equation of +the third degree, so that the resolution of equations +\index{Equations!fifth@of the fifth degree}% +of the fourth degree depends upon the resolution of +equations of the third and is therefore subject to the +same drawbacks of the irreducible case. + +The \textit{Algebra} of Bombelli was printed in Bologna +\index{Algebra!Italy@in Italy}% +in~1579\footnote + {This was the second edition. The first edition appeared in Venice in~1572.---\textit{Trans.}} +in the Italian language. It contains not only +the discovery of Ferrari but also divers other important +remarks on equations of the second and third +degree and particularly on the theory of radicals by +means of which the author succeeded in several cases +in extracting the imaginary cube roots of the two +binomials of the formula of the third degree in the irreducible +case, so finding a perfectly real result and +furnishing thus the most direct proof possible of the +reality of this species of expressions. + +Such is a succinct history of the first progress of +algebra in Italy. The solution of equations of the +\index{Italy, cradle of algebra in Europe}% +third and fourth degree was quickly accomplished. +But the successive efforts of mathematicians for over +two centuries have not succeeded in surmounting the +difficulties of the equation of the fifth degree. +\PageSep{65} + +Yet these efforts are far from having been in vain. +They have given rise to the many beautiful theorems +which we possess on the formation of equations, on +\MNote{Theory of equations.} +\index{Equations!theory of}% +the character and signs of the roots, on the transformation +of a given equation into others of which the +roots may be formed at pleasure from the roots of the +given equation, and finally, to the beautiful considerations +concerning the metaphysics of the resolution +of equations from which the most direct method of +arriving at their solution, when possible, has resulted. +All this has been presented to you in previous lectures +and would leave nothing to be desired if it were +but applicable to the resolution of equations of higher +degree. + +Vieta and Descartes in France, Harriot in England, +\index{Descartes}% +\index{Harriot}% +\index{Vieta}% +and Hudde in Holland, were the first after the +\index{Hudde}% +Italians whom we have just mentioned to perfect the +theory of equations, and since their time there is +scarcely a mathematician of note that has not applied +himself to its investigation, so that in its present state +this theory is the result of so many different inquiries +that it is difficult in the extreme to assign the author +of each of the numerous discoveries which constitute it. + +I promised to revert to the irreducible case. To +\index{Irreducible case}% +this end it will be necessary to recall the method +which seems to have led to the original resolution of +equations of the third degree and which is still employed +in the majority of the treatises on algebra. +\PageSep{66} +Let us consider the general equation of the third degree +deprived of its second term, which can always be +removed; in a word, let us consider the equation +\MNote{Equations of the third degree.} +\index{Equations!third@of the third degree}% +\[ +x^{3} + px + q = 0. +\] +Suppose +\[ +x = y + z, +\] +where $y$~and~$z$ are two new unknown quantities, of +which one consequently may be taken at pleasure and +determined as we think most convenient. Substituting +this value for~$x$, we obtain \emph{the transformed equation} +\[ +y^{3} + 3y^{2}z + 3yz^{2} + z^{3} + p(y + z) + q = 0. +\] +Factoring the two terms $3y^{2}z + 3yz^{2}$ we get +\[ +3yz(y + z), +\] +and the transformed equation may be written as follows: +\[ +y^{3} + z^{3} + (3yz + p)(y + z) + q = 0. +\] +Putting the factor multiplying $y + z$ equal to zero,---which +is permissible owing to the two undetermined +quantities involved,---we shall have the two equations +\[ +3yz + p = 0\Typo{.}{} +\] +and +\[ +y^{3} + z^{3} + q = 0\Typo{.}{,} +\] +from which $y$~and~$z$ can be determined. The means +which most naturally suggests itself to this end is to +take from the first equation the value of~$z$, +\[ +z = -\frac{p}{3y}, +\] +and to substitute it in the second equation, removing +the fractions by multiplication. So proceeding, we +\PageSep{67} +obtain the following equation of the sixth degree in~$y$, +called \emph{the reduced equation}, +\MNote{The reduced equation.} +\[ +y^{6} + qy^{3} - \frac{p^{3}}{27} = 0, +\] +which, since it contains two powers only of the unknown +quantity, of which one is the square of the +other, is resolvable after the manner of equations of +the second degree and gives immediately +\[ +y^{3} = -\frac{q}{2} + \sqrt{\frac{q^{2}}{4} + \frac{p^{3}}{27}}, +\] +from which, by extracting the cube root, we get +\[ +y = \sqrt[3]{-\frac{q}{2} + \sqrt{\frac{q^{2}}{4} + \frac{p^{3}}{27}}}, +\] +and finally, +\[ +x = y + z = y - \frac{p}{3y}\Add{.} +\] +This expression for~$x$ may be simplified by remarking +that the product of~$y$ by the radical +\[ + \sqrt[3]{-\frac{q}{2} - \sqrt{\frac{q^{2}}{4} + \frac{p^{3}}{27}}}\Add{,} +\] +supposing all the quantities under the sign to be multiplied +together, is +\[ +\sqrt[3]{-\frac{p^{3}}{27}} = -\frac{p}{3}. +\] +The term $\dfrac{p}{3y}$, accordingly, takes the form +\[ +-\sqrt[3]{-\frac{q}{2} - \sqrt{\frac{q^{2}}{4} + \frac{p^{3}}{27}}}, +\] +and we have +\[ +x = \sqrt[3]{-\frac{q}{2} + \sqrt{\frac{q^{2}}{4} + \frac{p^{3}}{27}}} + + \sqrt[3]{-\frac{q}{2} - \sqrt{\frac{q^{2}}{4} + \frac{p^{3}}{27}}}, +\] +\PageSep{68} +an expression in which the square root underneath the +cubic radical occurs in both its plus and minus forms +and where consequently there can, on this score, be +no occasion for ambiguity. + +This last expression is known as the \emph{Rule of Cardan}, +\index{Cardan}% +\index{Rule!Cardan's}% +\MNote{Cardan's rule.} +and there has hitherto been no method devised +for the resolution of equations of the third degree +which does not lead to it. Since cubic radicals naturally +present but a single value, it was long thought +that Cardan's rule could give but one of the roots of +the equation, and that in order to find the two others +we must have recourse to the original equation and divide +it by~$x - a$, $a$~being the first root found. The +resulting quotient being an equation of the second degree +may be resolved in the usual manner. The division +in question is not only always possible, but it is +also very easy to perform. For in the case we are +considering the equation being +\[ +x^{3} + px + q = 0, +\] +if $a$~is one of the roots we shall have +\[ +a^{3} + pa + q = 0, +\] +which subtracted from the preceding will give +\[ +x^{3} - a^{3} + p(x - a) = 0, +\] +a quantity divisible by~$x - a$ and having as its resulting +quotient +\[ +x^{2} + ax + a^{2} + p = 0; +\] +so that the new equation which is to be resolved for +finding the two other roots will be +\PageSep{69} +\[ +x^{2} + ax + a^{2} + p = 0, +\] +from which we have at once +\[ +x = -\frac{a}{2} ± \sqrt{-p - \frac{3a^{2}}{4}}. +\] + +I see by the \textit{Algebra} of Clairaut, printed in~1746, +\index{Clairaut}% +and by D'Alem\-bert's article on the \emph{Irreducible Case} in +\index{Irreducible case}% +\MNote{The generality of algebra.} +the first \textit{Encyclopædia} that the idea referred to prevailed +even in that period. But it would be the height +of injustice to algebra to accuse it of not yielding results +\index{Algebra!generality@the generality of}% +which were possessed of all the generality of +which the question was susceptible. The sole requisite +is to be able to read the peculiar hand-writing +\index{Algebra!hand-writing of}% +\index{Hand-writing of algebra}% +of algebra, and we shall then be able to see in it everything +which by its nature it can be made to contain. +In the case which we are considering it was forgotten +that every cube root may have three values, as every +square root has two. For the extraction of the cube +root of~$a$ for example is merely equivalent to the resolution +of the equation of the third degree $x^{3} - a = 0$. +Making $x = y\sqrt[3]{a}$, this last equation passes into the +simpler form $y^{3} - 1 = 0$, which has the root $y = 1$. +Then dividing by~$y - 1$ we have +\[ +y^{2} + y + 1 = 0, +\] +from which we deduce directly the two other roots +\[ +y = \frac{-1 ± \sqrt{-3}}{2}. +\] +These three roots, accordingly, are the three cube +roots of unity, and they may be made to give the three +cube roots of any other quantity~$a$ by multiplying +\PageSep{70} +them by the ordinary cube root of that quantity. It +is the same with roots of the fourth, the fifth, and all +the following degrees. For brevity, let us designate +the two roots +\MNote{The three cube roots of a quantity.} +\index{Cube roots of a quantity, the three}% +\[ +\frac{-1 + \sqrt{-3}}{2} \quad\text{and}\quad \frac{-1 - \sqrt{-3}}{2}\Typo{,}{} +\] +by $m$~and~$n$. It will be seen that they are imaginary, +although their cube is real and equal to~$1$, as we may +readily convince ourselves by raising them to the +third power. We have, therefore, for the three cube +roots of~$a$, +\[ +\sqrt[3]{a},\quad m\sqrt[3]{a},\quad n\sqrt[3]{a}. +\] + +Now, in the resolution of the equation of the third +degree above considered, on coming to the reduced +expression $y^{3} = A$, where for brevity we suppose +\[ +A = -\frac{q}{2} + \sqrt{\frac{q^{2}}{4} + \frac{p^{3}}{27}}, +\] +we deduced the following result only: +\[ +y = \sqrt[3]{A}. +\] +But from what we have just seen, it is clear that we +shall have not only +\[ +y = \sqrt[3]{A}, +\] +but also +\[ +y = m\sqrt[3]{A} \quad\text{and}\quad y = n\sqrt[3]{A}. +\] +The root~$x$ of the equation of the third degree which +we found equal to +\[ +y - \frac{p}{3y}, +\] +will therefore have the three following values +\PageSep{71} +\[ +\sqrt[3]{A} - \frac{p}{3\sqrt[3]{A}},\quad +m\sqrt[3]{A} - \frac{p}{3m\sqrt[3]{A}},\quad +n\sqrt[3]{A} - \frac{p}{3n\sqrt[3]{A}}, +\] +\MNote{The roots of equations of the third degree.} +\index{Roots!equations@of equations of the third degree}% +\index{Third degree, equations of the}% +which will be the three roots of the equation proposed. +But making +\[ +B = -\frac{q}{2} - \sqrt{\frac{q^{2}}{4} + \frac{p^{3}}{27}}, +\] +it is clear that +\[ +AB = -\frac{p^{3}}{27}, +\] +whence +\[ +\sqrt[3]{A} × \sqrt[3]{B} = -\frac{p}{3}. +\] +Substituting $\sqrt[3]{B}$ for $-\dfrac{p}{3\sqrt[3]{A}}$, and remarking that +$mn = 1$, and that consequently +\[ +\frac{1}{m} = n,\quad \frac{1}{n} = m, +\] +the three roots which we are considering will be expressed +as follows: +%[** TN: Set on two lines in the original] +\[ +x = \sqrt[3]{A} + \sqrt[3]{B},\quad +x = m\sqrt[3]{A} + n\sqrt[3]{B},\quad +x = n\sqrt[3]{A} + m\sqrt[3]{B}. +\] + +We see, accordingly, that when properly understood +the ordinary method gives the three roots directly, +and gives three only. I have deemed it necessary +to enter upon these slight details for the reason +that if on the one hand the method was long taxed +with being able to give but one root, on the other +hand when it was seen that it really gave three it was +thought that it should have given six, owing to the +\PageSep{72} +false employment of all the possible combinations of +the three cubic roots of unity, viz., $1$,~$m$,~$n$, with the +\index{Unity, three cubic roots of}% +two cubic radicals $\sqrt[3]{A}$~and~$\sqrt[3]{B}$. + +We could have arrived directly at the results which +\MNote{A direct method of reaching the roots.} +we have just found by remarking that the two equations +\[ +y^{3} + z^{3} + q = 0 \quad\text{and}\quad 3yz + p = 0 +\] +give +\[ +y^{3} + z^{3} = -q \quad\text{and}\quad y^{3}z^{3} = -\frac{p^{3}}{27}; +\] +where it will be seen at once that $y^{3}$~and~$z^{3}$ are the +roots of an equation of the second degree of which +the second term is~$q$ and the third~$-\dfrac{p^{3}}{27}$. This equation, +which is called \emph{the reduced equation}, will accordingly +have the form +\[ +u^{2} + qu - \frac{p^{3}}{27} = 0; +\] +and calling $A$~and~$B$ its two roots we shall have immediately +\[ +y = \sqrt[3]{A},\quad z = \sqrt[3]{B}, +\] +where it will be observed that $A$~and~$B$ have the same +values that they had in the previous discussion. Now, +from what has gone before, we shall likewise have +\[ +y = m\sqrt[3]{A} \quad\text{or}\quad y = n\sqrt[3]{A}, +\] +and the same will also hold good for~$z$. But the equation +\[ +zy = -\frac{p}{3}, +\] +of which we have employed the cube only, limits these +\PageSep{73} +values and it is easy to see that the restriction requires +the three corresponding values of~$z$ to be +\[ +\sqrt[3]{B},\quad m\sqrt[3]{B},\quad n\sqrt[3]{B}; +\] +whence follow for the value of~$x$, which is equal to~$y + z$, +the same three values which we found above. + +As to the form of these values it is apparent, first, +that so long as $A$~and~$B$ are real quantities, one only +\MNote{The form of the roots\Add{.}} +of them can be real, for $m$~and~$n$ are imaginary. They +can consequently all three be real only in the case +where the roots $A$~and~$B$ of the reduced equation are +imaginary, that is, when the quantity +\[ +\frac{q^{2}}{4} + \frac{p^{3}}{27} +\] +beneath the radical sign is negative, which happens +only when $p$~is negative and greater than +\[ +3\sqrt[3]{\frac{q^{2}}{4}}. +\] +And this is the so-called \emph{irreducible case}. +\index{Irreducible case}% + +Since in this event +\[ +\frac{q^{2}}{4} + \frac{p^{3}}{27} +\] +is a negative quantity, let us suppose it equal to~$-g^{2}$, +$g$~being any real quantity whatever. Then making, +for the sake of simplicity, +\[ +-\frac{q}{2} = f, +\] +the two roots $A$~and~$B$ of the reduced equation assume +the form +\[ +A = f + g\sqrt{-1},\quad B = f - g\sqrt{-1}. +\] +\PageSep{74} + +Now I say that if $\sqrt[3]{A} + \sqrt[3]{B}$, which is one of the +\MNote{The reality of the roots\Add{.}} +\index{Roots!reality@the reality of the}% +roots of the equation of the third degree, is real, then +the two other roots, expressed by +\[ +m\sqrt[3]{A} + n\sqrt[3]{B} \quad\text{and}\quad n\sqrt[3]{A} + m\sqrt[3]{B}, +\] +will also be real. Put +\[ +\sqrt[3]{A} = t,\quad \sqrt[3]{B} = u; +\] +we shall have +\[ +t + u = h, +\] +where $h$~by hypothesis is a real quantity. Now, +\[ +tu = \sqrt[3]{AB} \quad\text{and}\quad AB = f^{2} + g^{2}, +\] +therefore +\[ +tu = \sqrt[3]{f^{2} + g^{2}}; +\] +squaring the equation $t + u = h$ we have +\[ +t^{2} + 2tu + u^{2} = h^{2}; +\] +from which subtracting~$4tu$ we obtain +\[ +(t - u)^{2} = h^{2} - 4\sqrt[3]{f^{2} + g^{2}}. +\] +I observe that this quantity must necessarily be negative, +for if it were positive and equal to~$k^{2}$ we should +have +\[ +(t - u)^{2} = k^{2}, +\] +whence +\[ +t - u = k. +\] +Then since +\[ +t + u = h, +\] +it would follow that +\[ +t = \frac{h + k}{2} \quad\text{and}\quad u = \frac{h - k}{2}, +\] +\PageSep{75} +both of which are real quantities. But then $t^{3}$~and~$u^{3}$ +would also be real quantities, which is contrary to +our hypothesis, since these quantities are equal to $A$~and~$B$, +both of which are imaginary. + +The quantity +\[ +h^{2} - 4\sqrt[3]{f^{2} + g^{2}} +\] +therefore, is necessarily negative. Let us suppose it +equal to~$-k^{2}$; we shall have then +\[ +(t - u)^{2} = -k^{2}, +\] +and extracting the square root +\[ +t - u = k\sqrt{-1}; +\] +\MNote{The form of the two cubic radicals.} +\index{Cubic radicals}% +\index{Radicals, cubic}% +whence +\[ +t = \frac{h + k\sqrt{-1}}{2} = \sqrt[3]{A},\quad +u = \frac{h - k\sqrt{-1}}{2} = \sqrt[3]{B}. +\] + +Such necessarily will be the form of the two cubic +radicals +\[ +\sqrt[3]{f + g\sqrt{-1}} \quad\text{and}\quad \sqrt[3]{f - g\sqrt{-1}}, +\] +a form at which we can arrive directly by expanding +these roots according to the Newtonian theorem into +series. But since proofs by series are apt to leave +some doubt in the mind, I have sought to render the +preceding discussion entirely independent of them. + +If, therefore, +\[ +\sqrt[3]{A} + \sqrt[3]{B} = h, +\] +we shall have +\[ +\sqrt[3]{A} = \frac{h + k\sqrt{-1}}{2} \quad\text{and}\quad +\sqrt[3]{B} = \frac{h - k\sqrt{-1}}{2}. +\] +Now we have found above that +\[ +m = \frac{-1 + \sqrt{-3}}{2},\quad n = \frac{-1 - \sqrt{-3}}{2}; +\] +\PageSep{76} +wherefore, multiplying these quantities together, we +have +\begin{align*} +m\sqrt[3]{A} + n\sqrt[3]{B} &= \frac{-h + k\sqrt{-3}}{2} \\ +\intertext{and} +n\sqrt[3]{A} + m\sqrt[3]{B} &= \frac{-h - k\sqrt{-3}}{2}, +\end{align*} +which are real quantities. Consequently, if the root~$h$ +\MNote{Condition of the reality of the roots.} +\index{Reality of roots}% +\index{Roots!reality@the reality of the}% +is real, the two other roots also will be real in the +irreducible case and they will be real in that case only. + +But the invariable difficulty is, to demonstrate directly +that +\[ +\sqrt[3]{f + g\sqrt{-1}} + \sqrt[3]{f - g\sqrt{-1}}, +\] +which we have supposed equal to~$h$, is always a real +quantity whatever be the values of $f$~and~$g$. In particular +cases the demonstration can be effected by the +extraction of the cube root, when that is possible. For +example, if $f = 2$, $g = 11$, we shall find that the cube +root of~$2 + 11\sqrt{-1}$ will be~$2 + \sqrt{-1}$, and similarly +that the cube root of~$2 - 11\sqrt{-1}$ will be~$2 - \sqrt{-1}$, +and the sum of the radicals will be~$4$. An infinite +number of examples of this class may be constructed +and it was through the consideration of such instances +that Bombelli became convinced of the reality of the +imaginary expression in the formula for the irreducible +case. But forasmuch as the extraction of cube roots +is in general possible only by means of series, we cannot +arrive in this way at a general and direct demonstration +of the proposition under consideration. +\PageSep{77} + +It is otherwise with square roots and with all roots +of which the exponents are powers of~$2$. For example, +\MNote{Extraction of the square roots of two imaginary binomials.} +\index{Binomials, extraction of the square roots of two imaginary}% +\index{Imaginary binomials, square roots of}% +if we have the expression +\[ +\sqrt{f + g\sqrt{-1}} + \sqrt{f - g\sqrt{-1}}, +\] +composed of two imaginary radicals, its square will be +\[ +2f + 2\sqrt{f^{2} + g^{2}}, +\] +a quantity which is necessarily positive. Extracting +the square root, so as to obtain the equivalent expression, +we have +\[ +\sqrt{2f + 2\sqrt{f^{2} + g^{2}}}, +\] +for the real value of the imaginary quantity we started +with. But if instead of the sum we had had the difference +between the two proposed imaginary radicals +we should then have obtained for its square the following +expression +\[ +2f - 2\sqrt{f^{2} + g^{2}}, +\] +a quantity which is necessarily negative; and, taking +the square root of the latter, we should have obtained +the simple imaginary expression +\[ +\sqrt{2f - 2\sqrt{f^{2} + g^{2}}}. +\] + +Further, if the quantity +\[ +\sqrt[4]{f + g\sqrt{-1}} + \sqrt[4]{f - g\sqrt{-1}} +\] +were given, we should have, by squaring, the form +\begin{multline*} +%[** TN: Moved equality sign to second line] +\sqrt{f + g\sqrt{-1}} + \sqrt{f - g\sqrt{-1}} + 2\sqrt[4]{f^{2} + g^{2}} \\ += \sqrt{2f + 2\sqrt{f^{2} + g^{2}}} + 2\sqrt[4]{f^{2} + g^{2}}, +\end{multline*} +a real and positive quantity. Extracting the square +\PageSep{78} +root of this expression we should obtain a real value +for the original quantity; and so on for all the other +remaining even roots. But if we should attempt to +apply the preceding method to cubic radicals we +should be led again to equations of the third degree +in the irreducible case. + +For example, let +\MNote{Extraction of the cube roots of two imaginary binomials.} +\[ +\sqrt[3]{f + g\sqrt{-1}} + \sqrt[3]{f - g\sqrt{-1}} = x. +\] +Cubing, we get +\[ +2f + 3\sqrt[3]{f^{2} + g^{2}}\left( +\sqrt[3]{f + g\sqrt{-1}} + \sqrt[3]{f - g\sqrt{-1}} +\right) = x^{3}; +\] +that is +\[ +2f + 3x\sqrt[3]{f^{2} + g^{2}} = x^{3}, +\] +or, with the terms properly arranged, +\[ +x^{3} - 3x\sqrt[3]{f^{2} + g^{2}} - 2f = 0, +\] +the general formula of the irreducible case, for +\[ +\frac{1}{4}(2f)^{2} + \frac{1}{27}\bigl(-3\sqrt[3]{f^{2} + g^{2}}\bigr)^{3} + = -g^{2}. +\] +If $g = 0$ we shall have $x = 2\sqrt[3]{f}$. The sole \textit{desideratum}, +therefore, is to demonstrate that if $g$~have any value +whatever, $x$~has a corresponding real value. Now the +second last equation gives +\[ +\sqrt[3]{f^{2} + g^{2}} = \frac{x^{3} - 2f}{3x}\Add{,} +\] +and cubing we get +\[ +f^{2} + g^{2} = \frac{x^{9} - 6x^{6}f + 12x^{3}f^{2} - 8f^{3}}{27x^{3}}, +\] +whence +\[ +g^{2} = \frac{x^{9} - 6x^{6}f - 15x^{3}f^{2} - 8f^{3}}{27x^{3}}, +\] +\PageSep{79} +an equation which may be written as follows +\[ +g^{2} = \frac{(x^{3} - 8f)(x^{3} + f)^{2}}{27x^{3}}, +\] +or, better, thus: +\[ +g^{2} = \frac{1}{27}\left(1 - \frac{8f}{x^{3}}\right)(x^{3} + f)^{2}. +\] + +It is plain from the last expression that $g$~is zero +when $x^{3} = 8f$; further, that $g$~constantly and uninterruptedly +\MNote{General theory of the reality of the roots\Add{.}} +\index{Roots!reality@the reality of the}% +increases as $x$~increases; for the factor +$(x^{3} + f)^{2}$ augments constantly, and the other factor +$1 - \dfrac{8f}{x^{3}}$ also keeps increasing, seeing that as the denominator~$x^{3}$ +increases the negative part~$\dfrac{8f}{x^{3}}$, which is +originally equal to~$1$, keeps constantly growing less +than~$1$. Therefore, if the value of~$x^{3}$ be increased by +insensible degrees from~$8f$ to infinity, the value of~$g^{2}$ +will also augment by insensible and corresponding +degrees from zero to infinity. And therefore, reciprocally, +to every value of~$g^{2}$ from zero to infinity there +must correspond some value of~$x^{3}$ lying between the +limits of~$8f$ and infinity, and since this is so whatever +be the value of~$f$ we may legitimately conclude that, +be the values of $f$~and~$g$ what they may, the corresponding +value of~$x^{3}$ and consequently also of~$x$ is +always real. + +But how is this value of~$x$ to be assigned? It would +\index{Imaginary expressions|EtSeq}% +seem that it can be represented only by an imaginary +expression or by a series which is the development of +an imaginary expression. Are we to regard this class +of imaginary expressions, which correspond to real +\PageSep{80} +values, as constituting a new species of algebraical expressions +which although they are not, like other expressions, +\MNote{Imaginary expressions\Add{.}} +susceptible of being numerically evaluated +in the form in which they exist, yet possess the indisputable +advantage---and this is the chief requisite---that +they can be employed in the operations of algebra +exactly as if they did not contain imaginary expressions. +They further enjoy the advantage of having a +wide range of usefulness in geometrical constructions, +as we shall see in the theory of angular sections, so +\index{Angular sections, theory of}% +that they can always be exactly represented by lines; +while as to their numerical value, we can always find +it approximately and to any degree of exactness that +we desire, by the approximate resolution of the equation +on which they depend, or by the use of the common +trigonometrical tables. + +It is demonstrated in geometry that if in a circle +having the radius~$r$ an arc be taken of which the chord +is~$c$, and that if the chord of the third part of that arc +be called~$x$, we shall have for the determination of~$x$ +the following equation of the third degree +\[ +x^{3} - 3r^{2}x + r^{2}c = 0, +\] +an equation which leads to the irreducible case since +$c$~is always necessarily less than~$2r$, and which, owing +to the two undetermined quantities $r$~and~$c$, may be +taken as the type of all equations of this class. For, +if we compare it with the general equation +\[ +x^{3} + px + q = 0, +\] +we shall have +\PageSep{81} +\[ +r = \sqrt{-\frac{p}{3}} \quad\text{and}\quad c = -\frac{3q}{p} +\] +so that by trisecting the arc corresponding to the +chord~$c$ in a circle of the radius~$r$ we shall obtain at +\MNote{Trisection of an angle.} +\index{Angle, trisection of an}% +\index{Trisection of an angle}% +once the value of a root~$x$, which will be the chord of +the third part of that arc. Now, from the nature of a +circle the same chord~$c$ corresponds not only to the +arc~$s$ but (calling the entire circumference~$u$) also to +the arcs +\[ +u - s,\quad 2u + s,\quad 3u - s, \dots\Add{.} +\] +Also the arcs +\[ +u + s,\quad 2u - s,\quad 3u + s, \dots +\] +have the same chord, but taken negatively, for on +completing a full circumference the chords become +zero and then negative, and they do not become positive +again until the completion of the second circumference, +as you may readily see. Therefore, the values +of~$x$ are not only the chord of the arc~$\dfrac{s}{3}$ but also +the chords of the arcs +\[ +\frac{u - s}{3},\quad \frac{2u + s}{3}, +\] +and these chords will be the three roots of the equation +proposed. If we were to take the succeeding arcs +which have the same chord~$c$ we should be led simply +to the same roots, for the arc~$3u - s$ would give the +chord of~$\dfrac{3u - s}{3}$, that is, of~$u - \dfrac{s}{3}$, which we have already +seen is the same as that of~$\dfrac{s}{3}$, and so with the +rest. +\PageSep{82} + +Since in the irreducible case the coefficient~$p$ is +\index{Irreducible case}% +necessarily negative, the value of the given chord~$c$ +\MNote{Trigonometrical solution.} +will be positive or negative according as $q$~is positive +or negative. In the first case, we take for~$s$ the arc +subtended by the positive chord $c = -\dfrac{3q}{p}$. The second +case is reducible to the first by making $x$~negative, +whereby the sign of the last term is changed; so +that if again we take for~$s$ an arc subtended by the +positive chord~$\dfrac{3q}{p}$, we shall have simply to change +the sign of the three roots. + +Although the preceding discussion may be deemed +sufficient to dispel all doubts concerning the nature +of the roots of equations of the third degree, we propose +\index{Equations!third@of the third degree}% +\index{Third degree, equations of the}% +adding to it a few reflexions concerning the +method by which the roots are found. The method +which we have propounded in the foregoing and which +is commonly called \emph{Cardan's method}, although it seems +\index{Cardan}% +to me that we owe it to Hudde, has been frequently +\index{Hudde}% +criticised, and will doubtless always be criticised, for +giving the roots in the irreducible case in an imaginary +form, solely because a supposition is here made which +is contradictory to the nature of the equation. For +the very gist of the method consists in its supposing +\index{Undetermined quantities}% +the unknown quantity equal to two undetermined +quantities $y + z$, in order to enable us afterwards to +separate the resulting equation +\[ +y^{3} + z^{3} + (3yz + p)(y + z) + q = 0 +\] +into the two following: +\PageSep{83} +\[ +3yz + p = 0 \quad\text{and}\quad y^{3} + z^{3} + q = 0. +\] +Now, throwing the first of these into the form +\MNote{The method of indeterminates.} +\index{Indeterminates, the method of}% +\[ +y^{3}z^{3} = -\frac{p^{3}}{27} +\] +it is plain that the question reduces itself to finding +two numbers $y^{3}$~and~$z^{3}$ of which the sum is~$-q$ and +the product~$-\dfrac{p^{3}}{27}$, which is impossible unless the +square of half the sum exceed the product, for the +difference between these two quantities is equal to the +square of half the difference of the numbers sought. + +The natural conclusion was that it was not at all +astonishing that we should reach imaginary expressions +\index{Imaginary expressions}% +when proceeding from a supposition which it +was impossible to express in numbers, and so some +writers have been induced to believe that by adopting +a different course the expression in question could be +avoided and the roots all obtained in their real form. +\index{Reality of roots}% +\index{Roots!reality@the reality of the}% + +Since pretty much the same objection can be advanced +against the other methods which have since +been found and which are all more or less based upon +the method of indeterminates, that is, the introduction +of certain arbitrary quantities to be determined +so as to satisfy the conditions of the problem,---we +propose to consider the question of the reality of the +roots by itself and independently of any supposition +whatever. Let us take again the equation +\[ +x^{3} + px + q = 0; +\] +and let us suppose that its three roots are $a$,~$b$,~$c$. +\PageSep{84} + +By the theory of equations the left-hand side of +\index{Equations!theory of}% +the preceding expression is the product of three quantities +\MNote{An independent consideration.} +\[ +x - a,\quad x - b,\quad x - c, +\] +which, multiplied together, give +\[ +x^{3} - (a + b + c)x^{2} + (ab + ac + bc)x - abc; +\] +and comparing the corresponding terms, we have +\[ +a + b + c = 0,\quad +ab + ac + bc = p,\quad +abc = -q. +\] +As the degree of the equation is odd we may be certain, +as you doubtless already know and in any event +will clearly see from the lecture which is to follow, +that it has necessarily one real root. Let that root +be~$c$. The first of the three equations which we have +just found will then give +\[ +c = -a - b, +\] +whence it is plain that $a + b$ is also necessarily a real +quantity. Substituting the last value of~$c$ in the second +and third equations, we have +\[ +ab - a^{2} - ab - ab - b^{2} = p,\quad -ab(a + b) = -q, +\] +or +\[ +a^{2} + ab + b^{2} = -p,\quad ab(a + b) = q, +\] +from which are to be found $a$~and~$b$. The last equation +gives $ab = \dfrac{q}{a + b}$ from which I conclude that $ab$ +also is necessarily a real quantity. Let us consider +now the quantity $\dfrac{q^{2}}{4} + \dfrac{p^{3}}{27}$ or, clearing of fractions, the +quantity $27q^{2} + 4p^{3}$, upon the sign of which the irreducible +case depends. Substituting in this for $p$~and~$q$ +their value as given above in terms of $a$~and~$b$, +\PageSep{85} +we shall find that when the necessary reductions are +made the quantity in question is equal to the square of +\MNote{New view of the reality of the roots.} +\index{Reality of roots}% +\index{Roots!reality@the reality of the}% +\[ +2a^{3} - 2b^{3} + 3a^{2}b - 3ab^{2} +\] +taken negatively; so that by changing the signs and +extracting the square root we shall have +\[ +2a^{3} - 2b^{3} + 3a^{2}b - 3ab^{2} = \sqrt{-27q^{2} - 4p^{3}}, +\] +whence it is easy to infer that the two roots $a$~and~$b$ +cannot be real unless the quantity $27q^{2} + 4p^{3}$ be negative. +But I shall show that in that case, which is as +we know the irreducible case, the two roots $a$~and~$b$ +are necessarily real. The quantity +\[ +2a^{3} - 2b^{3} + 3a^{2}b - 3ab^{2} +\] +may be reduced to the form +\[ +(a - b)(2a^{2} + 2b^{2} + 5ab), +\] +as multiplication will show. Now, we have already +seen that the two quantities $a + b$ and $ab$ are necessarily +real, whence it follows that +\[ +2a^{2} + 2b^{2} + 5ab = 2(a + b)^2 + ab +\] +is also necessarily real. Hence the other factor~$a - b$ +is also real when the radical $\sqrt{-27q^{2} - 4p^{3}}$ is real. +Therefore $a + b$ and $a - b$ being real quantities, it follows +that $a$~and~$b$ are real. + +We have already derived the preceding theorems +from the form of the roots themselves. But the present +demonstration is in some respects more general +and more direct, being deduced from the fundamental +principles of the problem itself. We have made no +\PageSep{86} +suppositions, and the particular nature of the irreducible +case has introduced no imaginary quantities. + +\MNote{Final solution on the new view.} +But the values of $a$~and~$b$ still remain to be found +from the preceding equations. And to this end I observe +that the left-hand side of the equation +\[ +a^{3} - b^{3} + \frac{3}{2}(a^{2}b - ab^{2}) + = \frac{1}{2}\sqrt{-27q^{2} - 4p^{3}} +\] +can be made a perfect cube by adding the left-hand +side of the equation +\[ +ab(a + b) = q, +\] +multiplied by $\dfrac{3\sqrt{-3}}{2}$, and that the root of this cube is +\[ +\frac{1 - \sqrt{-3}}{2}b - \frac{1 + \sqrt{-3}}{2}a +\] +so that, extracting the cube root of both sides, we +shall have the expression +\[ +\frac{1 - \sqrt{-3}}{2}b - \frac{1 + \sqrt{-3}}{2}a +\] +expressed in known quantities. And since the radical +$\sqrt{-3}$ may also be taken negatively, we shall also +have the expression +\[ +\frac{1 + \sqrt{-3}}{2}b - \frac{1 - \sqrt{-3}}{2}a +\] +expressed in known quantities, from which the values +of $a$~and~$b$ can be deduced. And these values will +contain the imaginary quantity~$\sqrt{-3}$, which was introduced +by multiplication, and will be reducible to +the same form with the two roots +\PageSep{87} +\[ +m\sqrt[3]{A} + n\sqrt[3]{B} \quad\text{and}\quad n\sqrt[3]{A} + m\sqrt[3]{B}, +\] +which we found above. The third root +\MNote{Office of imaginary quantities.} +\[ +c = -a - b +\] +will then be expressed by $\sqrt[3]{A} + \sqrt[3]{B}$. + +By this method we see that the imaginary quantities +\index{Imaginary quantities, office of the}% +employed have simply served to facilitate the extraction +of the cube root without which we could not +determine separately the values of $a$~and~$b$. And since +it is apparently impossible to attain this object by a +different method, we may regard it as a demonstrated +truth that the general expression of the roots of an +equation of the third degree in the irreducible case +cannot be rendered independent of imaginary quantities. + +Let us now pass to \emph{equations of the fourth degree}. +\index{Equations!fourth@of the fourth degree}% +We have already said that the artifice which was originally +employed for resolving these equations consisted +in so arranging them that the square root of +the two sides could be extracted, by which they were +reduced to equations of the second degree. The following +is the procedure employed. Let +\[ +x^{4} + px^{2} + qx + r = 0 +\] +be the general equation of the fourth degree deprived +of its second term, which can always be eliminated, +as you know, by increasing or diminishing the roots +by a suitable quantity. Let the equation be put in +the form +\[ +x^{4} = -px^{2} - qx - r, +\] +\PageSep{88} +and to each side let there be added the terms $2x^{2}y + y^{2}$, +which contain a new undetermined quantity~$y$ but +\MNote{Biquadratic equations.} +\index{Biquadratic equations}% +\index{Equations!biquadratic}% +which still leave the left-hand side of the equation a +square. We shall then have +\[ +(x^{2} + y)^{2} = (2y - p)x^{2} - qx + y^{2} - r. +\] +We must now make the right-hand side also a square. +To this end it is necessary that +\[ +4(2y - p)(y^{2} - r) = q^{2}, +\] +in which case the square root of the right-hand side +will have the form +\[ +x\sqrt{2y - p} - \frac{q}{2\sqrt{2y - p}}. +\] +Supposing then that the quantity~$y$ satisfies the equation +\[ +4(2y - p)(y^{2} - r) = q^{2}, +\] +which developed becomes +\[ +y^{3} - \frac{py^{2}}{2} - ry + \frac{pr}{2} - \frac{q^{2}}{8} = 0, +\] +and which, as we see, is an equation of the third degree, +the equation originally given may be reduced to +the following by extracting the square root of its two +members,~viz.: +\[ +x^{2} + y = x\sqrt{2y - p} - \frac{q}{2\sqrt{2y - p}}, +\] +where we may take either the plus or the positive +value for the radical $\sqrt{2y - p}$, and shall consequently +have two equations of the second degree to which the +given equation has been reduced and the roots of +which will give the four roots of the original equation. +\PageSep{89} +All of which furnishes us with our first instance of the +decomposition of equations into others of lower degree. + +The method of Descartes which is commonly followed +\index{Descartes}% +in the elements of algebra is based upon the +\MNote{The method of Descartes.} +same principle and consists in assuming at the outset +that the proposed equation is produced by the multiplication +of two equations of the second degree, as +\[ +x^{2} - ux + s = 0 \quad\text{and}\quad x^{2} + ux + t = 0, +\] +where $u$,~$s$, and~$t$ are indeterminate coefficients. Multiplying +\index{Coefficients!indeterminate}% +\index{Indeterminate coefficients}% +them together we have +\[ +x^{4} + (s + t - u^{2})x + (s - t)ux + st = 0, +\] +comparison of which with the original equation gives +\[ +s + t - u^{2} = p,\quad (s - t)u = q \quad\text{and}\quad st = r. +\] +The first two equations give +\[ +2s = p + u^{2} + \frac{q}{u},\quad 2t = p + u^{2} - \frac{q}{u}. +\] +And if these values be substituted in the third equation +of condition $st = r$, we shall have an equation of +the sixth degree in~$u$, which owing to its containing +only even powers of~$u$ is resolvable by the rules for +cubic equations. And if we substitute in this equation +$2y - p$ for~$u^{2}$, we shall obtain in~$y$ the same reduced +equation that we found above by the old method. + +Having the value of~$u^{2}$ we have also the values of +$s$~and~$t$, and our equation of the fourth degree will be +decomposed into two equations of the second degree +which will give the four roots sought. This method, +as well as the preceding, has been the occasion of some +\PageSep{90} +hesitancy as to which of the three roots of the reduced +cubic equation in $u^{2}$ or~$y$ should be employed. +\MNote{The determined character of the roots\Add{.}} +The difficulty has been well resolved in Clairaut's +\index{Clairaut}% +\textit{Algebra}, where we are led to see directly that we always +obtain the same four roots or values of~$x$ whatever +root of the reduced equation we employ. But +this generality is needless and prejudicial to the simplicity +which is to be desired in the expression of +the roots of the proposed equation, and we should +prefer the formulæ which you have learned in the +principal course and in which the three roots of the +reduced equation are contained in exactly the same +manner. + +The following is another method of reaching the +same formulæ, less direct than that which has already +been expounded to you, but which, on the other hand +has the advantage of being analogous to the method +of Cardan for equations of the third degree. +\index{Cardan}% + +I take up again the equation +\[ +x^{4} + px^{2} + qx + r = 0, +\] +and I suppose +\[ +x = y + z + t. +\] +Squaring I obtain +\[ +x^{2} = y^{2} + z^{2} + t^{2} + 2(yz + yt + zt). +\] +Squaring again I have +\[ +%[** TN: Set on two lines in original] +x^{4} = (y^{2} + z^{2} + t^{2})^{2} + 4(y^{2} + z^{2} + t^{2})(yz + yt + zt) ++ 4(yz + yt + zt)^{2}; +\] +but +\begin{align*} +%[** TN: Re-broken] +(yz + yt + zt)^{2} + &= y^{2}z^{2} + y^{2}t^{2} + z^{2}t^{2} + + 2y^{2}zt + 2yz^{2}t + 2yzt^{2} \\ + &= y^{2}z^{2} + y^{2}t^{2} + z^{2}t^{2} + 2yzt(y + z + t). +\end{align*} +\PageSep{91} +Substituting these three values of $x$,~$x^{2}$, and~$x^{4}$ in the +original equation, and bringing together the terms +multiplied by~$y + z + t$ and the terms multiplied by~$yz + yt + zt$, +\MNote{A third method.} +I have the transformed equation +\begin{gather*} +%[** TN: Re-broken] +(y^{2} + z^{2} + t^{2})^{2} + p(y^{2} + z^{2} + t^{2}) \\ + + \bigl[4(y^{2} + z^{2} + t^{2}) + 2p\bigr](yz + yt + zt) \\ + + 4(y^{2}z^{2} + y^{2}t^{2} + z^{2}t^{2}) + + (8yzt + q)(y + z + t) + r = 0. +\end{gather*} +We now proceed as we did with equations of the third +degree, where we caused the terms containing $y + z$ +to vanish, and in the same manner cause here the +terms containing $y + z + t$ and $yz + yt + zt$ to disappear, +which will give us the two equations of condition +\[ +8yzt + q = 0 \quad\text{and}\quad 4(y^{2} + z^{2} + t^{2}) + 2p = 0. +\] + +There remains the equation +\[ +(y^{2} + z^{2} + t^{2})^{2} + p(y^{2} + z^{2} + t^{2}) + + 4(y^{2}z^{2} + y^{2}t^{2} + z^{2}t^{2}) + r = 0; +\] +and the three together will determine the quantities +$y$,~$z$, and~$t$. The second gives immediately +\[ +y^{2} + z^{2} + t^{2} = -\frac{p}{2}, +\] +which substituted in the third gives +\[ +y^{2}z^{2} + y^{2}t^{2} + z^{2}t^{2} = \frac{p^{2}}{16} - \frac{r}{4}\Add{.} +\] +The first, raised to its square, gives +\[ +y^{2}z^{2}t^{2} = \frac{q^{2}}{64}. +\] +Hence, by the general theory of equations the three +\PageSep{92} +quantities $y^{2}$,~$z^{2}$,~$t^{2}$ will be the roots of an equation of +the third degree having the form +\MNote{The reduced equation.} +\[ +u^{3} + \frac{p}{2} u^{2} + + \left(\frac{p^{2}}{16} - \frac{r}{4}\right)u + - \frac{q^{2}}{64} = 0; +\] +so that if the three roots of this equation, which we +will call \emph{the reduced equation}, be designated by $a$,~$b$,~$c$, +we shall have +\[ +y = \sqrta,\quad z = \sqrt{b},\quad t = \sqrtc, +\] +and the value of~$x$ will be expressed by +\[ +\sqrta + \sqrt{b} + \sqrtc. +\] +Since the three radicals may each be taken with the +plus sign or the minus sign, we should have, if all +possible combinations were taken, eight different values +for~$x$. It is to be observed, however, that in the +preceding analysis we employed the equation $y^{2}z^{2}t^{2} = \dfrac{q^{2}}{64}$, +whereas the equation immediately given is $yzt = -\dfrac{q}{8}$. +Hence the product of the three quantities $y$,~$z$,~$t$, +that is to say of the three radicals +\[ +\sqrta,\quad \sqrt{b}, \quad \sqrtc, +\] +must have the contrary sign to that of the quantity~$q$. +Therefore, if $q$~be a negative quantity, either three +positive radicals or one positive and two negative radicals +must be contained in the expression for~$x$. And +in this case we shall have the following four combinations +only: +\begin{alignat*}{2} + &\sqrta + \sqrt{b} + \sqrtc,\qquad && \sqrta - \sqrt{b} - \sqrtc,\\ +-&\sqrta + \sqrt{b} - \sqrtc, &\Typo{}{-}&\sqrta - \sqrt{b} + \sqrtc, +\end{alignat*} +\PageSep{93} +which will be the four roots of the proposed equation +of the fourth degree. But if $q$~be a positive quantity, +either three negative radicals or one negative and two +\MNote{Euler's formulæ.} +positive radicals must be contained in the expression +for~$x$, which will give the following four other combinations +as the roots of the proposed equation:\footnote + {These simple and elegant formulæ are due to Euler. But M.~Bret, Professor + \index{Bret, M.|FN}% + \index{Euler}% + of Mathematics at Grenoble, has made the important observation (see + the \textit{Correspondance sur l'\Typo{Ecole}{École} Polytechnique}, t.~II., 3\ieme~Cahier, p.~217) that + they can give false values when imaginary quantities occur among the four + roots. + + In order to remove all difficulty and ambiguity we have only to substitute + for one of these radicals its value as derived from the equation $\sqrta\sqrt{b}\sqrtc = -\dfrac{q}{8}$. + Then the formula + \[ + \sqrta + \sqrt{b} - \frac{q}{8\sqrta\sqrt{b}} + \] + will give the four roots of the original equation by taking for $a$~and~$b$ any two + of the three roots of the reduced equation, and by taking the two radicals + successively positive and negative. + + The preceding remark should be added to article~777 of Euler's \textit{Algebra} + and to article~37 of the author's Note~XIII of the \textit{Traité de la résolution des + équations numériques}.} +\begin{alignat*}{2} +-&\sqrta - \sqrt{b} - \sqrtc,\qquad & -&\sqrta + \sqrt{b} + \sqrtc, \displaybreak[1] \\ + &\sqrta - \sqrt{b} + \sqrtc, &&\sqrta + \sqrt{b} - \sqrtc. +\end{alignat*} + +Now if the three roots $a$,~$b$,~$c$ of the reduced equation +\index{Reality of roots}% +\index{Roots!reality@the reality of the}% +\index{Three roots, reality of the}% +of the third degree are all real and positive, it is +evident that the four preceding roots will also all be +real. But if among the three real roots $a$,~$b$,~$c$, any +are negative, obviously the four roots of the given +biquadratic equation will be imaginary. Hence, besides +the condition for the reality of the three roots of +the reduced equation it is also requisite in the first +case, agreeably to the well-known rule of Descartes, +\index{Descartes}% +\PageSep{94} +that the coefficients of the terms of the reduced equation +should be alternatively positive and negative, and +\MNote{Roots of a biquadratic equation.} +\index{Biquadratic equations}% +\index{Roots!biquadratic@of a biquadratic equation}% +consequently that $p$~should be negative and $\dfrac{p^{2}}{16} - \dfrac{r}{4}$ +positive, that is, $p^{2} > 4r$. If one of these conditions +is not realised the proposed biquadratic equation cannot +have four real roots. If the reduced equation have +but one real root, it will be observed, first, that by +reason of its last term being negative the one real root +of the equation must necessarily be positive. It is +then easy to see from the general expressions which +we gave for the roots of cubic equations deprived of +their second term,---a form to which the reduced equation +in~$u$ can easily be brought by simply increasing +all the roots by the quantity~$\dfrac{p}{6}$,---it is easy to see, I +say, that the two imaginary roots of this equation will +be of the form +\[ +f + g\sqrt{-1} \quad\text{and}\quad f - g\sqrt{-1}. +\] +Therefore, supposing $a$~to be the real root and $b$,~$c$ the +two imaginary roots, $\sqrta$~will be a real quantity and +$\sqrt{b} + \sqrtc$ will also be real for reasons which we have +given above; while $\sqrt{b} - \sqrtc$ on the other hand will +be imaginary. Whence it follows that of the four +roots of the proposed biquadratic equation, the two +first will be real and the two others will be imaginary. + +As for the rest, if we make $u = s - \dfrac{p}{6}$ in the reduced +equation in~$u$, so as to eliminate the second +term and to reduce it to the form which we have above +\PageSep{95} +examined, we shall have the following transformed +equation in~$s$: +\[ +s^{3} - \left(\frac{p^{2}}{48} + \frac{r}{4}\right)s + - \frac{p^{3}}{864} + \frac{pr}{24} - \frac{q^{2}}{64} = 0; +\] +and the condition for the reality of the three roots of +the reduced equation will be +\[ +4\left(\frac{p^{2}}{48} + \frac{r}{4}\right)^{3} + > 27\left(\frac{p^{3}}{864} - \frac{pr}{24} + \frac{q^{2}}{64}\right)^{2}. +\] +\PageSep{96}%XXXX + + +\Lecture{IV.}{On the Resolution of Numerical Equations.} +\index{Numerical equations!resolution of|(}% + +\First{We} have seen how equations of the second, the +third, and the fourth degree can be resolved. +\MNote{Limits of the algebraical resolution of equations.} +\index{Algebraical resolution of equations!limits of the}% +\index{Equations!limits of the algebraical resolution of}% +The fifth degree constitutes a sort of barrier to analysts, +\index{Equations!fifth@of the fifth degree}% +\index{Fifth degree, equations of the}% +which by their greatest efforts they have never +yet been able to surmount, and the general resolution +of equations is one of the things that are still to be +desired in algebra. I say in algebra, for if with the +third degree the analytical expression of the roots is +insufficient for determining in all cases their numerical +value, \textit{a~fortiori} must it be so with equations of a +higher degree; and so we find ourselves constantly +under the necessity of having recourse to other means +for determining numerically the roots of a given equation,---for +to determine these roots is in the last resort +the object of the solution of all problems which +necessity or curiosity may offer. + +I propose here to set forth the principal artifices +which have been devised for accomplishing this important +object. Let us consider any equation of the +\index{Equations!mth@of the $m$th degree}% +$m$th~degree, represented by the formula +\PageSep{97} +\[ +x^{m} + px^{m-1} + qx^{m-2} + rx^{m-3} + \dots + u = 0, +\] +in which $x$~is the unknown quantity, $p$,~$q$,~$r$,~$\dots$ the +known positive or negative coefficients, and $u$~the +\MNote{Conditions of the resolution of numerical equations.} +\index{Numerical equations!conditions of the resolution of}% +last term, not containing~$x$ and consequently also a +known quantity. It is assumed that the values of +these coefficients are given either in numbers or in +lines; (it is indifferent which, seeing that by taking a +given line as the unit or common measure of the rest +we can assign to all the lines numerical values;) and it +is clear that this assumption is always permissible +when the equation is the result of a real and determinate +problem. The problem set us is to find the value, +or, if there be several, the values, of~$x$ which satisfy the +equation, i.e.\Add{,} which render the sum of all its terms +zero. Now any other value which may be given to~$x$ +will render that sum equal to some positive or negative +quantity, for since only integral powers of~$x$ enter +the equation, it is plain that every real value of~$x$ +will also give a real value for the quantity in question. +The more that value approaches to zero, the more +will the value of~$x$ which has produced it approach to +a root of the equation. And if we find two values of~$x$, +of which one renders the sum of the terms equal to +a positive quantity and the other to a negative quantity, +we may be assured in advance that between these +two values there will of necessity be at least one value +which will render the expression zero and will consequently +be a root of the equation. + +Let $P$~stand for the sum of all the terms of the +\PageSep{98} +equation having the sign~$+$ and $Q$~for the sum of all +the terms having the sign~$-$; then the equation will +be represented by +\[ +P - Q = 0. +\] +Let us suppose, for further simplicity, that the two +\MNote{Position of the roots of numerical equations.} +\index{Numerical equations!position of the roots of}% +values of~$x$ in question are positive, that $A$~is the +smaller, $B$~the greater, and that the substitution of~$A$ +for~$x$ gives a negative result and the substitution of~$B$ +for~$x$ a positive result; i.e., that the value of~$P - Q$ +is negative when $x = A$, and positive when $x = B$. + +Consequently, when $x = A$, $P$~will be less than~$Q$, +and when $x = B$, $P$~will be greater than~$Q$. Now, +from the very form of the quantities $P$~and~$Q$, which +contain only positive terms and whole positive powers +of~$x$, it is clear that these quantities augment continuously +as $x$~augments, and that by making $x$ augment by +insensible degrees through all values from $A$~to~$B$, they +also will augment by insensible degrees but in such +wise that $P$~will increase more than~$Q$, seeing that +from having been smaller than~$Q$ it will have become +greater. Therefore, there must of necessity be some +expression for the value of~$x$ between $A$~and~$B$ which +will make $P = Q$; just as two moving bodies which +\index{Moving bodies, two}% +we suppose to be travelling along the same straight +line and which having started simultaneously from +two different points arrive simultaneously at two other +points but in such wise that the body which was at first +in the rear is now in advance of the other,---just as +two such bodies, I say, must necessarily meet at some +\PageSep{99} +point in their path. That value of~$x$, therefore, which +will make $P = Q$ will be one of the roots of the equation, +and such a value will lie of necessity between $A$~and~$B$. + +The same reasoning may be employed for the +\MNote{Position of the roots of numerical equations.} +other cases, and always with the same result. + +The proposition in question is also demonstrable +by a direct consideration of the equation itself, which +may be regarded as made up of the product of the +factors, +\[ +x - a,\quad x - b,\quad x - c,\dots, +\] +where $a$,~$b$,~$c$,~$\dots$ are the roots. For it is obvious +that this product cannot, by the substitution of two +different values for~$x$, be made to change its sign, unless +at least one of the factors changes its sign. And +it is likewise easy to see that if more than one of the +factors changes its sign, their number must be odd. +Thus, if $A$~and~$B$ are two values of~$x$ for which the +factor $x - b$, for example, has opposite signs, then if +$A$~be larger than~$b$, necessarily $B$~must be smaller +than~$b$, or \textit{vice versa}. Perforce, then, the root~$b$ will +fall between the two quantities $A$~and~$B$. + +As for imaginary roots, if there be any in the equation, +\index{Imaginary roots, occur in pairs}% +since it has been demonstrated that they always +occur in pairs and are of the form +\[ +f + g\sqrt{-1},\quad f - g\sqrt{-1}, +\] +therefore if $a$~and~$b$ are imaginary, the product of the +factors $x - a$ and $x - b$ will be +\PageSep{100} +\[ +(x - f - g\sqrt{-1})(x - f + g\sqrt{-1}) = (x - f)^{2} + g^{2}, +\] +a quantity which is always positive whatever value be +given to~$x$. From this it follows that alterations in +the sign can be due only to real roots. But since the +theorem respecting the form of imaginary roots cannot +be rigorously demonstrated without employing the +other theorem that every equation of an odd degree +has necessarily one real root, a theorem of which the +general demonstration itself depends on the proposition +which we are concerned in proving, it follows +that that demonstration must be regarded as a sort of +vicious circle, and that it must be replaced by another +which is unassailable. + +But there is a more general and simpler method +\MNote{Application of geometry to algebra.} +\index{Algebra!application of geometry to|EtSeq}% +\index{Geometry!application of to algebra|EtSeq}% +of considering equations, which enjoys the advantage +\index{Equations!constructions for solving|EtSeq}% +of affording direct demonstration to the eye of the +principal properties of equations. It is founded upon +a species of application of geometry to algebra which +is the more deserving of exposition as it finds extended +employment in all branches of mathematics. + +Let us take up again the general equation proposed +above and let us represent by straight lines all +the successive values which are given to the unknown +quantity~$x$ and let us do the same for the corresponding +values which the left-hand side of the equation +assumes in this manner. To this end, instead of supposing +the right-hand side of the equation equal to +zero, we suppose it equal to an undetermined quantity~$y$. +We lay off the values of~$x$ upon an indefinite +\PageSep{101} +straight line~$AB$ (Fig.~1), starting from a fixed point~$O$ +at which $x$~is zero and taking the positive values of~$x$ +in the direction~$OB$ to the right of~$O$ and the negative +values of~$x$ in the opposite direction to the left of~$O$. +Then let~$OP$ be any value of~$x$. To represent +the corresponding value of~$y$ we erect at~$P$ a perpendicular +to the line~$OB$ and lay off on it the value of~$y$ +in the direction~$PQ$ above the straight line~$OB$ if it is +positive, and on the same perpendicular below~$OB$ if +it is negative. We do the same for all the values of~$x$, +\MNote{Representation of equations by curves.} +\index{Curves!representation of equations by|EtSeq}% +\Figure{1}{0.8\textwidth} +positive as well as negative; that is, we lay off +corresponding values of~$y$ upon perpendiculars to the +straight line through all the points whose distance +from the point~$O$ is equal to~$x$. The extremities of all +these perpendiculars will together form a straight line +or a curve, which will furnish, so to speak, a picture +of the equation +\[ +x^{m} + px^{m-1} + qx^{m-2} + \dots + u = y. +\] +The line~$AB$ is called the axis of the curve, $O$~the origin +of the abscissæ, $OP = x$ an abscissa, $PQ = y$ the corresponding +\PageSep{102} +ordinate, and the equations in $x$~and~$y$ the +\index{Equations!general remarks upon the roots of|EtSeq}% +equations of the curve. A curve such as that of Fig.~1 +having been described in the manner indicated, it is +clear that its intersections with the axis~$AB$ will give +the roots of the proposed equation +\MNote{Graphic resolution of equations.} +\index{Equations!graphic resolution of}% +\index{Intersections, with the axis give roots|EtSeq}% +\[ +x^{m} + px^{m-1} + qx^{m-2} + \dots + u = 0. +\] +For seeing that this equation is realised only when in +the equation of the curve $y$~becomes zero, therefore +those values of~$x$ which satisfy the equation in question +and which are its roots can only be the abscissæ +\ifthenelse{\not\boolean{ForPrinting}}{% +\Figure{1}{0.8\textwidth} %[** TN: [sic], figure repeated] +}{}% [Discard second copy if formatting for printing] +that correspond to the points at which the ordinates +are zero, that is, to the points at which the curve cuts +the axis~$AB$. Thus, supposing the curve of the equation +in $x$~and~$y$ is that represented in Fig.~1, the roots +of the proposed equation will be +\[ +OM,\quad ON,\quad OR,\dots \quad\text{and}\quad -OI,\quad -OG,\dots. +\] +I give the sign~$-$ to the latter because the intersections +$I$,~$G$,~$\dots$ fall on the other side of the point~$O$. +The consideration of the curve in question gives rise +to the following general remarks upon equations: +\PageSep{103} + +(1) Since the equation of the curve contains only +whole and positive powers of the unknown quantity~$x$ +it is clear that to every value of~$x$ there must correspond +\MNote{The consequences of the graphic resolution.} +a determinate value of~$y$, and that the value in +question will be unique and finite so long as $x$~is finite. +But since there is nothing to limit the values of~$x$ they +may be supposed infinitely great, positive as well as +negative, and to them will correspond also values of~$y$ +which are infinitely great. Whence it follows that +the curve will have a continuous and single course, +and that it may be extended to infinity on both sides +of the origin~$O$. + +(2) It also follows that the curve cannot pass from +one side of the axis to the other without cutting it, +and that it cannot return to the same side without +having cut it twice. Consequently, between any two +points of the curve on the same side of the axis there +will necessarily be either no intersections or an even +number of intersections; for example, between the +points $H$~and~$Q$ we find two intersections $I$~and~$M$, +and between the points $H$~and~$S$ we find four, $I$, $M$ +$N$, $R$, and so on. Contrariwise, between a point on +one side of the axis and a point on the other side, the +curve will have an odd number of intersections; for +example, between the points $L$~and~$Q$ there is one intersection~$M$, +and between the points $H$~and~$K$ there +are three intersections, $I$, $M$, $N$, and so on. + +For the same reason there can be no simple intersection +unless on both sides of the point of intersection, +\PageSep{104} +above and below the axis, points of the curve are +situated as are the points $L$,~$Q$ with respect to the intersection~$M$. +\MNote{Intersections indicate the roots.} +But two intersections, such as $N$~and~$R$, +may approach each other so as ultimately to coincide +at~$T$. Then the branch~$QKS$ will take the form +of the dotted line~$QTS$ and touch the axis at~$T$, and +will consequently lie in its whole extent above the +axis; this is the case in which the two roots $ON$,~$OR$ +are equal. If three intersections coincide at a point,---a +coincidence which occurs when there are three +equal roots,---then the curve will cut the axis in one +additional point only, as in the case of a single point +of intersection, and so on. + +Consequently, if we have found for~$y$ two values +having the same sign, we may be assured that between +the two corresponding values of~$x$ there can fall only +an even number of roots of the proposed equation; +that is, that there will be none or there will be two, or +there will be four, etc. On the other hand, if we have +found for~$y$ two values having contrary signs, we may +be assured that between the corresponding values of~$x$ +there will necessarily fall an odd number of roots of +the proposed equation; that is, there will be one, or +there will be three, or there will be five, etc.; so that, +in the case last mentioned, we may infer immediately +that there will be at least one root of the proposed +equation between the two values of~$x$. + +Conversely, every value of~$x$ which is a root of the +equation will be found between some larger and some +\PageSep{105} +smaller value of~$x$ which on being substituted for~$x$ in +the equation will yield values of~$y$ with contrary signs. + +This will not be the case, however, if the value of~$x$ +is a double root; that is, if the equation contains +\MNote{Case of multiple roots.} +\index{Multiple roots}% +\index{Roots!multiple}% +two roots of the same value. On the other hand, if +the value of~$x$ is a triple root, there will again exist +a larger and a smaller value for~$x$ which will give to +the corresponding values of~$y$ contrary signs, and so +on with the rest. + +If, now, we consider the equation of the curve, it +is plain in the first place, that by making $x = 0$ we +shall have $y = u$; and consequently that the sign of +the ordinate~$y$ will be the same as that of the quantity~$u$, +the last term of the proposed equation. It is also +easy to see that there can be given to~$x$ a positive or +negative value sufficiently great to make the first term~$x^{m}$ +of the equation exceed the sum of all the other +terms which have the opposite sign to~$x^{m}$; with the +result that the corresponding value of~$y$ will have the +same sign as the first term~$x^{m}$. Now, if $m$~is odd $x^{m}$~will +be positive or negative according as $x$~is positive +or negative, and if $m$~is even, $x^{m}$~will always be positive +whether $x$~be positive or not. + +Whence we may conclude: + +(1) That every equation of an odd degree of which +\index{Equations!odd@of an odd degree, roots of}% +the last term is negative has an odd number of roots +between $x = 0$ and some very large positive value of~$x$, +and an even number of roots between $x = 0$ and +some very large negative value of~$x$, and consequently +\PageSep{106} +that it has at least one real positive root. That, contrariwise, +if the last term of the equation is positive it +\MNote{General conclusions as to the character of the roots.} +will have an odd number of roots between $x = 0$ and +some very large negative value of~$x$, and an even +number of roots between $x = 0$ and some very large +positive value of~$x$, and consequently that it will have +at least one real negative root. + +(2) That every equation of an even degree, of +\index{Equations!even@of an even degree, roots of}% +which the last term is negative, has an odd number of +roots between $x = 0$ and some very large positive value +of~$x$, as well as an odd number of roots between $x = 0$ +and some very large negative value of~$x$, and consequently +that it has at least one real positive root and +one real negative root. That, on the other hand, if +the last term is positive there will be an even number +of roots between $x = 0$ and some very large positive +value of~$x$, and also an even number of roots between +$x = 0$ and some very large negative value of~$x$; with +the result that in this case the equation may have no +real root, whether positive or negative. + +We have said that there could always be given to~$x$ +a value sufficiently great to make the first term~$x^{m}$ of +the equation exceed the sum of all the terms of contrary +sign. Although this proposition is not in need +of demonstration, seeing that, since the power~$x^{m}$ is +higher than any of the other powers of~$x$ which enter +the equation, it is bound, as $x$~increases, to increase +much more rapidly than these other powers; nevertheless, +in order to leave no doubts in the mind, we +\PageSep{107} +shall offer a very simple demonstration of it,---a demonstration +which will enjoy the collateral advantage +of furnishing a limit beyond which we may be certain +no root of the equation can be found. + +To this end, let us first suppose that $x$~is positive, +\index{Limits of roots|(}% +and that $k$~is the greatest of the coefficients of the +\index{Coefficients!greatest negative|EtSeq}% +\MNote{Limits of the real roots of equations.} +\index{Equations!real roots of, limits of the|EtSeq}% +negative terms. If we make $x = k + 1$ we shall have +\[ +x^{m} = (k + 1)^{m} = k(k + 1)^{m-1} + (k + 1)^{m-1}. +\] +Similarly, +\begin{align*} +(k + 1)^{m-1} &= k(k + 1)^{m-2} + (k + 1)^{m-2}, \\ +(k + 1)^{m-2} &= k(k + 1)^{m-3} + (k + 1)^{m-3} +\end{align*} +and so on; so that we shall finally have +\[ +(k + 1)^{m} + = k(k + 1)^{m-1} + + k(k + 1)^{m-2} + + k(k + 1)^{m-3} + \dots + k + 1. +\] +Now this quantity is evidently greater than the sum +of all the negative terms of the equation taken positively, +on the supposition that $x = k + 1$. Therefore, +the supposition $x = k + 1$ necessarily renders the first +term~$x^{m}$ greater than the sum of all the negative terms. +Consequently, the value of~$y$ will have the same sign +as~$x$. + +The same reasoning and the same result hold good +when $x$~is negative. We have here merely to change~$x$ +into~$-x$ in the proposed equation, in order to change +the positive roots into negative roots, and \textit{vice versa}. + +In the same way it may be proved that if any value +be given to~$x$ greater than~$k + 1$, the value of~$y$ will +still have the same sign. From this and from what +has been developed above, it follows immediately that +\PageSep{108} +the equation can have no root equal to or greater than~$k + 1$. + +Therefore, in general, if $k$~is the greatest of the +\MNote{Limits of the positive and negative roots.} +coefficients of the negative terms of an equation, and +changing the unknown quantity~$x$ into~$-x$, $h$~is +the greatest of the coefficients of the negative terms +of the new equation,---the first term always being supposed +positive,---then all the real roots of the equation +will necessarily be comprised between the limits +\[ +k + 1 \quad\text{and}\quad -h - 1. +\] + +But if there are several positive terms in the equation +preceding the first negative term, we may take +for~$k$ a quantity less than the greatest negative coefficient. +In fact it is easy to see that the formula given +above can be put into the form +\[ +(k + 1)^{m} + = k(k + 1)(k + 1)^{m-2} + + k(k + 1)(k + 1)^{m-3} + \dots + (k + 1)^{2} +\] +and similarly into the following +\[ +(k + 1)^{m} + = k(k + 1)^{2}(k + 1)^{m-3} + + k(k + 1)^{2}(k + 1)^{m-4} + \dots + (k + 1)^{3} +\] +and so on. + +Whence it is easy to infer that if $m - n$ is the exponent +of the first negative term of the proposed equation +of the $m$th~degree, and if $l$~is the largest coefficient +of the negative terms, it will be sufficient if $k$~is +so determined that +\[ +k(k + 1)^{n-1} = l. +\] +And since we may take for~$k$ any larger value that we +please, it will be sufficient to take +\PageSep{109} +\[ +k^{n} = l,\quad\text{or}\quad k = \sqrt[n]{l}. +\] +And the same will hold good for the quantity~$h$ as the +limit of the negative roots. +\index{Positive roots, superior and inferior limits of the}% +\index{Roots!superior and inferior limits of the positive}% + +If, now, the unknown quantity~$x$ be changed into~$\dfrac{1}{z}$, +the largest roots of the equation in~$x$ will be converted +\MNote{Superior and inferior limits of the positive roots.} +into the smallest in the new equation in~$z$, and +conversely. Having effected this transformation, and +having so arranged the terms according to the powers +of~$z$ that the first term of the equation is~$z^{m}$, we may +then in the same manner seek for the limits $K + 1$ and +$-H - 1$ of the positive and negative roots of the +equation in~$z$. + +Thus $K + 1$ being larger than the largest value of~$z$ +or of~$\dfrac{1}{x}$, therefore, by the nature of fractions, $\dfrac{1}{K + 1}$ +will be smaller than the smallest value of~$x$ and similarly +$\dfrac{1}{H + 1}$ will be smaller than the smallest negative +value of~$x$. + +Whence it may be inferred that all the positive +real roots will necessarily be comprised between the +limits +\[ +\frac{1}{K + 1} \quad\text{and}\quad k + 1, +\] +and that the negative real roots will fall between the +limits +\[ +-\frac{1}{H + 1} \quad\text{and}\quad -h - 1. +\] + +There are methods for finding still closer limits; +but since they require considerable labor, the preceding +\PageSep{110} +method is, in the majority of cases, preferable, as +being more simple and convenient. + +For example, if in the proposed equation $l + z$ be +\MNote{A further method for finding the limits.} +\index{Roots!method for finding the limits of}% +substituted for~$x$, and if after having arranged the +terms according to the powers of~$z$, there be given to~$l$ +a value such that the coefficients of all the terms +become positive, it is plain that there will then be no +positive value of~$z$ that can satisfy the equation. The +equation will have negative roots only, and consequently +$l$~will be a quantity greater than the greatest +value of~$x$. Now it is easy to see that these coefficients +will be expressed as follows: +\begin{gather*} +%[** TN: Re-broken] +p + ml, \\ +q + (m - 1)pl + \frac{m(m - 1)}{2}\, l^{2}, \\ +r + (m - 2)ql + \frac{(m - 1)(m - 2)}{2}\, pl^{2} + + \frac{m(m - 1)(m - 2)}{2·3}\, l^{3}, +\end{gather*} +and so on. Accordingly, it is only necessary to seek +by trial the smallest value of~$l$ which will render them +all positive. + +But in the majority of cases it is not sufficient to +\index{Problems}% +know the limits of the roots of an equation; the thing +necessary is to know the values of those roots, at +least as approximately as the conditions of the problem +require. For every problem leads in its last analysis +to an equation which contains its solution; and +if it is not in our power to resolve this equation, all +\PageSep{111} +the pains expended upon its formulation are a sheer +loss. We may regard this point, therefore, as the +most important in all analysis, and for this reason I +\MNote{The real problem, the finding of the roots.} +have felt constrained to make it the principal subject +of the present lecture. + +From the principles established above regarding +\index{Substitutions|EtSeq}% +the nature of the curve of which the ordinates~$y$ represent +all the values which the left-hand side of an +equation assumes, it follows that if we possessed +some means of describing this curve we should obtain +at once, by its intersections with the axis, all the roots +of the proposed equation. But for this purpose it is +not necessary to have all of the curve; it is sufficient +to know the parts which lie immediately above and +below each point of intersection. Now it is possible +to find as many points of a curve as we please, and as +near to one another as we please by successively substituting +for~$x$ numbers which are very little different +from one another, but which are still near enough for +our purpose, and by taking for~$y$ the results of these +substitutions in the left-hand side of the equation. If +among the results of these substitutions two be found +having contrary signs, we may be certain, by the principles +established above, that there will be between +these two values of~$x$ at least one real root. We can +then by new substitutions bring these two limits still +closer together and approach as nearly as we wish to +the roots sought. + +Calling the smaller of the two values of~$x$ which +\PageSep{112} +have given results with contrary signs,~$A$, and the +larger~$B$, and supposing that we wish to find the +\MNote{Separation of the roots.} +\index{Roots!separation of the}% +\index{Roots!arithmetical@the arithmetical progression revealing the|EtSeq}% +value of the root within a degree of exactness denoted +by~$n$, where $n$~is a fraction of any degree of smallness +we please, we proceed to substitute successively for~$x$ +the following numbers in arithmetical progression: +\index{Arithmetical progression revealing the roots|EtSeq}% +\[ +A + n,\quad A + 2n,\quad A + 3n, \dots, +\] +or +\[ +B - n,\quad B - 2n,\quad B - 3n, \dots, +\] +until a result is reached having the contrary sign to +that obtained by the substitution of~$A$ or of~$B$. Then +one of the two successive values of~$x$ which have given +results with contrary signs will necessarily be larger +than the root sought, and the other smaller; and since +by hypothesis these values differ from one another +only by the quantity~$n$, it follows that each of them +approaches to within less than~$n$ of the root sought, +and that the error is therefore less than~$n$. + +But how are the initial values substituted for~$x$ to +be determined, so as on the one hand to avoid as +many useless trials as possible, and on the other to +make us confident that we have discovered by this +method all the real roots of this equation. If we examine +the curve of the equation it will be readily seen +that the question resolves itself into so selecting the +values of~$x$ that at least one of them shall fall between +two adjacent intersections, which will be necessarily +the case if the difference between two consecutive values +\PageSep{113} +is less than the smallest distance between two +adjacent intersections. + +Thus, supposing that $D$~is a quantity smaller than +the smallest distance between two intersections immediately +\MNote{To find a quantity less than the difference between any two roots.} +\index{Roots!quantity less than the difference between any two}% +following each other, we form the arithmetical +progression +\[ +0,\quad D,\quad 2D,\quad 3D,\quad 4D,\dots, +\] +and we select from this progression only the terms +which fall between the limits +\[ +\frac{1}{K + 1} \quad\text{and}\quad k + 1, +\] +as determined by the method already given. We obtain, +in this manner, values which on being substituted +for~$x$ ultimately give us all the positive roots of +the equation, and at the same time give the initial +limits of each root. In the same manner, for obtaining +the negative roots we form the progression +\[ +0,\quad -D,\quad -2D,\quad -3D,\quad -4D,\dots, +\] +from which we also take only the terms comprised +between the limits +\[ +-\frac{1}{H + 1} \quad\text{and}\quad -h - 1. +\] + +Thus this difficulty is resolved. But it still remains +to find the quantity~$D$,---that is, a quantity +smaller than the smallest interval between any two adjacent +intersections of the curve with the axis. Since +the abscissæ which correspond to the intersections are +\index{Intersections, with the axis give roots}% +the roots of the proposed equation, it is clear that the +question reduces itself to finding a quantity smaller +\PageSep{114} +than the smallest difference between two roots, neglecting +the signs. We have, therefore, to seek, by the +methods which were discussed in the lectures of the +principal course, the equation whose roots are the differences +between the roots of the proposed equation. +And we must then seek, by the methods expounded +above, a quantity smaller than the smallest root of +this last equation, and take that quantity for the value +of~$D$. + +This method, as we see, leaves nothing to be desired +\MNote{The equation of differences.} +\index{Differences, the equation of|EtSeq}% +as regards the rigorous solution of the problem, +but it labors under great disadvantage in requiring +extremely long calculations, especially if the proposed +equation is at all high in degree. For example, if $m$~is +the degree of the original equation, that of the equation +of differences will be~$m(m - 1)$, because each root +can be subtracted from all the remaining roots, the +number of which is~$m - 1$,---which gives $m(m - 1)$ +differences. But since each difference can be positive +or negative, it follows that the equation of differences +must have the same roots both in a positive and in a +negative form; that consequently the equation must +be wanting in all terms in which the unknown quantity +is raised to an odd power; so that by taking the +square of the differences as the unknown quantity, this +unknown quantity can occur only in the $\dfrac{m(m - 1)}{2}$th +degree. For an equation of the $m$th~degree, accordingly, +there is requisite at the start a transformed +\PageSep{115} +equation of the $\dfrac{m(m - 1)}{2}$th degree, which necessitates +an enormous amount of tedious labor, if $m$~is at all +large. For example, for an equation of the $10$th~degree, +\MNote{Impracticability of the method.} +the transformed equation would be of the~$45$th. +And since in the majority of cases this disadvantage +renders the method almost impracticable, it is of great +importance to find a means of remedying it. + +To this end let us resume the proposed equation of +the $m$th~degree, +\[ +x^{m} + px^{m-1} + qx^{m-2} + \dots + u = 0, +\] +of which the roots are $a$,~$b$,~$c$,~$\dots$. We shall have +then +\[ +a^{m} + pa^{m-1} + qa^{m-2} + \dots + u = 0 +\] +and also +\[ +b^{m} + pb^{m-1} + qb^{m-2} + \dots + u = 0. +\] +Let $b - a = i$. Substitute this value of~$b$ in the second +equation, and after developing the different powers of~$a + i$ +according to the well known binomial theorem, +\index{Binomial theorem}% +arrange the resulting equation according to the powers +of~$i$, beginning with the lowest. We shall have the +transformed equation +\[ +P + Qi + Ri^{2} + \dots + i^{m} = 0, +\] +in which the coefficients $P$,~$Q$,~$R$,~$\dots$ have the following +values +\begin{align*} +P &= a^{m} + pa^{m-1} + qa^{m-2} + \dots + u, \displaybreak[1] \\ +Q &= ma^{m-1} + (m - 1)pa^{m-2} + (m - 2)qa^{m-3} + \dots\Add{,} \displaybreak[1] \\ +\PageSep{116} +R &= \begin{aligned}[t] + \frac{m(m - 1)}{2}\, a^{m-2} + &+ \frac{(m - 1)(m - 2)}{2}\, pa^{m-3} \\ + &+ \frac{(m - 2)(m - 3)}{2}\, qa^{m-4} + \dots\Add{,} +\end{aligned} +\end{align*} +\MNote{Attempt to remedy the method.} +and so on. The law of formation of these expressions +is evident. + +Now, by the first equation in~$a$ we have~$P = 0$. +Rejecting, therefore, the term~$P$ of the equation in~$i$ +and dividing all the remaining terms by~$i$, the equation +in question will be reduced to the $(m - 1)$th~degree, +and will have the form +\[ +Q + Ri + Si^{2} + \dots + i^{m-1} = 0. +\] + +This equation will have for its roots the $m - 1$~differences +between the root~$a$ and the remaining roots +$b$,~$c$,~$\dots$\Add{.} Similarly, if $b$~be substituted for~$a$ in the expressions +for the coefficients $Q$,~$R$,~$\dots$, we shall obtain +an equation of which the roots are the difference +between the root~$b$ and the remaining roots $a$,~$c$,~$\dots$, +and so on. + +Accordingly, if a quantity can be found smaller +\index{Roots!smallest|EtSeq}% +than the smallest root of all these equations, it will +possess the property required and may be taken for +the quantity~$D$, the value of which we are seeking. + +If, by means of the equation $P = 0$, $a$~be eliminated +from the equation in~$i$, we shall get a new equation in~$i$ +which will contain all the other equations of which +we have just spoken, and of which it would only be +necessary to seek the smallest root. But this new +\PageSep{117} +equation in~$i$ is nothing else than the equation of differences +which we sought to dispense with. + +\MNote{Further improvement.} +In the above equation in~$i$ let us put it $i = \dfrac{1}{z}$. We +shall have then the transformed equation in~$z$, +\[ +z^{m-1} + + \frac{R}{Q}\, z^{m-2} + + \frac{S}{Q}\, z^{m-3} + \dots + \frac{1}{Q} = 0, +\] +and the greatest negative coefficient of this equation +will, from what has been demonstrated above, give a +value greater than its greatest root; so that calling~$L$ +this greatest coefficient, $L + 1$~will be a quantity +greater than the greatest value of~$z$. Consequently, +$\dfrac{1}{L + 1}$ will be a quantity smaller than the smallest +positive value of~$i$; and in like manner we shall find +a quantity smaller than the smallest negative value +of~$i$. Accordingly, we may take for~$D$ the smallest of +these two quantities, or some quantity smaller than +either of them. + +For a simpler result, and one which is independent +of signs, we may reduce the question to finding a +quantity~$L$ numerically greater than any of the coefficients +\index{Coefficients!greatest negative}% +of the equation in~$z$, and it is clear that if we +find a quantity~$N$ numerically smaller than the smallest +value of~$Q$ and a quantity~$M$ numerically greater +than the greatest value of any of the quantities $R$, +$S$,~$\dots$, we may put $L = \dfrac{M}{N}$. + +Let us begin with finding the values of~$M$. It is +not difficult to demonstrate, by the principles established +above, that if $k + 1$~is the limit of the positive +\PageSep{118} +roots and $-h - 1$~the limit of the negative roots of +the proposed equation, and if for~$a$, $k + 1$~and~$-h - 1$ +\MNote{Final resolution.} +be successively substituted in the expressions for $R$, +$S$,~$\dots$, considering only the terms which have the +same sign as the first,---it is easy to demonstrate that +we shall obtain in this manner quantities which are +greater than the greatest positive and negative values +of $R$, $S$,~$\dots$ corresponding to the roots $a$,~$b$, $c$\Add{,}~$\dots$ of +the proposed equation; so that we may take for~$M$ +the quantity which is numerically the greatest of +these. + +It accordingly only remains to find a value smaller +than the smallest value of~$Q$. Now it would seem +that we could arrive at this in no other way than by +employing the equation of which the different values +of~$Q$ are the roots,---an equation which can only be +reached by eliminating~$a$ from the following equations: +\begin{gather*} +a^{m} + pa^{m-1} + qa^{m-2} + \dots + u = 0, \\ +ma^{m-1} + (m - 1)pa^{m-2} + (m - 2)qa^{m-3} + \dots = Q. +\end{gather*} + +It can be easily demonstrated by the theory of +elimination that the resulting equation in~$Q$ will be of +the $m$th~degree, that is to say, of the same degree with +the proposed equation; and it can also be demonstrated +from the form of the roots of this equation +that its next to the last term will be missing. If, accordingly, +we seek by the method given above a quantity +numerically smaller than the smallest root of this +equation, the quantity found can be taken for~$N$. The +\PageSep{119} +problem is therefore resolved by means of an equation +of the same degree as the proposed equation. + +The upshot of the whole is \Typo{a}{as} follows,---where for +\MNote{Recapitulation.} +the sake of simplicity I retain the letter~$x$ instead of +the letter~$a$. + +Let the following be the proposed equation of the +$m$th~degree: +\[ +x^{m} + px^{m-1} + qx^{m-2} + rx^{m-3} + \dots = 0; +\] +let $k$~be the largest coefficient of the negative terms, +and $m - n$~the exponent of~$x$ in the first negative term. +Similarly, let $h$ be the greatest coefficient of the terms +having a contrary sign to the first term after $x$~has +been changed into~$-x$; and let $m - n'$ be the exponent +of~$x$ in the first term having a contrary sign to +the first term of the equation as thus altered. Putting, then, +\[ +f = \sqrt[n]{k} + 1 \quad\text{and}\quad g = \sqrt[n]{h} + 1, +\] +we shall have $f$~and~$-g$ for the limits of the positive +and negative roots. These limits are then substituted +\index{Roots!limits of the positive and negative}% +successively for~$x$ in the following formulæ, neglecting +the terms which have the same sign as the first +term: +\begin{gather*} +%[** TN: Re-broken] +\begin{aligned} +\frac{m(m - 1)}{2}\, x^{m-2} + &+ \frac{(m - 1)(m - 2)}{2}\, px^{m-3} \\ + &+ \frac{(m - 2)(m - 3)}{2}\, qx^{m-4} + \dots, +\end{aligned} \\ +\frac{m(m - 1)(m - 2)}{2·3}\, x^{m-3} + + \frac{(m - 1)(m - 2)(m - 3)}{2·3}\, px^{m-4} + \dots, +\end{gather*} +\PageSep{120} +and so on. Of these formulæ there will be~$m - 2$. Let +the greatest of the numerical quantities obtained in +this manner be called M. We then take the equation +\MNote{The arithmetical progression revealing the roots.} +\index{Arithmetical progression revealing the roots}% +\index{Roots!arithmetical@the arithmetical progression revealing the}% +\[ +mx^{m-1} + (m - 1)px^{m-2} + (m - 2)qx^{m-3} + (m - 3)rx^{m-4} + \dots = y +\] +and eliminate~$x$ from it by means of the proposed +equation,---which gives an equation in~$y$ of the $m$th~degree +with its next to the last term wanting. Let $V$~be +the last term of this equation in~$y$, and $T$~the largest +coefficient of the terms having the contrary sign +to~$V$, supposing $y$~positive as well as negative. Then +taking these two quantities $T$~and~$V$ positive, $N$~will +be determined by the equation +\[ +\frac{N}{1 - N} = \sqrt[n]{\frac{V}{T}} +\] +where $n$~is equal to the exponent of the last term having +the contrary sign to~$V$. We then take $D$ equal to +or smaller than the quantity~$\dfrac{N}{M + N}$, and interpolate +the arithmetical progression: +\[ +0,\quad D,\quad 2D,\quad 3D,\dots,\quad +-D,\quad -2D,\quad -3D, \dots +\] +between the limits $f$~and~$-g$. The terms of these +progressions being successively substituted for~$x$ in +the proposed equation will reveal all the real roots, +positive as well as negative, by the changes of sign +in the series of results produced by these substitutions, +and they will at the same time give the first +limits of these roots,---limits which can be narrowed +as much as we please, as we already know. +\index{Limits of roots|)}% +\PageSep{121} + +If the last term~$V$ of the equation in~$y$ resulting +from the elimination of~$x$ is zero, then $N$~will be zero, +and consequently $D$~will be equal to zero. But in +\MNote{Method of elimination\Add{.}} +\index{Elimination!method of}% +this case it is clear that the equation in~$y$ will have +one root equal to zero and even two, because its next +to the last term is wanting. Consequently the equation +\[ +mx^{m-1} + (m - 1)px^{m-2} + (m - 2)qx^{m-3} + \dots = 0\Typo{.}{} +\] +will hold good at the same time with the proposed +equation. These two equations will, accordingly, have +\index{Common divisor of two equations}% +\index{Equations!common divisor of two}% +a common divisor which can be found by the ordinary +method, and this divisor, put equal to zero, will give +one or several roots of the proposed equation, which +roots will be double or multiple, as is easily apparent +from the preceding theory; for if the last term~$Q$ of +the equation in~$i$ is zero, it follows that +\[ +i = 0 \quad\text{and}\quad a = b. +\] +The equation in~$y$ is reduced, by the vanishing of its +last term, to the $(m - 2)$th~degree,---being divisible +by~$y^{2}$. If after this division its last term should still +be zero, this would be an indication that it had more +than two roots equal to zero, and so on. In such a +contingency we should divide it by~$y$ as many times +as possible, and then take its last term for~$V$, and the +greatest coefficient of the terms of contrary sign to~$V$ +for~$T$, in order to obtain the value of~$D$, which will +enable us to find all the remaining roots of the proposed +equation. If the proposed equation is of the +third degree, as +\PageSep{122} +\[ +x^{3} + qx + r = 0, +\] +we shall get for the equation in~$y$, +\[ +y^{3} + 3qy^{2} - 4q^{3} - 27r^{2} = 0. +\] + +If the proposed equation is +\[ +x^{4} + qx^{2} + rx + s = 0 +\] +we shall obtain for the equation in~$y$ the following +\begin{multline*} +%[** TN: Re-broken] +y^{4} + 8ry^{3} + (4q^{3} - 16qs + 18r^{2})y^{2} \\ + + 256s^{3} - 128s^{2}q^{2} + 16sq^{4} + 144r^{2}sq - 4r^{2}q^{3} - 27r^{4} + = 0 +\end{multline*} +and so on. + +Since, however, the finding of the equation in~$y$ by +\MNote{General formulæ for elimination.} +\index{Elimination!general formulæ for}% +the ordinary methods of elimination may be fraught +with considerable difficulty, I here give the general +formulæ for the purpose, derived from the known +properties of equations. We form, first, from the coefficients +$p$,~$q$,~$r$ of the proposed equation, the quantities +$x_{1}$,~$x_{2}$,~$x_{3}$,~$\dots$, in the following manner: +\[ +\begin{array}{r@{\,}l} +x_{1} &= -p, \\ +x_{2} &= -px_{1} - 2q, \\ +x_{3} &= -px_{2} - qx_{1} - 3r, \\ +\hdotsfor{2}. +\end{array} +\] +We then substitute in the expressions for $y$,~$y^{2}$,~$y^{3}$,~$\dots$ +up to~$y^{m}$, after the terms in~$x$ have been developed +the quantities $x_{1}$~for~$x$, $x_{2}$~for~$x^{}$, $x_{3}$~for~$x^{3}$, and so forth, +and designate by $y_{1}$,~$y_{2}$, $y_{3}$,~$\dots$ the values of $y$,~$y^{2}$, $y^{3}$,~$\dots$ +resulting from these substitutions. We have then +simply to form the quantities $A$,~$B$,~$C$ from the formulæ +\PageSep{123} +\index{Differences, the equation of}% +\[ +\begin{array}{r@{\,}l} +A &= y_{1}, \\ +B &= \dfrac{Ay_{1} - y_{2}}{2}, \\ +C &= \dfrac{By_{1} - Ay_{2} + y_{3}}{3}, \\ +\hdotsfor{2}, +\end{array} +\] +and we shall have the following equation in~$y$: +\[ +y^{m} - Ay^{m-1} + By^{m-2} - Cy^{m-3} + \dots = 0. +\] + +The value, or rather the limit of~$D$, which we find +by the method just expounded may often be much +\MNote{General result.} +smaller than is necessary for finding all the roots, but +there would be no further inconvenience in this than +to increase the number of successive substitutions for~$x$ +\index{Substitutions}% +in the proposed equation. Furthermore, when there +are as many results found as there are units in the +highest exponent of the equation, we can continue +these results as far as we wish by the simple addition +of the first, second, third differences, etc., because +the differences of the order corresponding to the degree +of the equation are always constant. + +We have seen above how the curve of the proposed +equation can be constructed by successively giving +different values to the abscissæ~$x$ and taking for the +ordinates~$y$ the values resulting from these substitutions +in the left-hand side of the equation. But these +values for~$y$ can also be found by another very simple +construction, which deserves to be brought to your +notice. Let us represent the proposed equation by +\[ +a + bx + cx^{2} + dx^{3} + \dots = 0 +\] +\PageSep{124} +where the terms are taken in the inverse order. The +equation of the curve will then be +\[ +y = a + bx + cx^{2} + dx^{3} + \dots\Add{.} +\] +Drawing (Fig.~2) the straight line~$OX$, which we take +\MNote{A second construction for solving equations.} +\index{Equations!constructions for solving}% +\index{Machine for solving equations|(}% +as the axis of abscissæ with $O$~as origin, we lay off on +this line the segment~$OI$ equal to the unit in terms of +which we may suppose the quantities $a$,~$b$,~$c$\Add{,}~$\dots$, to +be expressed; and we erect at the points~$OI$ the perpendiculars +\Figure{2}{0.5\textwidth} +$OD$,~$IM$. We then lay off upon the line~$OD$ +the segments +\[ +OA = a,\quad AB = b,\quad BC = c,\quad CD = d, \dots, +\] +and so on. Let $OP = x$, and at the point~$P$ let the +perpendicular~$PT$\Typo{}{ }be erected. Suppose, for example, +that $d$~is the last of the coefficients $a$,~$b$,~$c$,~$\dots$, so that +the proposed equation is only of the third degree, and +that the problem is to find the value of +\[ +y = a + bx + cx^{2} + dx^{3}. +\] +The point~$D$ being the last of the points determined +upon the perpendicular~$OD$, and the point~$C$ the next +\PageSep{125} +to the last, we draw through~$D$ the line~$DM$ parallel +to the axis~$OI$, and through the point~$M$ where this +line cuts the perpendicular~$IM$ we draw the straight +\MNote{The development and solution.} +line~$CM$ connecting $M$ with~$C$. Then through the +point~$S$ where this last straight line cuts the perpendicular~$PT$, +we draw $HSL$ parallel to~$OI$, and through +the point~$L$ where this parallel cuts the perpendicular~$IM$ +we draw to the point~$B$ the straight line~$BL$. +Similarly, through the point~$R$, where this last line +cuts the perpendicular~$PT$, we draw $GRK$ parallel to~$OI$, +and through the point~$K$, where this parallel cuts +the perpendicular~$IM$ we draw to the first division +point~$A$ of the perpendicular~$DO$ the straight line~$AK$. +The point~$Q$ where this straight line cuts the perpendicular~$PT$ +will give the segment $PQ = y$. + +Through $Q$ draw the line $FQ$ parallel to the axis~$OP$. +The two similar triangles $CDM$~and~$CHS$ give +\[ +DM(1) : DC(d) = HS(x) : CH(= dx). +\] +Adding $CB(c)$ we have +\[ +BH = c + dx. +\] +Also the two similar triangles $BHL$~and~$BGR$ give +\[ +HL(1) : HB(c + dx)= GR(x) : BG(= cx + dx^{2}). +\] +Adding $AB(b)$ we have +\[ +AG = b + cx + dx^{2}. +\] +Finally the similar triangles $AGK$~and~$AFQ$ give +\[ +%[** TN: Set on two lines in original] +GK(1) : GA(b + cx + dx^{2}) = FQ(x) : FA(= bx + cx^{2} + dx^{3}), +\] +and we obtain by adding $OA(a)$ +\[ +OF = PQ = a + bx + cx^{2} + dx^{3} = y. +\] +\PageSep{126} + +The same construction and the same demonstration +hold, whatever be the number of terms in the +proposed equation. When negative coefficients occur +among $a$,~$b$, $c$,~$\dots$, it is simply necessary to take +them in the opposite direction to that of the positive +coefficients. For example, if $a$~were negative we +should have to lay off the segment~$OA$ below the axis~$OI$. +Then we should start from the point~$A$ and add +to it the segment $AB = b$. If $b$~were positive, $AB$~would +be taken in the direction of~$OD$; but if $b$~were +negative, $AB$~would be taken in the opposite direction, +and so on with the rest. + +With regard to~$x$, $OP$~is taken in the direction of~$OI$, +which is supposed to be equal to positive unity, +when $x$~is positive; but in the opposite direction when +$x$~is negative. + +It would not be difficult to construct, on the foregoing +\MNote{A machine for solving equations.} +\index{Equations!machine@a machine for solving}% +model, an instrument which would be applicable +to all values of the coefficients $a$,~$b$, $c$,~$\dots$, and which +by means of a number of movable and properly jointed +rulers would give for every point~$P$ of the straight +line~$OP$ the corresponding point~$Q$, and which could +be even made by a continuous movement to describe +the curve. Such an instrument might be used for +solving equations of all degrees; at least it could be +used for finding the first approximate values of the +roots, by means of which afterwards more exact values +could be reached. +\index{Machine for solving equations|)}% +\index{Numerical equations!resolution of|)}% +\PageSep{127} + + +\Lecture[The Employment of Curves.] +{V.}{On the Employment of Curves in the Solution +of Problems.} +\index{Curves!employment of in the solution of problems|(}% +\index{Problems!employment of curves in the solution of|(}% + +\First{As long} as algebra and geometry travelled separate +\index{Algebra!application of geometry to|EtSeq}% +\index{Geometry!application of to algebra|EtSeq}% +paths their advance was slow and their +\MNote{Geometry applied to algebra.} +applications limited. But when these two sciences +joined company, they drew from each other fresh vitality +and thenceforward marched on at a rapid pace +towards perfection. It is to Descartes that we owe +\index{Descartes}% +the application of algebra to geometry,---an application +which has furnished the key to the greatest discoveries +in all branches of mathematics. The method +which I last expounded to you for finding and demonstrating +divers general properties of equations by considering +the curves which represent them, is, properly +speaking, a species of application of geometry to algebra, +and since this method has extended \Typo{applicacations}{applications}, +and is capable of readily solving problems +whose direct solution would be extremely difficult or +even impossible, I deem it proper to engage your attention +in this lecture with a further view of this subject,---especially +\PageSep{128} +since it is not ordinarily found in +elementary works on algebra. + +You have seen how an equation of any degree +\MNote{Method of resolution by curves.} +whatsoever can be resolved by means of a curve, of +which the abscissæ represent the unknown quantity +of the equation, and the ordinates the values which +the left-hand member assumes for every value of the +unknown quantity. It is clear that this method can be +applied generally to all equations, whatever their form, +and that it only requires them to be developed and +arranged according to the different powers of the unknown +quantity. It is simply necessary to bring all +the terms of the equation to one side, so that the other +side shall be equal to zero. Then taking the unknown +quantity for the abscissa~$x$, and the function of the +unknown quantity, or the quantity compounded of +that quantity and the known quantities, which forms +one side of the equation, for the ordinate~$y$, the curve +described by these co-ordinates $x$~and~$y$ will give by +its intersections with the axis those values of~$x$ which +are the required roots of the equation. And since +most frequently it is not necessary to know all possible +values of the unknown quantity but only such as +solve the problem in hand, it will be sufficient to describe +that portion of the curve which corresponds to +these roots, thus saving much unnecessary calculation. +We can even determine in this manner, from the shape +of the curve itself, whether the problem has possible +solutions satisfying the proposed conditions. +\PageSep{129} + +Suppose, for instance, that it is required to find on +\index{Light, law of the intensity of}% +\index{Lights, problem of the two|EtSeq}% +the line joining two luminous points of given intensity, +the point which receives a given quantity of light,---the +\MNote{Problem of the two lights.} +law of physics being that the intensity of light decreases +with the square of the distance. + +Let $a$~be the distance between the two lights and +$x$~the distance between the point sought and one of +the lights, the intensity' of which at unit distance is~$M$, +the intensity of the other at that distance being~$N$. +The expressions $\dfrac{M}{x^{2}}$ and $\dfrac{N}{(a - x)^{2}}$, accordingly, +give the intensity of the two lights at the point in +question, so that, designating the total given effect by~$A$, +we have the equation +\[ +\frac{M}{x^{2}} + \frac{N}{(a - x)^{2}} = A\Add{,} +\] +or +\[ +\frac{M}{x^{2}} + \frac{N}{(a - x)^{2}} - A = 0. +\] + +We will now consider the curve having the equation +\[ +\frac{M}{x^{2}} + \frac{N}{(a - x)^{2}} - A = y +\] +in which it will be seen at once that by giving to~$x$ a +very small value, positive or negative, the term~$\dfrac{M}{x^{2}}$, +while continuing positive, will grow very large, because +a fraction increases in proportion as its denominator +decreases, and it will be infinite when $x = 0$. +Further, if $x$~be made to increase, the expression~$\dfrac{M}{x^{2}}$ +will constantly diminish; but the other expression~$\dfrac{N}{(a - x)^{2}}$, +\PageSep{130} +which was $\dfrac{N}{a^{2}}$ when $x = 0$, will constantly increase +until it becomes very large or infinite when $x$ +has a value very near to or equal to~$a$. + +Accordingly, if, by giving to~$x$ values from zero to~$a$, +\MNote{Various solutions.} +the sum of these two expressions can be made to +become less than the given quantity~$A$, then the value +of~$y$, which at first was very large and positive, will +become negative, and afterwards again become very +large and positive. Consequently, the curve will cut +the axis twice between the two lights, and the problem +will have two solutions. These two solutions will +be reduced to a single solution if the smallest value of +\[ +\frac{M}{x^{2}} + \frac{N}{(a - x)^{2}} +\] +is exactly equal to~$A$, and they will become imaginary +if that value is greater than~$A$, because then the value +of~$y$ will always be positive from $x = 0$ to $x = a$. +Whence it is plain that if one of the conditions of the +problem be that the required point shall fall between +the two lights it is possible that the problem has no +solution. But if the point be allowed to fall on the +prolongation of the line joining the two lights, we +shall see that the problem is always resolvable in two +ways. In fact, supposing $x$~negative, it is plain that +the term~$\dfrac{M}{x^{2}}$ will always remain positive and from being +very large when $x$~is near to zero, it will commence +and keep decreasing as $x$~increases until it grows very +small or becomes zero when $x$~is very great or infinite. +\PageSep{131} +The other term~$\dfrac{N}{(a - x)^{2}}$, which at first was equal to~$\dfrac{N}{a^{2}}$, +also goes on diminishing until it becomes zero +when $x$~is negative infinity. It will be the same if $x$~is +positive and greater than~$a$; for when $x = a$, the +expression $\dfrac{N}{(a - x)^{2}}$ will be infinitely great; afterwards +it will keep on decreasing until it becomes zero when $x$~is +infinite, while the other expression $\dfrac{M}{x^{2}}$ will first be +equal to $\dfrac{M}{a^{2}}$ and will also go on diminishing towards +zero as $x$~increases. + +Hence, whatever be the value of the quantity~$A$, +it is plain that the values of~$y$ will necessarily pass +\MNote{General solution.} +from positive to negative, both for $x$~negative and for +$x$~positive and greater than~$a$. Accordingly, there +will be a negative value of~$x$ and a positive value of~$x$ +greater than~$a$ which will resolve the problem in all +cases. These values may be found by the general +method by successively causing the values of~$x$ which +give values of~$y$ with contrary signs, to approach +nearer and nearer to each other. + +With regard to the values of~$x$ which are less than~$a$ +we have seen that the reality of these values depends +on the smallest value of the quantity +\[ +\frac{M}{x^{2}} + \frac{N}{(a - x)^{2}}. +\] +Directions for finding the smallest and greatest values +of variable quantities are given in the Differential Calculus. +\index{Differential Calculus}% +We shall here content ourselves with remarking +\PageSep{132} +that the quantity in question will be a minimum +when +\MNote{Minimal values.} +\index{Minimal values}% +\index{Values!minimal}% +\[ +\frac{x}{a - x} = \sqrt[3]{\frac{M}{N}}; +\] +so that we shall have +\[ +x = \frac{a\sqrt[3]{M}}{\sqrt[3]{M} + \sqrt[3]{N}}, +\] +from which we get, as the smallest value of the expression +\[ +\frac{M}{x^{2}} + \frac{N}{(a - x)^{2}}, +\] +the quantity +\[ +\frac{(\sqrt[3]{M} + \sqrt[3]{N})^{3}}{a^{2}}. +\] +Hence there will be two real values for~$x$ if this quantity +is less than~$A$; but these values will be imaginary +if it is greater. The case of equality will give two +equal values for~$x$. + +I have dwelt at considerable length on the analysis +of this problem, (though in itself it is of slight importance,) +for the reason that it can be made to serve +as a type for all analogous cases. + +The equation of the foregoing problem, having +been freed from fractions, will assume the following +form: +\[ +Ax^{2}(a - x)^{2} - M(a - x)^{2} - Nx^{2} = 0. +\] +With its terms developed and properly arranged it +will be found to be of the fourth degree, and will consequently +have four roots. Now by the analysis which +we have just given, we can recognise at once the character +\PageSep{133} +of these roots. And since a method may spring +from this consideration applicable to all equations of +\index{Equations!fourth@of the fourth degree}% +\index{Fourth degree, equations of the}% +the fourth degree, we shall make a few brief remarks +\MNote{Preceding analysis applied to bi-quadratic equations.} +\index{Biquadratic equations}% +upon it in passing. Let the general equation be +\[ +x^{4} + px^{2} + qx + r = 0. +\] +We have already seen that if the last term of this +equation be negative it will necessarily have two real +roots, one positive and one negative; but that if the +last term be positive we can in general infer nothing +as to the character of its roots. If we give to this +equation the following form +\[ +(x^{2} - a^{2})^{2} + b(x + a)^{2} + c(x - a)^{2} = 0, +\] +a form which developed becomes +\[ +x^{4} + (b + c - 2a^{2})x^{2} + 2a(b - c)x + a^{4} + a^{2}(b + c) = 0, +\] +and from this by comparison derive the following +equations of condition +\[ +b + c - 2a^{2} = p,\quad 2a(b - c) = q,\quad a^{4} + a^{2}(b + c) = r, +\] +and from these, again, the following, +\[ +b + c = p + 2a^{2},\quad b - c = \frac{q}{2a},\quad 3a^{4} + pa^{2} = r, +\] +we shall obtain, by resolving the last equation, +\[ +a^{2} = -\frac{p}{6} + \sqrt{\frac{r}{3} + \frac{p^{2}}{36}}. +\] +If $r$~be supposed positive, $a^{2}$~will be positive and real, +and consequently $a$~will be real, and therefore, also, +$b$~and~$c$ will be real. + +Having determined in this manner the three quantities +$a$,~$b$,~$c$, we obtain the transformed equation +\[ +(x^{2} - a^{2})^{2} + b(x + a)^{2} + c(x - a)^{2} = 0. +\] +\PageSep{134} + +Putting the right-hand side of this equation equal +to~$y$, and considering the curve having for abscissæ +\MNote{Consideration of equations of the fourth degree.} +the different values of~$y$, it is plain, that when $b$~and~$c$ +are positive quantities this curve will lie wholly +above the axis and that consequently the equation +will have no real root. Secondly, suppose that $b$~is a +negative quantity and $c$~a positive quantity; then $x = a$ +will give $y = 4ba^{2}$,---a negative quantity. A very +large positive or negative~$x$ will then give a very large +positive~$y$,---whence it is easy to conclude that the +equation will have two real roots, one larger than~$a$ +and one less than~$a$. We shall likewise find that if +$b$~is positive and $c$~is negative, the equation will have +two real roots, one greater and one less than~$-a$. +Finally, if $b$~and~$c$ are both negative, then $y$~will become +negative by making +\[ +x = a \quad\text{and}\quad x = -a +\] +and it will be positive and very large for a very large +positive or negative value of~$x$,---whence it follows +that the equation will have two real roots, one greater +than~$a$ and one less than~$-a$. The preceding considerations +might be greatly extended, but at present we +must forego their pursuit. + +It will be seen from the preceding example that +the consideration of the curve does not require the +equation to be freed from fractional expressions. The +\index{Fractional expressions in equations}% +\index{Radical expressions in equations}% +same may be said of radical expressions. There is +an advantage even in retaining these expressions in +\PageSep{135} +the form given by the analysis of the problem; the +advantage being that we may in this way restrict our +attention to those signs of the radicals which answer +\MNote{Advantages of the method of curves.} +\index{Curves!advantages of the method of}% +to the special exigencies of each problem, instead of +causing the fractions and the radicals to disappear +and obtaining an equation arranged according to the +different whole powers of the unknown quantity in +which frequently roots are introduced which are entirely +foreign to the question proposed. It is true that +these roots are always part of the question viewed in +its entire extent; but this wealth of algebraical analysis, +although in itself and from a general point of view +extremely valuable, may be inconvenient and burdensome +in particular cases where the solution of which +we are in need cannot by direct methods be found independently +of all other possible solutions. When +the equation which immediately flows from the conditions +of the problem contains radicals which are essentially +ambiguous in sign, the curve of that equation +(constructed by making the side which is equal to +zero, equal to the ordinate~$y$) will necessarily have as +many branches as there are possible different combinations +of these signs, and for the complete solution it +would be necessary to consider each of these branches. +But this generality may be restricted by the particular +conditions of the problem which determine the branch +on which the solution is to be sought; the result being +that we are spared much needless calculation,---an +advantage which is not the least of those offered by +\PageSep{136} +the method of solving equations from the consideration +of curves. + +But this method can be still further generalised +\MNote{The curve of errors.} +\index{Errors, curve of|EtSeq}% +and even rendered independent of the equation of the +problem. It is sufficient in applying it to consider +the conditions of the problem in and for themselves, +to give to the unknown quantity different arbitrary +values, and to determine by calculation or construction +the errors which result from such suppositions +according to the original conditions. Taking these +errors as the ordinates~$y$ of a curve having for abscissæ +the corresponding values of the unknown quantity, +we obtain a continuous curve called \emph{the curve of errors}, +which by its intersections with the axis also gives all +solutions of the problem. Thus, if two successive errors +be found, one of which is an excess, and another +a defect, that is, one positive and one negative, we +may conclude at once that between these two corresponding +values of the unknown quantity there will +be one for which the error is zero, and to which we +can approach as near as we please by successive substitutions, +or by the mechanical description of the +curve. + +This mode of resolving questions by curves of errors +\index{Astronomy, mechanics, and physics, curves of errors in}% +\index{Mechanics, astronomy, and physics, curves of errors in}% +\index{Physics, astronomy, and mechanics, curves of errors in}% +is one of the most useful that have been devised. +It is constantly employed in astronomy when direct +solutions are difficult or impossible. It can be employed +for resolving important problems of geometry +and mechanics and even of physics. It is properly +\PageSep{137} +speaking the \textit{regula falsi}, taken in its most general +\index{False, rule of}% +\index{Regula@\textit{Regula falsi}}% +\index{Rule!false@of false}% +sense and rendered applicable to all questions where +there is an unknown quantity to be determined. It +\MNote{Solution of a problem by the curve of errors.} +can also be applied to problems that depend on two +or several unknown quantities by successively giving +to these unknown quantities different arbitrary values +and calculating the errors which result therefrom, afterwards +linking them together by different curves, or +reducing them to tables; the result being that we may +\index{Tables}% +by this method obtain directly the solution sought +\Figure{3}{0.4\textwidth} +without preliminary elimination of the unknown quantities. + +We shall illustrate its use by a few examples. + +\textit{Required a circle in which a polygon of given sides can +be inscribed.} + +This problem gives an equation which is proportionate +in degree to the number of sides of the polygon. +To solve it by the method just expounded we +describe any circle~$ABCD$ (Fig.~3) and lay off in this +circle the given sides $AB$,~$BC$, $CD$, $DE$,~$EF$ of the +\PageSep{138} +polygon, which for the sake of simplicity I here suppose +to be pentagonal. If the extremity of the last +\MNote{Problem of the circle and inscribed polygon.} +\index{Circle!and inscribed polygon, problem of the}% +\index{Polygon, problem of the circle and inscribed}% +side falls on~$A$, the problem is solved. But since it +is very improbable that this should happen at the first +trial we lay off on the straight line~$PR$ (Fig.~4) the +radius~$PA$ of the circle, and erect on it at the point~$A$ +the perpendicular~$AF$ equal to the chord~$AF$ of the +arc~$AF$ which represents the error in the supposition +\index{Supposition, rule of}% +\index{Trial and error, rule of}% +made regarding the length of the radius~$PA$. Since +this error is an excess, it will be necessary to describe +\Figure{4}{0.3\textwidth} +a circle having a larger radius and to perform the +same operation as before, and so on, trying circles of +various sizes. Thus, the circle having the radius~$PA$ +gives the error~$F'A'$ which, since it falls on the hither +side of the point~$A'$, should be accounted negative. It +will consequently be necessary in Fig.~4 in applying +the ordinate~$A'F'$ to the abscissa~$PA'$ to draw that +ordinate below the axis. In this manner we shall obtain +several points $F$,~$F'$,~$\dots$, which will lie on a +curve of which the intersection~$R$ with the axis~$PA$ +\PageSep{139} +will give the true radius~$PR$ of the circle satisfying +the problem, and we shall find this intersection by +successively causing the points of the curve lying on +\MNote{Solution of a second problem by the curve of errors.} +the two sides of the axis as $F$,~$F'$,~$\dots$ to approach +nearer and nearer to one another. + +\textit{From a point, the position of which is unknown, three +\index{Point in space, position of a}% +objects are observed, the distances of which from one another +are known. The three angles formed by the rays of +light from these three objects to the eye of the observer are +also known. Required the position of the observer with +respect to the three objects.} + +If the three objects be joined by three straight +lines, it is plain that these three lines will form with +the visual rays from the eye of the observer a triangular +pyramid of which the base and the three face angles +forming the solid angle at the vertex are given. +And since the observer is supposed to be stationed at +the vertex, the question is accordingly reduced to determining +the dimensions of this pyramid. + +Since the position of a point in space is completely +determined by its three distances from three given +points, it is clear that the problem will be resolved, if +the distances of the point at which the observer is +stationed from each of the three objects can be determined. +Taking these three distances as the unknown +quantities we shall have three equations of the second +degree, which after elimination will give a resultant +equation of the eighth degree; but taking only one of +these distances and the relations of the two others to it +\PageSep{140} +for the unknown quantities, the final equation will be +only of the fourth degree. We can accordingly rigorously +\MNote{Problem of the observer and three objects.} +solve this problem by the known methods; but +the direct solution, which is complicated and inconvenient +in practice, may be replaced by the following +which is reached by the curve of errors. + +Let the three successive angles $APB$, $BPC$, $CPD$ +\index{Observer, problem of the, and three objects}% +(Fig.~5) be constructed, having the vertex~$P$ and +respectively equal to the angles observed between the +first object and the second, the second and the third, +\Figure{5}{0.4\textwidth} +the third and the first; and let the straight line~$PA$ +be taken at random to represent the distance from the +observer to the first object. Since the distance of +that object to the second is supposed to be known, +let it be denoted by~$AB$, and let it be laid off on the +line~$AB$. We shall in this way obtain the distance~$BP$ +of the second object to the observer. In like manner, +let $BC$, the distance of the second object to the +third, be laid off on~$BC$, and we shall have the distance~$PC$ +of that object to the observer. If, now, the +\PageSep{141} +distance of the third object to the first be laid off on +the line~$CD$, we shall obtain~$PD$ as the distance of +the first object to the observer. Consequently, if the +\MNote{Employment of the curve of errors.} +distance first assumed is exact, the two lines $PA$~and~$PD$ +will necessarily coincide. Making, therefore, on +the line~$PA$, prolonged if necessary, the segment +$PE = PD$, if the point~$E$ does not fall upon the point~$A$, +the difference will be the error of the first assumption~$PA$. +Having drawn the straight line~$PR$ (Fig.~6) +we lay off upon it from the fixed point~$P$, the abscissa~$PA$, +and apply to it at right angles the ordinate~$EA$; +we shall have the point~$E$ of the curve of errors~$ERS$. +\Figure{6}{0.4\textwidth} +Taking other distances for~$PA$, and making the same +construction, we shall obtain other errors which can be +similarly applied to the line~$PR$, and which will give +other points in the same curve. + +We can thus trace this curve through several +points, and the point~$R$ where it cuts the axis~$PR$ will +give the distance~$PR$, of which the error is zero, and +which will consequently represent the exact distance +of the observer from the first object. This distance +being known, the others may be obtained by the same +construction. + +It is well to remark that the construction we have +been considering gives for each point~$A$ of the line~$PA$, +\PageSep{142} +two points $B$~and~$B'$ of the line~$PB$; for, since +the distance~$AB$ is given, to find the point~$B$ it is only +\MNote{Eight possible solutions of the preceding problem.} +necessary to describe from the point~$A$ as centre and +with radius~$AB$ an arc of a circle cutting the straight +line~$PB$ at the two points $B$~and~$B'$,---both of which +points satisfy the conditions of the problem. In the +same manner, each of these last-mentioned points will +give two more upon the straight line~$PC$, and each of +the last will give two more on the straight line~$PD$. +Whence it follows that every point~$A$ taken upon the +straight line~$PA$ will in general give eight upon the +straight line~$PD$, all of which must be separately and +successively considered to obtain all the possible solutions. +I have said, \emph{in general}, because it is possible +(1)~for the two points $B$~and~$B'$ to coincide at a single +point, which will happen when the circle described +with the centre~$A$ and radius~$AB$ touches the straight +line~$PB$; and (2)~that the circle may not cut the +straight line~$PB$ at all, in which case the rest of the +construction is impossible, and the same is also to be +said regarding the points $C$,~$D$. Accordingly, drawing +the line~$GF$ parallel to~$BP$ and at a distance from it +equal to the given line~$AB$, the point~$F$ at which this +line cuts the line~$PE$, prolonged if necessary, will be +the limit beyond which the points~$A$ must not be taken +if we desire to obtain possible solutions. There exist +also limits for the points $B$~and~$C$, which may be employed +in restricting the primitive suppositions made +with respect to the distance~$PA$. +\PageSep{143} + +The eight points~$D$, which depend in general on +each point~$A$, answer to the eight solutions of which +the problem is susceptible, and when one has no special +\MNote{Reduction of the possible solutions in practice.} +datum by means of which it can be determined +which of these solutions answer best to the case proposed, +it is indispensable to ascertain them all by employing +for each one of the eight combinations a special +curve of errors. But if it be known, for example, +that the distance of the observer to the second object +is greater or less than his distance to the first, it will +then be necessary to take on the line~$PB$ only the +point~$B$ in the first case and the point~$B'$ in the second,---a +course which will reduce the eight combinations +one-half. If we had the same datum with regard +to the third object relatively to the second, and with +regard to the first object relatively to the third, then +the points $C$~and~$D$ would be determined, and we +should have but a single solution. + +These two examples may suffice to illustrate the +uses to which the method of curves can be put in solving +\index{Curves!method of, submitted to analysis|EtSeq}% +problems. But this method, which we have presented, +so to speak, in a mechanical manner, can also +be submitted to analysis. + +The entire question in fact is reducible to the description +of a curve which shall pass through a certain +number of points, whether these points be given by +calculation or construction, or whether they be given +by observation or single experiences entirely independent +of one another. The problem is in truth indeterminate, +\PageSep{144} +for strictly speaking there can be made +to pass through a given number of points an infinite +\MNote{General conclusion on the method of curves.} +\index{Curves!advantages of the method of}% +number of different curves, regular or irregular, that +is, subject to equations or arbitrarily drawn by the +hand. But the question is not to find any solutions +whatever but the simplest and easiest in practice. + +Thus if there are only two points given, the simplest +solution is a straight line between the two points. +\index{Straight line}% +If there are three points given, the arc of a circle is +\index{Circle}% +drawn through these points, for the arc of a circle +after the straight line is the simplest line that can be +described. + +But if the circle is the simplest curve with respect +to description, it is not so with respect to the equation +between its abscissæ and rectangular ordinates. +In this latter point of view, those curves may be regarded +as the simplest of which the ordinates are expressed +by an integral rational function of the abscissæ, +as in the following equation +\[ +y = a + bx + cx^{2} + dx^{3} + \dots, +\] +where $y$~is the ordinate and $x$~the abscissa. Curves +of this class are called in general \emph{parabolic}, because +\index{Parabolic@\textit{Parabolic} curves|EtSeq}% +they may be regarded as a generalisation of the parabola,---a +curve represented by the foregoing equation +when it has only the first three terms. We have already +illustrated their employment in resolving equations, +and their consideration is always useful in the +approximate description of curves, for the reason that +a curve of this kind can always be made to pass +\PageSep{145} +through as many points of a given curve as we please,---it +being only necessary to take as many undetermined +coefficients $a$,~$b$,~$c$,~$\dots$ as there are points given, +\MNote{Parabolic curves.} +and to determine these coefficients so as to obtain the +abscissæ and ordinates for these points. Now it is +clear that whatever be the curve proposed, the parabolic +curve so described will always differ from it by +less and less according as the number of the different +points is larger and larger and their distance from +one another smaller and smaller. + +Newton was the first to propose this problem. The +\index{Newton, his problem}% +following is the solution which he gave of it: + +Let $P$,~$Q$, $R$,~$S$,~$\dots$ be the values of the ordinates~$y$ +corresponding to the values $p$,~$q$, $r$,~$s$,~$\dots$ of +the abscissæ~$x$; we shall have the following equations +\[ +\begin{array}{r@{\,}*{3}{l@{\,}}l} +P &= a + bp &+ cp^{2} &+ dp^{3} &+ \dots, \\ +Q &= a + bq &+ cq^{2} &+ dq^{3} &+ \dots, \\ +R &= a + br &+ cr^{2} &+ dr^{3} &+ \dots, \\ +\hdotsfor{5}\Add{.} +\end{array} +\] +The number of these equations must be equal to the +number of the undetermined coefficients $a$,~$b$,~$c$,~$\dots$. +Subtracting these equations from one another, the remainders +will be divisible by $q - p$, $r - q$,~$\dots$, and +we shall have after such division +\[ +\begin{array}{r@{\,}*{2}{l@{\,}}l} +\dfrac{Q - P}{q - p} &= b + c(q + p) &= d(q^{2} + qp + p^{2}) &+ \dots, \\[8pt] +\dfrac{R - Q}{r - q} &= b + c(r + q) &= d(r^{2} + rq + q^{2}) &+ \dots, \\ +\hdotsfor{4}\Add{.} +\end{array} +\] +\PageSep{146} + +Let +\[ +\frac{Q - P}{q - p} = Q_{1},\quad +\frac{R - Q}{r - q} = R_{1},\quad +\frac{S - R}{s - r} = S_{1},\dots\Add{.} +\] +\MNote{Newton's problem.} +We shall find in like manner, by subtraction and division, +the following: +\[ +\begin{array}{r@{\,}l@{\,}l} +\dfrac{R_{1} - Q_{1}}{r - p} &= c + d(r + q + p) &+ \dots, \\[8pt] +\dfrac{S_{1} - R_{1}}{s - q} &= c + d(s + r + q) &+ \dots, \\ +\hdotsfor{3}\Add{.} +\end{array} +\] + +Further let +\[ +\frac{R_{1} - Q_{1}}{r - p} = R_{2},\quad +\frac{S_{1} - R_{1}}{s - q} = S_{2},\dots. +\] +We shall have +\[ +\frac{S_{2} - R_{2}}{s - p} = d + \dots, +\] +and so on. + +In this manner we shall find the value of the coefficients +$a$,~$b$,~$c$,~$\dots$ commencing with the last; and, +substituting them in the general equation +\[ +y = a + bx + cx^{2} + dx^{3} + \dots, +\] +we shall obtain, after the appropriate reductions have +been made, the formula +\[ +y = P + + Q_{1}(x - p) + + R_{2}(x - p)(x - q) + + S_{3}(x - p)(x - q)(x - r) + \dots, +\Tag{(1)} +\] +which can be carried as far as we please. + +But this solution may be simplified by the following +consideration. + +Since $y$~necessarily becomes $P$,~$Q$,~$R$\Add{,}~$\dots$, when $x$~becomes +\PageSep{147} +$p$,~$q$,~$r$, it is easy to see that the expression +for~$y$ will be of the form +\MNote{Simplification of Newton's solution.} +\[ +y = AP + BQ + CR + DS + \dots +\Tag{(2)} +\] +where the quantities $A$,~$B$, $C$,~$\dots$ are so expressed in +terms of~$x$ that by making $x = p$ we shall have +\[ +A = 1,\quad B = 0,\quad C = 0,\dots, +\] +and by making $x = q$ we shall have +\[ +A = 0,\quad B = 1,\quad C = 0,\quad D = 0,\dots, +\] +and by making $x = r$ we shall similarly have +\[ +A = 0,\quad B = 0,\quad C = 1,\quad D = 0,\dots\ \text{etc.} +\] +Whence it is easy to conclude that the values of $A$, +$B$, $C$,~$\dots$ must be of the form +\begin{align*} +A &= \frac{(x - q)(x - r)(x - s)\dots}{(p - q)(p - r)(p - s)\dots}, \\ +B &= \frac{(x - p)(x - r)(x - s)\dots}{(q - p)(q - r)(q - s)\dots}, \\ +C &= \frac{(x - p)(x - q)(x - s)\dots}{(r - p)(r - q)(r - s)\dots}, +\end{align*} +where there are as many factors in the numerators +and denominators as there are points given of the +curve less one. + +The last expression for~$y$ (see equation~2), although +different in form, is the same as equation~1. To show +this, the values of the quantities $Q_{1}$,~$R_{2}$, $S_{3}$,~$\dots$ need +only be developed and substituted in equation~1 and +the terms arranged with respect to the quantities $P$, +$Q$, $R$,~$\dots$\Add{.} But the last expression for~$y$ (equation~2) +is preferable, partly because of the simplicity of the +\PageSep{148} +analysis from which it is derived, and also because of +its form, which is more convenient for computation. + +\MNote{Possible uses of Newton's problem.} +Now, by means of this formula, which it is not +difficult to reduce to a geometrical construction, we +are able to find the value of the ordinate~$y$ for any abscissa~$x$, +because the ordinates $P$,~$Q$, $R$,~$\dots$ for the +given abscissæ $p$,~$q$, $r$,~$\dots$ are known. Thus, if we +have several of the terms of any series, we can find +any intermediate term that we wish,---an expedient +which is extremely valuable for supplying lacunæ +which may arise in a series of observations or experiments, +\index{Experiments!expedient@an expedient for supplying lacunæ in a series of}% +\index{Observations, expedient for supplying lacunæ in series of}% +or in tables calculated by formulæ or in given +\index{Tables!expedient for supplying lacunæ in}% +constructions. + +If this theory now be applied to the two examples +\index{Regula@\textit{Regula falsi}}% +\index{Supposition, rule of}% +\index{Trial and error, rule of}% +discussed above and to similar examples in which we +have errors corresponding to different suppositions, we +can directly find the error~$y$ which corresponds to any +intermediate supposition~$x$ by taking the quantities +$P$,~$Q$, $R$,~$\dots$, for the errors found, and $p$,~$q$, $r$,~$\dots$ for +the suppositions from which they result. But since +in these examples the question is to find not the error +which corresponds to a given supposition, but the +supposition for which the error is zero, it is clear that +the present question is the opposite of the preceding +and that it can also be resolved by the same formula +by reciprocally taking the quantities $p$,~$q$, $r$,~$\dots$ for +the errors, and the quantities $P$,~$Q$, $R$,~$\dots$ for the +corresponding suppositions. Then $x$~will be the error +for the supposition~$y$; and consequently, by making +\PageSep{149} +$x = 0$, the value of~$y$ will be that of the supposition +for which the error is zero. + +Let $P$,~$Q$, $R$,~$\dots$ be the values of the unknown +quantity in the different suppositions, and $p$,~$q$, $r$\Add{,}~$\dots$ +\MNote{Application of Newton's problem to the preceding examples.} +the errors resulting from these suppositions, to which +the appropriate signs are given. We shall then have +for the value of the unknown quantity of which the +error is zero, the expression +\[ +AP + BQ + CR + \dots, +\] +in which the values of $A$,~$B$,~$C$\Add{,}~$\dots$ are +\begin{align*} +A &= \frac{q}{q - r} × \frac{r}{r - p} × \dots, \displaybreak[1] \\ +B &= \frac{P}{p - q} × \frac{r}{r - q} × \dots, \displaybreak[1] \\ +C &= \frac{p}{p - r} × \frac{q}{q - r} × \dots, +\end{align*} +where as many factors are taken as there are suppositions +less one. +\index{Curves!employment of in the solution of problems|)}% +\index{Problems!employment of curves in the solution of|)}% +\PageSep{150} +%[Blank page] +\PageSep{151} + + +\Appendix{Note on the Origin of Algebra.} +\PgLabel{151} +\index{Algebra!history of}% + +\First{The} impression (\PgRef{54}) that Diophantus was the +\index{Diophantus}% +``inventor'' of algebra, which sprang, in its Diophantine +form, full-fledged from his brain, was a widespread +one in the eighteenth and in the beginning of +the nineteenth century. But, apart from the intrinsic +improbability of this view which is at variance with +the truth that science is nearly always gradual and +organic in growth, modern historical researches have +traced the germs and beginnings of algebra to a much +remoter date, even in the line of European historical +continuity. The Egyptian book of Ahmes contains +\index{Ahmes}% +examples of equations of the first degree. The early +Greek mathematicians performed the partial resolution +\index{Greeks, mathematics of the}% +of equations of the second and third degree +by geometrical methods. According to Tannery, an +\index{Tannery, M. Paul}% +embryonic indeterminate analysis existed in Pre-Christian +times (Archimedes, Hero, Hypsicles). But +\index{Archimedes}% +\index{Hero}% +\index{Hypsicles}% +the merit of Diophantus as organiser and inaugurator +of a more systematic short-hand notation, at +least in the European line, remains; he enriched +whatever was handed down to him with the most +manifold extensions and applications, betokening his +\PageSep{152} +originality and genius, and carried the science of algebra +\index{Algebra!among the Arabs}% +\index{Algebra!India@in India}% +to its highest pitch of perfection among the +\PgLabel{152} +Greeks. (See Cantor, \textit{Geschichte der Mathematik}, second +\index{Cantor}% +edition, Vol.~I., p.~438, et~seq.; Ball, \textit{Short Account +\index{Ball}% +of the History of Mathematics}, second edition, p.~104 +et~seq.; Fink, \textit{A Brief History of Mathematics}, pp.~63 +\index{Fink}% +et~seq., 77~et~seq. (Chicago: The Open Court +Publishing~Co.) + +The development of Hindu algebra is also to be +noted in connexion with the text of \PgRange{59}{60}. The +Arabs, who had considerable commerce with India, +\index{Arabs!Algebra among the}% +drew not a little of their early knowledge from the +works of the Hindus. Their algebra rested on both +that of the Hindus and the Greeks. (See Ball, \textit{op.~cit.}, +p.~150 et~seq.; Cantor, \textit{op.~cit.}, Vol.~I., p.~651 et~seq.).---\textit{Trans.} +\PageSep{153} +\BackMatter +\printindex +\iffalse +INDEX. + +Academies, rise of 62, 63 + +Ahmes 151 + +Algebra + definition of 2 + history of|EtSeq#history 54 % et seq., + history of 151 + essence of 55 + name@the name of 59 + among the Arabs|EtSeq 59 % et seq, + among the Arabs 152 + Europe@in Europe 60 + Italy@in Italy 64 + India@in India 152 + generality@the generality of 69 + hand-writing of 69 + application of geometry to|EtSeq 100, 127 % et seq. + +Algebraical resolution of equations + limits of the 96 + +Alligation + generally|EtSeq 44 % et seq.; + alternate 47 + +Analysis + indeterminate|EtSeq 47 % et seq., + indeterminate 55 + +Angle, trisection of an 62, 81 + +Angular sections, theory of 80 + +Annuities 16 + +Apollonius 54, 59 + +Arabs + Algebra among the|EtSeq 59 % et seq., + Algebra among the 152 + +Archimedes 54, 151 + +Archimedes|FN 58 % footnote + +Arithmetic + universal|EtSeq 2 % et seq.; + operations of|EtSeq 24 % et seq. + +Arithmetical progression revealing the roots 120 + +Arithmetical progression revealing the roots|EtSeq 112 % et seq. + +Arithmetical proportion 12 + +Astronomy, mechanics, and physics, curves of errors in 136 + +Average life|EtSeq 45 % et seq. + +Bachet de Méziriac 58 + +Ball 152 + +Binomial theorem 115 + +Binomials, extraction of the square roots of two imaginary 77 + +Biquadratic equations 63, 88, 94, 133 + +Bombelli 63, 64 + +Bret, M.|FN 93 % footnote. + +Briggs 20 + +Buteo 61 + +Cantor|FN 54, 60 % footnote, + +Cantor 152 + +Cardan 60, 61, 68, 82, 90 + +Checks on multiplication and division 39 + +Circle 144 + squaring of the 62 + and inscribed polygon, problem of the 138 + +Clairaut 69, 90 + +Coefficients + indeterminate 89 + greatest negative|EtSeq 107 % et seq., + greatest negative 117 + +Common divisor of two equations 121 + +Complements, subtraction by 26 + +Constantinople 58 + +Continued fractions, solution of alligation by|EtSeq 50 % et seq. + +Convergents 7 + +Cube, duplication of the 62 + +Cube roots of a quantity, the three 70 + +Cubic radicals 75 + +Curves + representation of equations by|EtSeq 101 % et seq; + employment of in the solution of problems 127-149 + method of, submitted to analysis|EtSeq 143 % et seq.; + advantages of the method of 135, 144 + +Decimal + fractions 9 + numbers|EtSeq 27 % et seq. + +Decimals + multiplication of 30 + division of 31 +\PageSep{154} + +DeMorgan@{\Typo{DeMorgan}{De Morgan}} v + +Descartes viii, 60, 65, 89, 93, 127 + +Differences, the equation of|EtSeq 114 % et seq., + +Differences, the equation of 123 + +Differential Calculus 131 + +Diophantine problems 55 + +Diophantus|EtSeq 54 % et seq + +Diophantus 151 + +Division + nine@by \textit{nine} 34 + eight@by \textit{eight} 34 + seven@by \textit{seven}|EtSeq 34 % et seq.; + decimals@of decimals 31 + +Divisor, greatest common|EtSeq 2 % et seq. + +Duhring@{Dühring, E.} v + +Duodecimal system 32 + +Ecole@{\Typo{Ecole}{École} Normale} v, xi, 12 + +Economy of thought vii + +Efflux, law of 42 + +Eleven, the number, test of divisibility by 37 + +Elimination + method of 121 + general formulæ for 122 + +Equations + second@of the second degree 56 + third@of the third degree 60, 66, 82 + fourth@of the fourth degree 63, 87, 133 + fifth@of the fifth degree 64 + theory of 65, 84 + biquadratic 88 + limits of the algebraical resolution of 96 + fifth@of the fifth degree 96 + mth@of the $m$th degree 96 + general remarks upon the roots of|EtSeq 102 % et seq.; + graphic resolution of 102 + odd@of an odd degree, roots of 105 + even@of an even degree, roots of 106 + real roots of, limits of the|EtSeq 107 % et seq.; + common divisor of two 121 + constructions for solving|EtSeq 100 % et seq. + constructions for solving 124 + machine@a machine for solving 126 + +Equi-different numbers 13 + +Errors, curve of|EtSeq 136 % et seq. + +Euclid 2, 57 + +Euler viii, x, 93 + +Europe, algebra in 60 + +Evolution 11, 40 + +Experiments + average of 46 + expedient@an expedient for supplying lacunæ in a series of 148 + +Falling stone, spaces traversed by a 42 + +False, rule of 137 + +Fermat 58 + +Ferrari, Louis 64 + +Ferrous, Scipio|EtSeq 60 % et seq. + +Fifth degree, equations of the 96 + +Fink 152 + +Fourth degree, equations of the 133 + +Fractional expressions in equations 134 + +Fractions|EtSeq 2 % et seq.; + +Fractions + continued|EtSeq 3 % et seq.; + converging 6 + decimal 9 + origin of continued 10 + +France 58, 61 + +Galileo ix + +Geometers, ancient|EtSeq 54 % et seq. + +Geometers, ancient 58, 59 + +Geometrical + proportion 13 + calculus 24 + +Geometry 24, 60 + application of to algebra|EtSeq 100, 127 % et seq. + +Germany 61 + +Girard, Albert 62 + +Grain, of different prices 44 + +Greeks, mathematics of the vii, 151 + +Greeks, mathematics of the|EtSeq 54 % et seq. + +Hand-writing of algebra 69 + +Harriot 65 + +Hero 59, 151 + +Horses 43 + +Hudde 65, 82 + +Huygens ix, 10 + +Hypsicles 151 + +Imaginary binomials, square roots of 77 + +Imaginary expressions|EtSeq 79 % et seq. + +Imaginary expressions 83 + +Imaginary quantities, office of the 87 + +Imaginary roots, occur in pairs 99 + +Indeterminate analysis|EtSeq 47 % et seq. + +Indeterminate analysis 55 + +Indeterminate coefficients 89 + +Indeterminates, the method of 83 + +Ingredients 48 + +Interest 15 + +Intersections, with the axis give roots|EtSeq 102 % et seq , + +Intersections, with the axis give roots 113 + +Inventors, great 22 + +Involution and evolution 11 + +Irreducible case 61, 65, 69, 73, 82 + +Italy, cradle of algebra in Europe 61, 64 + +Laborers, work of 41 + +Lagrange, J. L.#Lagrange v + +Lagrange, J. L.|EtSeq#Lagrange vii % et seq. +\PageSep{155} + +Laplace v, xi + +Lavoisier xii + +Leibnitz viii + +Life insurance|EtSeq 45 % et seq. + +Life, probability of 46 + +Light, law of the intensity of 129 + +Lights, problem of the two|EtSeq 129 % et seq. + +Limits of roots 107-120 + +Logarithms|EtSeq 16 % et seq. + +Logarithms 40 + advantages in calculating by 28 + origin of 19 + tables of 20 + +Machine for solving equations 124-126 + +Mathematics + wings of 24 + exactness of 43 + evolution of vii + +Mean values|EtSeq 45 % et seq. + +Mechanics, astronomy, and physics, curves of errors in 136 + +Metals, mingling of, by fusion 44 + +Meziriac@Méziriac, Bachet de 58 + +Minimal values 132 + +Mixtures, rule of|EtSeq 44 % et seq. + +Mixtures, rule of 49 + +Monge v, xi + +Mortality, tables of 45 + +Moving bodies, two 98 + +Multiple roots 105 + +Multiplication + abridged methods of|EtSeq 26 % et seq.; + inverted 28 + approximate 29 + decimals@of decimals 30 + +Music 22 + +Napier|EtSeq 17 % et seq. + +Napoleon xii + +Negative roots 60 + +Newton, his problem 145, viii + +Nine + property of the number|EtSeq 31 % et seq.; + property of the number generalised 33 + +Nizze|FN 58 % footnote. + +Numeration, systems of 1 + +Numerical equations |See Equations 0 + +Numerical equations + resolution of 96-126 + conditions of the resolution of 97 + position of the roots of 98 + +Observations, expedient for supplying lacunæ in series of 148 + +Observer, problem of the, and three objects 140 + +Oughtred 30 + +Paciolus, Lucas 59, 60 + +Pappus 59 + +Parabolic@\textit{Parabolic} curves|EtSeq 144 % et seq. + +Peletier 61 + +Peyrard 58 + +Physics, astronomy, and mechanics, curves of errors in 136 + +Planetarium 9 + +Point in space, position of a 139 + +Polygon, problem of the circle and inscribed 138 + +Polytechnic School v, xi + +Positive roots, superior and inferior limits of the 109 + +Powers|EtSeq 10 % et seq. + +Practice, theory and 43 + +Present value 15 + +Printing, invention of 59 + +Probabilities, calculus of|EtSeq 45 % et seq. + +Problems 110 + solution@for solution 62 + employment of curves in the solution of 127-149 + +Proclus 59 + +Progressions, theory of 12, 14 + +Proportion|EtSeq 11 % et seq. + +Ptolemy 59 + +Radical expressions in equations 134 + +Radicals, cubic 75 + +Ratios, constant 42 + +Ratios, constant|EtSeq 2, 11 % et seq. + +Reality of roots 76, 83, 85, 93 + +Regula@\textit{Regula falsi} 137, 148 + +Remainders + theory of|EtSeq 34 % et seq. + theory of 38 + negative|EtSeq 35 % et seq. + +Romans, mathematics of the 54 + +Roots + negative 60 + equations@of equations of the third degree 71 + reality@the reality of the 74, 76, 79, 83, 85, 93 + biquadratic@of a biquadratic equation 94 + multiple 105 + superior and inferior limits of the positive 109 + method for finding the limits of 110 + separation of the 112 + arithmetical@the arithmetical progression revealing the|EtSeq 112 % et seq. + arithmetical@the arithmetical progression revealing the 120 + quantity less than the difference between any two 113 + smallest|EtSeq 116 % et seq.; + limits of the positive and negative 119 + +Rule + Cardan's 68 + false@of false 137 + mixtures@of mixtures|EtSeq 44 % et seq.; + three@of three|EtSeq 11, 40 % et seq. +\PageSep{156} + +Science + history of 22 + development of|EtSeq vii % et seq. + +Seven, tests of divisibility by 35 + +Short-mind symbols|EtSeq vii % et seq. + +Signs $+$ and $-$ 57 + +Squaring of the circle 62 + +Stenophrenic symbols|EtSeq vii % et seq. + +Straight line 144 + +Substitutions|EtSeq 111 % et seq. + +Substitutions 123 + +Subtraction, new method of|EtSeq 25 % et seq. + +Sum and difference, of two numbers 56 + +Supposition, rule of 137, 148 + +Symbols|EtSeq vii % et seq. + +Tables 137 + expedient for supplying lacunæ in 148 + +Tannery, M. Paul|FN 58 % footnote + +Tannery, M. Paul 151 + +Tartaglia 60, 61 + +Temperament, theory of 23 + +Theon 59 + +Theory and practice 43 + +Theory of remainders, utility of the 38 + +Third degree, equations of the 71, 82 + +Three roots, reality of the 93 + +Trial and error, rule of 137, 148 + +Trisection of an angle 62, 81 + +Turks 58 + +Undetermined quantities 82 + +Unity, three cubic roots of 72 + +Unknown quantity 55 + +Values + mean|EtSeq 45 % et seq.; + minimal 132 + +Variations, calculus of x + +Vatican library 58 + +Vieta viii, 62, 65 + +Vlacq 20 + +Wallis viii + +Wertheim, G.|FN 58 % footnote. + +Woodhouse x + +Xylander 58 +\fi +\PageSep{157} + +\Catalog +%[** TN: Macro prints the following text] +% Catalogue of Publications +% of the +% Open Court Publishing Co. + +\begin{Author}{COPE, E. D.} +\Title{THE PRIMARY FACTORS OF ORGANIC EVOLUTION.} +{121~cuts. Pp.~xvi,~547. Cloth,~\$2.00 (10s.).} +\end{Author} + +\begin{Author}{MÜLLER, F. MAX.} +\Title{THREE INTRODUCTORY LECTURES ON THE SCIENCE OF +THOUGHT.} +{128~pages. Cloth,~75c (3s.\ 6d.).} + +\Title{THREE LECTURES ON THE SCIENCE OF LANGUAGE.} +{112~pages. 2nd~Edition. Cloth,~75c (3s.\ 6d.).} +\end{Author} + +\begin{Author}{ROMANES, GEORGE JOHN.} +\Title{DARWIN AND AFTER DARWIN.} +{Three Vols., \$4.00. Singly, as follows:}{} + +%[** TN: Next three extries get a bit less hanging indentation] +\Title[3\parindent]{}{1.~\textsc{The Darwinian Theory.} 460~pages. 125~illustrations. Cloth, \$2.00\Add{.}} + +\Title[3\parindent]{}{2.~\textsc{Post-Darwinian Questions.} Heredity and Utility. Pp.~338. \$1.50\Add{.}} + +\Title[3\parindent]{}{3.~\textsc{Post-Darwinian Questions.} Isolation and Physiological Selection +Pp.~181. \$1.00.} + +\Title{AN EXAMINATION OF WEISMANNISM.} +{236~pages. Cloth, \$1.00.} + +\Title{THOUGHTS ON RELIGION.} +{Third Edition, Pages,~184. Cloth, gilt top, \$1.25.} +\end{Author} + +\begin{Author}{SHUTE, DR. D. KERFOOT.} +\Title{FIRST BOOK IN ORGANIC EVOLUTION.} +{9~colored plates, 39~cuts. Pp.~xvi+285. Price, \$2.00 (7s.\ 6d.).} +\end{Author} + +\begin{Author}{MACH, ERNST.} +\Title{THE SCIENCE OF MECHANICS.} +{Translated by \textsc{T. J. McCormack.} 250~cuts. 534~pages. \$2.50 (12s.\ 6d.)} + +\Title{POPULAR SCIENTIFIC LECTURES.} +{Third Edition. 415~pages. 59~cuts. Cloth, gilt top. \$1.50 (7s.\ 6d.).} + +\Title{THE ANALYSIS OF THE SENSATIONS.} +{Pp.~208. 37~cuts. Cloth, \$1.25 (6s.\ 6d.).} +\end{Author} + +\begin{Author}{LAGRANGE, JOSEPH LOUIS.} +\Title{LECTURES ON ELEMENTARY MATHEMATICS.} +{With portrait of the author. Pp.~172. Price, \$1.00 (5s.).} +\end{Author} + +\begin{Author}{DE MORGAN, AUGUSTUS.} +\Title{ON THE STUDY AND DIFFICULTIES OF MATHEMATICS.} +{New Reprint edition with notes. Pp.~viii+288. Cloth, \$1.25 (5s.).} + +\Title{ELEMENTARY ILLUSTRATIONS OF THE DIFFERENTIAL AND +INTEGRAL CALCULUS.} +{New reprint edition. Price, \$1.00 (5s.).} +\end{Author} + +\begin{Author}{FINK, KARL.} +\Title{A BRIEF HISTORY OF MATHEMATICS.} +{Trans.\ by W. W. Beman and D. E. Smith. Pp.\Typo{,}{}~333. Cloth, \$1.50 (5s.\ 6d.)} +\end{Author} + +\begin{Author}{SCHUBERT, HERMANN.} +\Title{MATHEMATICAL ESSAYS AND RECREATIONS.} +{Pp.~149. Cuts,~37. Cloth, 75c (33.\ 6d.).} +\end{Author} + +\begin{Author}{HUC AND GABET, MM.} +\Title{TRAVELS IN TARTARY, THIBET AND CHINA.} +{100~engravings. Pp\Add{.}~28+660. 2~vols. \$2.00 (10s.). One vol., \$1.25 (5s.)} +\end{Author} +\PageSep{158} + +\begin{Author}{CARUS, PAUL.} +\Title{THE HISTORY OF THE DEVIL, AND THE IDEA OF EVIL.} +{311~Illustrations. Pages,~500. Price, \$6.00 (30s.).} + +\Title{EROS AND PSYCHE.} +{Retold after Apuleius. With Illustrations by Paul Thumann. Pp.~125. +Price, \$1.50 (6s.).} + +\Title{WHENCE AND WHITHER?} +{An Inquiry into the Nature of the Soul. 196~pages. Cloth, 75c (3s.\ 6d.)} + +\Title{THE ETHICAL PROBLEM.} +{Second edition, revised and enlarged. 351~pages. Cloth, \$1.25 (6s.\ 6d.)} + +\Title{FUNDAMENTAL PROBLEMS.} +{Second edition, revised and enlarged. 372~pp.\ Cl., \$1.50 (7s.\ 6d.).} + +\Title{HOMILIES OF SCIENCE.} +{317~pages. Cloth, Gilt Top, \$1.50 (7s.\ 6d.).} + +\Title{THE IDEA OF GOD.} +{Fourth edition. 32~pages. Paper, 15c (9d.).} + +\Title{THE SOUL OF MAN.} +{2nd~ed. 182~cuts. 482~pages. Cloth, \$1.50 (6s.).} + +\Title{TRUTH IN FICTION. \textsc{Twelve Tales with a Moral.}} +{White and gold binding, gilt edges. Pp.~111. \$1.00 (5s.).} + +\Title{THE RELIGION OF SCIENCE.} +{Second, extra edition. Pp.~103. Price, 50c (2s.\ 6d.).} + +\Title{PRIMER OF PHILOSOPHY.} +{240~pages. Second Edition. Cloth, \$1.00 (5s.).} + +\Title{THE GOSPEL OF BUDDHA. According to Old Records.} +{Fifth Edition. Pp.~275. Cloth, \$1.00 (5s.). In German, \$1.25 (6s.\ 6d.)\Add{.}} + +\Title{BUDDHISM AND ITS CHRISTIAN CRITICS.} +{Pages,~311. 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Price, \$1.75 (9s.).} +\end{Author} + +\begin{Author}{LÉVY-BRUHL, PROF. L.} +\Title{HISTORY OF MODERN PHILOSOPHY IN FRANCE.} +{23 Portraits. Handsomely bound. Pp. 500. Price, \$3.00 (12s.).} +\end{Author} + +\begin{Author}{TOPINARD, DR. PAUL.} +\Title{SCIENCE AND FAITH, \textsc{or Man as an Animal and Man as a Member +of Society.}} +{Pp.~374. Cloth, \$1.50 (6s.\ 6d.).} +\end{Author} + +\begin{Author}{BINET, ALFRED.} +\Title{THE PSYCHOLOGY OF REASONING.} +{Pp.~193. Cloth, 75c (3s.\ 6d.).} + +\Title{THE PSYCHIC LIFE OF MICRO-ORGANISMS.} +{Pp.~135. Cloth, 75 cents.} + +\Title{ON DOUBLE CONSCIOUSNESS.} +{See No.~8, Religion of Science Library.} +\end{Author} + +\begin{Author}{THE OPEN COURT.} +\Title{A Monthly Magazine Devoted to the Science of Religion, the Religion of +Science, and the Extension of the Religious Parliament Idea.} +{Terms: \$1.00 a year; 5s.\ 6d.\ to foreign countries in the Postal Union. +Single Copies, 10~cents (6d.).} +\end{Author} + +\begin{Author}{THE MONIST.} +\Title{A Quarterly Magazine of Philosophy and Science.} +{Per copy, 50~cents; Yearly, \$2.00. 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Candlin.} 15c (9d.). + +\Item{34.} \textit{Mathematical Essays and Recreations.} By \textsc{H. Schubert.} 25c (1s.\ 6d.) + +\Item{35.} \textit{The Ethical Problem.} By \textsc{Paul Carus.} 50c (2s.\ 6d.). + +\Item{36.} \textit{Buddhism and Its Christian Critics.} By \textsc{Paul Carus.} 50c (2s.\ 6d.). + +\Item{37.} \textit{Psychology for Beginners.} By \textsc{Hiram M. Stanley.} 20c (1s.). + +\Item{38.} \textit{Discourse on Method.} By \textsc{Descartes.} 25c (1s.\ 6d.). + +\Item{39.} \textit{The Dawn of a New Era.} By \textsc{Paul Carus.} 15c (9d.). + +\Item{40.} \textit{Kant and Spencer.} By \textsc{Paul Carus.} 20c (1s.). + +\Item{41.} \textit{The Soul of Man.} By \textsc{Paul Carus.} 75c (3s.\ 6d.). + +\Item{42.} \textit{World' s Congress Addresses.} By \textsc{C. C. Bonney.} 15c (9d.). + +\Item{43.} \textit{The Gospel According to Darwin.} By \textsc{Woods Hutchinson.} 50c (2s.\ 6d.) + +\Item{44.} \textit{Whence and Whither.} By \textsc{Paul Carus.} 25c (1s.\ 6d.). + +\Item{45.} \textit{Enquiry Concerning Human Understanding.} By \textsc{David Hume.} 25c +(1s.\ 6d.). + +\Item{46.} \textit{Enquiry Concerning the Principles of Morals.} By \textsc{David Hume.} +25c (1s.\ 6d.) + +\normalsize +\tb +\vfill +\begin{center} +\makebox[\textwidth][s]{\Large THE OPEN COURT PUBLISHING CO.,} \\[4pt] +\normalsize CHICAGO: 324 \textsc{Dearborn Street.} \\[4pt] +\footnotesize \textsc{London}: Kegan Paul, Trench, Trübner \&~Company, Ltd. +\end{center} +%%%%%%%%%%%%%%%%%%%%%%%%% GUTENBERG LICENSE %%%%%%%%%%%%%%%%%%%%%%%%%% +\PGLicense +\begin{PGtext} +End of the Project Gutenberg EBook of Lectures on Elementary Mathematics, by +Joseph Louis Lagrange + +*** END OF THIS PROJECT GUTENBERG EBOOK LECTURES ON ELEMENTARY MATHEMATICS *** + +***** This file should be named 36640-pdf.pdf or 36640-pdf.zip ***** +This and all associated files of various formats will be found in: + http://www.gutenberg.org/3/6/6/4/36640/ + +Produced by Andrew D. 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Anyone seeking to utilize +this eBook outside of the United States should confirm copyright +status under the laws that apply to them. diff --git a/README.md b/README.md new file mode 100644 index 0000000..ad2e238 --- /dev/null +++ b/README.md @@ -0,0 +1,2 @@ +Project Gutenberg (https://www.gutenberg.org) public repository for +eBook #36640 (https://www.gutenberg.org/ebooks/36640) |
