Mathematics In the Enlightenment A Study of Algebra, 1685-1800

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8113173

Rid e r , R o b in E la in e

MATHEMATICS IN THE ENLIGHTENMENT: A STUDY OF ALGEBRA, 1685*1800

PhJD.

University o f California. Berkeley

University Microfilms International

1980

300 K. Zceb Road. Ann Artxsr. Ml 48106

Copyright 1980 by Rider, Robin Elaine All Rights Reserved

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Mathematics in the Enlightenment: A Study of Algebra, 1685-1800 By

Robin Elaine Rider B.S. (Stanford University) 1972 M.A. (University of California) 1973 C.Phll. (University of California) 1977 DISSERTATION

Submitted in partial satisfaction of the requirements for the deqree of DOCTOR OF PHILOSOPHY in

History in the GRADUATE DIVISION OF THE UNIVERSITY OF CALIFORNIA, BERKELEY

Approved:

Chairman

DOCTORAL DECREE CONFERRED

...OfCEWER 6.1980_ _

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1 Mathematics In the Enlightenment: A Study of Algebra, 1685-1800 by Robin Elaine Rider ABSTRACT This work describes the development of algebraic research and practice between Wallis' Treatise on algebra and Gauss' 1799 doctoral dissertation. In the period 1685-1740 mathematicians refined their algebraic heritage, mindful of the versatility of algebra in p"oblem-solving. Even the key theoretical advance of this period — the Nevton-Raphson method for approximating roots — aimed ac improving a problem-solving technique.

Other research sharpened the wording of Descartes' role

and the fundamental theorem (for factoring polynomials) without providing proof, and set the problems of higher-degree equations and multi-variable equation systems. During these years mathematicians also recast and extended the calculus in algebraic terms, a process accelerated by Euler's elaboration of an algebraically-based infinitesimal analysis in the 1740s.

This

confirmed the mathematical potential of algebra. The need to secure the logical foundations of algebra, on which a growing proportion of mathematics was based, and deepening concern with generality, proof, and elegance guided algebraic research in the years aftet 1740.

Because the theorem guaranteeing that each polynomial

equation had a root of form

a + b/^1

played a key role in the integral

calculus as well as algebra, it stimulated a series of proofs.

The

Importance of imaginary numbers also helped focus attention on other algebraic Issues:

the unsolved problem of higher-degree equations,

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2 elimination of unknowns from equation systems, demonstration of Descartes' rule, remaining theoretical weaknesses in approximation methods. These questions drew the attention of such notables as Euler, d'Alembert, Lagrange, Laplace, Waring, Vandermonde, Bezout, and Causs, as well as the efforts of lesser-known mathematicians.

Their research,

especially when guided by the desiderata of generalization and elegance, addressed newer questions of algebraic structures.

The structural

thinking that would come to characterize 19th-century algebra thus had roots in the 18th century. The efforts of these researchers, combined with the activity of hundreds of algebra teachers and textbook writers, secured the place of algebra as the fundamental language of mathematics and inculcated its principles In a variety of settings.

Condillac, Hutcheson, and other

Enlightenment thinkers also appropriated algebraic concepts with ease, cognizant of algebra's philosophical virtues as a well-made language and a versatile analytic tool.

The result was active pursuit of

algebraic knowledge in many quarters, consonant with the values of an enlightened age equipped with utilitarian aspirations, concerned with the structure of thought, and convinced of the Importance of mathematics.

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i

Ac k n o w l e d g m e n t s

This study grew out of research on Lagrange initially suggested by J. Heilbron and guided by R. Hahn.

I am grateful for the caliber

of their example as historians and for the care with which they and M. Rieffel read the entire manuscript. For guidance and assistance with source material, thanks are due to L. Charbonneau, P. Costabel, R. Hahn, J. Heilbron, I. Schneider, P. Speziali, R. Taton, B. Wheaton, A. Yushkevich.

I also profited

from consultation of the rare book and manuscript collections of the Academie des Sciences, Bibliotheque de l'lnscitut, Bibliotheque de 1'Arsenal, Ecole des Ponts ec Chaussees, and Bibliotheque Mazarine in Pari-?; the Royal Society, London; Trinity College, Cambridge; the Bodleian Library, Oxford; Deutsches Museum, Munich; and the university libraries of Geneva, Basel, and Lund.

The resources of the British

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ii

Library (BL) and the Bibliotheque Nationale (BN), as well as those of American libraries, were extensive and valuable.

The assistance of

the staffs of these institutions is gratefully acknoweldged. Finally, I owe special thanks to friends and colleagues — particularly G. Hamburg, J. Palumbo, and B. Wheaton — for their help and encouragement.

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Ta b l e

1.

of

contents

INTRODUCTION

1

A matter of definition The progress of algebraic research Extent of the enterprise Algebraic styles 2.

ALGEBRA BEFORE 1740

19

Approximation methods and formulas Newton's method Factorization Integration of rational fractions Descartes' rule Elimination 3.

A SHIFT IN INTEREST:

3 8 10 14

THE FUNDAMENTALTHEOREM

21 23 27 30 32 35 41

The issue of imaginazh.es

45

The fundamental theorem of algebra

54

D'Alembert's proof Euler's algebraic proof Lagrange Daviet de Foncenex Laplace Gauss' first proof

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55 58 65 68 70

72

4. HIGHER-DEGREE EQUATIONS

76

The form of the root The resolvenc and eliminacion theory A new approach Lagrange, Vandermonde, Waring Doubts about the enterprise Ruffini's statement of insolvability 5.

79 82 88 90 97 102

ELIMINATION THEORY

107

Linear syatemr

108

Cramer's rule The determinant for its own sake Solution of equations in a new light

108 109 113

IIon-linear systems Attempts at a solution: Clairaut,Cramer, Euler Problems with existing methods; degree of the resultant Bezout's general theorem

117 117

6. DESCARTES' RULE A geometrical approach The algebraic route Fourier's and Ruffini's proofs 7. APPROXIMATION The language of function theory Shortcomings in Newton’s method; their remedies Rival methods More difficulties in deployment 8. TRANSMISSION OF ALGEBRA Academies Publications Setting norms Higher education Textbooks Enforcing standards A bag of mathematical tricks Textbooks A variety of settings

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119 123 130 131 134 137 142 143 144 150 152 156 157 158 160 166 167 179 181 182 185

9.

CONCLUSION:

ALGEBRA AND THEENLIGHTENMENT

A universal language An enlightened view of algebra A moral algebra The ordered relations of algebra A congenial device NOTES

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191 i91 193 196 197 200 202

1 Algebra when viewed ac a distance Is a frightful phantom. But when we have courage to approach, its terrors vanish. — Charles Butler Que 1'Arithmetique nous fait aller a pied, que la Geometrie nous fait voyager 5 cheval et que l'Algebre nous donne des ailes pour planer dans les airs. — J.-J. Rossignol

1 In t r o d u c t i o n

In the latest installment of a long debate over the nature of Greek mathematics, Sabetal Unguru has reminded historians of mathematics that their task is to write history, not mathematics.

The central

question in the dispute has been whether Greek mathematics, and particularly the solution of geometric problems, had an algebraic component.

Unguru, a partisan of the "nay" school, accuses his

eminent opponents of reading into essentially geometric procedures the mathematically equivalent (but historically absent) algebraic techniques for p'.uuiea-solving.1 The dispute thus turns on a fundamental difference in attitude between mathematicians and historians.

The latter do not

doubt that past mathematical concepts and constructs contained, in votentia, their logical equivalents and consequences.

They do question,

however, the wisdom of rewriting the historical record in quest of such equivalents and consequences.

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2 The dispute also hinges on the proper definition of algebra.

Is

it a body of mathematical results, or a method for solving a variety of mathematical problems?

Is it coextensive with the use of literal

variables and numerical expressions arrayed in equations?

Modern

definitions of algeora might encompass all these notions, plus a good deal more, whereas only the definition of aigebra ?.? a problem-solving technique could serve in the argument that Unguru is countering.

Greek

mathemaLics clearly provided solutions to a number of geometric problems.

Once the problems are translated into an algebraic idiom,

they look like standard algebraic equations, whose solutions, rendered back into geometrical fora, are the same as thosi: offered by the Greeks. Explicit equation theory was lacking, however, and the texts nowhere use the algebraic symbolism we have come to expect. Mathematicians active in the loth and early 17th centuries created the kernel of what we know as algebra — with rudimentary equation theory, useful results, and recognizable symbolization.

Their successors improved

on their efforts, so that by the beginning of the 18th century algebra was an established mathematical field, encompassing theory and problem­ solving and expressed in distinctive mathematical style. The proper characterization cf 18th-century aigebra has still posed significant difficulties for historians. persistent definitional problem: teristics of algebra.

The first is the

to distinguish between dual charac­

Eighteenth-century mathematicians used and studied

algebra both as a body of research of intrinsic merit and as a mathe­ matical language of broad utility.

To concentrate on the internal

developments of the former is to miss the importance of the latter to mathematics and to insulate it from cognate developments in

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3 Enlightenment philosophy.

Second, the usual historiographical

attention to only a part of the ongoing algebraic research has obscured general trends in its 18th-century development.

Moreover, this narrow

focus has led historians to underestimate the size of the algebraic enterprise — the number of participants and the range of their activities.

Finally, 18th-century algebraists exhibited a variety of

algebraic styles, but the existence and effect of such stylistic dif­ ferences are most evident only when the full extent of algebraic activity is considered.

The remainder of this chapter will address

the issues of contemporary definitions of algebra, the progress of research, the size of the research and teaching enterprise, and the existence of different algebraic styles.

This will furnish a contextual

framework in which to fit the algebraic problems and practices explored in subsequent chapters.

A matter of definition L'esprit encyclopSdique, which enjoyed more popularity in the Enlightenment than did the already wide circulation of Diderot and d'Alembert's Encyclopddie. suggests a fitting point of departure for this study.

The 18th century produced a wealth of encyclopedias and

dictionaries, many with a particular emphasis on science, that key element of Enlightenment thought.

Their definitions of algebra, as

refined in practice and teaching, can serve as first approximations for a historical characterization of the field of algebra in the 18th century. From the outset algebra was defined as a method for the solution of mathematical problems, a characterization that persisted throughout

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4 Che 18ch century.

Ephraim Chambers' definition of algebra in his

Cyclopedia — "the art of resolving all problems which are resolvible" — was an extravagant expression of this attitude, which was echoed throughout the century.

In Charles Hutton's A mathematical and

philosophical dictionary, for example, algebra was held to be "a general method of resolving mathematical problems by means of equations."2 This sort of definition figured most commonly among chose for whom the proper business of mathematics, and especially its teaching, was the cultivation of problem-solving abilities.

Authors of elementary text­

books throughout the period sustained this opinion, untouched by the vicissitudes of research mathematics; it held particular

sway among

English mathematicians and teachers, for whom problem-solving was on important concern all during the 18th century. In an era with a pronounced utilitarian bent, such versatile and effective techniques assumed great importance.

The use of algebra was

seen to be appropriate and expedient, for example, in commercial matters from the calculation of interest and annuities to the business of lotteries and other games of chance. careers also learned algebra.

Those destined for military

The French military educator B£lidor,

for example, recognized that a knowledge of quadratic equations, arithmetic progressions, and conic sections was useful in studying ballistics, and that algebraic analysis applied as well to the determination of the metal content of bullets and estimates of piles of shells.

Similar problems also arose in Hutton's mathematics

course for the Royal Military Academy at Woolwich.

Although mathe­

matics was deemed militarily useful primarily as training and discipline for the mind, military and naval schools in revolutionary

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5 France felt the need for algebraic knowledge keenly enough co request numerous advanced algebra books for their libraries, and Marguerie pursued his studies of elimination theory in part because of their importance to naval science.

The mathematician-physicists of tne 18th

century also agreed that "mathematical inquiries may be instituted also concerning any [measurable] physical quantities.... [The measures of such physical quantities} may be reasoned upon according to the principles of algebra, and from such reasonings, new relations of the quantities which they represent, may be discovered."3 Other definitions of algebra operative in the 18th century reinforced the sense of its utility in the solution of mathematical problems.

The

concept of algebra as a universal arithmetic was hardly a new notion in the 18th century; its origins coincided with the formulation of algebra as an abstraction from numerical problems, a oalculua litteraii3 of rules as derived from arithmetic.

Newton so defined algebra, and the

Encyclopedia britannica. following his lead, called attention to both the arithmetic origins of algebra and its praiseworthy generality.

It is

a general method of computation by certain signs and symbols, which have been contrived for this purpose, and found convenient. It is called an Universal Arithmetic, and proceeds by operations and rules similar to those in common arithmetic, founded upon the same principles.14 The utility of algebra for solving numerical problems followed directly. From Descartes on, algebra had also demonstrated its applicability to the geometrical realm.

The construction of equations and the

analysis of geometrical questions proved that algebraic rules ard operations applied to geometrical magnitudes as well as numbers, and the incorporation of analytic geometry in 18th-century algebra texts

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reflected this fact.

No longer was geometry presented as the exclusive

domain of synthesis; analytical reasoning, clearly in algebraic dress, was restored to geometry through the efforts of Descartes and his successors.

This conception of algebra as a literal calculus suited

to any type of quantity, numerical or geometrical, gave way in the 18th century to a still broader one:

the science of quantity in general.

This definition, operative throughout the century, was expressed forcefully by Bossut in 1773:

"Algebra is a science that considers

abstract quantities in general, and that compares them in the same way that arithmetic compares numbers."5 This generous definition of algebra helped guarantee the diffusion of its techniques and concepts throughout the mathematical domain.

As

the gentlemen who compiled the Encyclopedia britannica explained, the "extent and generality" of algebra are "its greatest excellence." The Oratorian Bernard Lamy noted that grandeur — that which could be agumented or diminished — comprised both material and spiritual entities, since the company of angels could be augmented or diminished. Surely then, any mundane geometrical or arithmetical entity was subject to algebra.

In De usu algebrae Kullin quoted Richard Sault to the

effect "that no one can be a good Geometrician, that is not a good Algebraist, by consequence no tolerable Mathematician without it."6 And certainly all questions of number theory fell within the purview of the universal arithmetic.

Although neither the geometrical nor the

arithmetical applications of algebra were unique to the 18th century, its teaching and research stressed the general mathematical utility of algebra. Consequently Kullin, a student at the University of Uppsala,

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7 spoke for many European mathematicians in 1743 when he defended the thesis:

”He who neglects algebra does not deserve the name of

mathematician."

Early in the century Varignon had listed the uses

of algebra to demonstrate all sorts of theorems in all parts of mathematics, "to solve problems and make discoveries that seem far beyond the feeble reach of the human spirit."

Sauvin and others

reiterated the point, speaking of the "great progress" that geometry owed to algebra, and the ease algebra conferred on otherwise difficult mathematical concepts and questions.

Indeed, "almost every branch of

mathematics" made successful appeals to algebra.7 A notable success was the recasting and extension of the infinitesimal calculus in algebraic terms.

Progress in this direction began in the

late 17th and early 18th centuries with the work of Leibnizian mathe­ maticians.

In 1704 the physician-physicist Polini&re could already

refer to "authentic proofs of the excellence of algebra" in "this new infinitesimal geometry."8

Progress accelerated with Euler's elabo­

ration of an algebraically-based infinitesimal analysis in the 1740s, a dramatic confirmation of the mathematical potential of algebra and an influence on mathematicians throughout the remainder of the century. As a result of these varied applications, algebra had emerged as the fundamental language of mathematics by the close of the 18th century.

This operational definition, overlaid on algebra as the

art of problem-solving, was fortified by the widespread Enlightenment concern for the relation of language to analytic thought.

It was

stated in many quarters, from Newton's Arithmetics universalis to Condillac's La langue des calculs, and exemplified in much mathe­ matical research, especially after mid-century.

Together with the

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8 enthusiastic Enlightenment faith in the utility of mathematics, this definition helped imbue algebra with a broader cultural significance as a model of analytic thought and fostered the application of algebra to a variety of non-mathematical subjects.

As Baselli characterized

the situation in the 1780s, the "generality and fecundity of the algebraic language" had assured the application of algebraic rules "to every subject into which mathematical reasoning can be introduced."9

The progress of algebraic research At the same time the body of algebraic theory expanded and changed. Throughout the 17th and 18th centuries the theory of equations remained the "soul of algebra,"10 but to the emphasis on improving problem­ solving capabilities was added the need to shore up the logical basis of algebra, in recognition of its fundamental importance in all of mathematics. I shall begin with Wallis' Treatise on algebra, both historical and theoretical (1685).11

Wallis' work reflected the assimilation of

algebraic theory and practice as advanced by Vifete, Harriot, and Descartes, confirmed the mathematical worth of the analytic art, and helped guide subsequent research by his introduction of some of Newton's algebraic innovations.

The theoretical efforts of the half

century following Wallis' Treatise were directed toward working out this heritage.

A preoccupation with algebra as problem-solving

operated, however, to limit the accomplishments of algebraists before 1740 to the elaboration of Newton's technique for approximate solution of numerical equations and to the identification of other crucial questions in algebraic theory without evident concern for proving

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9 key results.

The algebraic work in the latter category, just because

of its inconclusive character, has been largely ignored in the historiography of mathematics, but is quite consistent with the work on approximation before 1740, given the focus on problem-solving and particular cases. A shift in mathematics, and in algebraic research in particular, began to be evident beginning in the 1740s, due in large measure to the Eulerian efforts to build a mathematical analysis, both finite and infinitesimal, on an algebraic foundation.

This served to focus

attention on flaws and gaps in algebraic theory, and encouraged both the generalization and the logical demonstration of the theory of equations.

Such stimuli produced a number of innovative and influential

pieces of research after mid-century, although some of the critical problems still lacked complete or convincing solution by 1800. Most historians of 18th-century mathematics, when they have looked past the dramatic accomplishments in infinitesimal analysis, have seen only the attacks on the principal unsolved algebraic problem, the formulaic solution (in radicals) of the general fifth-degree equation.

In the imaginative but ultimately unsuccessful research of

Lagrange, Vandermonde, and Waring (all writing in the early 1770s), historians have sought, with the assurance of mathematicians, the roots of group theory, since the eventual solution of the problem in the 19th century relied on just such a theory.12 approach has had several consequences.

This historiographical One is the focus on 1770 as a

watershed, rather than the consideration of a longer period of devel­ opment.

The second is lowered esteem for the accomplishments of a

number of adept and innovative algebraists, stumped by a notably

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difficult problem.

Studies of one or anothor-problem in 18th-century

algebra have also suffered from a lack of peripheral vision.

Extent of the enterprise The algebraic enterprise, defined to include both research and publication of textbooks treating algebraic topics, engaged over threehundred individuals between 1685 and 1800.

The hundred who qualified

as research algebraists did so by virtue of their publications on alge­ braic topics, centering on the theory of equations.

These publications

included memoirs, in academic collections and other learned journals, mathematical monographs, plus ocher scholarly publications not specif­ ically intended for textbook use.

An occasional textbook might have

contained an account of the author's algebraic research, and such texts have been classified here as research publications; for the most part, however, authorship of a textbook covering algebraic topics (but con­ taining no original research) assigned an individual to another category than research.

Textbook authors swelled the full cast to some three

hundred, and their activity did much to secure algebra's position as the fundamental language of mathematics. The variety among the algebraists; 13 striking, although scarcely surprising.

The intellectual prestige of mathematics in the Age of

Enlightenment encouraged the pursuit of mathematical knowledge in many occupational settings, especially since the scientist's role still lacked precise definition.

Most often, research algebraists were

affiliated with teaching institutions as members of the professoriate and with learned societies or academies.

Some could be considered

professional mathematicians, their research supported by the

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11 cnlighcened crowns of Europe.

In Che professorlace were chairholders

in logic, history, belles letcres, and hydrography as well as (more predictably) professors of mathematics, physics, philosophy, and astronomy.

Other researchers maintained themselves in a variety of

occupations.

Mourraille, active in the academic life of Marseille,

eventually held the post of mayor; Hulbe was secretary of the royal lottery in Berlin; Marguerie served in the French navy; Hellins assisted the royal astronomer in England. religious orders:

A number were members of

besides the Jesuits (notably V. Riccati and Jacobs),

whose research and teaching helped shaped scientific life in 18thcentury Europe, there were the Minim priests Leseur and Jacquier in Rome, the Benedictine bishop Walmesley in England, the parsons Rovning and Campbell.

Some algebraists occupied exalted positions — Baron

Maseres of the Exchequer in England, che two marquis Fagnauu and Courtivron on the Continent — while another, E. Stone, withdrew from che Royal Society and died in poverty.

Many worked in the capital

cities of Europe; others, like Micoion of che Clermont academy, remained in the provinces.

Not only mature scholars, but also che

young, like Fourier, Poisson, Marguerie, and Gauss, produced noteworthy and innovative scholarship. The fruits of their research ranged from short notes to major monographs, and quality varied as much as length.

Some work was

pioneering; ocher results were familiar despite ill-informed claims for their novelty.

Some approaches led to the same mistakes and

impasses as they had in earlier centuries; ocher research found new routes to failure, or to marked success. Research was dominated by five outstanding problems.

An exact

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12 formulaic solution of the fifth-degree (or more generally the degree) equation had long been a desideratum.

nth-

Without such formulas

the approximation of roots of numerical equations had assumed greater importance, and in the early years of the 18th century algebraists directed considerable effort to devising accurate and expedient approximation techniques.

Especially from the 1740s on, attention

turned to other algebraic problems:

the proof of Descartes' rule

(which related the number of positive and negative roots of an equation to the signs of its coefficients), the elimination of variables in the solution of multi-variable equation systems, the fundamental theorem of algebra as it is now called (on the factorization of polynomials), and the still unsolved fifth-degree equation.

Of the research

algebraists, 60 per cent worked on one or more of these five problems. The academicians among them were especially concerned with the five problems:

two-thirds of those with academy credentials attacked one

or more, and their research on these problems predominated.

If it was

more likely that the perceptive and adept mathematicians were honored with academic membership, it seems evident that che best mathematical judgment of che day deemed che five problems significant and worthy of attention. Algebraists participated in an upsurge of research interest beginning in the 1740s.

Nearly three-quarters published their first

algebraic work after 1740, three times the number active in the fifty-five years between 1685

and 1740.

Novj?fs explanation of the growth —

interest in solving equations increased only with the "pronounced" quantitative increase in mathematical scholarship in mid-century13 — misstates che case.

Interest in che solution of equations, defined to

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13 Include che exploration of cubics and biquadratics as well as

nrh-

degree equations, gradually increased between 1685 and 1800, but the formulaic solution of lover-degree equations attracted considerable attention even before 1750.

The evidence of vigorous activity in

numerical solution early in the century also shakes Novy's claim. Research patterns also link algebraic research and the 18thcentury investigations of che potent infinitesimal calculus.

Forty

per cent of che researchers also examined one or another facet of infinitesimal analysis.

The connection was particularly pronounced

beginning in the 1740s and throughout the rest of the century, reflecting che expanding recognition of algebra's fundamental role in infinitesimal analysis.

This development owed much to the Eulerian

transformation of che calculus and che growth of an algebraically-based function theory.

Both changes accelerated in che 1740s.

Far fewer

researchers, by contrast, combined their interest in algebraic topics with the older geometrical tradition of the analysis of curves to the exclusion of the calculus.

The correlation between research on che

five key algebraic problems and Chat in infinitesimal analysis was particularly strong, reinforcing the argument that che needs of che calculus prompted attention to fundamental unsolved algebraic questions. Algebraic ideas were not the private domain of 18th-century research mathematicians, however central they were in che expansion of mathematical knowledge.

Partly as a consequence of their mathematical

importance, algebraic notions spread beyond the bounds of research, but this diffusion fortified their significance in a broader context.

The

propagation of algebraic ideas was achieved in part by textbooks, both those dealing solely with algebra and Chose more general texts including

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14

algebraic sections.

There were over two hundred such textbooks, plus

numberous re-editions and translations.

The 18th-century expansion

and development of mathematical culture induced changes in the teaching of algebra.

The appearance of mere and more algebra texts

both reflected and helped propagate the algebraicization of mathematics. Especially as research mathematicians applied their talents to the business of instruction, the quality of algebraic education also rose. The growth of mathematical education thus provides additional evidence of the vitality of 18th-century algebra and the value of algebraic skills and concepts.

Algebraic styles A variety of algebraic styles marked che broad expanse of algebraic research and practice.

The examples of Lagny, de Gua, Bezout,

Uaring, and Ruffini illustrate this diversity, although in choosing them I neither claim that they exhaust che possibilities nor propose firm definitions of algebraic styles. Lagny honored che problem-solving tradition in algebra during his years at the Paris Academy of Sciences.

He devoted much attention

to problems of approximating che roots of numerical equations, and applied impressive, if ad hoc, ingenuity to their solution. was a mass of techniques, formulas, and examples.

The result

For instance, he

offered both rational and irrational formulas for che solution of z

- pz - q,

with variations according to the signs of

p

and

q.lu

Many ocher algebraists throughout the period shared Lagny's preference for particular problems and special cases.

Nicole, one of Lagny's

successors at the Paris Academy, attacked repeatedly che problem of

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15 che irreducible cubic — a special case of che cubic equation wich all real roccs.

Ic had che annoying propercy chac no finice expression

free of imaginary quancicies could be found for ics real roocs, alchough an exacc solucion involving lmaginarles had long been available. Icallan mathematicians resumed che quesc for a "sacisfaccory" solucion in Che 1780s, crue Co che problem-solving cradicion in Icalian algebra chac had produced Che original solucion.

Many English algebraiscs also

subscribed co che problem-solving cradicion:

wlcness de Moivre's scudy

of parclcular forms of higher-degree equacions admicclng formulaic solucion, Stanhope's proposal, wlchouc explanacion or generalizacion, for a recurring series solucion of one parclcular fora of quadracic equation,15 and che space of English algebra cexcbooks emphasizing praccical appllcaclons and parclcular cases. A second scyle drew insplracion and insight from geomecry.

In

che 18ch century, che word geomecer meant machematiclan, noc merely geometrician.

Noc all mathematicians, and cercainly noc all algebraiscs,

were geometricians at heart, but some, including the Paris academician and onetime editor of che Encyclopedic, che Abbe de Gua, clearly approached algebraic research from che side of geomecry.

In his proofs

of Descarces* rule for che number and nature of an equation's (real) roocs, de Gua could draw frequencly on geomecrical resulcs, since he conceived of polynomial equacions as curves in a Cartesian analytic geometry.16

Similarly, recourse co che geomecrical representation of

polynomials allowed Fourier and Mourraille to refine che NewtonRaphson approximation method and thereby guarantee greacer accuracy. This view of the merits of geomecry differed markedly from that of Lagrange, whose avoidance of geomecrical diagrams was widely known.

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A

preference for the geomccrical and a faith In the primacy of geometrical results could affect the content as well as the presentation of algebraic results.

De Gua ignored Imaginary roots in his treatment

of Descartes' rule, consistent with the assumption chat only chose roots at the intersection of curve and axis really mattered. The third style mediated between the demands of pure theory and of application.

Its hallmarks were attention to theoretical generality

and demonstration as well as a sensitivity to the needs of chose who applied algebraic methods in problem-solving.

It was an attitude

shared by a number of applied mathematicians in the 18th century and exemplified by Etienne Bezouc, a Paris academician and influential figure in French mathematical pedagogy.

His concern for the practical

side of algebra issued from his involvement in the technical and military education structure in France, and prompted him to supplement the general methods for the elimination of variables with special methods more expeditious in practice, if not fully general.17

In all

he displayed a great facility for the fruitful manipulation of algebraic quantities, a talent possessed by many of his contemporaries, but he combined computational patience and intuition with a keen theoretical eye.

Like the work of Vandermonde and others, Bezout's

algebraic publications — with their strings of algebraic manipulations and elaborate notation — try the patience of modern readers.

As such,

they also demonstrate the vitality of the computational, problemoriented tradition in conjunction with the impulse toward general theory. A fourth style hearkened back to the arithmetic origins of the algebraic art by linking algebraic theory with what we now call number

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17 theory.

For example, Waring made substantial contributions to number

theory, an Interest which also informed his algebraic research. Bashmakova has drawn attention to Waring's exploration of numerical realms in connection with his extension of Newton's work on the factori­ zation of polynomials.

In generalizing Newton's algorithm for finding

coefficients of a given form. Waring continued the discussion of what are now called extension fields, a concept central to Galois theory. Waring's work in algebra bore the unmistakable imprint of number theory, as did Gauss' research.18 to arithmetic insights.

Theirs was a sophisticated and fecund appeal

The same was not true of the reactionary

algebra of Baron Maseres and his mathematical coterie.

Their rejection

of negative and imaginary numbers as non-arlthmetlcal and therefore nonsensical caused them to reject such fundamental propositions as equating an equation's degree and the number of its roots as "all jargon, at which common sense recoils."

They condemned those algebraists

who permitted negative or imaginary numbers as grave men, imitating the philosophers of a well-known region, who were extracting sun-beams from cucumbers, wasting the midnight oil just as profitably in settling the rights and privileges of impossible quantities.19 In their own algebra textbooks, their prejudice against such mathematical expedients forced them to long and clumsy derivations and a proliferation of special cases — a waste of algebra's potential. In still another mathematical style (not confined to algebra) the goals were elegance and generality in a comprehensive presentation of the subject matter.

An early example of the tradition was Reyneau's

Analyse dSmontrge.23 an attempt to united the analytical techniques and results of the late 17th century in one general treatise.

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By the

18 lacter part of the 18th century Reynea's ?,oal was best accomplished by the invocation of unifying concepts such as function theory.

In

the appeal to functions, especially functions of an equation's roots, Euler's work guided those algebraists who sought to systematize and secure the results of equation theory.

Two outstanding practitioners

of this style were Lagrange and Ruffinl.

At the same time educators

appropriated these goals in their construction of algebra courses and textbooks, improving the caliber of mathematical instruction as they fortified the claims of algebra to a fundamental position in mathematics itself. The variety of algebraic styles in the 18th century was built on a strong problem-solving tradition, in which algebra had proven its value in many mathematical questions.

As the integration of algebra

in mathematics progressed, algebraists, especially those after 1740, added interests in theory and generality to these varied styles, and, toward the end of the century, the polish of mathematical elegance. The results of these efforts was a body of algebraic knowledge enriched by new results, more firmly grounded in theory, and widely accepted as useful.

They also hinted at a new brand of algebra.

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19

2 Algebra before

1749

John Wallis' 1685 Treatise on algebra, both historical and practical allows us to survey the algebraic legacy inherited by 18th-century mathematicians.

Wallis' strident nationalism, which denied Descartes

credit for anything, resulted in a less than judicious account of European algebra, but his shortcomings in the fine art of history do not diminish the historical value of his work.

His was an attempt to

impose some order on the algebraic output of his predecessors, since the algebraists of the 16th and 17th centuries had bequeathed to their successors a "piecemeal collection of methods and results."1 Wallis began with the canons of algebraic manipulations, calling attention to the development of appropriate notation and presenting the extraction of roots and the study of proportions in algebraic garb. In assessing the history of the algebraic art, he also dealt with available theory on the behavior of polynomial equations and their

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20 roots.

He favored Harriot's conception of the formation of polynomials

out of linear factors, which facilitated the construction of canonical 2 equations such as (x - a)(x + b) ■ x + (b - a)x - ab and (x - a)(x - b) ■ x

2

+ (-a - b)x + ab,

and illustrated neatly the

relation between roots and coefficients (e.g., the constant term equalled the product of the roots).

Wallis' presentation of equation

theory also contained the fundamental if unproven tenets that an equation of degree

n

had

n

roots, that imaginary roots occurred in

pairs, and that the sequence of positive and negative coefficients in the equation determined the number of positive and negative roots.

He

then devoted considerable attention to the actual solution of poly­ nomial equations, presenting the various known solutions for degrees two through four (including Descartes’ method for the decomposition of quartic equations into two quadratic factors, explained by reference to Harriot's principles).

Further attention to the problem of estab­

lishing limits for the roots was recommended, as one means toward greater facility in the solution of numerical equations.

Wallis also

provided the first published explanation of Newton's new method for approximate solution of numerical equations, as well as elucidating an older method that proceeded by analogy to the arithmetical extraction of square and cube roots. Wallis' treatise revealed a cognizance of the broad spectrum of problems to which algebra could be applied.

Wallis, like Newton and

many other mathematicians (especially the English) emphasized the interpretation of algebra as a universal arithmetic and cited its utility in arithmetic and number-theoretical questions.

But this

preference for the arithmetic (and Wallis' prejudice against jescartes)

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21 did not preclude a demonstration of the contributions of algebra to geometry.

Wallis also recognized the potential advantages of using

algebra in infinitesimal mathematics, and Included algebraic treat­ ments of both the "arithmetick of infinites" and series. But Wallis did not confine

himself to problem-solving; he also

sought greater generality in algebra and the demonstration of theor­ etical results.

He called, for example, for a generally effective

method to determine the divisors of a polynomial, and rejected the concept of algebra as a curiosity cabinet of unrelated applications and problems. of

(a + e),

In his explanation of the binomial theorem for powers these attitudes, along with a fairly lax standard of

proof, emerged: I always pursue so far...till[induction] lead me into a regular orderly process...in which I acquiesce as a sufficient evidence, when there is no colour of pretence why it should be thought not to proceed onward in like manner. And without this, we must be content to rest at particulars (in all such kind of process) without proceeding to the Generals. 2 The two approaches to algebra in tension in Wallis' work — one focused on particular applications and the other attempting to construct a coherent, general, and demonstrable theory — would both reappear as algebra developed between 1685 and 1800.

Approximation methods and formulas Throughout the period, mathematicians recognized clearly that "in the practice of algebra, all equations reduce finally to numerical equations."

But for degrees above four, there was no general algebraic

formula into which a practitioner could plug numerical values, for here practical needs had outstripped theory.

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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 solutionof all problems which necessity or curiosity may offer. All during the 18th century there were thus mathematicians who considered "the business of resolving equations tant branch of algebra. " 3

by approximation...themost impor­

Certainly in the late 17th and early 13th

centuries the approximation of roots dominated algebraic research; notable innovations resulted. When Wallis' treatise was published in 1685, two arirhmetical methods for numerical solution had long been available.

The first

general techniques promising any success had involved double false position and interpolation, that is, positing two values on either side of the desired root and then defining an approximate value in between. values

For the equation a and

3

f(x) ■ x° +■ axn ^ + ... + p ■ 0,

two

could be found by trial and error such that

was smaller than the desired root and

x ■ 3 was greater.

x ■ a

Mathe­

maticians developed various schemes to find intermediate values between

a

and

S

that would seem to converge to the true root.

One could, for example, choose the mean value lying between the two posited quantities

a ■

(a^ + a2 >/(b^ + b0) and

3

■

Another approach to the problem of numerical solution proceeded by analogy to the standard arithmetical method of extracting square roots. Even late in the 17th century Rolle would offer a method of approxi­ mation that "strongly resembles that used ordinarily for the extraction of roots."1* Other mathematicians took a more algebraic approach to what Lagny, a champion of approximation studies, called the resolution r4gulikre

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23 of numerical equations: tc proceed by a "universal and infallible" method.

This method would be made more perfect as it was made shorter

and simpler.2 Although none would dispute Lagny's goal of resolution reg-uli^re, his work was a center of controversy in France.

Not only did he fall

under attack in the 1690s on a priority issue; the generality of his technique was also questioned.

He, as well as his opponents, emphasized

approximation formulas and presented a profusion of special cases lacking a unifying principle or procedure.

The emphasis on formula

rather than method was the key weakness of the Parisian appToech to approximation.

£ven Richer, the editor of Lagny's work, had difficulty

in casting the thirty folio volumes of memoirs (unpublished and pub­ lished) in coherent form in Analyse qgnerale. This work did not appear until 1733 (although many of Lagny's papers had been published separately during his lifetime);^ by this time the issue of approximation method had been settled in favor of a more general Newtonian method.

Newtor.'s method Newton had developed a new approximation technique in the context of his integration work, drawing inspiration from research on infinite series.

He described the method in a manuscript shown to Barrow and

Collins in 1669, but it first appeared in print in Wallis' treatise 3

in 1685.

For the example

procedure, set root.

Thus

y ■

2

+ p,

-1 + lOp +

p ■

0.1

+ q

- 2y - 5 ■ 0,

where 2

6p

above the first, yielding substituted

y

+ p

3

2

* 0.

Wallis, following Newton's

was the closest integer to the He suppressed powers of

- 1 + lOp *0,

or

p » 0.1.

into the cubic equation in

p,

p

Then he arriving at

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24 2 3 0.061 + 11.23 q + 6.3 qw + q - 0, or (neglecting the small quantities 2 q“

3 q ) 0.061 + 11.23 q ■ 0.

and

The process could be continued at

will, resulting in a series expression value of

y.

2 + p + q + r +

...

for the

The option also remained, at least in Newton's later

published account of the method, to utilize the quadratic term of the equations for one couid solve

p, q, r, etc. Thus, rather than solve

-1 + lOp + 6p

2

*0

-1 + lOp * 0,

for the smaller real root.

This

could presumably afford greater accuracy or quicker convergence.7 The first significant modification of Newton's method was achieved by the English mathematician Raphson.

Nordgaard, author of the principal

account of the development of approximation methods, goes so far as to claim that Raphson was the first to expand Newton's method into a system. Certainly the changes Raphson introduced shortened and simplified tl.e procedure.

They were contained in his Analysis aequationum universalis,

first published in 1690.

While Newton had solved only one cubic

example, Raphson solved thirty-four problems

of varying degrees using

"theorems" or approximation formulas arising from the substitution of a + p

for the variable

y

and suppression of the second and higher

powers of the small quantity

p.

Following the formulaic tradition,

he aiso provided canone3 airectoTri, — a collection of forty-six cases of quadratic, cubic, and quartic equations with formulas for the proper supplement p to the initial approximation

a.9

Raphson's principal innovation, however, was the recourse, at each stage of the approximation, to the original equation, rather than the use of additional equations in procedure. y ■ 2.1

p, q, r, etc.

throughout the

In Newton'3 example, Raphson would have substituted

+> q

into

y

3

- 2y - 5 ■ 0,

not

p - 0.1 + q

into the

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25 derived equation

-1 + lOp +

6

p"* + p

3

0.

His contribution was thus

to simplify Newton's method by eliminating the need for auxiliary equations. Hailey's contribution to approximation research also facilitated the application of Newton's method.

In considering one of Lagny's

memoirs on approximation formulas, Hailey explained, he "fell upon a general method of forms for any power whatever, which is elegant enough...."

These forms consisted of a table of powers

cz, dz“, ez3,

etc.

to be plugged in when making the substitution called for in 2 Newton's method. For the polynomial equation b + cz +• dz~ + ... * 0, for example, the substitution

z ■ a + e was to be facilitated by

Hailey's forms + ce

cz

- ca

dz2

- da2 +

2 dae

fz3

- fa3

3fa“e + 3fae

4 82

4 . 3 + , 2 2 , 3 - ga + 4ga e 6 ga e + 4gae

9

+ de“ 2

+ fe

3 4 + ge , etc.

What Hailey called his "general analytical speculum" was hardly a theoretical innovation, considering Newton's more general statement of the binomial theorem for the expansion of power.

(a + e)

to any real

But Hailey's use of this arrangement not only obviated the

need for Lagny's elaborate tables and formulas; it allowed Taylor to recognize

the similarity between Hailey's result and his own work on the

expansion that now bears his name.

With this recognition, the approxi­

mation of roots could be cast in more elegant form and related to ongoing research on the infii .tesimal calculus. In 1717 Taylor published "An attempt towards the improvement of the method of approximating, in the extraction of the roots of equations

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26 in numbers" in which he re-examined the entries in Hailey's general analytical speculum. ca + da

2

He collected the coefficients of

3 + fa + ... ;

chose of

"> c + 2da + 3 fa“ + ... ; etc.

e,

and recognized their identitywith his own expansion f(a + e) - f(a) + f'(a)e + for the original polynomial expression f(z) -0,

(a + e) f(a+e)

of

£(z) =

e 2 + ...

-> 3 f(z) ■ b + cz + dz“ + fz + ... . For the

to qualify as an approximation to the root of would need to approach

approximation of

£(a + e),

0,

e ■ -f (a)/f'(a).

Taylor found

e^,

f(a) + f'(a)e,

0.

Taking the first

and setting it equal to

This preliminary estimate of

implied the simple approximation formula

a -

I

/

e

for che root.

When Taylor took instead f (a) + f ’(a)e +

e2 -

[ f'(a) + e -fy a'

j

0

- - f(a)

- f (a) f.(a) + eTia}

and substituted the first value

e ■

-f (a) f (a)

in the denominator, he

arrive at a more complicated expression - f (a) e " £ u;

. f(a)f"(a) 2f'(a)

to serve in a second approximation

(a + e)

to the root.10

The key

innovation here was not the derivation of better formulas, but Taylor's appeal to che infinitesimal calculus in his algebraic research.

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27 Factorization 3y 1685 the factorization of polynomials had long been a topic of analytic concern.

In his work of 1632, for example, Harriot spoke

to the question of '.he formation of equations; he understood the biquadratic

4 3 2 x + ax + 3x + yx + 5 ■ 0

plication of the four factors

to be formed by the multi­

(x - b)(x - c)(x - d)(x - f) ■ 0.

Descartes' solution of the biquadratic depended on its factorization into two quadratic factors with coefficients to be determined in the course of solution.

In 1657 Hudde sought rules for reducing equations,

even those offifthand sixth degree, to factors, and provided tables of workable factors. Wallis' work.

Similar tables compiled by Merry ended up in

All these investigations proceeded with the intent of

solving, or simplifying the solution of, specific equations or classes of equations.** This trend continued in the work of Newton and Maclaurin.

The

simplest approach to factorization was to find the rational divisors of the constant term

of the polynomial.

tuted for che variable (x - d)

x,

If a divisor

d,

when substi­

caused the polynomial to vanish, then

was a linear factor of che polynomial.

This approach relied

on Harriot's notion of che formation of equations.

Greater complexity

attended the question of non-linear factors, and Newton's algorithm for finding surd divisors reflected chat complexity. 12 In devising the algorithm, Newton approached two general problems pertaining to che general polynomial of even degree denote the polynomial by (following Bashmakova) . 13 (1)

t^m m

2m.

We may

+ p^x^m ^ + ... + P2 m * 0

The two problems were

Determine where there existed an integral, non-square number

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28 n

such chat

f2 m

could be factored into

where the coefficients of a +• b»'n,

for

a

p., p_,..., p_ jl i ~m (2)

If so, find

n

and

b

5

31

and

v

31

m

and

were of the fora

rational numbers (just as

were defined to be). and the coefficients.

The examination of the case of degree

2n « 4 will illustrate

the pat tern Newton followed in establishing an algorithm. quantities

k, £,

n

in order to complete

would then equal the difference of

factorization would be

f^ - ?2* v,

He sought

f f^to a square: •4

f^ + n(kx + i) 2 » (x2 + 4 px Sincef^

v_,

+ Q)2.

two squares,

the desired

where

t1, ■ (x2 + h px + Q) + v'ii (kx + 1) -x

2

+ x(4p

+

r~ r k«n)+ *ni + Q

and > _ - (x“ + 4 px + Q) - /n (kx + 1) ■x

2

+x(4p

r-k ^r.) - •> 2_ (x - a*V-l)(x“ + a /-l).

He did not, however, see how it could be

further decomposed into real quadratic and/or linear factors.

The

answer to Leibniz's challenge was published by Nicolas Bernoulli in 1719, when he noted that x* + a** ■ (a 3ut the general

+x ) -

2 a"x“

* (a + x*’ + ax^T) (a“ + x" - ax»I) . 1 '

problem remained of interest.

In 1719Taylor,

evidently in possession of the same general result, challenged the geometers of Europe with similar examples of the integrat_on of rational fractions.

His challenge was conveyed to Jean I Bernoulli

through the agency of Montmort.

Bernoulli responded in two forms.

In

a piece in the Acta eruditorum he cited his own 1702 method for inte­ gration, and produced many specific propositions toward the reduction of integrals to the quadrature of circles or hyperboles.

He also

wrote Montmort in a letter intended for Taylor: If you would like to take the trouble to read what I communicated in the 1702 [Paris] memoirs and in the Leipzig acts in 1703 on the integration of rational fractions (which pleased you extremely if I recall correctly), you will agree that Taylor used no other method than mine. If any doubt remained in Taylor's mind, Bernoulli suggested an integration context enlivened by a wager of fifty guineas apiece, with Montmort as adjudicator.

This move, predictably, succeeded in arousing Taylor's ire.

He rejected Bernoulli's mathematical challenge, but did not forbear a

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32 personal actack: It is impossible to express admiration enough at Mr. Sernouilli's [sic] excess of vanity, which makes him imagine himself to be the sole instructor of all the world. I can easily forgive his pretensions to have instructed me, because he has now lately pretended to have taught Sir Isaac Newton himself....'" The disputants missed a crucial and debatable point.

In allowing

nationalistic and personal pride to dominate the discussion, they over­ looked the logical necessity of proving the general claim.

Their

assertion of the result was not mistaken, but no proof of the possibility of factorization appeared before the 1740s.

Descartes' rule A key — and controversial — element in the algebraic legacy of the 17th century was Descartes' rule of signs for determining the number of positive and negative roots of an equation.

Descartes had offered

the rule without proof in the third book of his Geometry, published in 1637: We can determine also the number of true and false [negative] roots that any equation can have, as follows: An equation can have as many true roots asit contains changes of sign, from + to - or from to + ; and as many false roots as the number of times two + signs or two - signs are found in succession. In the equation

4 3 2 x - 4x - 19x + 106x - 120 ■ 0,

could be three positive and one negative root.

for instance, there

The rule had also

appeared in the Artis analvticae praxis, a 1631 work by the English mathematician Harriot, and had perhaps been known to Cardano in the 16th century as well. 19 The late 17th-century controversies surrounding the rule were of two sorts.

First, the question of priority:the French predictably

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33 defended Descartes' claim to the rule, while the English gave credit to Harriot.

3 nd

Germans

More than priority was at issue, however.

Descartes' formulation failed to mention the case of imaginary roots. But if an equation had imaginary roots, the rule equating the number of sign changes and the number of positive roots could fail to hold. The issue was raised by Rolle, among others.

He presented examples

where the strict formulation of the rule failed.

The equation

2 1 U 3 1 (.z" - 2z - 3) (z" + z + 6) ■ z - z + z“ - 15z - 18 * 0,

for instance,

had one positive, one negative, and two imaginary roots, although the rule predicted three positive roocs.

At the same time, Ozanam defended

Descartes’ formulation of the rule as infaiZlible.

In particular, he

thought the original form of che theorem held true even for equations with imaginary roocs.

This required the proviso chat an imaginary root,

assumed by Ozanam to be of the form positive if

A

were positive.

A + B»-l,

would be classified as

Ozanam demonstrated his contention for

degree two without difficulty, using che well-known fact that imaginary roots occur in conjugate pairs.

The demonstration for degree three

went less smoothly, and Ozanam stopped there without completing the induction argument.

Prescet entered che fray, maintaining an inter­

mediate position between Ozanam and Rolle, defending Descartes' wording of che rule while acknowledging that imaginary roots presented exceptions to it.

Prestet argued that Descartes did not mean che rule

to be general, and offered as evidence Descartes' wording, "peut avoir autant de vrayes fracines], que les signes fois estre changes."

+

et

-

s ’y trouvent de

This position was unacceptable to Rolle, who

maintained that Descartes had "himself declared chat che rule is not general. " 2 0

At stake was the proper formulation of an important theorem

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34 in equation theory, taking into account the possible complication of imaginary roots. The vexing question of the number of imaginary roots also interested Newton.

After illustrating the ambiguities involved in equations with

some imaginary roots, he offered a complicated and "tedious" rule for determining the number of imaginary roots, although without much explanation or motivation.

Maclaurin elucidated Newton's results by

referring to the proposition (known to Rolle as well as Newton) chat the roots of a polynomial equation

xn

limits of the roots of che equation

Axn ^ ■+• ... = 0

nxn-i

+ (n-l)Axn

2

were che + ... - 0

and che roots of che latter separated chose of the former.

Were any

roocs of the second equation impossible, some roots of che proposed equation would necessarily be impossible.

By successive differentiation

(although Maclaurin did not use che term) one arrived at a quadratic equation, che character of whose roocs was readily determined.

The

roots of che quadratic limited che roots of che preceding cubic and so on.

The rule could fail, however (notably when all roots of the

limit equation were real), since che number of limits (roocs of the second equation) was one less than the number of roots in the given equation.

Thus the limits could no more "fix che relations" of the

roots than "equation'; fewer in number than the quantities sought can furnish a determinate solution of a problem."

Campbell and Stirling

also attempted to refine Newton's results, but the product was a set of rules Agnesi, for one, founa "very perplexed and prolix. " 2 1

These

rules, like the early research on Descartes' rul* itself, were directed first to the task of problem-solving.

Once Descartes' rule was properly

formulated, however, it could be deployed with success in the attack on

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35 polynomial equacions, wherever they mighc occur.

It could reveal

with some certainty the distribution of real roots, and in conjunction with the Newtonian rules, the existence of imaginary roots as well. 3ut such results rested on shaky foundations without a proof of Descartes' rule.

Efforts toward the latter began in earnest only in

the 1740s.

Eliminate on Although uuch algebraic research before 1740 addressed che issues surrounding che solution of a single polynomial equation, systems of equations, both linear and non-linear, raised different problems.

The

simultaneous solution of two or more equations was required in various contexts.

The simplest problem consisted of two linear relations

between two unknowns, as in elementary number-theoretical questions, a problem readily generalized co more equations in more unknowns.

The

representation of curves by equations in a two-dimension coordinate system led co non-linear polynomial equacions in two unknowns, and co find che intersection cf two or more curves necessitated the simultaneous solucion of che equations representing them.

From che outset (and until

the late 18th century) algebraists attacked the linear and non-linear cases separately. As an illustration of his improvement of algebraic notation, in which numbers stood for the letters generally used in algebraic compu­ tations, Leibniz sent L'Hdpital a discussion of three equacions in two unknowns + + 30 +

10 20

llx+ 1 2 y - 0 2 1 x+ 2 2 y - 0 31x+ 32y - 0 .

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36 Here che first digit tor the coefficient signified che number of che equation; the second, the variable to which it pertained. sought to eliminate both

x

and

y

Leibniz

from the system (hence the term

"elimination") in order to exhibit a necessary relationship among the coefficients.

The relationship turned out co be + 10.21.32 - 10.22.31 - 11.20.32 + 11.22.30 + 12.20.31 - 12.21.30 - 0 .

Leibniz had also conceived a rule for che formulation of the six-term expression.

Its terms consisted of all possible combinations of the

coefficients taken three at a time, with an awkward lex sigr.crur. applied:

Assign an arbitrary sign co one combination, and co

3

ll

combinations differing from the first in an even number of factors, assign the opposite sign. 10.22.31

negative.

Thus

10.21.32

would be positive and

Although Leibniz wrote L'Hopital in 1693, his rule

was not published until 1850, so that its impact on 18th-century mathematical developments was restricted. 22 Maclaurin seems to have conceived a general elimination rule for linear systems.

Already in 1720 he described his projected algebra

textbook to Folkes, although the Treatise of algebra, "containing some General Theorems for exterminating unknown quantities," would only be published posthumously in 1748.23 Maclaurin began with the simple case of two equacions in two unknowns

ax + by ■ c

c *“ by x » — "a 1 ;

and

dx + ey ■ f.

from the second,

x ■

From the first he found

f — gy — — . Equating che two and

solving for y, he found af - cd y “ ae - db

‘

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37

Calling

a, e

and

d, b

pairs of "opposite" coefficients, he observed

that the numerator was the difference of the products of the opposite coefficients for the terms not the difference of the

containing

y,and the denominator was

products of the opposite coefficients from the

terms involving the two unknowns

x

and

y.

A similar, if more

tedious, derivation for the case of three equations ax + by +•cz ■ a dx + ey + fz ■ n gx + hy + kz ■ p yielded aep - ahn + dhta - dbp + gbn - gem aek - ahf +• dhc - dbk + gbf - gee Maclaurin also described this result using the deviceof opposite

coef­

ficients butwithout accounting for the proper sequence of plusand minus signs.

His summary instructions for the case of four equations

did suggest, however,the trend of his thinkingabout

asign

rule:

If four equations are given , involving fourunknownquantities, their values may be found much after thesame manner, by taking all the products that can be made of fouropposite coefficients, and always prefixing contrary signs to those that involve the products of two opposite coef ficients.2!* The relation between the problems Leibniz and Maclaurin set them­ selves is worth noting.

Leibniz began with a simpler version of

Maclaurin's system, equivalent to letting

z ■ 1,

m - n - p ■ 0.

then sought the equation (now expressed as the determinant ■ 0)

He resulting

from the elimination of all

the unknowns in his system.

Maclaurin, with

a clearer intent of solving

the equation system, wanted expressions for

the unknowns satisfying all

the equations.

In so doing he made use of

the same expression (the determinant) as the common denominator of the unknown values he sought.

Neither Maclaurin nor Leibniz offered, however,

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38

a particularly edifying version of the rule for assigning signs to the terms of the detemrinant. enunciated rule for solving

The honor of a general and nore clearly n

equations in

n

unknowns would go to

Cramer for his investigations of elimination theory in the 1740s.-5 The elimination of variables from non-linear equacions proved a much less tractable problem, since the least complication in che equacions resulted in "such tangled formulas that one completely loses patience in pursuing the calculations."

Given such formidable diffi­

culties, it is hardly surprising that early research on che non-linear case was the province of che notables Newton and Leibniz.

Because

Leibniz's work on this question was not widely known during che 18ch century, however, Newton was credited with che first general methods for eliminating unknowns.

In his Arichcetica universalis Newton

explained che "Extermination of an unknown Quantity, by substituting its Value for it," in both linear and non-linear e x a m p l e s . F o r the latter, the value of che greatest power was to be sought, which could necessitate che introduction of an extraneous factor.

For che example

of y

3

y

2

■ xy - x

2

">

+ 3x

y

3

,

- xy - 3 ,

Newton multiplied equation (2) by of

(1 )

y

and thereby to eliminate the

(2 )

in order to equate the two values 3 y

term.

Since

y*y

2

. 2

“ y(x“ - xy - 3)

the equation of (1 ) and (3) yielded xy

2

2

+ 3x ■ x y - xy

2

-3y,

or

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(3)

39 2

2

2v x + 3y - x y + 3x - 0 . Newton then substituted

y“ * x

2

- xy - 3

(4)

from (2) in equation (4) to

produce ■> 2 2x(x“ - xy - 3) + 3 y - x y + 3 x * 0 *>

3 2x -

2 y -

2x

6x

7

+ 3y - x“y + 3x - 0

y(- 3x^ + 3) + 2x^ - 3x from which

■

0 ,

_ - 2x y =

+ 3x

3 - 3x2

yielding a high-degree equation in of the roots of this equation in

x x

’ when substituted in (1).

Some

would be extraneous because of che

early multiplication of equation (2) by

y.

Newton's expedient would

later prompt criticism because the introduction ofuseless roots gave rise to two difficulties: "mixing

useful roots with useless ones and

being obliged often to abandon the calculations as interminable. " 2 7 In thefaceof these

stumbling blocks, the matter wasallowed to rest

until arevival ofinterest in

elimination theory beginningin the *

*

17403.

*

As these cases illustrate, the call before 1740 was to devise and refine useful algebraic techniques and results.

Algebraists responded

to the paramount needs of problem-solving with the development of new approximation methods. special cases.

They also directed their other research toward

The factorization assertion, although enunciated as a

general proposition, was controverted with particular "counterexamples." Even in formulating the general claim, mathematicians aimed at its application to a specific class of integrable expressions.

In the

dispute over Descartes' rule, the consideration of special cases

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predominated, and for those cases complicated by imaginary roots a second and still indecisive rule was proposed.

Techniques for the

elimination of variables from both linear and non-linear systems also answered the specific needs of problem-solving, but fell short of general, easily applied methods.

Algebraic research outside these

four problems evinced the same characteristics; witness the investi­ gation of Newton's binomial theorem for negative and rational exponents, re-examination of known solutions to third-and fourth-degree equations, ana expansion of the class of solvable equations of higher degrees. A theoretical component or an element of generality occasionally intervened in the applied algebraic research of the period 1685-1740, but a firmly anchored theoretical structure was absent.

The long period

of noteworthy success in extending the application of algebraic methods, however, spoke to the importance of algebra.

It made more acute the

need to remedy the theoretical weaknesses and absence of proof in the analytic art.

Beginning in the 1740s algebraists applied themselves

to this difficult task.

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41

3 A SHIf-T IN INTEREST: Th e

fundamental

theorem

The algebraic heritage as elaborated by mathematicians between 1685 and 1740 left much undone:

the unsolved problem of the fifth-

degree equation, unrefined methods for solving equation systems, such undemonstrated theoretical results as Descartes' rule and factorization of polynomials.

The Importance subsequently conferred on these casks

and the direction of algebraic research after 1740 were conditioned by three trends in 18th-century mathematics — the algebraic formulation of the calculus, the elaboration and extension of the calculus, and the algebraic definition of a function. Euler played a pivotal role in algebraic research as he did in much of 18th-century mathematics.

He united and transformed these

three trends, achieving a redefinition and extension of mathematical analysis encompassing both finite and infinitesimal components, the whole grounded in algebra and centered on the concept of function.

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42 This confluence of trends, fortified by the successes of mathematical analysis, affected algebraic research by focusing attention on algebra's foundations.

That much depended on algebra redoubled the need to

construct and demonstrate algebraic theory, as well as reinforced the need for efficacious solution of the growing number of problems expressed in algebraic form.

Reciprocally, algebraic research profited

from application of the Eulerian function concept to algebraic questions. One of the capital features in late 17th-and early 18th-century mathematics was the transformation of calculus from a sort of infini­ tesimal geometry into a potent analytical tool that employed the language and canons of algebra.

The methods constituting the calculus

were consolidated in the years 1665 to 1690, with much attention paid to the derivation of basic algorithms.

One result was the Leibnizian

calculus, conceived by Continental mathematicians of the 17th and 18th centuries as a calculus of operators on algebraic expressions.

Although

geometry definitely conditioned the development of fluxions in England, by the 1740s Maclaurin's influential voice in English mathematics echoed Continental opinion on the advantages of basing the calculus cn algebra: The improvements that have been made by (the doctrine of fluxions], either in geometry or in philosophy, are in great measure owing to the facility, conciseness, and great extent of the method of computation, or algebraic part. 1 The 18th-century elaboration of the calculus extended the comple­ mentary questions of differentiation and integration to the consideration of differential equations.

The application of the calculus to mechanical

problems created the new mathematical field of rational mechanics as mathematicians invented and refined the mathematical techniques st its

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43 heart.

In all these areas mathematicians incorporated the growing

knowledge of series and recognized their analytical potential. At the same time mathematicians began to polish the concept of function and imparted to it an algebraic definition.

During the 17th

century few particular functions were known, and general functional methods were largely confined to geometric models, such as defining as functions the areas and arc-lengths of conics.

The precedent

existed for an analytic understanding of functions, however, in the treatment of logarithms as analytical as well as geometrical entities. By 1718 Jean I Bernoulli offered an explicit definition of the function as an analytic expression,

and thereby opened what one historian of

mathematics has labeled the Eulerlan period in function theory, during which the function was liberated from geometrical and mechanical interpretations. 2 Euler's Introductio in analy3 in infinitorum (1748) was the first work on mathematical analysis in which the function concept was made primary.

He Introduced and used functions in neither a geometrical nor

a kinematic context, thus rendering his analysis fully algebraic. Through the force of Euler's example and through the recognized excel­ lence of his Introductio. called in this century the foremost mathematics textbook of modern times, he made his conception of mathematical analysis stick.

"In the form of an analytic expression the notion of function

eventually became the principal object of mathematical research in the 18th century, around which revolved, as Euler said in 1748, all analysis of infinitesimal quantities."

Euler's influence in this development, as

Condorcet explained, was decisive:

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44

[He] sensed chat algebraic analysis was Che moot comprehensive and certain Instrument one can employ In all sciences, and he sought to render Its usage general. This revolution, which capped chat which Descartes had begun and which was accelerated by the discovery of new co.Lcu.Iq , was Euler's work, and earned him the unique honor of having as many disciples as Europe has mathematicians. 3 Among Euler's distinguished disciples was Lagrange, who would play a central role in the development of algebra, and would continue Euler's work in building mathematical analysis on the foundation of algebraically defined functions.

In a 1772 memoir Lagrange presented "a new sort of

calculus relative to differentiation and integration" in which he main­ tained chat "the differential calculus, considered in all its generality, consists of finding directly, by simple procedures, the...derivative functions of the [given] function consists of recovering the function

u;

and the integral calculus u by means of these functions."

He pursued this end in a 1797 treatise whose full title reflected his commitment to the algebraic brand of function theory:

Theorie des

fonctions analvtiques, ccntenant les.principes du calcul diffgrentiel ddgagds de toute consideration d'infiniment petits, d'^vanouissants. de llmites et de fluxions, et rddults S

1

'analyse alg^brlque des

quantitds flnles. The full story of the development of function theory and mathematical analysis lies outside the scope of this study, but Lagrange's work on fonctiono analytiques, like that of Condorcet, Fourier, and others, illustrated the extent and significance of the interplay among the calculus, function theory, and algebra.u As much of mathematics came to rest on an algebraic foundation, the validity of results in Infinitesimal analysis was seen to depend in part on algebra's logical integrity, and an increase in the vitality and power of algebra could augur well for a variety of mathematical

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45 problems. The coincidence of che mathematical developments just before mid-century, in which Euler played a leading role, and the renewal of interest in algebraic topics is therefore no surprise.

Beginning in

the 1740s mathematicians moved to fortify the theoretical basis of algebra and to prepare for a new attack on the problems that their predecessors had failed to solve completely.

One of the most pressing

questions was the nature of imaginary numbers and '’heir behavior under algebraic and transcendental operations.

THE ISSUE OF IMAGDIARIES By the 1740s the imaginary number had troubled mathematicians for generations.

Even the conservative camp generally conceded a limited

argument in favor of the utility of imaginary numbers.

If a problem

yielded an Imaginary solution, this was a sure sign that the problem, as framed, was impossible.

As Playfair, a geometer at heart, would

explain it, The natural office of imaginary expressions is, therefore, to point out when the conditions, from which a general formula is derived, become inconsistent with each other.... But more was at stake.

Although these "impossible numbers" were a

logical consequence of the effective methods developed centuries before for the solution of equations, they posed specific difficulties for Algebraic practitioners.

They complicated the proper formulation of

Descartes' rule, as they would complicate its proof.

In the case of

the irreducible cubic, mathematicians could not rid the finite expression for its real roots of troublesome imaginary quantities.

The possibility

of decomposing polynomials into real linear and quadratic factors hinged on the nature of imaginarles as well.5

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46 Outside the realm of algebra itself, the issue of Imaginary numbers had also arisen.

First, in the elaboration of the integral

calculus, Jean Bernoulli's assertion that the Integration of rational fractions required only logarithmic and trigonometric expressions was an easy consequence of the assumption that all polynomials could be factored into linear and quadratic factors.

And the latter followed

directly from the notion that all imaginary (that is, non-real) roots of an equation had the form

a + b/HT.

As a result, it was claimed

that ''the arithmetic of impossible quantities is nowhere of greater use than in the investigation of fluents...."

But by the 1740s the

question of imaginary numbers intersected with other issues in mathe­ matical analysis as well.

As the function concept gained adherents,

attempts were made to enlarge its compass, necessitating the definition of functions with imaginary arguments or values.

What, for example,

would be the meaning of the logarithm of a negative or imaginary number?

Or as d'Alembert put the problem, what form would urte fonation

queIconque of imaginary numbers take?® All of these questions served, in the 1740s and thereafter, to confer mathematical importance on the behavior of imaginary numbers. While they were not always discussed in a strictly algebraic context, the discussions focused interest on algebraic entities and issues and Inaugurated a new period in algebraic research.

Although the definition

of lmaginarles was linked to the proposition now known as the fundamental theorem of algebra (which guarantees at least one real or complex root for any polynomial equation with real coefficients),

the exploration

of the form and behavior of lmaginarles was undertaken for its own sake as well.

A leading figure in this enterprise was d'Alembert.

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47 From 1739 to 1741 the young d'Alembert bombarded the Paris Academy with memoirs on a variety of mathematical and mechanical topics, with the Intent of proving his worth and promoting his campaign for membership. By mid-1741 his tactic had succeeded, and he was elected adjoint.

His success was due in part to a promising paper submitted in

late 1740 and entitled "Recherches sur 1'integration des fractions rationelles."

In it d'Alembert took off from Bernoulli's 1702 memoir

on rational fractions, but also attacked directly the problem of the form of imaginary numbers.

So at least we may reason from the report of

ccnyni83aire8 assigned to review the manuscript: He makes on this subject several curious remarks about imaginary roots; for example, he subjects real and imaginary quantities to an algebraic form. The whole memoir supposes much knowledge, and the author's discovery agrees perfectly with the impression of sagacity he has conveyed. . . . 7 The question of imaginary numbers also arose in d'Alembert's prizewinning essay for the 1746 competition sponsored by the Berlin Academy. In Reflexions sur la cause generale des vents he maintained that "it is certain that an arbitrary algebraic quantity, composed of as many lmaginarles as one wishes, can always be reduced to A + B/-i, for and

B

real numbers. " 0

a + b/^T

In his demonstration he showed first that

had the desired form.

by the same quantity

A

He multiplied numerator and denominator

g - h/^if to find the product ag + bh 4- (b - h) J-l 2

g Thus for

A ■ (ag + bh)/(g^ + h^),

took the

form A + B/-1.

The

+

2

B - (b - h)/(g^ + h^), the quotient

case of an imaginary quantity raised to an imaginary quantity

was not straightforward, nor could d'Alembert rely on strictly algebraic

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48 manipulations.

He began by equating

for

to be determined.

A

and

he found

B

(a + b/-I) ■ ln(A + Br^l). He differentiated

each side, assuming

(g + h*^-I) (g + h^-1)

constant throughout, da + db^l

to produce

dA + dB/-l A + B/-1 .

+ b*-T a _

k /TT

After multiplying the left hand side by ^ _ b A l Snd Che right hand A — b /—1 side by — ~~Y' /Sf collecting terms, he arrived at g(a da + b db) - h(a db - b da) + /-ifg(a db - b da) + h(a da +•b db)l 2 . ,2 a + b _ A dA + B dB 4dB - B dA) 2 2 A + B But since

m + n^-T - r + s»^r implied that

m » r

and

n - s,

d'Alembert was able to equate the real parts of each side of the equation: g(a da + b db) _ h(a db - b da) 2 2 ~ 2 2 * a + b a +■ b

A dA + B dB 2 2 A + B

But a da + b db a2 + b2

d/a^ + b^

g d/a^ + b^

/a2 + b2

Integrating, and noting that

h(a db — b da) _ d^A^ + B^

/a* + bJ

a2 + b2

' 7 7 7 V

In e - 1, he found

, / I i T2" , / 2 , ,2 8 f, r a db - b da ^ . In*'A + B - ln/a + b - [h • J — r--- — In e . a + b

J

Thus c a qd p da oa db - b

A2 + B2 - (a2 + b2)8 • e

1 a2 + b"

.

D'Alembert also equated the imaginary terms of his equation and derived

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49

che result r A dB - B dA 1 ~ A1 ~+ VB ~

, , / $ , 1

.

r a db - b da g/ k• * aC + b

'

The tvo integrals were known co be expressions of angles with tangents ?/A

and

b/a.

B

and

with hypotenuse

A were thus che sine and cosine of a triangle

r

r a db - b da 2 .,2*

(a" + b**) and hence real.

*e

~h J

a^+

b2 '

The cases of addition, subtraction, and muliplication

of imaginary quantities were easily disposed of. By December 1746 d'Alembert had reflnea his treatment of the problem in memoirs submitted co che Berlin Academy via Euler.

In

presenting the somewhat simpler case of

(a + b*'/-l)^C* , supposed equal

to

A - cos 9,

A + Br'^T,

a ■ cos u,

d'Alembert observed that

b ■ sin u,

where

B ■ sin 9,

9 was an angle such chat

9/u - i/d.

If one could divide the angle (determined by che known quantities and

b)

into

d

equal parts by geometrical means, then

would have analytic expressions. of quantities like

A + b /-1"

A

and

a B

More than that, there were "a number

such that

(a + b^-1)^** - A + B^-l.

D'Alembert also remarked chat If one has a quantity under che sign /, composed of as many real or imaginary variables as one wants, raised co real or imaginary powers, one can always suppose chat this [integral] equals p + q^-I, although it is often impossible to determine the analytic values of p and q. This additional property of lmaginarles was based on the assumption (for which d'Alembert provided no justification): Since the quantity under the sign / is a differential, one can always divide it into two parts or factors, one infinitely small — that reduces to dx + dyv^-T, the other “ finite — that reduces to r + s^-l, and their product can be supposed equal to dp + dq/^-1, for which the integral is + q *-1.

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50 In sum, these derivations and assertions permitted d'Alembert the conclusion Thus an arbitrary function of whatever imaginary magnitudes one wishes can always be supposed equal to p + q*/-l, for p and q real quantities. Two characteristics of d'Alembert's work on the behavior of imaginaries deserve note.

First, he sometimes left the strictly algebraic

realm in pursuit of proof or result.

For example, he resorted to dif­

ferentiation, integration, logarithms, exponentials, and trigonometric notions in his treatment of imaginaries raised to imaginary powers.

In

this his faith in the power and validity of infinitesimal processes was evident.

So too was his debt

to Cotes and de Moivre.

Without attri­

buting the result, or expressing it so concisely, he presented what is now known as de Moivre's theorem for

(a +• bi)®:

(r(cos 6 +■ i sin 0))® ■ r® (cos g6 + i sin g0) for

2 2 r ■ (a + b )

and

tan 9

■ b/a.

Second, he assumed throughout that any function composed of imaginary quantities was in fact composed only of quantities of the form

a + b*^-T.

This is not to minimize his accomplishment in reducing many complicated expressions to just such a form, nor does it suggest that d'Alembert Ignored any imaginary quantities already encountered by mathematicians of his day.

But he was necessarily guided by the limited experience of

algebraists in equation theory.

An English mathematician later explained:

As far as the actual solution of equations was carried, viz. in cubics ana biquadratics, the imaginary roots were found to be of this form, a + «MvZ; and subsequent writers, from this imperfect induction, concluded in general, chat every equation has as many roots, of the form a /-b^, as it has dimensions.10

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51 Moreover, d'Alembert evidently concluded that this form accounted for all non-real quantities wherever they might arise. Other

18th-century mathematicians took the same tack in framing

the problem of Imaginaries.

In his Introduction to the Traitd du

calcul integral, the French mathematician Bougainville referred to d'Alembert's laconic results in the 1746 Berlin memoir:

"I have

extended his demonstrations and have given them the form I believe most appropriate to adapt them to general use."

Although Bougainville's

exposition of the proofs was more complete, and his statement of the proposition more concise (any algebraic expression composed of imaginary numbers could be reduced to the form outran his evidence: A + a/-!

A +■ B*/-l) , his conclusion still

"Every imaginary quantity can be reduced to

"n

Bougainville also mentioned the 1749 work of Euler's d'Alembert's principal rival on this issue (as on others).

According to Bougainville,

Euler's memoir Included a method — essentially the same as d'Alembert's — for changing imaginary numbers into the desired form.

It also extended

the coverage to logarithmic and other transcendental functions.12 Other mathematicians also followed this line of march.

Paoli

offered a more complete demonstration of the reduction of functions of lmaginarles, including logarithms, logarithms of logarithms, sines, tangents, secants, log sin,

arc sin, etc.

also Include his reduction of

ea +

Condorcet's manuscripts same form.

All of

them, Including d'Alembert, seemed to make the same assumption for which d'Alembert criticized Euler in 1748 — that all Imaginary quantities were necessarily composed of quantities of the form

a + b/-T.13 Their

attention to reducing complicated expressions to a uniform form and the

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52 vigor of che priority disputes involved, however, reveal a more incense impulse toward proof than in earlier decades.

This is significant in

spite of che fact that Che result they sought to prove was generally granted and chat their proofs actually established a more limited proposition (all expressions composed of quantities of che form a + b*-l

are themselves of this form) than that for which they claimed

credit. Their approach to the problem of imaginaries had its advantages and its drawbacks.

D'Alembert et al. manipulated mathematical char­

acters according to pre-established rules and explored che results. In this regard they were true to the tradition defining algebra as tl.e sec of canons governing literal calculus.

In the Encyclopedic. for

example, d'Alembert applied the term algbbre to the literal arithmetic of computing with letters, not co the solution of problems by means of equations, or analyse.

In this manipulation of characters within

a mo’-e or less defined structure were roots of che formalism chat would prevail when mathematics would come to have "as its subject matter the structure of systems of unlnterpreced signs."11* In che case of imaginary numbers, 18th-century mathematicians worked with a negative definition: Because all conceivable numbers are either greater than zero or less than 0 or equal to 0, then it is clear that the square roots of negative numbers cannot be Included among the possible numbers [real numbers]. Consequently we must say that these are impossible numbers. And this circumstance lead3 us to the concept of such numbers, which by their nature are impossible, and ordinarily are called imaginary or fancied numbers, because they exist only in the Imagination. Despite their fanciful character, imaginary numbers were assigned a form

(a + b/^1); in this, algebraic experience guided mathematicians.

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53 Imaginaries were also subjected to a set of rules, as Bossut explained: Although these sorts of roots represent nothing in existence, they can nevertheless be subjected to the same rules of calculation as real quantities, because algebra operates on unknown quantities as it does on known ones, and often one sees only at the end of the calculations whether or not imaginary quantities remain.15 These rules were not particularly abstract, in the sense of being far removed from ordinary mathematical experience. exception — /-I •

M' - A/2

B - N + N' +■ MM' - N + N' + A2/4 C ■* MN' + M'N D - NN*

->

N' - D/N .

Thus

But if the coefficients

A, B, C, D

were such that

A 4

B

then - 4D -

implying that — - N

N2

2 + 2D + N2 - 4D < 0

and hence

N

would be imaginary.

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- 4D < 0

67

Thus even if

M

assumed a real value such as

imaginary if the coefficients were such that this case

A/2

A2

A/2, 2

B

N

would be

- 4D < 0.

In

was a double root of the sixth-degree equation in M.

As Lagrange had already determined, it was a straightforward matter to find a rational expression defining one function of the roots in terms of another such function unless the equation determining the first function (in this instance, the equation determining the coefficient had equal roots of even multiplicity.

M)

For the case at hand, this

problem arose.31 To avoid the problem Lagrange took a slightly different tack. Taking again the proposed equation of degree looked at the two factors

for

m • 2r, he

x11 - Mxn ^ + i«xn ^ - ... * 0

xa - M'xn ^ + N'xn ^ - ... * 0. r—1 tjm - 2 .)

m,

(Here n

and

was supposed equal to

As in the fourth-degree cate, he considered the

equations resulting from multiplying out the two factors of degree and equating coefficients with the proposed equation. with the equation system he defined a new set of

and so forth.

d-N+N'.y-P+P',

u-M-M',

v-N-N'.u-P-P', M, M', N, N', etc.

recast in terms of the new variables

a, 6, etc.

Lagrange then introduced a new variable t ■ au + for arbitrary coefficients from the system of and the new equation

2n

a, b,

bv c,

equations in t

t

nth To deal

variables

a-M+M',

The 2n equations in

in the new variable

2n

2n

and

were readily u» v, etc.

t via + cu) + ... etc.

After eliminationof

a, B, Y,

etc.

■ au + bv + cw + ...,he arrived (and the arbitrary coefficients

andu,v,

variables u, etc.

at an equation a, b, c, etc.).

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68

This equacion had che some degree as if Lagrange had used the straight­ forward Eulerian approach. a, b, c, etc.

3y changing the arbitrary coefficients

if necessary, Lagrange could avoid the difficulty of

multiple roots for

t;

the values for

M, M', etc.

could then be

recovered easily.

Daviet de Foncenex Euler's attack on che fundamental theorem also prompted che algebraic efforts of Daviet de Foncenex.

Foncenex was sent to Turin

at an early age, where he became Lagrange's atudent and procggd.

His

papers on algebra, transcendental analysis, and mechanics began to appear in the Turin Academy memoirs in 1759, and his mathematical talents earned him membership in the Turin Academy in 1778 as well as Invitations to Berlin and St. Petersburg.

His approach to the funda­

mental theorem differed markedly from Lagrange's.

They did agree,

however, on the virtues of an algebraic demonstration and the vulner­ ability of Euler's proof over the issue of che reality of the constant term in the equation defining the first coefficient usage,

M

in Lagrange's).

(u

in Euler's

Foncenex observed the Euler said only chat

this term, being determined by che coefficient of che original equacion, could not be imaginary.

But Foncenex perceived the need to guarantee

that the constant term was in fact equal to a rational function of the original coefficients, so that the proposition on rational functions could be Invoked.

"This circumstance, without which the [main] theorem

loses all Its force, seemed quite difficult to demonstrate."32 Foncenex cook a different route. the roots of

2

x

As a preliminary step, he examined

+ Ax + B • 0, where — in a significant generalization —

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A

and

on che

B

were imaginary numbers of che form

r(cos

0

+ /-I sin 0)

a + b«^T.

By relying

represencacion of imaginaries, he was

able Co show chac che roots would cake che same form.

In this manner

he helped co pave the way for consideracion of a broader class of polynomials with non-real coefficients. But his principal intent was che same as Euler's ultimate objective:

co find, by an iterative

process, a quadratic factor with real coefficients for the given equacion of degree

2

(for

P

odd), also with ail real coefficients.

He assumed the given equation was exactly divisible by for indeterminate

u, M,

yielding an equation in

u

x

2

+ ux + M.

of degree

- 2”-1P(2"P - 1).

If this degree was odd, the equacion in real root. trinomial

u would necessarily have a

If not, Foncenex divided the equacion in u^ + u’u + M',

leading to an equacion in

the degree of this new equation in cease, since a real value for che root of

u'

2 u * u'u + M ■ 0.

u

by the second

u'.

Again, if

u ’ was odd, che procedure could would yield a complex value for

Then

2 2

+ uz + M ■ 0,

a quadratic

equation with complex coefficients, would have a complex root by Foncenex’s preliminary result, implying that x - (A - b /=1)

would divide the given equacion.

u,

A + b /^1

x - (A + B*^l)

and

The desired factor

of the given equation would therefore be che product (x - (A + B^l)) (x - (A - B/^I)) 2

- x But if che equation in be continued.

u'

Ac the mth

+ 2Ax + A

2

+ B“ .

was of even degree, the procedure had to step, an equation of odd degree would result.

with permission of the copyright owner. Further reproduction prohibited without permission.

70 Although Foncenex concluded that this last equacion had ac lease one real root by virtue of its odd degree, he had overlooked a difficulty: how co guarantee that the

mth

equation itself had real coefficients

so that the standard theorem on equations of odd degree could apply. The solution would eventually be supplied by Laplace, who extended Foncenex*s fecund idea of depressing the power of

2

in the degree

of the auxiliary equation.

Laplace Laplace's principal venture into algebraic research cook place in connection with his pedagogical duties in revolutionary France.

Charged

in 1795 with teaching algebra and mathematical analysis co che scudencs of the short-lived Ecole Normale (following Lagrange's elementary lessons), he sought a sounder proof of the fundamental theorem on which so many results in finite and infinitesimal analysis rested. The result, published first in the Journal de 1'Ecole Polytechnique, was based on finding theequacion that determined a function of che roots of the givenequation.33 unmistakable.

The influence of Euler ani Lagrange was

Laplace differed, however, in seeking directly a real

quadratic factor of che original equation. degree of the given equation was and Chat che roots were equal co

(a + b +

2*s,

a, b, c, etc.

for

He assumed that che s odd

i

an integer,

He sought an equacion with roots

mab),(a +c + mac), etc.,

equation was necessarily of degree

and

2* ^s',

for any real

for some odd

the degree would equal the number of combinations of

2^s

s',

m.

The

since

things taken

two ac a time, or

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where

s(2

- 1) ■ s',

the product of two odd numbers, was itself odd.

Laplace first considered the case where 2* 1s'

i - 1.

The degree

would thus be odd, so that the equation defining

(a + b + mab),

which Laplace assumed would have real coefficients, would have at least one real root.

Again, the tacit assumption that the equations had real

coefficients was operative.

Since

m

was chosen arbitrarily, Laplace

could conclude that there existed an infinite number of functions of the form

(a

b + mab)

two real quantities

m

that would assume real values. and

Taking any

m', he subtracted

(a + b + mab) - (a + b + m'ab) ■ (m - ra')ab . The closure of the real numbers under multiplication and addition meant that

ab

had to be real. By the same token a + b had to be real, 2 implying that x - (a + b)x + ab • (x - a)(x - b) was the desired real quadratic factor. In the second part of his induction argument, Laplace assumed that every equation of degree

2* ^s' had a real quadratic factor

and therefore imaginary roots of the form

e + g.

the constant Then

-■ yo * - 36

which result could be plugged into the first equation to produce the desired resultant equation in

y

alone.11

Cramer's method, presented in an appendix to his Introduction, was not dissimilar, although he treated equations of higher degree and drew conclusions about the degree of the final equation in one variable, as Clairaut had not.

He considered the problem of eliminating

x

from

0 - xn - (ljx0”1 + {l2) xn"2 - ... (ln)

(1)

0 - (0)x° + (l)x1 + ... + (m)xm ,

(2)

where the coefficients

(I*4) , (k)

were actually rational functions of

y.12 What Cramer sought was a third equation in functions

(l^), (k)

of

y.

He let

y

a, b,..., n

equation (1), viewed as an equation in

x.

composed of the be the roots of

When the values

a, b,..., n

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118 (functions of was

n

y)

were substituted into equation (2), the result

equations in

The product

y;

call them

a, 3, y, etc.

(0)a° +

(l)a1 + ... + (m)am - 0

(a)

(O)b^ +

(l)b^ + ... + (m)bm ■ 0 , etc.

(3)

a6y... ■ 0

was the desired resultant equation in If If since all its terms were composed of (0),...,(m) and a ,b themselves all functions of

y.

y, n ,

Cramer expressed the resultant as the

sum of terms, each composed of a faateur-premier (the product of several coefficients of equation (2) ) and a facteur-second. of

the roots, a,b,...,n

n ■ 4,

of equation (1) ). For the case of

(afunction m • 3,

Cramer provided a table of the facteura-premiera . The facteurs-

aeconda were more difficult to express concisely without solving equation (1) explicitly.

The well-known relations between the roots

of an equation and its coefficients came to Cramer's aid and allowed him to provide an elaborate theorem for forming products of the facteura-aeconda. He offered no general proof. By considering the formation of the facteura Cramer demonstrated, or at least argued the plausibility,that the resultant equation was of degree

mn.

For the example of x

3

- 2ax

2

3 + 4ayx - y - 0

2 2 2 ax + y x -ay ■ 0 , Cramer found that the resultant equation in

y was of degree nine.

This result was of immediate use in ti:e geometry of plane curves, where it implied that two curves of orders most

mn

points.

of curves.13

m

and

n

could intersect in at

Indeed, the result was embedded in Cramer's analysis

The geometric problem of intersecting curves had focused

his attention on the algebraic question, just as the limitation of

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119 plane curves accounted for the restriction of his method to two equations in two unknowns. Euler also concentrated cn the problem of two equations in two unknowns in connection with the intersection of two curves.

His

Introductio shared aims if not methods with Cramer's work, and offered a technique for eliminating

y

from two equations in

x

and

y.

Euler, like Cramer, considered the coefficients to be rational functions of one of the variables (in Euler's case, x). He multiplied both If equations by constants and perhaps by y (for k ■ the difference in the highest exponents of

y

in the two equations), so that

subtraction would cause the highest power of an equation in

y

of degree one less (say

y

to vanish, leaving

(m - 1) ) . Similarly, he

multiplied the two equations by other constants in order to equate and eliminate the

y^

terms.

dividing through by y.

y

Subtracting one equation from the ocher and

again yield an equation of degree

(m - 1)

in

The procedure could then be repeated with the two equations of

degree

(m - 1),

with the eventual outcome an equation only in

x. ^

Problems with existing methods; degree of the resultant We owe to Bezout the clearest statements of the flaws in these methods of eliminating variables from non-linear equation systems. The first difficulty was the variety of forms the final equation (in one variable) could take when the procedure of successive substitutions was employed, depending on the order in which the equations were combined.

But the problem did not stop there.

equations two at a time involved

The combination of

"labor much more painful than is

necessary," and led to a final equation involving factors foreign to

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120

che question.

Similarly, methods such as Newton's and Euler's, which

involved multiplication by

y

(a power of one of the variables), could

introduce superfluous roots into che final solution.’5 In 1764 the Berlin and Paris academies heard memoirs that tackled these problems by reconsidering the general theory of eliminating variables.

In his 1764 Berlin paper Euler returned to Newton's method

of elimination, as applied to two cubic equations in several variables, and discovered that, upon eliminating one variable, the resultant equation was "coo complicated" and involved "completely useless factors." This circumstance prompted him to seek another route to solution. In n 3 o the system z“ + Pz + Q ■ 0 and z + pz“ + qz + r » 0, for there to exist a simultaneous solution, the two equations had to share a common •> root z ■ w. Thus (z - w) divided Z“ + Pz + Q, so chat 2 z + Pz + Q ■ (z - w)(z + A). Likewise product of

(z - w)

2

and

(z

remained to be determined. by

(z

2

+ az + b)

3 2 z + pz + qz + r

+ az + b). The quantities

was the

A, a, b

Euler could Chen multiply che first equation

and the second by

factors were thereby introduced.

(z + A), sure that no superfluous

Equating the two products (both equal

to zero) and equating coefficients, he found a sec of four linear equations in

A, a, b.

The first three, solved by any standard method

for linear equations, yielded expressions for original coefficients

P, Q, p, q, r,

This was the desired resultant.

in terms of che

which when substituted in the

fourth equation produced an equation in z.

A, a, b

P, Q, p, q, r

and

free of

Euler said nothing about its

degree, either in this particular instance or in che general case of two equation of arbitrary orders.16 Even though Euler's method, applied to two equations in two

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121 unknowns, resulted in a final equation "free of any superfluous factor," its extension to more complex systems was not free of difficulty. Bezout would explain, for four equations, each of degree

t,

As

successive

elimination following Euler's method would lead to a final equation of degree

8

t . But the true degree of the final equation, as Bezout would 4 demonstrate in 1779, was only t . Already in 1764 he was able to

establish a maximum value for the degree of the resultant by means of a new method of elimination.-7

In so doing he provided the grounds

for the rejection of Euler's approach. Bezout's new method reduced the problem to a linear one by the introduction of multiplier polynomials with indeterminate coefficients. For the example

ax

2

■> + bxy + cy“ + d x + e y + f “ 0

and

d y + e'y + f' ■ 0,

to eliminate second by

v Bezout would multiply the first equation by C, the 2 A'x + B'x + C' and add the two products. For this sum to

serve as the final equation in to vanish.

x,

all coefficients of

y

terms had

The result was a system of linear equations.

Bezout's and Euler's ideas bear striking similarities, made more striking by their simultaneity.

Each recognized the need to avoid

superfluous factors in the resultant; each introduced multiplier poly­ nomials with indeterminate coefficients and reduced system of linear equationsin those indeterminates. a more general question.

the problem to a But Bezout tackled

Using results based on the rules of permuta­

tions and arithmetic progressions, he sought to determine a maximum value for the degree of the resultant for systems of two or more equations.

Knowing an upper bound could then prove useful as a

criterion for the rejection of elimination methods (like those comparing equations two by

two) leading to a degreebeyond thisbound.18

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122 Bezout confronted the general case of more than two equations directly. The work of Marguerie took off from Bezout's memoir, and appeared as "Sur une operation d'algSbre appelde 1'Elimination des inconnues" (read 21 June 1770) in the MEmoires de 1'AcadEmie de Marine. Noting that the foundation of 3ezout's 1764 work was the degree of the resultant, Marguerie offered a method for finding the resultant of two equations. Whatdistinguished hiswork, according to to review his paper, was

the navalofficers

assigned

his provision of a well-defineailgorithm.

Concerned with the applicability of any such method, they pointed to the importance of reducing the labor involved in calculation.

Marguerie

set up a notational system where, for instance, *1 = 3la2 - V 2 al " ala3 " ala3 * Then the elimination of yielded

a^a^ - a1a2 "

He reduced the problem of

x

from

a^x +

* 0

and

a^x + a0 ■ 0

+ a,x + a^ ■ 0

and

a^x

or

2

a^x

2

+ o^s +

■ 0

to 1 2 a^x + a^ - 0

and

2 1 a^x + a2 ■ 0 by equating the

2

x

terms, and proceeded by a rough induction to a general

form, using the notation introduced in Fontaine's second method of integral calculus.

(Marguerie had encountered Fontaine's system while

a protege of that eccentric academician.)

After the construction of

elaborate tables and more notation Marguerie concluded that the resultant of two equations of order two was of degree four, for order three, of

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123 degree nine, etc.

Without really explaining why, Marguerie also noted

that three equations of order

a

would have a resultant of degree

m^,

anticipating (without demonstration) Be-'.out's s~neral theorem of 1779. At about the same time Waring advanced the same proposition, though again without proof.1- The level of generality in elimination theory had certainly risen since 1740, and the practical methods themselves were more refined.

But the theory lacked general , firmly-established

statements on the degree of the resolvent, viewed as a function of intrinsic interest.

bezout*s general theorem The principal accomplishment of Bezout's Thgorie gendrale was the enunciation and demonstration of a general theorem for determining the degree of the resultant.

Bezout considered

(that is, with no terms lacking) in etc.

n

n

complete polynomials

unknowns, of degrees

t, t', t",

(In place of Bezout's clumsy notation, let us denote them by

P^, P.,,..., ?n>) of degree

T

Then Bezout introduced another complete polynomial

in the same number

of variables.

would then be an equation of degree all terms divisible by

x

t + T.

t' t" , y , etc.

from

The product

Q

P^Q

After the expulsion of Z

and

P-^Q»

ana using

formulas derived from finite difference theory, Bezout was able to define

N - the number of terms remaining in

of terms remaining in

Q.

P^Q

and

He wanted ultimately for

M - the number

P^Q

to have only

one unknown, which would therefore qualify it to be the resultant of the elimination of all other variables. degree D + 1

D

If the resultant were of

(with only one unknown), Bezout saw it that it could only have

terms.

Thus for P^Q

to qualify as the resultant, it could have

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124 only

D + 1

terms.

Bezout therefore contrived the coefficients of

Q

such that

N - (D + 1)

coefficients of the reduced

P^Q

Since

for any

would accomplish this if

could, one coef­

kQ,

ficient of M - 1

Q

k,

could be arbitrary.

coefficients in

to solve

Q

Q

Thus Bezout had only to determine

to accomplish his purpose.

coefficients equal zero) in

coefficients of resultant.

0)

M - 1 variables (the necessary

in order to render the reduced

P^Q

D ■ N - M.

N

and

the desired

This was possible, or so 3ezout tacitly assumed, if the

number of equations equalled the number of variables:

for

He thus needed

N - (D + 1) linear equations (from the requirement that

N - (D +■ 1)

or

would vanish.

N -(D + l) ■ M-l,

Appealing to che results of finite difference theory

M, Bezout found that

N - M ■ t-t'.t"... .23

Several characteristics of Bezout's demonstration are notable. His facility with difference theory and intuition about the most fruitful manipulations of complicated expressions allowed him to overcome the difficulties of a clumsy notation, and he used to great advantage che simplification of reducing che problem to a system of linear equations. His was more an existence than a problem-solving approach, since he considered first che degree of the resulting equation before specifying it.

But his concern was not exclusively general.

He discussed che

symptoms in the final equation (namely, a common factor) of the possibility of reduction of the degree of che final equation, dependent on the particular relations among the coefficients of che system.

He

offered two methods for getting at the resultant, the first general and the second allowing for reduction of the final degree.

In so

doing he introduced situations where there were fewer equations Chan unknowns, and offered procedures for shortening the calculations.

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125

The combination of theory and practice as motives for Bezout'a efforts was particularly important, and in keeping with the late Enlightenment concern for theoretical completeness and utility in mathematics. 3ezout's approach was to give a general result and then to consider ways to shorten the calculations for particular cases. His goal throughout was a rigorous and general result, but he was a sufficiently adept practicing mathematician to reject some otherwise acceptable methods because of their impracticality for calculations. What emerged from his work was a comprehensive, dense, and influential presentation of elimination theory, solving the outstanding theoretical problem for systems of non-linear equations and laying the groundwork for the branch of modern mathematics known as algebraic geometry.’1 At the end of the century Bezout*s methods for find che resultant were still recomaended as practicable.

His proof of the theorem that

bears his name, although adequately general, was far from elegant.

This

was to remedied by a student, Chen not twenty, at the Ecolc Polytechnique. Poisson had clearly absorbed the lessons of fin-de-siiicle French mathe­ matics, with its premium on elegance and felicity of expression, and recast Bezout*s proof in more direct and concise fashion-22 His was an induction argument, valid for general systems of non­ linear equations but explained for four complete equations'(a), (b), (c), (m)

of degrees

x, y, z, u. y, z

a, b, c, o

respectively, in che four unknowns

He considered the first three equations and eliminated

from them, yielding an equation in

elimination of

x, z

led to an equation in

(Poisson asserted), and finally one in these equacionsgave

x, u

n

values for

x

y, u,

z, u (y,

of degree

z)

Likewise

also of degree

of degree

or

n.

n.

Each of

in terms of

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u.

n

126 Arranged in triplets,

n

combinations

(aysv&nes) of these values,

denoted by

(a , b , c ), satisfied (a), (b), (c) simultaneously. P P P Substituting these triplets for x, y, z in the first member of (m), supposed equal to zero, gave rise to

n

functions of u.

There

would then be a one-to-one correspondence between values causing one of these functions to vanish and values for the unknown given system.

The product of these functions of

u,

u

in the

set equal to

zero, would be the resultant equation of the whole system.

The task

then arose to find its degree. Equation A, 3,..., K

(m)

was of the form

polynomials in x, y, z

respectively (since resultant

R

evaluated at term would be

(m)

ua + Au0-^ + ... + K ■ 0, of degrees

the n

1, 2,..., m

was assumed complete).

was defined to be the product of

Because the

u3 + Au*8""^ +

... + K

triplets

(a ,b , c ), p ■ 1,..., n, its first P P P u081. Likewise the sum of exponents of u, x, y, z in

each of the other terms was also no greater than

mn. The product

would be unaffected by exchanging some pair of indices (that is, exchanging R

for

a and p contained a term such as

a , b q p

and

b , c q p

p

and

and c ). q

R

q Since

k a,8 y a' 3' y* u ap bp cp a p , b p, cp, ... such that the sum of exponents where

p, p', p", etc.

k + a + S + y + a' + 3' + y' + ... 0,

F'(3) < 0, F'(x) ■ 0

3,

etc.

F'(x) ■ 0

3 and

y,

b.

F(a)

and

would necessarily

Then Lagrange con­

with real roots

by Rolle's theorem the equation another between

F(x) - 0

and

F(x) ■ 0

such chac

F'(y) > 0,

a > 3 > y, etc.

Then

had a root between

a

and

By the same token, the roots of

served to separate the roots of

forward approach had drawbacks, however.

F(x) ■ 0.

This straight­

The existence of imaginary

roots at any stage of the process rendered the results less than absolute, and Lagrange based his demonstration on the common but as yet unproven assumption that all imaginary roots had to be of the form a + b/^1.

Moreover, to use the method in practice meant, at least for

higher degrees, solving a succession of derivative eauations for which no exact solution was available.

The length of the calculations and the

uncertainty induced by imaginary roots made the method unfeasible.3 Lagrange proposed instead to use the difference equation, whose roots were the differences between the roots of the given equation. Given an equation with roots would have

a, 3, y, etc., the difference equation

a - 3, a - y,..., 3 - y , etc.

found a lower bound

A

would have a quantity series of numbers

as roots.

Once Lagrange

for the roots of the difference equation, he A

sufficiently small with which to form a

..., -2A, -A, 0, A, 2A,...;

their successive

substitution would locate with certainty all real roots of the given equation.

Once Lagrange was armed with the coefficients of the dif­

ference equation (or the equation whose roots were the reciprocals of the differences), it was an easy matter, using rules known to Newton,

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146 Rolle, Maclaurin, and others, to establish a lower limit for Its roots. Finding the coefficients posed practical difficulties because of the tedious calculations involved.

Lagrange suggested a somewhat shorter

alternative, but it remained long-winded. x3 - 7x + 7 - 0,

For the example

it required four p»ges of explanation.14

The same idea of the difference equation had also occurred to Eduard Waring before 1762 in his study of the separation of roots, as well as the complication chat the separation of

t

roots required the solution of an equation of degree

nearly equal c.

According to

Lagrange's testimony, he was unaware of Waring's 1762 Miscellanea analvtica at the time he was writing his own first paper on numerical solution.

Besides, Lagrange continued, Warlng's idea, which appeared

in an "isolated manner" in the Miscellanea, was rescued from sterility by Lagrange's own work.

Although Lagrange was right in thinking chat

Warlng's books had not enjoyed wide circulation on the Continent, his dismissal of Warlng's influence was hardly just.

Over three hundred

Englishmen listed themselves as subscribers to the Miscellanea. And reversing the attack, Canovai later took Lagrange to cask for not conceding priority to Waring.^

The utility of the difference equation,

however, was not to be doubted. Lagrange used the quantity

A

not only as a means to separate the

roots of a given equation, but also to establish the first approximate value from which Newton's method could proceed.

Newton and others had

offered no prescriptions for finding the closest integer to the desired root.

Lagrange let

1/k 0.The requirement

convexity in more economical form chan in Mourraille's work.

that

Despite

the striking similarity between Mourraille's and Fourier's ideas,

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150 Fourier said chac he was unacquainted with his predecessor's work when he first conceived his idea: When I discovered these methods, I did not know at all everything chat had been cried for the perfection of approxioaclon methods and even che use of parabolic curves. Cercainly che ulcimace printed version of Fourier's Analyse made no reference to Mourraille, and Mourraille's anticipations of Fourier's improvements remained largely unappreciated until Cajorl's research in 1911.11

Rival methods Through che efforts of Rolle, Lagrange, Waring, and Fourier in the separation of roots, and chose of Mourraille and Fourier in securing convergence, Newton's approximation method was mathematically sound as well as popular by the end of the 18th century. however, lack rivals.

The one closest in spirit to Newton's method

was Lagrange's use of continued fractions. variable by

p + y,

substituted

p +■ ^

was

q,

It did not,

for

p

che closest Integer to the root, Lagrange

and solved for

he substituted

Rather than replace the

q + ~

y.

If the integer closest to

and solved for

z,

y

resulting in the

continued fraction expression x ■ p +

1 r+ etc.

It offered the advantage chat the expression halted in a finite number of steps if che root sought turned out to be rational. Lagrange could convert a continued fraction representation to a series of ordinary

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151 fraccions chat necessarily converged to che true value of

x;

these

fractions alternated, greater and less chan the true value, and were of che form

a u

where a ■ P.

a

3 ■ qa + 1

3’ - qa' - q.

Y

■ rS + a,

-5 ■ s y + 3

y'

- 1

* rS' + a'

5' • s y ' + 3',

etc

Among those Influenced by Lagrange's innovation were Caluso, whose article in the memoirs of the Turin Academy reflected a careful reading of che Traitg. But Canovai, who consented on che continued fraction method in che Siena memoirs, found Lagrange's approach impractical. He also noted that che mathematical basis of the continued fraction method derived from Newton's still practicable method of numerical solution.12 Numerous mathematicians also developed che theory of recurrent series and their application to the approximation of roots.

The first

conception of recurring series — so called because a recursion formula relates each coefficient to certain of the preceding coefficients — was the work of de Moivre.

Not long afterwards Daniel Bernoulli

pioneered in their use for approximating roots.

Analytical proofs for

the validity of his procedure, however, were long delayed, appearing first in the work of Euler and Lagrange.

Euler offered the most

straightforward explanation of approximation by recurring series in his 1770 algebra text. such that

q/p,

r/q,

Beginning with a series s/r, etc.

approximated the desired root

the given equation, Euler observed chac 2

r/p ■ x . But

s/r - x

p, q, r, s, c, etc.

z/p ■ x

as well, implying

and

r/q - x

3

x

implied

s/p » x , and similarly

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of

152 4 c/p - x , etc. q/p

for

x

and

r/p

for

x

2

s ■ r + q,

valuesfor powers

of x

p, q, etc.

** q + p ■ r. 1 *

t ■ s + r, etc.

In this manner che assumed

yielded arecursion

the series, intowhicharbitrary could be plugged.

che substitutions

produced

— • +1 P P Similarly

2 x - x + 1,

Thus for the example

choices

formula for theterms in

for che initial values

of

The result would be a series chat by

definition converged toward the value of the root.

The arbitrary choice

of initial values, however, affected seriously che speed of convergence, and the method of recurrent series failed to generate substantial enthusiasm in approximation studies.13

More difficulties in deployment The deployment of approximation methods involved two additional problems:

che nature and extent of numerical error for each method

and che thorny imaginary root.

Only a few mathematicians attempted to

come to terms with these difficulties. Early on Lagny had declared chat "Any approximation method is useless if unaccompanied by another method giving the limits of error." But his warning was buried in che myriad details of one of his illreceived approximation methods.

The divergence of his path from the

development of the Newtonian method robbed his insistence on che importance of error of most of its force.

Likewise, Mourraille's

attention to relative accuracy passed unnoticed.

He questioned che

assumption chac solving a quadratic equation would afford greater accuracy in finding

z

for the Newtonian approximation

A + z,

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and

showed chat it was not always true that closer to

0

than

f(A) + f'(A)z.

f(A) + C’(A)z + f"(A)z

2

lay

In other words, he showed chac it

was not necessarily safer to neglect che terms of exponent greater than two in che Taylor expansion than to neglect all terms after f'(A)z.

He offered no specific numerical example.

But Lagrange, in

his exposition of the method of continued fractions, gave both an algebraic expression for maximum error and applied it to a specific equation. root

x,

For

a 3 -p , rp, etc. a o

converging to the true value of che

with the first term greater thau

x,

the second less, etc.

the quantity 3 a S' ~ o' limited the error of the approximation — p . But the conditions on a a, a', 3, 3', etc.

implied that 3 — a 3' a’

a

^ • a* 3' ' 8 ° a

V

'

1 > CaTjZ ' a' 3 ' Since this held also for succeeding pairs of fractions in che sequence, "the error for each fraction will always be less than one divided by the square of the denominator of that fraction." example of

x

- 2x - 5 ■ 0,

at che tenth approximation

Applied to Newton's

this meant an error less than L/(7837)2 2.0945514865.

At the end of the century,

che estimation of error in numerical solution could have also drawn upon the work of Massabiau, whose Essai sur les nombres approximatifs sought a limit on error in che performance of algebraic operations on numerical quantities.

The cumbersome descriptions of his procedure

probably blunted his influence, already limited by his distance from the mathematical establishment.

The net result was chac the century

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154

closed without much attention to the question of approximation error. Pae approximation of imaginary roots was also a subject of little concern.

Fontaine supplied an elaborate table of formulas for

combinations of real and imaginary roots of polynomial equations, with associated conditions on the coefficients of the equations.

In a

limited number of cases, manipulation of the quantities involved could produce a numerical expression of the form

a + b