I wrote the following two years ago, and I'm not sure I'd write it today, partly because I've forgotten more about analysis than I ever thought I knew, and partly because my writing style has changed.
A discussion of the developments made in the analysis of functions by Cauchy, attempting to gauge the importance of these developments for subsequent mathematicians.
Introduction
The value of the rôle of Augustin-Louis Cauchy in the development of analysis has been estimated very differently by those who have written about him. It is widely agreed that his first biography, C A Valson's La Vie et les Travaux du Baron Cauchy is, in the words of the much more recent biographer Professor Bruno Belhoste, 'more hagiography than history'1. Intent on glorifying its subject, Valson's work is consequently unforthcoming on some of his most interesting characteristics and the work which brings them to light. Ivor Grattan-Guinness states that 'a legend has grown that {Cauchy} swept all before him in a revolutionary program of new analysis which has remained standard ever since'2 and describes Cauchy as 'selfish'3 and 'unpleasant'4. Professor Belhoste is more charitable. In this essay, I shall be reviewing the work of Cauchy in connection with the rigorisation of analysis, an area in which he did much important work, whilst aggravating many of his contemporaries. The parts of this discussion are as follows:
- Background: A description of the difficulties in calculus and analysis as they stood before the time of Cauchy.
- Cauchy and the École Polytechnique: An account of Cauchy's early life and employment before his self-imposed exile of 1830.
- Cauchy's Textbooks: A summary of the impact of the Analyse Algébrique and Calcul Infinitésimal, the texts Cauchy wrote while at the École Polytechnique.
- Conclusion: An assessment of the lasting importance of Cauchy's work as discussed above.
Background
In 1734, seven years after the death of Isaac Newton, George Berkeley, Bishop of Cloyne, published a tract entitled The Analyst, in which he attacked the existing state of the calculus. Berkeley's claim was that the use of infinitesimals, which was fundamental to the calculus of both Newton and Leibniz, could not be justified. Other mathematicians at the time were aware that the methods of infinitesimals led to difficulties in certain cases, and were not properly rigorous, but Berkeley contested that the methods themselves could not be justified. Without suggesting an alternative to infinitesimals or disputing the usefulness of the results of Newton or Leibniz, Berkeley demonstrated that one of the basic principles of proof was violated by the use of these infinitely small quantities. As part of any argument using these quantities, it was to be assumed that such quantities existed, usually as increments to variables (Newton called these the moments of the fluents5), and then subsequently declaring these increments to vanish. "Certainly" wrote Berkeley, "when we suppose the increments to vanish, we must suppose their proportions, their expressions, and every thing else derived from the supposition of their existence to vanish with them."6 But it is essential to proofs made by assumptions that the truth of the proof is dependent on the reality of the assumption. By declaring the infinitesimal variations to be zero, Newton, Leibniz and their respective followers were breaking this principle of proof, since the initial assumption is that an infinitesimal is very small, but non-zero. So among the things vanishing along with the increments is the proof itself!
This attack did not go unanswered, and among the mathematicians who responded and attempted to reformulate calculus on a non-infinitesimal basis was Jean-le-Rond d'Alembert, the nephew of a French cardinal. In 1754, d'Alembert suggested, but was not himself able to implement, the development of a proper theory of limits to provide this basis. His suggestion was ignored for half a century. So too was Euler's promotion in his Introductio in analysin infinitorum (Introduction to the Analysis of Infinities) of 1748 of the use of functions to formalise the raw materials, as it were, of analysis, where d'Alembert had concerned himself with the method. Euler was also responsible for moving analysis, and with it calculus, firmly from the domain of geometry to that of pure algebra7. But the problems with the basis of analysis were not resolved for another hundred and fifty years. This resolution began with the work of Augustin-Louis Cauchy.
Cauchy and the École Polytechnique
Cauchy was born on the twenty-first of August 1789, less than a month after his father's job as principal commis of the Lieutenant Général de Police of Paris had come to an end in the events that were to become the French Revolution. Many mathematicians were to become involved in the politics of the following turbulent years. Following the Reign of Terror in 1794, Augustin-Louis' father, Louis-François, returned to Paris and got a new administrative job. The arrival of Napoleon Bonaparte in 1799 led to the promotion of Louis-François to the post of Secretary-General of the Senate, of which the mathematician Count Laplace was the Chancellor and Lagrange a member. Laplace was very taken with the young Cauchy, and encouraged him to pursue an intellectual career. As a consequence of this, Augustin-Louis took the examination for the École Polytechnique in 1805. Although it had been founded by the revolutionaries, the École had just been reformed by Bonaparte, and was to be organised along military lines. Cauchy came second out of 293 examinees, and joined the Highways and Bridges department, in accordance with his choice for the technical field he intended to enter upon leaving. Whilst at the École, he studied analysis, geometry and mechanics under many of the great names of the day, including Sylvestre François Lacroix, who had written the standard textbook on calculus at the time and would later assess one of Cauchy's first published works. After leaving the École, Cauchy spent several years at Cherbourg, working on the construction of a new port facility. He returned to Paris in 1812 in ill health, but while at Cherbourg he had developed the idea which was to drive him forever after: that he would concentrate upon 'the search for truth'.8
In the years that followed, Cauchy did not receive any proper work as a mathematician. His attempts to enter the inner sancta of science in Napoleonic France were unsuccessful, and although he did much important work on group theory and related topics, the recognition he received did not pay in real terms. He was allowed a job working on the Ourcq Canal project, and his father's well-paid job also enabled him to spend time on mathematics, but it was not until after the fall of Bonaparte that a proper academic post was to become available to him. For political reasons, several of the teachers at the École Polytechnique resigned in late 1815, and the power-vacuum thus created opened the way for Cauchy. Where those who had left were Bonapartists, Cauchy was ultra-Catholic and a staunch supporter of the re-established Bourbon dynasty. At the same time as he became assistant professor of analysis to Poinsot at the École, he also gained admission at long last to the Institut de France for his work on the theory of waves, and for a proof of a conjecture of Fermat.
Officially, Cauchy's rôle at the École was as a teacher of mechanics with an analytical basis. However, he soon found that the officially mandated timetable did not allow enough time for him to present what he considered a sufficiently rigorous treatment of the analytical topics. On April 17, 1821, just a few months before the publication of his work Analyse Algébrique, Cauchy was criticised in a meeting of the École's governing body for requiring 66 lessons instead of the allotted 50 for the analytical content of the course, and thus compromising the teaching of mechanics, which was the object of the course9. The discussion, and a subsequent report critical of Cauchy, arose from an incident in one of his classes a few days previously where he was jeered and hissed at by some (unidentified) students. The Analyse Algébrique, which had been intended by Cauchy as a textbook for his course, was consequently never used in it.
Cauchy's Textbooks
Cauchy was accused, with some justification, of sacrificing his requirement to teach the syllabus in favour of writing his own works. It is ironic that the Analyse Algébrique should have so upset the École, since its full title was Cours d'Analyse de l'École Royale Polytechnique: Première Partie: Analyse Algébrique (Analysis Course of the Royal École Polytechnique: First Part: Algebraic Analysis). But Cauchy was dedicated to setting the analysis on a firmer footing than that conceived of by the authors of his syllabus. Cauchy's treatment of his topic resembles closely the modern rigorous approach to the subject; but this was the first time such a thing had been done. Beginning from simple definitions of numbers and operators, he established precisely what the subject matter was. But most crucially of all for the formalisation of the subject, the very first section - the Preliminaries - contains the definition of a limit.
When the successively attributed values of one variable indefinitely approach a fixed value in such a way that they finally differ from it by as little as desired, then that fixed value is called the limit of all the others.10
This is a rather clearer and more general version of the same definition given in {3}, p 70, over 170 years later:
A sequence (an) converges to a limit L if and only if for every epsilon>0 there exists a natural number N such that n > N ⇒ |an - L| < ε.
Cauchy uses this definition to define infinitesimals as quantities with zero limits - a definition which both removes the difficulties with such quantities and renders them much less significant, as his subsequent discussion of the analysis shows. The attitude to limits was not an entirely new approach, and indeed, the Conseil d'Instruction of the École had declared it to be lacking in rigour and had endorsed infinitesimals instead in 181111. But Cauchy had given it a brilliantly clear exposition, and had moreover brought together the fundamentals of the subject in a manner never previously done. The definition of an infinitesimal is characteristic of this. The definition of continuity in terms of limits followed, but Cauchy failed to distinguish between simple and uniform continuity, often using the former in the proof of theorems only in fact true of the latter12. Later in the same book, Cauchy gave a treatment of series and sequences, drawing on the basis of limits once again, which is unsurpassed today, including the most general and definitive test for the convergence of a sequence - Cauchy's test, as it is known today. This test uses as its criterion the convergence to zero of the differences of arbitrary terms of the sequence, and was new to Cauchy's audience. He also stated that the sum of a series of continuous functions was itself continuous, which Niels Henrik Abel, who had been a student of Cauchy's, disproved in 1826 with the counterexample
∞ sin(2n+1)x
f(x) = Σ ----------
n=0 2n+1
which is discontinuous at π, while its individual terms are all continuous13. After the section on series and sequences, the Analyse Algébrique concludes with four chapters on complex numbers, where the form which much of the material takes is close to, but not identical to, the equivalent today. In particular, the symbol i for √-1 had been coined by Euler, but Cauchy treated √-1 as an essentially symbolic concept, with no meaning on its own, but only as a tool, or a feature in functions of real numbers. Nevertheless, his work on the subject was wide-ranging and included the definition of functions of complex numbers such as ez.
The Conseil d'Instruction were not happy with Cauchy's conduct, and required him to abandon work on further parts of his Cours d'Analyse and instead to compile (along with André-Marie Ampère, his fellow-lecturer for the course) a collection of lecture notes for the direct consumption of the students. The result of this was the publication in August 1823 of the Résumé des Leçons Données à l'École Royale Polytechnique sur le Calcul Infinitésimal (Summary of Classes Given to the Royal École Polytechnique on the Infinitesimal Calculus). As the title suggests, the methods described in this work made concessions to the then orthodox means of proceeding. But this was not the principal problem. Cauchy had again run into difficulties over the meaning of continuity. The definition of a differential in the third lesson of the Calcul Infinitésimal is as follows:
When the function y=f(x) remains continuous between two limits given to the variable x, and when one assigns to that variable a value contained between the two limits in question, an infinitely small increment, added to the variable, produces an infinitely small increment in the function itself. Consequently, if you then put Δx = i , the two terms of the ratio of the differences
Δx f(x+i) - f(x)
-- = --------------
Δy i
are infinitely small quantities. But, whereas both these terms approach the limit zero indefinitely and simultaneously, the ratio itself may converge to another limit, either positive or negative. This limit, where it exists, has a determined value for each particular value of x; but it varies with x. {Here he gives the example f(x)=xm, and states the limit of the ratio of the differences as a new function of x.} It will be the same in general; only, the form of the new function which serves as the limit of the ratio f(x+i) - f(x) / i will depend on the form of the proposed function y=f(x). To indicate the dependence, one gives the new function the name of derived function, and one designates it, with the help of an accent, by the notation y' or f'(x).14
Here we see many of the best characteristics of Cauchy's rigorist attitude, and also most of the remaining flaws in his treatment of calculus. Aside from the (to our eyes) superfluous use of infinitesimals, he is still using his slightly flawed definition of continuity, and he states without proof, for he thought it self-evident, that all continuous functions are differentiable. The seventh lesson of the Calcul Infinitésimal features the first use of the symbolic means - now commonly employed with epsilons and deltas - to define a limit. However, the proof in which this step is taken is itself flawed, again because of the lack of understanding on the point of uniform continuity. The rest of the book dealt with integral calculus, but as Professor Belhoste observes15, the later material, especially on Maclaurin and Taylor series, is placed after the fundamental material on limits quite deliberately. Even when jumping through the Conseil d'Instruction's hoops, Cauchy put his own ideas first.
Conclusion
Cauchy would go on to do other important work in many fields of mathematics and mathematical physics. But his philosophical and intellectual contribution to his fields of interest is best exemplified by the material discussed above. In The Analyst, Bishop Berkeley accused the mathematicians, especially those who attacked religion for its lack of rigour and demonstration, of hypocrisy. He questioned "{w}hether such mathematicians as cry out against mysteries have ever examined their own principles?"16 In Cauchy, the Bishop's objection was met from both sides. On the one hand, Cauchy was largely responsible for demystifying the field of mathematics upon which the Bishop had written, and setting in motion the subsequent completion of this process. On the other, Cauchy was himself a man of profound faith whose philosophy was founded on a trust in God and the reality of absolute truth. Indeed, he wrote, "the joy of heaven itself is no more than the full and complete possession of immortal truth"17. It would be left to Weierstrauss in 1874 to produce the counterexample to Cauchy's assertion that all continuous functions possessed at least some derivatives, and thus precipitate the arithmetisation of analysis which gave the subject its present form18. But Cauchy, with his belief in a divine scheme that made truth attainable, had first set down the principles that analysis follows to this day. I suppose a few words had better be said here concerning Cauchy's character. It is true that he was an opinionated hard-liner, and he frequently did not do as he was asked with material presented to him. This did not apply only to syllabuses; Both Abel and Evariste Galois died, tragically young, without Cauchy having fulfilled long overdue commitments to review and present their work to his colleagues. But I believe Cauchy did not act out of malice, but merely a conviction that whatever he felt the search for truth led him to consider took precedence over the individual concerns of those around him. Nor was he always heartless. Mikhail Ostrogradski, a talented student of Cauchy's, was helped both academically and in the very practical matter of being released from debtor's prison by his tutor. While he may have been an awkward person, Cauchy does not deserve the harsh treatment he sometimes received, both in his lifetime and afterward (in {4}, for example), for his contribution to his field should outlast the individual unhappiness of men.
Cauchy's legacy is with us today in every analysis text, as in many other fields of mathematics, but perhaps it is not as widely felt as it ought to be. It is a source of continuing amazement to me that pupils in secondary schools are expected to appreciate the complexities of the calculus without ever learning of limits or appreciating the true definitions of the concepts - differentials and integrals - which they are expected to manipulate fluently in their exercises and exams. As Cauchy himself wrote in 1844: "Unless it be accompanied by a good education, instruction can become more troublesome than useful."19 I would hope that in coming years, educators realise the value of rigour in all subjects, and thus make education as a whole more useful and less troublesome.
Notes:
- {1}, p viii
- {4}, p 24
- {4}, p 26
- (4}, p 28
- Moment indicates an infinitesimal increment or decrement to a variable, while the variable itself is referred to by Newton as the fluent.
- {2}, p 557.
- Details on the life and work of d'Alembert and of Euler are taken from {3}, pp. 1, 13-14 and 111-112
- {1}, p 31. The same source furnishes the bulk of the biographical detail in this essay, the remainder coming from 4.
- {1}, p 72.
- {1}, p 66.
- {1}, p 69.
- For example, {1}, p 69, first paragraph.
- {1}, p 76. Translation mine.
- {1}, p 79.
- {2}, p 557.
- {1}, p 216.
- {3}, p 15
- {1}, p 217
Bibliography
{1} Augustin-Louis Cauchy: A Biography, Professor Bruno Belhoste, tr. Frank Ragland, Spring-Verlag, 1991. Originally published in French as Cauchy: 1789-1857, written 1989. An excellent source.
{2} The History of Mathematics: A Reader, eds. John Fauvel and Jeremy Gray, Macmillan Press, 1987.
{3} Fundamentals of Mathematical Analysis, Rod Haggarty, Addison-Wesley Publishing, 1993.
{4} The Development of the Foundations of Mathematical Analysis from Euler to Riemann, Ivor Grattan-Guinness, MIT Press, 1970. Aside from being unduly harsh to Cauchy, this book also makes much less easy reading than Ragland's translation of {1}.