Jules Henri Poincaré

Born: 29 April 1854 in Nancy, Lorraine, France
Died: 17 July 1912 in Paris, France

Henri Poincaré's father was Léon Poincaré and his mother was Eugénie Launois. They were 26 and 24 years of age, respectively, at the time of Henri's birth. Henri was born in Nancy where his father was Professor of Medicine at the University. Léon Poincaré's family produced other men of great distinction during Henri's lifetime. Raymond Poincaré, who was prime minister of France several times and president of the French Republic during World War I, was the elder son of Léon Poincaré's brother Antoine Poincaré. The second of Antoine Poincaré's sons, Lucien Poincaré, achieved high rank in university administration.

Henri was :-

... ambidextrous and was nearsighted; during his childhood he had poor muscular coordination and was seriously ill for a time with diphtheria. He received special instruction from his gifted mother and excelled in written composition while still in elementary school.

In 1862 Henri entered the Lycée in Nancy (now renamed the Lycée Henri Poincaré in his honour). He spent eleven years at the Lycée and during this time he proved to be one of the top students in every topic he studied. Henri was described by his mathematics teacher as a "monster of mathematics" and he won first prizes in the concours général, a competition between the top pupils from all the Lycées across France.

Poincaré entered the École Polytechnique in 1873, graduating in 1875. He was well ahead of all the other students in mathematics but, perhaps not surprisingly given his poor coordination, performed no better than average in physical exercise and in art. Music was another of his interests but, although he enjoyed listening to it, his attempts to learn the piano while he was at the École Polytechnique were not successful. Poincaré read widely, beginning with popular science writings and progressing to more advanced texts. His memory was remarkable and he retained much from all the texts he read but not in the manner of learning by rote, rather by linking the ideas he was assimilating particularly in a visual way. His ability to visualise what he heard proved particularly useful when he attended lectures since his eyesight was so poor that he could not see the symbols properly that his lecturers were writing on the blackboard.

After graduating from the École Polytechnique, Poincaré continued his studies at the École des Mines. His :-

... meticulous notes taken on field trips while a student there exhibit a deep knowledge of the scientific and commercial methods of the mining industry; a subject that interested him throughout his life.

After completing his studies at the École des Mines Poincaré spent a short while as a mining engineer at Vesoul while completing his doctoral work. As a student of Charles Hermite, Poincaré received his doctorate in mathematics from the University of Paris in 1879. His thesis was on differential equations and the examiners were somewhat critical of the work. They praised the results near the beginning of the work but then reported that the :-

... remainder of the thesis is a little confused and shows that the author was still unable to express his ideas in a clear and simple manner. Nevertheless, considering the great difficulty of the subject and the talent demonstrated, the faculty recommends that M Poincaré be granted the degree of Doctor with all privileges.

Immediately after receiving his doctorate, Poincaré was appointed to teach mathematical analysis at the University of Caen. Reports of his teaching at Caen were not wholly complimentary, referring to his sometimes disorganised lecturing style. He was to remain there for only two years before being appointed to a chair in the Faculty of Science in Paris in 1881. In 1886 Poincaré was nominated for the chair of mathematical physics and probability at the Sorbonne. The intervention and the support of Hermite was to ensure that Poincaré was appointed to the chair and he also was appointed to a chair at the École Polytechnique. In his lecture courses to students in Paris :-

... changing his lectures every year, he would review optics, electricity, the equilibrium of fluid masses, the mathematics of electricity, astronomy, thermodynamics, light, and probability.

Poincaré held these chairs in Paris until his death at the early age of 58.

Before looking briefly at the many contributions that Poincaré made to mathematics and to other sciences, we should say a little about his way of thinking and working. He is considered as one of the great geniuses of all time and there are two very significant sources which study his thought processes. One is a lecture which Poincaré gave to l'Institute Général Psychologique in Paris in 1908 entitled Mathematical invention in which he looked at his own thought processes which led to his major mathematical discoveries. The other is the book [30] by Toulouse who was the director of the Psychology Laboratory of l'École des Hautes Études in Paris. Although published in 1910 the book recounts conversations with Poincaré and tests on him which Toulouse carried out in 1897.

In [30] Toulouse explains that Poincaré kept very precise working hours. He undertook mathematical research for four hours a day, between 10 am and noon then again from 5 pm to 7 pm. He would read articles in journals later in the evening. An interesting aspect of Poincaré's work is that he tended to develop his results from first principles. For many mathematicians there is a building process with more and more being built on top of the previous work. This was not the way that Poincaré worked and not only his research, but also his lectures and books, were all developed carefully from basics. Perhaps most remarkable of all is the description by Toulouse in  of how Poincaré went about writing a paper. Poincaré:-

... does not make an overall plan when he writes a paper. He will normally start without knowing where it will end. ... Starting is usually easy. Then the work seems to lead him on without him making a wilful effort. At that stage it is difficult to distract him. When he searches, he often writes a formula automatically to awaken some association of ideas. If beginning is painful, Poincaré does not persist but abandons the work.

Toulouse then goes on to describe how Poincaré expected the crucial ideas to come to him when he stopped concentrating on the problem:-

Poincaré proceeds by sudden blows, taking up and abandoning a subject. During intervals he assumes ... that his unconscious continues the work of reflection. Stopping the work is difficult if there is not a sufficiently strong distraction, especially when he judges that it is not complete ... For this reason Poincaré never does any important work in the evening in order not to trouble his sleep.

As Miller notes in :-

Incredibly, he could work through page after page of detailed calculations, be it of the most abstract mathematical sort or pure number calculations, as he often did in physics, hardly ever crossing anything out.

Let us examine some of the discoveries that Poincaré made with this method of working. Poincaré was a scientist preoccupied by many aspects of mathematics, physics and philosophy, and he is often described as the last universalist in mathematics. He made contributions to numerous branches of mathematics, celestial mechanics, fluid mechanics, the special theory of relativity and the philosophy of science. Much of his research involved interactions between different mathematical topics and his broad understanding of the whole spectrum of knowledge allowed him to attack problems from many different angles.

Before the age of 30 he developed the concept of automorphic functions which are functions of one complex variable invariant under a group of transformations characterised algebraically by ratios of linear terms. The idea was to come in an indirect way from the work of his doctoral thesis on differential equations. His results applied only to restricted classes of functions and Poincaré wanted to generalise these results but, as a route towards this, he looked for a class functions where solutions did not exist. This led him to functions he named Fuchsian functions after Lazarus Fuchs but were later named automorphic functions. The crucial idea came to him as he was about to get onto a bus, as he relates in Science and Method (1908):-

At the moment when I put my foot on the step the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformation that I had used to define the Fuchsian functions were identical with those of non-euclidean geometry.

In a correspondence between Klein and Poincaré many deep ideas were exchanged and the development of the theory of automorphic functions greatly benefited. However, the two great mathematicians did not remain on good terms, Klein seeming to become upset by Poincaré's high opinions of Fuchs' work. Rowe examines this correspondence in .

Poincaré's Analysis situs , published in 1895, is an early systematic treatment of topology. He can be said to have been the originator of algebraic topology and, in 1901, he claimed that his researches in many different areas such as differential equations and multiple integrals had all led him to topology. For 40 years after Poincaré published the first of his six papers on algebraic topology in 1894, essentially all of the ideas and techniques in the subject were based on his work. Even today the Poincaré conjecture remains as one of the most baffling and challenging unsolved problems in algebraic topology.

Homotopy theory reduces topological questions to algebra by associating with topological spaces various groups which are algebraic invariants. Poincaré introduced the fundamental group (or first homotopy group) in his paper of 1894 to distinguish different categories of 2-dimensional surfaces. He was able to show that any 2-dimensional surface having the same fundamental group as the 2-dimensional surface of a sphere is topologically equivalent to a sphere. He conjectured that this result held for 3-dimensional manifolds and this was later extended to higher dimensions. Surprisingly proofs are known for the equivalent of Poincaré's conjecture for all dimensions strictly greater than three. No complete classification scheme for 3-manifolds is known so there is no list of possible manifolds that can be checked to verify that they all have different homotopy groups.

Poincaré is also considered the originator of the theory of analytic functions of several complex variables. He began his contributions to this topic in 1883 with a paper in which he used the Dirichlet principle to prove that a meromorphic function of two complex variables is a quotient of two entire functions. He also worked in algebraic geometry making fundamental contributions in papers written in 1910-11. He examined algebraic curves on an algebraic surface F(x, y, z) = 0 and developed methods which enabled him to give easy proofs of deep results due to Emile Picard and Severi. He gave the first correct proof of a result stated by Castelnuovo, Enriques and Severi, these authors having suggested a false method of proof.

His first major contribution to number theory was made in 1901 with work on :-

... the Diophantine problem of finding the points with rational coordinates on a curve f(x, y) = 0, where the coefficients of f are rational numbers.

In applied mathematics he studied optics, electricity, telegraphy, capillarity, elasticity, thermodynamics, potential theory, quantum theory, theory of relativity and cosmology. In the field of celestial mechanics he studied the three-body-problem, and the theories of light and of electromagnetic waves. He is acknowledged as a co-discoverer, with Albert Einstein and Hendrik Lorentz, of the special theory of relativity. We should describe in a little more detail Poincaré's important work on the 3-body problem.

Oscar II, King of Sweden and Norway, initiated a mathematical competition in 1887 to celebrate his sixtieth birthday in 1889. Poincaré was awarded the prize for a memoir he submitted on the 3-body problem in celestial mechanics. In this memoir Poincaré gave the first description of homoclinic points, gave the first mathematical description of chaotic motion, and was the first to make major use of the idea of invariant integrals. However, when the memoir was about to be published in Acta Mathematica, Phragmen, who was editing the memoir for publication, found an error. Poincaré realised that indeed he had made an error and Mittag-Leffler made strenuous efforts to prevent the publication of the incorrect version of the memoir. Between March 1887 and July 1890 Poincaré and Mittag-Leffler exchanged fifty letters mainly relating to the Birthday Competition, the first of these by Poincaré telling Mittag-Leffler that he intended to submit an entry, and of course the later of the 50 letters discuss the problem concerning the error. It is interesting that this error is now regarded as marking the birth of chaos theory. A revised version of Poincaré's memoir appeared in 1890.

Poincaré's other major works on celestial mechanics include Les Méthodes nouvelles de la mécanique céleste in three volumes published between 1892 and 1899 and Leçons de mecanique céleste (1905). In the first of these he aimed to completely characterise all motions of mechanical systems, invoking an analogy with fluid flow. He also showed that series expansions previously used in studying the 3-body problem were convergent, but not in general uniformly convergent, so putting in doubt the stability proofs of Lagrange and Laplace.

He also wrote many popular scientific articles at a time when science was not a popular topic with the general public in France. As Whitrow writes in :-

After Poincaré achieved prominence as a mathematician, he turned his superb literary gifts to the challenge of describing for the general public the meaning and importance of science and mathematics.

Poincaré's popular works include Science and Hypothesis (1901), The Value of Science (1905), and Science and Method (1908). A quote from these writings is particularly relevant to this archive on the history of mathematics. In 1908 he wrote:-

The true method of foreseeing the future of mathematics is to study its history and its actual state.

Finally we look at Poincaré's contributions to the philosophy of mathematics and science. The first point to make is the way that Poincaré saw logic and intuition as playing a part in mathematical discovery. He wrote in Mathematical definitions in education (1904):-

It is by logic we prove, it is by intuition that we invent.

In a later article Poincaré emphasised the point again in the following way:-

Logic, therefore, remains barren unless fertilised by intuition.

McLarty  gives examples to show that Poincaré did not take the trouble to be rigorous. The success of his approach to mathematics lay in his passionate intuition. However intuition for Poincaré was not something he used when he could not find a logical proof. Rather he believed that formal arguments may reveal the mistakes of intuition and logical argument is the only means to confirm insights. Poincaré believed that formal proof alone cannot lead to knowledge. This will only follow from mathematical reasoning containing content and not just formal argument.

It is reasonable to ask what Poincaré meant by "intuition". This is not straightforward, since he saw it as something rather different in his work in physics to his work in mathematics. In physics he saw intuition as encapsulating mathematically what his senses told him of the world. But to explain what "intuition" was in mathematics, Poincaré fell back on saying it was the part which did not follow by logic:-

... to make geometry ... something other than pure logic is necessary. To describe this "something" we have no word other than intuition.

The same point is made again by Poincaré when he wrote a review of Hilbert's Foundations of geometry (1902):-

The logical point of view alone appears to interest [Hilbert]. Being given a sequence of propositions, he finds that all follow logically from the first. With the foundations of this first proposition, with its psychological origin, he does not concern himself.

We should not give the impression that the review was negative, however, for Poincaré was very positive about this work by Hilbert. In  Stump explores the meaning of intuition for Poincaré and the difference between its mathematically acceptable and unacceptable forms.

Poincaré believed that one could choose either euclidean or non-euclidean geometry as the geometry of physical space. He believed that because the two geometries were topologically equivalent then one could translate properties of one to the other, so neither is correct or false. for this reason he argued that euclidean geometry would always be preferred by physicists. This, however, has not proved to be correct and experimental evidence now shows clearly that physical space is not euclidean.

Poincaré was absolutely correct, however, in his criticism that those like Russell who wished to axiomatise mathematics; they were doomed to failure. The principle of mathematical induction, claimed Poincaré, cannot be logically deduced. He also claimed that arithmetic could never be proved consistent if one defined arithmetic by a system of axioms as Hilbert had done. These claims of Poincaré were eventually shown to be correct.

We should note that, despite his great influence on the mathematics of his time, Poincaré never founded his own school since he did not have any students. Although his contemporaries used his results they seldom used his techniques.

Poincaré achieved the highest honours for his contributions of true genius. He was elected to the Académie des Sciences in 1887 and in 1906 was elected President of the Academy. The breadth of his research led to him being the only member elected to every one of the five sections of the Academy, namely the geometry, mechanics, physics, geography and navigation sections. In 1908 he was elected to the Académie Francaise and was elected director in the year of his death. He was also made chevalier of the Légion d'Honneur and was honoured by a large number of learned societies around the world. He won numerous prizes, medals and awards.

Poincaré was only 58 years of age when he died :-

M Henri Poincaré, although the majority of his friends were unaware of it, recently underwent an operation in a nursing home. He seemed to have made a good recovery, and was about to drive out for the first time this morning. He died suddenly while dressing.

His funeral was attended by many important people in science and politics :-

The President of the Senate and most of the members of the Ministry were present, and there were delegations from the French Academy, the Académie des Sciences, the Sorbonne, and many other public institutions. The Prince of Monaco was present, the Bey of Tunis was represented by his two sons, and Prince Roland Bonaparte attended as President of the Paris Geographical Society. The Royal Society was represented by its secretary, Sir Joseph Larmor, and by the Astronomer Royal, Mr F W Dyson.

Let us end with a quotation from an address at the funeral:-

M Poincaré was a mathematician, geometer, philosopher, and man of letters, who was a kind of poet of the infinite, a kind of bard of science.

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Guillaume François Antoine Marquis de L'Hôpital

Born: 1661 in Paris, France
Died: 2 Feb 1704 in Paris, France

Guillaume De l'Hôpital served as a cavalry officer but resigned because of nearsightedness. From that time on he directed his attention to mathematics. L'Hôpital was taught calculus by Johann Bernoulli from the end of 1691 to July 1692.

L'Hôpital was a very competent mathematician and solved the brachystochrone problem. The fact that this problem was solved independently by Newton, Leibniz and Jacob Bernoulli puts l'Hôpital in very good company.

L'Hôpital's fame is based on his book Analyse des infiniment petits pour l'intelligence des lignes courbes (1696) which was the first text-book to be written on the differential calculus. In the introduction L'Hôpital acknowledges his indebtedness to Leibniz, Jacob Bernoulli and Johann Bernoulli but L'Hôpital regarded the foundations provided by him as his own ideas.

In this book is found the rule, now known as L'Hôpital's rule, for finding the limit of a rational function whose numerator and denominator tend to zero at a point.

L'Hôpital: Analyse des infiniment petits Preface

Guillaume, Marquis de L'Hôpital, published Analyse des infiniment petits pour l'intelligence des lignes courbes in 1696. This was the first text-book to be written on the differential calculus and it is interesting to examine the Preface of the work in which de L'Hôpital gives historical comments as well as describing the contents of the work:

The type of analysis we shall describe in this work presupposes an acquaintance with ordinary analysis, but is very different from it. Ordinary analysis deals only with finite quantities whereas we shall be concerned with infinite ones. We shall compare infinitely small differences with finite quantities; we shall consider the ratios of these differences and deduce those of the finite quantities, which, by comparison with the infinitely small quantities are like so many infinities. We could never say that our analysis takes us beyond infinity because we shall consider not only these infinitely small differences but also the ratios of the differences of these differences, and those of the third differences and the fourth differences and so on, without encountering any obstacle to our progress. So we shall not only deal with infinity but with an infinity of infinity or an infinity of infinities.

Only this kind of analysis is capable of giving a true insight into the properties of curves. For curves are merely polygons with an infinite number of sides, and curves differ from one another only because these infinitely small sides form different angles with one another. Only by using these methods of analysis of infinitely small quantities can we determine the positions of these sides and thus find the properties of the curve they form: properties such a the directions of tangents and normals to the curve, its points of inflection, turning points, how it reflects or refracts rays, etc.

It has long been realised that polygons inscribed within a curve or circumscribed about it become identical with the curve as the number of their sides is increased to infinity. But there the matter rested until the invention of the type of analysis we are about to describe at last showed the scope and implications of such an idea.

The work done in this field by ancient scholars, particularly Archimedes, is certainly worthy of admiration. But they only considered a very few curves and those only cursorily. We find no more than a succession of special cases, in no particular order: and they provide no indication of any general and consistent method of procedure. We cannot reasonably blame the ancient scholars for this: it required a genius of the first order to find a way through so many difficulties and do the very first work in this hitherto completely unknown field. They did not go far, and they proceeded by roundabout ways, but, whatever Vieta says, they did not lose their way: and the more difficult and thorny the paths they trod the more we must admire these ancient mathematicians for having succeeded so well. To put it briefly: it does not appear that the Ancients could, at the time, have done better than they did. They did what our mathematicians would have done in their place, and if they were in ours it is likely that they would do as we do now. All this is a consequence of the unchanging nature of the human mind and the fact that discoveries can only be made in an orderly succession.

It is therefore not surprising that the Ancients did not get any further: but it is a matter of great astonishment that great men, some assuredly as great as the Ancients, should have failed so long to make any further progress, confining themselves to merely reading ancient authors and writing commentaries on their work, turning the knowledge they acquired to no use other than continuing to read, without daring to commit any such crime as to think for themselves and look beyond what Ancients had discovered. Thus it was that though there were many scholars who wrote a great deal, so that the number of books multiplied, yet for all this activity there was no progress made: all the work of several centuries merely served to supply the world with respectful commentaries and repeated translations, often of quite uninteresting works.

Such was the state of mathematics, and, above all, of philosophy, until the time of M Descartes, whose genius and self-confidence led him to abandon this study of ancient authorities and turn instead to reason, the authority to which these same Ancients had appealed. His boldness was considered to be merely a revolt but it led to a number of new and useful insights in physics and geometry. Then it was that men opened their eyes and began to think.

In mathematics, which is what concerns us here, M Descartes began where the ancients had left off, by solving a problem which Pappus said that no-one had been able to solve. It is well-known that he made great advances in analysis and in geometry, and that techniques he evolved from combining the two make it possible to solve many problems which had previously been completely intractable. But since he was mainly concerned with the solution of equations he was interested in curves only as a way to finding roots. He was therefore completely satisfied with using ordinary analysis, and did in fact use this new technique successfully in constructing tangents, and was so pleased with his method of solving this problem that he said it was the most useful and general problem he knew, or ever wanted to know, in all geometry.

M Descartes' Geometry made it fashionable to solve geometrical problems by means of equations, and opened up many possibilities of obtaining such solutions. Geometers applied themselves to the task and soon made new discoveries. Indeed, more and better new results are still being obtained. M Pascal was concerned with quite different matters: he studied curves as curves, and also as polygons. He found the lengths of some of them, the areas they enclosed, the volumes swept out by these areas, the centres of gravity of these areas and volumes etc etc. By considering only elements, that is infinitely small portions of the curve, he discovered methods which were of general application, and which, moreover, we must find the more surprising in that he seems to have arrived at them merely by the power of his imaginative grasp and not by means of analysis.

Soon after M Descartes had published his method for finding tangents, M Fermat discovered a different method, which M Descartes himself finally admitted was in many ways better than his own. At the time, however, this method was not as simple as M Barrow has since made it, by having paid closer attention to the properties of polygons, which naturally suggest that one consider the small triangles each made up of a part of the curve cut off between two infinitely close ordinates, the difference between these ordinates and the difference between the corresponding abscissae. This triangle is similar to that formed by the tangent, the ordinate and the subtangent, so that this method of finding the tangent uses straightforward similarity instead of the calculations which were required by M Descartes's method, and which this new method previously required. Barrow's work did not stop there: he also invented a kind of calculus based on this method, but this calculus, like that of Descartes, could only be used once all fractions and roots had been removed.

Barrow's calculus was replaced by that of M Leibniz, an accomplished geometer who started his own work where Barrow and others had ended theirs. His calculus led him into domains hitherto unknown and the discoveries he made amazed the most brilliant mathematicians of Europe. The Bernoullis were the first to recognise the elegance of Leibniz's method, and they in turn developed his calculus to a degree which enabled them to solve problems which had previously seemed too difficult to attempt.

This calculus is of immense scope: it can be used for the curves which occur in mechanics, transcendental curves such as the catenary, as well as for purely geometrical curves, squares or other roots do not cause any difficulty (and may even be an advantage), any number of variables may be considered, and it is equally easy to compare infinitely small quantities of any type. An endless number of interesting results can be obtained concerning tangents (including tangent curves) concerning problems connected with maxima and minima, points of inflection and turning points, evolutes, caustics derived by reflection or refraction, and so on, as we shall see in the work that follows.

I shall divide the work into ten sections. The first describes the principles of the calculus of differences [the differential calculus]. The second describes how it is used to find the tangent to any curve whatever the number of variables in the equation of the curve: though M Craig did not believe this method could be used for the transcendental curves which occur in mechanics. The third shows how the calculus is used in problems connected with maxima and minima. The fourth shows how it is used to find points of inflection and turning points. The fifth describes its use in finding M Huygen's evolutes for all kinds of curves. The sixth and seventh sections show how it is used to find the caustic curves discovered by the distinguished scholar M Tschirnhaus, both the type formed by reflection and that formed by refraction. The method is again applicable to all kinds of curves. The eighth section describes how the calculus is used to find the curves which touch an infinite number of given straight lines or curves. The ninth consists of solutions to various problems arising out of the earlier work. The tenth section describes a new way of using the differential calculus for geometrical curves: from which we can derive the method used by M Descartes and M Hudde, which is applicable only to this kind of curve. ...

I had intended to include an additional section which was to have described the marvellous use to which the calculus may be put in physics, what accuracy can thereby be obtained, and to show how useful the calculus would be in mechanics. Illness, however, prevented me. The public will, nevertheless, not lose this, since the work will eventually be published with additional material that has been accumulated in the meantime.

All this is only the first part of M Leibniz's work on calculus, which consists of working down from integral quantities to consider the infinitely small differences between them and comparing these infinitely small differences with each other, whatever their type: this part is called Differential Calculus. The other part of M Leibniz's work is called the Integral Calculus, and consists of working up from these infinitely small quantities to the quantities of totals of which they are the differences: that is, it consists of finding their sums. I had intended to describe this also. But M Leibniz wrote to me to say that he himself was engaged upon describing the integral calculus in a treatise he calls De Scientia infiniti, and I did not wish to deprive the public of such a work, which will deal with all the most interesting consequences of this inverse method of tangents, showing how it can be used to find the lengths of curves, to find the area they enclose, to find the volumes and surfaces of their solids of revolution, to find centres of gravity etc. I have said as much as this only because M Leibniz wrote and asked me to do so, and I myself think it necessary to prepare people's minds so that later they will be in a better position to understand all the results that are eventually obtained.

Finally, I am greatly indebted to the Bernoullis, particularly the younger Bernoulli [Johann Bernoulli], who is at present a professor at Groningen [Johann Bernoulli took up the appintment at Groningen in September 1695]. I have made free use of their work as well as that of M Leibniz. They may take the credit for as much of this work as they please, and I am quite content with what little they leave to me.

M Leibniz himself acknowledges his debt to M Newton, who, as it appears in his excellent work Philosophiae naturalis principis mathematica of 1687, had already invented a technque very like that of the differential calculus, which he uses throughout his book. But M Leibniz's use of the characteristic makes his calculus much simpler and quicker, and sometimes also proves very helpful.

As the last pages of this book were being printed, I came across M Nieuwentiit's book. Its title, Analysis infinitorum, excited my interest, but when I read it through I found that it was very different from the present work: not only does the author not use M Leibniz's characteristics but he also completely rejects second, third and further differences. Since he rejects what I have made the basis of most of my work I should feel obliged to reply to his objections, to show that they are unfounded, except that M Leibniz has himself already made a more than adequate reply in the Acta of Leipzig. Moreover, the two postulates or suppositions which I make at the beginning of this treatise, and which alone form the basis of what follows, seem to me to be so evidently true that no serious reader can reject them. I could, in fact, have proved them, in the manner of the Ancients, if I had not preferred to deal briefly with what was already well-known and enter into details only where the material itself was new.

Although this is the end of the Preface, it would be unfair to leave readers with de L'Hôpital's comments that "no serious reader can reject" his two postulates without giving these postulates which, as he says, come right at the beginning of the first chapter:

Postulate I: Any two quantities may be replaced by one another if they differ from each other by no more than an infinitely small amount. ...

Postulate II: We may consider a curve as an assemblage of an infinite number of straight lines each infinitely short, or (equivalently) as a polygon with an infinite number of sides, each infinitely small, which, by the angles they make with one another, determine the shape of the curve. ...

Jean Baptiste Joseph Fourier

Born: 21 March 1768 in Auxerre, Bourgogne, France
Died: 16 May 1830 in Paris, France

Joseph Fourier's father was a tailor in Auxerre. After the death of his first wife, with whom he had three children, he remarried and Joseph was the ninth of the twelve children of this second marriage. Joseph's mother died went he was nine years old and his father died the following year.

His first schooling was at Pallais's school, run by the music master from the cathedral. There Joseph studied Latin and French and showed great promise. He proceeded in 1780 to the École Royale Militaire of Auxerre where at first he showed talents for literature but very soon, by the age of thirteen, mathematics became his real interest. By the age of 14 he had completed a study of the six volumes of Bézout's Cours de mathématiques. In 1783 he received the first prize for his study of Bossut's Mécanique en général.

In 1787 Fourier decided to train for the priesthood and entered the Benedictine abbey of St Benoit-sur-Loire. His interest in mathematics continued, however, and he corresponded with C L Bonard, the professor of mathematics at Auxerre. Fourier was unsure if he was making the right decision in training for the priesthood. He submitted a paper on algebra to Montucla in Paris and his letters to Bonard suggest that he really wanted to make a major impact in mathematics. In one letter Fourier wrote

Yesterday was my 21st birthday, at that age Newton and Pascal had already acquired many claims to immortality.

Fourier did not take his religious vows. Having left St Benoit in 1789, he visited Paris and read a paper on algebraic equations at the Académie Royale des Sciences. In 1790 he became a teacher at the Benedictine college, École Royale Militaire of Auxerre, where he had studied. Up until this time there had been a conflict inside Fourier about whether he should follow a religious life or one of mathematical research. However in 1793 a third element was added to this conflict when he became involved in politics and joined the local Revolutionary Committee. As he wrote:-

As the natural ideas of equality developed it was possible to conceive the sublime hope of establishing among us a free government exempt from kings and priests, and to free from this double yoke the long-usurped soil of Europe. I readily became enamoured of this cause, in my opinion the greatest and most beautiful which any nation has ever undertaken.

Certainly Fourier was unhappy about the Terror which resulted from the French Revolution and he attempted to resign from the committee. However this proved impossible and Fourier was now firmly entangled with the Revolution and unable to withdraw. The revolution was a complicated affair with many factions, with broadly similar aims, violently opposed to each other. Fourier defended members of one faction while in Orléans. A letter describing events relates:-

Citizen Fourier, a young man full of intelligence, eloquence and zeal, was sent to Loiret. ... It seems that Fourier ... got up on certain popular platforms. He can talk very well and if he put forward the views of the Society of Auxerre he has done nothing blameworthy...

This incident was to have serious consequences but after it Fourier returned to Auxerre and continued to work on the revolutionary committee and continued to teach at the College. In July 1794 he was arrested, the charges relating to the Orléans incident, and he was imprisoned. Fourier feared the he would go to the guillotine but, after Robespierre himself went to the guillotine, political changes resulted in Fourier being freed.

Later in 1794 Fourier was nominated to study at the École Normale in Paris. This institution had been set up for training teachers and it was intended to serve as a model for other teacher-training schools. The school opened in January 1795 and Fourier was certainly the most able of the pupils whose abilities ranged widely. He was taught by Lagrange, who Fourier described as the first among European men of science, and also by Laplace, who Fourier rated less highly, and by Monge who Fourier described as having a loud voice and is active, ingenious and very learned.

Fourier began teaching at the Collège de France and, having excellent relations with Lagrange, Laplace and Monge, began further mathematical research. He was appointed to a position at the École Centrale des Travaux Publiques, the school being under the direction of Lazare Carnot and Gaspard Monge, which was soon to be renamed École Polytechnique. However, repercussions of his earlier arrest remained and he was arrested again and imprisoned. His release has been put down to a variety of different causes, pleas by his pupils, pleas by Lagrange, Laplace or Monge or a change in the political climate. In fact all three may have played a part.

By 1 September 1795 Fourier was back teaching at the École Polytechnique. In 1797 he succeeded Lagrange in being appointed to the chair of analysis and mechanics. He was renowned as an outstanding lecturer but he does not appear to have undertaken original research during this time.

In 1798 Fourier joined Napoleon's army in its invasion of Egypt as scientific adviser. Monge and Malus were also part of the expeditionary force. The expedition was at first a great success. Malta was occupied on 10 June 1798, Alexandria taken by storm on 1 July, and the delta of the Nile quickly taken. However, on 1 August 1798 the French fleet was completely destroyed by Nelson's fleet in the Battle of the Nile, so that Napoleon found himself confined to the land that he was occupying. Fourier acted as an administrator as French type political institutions and administration was set up. In particular he helped establish educational facilities in Egypt and carried out archaeological explorations.

While in Cairo Fourier helped found the Cairo Institute and was one of the twelve members of the mathematics division, the others included Monge, Malus and Napoleon himself. Fourier was elected secretary to the Institute, a position he continued to hold during the entire French occupation of Egypt. Fourier was also put in charge of collating the scientific and literary discoveries made during the time in Egypt.

Napoleon abandoned his army and returned to Paris in 1799, he soon held absolute power in France. Fourier returned to France in 1801 with the remains of the expeditionary force and resumed his post as Professor of Analysis at the École Polytechnique. However Napoleon had other ideas about how Fourier might serve him and wrote:-

... the Prefect of the Department of Isère having recently died, I would like to express my confidence in citizen Fourier by appointing him to this place.

Fourier was not happy at the prospect of leaving the academic world and Paris but could not refuse Napoleon's request. He went to Grenoble where his duties as Prefect were many and varied. His two greatest achievements in this administrative position were overseeing the operation to drain the swamps of Bourgoin and supervising the construction of a new highway from Grenoble to Turin. He also spent much time working on the Description of Egypt which was not completed until 1810 when Napoleon made changes, rewriting history in places, to it before publication. By the time a second edition appeared every reference to Napoleon would have been removed.

It was during his time in Grenoble that Fourier did his important mathematical work on the theory of heat. His work on the topic began around 1804 and by 1807 he had completed his important memoir On the Propagation of Heat in Solid Bodies. The memoir was read to the Paris Institute on 21 December 1807 and a committee consisting of Lagrange, Laplace, Monge and Lacroix was set up to report on the work. Now this memoir is very highly regarded but at the time it caused controversy.

There were two reasons for the committee to feel unhappy with the work. The first objection, made by Lagrange and Laplace in 1808, was to Fourier's expansions of functions as trigonometrical series, what we now call Fourier series. Further clarification by Fourier still failed to convince them. As is pointed out in:-

All these are written with such exemplary clarity - from a logical as opposed to calligraphic point of view - that their inability to persuade Laplace and Lagrange ... provides a good index of the originality of Fourier's views.

The second objection was made by Biot against Fourier's derivation of the equations of transfer of heat. Fourier had not made reference to Biot's 1804 paper on this topic but Biot's paper is certainly incorrect. Laplace, and later Poisson, had similar objections.

The Institute set as a prize competition subject the propagation of heat in solid bodies for the 1811 mathematics prize. Fourier submitted his 1807 memoir together with additional work on the cooling of infinite solids and terrestrial and radiant heat. Only one other entry was received and the committee set up to decide on the award of the prize, Lagrange, Laplace, Malus, Haüy and Legendre, awarded Fourier the prize. The report was not however completely favourable and states:-

... the manner in which the author arrives at these equations is not exempt of difficulties and that his analysis to integrate them still leaves something to be desired on the score of generality and even rigour.

With this rather mixed report there was no move in Paris to publish Fourier's work.

When Napoleon was defeated and on his way to exile in Elba, his route should have been through Grenoble. Fourier managed to avoid this difficult confrontation by sending word that it would be dangerous for Napoleon. When he learnt of Napoleon's escape from Elba and that he was marching towards Grenoble with an army, Fourier was extremely worried. He tried to persuade the people of Grenoble to oppose Napoleon and give their allegiance to the King. However as Napoleon marched into the town Fourier left in haste.

Napoleon was angry with Fourier who he had hoped would welcome his return. Fourier was able to talk his way into favour with both sides and Napoleon made him Prefect of the Rhône. However Fourier soon resigned on receiving orders, possibly from Carnot, that the was to remove all administrators with royalist sympathies. He could not have completely fallen out with Napoleon and Carnot, however, for on 10 June 1815, Napoleon awarded him a pension of 6000 francs, payable from 1 July. However Napoleon was defeated on 1 July and Fourier did not receive any money. He returned to Paris.

Fourier was elected to the Académie des Sciences in 1817. In 1822 Delambre, who was the Secretary to the mathematical section of the Académie des Sciences, died and Fourier together with Biot and Arago applied for the post. After Arago withdrew the election gave Fourier an easy win. Shortly after Fourier became Secretary, the Académie published his prize winning essay Théorie analytique de la chaleur in 1822. This was not a piece of political manoeuvring by Fourier however since Delambre had arranged for the printing before he died.

During Fourier's eight last years in Paris he resumed his mathematical researches and published a number of papers, some in pure mathematics while some were on applied mathematical topics. His life was not without problems however since his theory of heat still provoked controversy. Biot claimed priority over Fourier, a claim which Fourier had little difficulty showing to be false. Poisson, however, attacked both Fourier's mathematical techniques and also claimed to have an alternative theory. Fourier wrote Historical Précis as a reply to these claims but, although the work was shown to various mathematicians, it was never published.

Fourier's views on the claims of Biot and Poisson are given in the following, see:-

Having contested the various results [Biot and Poisson] now recognise that they are exact but they protest that they have invented another method of expounding them and that this method is excellent and the true one. If they had illuminated this branch of physics by important and general views and had greatly perfected the analysis of partial differential equations, if they had established a principal element of the theory of heat by fine experiments ... they would have the right to judge my work and to correct it. I would submit with much pleasure .. But one does not extend the bounds of science by presenting, in a form said to be different, results which one has not found oneself and, above all, by forestalling the true author in publication.

Fourier's work provided the impetus for later work on trigonometric series and the theory of functions of a real variable
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