Félix Savart

Felix Savart

---

Born: 30 June 1791 in Mézières, France
Died: 16 March 1841 in Paris, France

---

Félix Savart's father was Gérard Savart who was an engineer. The family had a long association with Mézières, Gérard's father (Félix's paternal grandfather), also a native of that city, had been involved with the founding of the engineering school there in 1748. Gérard Savart moved to Metz where he was in charge of draftsmen at the engineering school. Félix and his older brother Nicolas (born 1790) had their early schooling in Metz. With a strong family tradition of involvement with military engineering schools, one might have expected Félix Savart to also go down that path. In fact his early training did take him in that direction but in 1808, at the age of seventeen, he decided to train for a career in medicine.

Savart spent around two years from 1808 to 1810 studying at a hospital in Metz. Of course he had grown up in a period when France was enjoying military victories under Napoleon who had led French armies to victories over the armies of Austria, Prussia, Great Britain, Spain and the Netherlands between 1792 and 1797. Victories over three further coalitions set up to try to curb French power saw Napoleon at the height of his power in 1810. It was at this time, after training in the Metz hospital, that Savart became a regimental surgeon in Napoleon's army. He did follow certain family traditions by serving in the first battalion of engineers. However, during the years in which he served from 1810 to 1814, Napoleon suffered defeats in the Spanish and Russian campaigns. In 1814 Savart was discharged from the army and resumed his medical training.

It was to Strasbourg that Savart went in 1814 and, two years later, he graduated from the university with a medical degree. The topic for his thesis was varicose veins. During his medical studies Savart became interested in Aulus Cornelius Celsus (first century AD), one of the greatest Roman medical writers, author of De medicina. After completing his degree Savart remained in Strasbourg where he both gained further medical experience and also worked on a translation of Celsus' De medicina. After returning to Metz in 1817 where he set up a medical practice, Savart spent more time studying physics than treating patients. He set up an excellent physics laboratory to carry out experiments and became fascinated with a study of sound, in particular the acoustics of musical instruments such as the violin. He began to build violins trying to base the form of the instrument on mathematical principles.

With Savart showing little interest in his medical practice, and patients showing little interest in joining, he decided to go to Paris in 1819 and seek a publisher for his translation of Celsus' De medicina. He had another reason to go to Paris, and that was to see Biot so that he could discuss with him the acoustics of musical instruments that was by now fascinating Savart. As it happened, at the time that Savart reached Paris Biot was lecturing on acoustics at the Faculty of Science. He found Savart's work on the acoustics of bowed string instruments very interesting and he presented a memoir that Savart had written Mémoire des instruments à chordes et à archet to the Academy of Sciences; it was published in 1819. This memoir contained a design of a trapezoid violin which Savart claimed to have superior acoustic performance to the traditional violin. He used experimental results achieved using techniques similar to those of Chladni. It is reasonable to ask how successful he was with his trapezoid violin. Dostrovsky writes [1]:-

When the instrument was played before a committee that included Biot, the composer Cherubini, and other members of the Academy of Sciences and the Académie des Beaux-Arts, its tone was judged as extremely clear and even, but somewhat subdued.

When Savart arrived in Paris, Biot was undertaking research on electricity in addition to lecturing on acoustics. The two began a collaboration and when, early in 1820, Hans Christian Oersted reported that a compass needle placed near a wire carrying current pointed at right angles to the wire, they began to research more deeply into the field produced by the wire. Using the oscillation of a magnetic dipole to determine the strength of the field close to a wire carrying current, they discovered what today is called the Biot-Savart law. Magnetic fields produced by electric currents can be calculated using the Biot-Savart law which they presented to the Academy of Sciences on 30 October 1820. They took magnetism as the fundamental property rather than using the approach due to Ampère which treated it as derived from electric circuits. A joint Biot-Savart paper Note sur le magnétisme de la pile de Volta was published in the Annales de chemie et de physique in 1820.

Biot helped Savart find a teaching position in Paris and from 1820 he taught science in a private school there. On 5 November 1827, Savart was elected to the physics section of the Academy of Sciences to replace Fresnel who had died in July of that year. He taught at the Collège de France from 1828, becoming a professor of experimental physics there to succeed Ampère. He continued to hold this position until his death in 1841, a few months short of his fiftieth birthday.

In addition to the 1819 paper we mentioned above, Savart also carried out experiments on sound which became important for later students of acoustics. Among other papers he published on this topic we mention Mémoire sur la communication des mouvements vibratoires entre les corps solides (1820), Recherches sur les vibtarions de l'air (1823), and Mémoire sur les vibrations des corps solides, considérées en général (1824). His contributions to music are summed up in Grove's Dictionary of Music and Musicians [2]:-

In general he threw light on the nature of the complicated relation between a vibrating body which is the source of sound and other bodies brought into connection with it, by virtue of which the original sound is magnified in intensity and modified in quality, well-known examples of such an arrangement being furnished by the sounding board of the violin tribe and the pianoforte.

He also developed the Savart disk, a device which produced a sound wave of known frequency, using a rotating cog wheel as a measuring device. McKusick and Wiskind write [5]:-

About 1830 Savart invented a toothed wheel for determining the number of vibrations in a given musical tone. He attached tongues of pasteboard to a hoop of the wheel and arranged for these to strike a projecting object as the wheel was turned. Alternately, he had cogs of a wheel strike a tongue of pasteboard. Presumably he employed this instrument to determine the vibrations per second of the tones he elicited from his experimental models. he would speed up the rotation of the wheel until a tone which matched the experimental one was produced. Since the frequency of the tone produced by the wheel could be easily determined, the frequency of the unknown tone produced by the model was ascertained. by an extension of this method, Savart compounded musical notes. For example, he would use, in combination, wheels with numbers of teeth which bore a simple relationship to each other. he used this method to explore the physical basis of harmonious and discordant sounds.

There were other topics that interested Savart and on which he undertook research. For example he studied turbulence and the medical implications of his research is studied in detail in [5]. The authors write in summary:-

In the latter half of the nineteenth century, in medical and physiological writings on the genesis of heart murmurs, Savart's "fluid veins" were repeatedly referred to. When it was realised that, in the case of partial obstruction to the flow of blood, the murmur is produced not at, or proximal to, the site of obstruction but in the fluid beyond, it was presumed that a mechanism like that revealed by Savart was involved.

The paper [4] discusses a detailed experimental study of the existence of water bells when a cylindrical jet impacts with the velocity normal to a circular disc, which was first studied by Savart in 1833.

Let us end this biography with giving two postscripts. First let us note that although one of Savart's main aims in going to Paris was to publish his translation of Celsus' De medicina the work never appeared. It seems that Savart became diverted into more interesting directions. As a second postscript we note some facts about Savart's older brother Nicolas Savart. Unlike Félix, Nicolas studied at the École Polytechnique and then followed the family tradition of becoming an officer in the engineering corps. The reason for mentioning him in this postscript is that Nicolas, like his brother, also published papers on acoustics. For example he published Quelques faits résultant de la réflexion des ondes sonores (1839), and at least three further papers after the death of Félix.

Félix Savart was honoured by having a street in Mézières named after him.

Article by: J J O'Connor and E F Robertson

Jean Baptiste Biot

Jean Baptiste Biot

---

Born: 21 April 1774 in Paris, France
Died: 3 Feb 1862 in Paris, France

---

Jean-Baptiste Biot's father was Joseph Biot, whose ancestors were farmers in Lorraine, had achieved an important role working in the Treasury. Jean-Baptiste was educated at the college of Louis-le-Grand in Paris, where he specialised in classics. He completed his studies at Louis-le-Grand in 1791 following which, since his father wanted him to make a career in commerce, he took private lessons in mathematics from Antoine-Rene Mauduit who was professor of mathematics at the Collège de France. Joseph Biot then sent his son to Le Havre to become a clerical assistant to a merchant. His job there consisted of copying vast numbers of letters (we have to be thankful for photocopiers!) which bored Biot so much that he volunteered for the army. He joined the French army in September 1792, and served in the artillery at the Battle of Hondschoote in September 1793. In this battle the French defeated the British and Hanoverian soldiers besieging Dunkirk. After the battle Biot, suffering from an illness, decided to leave the army and return to his parents. As he was walking towards Paris, he was befriended by an important person who passed in his carriage. He took Biot in his carriage to Paris where he (Biot) was arrested as a deserter (he was still in uniform) and brought before a revolutionary committee. This would have had serious consequences for Biot had not the stranger intervened and he was set free. Despite Biot's efforts, he was not able to identify the important stranger and thank him.

It took Biot some months to recover from the illness during which time he continued with his studies of mathematics. He took the entrance examinations for the École des Ponts et Chausées and was accepted in January 1794. The École Polytechnique was founded later in 1794 (actually named 'École centrale des travaux publics' for its first year) and Biot transferred there in November of that year. Gaspard Monge, one of the founders of l'École Polytechnique who taught the first intake of students, quickly realised Biot's potential. Biot, however, quickly became involved in student politics and was made a section leader. There was an attempted insurrection by the royalists against the Convention and Biot took part. He was captured by government forces and taken prisoner. Had it not been for Monge, who could not see someone with such talents remain in jail, or even die, pleading successfully for his release his promising career might have ended. It is not surprising that for a second time he had come close to a death sentence, giving the dangerous times in Revolutionary France through which he lived. He returned to his studies at the École Polytechnique, befriending a fellow student, Siméon-Denis Poisson, and graduated in 1797. He had found a patron in Sylvestre-François Lacroix who was highly influential in helping Biot develop his career.

He became Professor of Mathematics at the École Centrale de l'Oise at Beauvais in 1797. It might seems remarkable that even someone as able as Biot could move from being an undergraduate straight into a professorship. However, things had worked in his favour [7]:-

The Convention, in a law of 25 February 1795 had called for a system of écoles centrales, one in each large town, to replace the collèges of the ancien régime as seats of secondary education. The curricula of the new schools was practical and modern and included two years devoted to mathematics, experimental physics and chemistry. These subjects had never been taught before on a high school level, so, inevitably, there was a shortage of qualified teachers.

Biot got to know of the vacant mathematics professorship through his friendship with Barnabé Brisson, the son of Antoine François Brisson de Beauvais, while studying at l'École Polytechnique. Through Barnabé Brisson, Biot had got to know his sister Gabrielle. Although Gabrielle was only sixteen years old, they married in 1797 soon after Biot took up his position at Beauvais. Biot taught Gabrielle mathematics and physics so that she might translate into French a German text by Ernst Gottfried Fischer. It was Claude Louis Berthollet who had asked Biot to make the translation which was published as Physique mécanique in 1806. Biot's wife was well educated and fluent in German, so her translation was excellent. However, in line with the practice of the time, the book records Biot himself as the translator rather than his wife. Biot and his wife had a son Edouard Constant Biot who was born in 1803.

It was largely through Lacroix that Biot had been appointed at Beauvais and Lacroix advised him frequently through the first years that he worked there. However, he also managed to get support in his career from Laplace. In fact in late 1799 Biot approached Laplace, who had taught him at l'École Polytechnique, and offered to proof-read the Mécanique céleste which at that time was with the publisher. He didn't give up when Laplace said "No thanks" but persisted in a polite way and eventually Laplace agreed. Biot was, by this time, an entrance examiner at the École Polytechnique so was frequently in Paris. He wrote [7]:-

From that time on, each time I went to Paris I brought my proof-reading work and personally presented it to M Laplace. He always received it with kindness, examined it and discussed it, and that gave me the opportunity to submit to him the difficulties that had stopped me. His willingness to explain them was boundless. But even he could not always do it without stopping to think for a long while. This usually occurred in places where he had used the expedient phrase 'it can easily be seen'.

Certainly the effort required by Biot was enormous. He wrote near the end of 1799 [7]:-

I spent all of my days at this work [proof-reading the 'Mécanique céleste'] which I felt to be important in several respects, and I hope I have succeeded. I worked like a devil to finish it on time, and 1 remained at the task eighteen hours in a row the last day, without eating or drinking. Finally, thank heaven, it is finished, and I can flatter myself that I understand the 'Mecanique celeste'. If this were to be the only benefit I would get from it, it would still be a lot.

Laplace was also interested in the research on mathematics which Biot was undertaking and gave him advice both on the material and on getting it published. In 1800 he was appointed Professor of Mathematical Physics at the Collège de France, an appointment which was due mainly to the influence of Laplace. In 1800 Biot was elected an associate to the First Class of the Institute, replacing Jean Montucla who died in December 1799. Again Laplace's support was important to Biot in this election. In January 1803 the Institut was reorganised and Delambre became perpetual secretary, creating a vacancy in the Mathematics Section. Although Biot had produced excellent mathematics memoirs in the years up to 1800, he had concentrated on experimental physics from that time on. However, presumably to improve his chances of gaining a place in the Mathematics Section, he submitted a memoir on the axes of tautochronous curves just before the election was to take place. It was sufficient to keep his mathematical reputation high and he was elected on 11 April 1803.

The first balloon ascent made for scientific purposes was by Biot and Joseph-Louis Gay-Lussac from the garden of the Conservatoire des Arts on 24 August 1804. They achieved a height of 4000 metres and measured magnetic, electrical, and chemical properties of the atmosphere at various heights. In 1806, again with the support of Laplace, Biot was appointed as an assistant astronomer at the Bureau de Longitudes in addition to his other roles. On 3 September of that year he set out with François Arago to Formentera, in the Balearic Islands, to complete earlier work begun there on calculating the measure of the arc of the meridian. They were still undertaking measurements when, in May 1808, Napoleon declared his brother Joseph Bonaparte as Spanish ruler and the War of Independence began. Biot and Arago must have looked extremely suspicious; two Frenchmen with sophisticated measuring instruments working on Spanish territory. Biot fled back to France immediately. Later in 1808, together with Claude-Louis Mathieu, he embarked on a series of measurements of the length of the seconds pendulum at different points on the meridian, in particular at Bordeaux and at Dunkirk. Both Mathieu and Biot received a prize from the Académie des Sciences in 1809 for this highly accurate work, and in 1812 they received a second prize from the Academy for their achievements. In 1809 Biot was appointed Professor of Physical Astronomy at the Faculty of Sciences. He held this position for over fifty years.

Biot studied a wide range of mathematical topics, mostly on the applied mathematics side. He made advances in astronomy, elasticity, electricity and magnetism, heat and optics on the applied side while, in pure mathematics, he also did important work in geometry. He collaborated with Arago on refractive properties of gases. Biot, together with Felix Savart, discovered that the intensity of the magnetic field set up by a current flowing through a wire varies inversely with the distance from the wire. This is now known as Biot-Savart's Law and is fundamental to modern electromagnetic theory. Morris Kline, reviewing [8], writes:-

Biot depended entirely upon experiments to determine the interaction of a magnet and a straight line of electrical current and then inferred a mathematical law concerning the force between current and magnet.

Light was the topic on which Biot devoted most time, making the major discovery of the laws of rotary polarization by crystalline bodies. Having discovered these laws he used them in analysis of saccharine solutions using an instrument called a polarimeter which he invented. For this work on the polarisation of light passing through chemical solutions he was awarded the Rumford Medal of the Royal Society of London in 1840. He had been elected as a foreign member of the Royal Society in 1815.

One of his important works was Mémoire sur la figure de la terre (1827) which describes the shape of the Earth. Among his other major works we mention: Analyse de la mécanique céleste de M Laplace (1801); Traité analytique des courbes et des surfaces du second degré (1802); Recherches sur l'intégration des équations différentielles partielles et sur les vibrations des surfaces (1803); Essai de géométrie analytique appliqué aux courbes et aux surfaces de second ordre (1806); Recherches expérimentales et mathématiques sur les mouvements des molécules de la lumière autour de leur centre de gravité (1814); Traité de physique experimentale et mathématique (1816); Precis de physique (1817); (with Arago) Recueil d'observations géodésiques, astronomiques et physiques exécutées en Espagne et Écosse (1821); Mémoire sur la vraie constitution de l'atmosphère terrestre (1841); Traité élémentaire d'astronomie physique (1805); Recherches sur plusieurs points de l'astronomie égyptienne (1823); Recherches sur l'ancienne astronomie chinoise (1840); Études sur l'astronomie indienne et sur l'astronomie chinoise (1862); Essai sur l'histoire générale des sciences pendant la Révolution (1803); Discours sur Montaigne (1812); Lettres sur l'approvisionnement de Paris et sur le commerce des grains (1835); Traite d'astronomie physique (1850); and Mélanges scientifiques et littéraires (1858).

He tried twice for the post of Secretary to the Académie des Sciences and to improve his chances for election to this post he wrote Essai sur l'Histoire Générale des Sciences pendant la Révolution. However he lost out in 1822 to Fourier for this post, then again when Fourier died in May 1830 he applied again for the post of Secretary, only to lose to Arago on this occasion. The rivalry between Arago and Biot was again evident in 1839 when different photographic processes were competing [10]:-

Arago and Biot, France's leading authorities in the field of optics, had spent the past thirty years disagreeing about the ability of optical instruments to represent the world. Photography became one more opportunity for them to disagree, and they did not shrink from the task. The debate extended beyond questions of assigning priority and apportioning credit. At its heart was the question of what the photographic surfaces showed, what relation the visible inscription bore to the real world.

Arago supported the Daguerre photographic process with silver plates while Biot championed an approach with paper soaked in a silver solution as developed by Henry Fox Talbot. This was not because he thought this process produced clearer photographs, rather it was because he believed that the images captured 'chemical radiation' invisible to the eye. We should note that the idea of 'chemical radiation' was widely believed at this time. Biot also believed that the photographic process should be one reserved for scientific use and not made available for public use. There was a period of collaboration between Biot and Talbot, the two exchanging letters frequently. Basically the photographic process became another tool for Biot to use in his investigations of light which had always been a major interest to him.

As well as interests in almost every branch of science, Biot was interested in the history of the subject. He published works on this topic such as: Essai sur l'histoire générale des sciences pendant la Révolution française (1803); Notice historique sur la vie et les ouvrages de Newton (1822); Recherches sur plusieurs points de l'astronomie égyptienne appliquées aux monuments astronomiques trouvés en Égypte (1823); Sur la manière de calculer les positions des étoiles relativement à l'équateur et à l'écliptique pour les époques anciennes (1823); Opuscule sur l'Astronomie ancienne des Chinois, des Indous et des Arabes (1840); Mémoire sur le Zodiaque de Denderah (1844); Précis de l'histoire de l'Astronomie planétaire (1847); and Études sur l'astronomie indienne et l'astronomie chinoise (1862).

In [4] St Beuve says that Biot was endowed to the highest degree with all the qualities of curiosity, finesse, penetration, precision, ingenious analysis, method, clarity, in short with all the essential and secondary qualities, bar one, genius, in the sense of originality and invention.

A contrasting comment by Olinthus Gregory in 1821 is:-

With regard to M Biot, I had an opportunity of pretty fully appreciating his character when we were together in the Zetland [= Shetland] Isles; and I do not hesitate to say that I never met so strange a compound of vanity, impetuosity, fickleness, and natural partiality, as is exhibited in his character.

In addition to the honours we mentioned above, Biot was also honoured by being made chevalier of the Legion of Honour in 1814 and commander in 1849. He was elected to the to the Academy of Inscriptions and Belles-Lettre in 1841 and to the French Academy in 1856.

Article by: J J O'Connor and E F Robertson

William John Macquorn Rankine

Born: 5 July 1820 in Edinburgh, Scotland Died: 24 Dec 1872 in Glasgow, Scotland

Wiiliam John Macquorn Rankine

 

William Rankine's mother was Barbara Grahame, the daughter of a Glasgow banker, and his father was David Rankine, a civil engineer and lieutenant in the rifle brigade. Although he was the second of his parents children, his older brother David died when young so William was brought up as an only child. It was a strict religious upbringing with his father teaching him not only arithmetical skills but also mechanics. William did not enjoy good health as a child and could only attend school for short periods. Most of his education took place at home with private tutors but he did attend Ayr Academy for about a year in 1828-29 and also for a short while Glasgow High School in 1830.

Rankine's interests were divided between music and mathematics. At first he was strongly attracted to number theory but when he was 14 years old one of his uncles gave him a Latin edition of Newton's Principia which he read eagerly. For two years from 1836 to 1838 Rankine studied at the University of Edinburgh, attending a wide range of lectures in science subjects, but choosing not to attend mathematics classes. He won a Gold Medal for an essay on The wave theory of light in 1836 and another Gold Medal for an essay on Methods in physical investigation two years later. He did not take a degree but chose to leave university in 1838 and become an apprentice to the engineer John Benjamin MacNeill. This was not Rankine's first experience of engineering for while he studied at Edinburgh University he had worked on the Edinburgh and Dalkeith Railway which his father was overseeing.

From 1839 to 1841 Rankine worked on numerous projects that John Benjamin MacNeill was involved with, including river improvements, waterworks, railways and both harbours. Some of the work took Rankine to Ireland. After his return to Edinburgh he undertook some investigative work with his father and they published An experimental inquiry into the advantages attending the use of cylindrical wheels on railways (1842). Further papers read to the Institution of Civil Engineers were highly thought of and several won Rankine prizes.

Rankine was appointed to the regius chair of civil engineering and mechanics at Glasgow in 1855. His [2]:-

... inaugural address espoused the harmony of theory with practice in mechanics, and outlined a tripartite theory of knowledge - theory, practice, and the application of theory to practice - which left room for a new breed of engineering scientists to bridge theoretical and practical domains.

He decided to found a Scottish version of the Institution of Civil Engineers and so he resigned from the London based Institution in 1857 and became the first president of the new Institution of Engineers in Scotland. As well as holding the presidency in 1857-59, he was elected for a second term in 1869-70.

Rankine's study of the applications of mathematics began quite early in his career as an engineer. While an apprentice engineer he made a mathematical analysis of the cooling of the earth (1840). He worked on heat, reading Clapeyron's works, and attempted to derive Sadi Carnot's law from his own hypothesis. R H Atkin, reviewing [14], describes Rankine's ideas on thermodynamics, and in particular compares his approach with that of Clausius:-

Rankine apparently regarded energy, as we do today, as being classified into two kinds, viz., kinetic and potential, and his thermodynamic theory was developed by considering the transformation of one into the other. He began with the hypothesis that matter was constituted by molecular vortices (without considering the cyclic process) and obtained the quantities "pressure", "specific heat", etc., from that consideration. His classification of energy was similar to, but not exactly the same as, that of Clausius. Both Rankine and Clausius approached the second law of thermodynamics from the point of view of the transformation from one kind of energy to the other. But whereas Clausius considered the conversion between heat and work and the flow of heat from high to low temperature in a cyclic process, Rankine concentrated on the change from kinetic (molecular) to potential energies, and related this change to heat flow by use of his "heat-potential" function.

Hutchison, in [9] and [10], looks at the entropy function which Rankine defined and its implications for the theory of thermodynamics which he developed. Rankine's work was extended by Maxwell. Rankine also wrote on fatigue in the metal of railway axles, on earth pressures in soil mechanics, and the stability of walls. He also developed methods to solve the force distribution in frame structures and worked on hydrodynamics and the design of ships. He was elected a fellow of the Royal Society of Edinburgh in 1849 and a fellow of the Royal Society of London in 1853. He was also elected to the American Academy of Arts and Sciences in 1856 and to the Royal Swedish Academy of Sciences in 1868. He was awarded an honorary degree from Trinity College, Dublin, in 1857.

Among his most important works are Manual of Applied Mechanics (1858), Manual of the Steam Engine and Other Prime Movers (1859), Civil Engineering (1862), Machinery and Millwork (1869), Useful Rules and Tables (1866), Mechanical Textbook (1873), and On the Thermodynamic Theory of Waves of Finite Longitudinal Disturbance.

As to his interests outside his professional studies, he was [2]:-

A keen cellist, pianist, and vocalist, his one published composition was a piano accompaniment to a song entitled the 'Iron Horse'; as a British Association red lion, hailed as lion-king in 1871, he penned quirky and humorous poems like 'The Mathematician in Love' and 'The Three-Foot Rule' (a protest against the metric system). These Songs and Fables (1874) appeared posthumously with illustrations by Jemima Blackburn, wife of Glasgow College's mathematics professor.

Rankine's health deteriorated rapidly during the final six months of his life. The first symptoms saw his vision become impaired, then his speech failed and finally he became partially paralysed.

Article by: J J O'Connor and E F Robertson

 

Joseph-Louis Lagrange

Born: 25 Jan 1736 in Turin, Sardinia-Piedmont (now Italy)Died: 10 April 1813 in Paris, France

Joseph-Louis Lagrange

Joseph-Louis Lagrange is usually considered to be a French mathematician, but the Italian Encyclopaedia [40] refers to him as an Italian mathematician. They certainly have some justification in this claim since Lagrange was born in Turin and baptised in the name of Giuseppe Lodovico Lagrangia. Lagrange's father was Giuseppe Francesco Lodovico Lagrangia who was Treasurer of the Office of Public Works and Fortifications in Turin, while his mother Teresa Grosso was the only daughter of a medical doctor from Cambiano near Turin. Lagrange was the eldest of their 11 children but one of only two to live to adulthood.

Turin had been the capital of the duchy of Savoy, but became the capital of the kingdom of Sardinia in 1720, sixteen years before Lagrange's birth. Lagrange's family had French connections on his father's side, his great-grandfather being a French cavalry captain who left France to work for the Duke of Savoy. Lagrange always leant towards his French ancestry, for as a youth he would sign himself Lodovico LaGrange or Luigi Lagrange, using the French form of his family name.

Despite the fact that Lagrange's father held a position of some importance in the service of the king of Sardinia, the family were not wealthy since Lagrange's father had lost large sums of money in unsuccessful financial speculation. A career as a lawyer was planned out for Lagrange by his father, and certainly Lagrange seems to have accepted this willingly. He studied at the College of Turin and his favourite subject was classical Latin. At first he had no great enthusiasm for mathematics, finding Greek geometry rather dull.

Lagrange's interest in mathematics began when he read a copy of Halley's 1693 work on the use of algebra in optics. He was also attracted to physics by the excellent teaching of Beccaria at the College of Turin and he decided to make a career for himself in mathematics. Perhaps the world of mathematics has to thank Lagrange's father for his unsound financial speculation, for Lagrange later claimed:-

If I had been rich, I probably would not have devoted myself to mathematics.

He certainly did devote himself to mathematics, but largely he was self taught and did not have the benefit of studying with leading mathematicians. On 23 July 1754 he published his first mathematical work which took the form of a letter written in Italian to Giulio Fagnano. Perhaps most surprising was the name under which Lagrange wrote this paper, namely Luigi De la Grange Tournier. This work was no masterpiece and showed to some extent the fact that Lagrange was working alone without the advice of a mathematical supervisor. The paper draws an analogy between the binomial theorem and the successive derivatives of the product of functions.

Before writing the paper in Italian for publication, Lagrange had sent the results to Euler, who at this time was working in Berlin, in a letter written in Latin. The month after the paper was published, however, Lagrange found that the results appeared in correspondence between Johann Bernoulli and Leibniz. Lagrange was greatly upset by this discovery since he feared being branded a cheat who copied the results of others. However this less than outstanding beginning did nothing more than make Lagrange redouble his efforts to produce results of real merit in mathematics. He began working on the tautochrone, the curve on which a weighted particle will always arrive at a fixed point in the same time independent of its initial position. By the end of 1754 he had made some important discoveries on the tautochrone which would contribute substantially to the new subject of the calculus of variations (which mathematicians were beginning to study but which did not receive the name 'calculus of variations' before Euler called it that in 1766).

Lagrange sent Euler his results on the tautochrone containing his method of maxima and minima. His letter was written on 12 August 1755 and Euler replied on 6 September saying how impressed he was with Lagrange's new ideas. Although he was still only 19 years old, Lagrange was appointed professor of mathematics at the Royal Artillery School in Turin on 28 September 1755. It was well deserved for the young man had already shown the world of mathematics the originality of his thinking and the depth of his great talents.

In 1756 Lagrange sent Euler results that he had obtained on applying the calculus of variations to mechanics. These results generalised results which Euler had himself obtained and Euler consulted Maupertuis, the president of the Berlin Academy, about this remarkable young mathematician. Not only was Lagrange an outstanding mathematician but he was also a strong advocate for the principle of least action so Maupertuis had no hesitation but to try to entice Lagrange to a position in Prussia. He arranged with Euler that he would let Lagrange know that the new position would be considerably more prestigious than the one he held in Turin. However, Lagrange did not seek greatness, he only wanted to be able to devote his time to mathematics, and so he shyly but politely refused the position.

Euler also proposed Lagrange for election to the Berlin Academy and he was duly elected on 2 September 1756. The following year Lagrange was a founding member of a scientific society in Turin, which was to become the Royal Academy of Sciences of Turin. One of the major roles of this new Society was to publish a scientific journal the Mélanges de Turinwhich published articles in French or Latin. Lagrange was a major contributor to the first volumes of the Mélanges de Turin volume 1 of which appeared in 1759, volume 2 in 1762 and volume 3 in 1766.

The papers by Lagrange which appear in these transactions cover a variety of topics. He published his beautiful results on the calculus of variations, and a short work on the calculus of probabilities. In a work on the foundations of dynamics, Lagrange based his development on the principle of least action and on kinetic energy.

In the Mélanges de Turin Lagrange also made a major study on the propagation of sound, making important contributions to the theory of vibrating strings. He had read extensively on this topic and he clearly had thought deeply on the works of Newton, Daniel Bernoulli, Taylor, Euler and d'Alembert. Lagrange used a discrete mass model for his vibrating string, which he took to consist of n masses joined by weightless strings. He solved the resulting system of n+1 differential equations, then let n tend to infinity to obtain the same functional solution as Euler had done. His different route to the solution, however, shows that he was looking for different methods than those of Euler, for whom Lagrange had the greatest respect.

In papers which were published in the third volume, Lagrange studied the integration of differential equations and made various applications to topics such as fluid mechanics (where he introduced the Lagrangian function). Also contained are methods to solve systems of linear differential equations which used the characteristic value of a linear substitution for the first time. Another problem to which he applied his methods was the study the orbits of Jupiter and Saturn.

The Académie des Sciences in Paris announced its prize competition for 1764 in 1762. The topic was on the libration of the Moon, that is the motion of the Moon which causes the face that it presents to the Earth to oscillate causing small changes in the position of the lunar features. Lagrange entered the competition, sending his entry to Paris in 1763 which arrived there not long before Lagrange himself. In November of that year he left Turin to make his first long journey, accompanying the Marquis Caraccioli, an ambassador from Naples who was moving from a post in Turin to one in London. Lagrange arrived in Paris shortly after his entry had been received but took ill while there and did not proceed to London with the ambassador. D'Alembert was upset that a mathematician as fine as Lagrange did not receive more honour. He wrote on his behalf [1]:-

Monsieur de la Grange, a young geometer from Turin, has been here for six weeks. He has become quite seriously ill and he needs, not financial aid, for the Marquis de Caraccioli directed upon leaving for England that he should not lack for anything, but rather some signs of interest on the part of his native country ... In him Turin possesses a treasure whose worth it perhaps does not know.

Returning to Turin in early 1765, Lagrange entered, later that year, for the Académie des Sciences prize of 1766 on the orbits of the moons of Jupiter. D'Alembert, who had visited the Berlin Academy and was friendly with Frederick II of Prussia, arranged for Lagrange to be offered a position in the Berlin Academy. Despite no improvement in Lagrange's position in Turin, he again turned the offer down writing:-

It seems to me that Berlin would not be at all suitable for me while M Euler is there.

By March 1766 d'Alembert knew that Euler was returning to St Petersburg and wrote again to Lagrange to encourage him to accept a post in Berlin. Full details of the generous offer were sent to him by Frederick II in April, and Lagrange finally accepted. Leaving Turin in August, he visited d'Alembert in Paris, then Caraccioli in London before arriving in Berlin in October. Lagrange succeeded Euler as Director of Mathematics at the Berlin Academy on 6 November 1766.

Lagrange was greeted warmly by most members of the Academy and he soon became close friends with Lambert and Johann(III) Bernoulli. However, not everyone was pleased to see this young man in such a prestigious position, particularly Castillon who was 32 years older than Lagrange and considered that he should have been appointed as Director of Mathematics. Just under a year from the time he arrived in Berlin, Lagrange married his cousin Vittoria Conti. He wrote to d'Alembert:-

My wife, who is one of my cousins and who even lived for a long time with my family, is a very good housewife and has no pretensions at all.

They had no children, in fact Lagrange had told d'Alembert in this letter that he did not wish to have children.

Turin always regretted losing Lagrange and from time to time his return there was suggested, for example in 1774. However, for 20 years Lagrange worked at Berlin, producing a steady stream of top quality papers and regularly winning the prize from the Académie des Sciences of Paris. He shared the 1772 prize on the three body problem with Euler, won the prize for 1774, another one on the motion of the moon, and he won the 1780 prize on perturbations of the orbits of comets by the planets.

His work in Berlin covered many topics: astronomy, the stability of the solar system, mechanics, dynamics, fluid mechanics, probability, and the foundations of the calculus. He also worked on number theory proving in 1770 that every positive integer is the sum of four squares. In 1771 he proved Wilson's theorem (first stated without proof by Waring) that n is prime if and only if (n -1)! + 1 is divisible by n. In 1770 he also presented his important workRéflexions sur la résolution algébrique des équations which made a fundamental investigation of why equations of degrees up to 4 could be solved by radicals. The paper is the first to consider the roots of an equation as abstract quantities rather than having numerical values. He studied permutations of the roots and, although he does not compose permutations in the paper, it can be considered as a first step in the development of group theory continued by Ruffini, Galois and Cauchy.

Although Lagrange had made numerous major contributions to mechanics, he had not produced a comprehensive work. He decided to write a definitive work incorporating his contributions and wrote to Laplace on 15 September 1782:-

I have almost completed a Traité de mécanique analytique, based uniquely on the principle of virtual velocities; but, as I do not yet know when or where I shall be able to have it printed, I am not rushing to put the finishing touches to it.

Caraccioli, who was by now in Sicily, would have liked to see Lagrange return to Italy and he arranged for an offer to be made to him by the court of Naples in 1781. Offered the post of Director of Philosophy of the Naples Academy, Lagrange turned it down for he only wanted peace to do mathematics and the position in Berlin offered him the ideal conditions. During his years in Berlin his health was rather poor on many occasions, and that of his wife was even worse. She died in 1783 after years of illness and Lagrange was very depressed. Three years later Frederick II died and Lagrange's position in Berlin became a less happy one. Many Italian States saw their chance and attempts were made to entice him back to Italy.

The offer which was most attractive to Lagrange, however, came not from Italy but from Paris and included a clause which meant that Lagrange had no teaching. On 18 May 1787 he left Berlin to become a member of the Académie des Sciences in Paris, where he remained for the rest of his career. Lagrange survived the French Revolution while others did not and this may to some extent be due to his attitude which he had expressed many years before when he wrote:-

I believe that, in general, one of the first principles of every wise man is to conform strictly to the laws of the country in which he is living, even when they are unreasonable.

The Mécanique analytique which Lagrange had written in Berlin, was published in 1788. It had been approved for publication by a committee of the Académie des Sciences comprising of Laplace, Cousin, Legendre and Condorcet. Legendre acted as an editor for the work doing proof reading and other tasks. The Mécanique analytique summarised all the work done in the field of mechanics since the time of Newton and is notable for its use of the theory of differential equations. With this work Lagrange transformed mechanics into a branch of mathematical analysis. He wrote in the Preface:-

One will not find figures in this work. The methods that I expound require neither constructions, nor geometrical or mechanical arguments, but only algebraic operations, subject to a regular and uniform course.

Lagrange was made a member of the committee of the Académie des Sciences to standardise weights and measures in May 1790. They worked on the metric system and advocated a decimal base. Lagrange married for a second time in 1792, his wife being Renée-Françoise-Adélaide Le Monnier the daughter of one of his astronomer colleagues at the Académie des Sciences. He was certainly not unaffected by the political events. In 1793 the Reign of Terror commenced and the Académie des Sciences, along with the other learned societies, was suppressed on 8 August. The weights and measures commission was the only one allowed to continue and Lagrange became its chairman when others such as the chemist Lavoisier, Borda, Laplace, Coulomb, Brisson and Delambre were thrown off the commission.

In September 1793 a law was passed ordering the arrest of all foreigners born in enemy countries and all their property to be confiscated. Lavoisier intervened on behalf of Lagrange, who certainly fell under the terms of the law, and he was granted an exception. On 8 May 1794, after a trial that lasted less than a day, a revolutionary tribunal condemned Lavoisier, who had saved Lagrange from arrest, and 27 others to death. Lagrange said on the death of Lavoisier, who was guillotined on the afternoon of the day of his trial:-

It took only a moment to cause this head to fall and a hundred years will not suffice to produce its like.

The École Polytechnique was founded on 11 March 1794 and opened in December 1794 (although it was called the École Centrale des Travaux Publics for the first year of its existence). Lagrange was its first professor of analysis, appointed for the opening in 1794. In 1795 the École Normale was founded with the aim of training school teachers. Lagrange taught courses on elementary mathematics there. We mentioned above that Lagrange had a 'no teaching' clause written into his contract but the Revolution changed things and Lagrange was required to teach. However, he was not a good lecturer as Fourier, who attended his lectures at the École Normale in 1795 wrote:-

His voice is very feeble, at least in that he does not become heated; he has a very pronounced Italian accent and pronounces the s like z ... The students, of whom the majority are incapable of appreciating him, give him little welcome, but the professors make amends for it.

Similarly Bugge who attended his lectures at the École Polytechnique in 1799 wrote:-

... whatever this great man says, deserves the highest degree of consideration, but he is too abstract for youth.

Lagrange published two volumes of his calculus lectures. In 1797 he published the first theory of functions of a real variable with Théorie des fonctions analytiques although he failed to give enough attention to matters of convergence. He states that the aim of the work is to give:-

... the principles of the differential calculus, freed from all consideration of the infinitely small or vanishing quantities, of limits or fluxions, and reduced to the algebraic analysis of finite quantities.

Also he states:-

The ordinary operations of algebra suffice to resolve problems in the theory of curves.

Not everyone found Lagrange's approach to the calculus the best however, for example de Prony wrote in 1835:-

Lagrange's foundations of the calculus is assuredly a very interesting part of what one might call purely philosophical study: but when it is a case of making transcendental analysis an instrument of exploration for questions presented by astronomy, marine engineering, geodesy, and the different branches of science of the engineer, the consideration of the infinitely small leads to the aim in a manner which is more felicitous, more prompt, and more immediately adapted to the nature of the questions, and that is why the Leibnizian method has, in general, prevailed in French schools.

The second work of Lagrange on this topic Leçons sur le calcul des fonctions appeared in 1800.

Napoleon named Lagrange to the Legion of Honour and Count of the Empire in 1808. On 3 April 1813 he was awarded the Grand Croix of the Ordre Impérial de la Réunion. He died a week later.

Article by: J J O'Connor and E F Robertson

Augustin Louis Cauchy

Born: 21 Aug 1789 in Paris, FranceDied: 23 May 1857 in Sceaux (near Paris), France

Augustin Louis Cauchy

Paris was a difficult place to live in when Augustin-Louis Cauchy was a young child due to the political events surrounding the French Revolution. When he was four years old his father, fearing for his life in Paris, moved his family to Arcueil. There things were hard and he wrote in a letter [4]:-

We never have more than a half pound of bread - and sometimes not even that. This we supplement with the little supply of hard crackers and rice that we are allotted.

They soon returned to Paris and Cauchy's father was active in the education of young Augustin-Louis. Laplace and Lagrange were visitors at the Cauchy family home and Lagrange in particular seems to have taken an interest in young Cauchy's mathematical education. Lagrange advised Cauchy's father that his son should obtain a good grounding in languages before starting a serious study of mathematics. In 1802 Augustin-Louis entered the École Centrale du Panthéon where he spent two years studying classical languages.

From 1804 Cauchy attended classes in mathematics and he took the entrance examination for the École Polytechnique in 1805. He was examined by Biot and placed second. At the École Polytechnique he attended courses by Lacroix, de Prony and Hachette while his analysis tutor was Ampère. In 1807 he graduated from the École Polytechnique and entered the engineering school École des Ponts et Chaussées. He was an outstanding student and for his practical work he was assigned to the Ourcq Canal project where he worked under Pierre Girard.

In 1810 Cauchy took up his first job in Cherbourg to work on port facilities for Napoleon's English invasion fleet. He took a copy of Laplace's Mécanique Céleste and one of Lagrange'sThéorie des Fonctions with him. It was a busy time for Cauchy, writing home about his daily duties he said [4]:-

I get up at four o'clock each morning and I am busy from then on. ... I do not get tired of working, on the contrary, it invigorates me and I am in perfect health...

Cauchy was a devout Catholic and his attitude to his religion was already causing problems for him. In a letter written to his mother in 1810 he says:-

So they are claiming that my devotion is causing me to become proud, arrogant and self-infatuated. ... I am now left alone about religion and nobody mentions it to me anymore...

In addition to his heavy workload Cauchy undertook mathematical researches and he proved in 1811 that the angles of a convex polyhedron are determined by its faces. He submitted his first paper on this topic then, encouraged by Legendre and Malus, he submitted a further paper on polygons and polyhedra in 1812. Cauchy felt that he had to return to Paris if he was to make an impression with mathematical research. In September of 1812 he returned to Paris after becoming ill. It appears that the illness was not a physical one and was probably of a psychological nature resulting in severe depression.

Back in Paris Cauchy investigated symmetric functions and submitted a memoir on this topic in November 1812. This was published in the Journal of the École Polytechnique in 1815. However he was supposed to return to Cherbourg in February 1813 when he had recovered his health and this did not fit with his mathematical ambitions. His request to de Prony for an associate professorship at the École des Ponts et Chaussées was turned down but he was allowed to continue as an engineer on the Ourcq Canal project rather than return to Cherbourg. Pierre Girard was clearly pleased with his previous work on this project and supported the move.

An academic career was what Cauchy wanted and he applied for a post in the Bureau des Longitudes. He failed to obtain this post, Legendre being appointed. He also failed to be appointed to the geometry section of the Institute, the position going to Poinsot. Cauchy obtained further sick leave, having unpaid leave for nine months, then political events prevented work on the Ourcq Canal so Cauchy was able to devote himself entirely to research for a couple of years.

Other posts became vacant but one in 1814 went to Ampère and a mechanics vacancy at the Institute, which had occurred when Napoleon Bonaparte resigned, went to Molard. In this last election Cauchy did not receive a single one of the 53 votes cast. His mathematical output remained strong and in 1814 he published the memoir on definite integrals that later became the basis of his theory of complex functions.

In 1815 Cauchy lost out to Binet for a mechanics chair at the École Polytechnique, but then was appointed assistant professor of analysis there. He was responsible for the second year course. In 1816 he won the Grand Prix of the French Academy of Sciences for a work on waves. He achieved real fame however when he submitted a paper to the Institute solving one of Fermat's claims on polygonal numbers made to Mersenne. Politics now helped Cauchy into the Academy of Sciences when Carnot and Monge fell from political favour and were dismissed and Cauchy filled one of the two places.

In 1817 when Biot left Paris for an expedition to the Shetland Islands in Scotland Cauchy filled his post at the Collège de France. There he lectured on methods of integration which he had discovered, but not published, earlier. Cauchy was the first to make a rigorous study of the conditions for convergence of infinite series in addition to his rigorous definition of an integral. His text Cours d'analyse in 1821 was designed for students at École Polytechnique and was concerned with developing the basic theorems of the calculus as rigorously as possible. He began a study of the calculus of residues in 1826 in Sur un nouveau genre de calcul analogue au calcul infinitésimal while in 1829 in Leçons sur le Calcul Différentiel he defined for the first time a complex function of a complex variable.

Cauchy did not have particularly good relations with other scientists. His staunchly Catholic views had him involved on the side of the Jesuits against the Académie des Sciences. He would bring religion into his scientific work as for example he did on giving a report on the theory of light in 1824 when he attacked the author for his view that Newton had not believed that people had souls. He was described by a journalist who said:-

... it is certain a curious thing to see an academician who seemed to fulfil the respectable functions of a missionary preaching to the heathens.

An example of how Cauchy treated colleagues is given by Poncelet whose work on projective geometry had, in 1820, been criticised by Cauchy:-

... I managed to approach my too rigid judge at his residence ... just as he was leaving ... During this very short and very rapid walk, I quickly perceived that I had in no way earned his regards or his respect as a scientist ... without allowing me to say anything else, he abruptly walked off, referring me to the forthcoming publication of his Leçons à 'École Polytechnique where, according to him, 'the question would be very properly explored'.

Again his treatment of Galois and Abel during this period was unfortunate. Abel, who visited the Institute in 1826, wrote of him:-

Cauchy is mad and there is nothing that can be done about him, although, right now, he is the only one who knows how mathematics should be done.

Belhoste in [4] says:-

When Abel's untimely death occurred on April 6, 1829, Cauchy still had not given a report on the 1826 paper, in spite of several protests from Legendre. The report he finally did give, on June 29, 1829, was hasty, nasty, and superficial, unworthy of both his own brilliance and the real importance of the study he had judged.

By 1830 the political events in Paris and the years of hard work had taken their toll and Cauchy decided to take a break. He left Paris in September 1830, after the revolution of July, and spent a short time in Switzerland. There he was an enthusiastic helper in setting up the Académie Helvétique but this project collapsed as it became caught up in political events.

Political events in France meant that Cauchy was now required to swear an oath of allegiance to the new regime and when he failed to return to Paris to do so he lost all his positions there. In 1831 Cauchy went to Turin and after some time there he accepted an offer from the King of Piedmont of a chair of theoretical physics. He taught in Turin from 1832. Menabrea attended these courses in Turin and wrote that the courses [4]:-

were very confused, skipping suddenly from one idea to another, from one formula to the next, with no attempt to give a connection between them. His presentations were obscure clouds, illuminated from time to time by flashes of pure genius. ... of the thirty who enrolled with me, I was the only one to see it through.

In 1833 Cauchy went from Turin to Prague in order to follow Charles X and to tutor his grandson. However he was not very successful in teaching the prince as this description shows:-

... exams .. were given each Saturday. ... When questioned by Cauchy on a problem in descriptive geometry, the prince was confused and hesitant. ... There was also material on physics and chemistry. As with mathematics, the prince showed very little interest in these subjects. Cauchy became annoyed and screamed and yelled. The queen sometimes said to him, soothingly, smilingly, 'too loud, not so loud'.

While in Prague Cauchy had one meeting with Bolzano, at Bolzano's request, in 1834. In [16] and [18] there are discussions on how much Cauchy's definition of continuity is due to Bolzano, Freudenthal's view in [18] that Cauchy's definition was formed before Bolzano's seems the more convincing.

Cauchy returned to Paris in 1838 and regained his position at the Academy but not his teaching positions because he had refused to take an oath of allegiance. De Prony died in 1839 and his position at the Bureau des Longitudes became vacant. Cauchy was strongly supported by Biot and Arago but Poisson strongly opposed him. Cauchy was elected but, after refusing to swear the oath, was not appointed and could not attend meetings or receive a salary.

In 1843 Lacroix died and Cauchy became a candidate for his mathematics chair at the Collège de France. Liouville and Libri were also candidates. Cauchy should have easily been appointed on his mathematical abilities but his political and religious activities, such as support for the Jesuits, became crucial factors. Libri was chosen, clearly by far the weakest of the three mathematically, and Liouville wrote the following day that he was:-

deeply humiliated as a man and as a mathematician by what took place yesterday at the Collège de France.

During this period Cauchy's mathematical output was less than in the period before his self-imposed exile. He did important work on differential equations and applications to mathematical physics. He also wrote on mathematical astronomy, mainly because of his candidacy for positions at the Bureau des Longitudes. The 4-volume text Exercices d'analyse et de physique mathématique published between 1840 and 1847 proved extremely important.

When Louis Philippe was overthrown in 1848 Cauchy regained his university positions. However he did not change his views and continued to give his colleagues problems. Libri, who had been appointed in the political way described above, resigned his chair and fled from France. Partly this must have been because he was about to be prosecuted for stealing valuable books. Liouville and Cauchy were candidates for the chair again in 1850 as they had been in 1843. After a close run election Liouville was appointed. Subsequent attempts to reverse this decision led to very bad relations between Liouville and Cauchy.

Another, rather silly, dispute this time with Duhamel clouded the last few years of Cauchy's life. This dispute was over a priority claim regarding a result on inelastic shocks. Duhamel argued with Cauchy's claim to have been the first to give the results in 1832. Poncelet referred to his own work of 1826 on the subject and Cauchy was shown to be wrong. However Cauchy was never one to admit he was wrong. Valson writes in [7]:-

...the dispute gave the final days of his life a basic sadness and bitterness that only his friends were aware of...

Also in [7] a letter by Cauchy's daughter describing his death is given:-

Having remained fully alert, in complete control of his mental powers, until3.30 a.m.. my father suddenly uttered the blessed names of Jesus, Mary and Joseph. For the first time, he seemed to be aware of the gravity of his condition. At about four o'clock, his soul went to God. He met his death with such calm that made us ashamed of our unhappiness.

Numerous terms in mathematics bear Cauchy's name:- the Cauchy integral theorem, in the theory of complex functions, the Cauchy-Kovalevskaya existence theorem for the solution of partial differential equations, the Cauchy-Riemann equations and Cauchy sequences. He produced 789 mathematics papers, an incredible achievement. This achievement is summed up in [4] as follows:-

... such an enormous scientific creativity is nothing less than staggering, for it presents research on all the then-known areas of mathematics ... in spite of its vastness and rich multifaceted character, Cauchy's scientific works possess a definite unifying theme, a secret wholeness. ... Cauchy's creative genius found broad expression not only in his work on the foundations of real and complex analysis, areas to which his name is inextricably linked, but also in many other fields. Specifically, in this connection, we should mention his major contributions to the development of mathematical physics and to theoretical mechanics... we mention ... his two theories of elasticity and his investigations on the theory of light, research which required that he develop whole new mathematical techniques such as Fourier transforms, diagonalisation of matrices, and the calculus of residues.

His collected works, Oeuvres complètes d'Augustin Cauchy (1882-1970), were published in 27 volumes.

Article by: J J O'Connor and E F Robertson

Michel Rolle

Born: 21 April 1652 in Ambert, Basse-Auvergne, FranceDied: 8 Nov 1719 in Paris, France

Michel Rolle

Michel Rolle's father was a shopkeeper. Michel had little formal education being largely self-educated after receiving some elementary schooling. He worked as a transcriber for a notary and then as an assistant to several attorneys in the district around his home town of Ambert. In 1675, probably seeking a better life, he went to Paris where he worked as a scribe and arithmetical expert. However, quite soon after he arrived in Paris he married and children quickly followed. His income was not sufficient to support his growing family but he had been studying higher mathematics on his own and it was the skill that he had developed in this discipline which provided the breakthrough.

In 1682 he achieved a certain fame by solving a problem which had been publicly posed by Jacques Ozanam. Jean-Baptiste Colbert, the controller general of finance and secretary of state for the navy under King Louis XIV of France, rewarded Rolle for this achievement. Colbert arranged a pension for Rolle which started him on the road to financial security, but there were other equally important consequences [6]:-

Of equal importance with the financial benefits to which Rolle's prize led, was the fact that it brought him to the notice of Louvois who happened to be looking for someone to teach mathematics to one of his sons. Rolle was hired by Louvois, who came to be greatly impressed by his pedagogic and mathematical skills. From 1783 onwards Louvois was in a position to confer further scientific patronage on his protégé and did so when he made Rolle a member of the Académie in 1685.

The problem which Rolle solved was posed in the Journal des sçavans on 31 August 1682 by Jacques Ozanam; it was the following:

Find four numbers the difference of any two being a perfect square, in addition the sum of the first three numbers being a perfect square.

Ozanam stated that the smallest of the four numbers with these properties would have at least 50 digits, but Rolle found four numbers satisfying the conditions with each number having seven digits. He made his solution known through publishing it in the Journal des sçavans. Louvois, who is referred to in the above quote, was François Michel le Tellier, Marquis de Louvois, the French Secretary of State for War. He employed Rolle to tutor his fourth son, Camille le Tellier (1675-1718). The Marquis de Louvois arranged for Rolle to have an administrative post in the Ministry of War, but Rolle disliked the work and soon resigned. Rolle was elected to the Académie Royale des Sciences in 1685 and the impressive mathematical work he produced following his election fully justified the Marquis de Louvois' faith in him.

Before going on to discuss the interesting mathematical contributions Rolle made, let us give a couple of further facts about his life. He became a Pensionnaire Géometre of the Académie des Sciences in 1699. In 1708 Rolle suffered a stroke. He recovered his health fairly well but his mental capacity was diminished and he made no further mathematical contributions after this stroke. He survived for eleven years after this first stroke but in 1719 he suffered a second stroke which proved fatal.

Let us now look at Rolle's important mathematical contributions. He worked on Diophantine analysis, algebra (using methods of Claude Gaspar Bachet de Méziriac involving the use of the Euclidean algorithm) and, to a lesser extent, on geometry. He published his most important work Traité d'algèbre in 1690 on the theory of equations. In this treatise, he invented the notation ⁿ√x for the nth root of x and, as a consequence, it became the standard notation; it is used today. In Traité d'algèbre Rolle used the Euclidean algorithm to find the greatest common divisor of two polynomials. He also used it to solve Diophantine linear equations. Perhaps the most significant part of the work, however, is where he introduces the notion of 'cascades'. Let us see how this idea worked: If P(x) = 0 is a given polynomial equation with real roots a and b then he constructs a polynomial P'(x), which he called the 'first cascade,' so that P'(b) = (b - a)Q(b) where Q(x) is a polynomial of lower degree. Of course in our terminology P'(x) is the first derivative of P(x). Rolle then constructs the 'second cascade' which is the second derivative, and continues in this fashion. Between any two consecutive roots of P(x) there is a root of P'(x), between any two consecutive roots of P'(x) there is a root of P''(x), etc. His method is to start with a given polynomial, make a linear transformation to obtain a polynomial all of whose roots are positive (he never proves that his transformation always works but it does), then to continue to construct the cascade of polynomials until a linear polynomial is reached. One can then move back up the cascade, finding approximately the roots of each polynomial. Julius Shain writes:-

The method of cascades has an important historical significance. Some basic principles of the calculus and the theory of equations can definitely be traced to their origin as incidental propositions of the method. It amplified the concepts of limits of roots of equations, provided the fundamentals from which Maclaurin derived his formula, began modern methods of series for determining roots, and discussed the relationship of imaginary roots in equations and their derivatives. Rolle's theorem, an important proposition of the calculus, also owes its origin to the method.

In fact Rolle is best remembered for 'Rolle's Theorem' which was published in Démonstration d'une Méthode pour resoudre les Egalitez de tous les degrez in 1691. This work was written to provide proofs of certain methods (in particular the method of cascades) which he gave without justification in Traité d'algèbre. His proofs were based on methods introduced by Johann van Waveren Hudde. The familiar Rolle's Theorem states:

If f (a) = f (b) = 0 then f '(x) = 0 for some x with a ≤ x ≤ b.

The name 'Rolle's Theorem' was given to this basic result by Giusto Bellavitis in 1846. In his 1691 work Rolle adopted the notion that if a > b then -b > -a. It seems strange today to realise that this was not the current practice at the time but was in opposition to the ordering of the real numbers used by Descartes and others. Other notation Rolle used in his 1691 work was the equals sign '=.' This notation was not invented by Rolle, rather it was invented by Robert Recorde, but Descartes had used a different notation and the sign = was not in common use. Rolle published another important work on solutions of indeterminate equations in 1699, Méthode pour résoudre les équations indéterminées de l'algèbre.

It might be assumed from what we have just written about Rolle's work that he was developing the infinitesimal calculus. This would be a serious error, for Rolle described the infinitesimal calculus as a collection of ingenious fallacies and he believed that the methods could lead to errors. He begins his memoir Du nouveau systême de l'infini (1703) as follows (see for example [4]):-

Geometry has always been considered as an exact science, and indeed as the source of the exactness which is widespread among other parts of mathematics. Among its principles one could only find true axioms and all the theorems and problems posed were either soundly demonstrated or capable of sound demonstration. And if any false or uncertain propositions were slipped into it they would immediately be banned from this science. But it seems that this feature of exactness doe not reign anymore in geometry since the new system of infinitely small quantities has been mixed to it. I do not see that this system has produced anything for the truth and it would seem to me that it often conceals mistakes.

Now we should note that this memoir, although only published by the Academy of Sciences in 1703, actually contained material which had been used in the early stages of a vigorous debate which took place in the Academy of Sciences in 1700-01. This vigorous disagreement was between Rolle and Pierre Varignon and it ended in uproar (some say that Rolle's lack of breeding showed in his bad behaviour). The Academy set up a commission to decide which of the two mathematicians was correct but it failed to come to a definite conclusion. Michel Blay [2] writes:-

This opposition, extremely active from 1700 on, was led by Michel Rolle. The burden of his critique rested on two arguments, one stressing the inadequacy and the lack of logical rigour of the fundamental concepts and principles of the new calculus, the other pretending to show (with the aid of cleverly selected examples) that the new calculus led to error, insofar as it did not yield the same results obtained in the classical, algebraically inspired methods of Fermat and, more especially, Hudde.

In particular Rolle suggested that there were difficulties associated with the following axiom given by de l'Hôpital in Analyse des infiniment petits pour l'intelligence des lignes courbes(1696):

Grant that two quantities whose difference is an infinitely small quantity may be taken (or used) indifferently for each other; or (which is the same thing) that a quantity which is increased or decreased only by an infinitesimally small quantity may be considered as remaining the same.

For example, Rolle addressed the Academy on 12 March and 16 March 1701. He asked his audience to consider the curve

y - b = (x² - 2ax + a² - b²)^(2/3)/a^(1/3).

He then applied the new method of the infinitesimally small and found that

dy = 4(x - a)dx/3(ax² - 2a²x + a³ - ab²)^(1/3)

so that this method showed that the curve has a turning point at x = a. Rolle then applied Hudde's method to show that the curve has turning points at three points, x = a, x = a - b andx = a + b. This is a cleverly constructed example, but Varignon was able to see the subtle error in Rolle's analysis. He later replied [2]:-

One will obtain first x = a by setting dy = 0; and second, x = a - b or x = a + b by making dy infinite in relation to dx or dx = 0. When one sees that if the desired curve has some maximum or minimum that meets a tangent parallel to the [x-axis], this can only be at the extremity of x = a; and that if it has one that meets ... [tangents] perpendicular to this axis, this can only be at the extremity of x = a - b and of x = a + b.

After the Academy decided that no further discussion of this topic was take place, Rolle continued the argument in the pages of the Journal des sçavans opposed by Joseph Saurin. To his eternal credit, Rolle eventually conceded that he was wrong. He acknowledged this to Varignon, Fontenelle and Malebranche. Jean Itard ends his article [1] with the following assessment:-

Rolle was a skillful algebraist who broke with Cartesian techniques; and his opposition to infinitesimal methods, in the final analysis, was beneficial.

Article by: J J O'Connor and E F Robertson