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 Ampere 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 Ecole 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 College 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 Ecole 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 infinitesimal while in 1829 in Lecons sur le Calcul Differentiel 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 Academie 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 Lecons a 'Ecole 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 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 Academie Helvetique 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:-

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 and there are discussions on how much Cauchy's definition of continuity is due to Bolzano, Freudenthal's view in [hat 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 College 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 College 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 Exercises d'analyse et de physique mathematique 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:-

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

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

Having remained fully alert, in complete control of his mental powers, until 3.30 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 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 completes d'Augustin Cauchy (1882-1970), were published in 27 volumes.

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

3.3.9. Carl Friedrich Gauss

Carl Friedrich Gauss was born in Braunschweig on April the 30th 1777 as the son of a poor worker. Already in his youth he was interested in mathematics. It is reported that when Gauss was a student at elementary school his teacher asked the students to add up all natural numbers from 1 to 100, hoping to keep his students busy for some time. Gauss however found the correct answer within a few minutes by cleverly rearranging the summands.

From 1792 to 1795 he was a student at the Collegium Carolinum in Braunschweig, which was made possible by a scholarship by the duke of Braunschweig. From 1795 to 1798 he studied at the university of Gottingen. 1807 he became professor for astronomy in Gottingen and director of the observatory. He remained in this position until his death in 1855.

During his time as a student in Gottingen he met the Hungarian mathematician Wolfgang Bolyai and they swore each other ``brotherhood und the banner of truth''.

The first important discovery he made was the construction of the 17-gon by circle and ruler in 1796. In 1799 he finished his dissertation, in which he proved the fundamental theorem of algebra. In 1801 his main work, the Disquisitiones Arithmeticae were published.

In the year 1801 the italian astronomer Joseph Piazzi discovered the planetoid Ceres, but could only watch it for a few days. Gaus predicted correctly the position at which it could be found again. It was rediscovered by Zach at 31st of December 1801 in Gotha and one day later by Olbers in Bremen. Zach said about this: ``Ohne die scharfsinnigen Bemuhungen und Berechnungen des Doktor Gauss hatten wir vielleicht die Ceres nicht wiedergefunden'' (Without the intelligent work and calculations of doctor Gauss we might not have found Ceres again.) By this and the discovery of the planetoid Pallas by Olbers in 1802 Gauss worked on a theory of the motion of planetoids disturbed by large planets. This work was published in 1809 under the name Theoria motus corporum coelestium in sectionibus conicis solem ambientum (Theory of motion of the celestial bodies moving in conic sections around the sun).

In 1816 he is asked to do land surveying in the kingdom of Hannover, which gave rise to several important scientific discoveries. In that time he developed the method of least squares independently of Legendre and the curvature of the earth made him think about noneuclidean geometries. Gauss also tried to measure whether earth's surface was euclidean, by measuring the sum of angles in a triangle formed by three mountains (Brocken, Inselberg and Hohen Hagen) but found that is was . He never published his works on geometry though. Later his friend Wolfgang Bolyai sent him his son's works about noneuclidean geometry Gauss said that he was unable to praise that work for he'd then have to praise himself, meaning that he had already found these things himself. But though Gauss wrote to Gerling that he considered the young Bolyai to be a genius, he never praised him in public and the young Bolyai turned away from mathematics.

Together with the physicist Wilhelm Weber he built the first electromagnetic telegraph in 1833, which connected the observatory with the institute for physics in Gottingen. Gauss also worked on the theory of magnetism and found a representation for the unit of magnetism by the units of mass, length and time. Inspired by Alexander von Humboldt he was interested in earth's magnetism. He ordered a magnetic observatory to be built in the garden of the observatory and founded the magnetischer Verein (magnetic club) together with Wilhelm Weber and this club supported measurements of earth's magnetic field in many regions of the world.

His private life however was not so happy. In 1809 died his first wife Johanna. Gauss mourned over her very much and it was more due to a feeling of duty and sympathy that he took his wife's best friend Minna as his second wife. From his first wife he had one daughter, Minna, whom he liked very much, and one son, Joseph, named after the discoverer of Ceres. From his second wife he had two sons and one daughter. He did not get on well with his children except for Minna, who was very much like her mother Johanna.

Gauss did not very much like to give public lectures. He hated it when he had to give lectures at university and it is said that he only attended a single scientific conference, which was in Berlin in 1828 when he was a guest of Humboldt's. He also did not have mathematical students.

However in other natural sciences he had good cooperation with many people for example with Olbers in Bremen, Schumacher in Altona (today Hamburg), Gerling in Marburg and Bessel in Koenigsburg.