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494 Karin Reich Meusnier Never Came Back to Mathematics, but He 494 Karin Reich Fig. 4. Meusnier’s figure 8, the catenoid Meusnier never came back to mathematics, but he contributed to var- ious other fields. For example, he was an aeronautical theorist. After the first flights of Montgolfier’s balloon he designed an elliptical shaped airship instead of the spherical balloon, but his balloon was never built. Meusnier also collaborated with Antoine de Lavoisier (1743-1794) to separate water into its constituents. Like Tinseau, Meusnier had an accomplished military career, but Meus- nier supported the revolution. He became even a g´en´eral de division. He was severely wounded during a battle between the French and the Prussians near Mainz. Goethe observed that battle and gave a detailed description [Goethe 1793]. 2.3.2. Graduate students from the Ecole´ Polytechnique In 1795, almost immediately after the foundation of the Ecole´ Poly- technique in Paris, Monge published his Feuilles d’Analyse appliqu´ee`ala G´eom´etrie `al’usage de l’Ecole´ Polytechnique [Monge 1805-1850]. It reap- peared in later editions, entitled Application de l’Analyse `ala G´eom´etrie [Monge 1795-1807]. The last edition from 1850 was prepared by Joseph Li- ouville (1809-1882), who remarkably also published Monge’s Application and Gauß’ Disquisitiones generales circa superficiem curvas. The lectures that Monge gave at the Ecole´ polytechnique in Paris had much greater influence than the lectures at M´ezi`eres. Taton mentioned the following students of Monge: Lacroix, 21 Hachette, Fourier, Lancret, Dupin, Livet, Brianchon, Amp`ere, Malus, Binet, Gaultier, Sophie Germain, Gergonne, O. Rodrigues, Poncelet, Berthot, Roche, Lam´e,Fresnel, Chasles, Olivier, Val´ee, Coriolis, Bobillier, Barr´ede Saint-Venant and many others [Taton 1951, p.235f]. Some of these worked on differential geometry. 21 Taton mentioned his name here, but Lacroix had not attended Monge’s courses at the Ecole´ Polytechnique. LOL-Ch24-P16 of 24 Euler’s Contribution to Differential Geometry and its Reception 495 Michel-Ange Lancret (1774-1807) Lancret began his studies in 1794. Before the official lectures at the Ecole´ Polythechnique began, Monge gave a special course to a small group of excellent students, including Lancret. [Taton 1951, p.39] He was among the first graduate students of Monge, who very highly respected Lancret. Later Monge, Lancret and other scientists accompanied Napoleon on his Egyptian expedition in 1799/1800. Lancret did not return to France until in 1802 when he became the secretary of the commission concerning the work of the Egyptian expedition. In 1806 and in 1811 Lancret published two papers on curve theory. Euler was only mentioned in an historical context [Lancret 1806, p.416]. Lancret worked mostly on the basis of the Monge’s various contributions. According to Struik, Lancret was the first to take up the systematic theory of space curves after Euler, but it seems in an independent way. The line of progress goes from Clairaut to Euler and then from Lancret to Cauchy and Frenet [Struik 1933, p.116]. Charles Dupin (1784-1873) Dupin’s name is still known in differential geometry. Under the guidance of Gaspard Monge, Dupin made his first discovery in 1801, what is now called the cyclid of Dupin. He graduated from the Ecole´ Polytechnique in 1803 as a naval engineer. For several years he worked abroad and came back to France only in 1813. In 1814 he became correspondent of the mechanical section and in 1818 he was elected member of the Acad´emie des Sciences. When Monge died in 1818, Dupin wrote a detailed Eloge´ for his former teacher. [Dupin 1819] Just a year later, in 1819, Dupin became professor at the Conservatoire des Arts et M´etiers in Paris, a position which he held until 1854. In 1834 he became minister of marine affairs. During the years 1836-1844 he was Vice-President of the Acad´emie des Sciences, and in 1838 he became peer and in 1852 he was appointed to the senate. Dupin’s D´eveloppements de g´eometrie [Dupin 1813] were his main con- tributions to differential geometry. Here one can find the introduction of conjugate and asymptotic lines on a surface, the so-called “indicatrix,” and Dupin’s theorem, which states that three families of orthogonal surfaces intersect in lines of curvature. When Hachette wrote the “Avertissement de l’Editeur,” he mentioned Euler, Biot, Monge and Clairaut. He made special mention of Euler’s Introductio [E101-102] and his contribution to surface curvature. [E333] Dupin and Hachette noted Euler’s Introductio for the work on the general equation of the surfaces of the second order and its classification of these kinds of surfaces into 5 different categories. LOL-Ch24-P17 of 24 496 Karin Reich They cited the second work because it gave the expression of the radius of curvature of a normal section of a surface and showed that the planes of sections of maximum and minimum radius were perpendicular on each other. Indeed, the “Article V” of Dupin’s D´eveloppements was devoted to the “D´emonstration de plusieurs th´eor`emes d’Euler sur la courbure des surfaces” [Dupin 1813, p.107-110]. Dupin, however, had achieved his own results independently from Euler. Augustin Louis Cauchy (1789-1857) Cauchy began his studies in 1805 at the Ecole´ Polytechnique and contin- ued at the Ecole´ des Ponts et des Chauss´ees. After several years working as an engineer in Cherbourg, Cauchy returned to Paris in 1813, where he became professor at the Ecole´ Polytechnique. In 1816 Cauchy became a member of the Acad´emie des sciences. While he was professor at the Ecole´ Polytechnique he wrote many papers, including some on differential ge- ometry and several textbooks, which became quite famous. Among these were his Le¸conssur les applications du calcul infinit´esimal`ala g´eometrie. [Cauchy 1826-1828] He used almost the same title as Monge. In his preface Cauchy emphasized the contributions of two physicists, Gaspard Gustave de Coriolis (1792-1843) and Andr´e-Marie Amp`ere(1775- 1836). The former had given a definition of the general radius of curvature of curves that was adopted by Cauchy in his 17th chapter. Both Amp`ere and Coriolis had been pupils of Monge as well. In this 17th chapter, “Du plan osculateur d’une courbe quelconque et de ses deux courbures. Rayon de courbure, centre de courbure et cercle osculateur,” Cauchy derived the first and the second of the Frenet formulas. Euler had only presented the first one in [E602]. In his 19th chapter Cauchy treated surface theory, especially the radius of curvature of principal sections and so on. Here Cauchy also quoted Euler: “We will not end this lesson without recalling that it was Euler who first established the theory of curvature of surfaces and showed the relations which exist between the radii of curvature of the different sections cut from a surface by its perpendicular planes. The discoveries of that illus- trious geometer on all these things have been published in the M´emoires of the Berlin Academy (1760)” 22 22 “Nous ne terminerons par cette Le¸con sans rappeler que c’est Euler qui le premier a ´etabli la th´eorie de la courbure des surfaces, et montr´eles relations qui existent entre les rayons de courbure des diverses sections faites dans une surface par des plans normaux. Les recherches de cet illustre g´eom`etre,sur l’objet dont il s’agit, ont ´et´eins´er´ees dans les M´emoires de l’Acad´emie de Berlin (ann´ee1760)” [Cauchy 1826-1828, p. 364]. LOL-Ch24-P18 of 24 Euler’s Contribution to Differential Geometry and its Reception 497 Olinde Rodrigues (1794-1851) Rodrigues was of Jewish origin, so he was not allowed to study at the Ecole´ Polytechnique. Instead he entered the Ecole´ normale where he was awarded a doctorate in mathematics in 1816. As Taton remarked, though Rodrigues was not a direct pupil of Monge, he should be counted among Monge’s students [Taton 1951, p.366]. In 1815 and 1816 Rodrigues pub- lished two papers on differential geometry in which he presented some work on the lines of curvature and simplified some of Monge’s results. Rodrigues he did not primarily take ideas from Euler; he mentioned mainly Monge and Dupin[Grattan-Guiness 2005, p. 100]. Euler’s influence on Rodrigues was perhaps more or less an indirect one. 23 Sylvestre Fran¸coisLacroix (1765-1843) Born in Paris in 1765, Lacroix also was a pupil of Gaspard Monge. Lacroix came from a very poor family. He began private lectures with Monge in 1780, and Monge continued to follow his education and his car- rier. In 1789 Lacroix became correspondent of the Parisian Academy and in 1794 he joined the Commission de L’instruction publique. He later became teacher at the Ecole´ Polytechnique, at the Ecole´ normale, at the Ecole´ Cen- trale des Quatre Nations, at the Facult´edes Sciences de Paris and finally at the Coll`ege de France. As a consequence of his many teaching positions, he became author of several textbooks, including his well-known Trait´edu calcul diff´erentiel et du calcul int´egral (2 vol., Paris 1797 and 1798). There was a second edition in three volumes (Paris 1810, 1814 and 1819) as well as a translation into German due to J.P.Gr¨uson (2 vol., Berlin 1799, 1800). Lacroix devoted his chapter 5 to surfaces and curves of double curva- ture. Here Lacroix mentioned in detail the contributions of Euler, Monge and Meunsier. Euler had been the first to recognize the importance of the principal curvatures and the analytical expression which allows a dis- tinction between developable surfaces and surfaces which do not have this property.
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