HADAMARD HADAMARD (5) Histoire des machines a vapeur . (Paris, 1830). honorary doctorates from many foreign universities . In addition, Hachette edited the first separately printed Hadamard's interest in pedagogy led him to write ed. of Monge's Geometrie descriptive (Paris, 1799), the articles about concepts in elementary mathematics reissues of that work, and the new ed . of 1811, as well that are introduced in the upper classes of the lycee . as the fifth and succeeding eds . of Monge's Traite elemen- His Legons de geometrie elementaire (1898, 1901) still taire de statique (Paris, 1809 ff.), and the third and fourth delight secondary school instructors and gifted pupils. eds. of his Application de l'analyse a la geometrie (1807, The extent of his grasp on all domains of advanced 1809). In addition Hachette edited Auguste Comte's French trans. of John Leslie, Elements of Geometry, Geometrical research in France and abroad was evident in his Analysis, and Trigonometry, 2nd ed . (London, 1811), as famous seminar at the College de France ; no branch Analyse geometrique (Paris, 1818) and published the French of mathematics was neglected . The world's most fa- trans. of Thomas Young, A Course of Lectures on Natural mous mathematicians came there to present their own Philosophy and the Mechanical Arts, 2 vols. (London, 1807), findings or those related to their specialty . But it was as Precis de mecanique et de la science des machines (Paris, always Hadamard who had the last word and the 1829). surest judgment concerning the significance or the . Hachette's life and works 11. SECONDARY LITERATURE potential of the research presented . were the subject of the following accounts (in chronological Hadamard's first important works were concerned order), some of which contain quite serious errors (e.g., with analytic functions, notably with the analytic date of birth, participation in the Egyptian expedition) : continuation of a Taylor series . Although Karl F. Arago and S.-D. Poisson, Funerailles de M. Hachette (Paris, 1834); A. F. Silvestre, Discours prononce sur la tombe Weierstrass and Charles Meray were the first to define de M. Hachette (Paris, 1834); C. Dupin and A. Quetelet, the meaning that must be attributed to the domain in Annuaire de l Academie royale de Bruxelles for 1836 of existence of the analytic continuation of a Taylor (1836), 71-77 ; V. Parisot, in Michaud, ed ., Biographie series, their reflections amounted to a theorem of universelle, LXVI (supp .), 339-341, also in new ed ., XVIII existence and uniqueness. Before Hadamard, little (Paris, 1857), 314-315 ; L. Louvet, in F . Hoefer, ed., was known about the nature and distribution of the Nouvelle biographie generale, XXIII (Paris, 1861), cols . singularities of the series in terms of the nature of 26-29; Poggendorff, I, cols . 985-986 ; A. Hannedouche, Les its coefficients, which define the function a priori . His illustrations ardennaises . (Sedan, 1880), pp . 81-82 ; thesis (1892), preceded by several notes in the 698- L. Sagnet, in Grande encyclopedie, XIX (Paris, n.d.), Comptes rendus of the Academy, is one of his most 699; and N. Nielsen, Geometres francais sous la Revolution beautiful works. For the first time an ensemble con- (Copenhagen, 1929), pp. 121-125 . cept was introduced into function theory . In fact the RENT TATON upper limit-made more precise, explained, and ap- plied to the ensemble composed of the coefficients- HADAMARD, JACQUES (b. Versailles, France, 8 permits the determination of the radius of conver- December 1865 ; d. Paris, France, 17 October 1963), gence (or, rather, its inverse) of the Taylor series . mathematics. Conditions affecting the coefficients enable one to Hadamard was the son of Amedee Hadamard, a characterize the singular points on the circle of con- Latin teacher in a noted Paris lycee ; his mother, vergence. Hadamard's famous theorem on lacunary Claude-Marie Picard, was a distinguished piano series ("lacunes a la Hadamard ") admitting the circle teacher. After studying at the Ecole Normale of convergence as a cut, the theorems on polar Superieure from 1884 to 1888, he taught at the Lycee singularities, the introduction of the concept of "ecart Buffon in Paris from 1890 to 1893 and received his fini" and of the "order" of a singular point, and the docteur es sciences degree in 1892 . Hadamard was a theorem on the composition of singularities (1898), lecturer at the Faculte des Sciences of Bordeaux from have remained fundamental in function theory . The 1893 to 1897, lecturer at the Sorbonne from 1897 to results have inspired generations of highly talented 1909, then professor at the College de France from mathematicians, especially those working at the be- 1909 to 1937, at the Ecole Polytechnique from 1912 ginning of the twentieth century and in the years to 1937, and at the Ecole Centrale des Arts et Manu- between the two world wars . His La serie de Taylor factures from 1920 to 1937 . et son prolongement analytique (1901) was the "Bible" Elected a member of the Academie des Sciences of all who were fascinated by the subject . in 1912, Hadamard was also an associate member of The year 1892 was one of the most fertile in the several foreign academies, including the National history of the theory of functions of a complex varia- Academy of Sciences of the United States, the Royal ble; it also marked the publication of a work by Society of London, the Accademia dei Lincei, and Hadamard that established the connection between the Soviet Academy of Sciences . In addition he held the decrease of the modulus of the coefficients of the 3 HADAMARD HADAMARD Taylor series of an integral function and the genus partial derivatives . The importance of analysis situs of the function . This work (which received the Grand in the theory of differential equations was shown by Prix of the Academie des Sciences) and the results Henri Poincare, whom Hadamard admired greatly of his thesis (especially those pertaining to polar and to whose work he devoted several memoirs and singularities), applied to Riemann's ~ function, monographs (1922, 1923, 1954) . enabled Hadamard in 1896 to solve the ancient and The memoirs in question (1898) treat surfaces of famous problem concerning the distribution of the negative curvature having a finite number of nappes prime numbers. He demonstrated (in a less explicit extending to infinity . All analytic description is form than is shown here but one easily reducible to abandoned. On these surfaces the geodesics behave it) that the function 7r(x) designating the number of in three different ways : (1) they are closed or prime numbers less than x is asymptotically equal asymptotic to other such geodesics; (2) they extend to x/log x. This is certainly the most important result to infinity on one of the nappes ; (3) entire segments ever obtained in number theory . Charles de La of these geodesics approach successively a series of Vallee-Poussin proved the theorem at the same time, closed geodesics, the length of these segments growing but his demonstration is much less simple than toward infinity . The striking thing is that the ensemble Hadamard's . The total result seems to indicate- E of tangents to the geodesics passing through a without, we believe, its ever having been mentioned point and remaining at a finite distance is perfect in writing by Hadamard-that the research in his and never dense ; and in each neighborhood of every thesis and in his work on integral functions was geodesic whose tangent belongs to E (neighborhood implicitly directed toward the ultimate goal of of directions) there exists a geodesic which extends indicating the properties of the function ~, in order to infinity in an arbitrarily chosen nappe . In each of to derive from it the theorem on the prime numbers . these neighborhoods there also exist geodesics of the Returning to analytic functions, one should men- third category . Hadamard states : "Any change, how- tion the 1896 theorem on the maximum modulus of ever small, carried in the initial direction of a geodesic an integral function (or of a holomorphic function which remains at a finite distance is sufficient to in a disk). And, while remaining close to the essential produce any variation whatsoever in the final aspect principles of analytic functions but leaving aside those of the curve, the disturbed geodesic being able to take with a complex variable, one must emphasize on any one of the forms enumerated above" (Oeuvres, Hadamard's introduction (1912, 1925) of the idea and p. 772). of the problem of quasi analyticity, which consists But in a physics problem a slight modification in in finding a relationship between the growth of the the circumstances at a certain moment ought to have maxima of the moduli of the derivatives of a function little influence on the solution, since one never on a segment and the fact of being determined in possesses conditions which are more than approxi- a unique way by the values that the function and mate. Hadamard concluded from this that the behav- its derivatives take at a point . It should be noted that ior of a trajectory might well depend on the arithmetic it was Albert Holmgren's considerations relating to character of the constants of integration . One already Augustin Cauchy's problem for the equation of heat sees here the genesis of the idea of the "problem that had led Hadamard to consider classes of correctly posed" which guided Hadamard in his infinitely differentiable functions, not necessarily researches on equations with partial derivatives . The analytic on a segment but nevertheless possessing the problem of geodesics on the surfaces studied by characteristic property of uniqueness on the segment . Hadamard is not a correctly posed mechanics prob- The idea of quasi analyticity plays a significant role lem.