April 19, 1877] 



NATURE 



535 



subjects of an algebraic character. If we except a comparatively 

 unimportant paper on Descartes' Rule of Signs, which appeared 

 in Crelle's yournal {1S2S), his only algebraical memoirs relate to 

 the theorem that every equation has a root. Of this he gave no 

 less than three distinct demonstrations, one in 1799, one in 1815, 

 and one in 1 816. That in 1799 formed the subject of his first pub- 

 lished paper: " Deraonstratio nova theorematis omnem functiu- 

 nem algebraicam, rationalem integram unius variabilis in factores 

 reales primi vel secundi gradus resolvi posse "— his inaugural 

 dissertation as a candidate for the degree of Doctor of Philo- 

 sophy in the University of Gottingen. This demonstration was 

 repeated over again in 1849 with certain changes and simplifica- 

 tions. These demonstrations are prior to any other ; ' for 

 various reasons those subsequently given by Cauchy have been 

 justly preferred for insertion in modern text-books. 



A new epoch in certain branches of analysis dates from the 

 publication of the "Disquisitiones Arithmeticse " (Leipzig, 1801),* 

 and from the researches with which some years later Gauss 

 supplemented or further developed the theories contained in that 

 work. We must bear in mind that he found the theory of num- 

 bers as Euler and Lagrange had left it. The former enriched it 

 with a multitude of results, relating to Diophantine problems, to 

 the theory of the residues of powers, and to binary quadratic 

 forms ; the latter had given the character of a general theory to 

 some at least of these results by his discovery of the reduction of 

 quadratic forms and of the true principles of the solution of in- 

 determinate equations of the second degree. Legendre (with 

 many additions of his own) had endeavoured to arrange as much 

 as possible of these scattered fragments of the science into a 

 systematic whole in his " Essai sur la Theorie des Nombres." 

 But the " D. Ar." was in the press when this important treatise 

 appeared, and what in it was new to others was already known 

 to Gauss. This grand work merits an analysis at our hands, but 

 lack of space compels us to pass on at once to the fourth section. 

 The greater portion of it is occupied with a research, which of 

 itself alone would have placed Gauss in the first rank of mathe- 

 maticians. " If / and (/ are positive uneven prime numbers, / 

 has the same quadratic character with regard to g that (/ has with 

 regard to p ; except when p and ^ are both of the form 4 « -H 3, 

 in which case the two characters are always opposite, instead of 

 identical." This is the celebrated Fundamental Theorem of 

 Gauss, known also as the Law of Quadratic Reciprocity of 

 T^gendre. Gauss discovered it (by induction) in March, 1795, 

 before he was eighteen ; the proof given of it in this section he 

 discovered in April of the year following."^ He cannot at the 

 earlier date have been aware that the theorem had been already 

 enunciated (though in a somewhat complex form) by Euler, and 

 that Legendre had attempted, though unsuccessfully, to prove it 

 in the Memoirs of the Academy of Paris for 1 784. The question 

 of priority of enunciation or of demonstrating by induction in 

 this case is a trifling one ; any rigorous demonstration of it 

 involved apparently insuperable difficulties. Gauss was not 

 content with once vanquishing the difficulty, he returns to it 

 again in the fifth section, and there obtains another demonstra- 

 tion reposing on entirely different, but perhaps still less elemen- 

 tary principles. In January, 1808, he submitted a third demon- 

 stration to the Royal Society of Gottingen ; a fourth in August 

 of the same year ; a fifth and sixth in February, 181 7. It is no 

 wonder he should have felt a sort of personal attachment to a 

 theorem which he had made so completely his own, and which 

 he used to call the " gem " of the higher arithmetic. His six 

 demonstrations remained for some time the only efforts in this 

 direction, but the subject subsequently attracted the attention of 

 other eminent mathematicians, and several proofs differing sub- 

 stantially from one another, and from those of Gauss, have been 

 given.* It would be impossible to exaggerate the important 



' This dissertation (Helmstadt, 1799), so little known that Lagrange 

 appears not to have been acquainted with it, and " Cauchy has received in 

 trance all the praise due to a first discoverer." — Larrouse. 



2 For an amusing notice of this -work see " Biographic nouvelle des Con- 

 temporains" (Paris, 1822) : " Cet ouvrage a obtenu un succes d'apres lequel 

 on serait tente de croire que le charlatanisme en vahit quelquefois jusqu'au do- 

 inaine des mathematiques"(see Roy. 5oc, Proceedings, pp 590, 591, nhi supra) 

 Twelve years Iater(" Biographie universelleet portative des Contemporains," 

 Paris, 1834) we read, "II suffit de dire qu'en general ses travaux sont estimes 

 des math^maticiens les plus distingues et qu'ils se recommandent autant par 

 leur exactitude que par la clartc, la precision et I'elcgance du style." 

 r.aplace's saying, quoted above, also shows in what estimation Gauss was 

 held at that date. Lalande speaks of his talent and zeal, " Histoire," p. 813 

 (1803). 



3 See Gauss's note (pp. 475, 476 of vol. i. of his " Werke," edited by 

 Schering). 



* By Jacobi and Eisenstein in Germany, M. Liouville in France ; perhaps 

 the simplest of all (one allied in its character to the third proof of Gauss) i:^ 



influence which this theorem has had on the subsequent develop- 

 ment of arithmetic, and the discovery of its demonstration by 

 Gauss must be certainly regarded (it was so regarded by him- 

 self), as one of his greatest scientific achievements. The fifth 

 section (" the^e marvellous pages ") abounds with subjects, each 

 of which has been the starting-point of long series of important 

 researches by subsequent mathematicians. In the Additamenta 

 to this section Gauss characteristically adds : "ex voto nobis sic 

 successit ut nihil amplius desiderandum supersit Nov. 30 — 

 Dec. 3, 1800." It is remarkable that he should never have 

 published the wonderful researches to which he here alludes. 

 They first saw the light sixty-three years later in the second 

 volume of the collected edition of his works. ' Till the time of 

 Jacobi, it is not too much to say, that the profound researches of 

 the fourth and fifth sections were passed over with almost uni- 

 versal neglect, but the seventh section at once made the reputa- 

 tion of the " D. Ar." The well-known theory of the division of 

 the circle, comprised ia this section, was received with great 

 and deserved enthusiasm as a memorable addition to the theory 

 of equations and to the geometry of the circle. Gauss's note on 

 §365 ("circulum in 17 partes divisibilem esse georaetrice, 

 deteximus 1796, Mart 30") -is interesting because'it shows that 

 he was not yet nineteen when he made this great discovery. 

 Even more remarkable, however, is a passage (§ 335), in which 

 he observes that the principles of his mt-thod are applicable to 



1777 



1877 



Carl Friedrich Gauss. 



many other functions beside the circular functions, and in par- 



y' dx 

 ^—^■ 



This almost casual remark shows (as Jacobi long since observed) 

 that Gauss at the date of the publication of the " D. Ar." had 

 already examined the nature and properties of the elliptic func- 

 tions and had discovered their fundamental property, that of 

 double periodicity. This observation of Jacobi's is amply con- 

 firmed by the papers on elliptic transcendents, now published in 

 the third volume of Gauss's collected works.^ 



The " D. Ar." were to have included an eighth section ; at 

 first it was intended to contain a complete theory of congruences, 

 but subsequently Gauss appears to have proposed to continue 

 the work by a more complete discussion of the theory of the 

 division of the circle. Manuscript drafts on each of these sub- 

 jects were found among his papers; the first of them is especially 

 interesting, as it treats of the general theory of congruences from 

 a point of view closely alliwi to that subsequently taken by 



one by M. Zeller(see Messenger of Matli£i>iatics, No. IviL, January, 1876 

 for an account by Prof. Paul Mansion). 



' The theorem to which they refer had, in the interval, been redisco»ereJ 

 and demonstrated by Lejeune Dirichlet This demonstration has been to a 

 certain extent simplified by M. Hermite, and the form of proof found io 

 Gauss's papers alter hi* death approaches very nearly to that adopted by 

 M. Hermite. . , ■ .- 



'^ Schering's edition, ubi supra. M. Chasles passes by this discovery with- 

 out any notice of it ; in this case the language could not be the barrier :_ 

 " Par suite de notre ignorance de la langue dans laquelle ils sont Merits." 

 "Aper^u Historique," p. 215. Delambrc gives an account in his " Rapport 

 Historique sur les progrfes des Sciences," Paris, 1810. In the Roy. Ast. S'oc. 

 Notice arc some pertinent remarks. 



3 On p. 593 Roy. Soc. Obituary Notice will be found the story of Gauss 

 and Jacobi. For every theorem in the subject of elliptic integrals produced 

 by the latter. Gauss could show its (cllow .-ioiong his manuscripts. 



