July 19, 1883] 



NA TURE 



273 



terminatis Secundi Gradus " — consists, as has been said 

 with great truth by Dirichlet, of two distinct parts. Of 

 these the first (Arts. 153-222) contains a complete exposi- 

 tion of the theory of binary quadratic forms, as far as it 

 was known from the researches of Euler and Lagrange ; 

 although even these known results are completed in 

 many respects and are exhibited from a new and inde- 

 pendent point of view. The second part (Arts. 223-305) 

 contains investigations which are entirely Gauss's own : 

 the distribution of the classes of binary forms into 

 genera ; the determination of the number of ambiguous 

 classes ; the demonstration that only one-half of the 

 genera possible a priori actually exist, and the proof of 

 the fundamental theorem deduced from this result ; a 

 disquisition on ternary quadratic forms, introduced as a 

 digression ; the theory of the decomposition of numbers 

 into three squares ; the solution of indeterminate equa- 

 tions of the second degree in rational numbers ; the de- 

 termination of the mean number of the genera and 

 classes ; the distinction between regular and irregular 

 determinants. Such is a brief list of the subjects treated 

 of in these marvellous pages, each of which has been the 

 starting-point of long series of important researches by 

 subsequent mathematicians. 



In the Additamenta to this section, Gauss intimates 

 that he had succeeded in determining the relations 

 between the determinant and the number of classes ; and 

 in a manuscript note he 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. These researches first saw the 

 light sixty-three years later in the second volume of the 

 collected edition of his works; but the theorem to which 

 they refer had in the interval been rediscovered and de- 

 monstrated by Lejeune Dirichlet. The demonstration 

 of Dirichlet had been to a certain extent simplified by M. 

 Hermite, and the form of demonstration found in Gauss's 

 papers after his death approaches very nearly to that 

 adopted by M. Hermite. 



The sixth section contains some applications of arithme- 

 tical principles to various practical questions. Of these the 

 first two are comparatively elementary, and relate to the 

 resolution of fractions into simpler fractions, and to the 

 conversion of vulgar into decimal fractions ; the others 

 consist in systematic methods of abbreviating certain 

 tentative processes such as the solution of quadratic 

 congruences, the decomposition of numbers into their 

 prime factors, the solution of certain indeterminate equa- 

 tions, &c. The methods of Gauss still remain the least 

 unsatisfactory that have been proposed for the indirect 

 treatment of these difficult problems, of which any direct 

 solution seems impossible. 



The seventh section, "De/EquationibusCirculi Sectiones 

 Definientibus,'' is that which at once made the reputation 

 of the "Disquisitiones Arithmetics." It is not too much 

 to say that till the time of Jacobi the profound researches 

 of the fourth and fifth sections were passed over with 

 almost universal neglect. But the well-known theory of 

 the division of the circle comprised in this section was 

 received with great and deserved enthusiasm as a memor- 

 able addition to the theory of equations and to the geo- 

 metry of the circle. One of Gauss's manuscript notes is 

 interesting, " Circulum in 17 partes divisibilem esse geo- 

 metrice, deteximus 1796, Mart. 30," because it shows that 

 he was not yet nineteen when he made this great dis- 

 covery. Even more remarkable, however, is a passage 

 in the first article of the section (Art. 335), in which 

 Gauss observes that the principles of his method are 

 applicable to many other functions besides the circular 

 functions, and in particular to the transcendents depen- 

 dent on the integral / .— . This almost casual remark 



J >Jl-x* 

 shows (as Jacobi long since observed) that Gauss, at the 



date of the publication of the " Disquisitiones Arithme- 

 ticae," had already examined the nature and properties of 

 the elliptic functions (the inverses of the elliptic integrals), 

 and had discovered their fundamental property, that of 

 double periodicity. This observation of Jacobi's is amply 

 confirmed by the papers on elliptic transcendents now 

 published in the third volume of Gauss's collected works. 



The "Disquisitiones Arithmetical " were to have in- 

 cluded an eighth section. This eighth section was at 

 first intended to contain a complete theory of congruences, 

 but subsequently Gauss appears to have proposed to con- 

 tinue the work by a more complete discussion of the 

 theory of the division of the circle. Manuscript drafts 

 on each of these subjects 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 allied to that subsequently Liken by Evariste 

 Galois and by MM. Serret and Dedekind. This draft 

 appears to belong to the years 1797 and 1798. 



To complete this hasty outline of the arithmetical 

 works of Gauss it only remains to mention (1) the remark- 

 able geometrical interpretation of the arithmetical theory of 

 positive binary and ternary quadratic forms, which will be 

 found in his review (1831) of the work of L. Seeber 

 (" YVerke," vol. ii. p. 188), and (2) the two important 

 memoirs on the theory of biquadratic residues (1825 and 

 1831). In the second of these memoirs Gauss introduces 

 into arithmetic complex numbers of the form a + bi. He 

 finds that in this complex theory every prime number of 

 the form \n + 1 is to be regarded as composite, because, 

 being the sum of two squares, e.g. p = a* -f- b", it is a product 

 of two conjugate factors, p = {a + bi) {a - bi). Thus 

 the true primes of the complex theory may be defined to 

 be the real primes of the form 4/1 -+- 3, and the imaginary 

 factors of real primes of the form pi -j- 1. Availing him- 

 self of this definition, Gauss discovered a theorem of 

 biquadratic reciprocity between any two prime numbers, 

 no less simple than the quadratic law, viz. "If p v and p 3 

 are two primary prime numbers, the biquadratic character 

 of/, with regard to p 2 is the same as that of p 2 with 

 regard to/,." 



Both this theorem of reciprocity itself and the intro- 

 duction of imaginary integers upon which it depends are 

 memorable in the history of arithmetic for the number 

 and variety of the researches to which they have given 

 rise. 



It may perhaps seem remarkable that Gauss should 

 have devoted so few memoirs to subjects of an algebrai- 

 cal character. If we except a comparatively unimportant 

 paper on Descartes' rule of signs which appeared in 

 Crc/lt-'s Journal in the year 1828, his only algebraical 

 memoirs relate to the theorem that every equation has a 

 root. Of this he gave no less than three distinct demon- 

 strations, one in 1799, one in 1S15, and one in 1S16 ; the 

 demonstration of 1799 was given in his first published 

 paper — his dissertation as a candidate for the degree of 

 Doctor of Philosophy in the University of Gottingen. 

 This demonstration he repeated over again in 1849, with 

 certain changes and simplification. There can be no 

 question that these three demonstrations are prior to 

 any other, though for various reasons those subsequently 

 given by Cauchy have been justly preferred for the 

 purpose of insertion in our modern text-books. 



ANTHROPOLOGY IN AMERICA 



WE cannot speak very highly of Prof. Otis T. Mason's 

 " Account of Progress in Anthropology in the Year 

 18S1," which was originally embodied in the Smithsonian 

 Report for that year, and is now issued in a separate 

 form. There is no comprehensive survey of the work 

 done in this wide field during the period indicated, and 

 the bibliography, of which the paper mainly consists, i 



