53^ 



NA TURE 



April 19, 1877 



Erariste Galois and by MM. Serret and Dedekind. This draft 

 appears to belong to the years 1797 and 1798. 



To complete our hasty sketch of the arithmetical works we 

 need only mention (i) the remarkable interpretation of the 

 arithmetical theory of positive binary and ternary quadratic 

 forms, which will be found in his review of the works of L. 

 Seebe'r [1831] ("Werke," vol. ii. p. 188) ; and (2) the two im- 

 portant memoirs on the theory of biquadratic residues (1825 and 

 1831). In the second of these memoirs he gives a theorem of 

 biquadratic reciprocity between any two prime numbers no less 

 important than the quadratic law, viz., " If /i and/j are two 

 primary prime numbers, the biquadratic character of p^ with 

 regard to/2 is the same as that of p^ with regard to p-^," This 

 theorem itself and the introduction 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.^ 



A writer remarks each work of Gauss is an event in the history 

 of science, a revolution, which, overturning the old theories and 

 methods, replaces them by new ones and advances science to a 

 height which no one had before dreamed of.^ We have gii-en 



proof of this in one branch of mathematics ; we shall see that the 

 witness is true as to other branches also. 



The discovery of the planet Ceres at Palermo on the first day 

 of the present century led to Gauss's taking up the subject of 

 astronomy. He did not come into possession of the requisite 

 data until the October following. In a few weeks he de- 

 termined the elements of its orbit with sufficient accuracy so 

 that the Baron de Zach was enabled to rediscover the planet at 

 the first attempt he made for that purpose on December 7. This 

 discovery was soon followed by that of three other small 

 planets. These discoveries supplied Gauss with the means of 

 further improving his solution of tlie problem, and in 1809 he 

 brought out at Hamburg his " Theoria motus corporum caeles- 

 tium in sectionibus conicis solem ambientium." This contains 

 an ' ' elaborate discussion of the various problems which present 

 themselves in the determination of the movements of planets 

 and comets from observations made on them under any circum- 

 stances." ' Gauss's other astronomical researches are chiefly con- 

 tained in De Zach's Monathche Correspondenz, the Transactions 

 of the Royal Society of Gottingen, and the Astronomische Nach- 

 richten ; all are contributions of the highest order of excellence. 



Gauss' Birthplace ia Brunswick. 



To astronomy Gauss joined geodesy, and the Hanoverian Govern- 

 ment charged him with the triangulation and measurement of an 

 arc of the meridian between Gottingen and Altona. This he 

 accomplished between the years 1821 and 1824. For carrying 

 out his purpose he invented many methods quite original. ^ It 

 was his intention to publish an extensive work upon geodesy, 

 but he did not accomplish his purpose. He contributed two 

 memoirs on the subject to the Royal Society of Berlin (1844, 

 1847). 



* In our account of Gauss's arithmetical work we have throughout reely 

 referred to Prof. H.J. S. Smith's presidential address (see above) and his 

 two reports on the theory of numbers (Brit. Assoc. Reports, 1859, pp. 

 228-267 ; i860, pp. 120-172). But we are still more deeply indebted to him 

 for references and criticisms most kindly given in the midst of the pressing 

 dauns of his other numerous engagements. 

 «< ^.^^Sener, in Michaud's " Biographic Universelle." Prot. Cayley writes. 



All that Gauss has written is first rate ; the interesting thing would be to 

 show the influence of his different memoirs in bringing to their present con- 

 dition the subjects to which they relate, but this is to write a History of 

 Mathematics from the year 1800." 



3 He invented the heliotrope to render angles visible at as g^-eat a dis- 

 tance as possible ; this he did by reflectmg the rays of the sun. He also 

 devised a method for the correction of the errors which occur in an exten- 

 sive system of triangulation.—" Gauss. Z. Ged.," pp. 51-53.; 



Mr. Todhunter in his " History of the Theories of Attraction," 

 devotes §§ 11 62-1 175 to an analysis of a memoir by Gauss, 

 *' Theoria attractionis corporum sphseroidicorum ellipticorum 

 homogeneorum methodo nova tractata" (Royal Society of Gottin- 

 gen, March 18, 1813). Mr. Todhunter says, "he completely 

 succeeds in his design ; his solution is both simple and elegant." " 

 He further remarks, ' ' Gauss's writings are distinguished for the 

 combination of mathematical ability with power of expression ; 

 in his hands Latin and German rival French itself for clearness 

 and precision." 



In another of Mr. Todhunter's works ("Calculus of Varia- 

 tions," 1861) he discusses in his third chapter (pp. 37-52) a memoir 



' Roy. Soc. Proceedings, p. 592 ; Roy. Ast. Soc. Monthly Notices (as above*. 

 A curious fact is recorded. The preface to this work is dated March 28, 1809, 

 just two centuries after Kepler's " Praefatio de Stella Martis," March 28. 

 1609; "Gauss. Z. Ged.," p. 40. After the publication of this work Gauss 

 became "a memher of all the learned societies from the Polar Circle to 

 the Tropics." 



2 Chasles (quoted by Todhunter) calls it "le beau m^moire de M. Gauss." 

 Another celebrated memoir, " AUgemeine Lehrsatze . . . Anziehungs- und 

 Abstossungs-krafte ' (Leipsic, 1840), is treated by Todhunter, § 1,253. In this 

 last Gauss uses the name Potential (apparently independent of Green) 

 § 790. See also Maxwell's " Electricity," § 70, 



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