272 



NA TURE 



[July 19, 1883 



octospores ; but he failed to interpret their true signifi- 

 cance as reproductive organs, and laid down his pen 

 under the firm conviction that the cystocarpic fruit was 

 entirely absent in Porphyra. Thuret's representation of 

 this kind of fruit proved that Janczewski was mistaken. 

 Dr. Berthold mentions that he was fortunate enough to 

 obtain by his researches at the Zoological Station at 

 Naples satisfactory proof that the reproductive processes 

 in the Bangiaceas correspond exactly with those of the 

 other Florideae; he further states (p. 21) that they are 

 true Floridea?, but that they undoubtedly occupy the very 

 lowest position in the class. 



The first part of the work describes at some length the 

 structure of the vegetative thallus of each of the three 

 genera. A minute description follows of the organs of fruc- 

 tification, namely, the tetraspores, cystocarps, and antheri- 

 dia, and of the mode in which the cystocarps are fertilised. 

 The fructification of all the genera is illustrated by a 

 plate containing twenty-five figures. We have then an 

 account of the germination of the spores and of their 

 development into plants, followed by observations on the 

 systematic position occupied by the Bangiaceas and their 

 relation to the Chlorosperms. To these are added de- 

 scriptions of the genera and species, with a notice of the 

 habitat and time of appearance of the several species. 



This very interesting work concludes with some remarks 

 on Goniotrichum, and short descriptions of the two spe- 

 cies G. elegans {Bangia elegans of the " Phyc. Brit.," PI. 

 ccxlvi.) and G. dichotomutn. Mary P. Merrifield 



GA USS AND THE LA TE PROFESSOR SMITH 

 TN the centenary notice of Gauss (Nature, vol. xv. 

 PP- 533-537) I more than once refer to notes placed 

 in my hands by the late Prof. Henry Smith. These took 

 the form of two MSS. (A), (B). The (firmer of these I 

 used in its entirety (p. 537), the latter I withheld, with 

 Prof. Smith's sanction, on account of the length to which 

 the article had already extended. Many mathematicians 

 may now like to read these further criticisms on Gauss by 

 such a kindred genius. R. Tucker 



We proceed to give brief references to some of 

 the most important points which have caused a new 

 epoch in certain branches of analysis to date from 

 the publication of the " Disquisitiones Arithmetical," 

 and from the researches with which, some years later, 

 Gauss supplemented or further developed the theories 

 contained in that work. It may be proper to premise 

 that Gauss found the theory of numbers as Euler and 

 Lagrange had left it. Of these the former had 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 indeterminate equa- 

 tions of the second degree. Legendre (with many addi- 

 tions 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 Thdorie des 

 Nombres." But the " Disquisitiones Arithmetics" was 

 in the press when this important treitise appeared, and 

 what in it was new to others was already known to 

 Gauss. 



The first section of the "Disquisitiones," " De Nume- 

 roium Augmentia in genere," occupies hardly more than 

 four pages of the quarto edition, and is of the most ele- 

 mentary character. Nevertheless, the definition and the 

 elementary properties of a congruence, which were for the 

 first time given in it, have exercised an immense influence 

 over all the branches of the higher arithmetic ; an in- 

 fluence which is perhaps surprising when we remember 

 that it is a question of notation only, and that (as Gauss 



has said himself in a letter to Schumacher) nothing can 

 be done with this notation which cannot (though less 

 conveniently) be done without it. 



The second section, " De Congruentiis Primi Gradus," 

 contains applications of the definition and of the ele- 

 mentary properties of congruences to linear congruences, 

 and to systems of such congruences. The problems solved 

 in it are of an elementary kind, and may be regarded as 

 either well known, or as lying within the scope of what 

 was well known, at the time of the publication of the 

 " Disquisitiones Arithmetical." 



The same remark applies to the third section, " De 

 Residuis Potestatum," which, notwithstanding the im- 

 mense advantage of clearness and simplicity obtained by 

 the use of the congruential notation, may be said to lie 

 almost wholly within the aid of ideas to be found in 

 Euler's memoirs. The demonstration of the existence of 

 primitive roots (a demonstration which Euler had failed 

 in rendering rigorous), is, however, a very noticeable 

 exception. 



The fourth section — "De Congruentiis Secundi Gradus" 

 — opens with an exposition of the elementary theorems 

 relating to quadratic residues and non-residues ; and so 

 far we are still entirely within the ground already 

 occupied by Euler. But the greater part of this section is 

 occupied with a research which of itself alone would have 

 placed Gauss in the first rank of mathematicians. " If/ 

 and q are positive uneven prime numbers, p has the 

 same quadratic character with regard to p that q has 

 with regard to p, except when / and q are both of the 

 form 4;/ -f- 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 Legendre. 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 Mimoires of the Academy of Paris for 1784. But 

 the question to whom the priority of the enunciation is 

 due is of even less moment than questions of priority 

 usually are ; for the discovery of the theorem by induc- 

 tion was easy, whereas any rigorous demonstration of it 

 involved apparently insuperable difficulties. Gauss was 

 not content with vanquishing these difficulties once for all 

 in the fourth section. In the fifth section he returns to 

 it again, and obtains another demonstration reposing on 

 entirely different, but perhaps still less elementary, prin- 

 ciples. In January of the year 1808 he submitted a third 

 demonstration to the Royal Society at Gottingen ; a fourth 

 in August of the same year ; a fifth and sixth in February, 

 1 8 17. It is no wonder that 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 

 substantially from one another and from those of Gauss, 

 have been given by Jacobi and Eisenstein in Germany, 

 and by M. Liouvillc in France, the simplest of all per- 

 haps being that which has been given by a Russian 

 mathematician, M. Zeller, and which is of the same 

 general character as the third proof of Gauss (see Mes- 

 senger of Mathematics, vol. v. pp. 140-3, 1S76). It would 

 certainly be impossible to exaggerate the important influ- 

 ence which this theorem has had on the subsequent 

 development of arithmetic, and the discoveiy of its 

 demonstration by Gauss must certainly be regarded 

 (indeed it was so regarded by himself) as one of his 

 greatest scientific achievements. 



The fifth section — " De Formis yEquationibusque Inde- 



