April i(), 1877] 



NATURE 



537 



by Gauss entitled " Princlpia gtneralia Theorise figurae Fluido- 

 rum in statu aequilibrii " (Royal Society of Gottingen, 1833).^ It 

 relates to the theory of capillary attraction, and demonstrates 

 in a new way some results which had already been obtained by 

 Laplace. The part analysed by Mr. Todhunter is that devoted 

 to the solution of a problem in the calculus of variations, "in- 

 volving the variation of a certain double integral, thelimits of the 

 integration being also variable ; it is the earliest example of the 

 solution of such a problem." In 1831 we find Gauss com- 

 mencing the study of crystallography ; in a few weeks he had 

 mastered the subject. We find that the question of the rational- 

 ity or irrationality of the ratios of the crystallographic coefficients 

 had attracted his attention.^ 



We can only touch upon Gauss's further contributions to 

 {;eometry.^ To him are due many fundamental theorems in the 

 theory of curve-surfaces ; also on the development of surfaces ; 

 thus it was he who found the equation to developable surfaces. 

 He was used to say " that he had laid aside several questions 

 which he had treated analytically, and hoped to apply to them 

 geometrical methods in a future state of existence, when his 

 conceptions of space should have become amplified and ex- 

 tended." ■* 



Those not acquainted with Gauss's writings would think we 

 must have exhausted our account of them. In 1831, however, 

 on Weber's arrival at Gottingen, physical questions took the 

 first place in Gauss's thoughts, and separately and in conjunction 

 many works were brought out by these two philosophers. There 

 is so full an account of Gauss's achievements in this direction in 

 the Royal Society's Obituary Notice, that we need only refer to 

 it." His contributions, we may briefly say, to the knowledge of 

 electro-magnetism and terrestrial magnetism were perhaps the 

 most considerable and important of his achievements. He in- 

 vented the magnetometer, and was one of the first to point out 

 the possibility of sending signals by galvanic currents, and so 

 contributed to the invention of the electric telegraph. 



'^ " If we except the great name of Newton (and the exception 

 is one which Gauss himself would have been delighted to make) 

 it is probable that no mathematician of any age or country has 

 ever surpassed Gauss in the combination of an abundant fertility 

 of invention with an absolute rigorousness in demonstration, 

 which the ancient Greeks themselves might have envied. It 

 may be admitted, without any disparagement to the eminence of 

 such great mathematicians as Euler and Cauchy that they were 

 so overwhelmed with the exuberant wealth of their own creations, 

 and so fascinated by the interest attaching to the results at which 

 they arrived, that they did not greatly care to expend their time 

 in arranging their ideas in a strictly logical order, or even in 

 establishing by irrefragable proof propositions which they instinc- 

 tively felt, and could almost see to be true. With Gauss the 



case was otherwise It may seem paradoxical, but it is 



probably nevertheless true that it is precisely the effort after a 

 logical perfection of form which has rendered the writings of 

 Gauss open to the charge of obscurity and unnecessary difficulty. 

 The fact is that there is neither obscurity nor difficulty in his 

 •writings, as long as we read them in the submissive spirit in 

 •which an intelligent schoolboy is made to read his Euclid. 



• Read Sept. 28, 1829. 



' See Gauss's review of Seeber's Untersuchungen iiber die Eigenschaften 

 der positiven ternaren quadratischen formen" in the Gottingen gclehrte 

 Anzdgen (1831) or Crcltc, vol. xx. p. 312. Prof. H. J. S. Smith " On the 

 Conditions of Perpendicularity in a Parallelopipedal System." (London Math. 

 Society's Procccdins^s, December, 1876). His method of drawing the crystals 

 w.-is essentially the same as that devised subsequently by Prof. Miller, of 

 ^Cambridge. "Gauss. Z. Ged..' p. 61. 



3 Disquisitiones generales circasuperficies curvas(7'r««.rrtc//o«.r, Gotiingen, 

 1827). 



' * " Gauss. Z. Ged.," p. 81, quoted by Prof Sylvester (?</'/ w/m). Gauss s 

 onnection with the so-called Gaussian logarithms is pointed out on p. 75 of 

 be Report of the Committee on Mathematical Tables (Brit. Assoc, 1873.) 

 Reporter, Mr. J. W. L. Glaisher. 

 5 "Gauss, as a member of the German Magnetic Union, brought his power- 

 Ill intellect to bear on the theory of magnetism and on the methods of 

 ^serving it, and he not only added greatly to our knowledge of the theory 

 [attractions, but reconstructed the whole of magnetic science as regards 

 c instruments used, the methods of observation, and the calculation of the 

 ^sults, so that his memoirs on Terrestrial Magnetism may be taken as 

 ttodels of physical research by all those who are engaged in the measure- 

 Bent of any of the forces in nature."— Prof. Clerk-Maxwell's " Electricity 

 dd Magnetism " (1873), p. viii. We may also refer for a statement of some 

 J Gauss's discoveries to §§ 14c, 144, 409, 421, 454, 706, and 744. Cf also 

 Prof. Maxwell's Address (Brit. As^oc , Liverpool, 1870). Pp. 594-598 for 

 Accounts of the memoir " Intensitas vis magneticae terrestris ad mensuram 

 absolutam revocata" (1832) and of the Theory of the Earth's Magnetism 

 ,(1830) : " Allgemeine 'Iheorie des Eidmagnetismus. ., ■ n r 



■6 We quote freely fiom notes placed at our service for this articl<? by Prof. 

 I. J. S. Smith. " Summus Newton," "Gauss. Z. Ged.," p. 84. 



Every assertion that is mafle is fully proved, and the assertions 

 succeed one another in a perfectly just analogical order ; there is 

 nothing so far of which we cin complain. But when we have 

 finished the perusal, we soon begin to feel that our work is but 

 begun, that we are still standing on the threshold of the temple, 

 and that there is a secret which lies behind the veil and is as yet 

 concealed from us .... no vestige appears of the process by 

 which the result itself was obtained, perhaps not even a trace of 

 the considerations which suggested the successive steps of the 

 demonstration. Gauss says more than once that, for brevity, he 

 only gives the synthesis, and suppresses the analysis of his pro- 

 positions. ^ Patica sed matttra' y/treiht words with which he 

 delighted to describe the character which he endeavoured to 



impress upon his mathematical writings If, on the other 



hand, we turn to a memoir of Euler's, there is a sort of free and 

 luxuriant gracefulness about the whole performance, which tells 

 of the quiet pleasure which Euler must have taken in each step 

 of his work ; but we are conscious nevertheless that we are at an 

 immense distance from the severe grandeur of design which is 

 characteristic of all Gauss's greater efforts. The preceding criti- 

 cism, if just, ought not to appear wholly trivial ; for though 

 it is quite true that in any mathematical work the sub- 

 stance is immeasurably more important than the form, yet 

 it cannot be doubted that many mathematical memoirs of 

 our own time suffer greatly (if we may dare to say so) 

 from a certain slovenliness in the mode of presentation ; and 

 that (whatever may be the value of their contents) they are 

 stamped with a character of slightness and perishableness, 

 which contrasts strongly with the adamantine solidity and 

 clear hard modelling, which (we may be sure) will keep the 

 writings of Gauss from being forgotten long after the chief results 

 and methods contained in them have been incorporated in trea- 

 tises more easily read, and have come to form a part of the 

 common patrimony of all working mathematicians. And we 

 must never forget (what in an age so fertile of new mathematical 

 conceptions as our own, we are only too apt to forget), that it is 

 the business of mathematical science not only to discover new 

 truths and new methods, but also to establish them, at whatever 

 cost of time and labour, upon a basis of irrefragable reasoning. 



" The fxadr^fiaTLKos TndavoKoywv has no more right to be listened 

 to now than he had in the days of Aristotle ; but it must be 

 owned that since the invention of the ' royal roads ' of analysis, 

 defective modes of reasoning and of proof have had a chance of 

 obtaining currency which they never had before. It is not the 

 greatest, but it is perhaps not the least, of Gauss's claims to the 

 admiration of mathematicians, that, while fully penetrated with 

 a sense of the vastness of the science, he exacted the utmost 

 rigorousness in every part of it, never passed over a difficulty, as 

 if it did not exist, and never accepted a theorem as true beyond 

 the limits within which it could actually be demonstrated." 



It will be evident to our readers that this notice has been 

 drawn up with a purpose. The town of Brunswick proposes to 

 celebrate the hundredth anniversary of Gauss's birthday, ar.d the 

 committee hope to have received before the 30th instant, suffi- 

 cient subscriptions to enable them to lay the foundaticm stone of 

 a memorial statue. We have endeavoured to present in a strong 

 light 1 the claims which this great mathematician has upon 

 mathematicians, not only in Germany, but on mathematicians 

 in this country.'-^ 



Gauss might himself have considered his works his best 

 monument (" exegi monumentum aere perennius,") and possibly if 

 sufficient funds flow in, the committee might see their way to 

 the bringing out a centenary edition of them. In this way they 

 would confer a great boon upon mathematicians everywhere, for 

 at present his writings are, as our great mathematical historian 

 writes, "very costly."^ R.Tucker 



I Our task has given us much pleasure ; it has been accomplished in the 

 midst of many interruptions. All our authorities have been given. We 

 close, as the author of the Book of Maccabees closes, with saying : If 

 I have done well and as is htting the story, it is that which I desired ; but 

 if slenderly and meanly, it is that which I could attain unto." Subscriptions 

 may be sent to the office of Nature up to the 26th instant. 



- ' Sein Tod wird nicht allcin in alien deutschcn Laiiden sondern auch 

 unter alien gebildeten Nationen der Welt die tiefste Trauer erzeugen." 

 Gelchrtc Anzcigm, No. 16, December 3, 1855. 



3 Carl t'rieclrich Gauss' Werke, Herausgegeben von der koniglichen 

 Cesellschaft der Wissenschaften zu Giktingen : vol. 1. ii. 1863, vol in. i8e6, 

 vol iv. 1873, vol. V. 1867, vol. vi. 1874, vol. vii. 1871. These are all wc 

 have seen. Che editor is Schering. Ihe house in which Gauss was born 

 bore the number 1550, and was situated on the west side of the Wenden- 

 graben. It now h.-is a memorial tablet. The house was sold in 1804, and 

 the family removed to another in the Miihlenstrasse, near St. Giles s 

 Church. "Z. Ged.," p. 8. 



