?62 



NATURE 



[StPTEMUER 22, IQIO 



LETTERS TO THE EDITOU. 



rhe Editor does not hold himself responsible for opinions 



' expressed by his correspondents. Neither can he undertake 



to return, or to correspond with the writers oj, rejected 



nianuscrtpts intended jor this or any other fori o/ NATURE. 



No notice is taken oj anonymous communications.] 



Gauss and Non-Euclidean Geometry. 

 Probablv someonr will before this have directed your 

 attention to a statement in Nature of June 30 regard- 

 ing Gauss's share in the discovery of non-Euclidean geo- 

 metry • but in case this mav have escaped notice, even 

 after' the lapse of three months, 1 venture to bring it again 

 before your readers. Speaking of , Mannoury s book— 

 •• Metho'dologisches. und Philosophisches zur Elementar- 

 mathematik "— " G. B. M." says "there is one remark- 

 able statement made which deserves mention. Ur. 

 Mannourv says that in December, 1818, F. K. Schweikart 

 sent to Gauss' a note asserting the existence of a geometry 

 in which the sum of the angles of a triangle is less than 

 two. right' angles, and in which the ahitude of an isosceles 

 triangle with -a finite base has a finite upper limit. This 

 «oes' far to demolish the claim made for Gauss that he 

 was the first to assert the possibility of a consistent system 

 of geometrv distinct from Euclid." 



The story of Gauss and the non-Euclidean geometry will 

 probablv alwavs be incomplete, as he never published his 

 investigations 'on this subject, and what is known of them 

 has been gleaned from his correspondence and some notes 

 only recently found among his papers (cf. Gauss, 

 ••Werkc," Bd. viii., Leipzig, 1900). But neither Engel 

 nor Stackel— to whom we owe much of what has been 

 written on the theory of parallels— nor any of the other 

 writers on this phase of non-Euclidean geometry, have 

 asserted that Gauss ever published any statement of his 

 theory, large or small. The most that has been claimed 

 for Gauss is that before Lobatschewsky, in 1826, and 

 Bolyai, in 1832, published their statement of the geometry 

 which vyill alwavs be associated with their names, also 

 even before Schweikart in 1818 had drawn up the note to 

 which reference is, made above. Gauss himself was con- 

 vinced of the logical possibility of a geometry independent 

 of the fifth postulate, and had mentioned many of his 

 conclusions to his friends, verbally or in writing. 



What happened with reference to Schweikart is well 

 known. The whole story is to be found in Gauss's letter 

 of 1819 to Gerling, by whom the memorandum had been 

 submitted to Gauss at the request of the author. Like 

 the subject of a recent political controversy, it could be 

 written on half a sheet of notepaper ; and it called forth 

 from Gauss the warmest praise. With it he fully agreed. 

 In fact, his results were exactly the same as those he 

 had already obtained. His own work, he added, he had 

 developed so far as to have fully solved all the problems 

 of the new geometry. Some of his results he sent to 

 Oerling to be communicated to Schweikart himself. _ 



It is not of much importance whether before this date 

 we have any reference to these investigations ; but such 

 is actually forthcoming in Wachter's letter to Gauss two 

 years earlier, where he speaks of their conversation at 

 Gottingen, and wonders " whether the anti-Euclidean geo- 

 metrv or your geometry is true." 



.'Vnd more valuable, as showing Gauss's real position, is 

 his well-known letter to Wolfgang Bolyai in 1832, when 

 Tie had received from him a copy of Johann's famous 

 work: — " If I begin by saying that I cannot praise this 

 Avork [of Johann's] you will assuredly be surprised for a 

 moment. But I cannot say anything else. To praise it 

 would be to praise myself.' In fact, the whole contents 

 of this work, the path' which your son has followed, and 

 the results to which he has been led, agree almost com- 

 pletely with my own meditations on this subject, some of 

 them as old as thirty to thirty-five years." 



This is but one of several statements of the same kind 

 -which we find in the correspondence of Gauss now avail- 

 able. Still, he would have been the last person to assert 

 any claim for himself in the matter. Indeed, it was " a 

 ■very great pleasure to him that it was actually the son 

 ■of 'his old friend who had made this advance upon him 

 in such a remarkable fashion." Yet there is ample 

 evidence that the ideas contained in Schweikart's memor- 



andum were already known to him, and that with much 

 of the work of Lobatschewsky and Bolyai he was familiar 

 long before they themselves had made these discoveries. 

 To them belongs the independent discovery of their geo- 

 metry, and its complete and systematic development. By 

 their names it will always be called. To Schweikart, to 

 a small extent, to Gauss to a much larger, can be given 

 the credit of having realised that, along the path which 

 Lobatschewsky and Bolyai travelled, complete success was 

 bound to be achieved. H. S. Carslaw. 



The University, Sydney, August 10. 



An Oblique Belt on Jupiter. 



Owing in the main part to the swift axial rotation of 

 the olanet luniler it is usual to find the dark belts, which 

 constitute the%rincipal configuration of his visible surface 

 lying both parallel to one another and to the planet s 

 equator. An instance of obliquity of one of the bands 

 relatively to the others is rare, and a most definite and 

 striking example of the kind was recorded in the northern 

 hemisphere in i860. A recent phenomenon akin to this 

 was observed in the spring months of the present year. 

 Although in this case the band was a faint one, yet the 

 marked trend which it exhibited called for special notice 

 and the more important facts relating to it might be briefly 

 recorded here. ■ . , . ^ ,■„ 



It attracted my attention, when engaged in a systematic 

 study of the planet, first on April i, and was subsequently 

 observed on the following nights -.-April 6, 8.. ^3, ^8. 

 Mav 2 and 7. After the last-mentioned date it was not 

 seen again, partly on account of its growing faintness, and 

 partly because ihe prevailing telescopic seeing was not 

 fnducive to a close scrutiny of the planet. During this 

 observed interval a number of careful drawings of, the 

 recion in question were executed, as well as a series of 

 central-meridian transits of spots situated in and around 

 the slanting belt. Some of these spots had been watched 

 several months', prior to the appearance of this belt, so 

 that the rotational velocity of the surface matter in this 

 particular region was, on the , whole, fairly accurately 



'"'Thl'^oMique bolt, which was a new formation, stretched 

 itself across the white zone between the S. temperate and 

 S S. temperate belts. Nowhere, however, did it coalesce 

 with these two belts, a point which can be better under- 

 stood from the accompanying drawings than from a ae- 

 scription alone. Its separate ^^'^^^<''^?'^ '"''"^J"?- .^'''^^J^^ 

 to .a repulsive action exerted upon it by the belts, which 

 seemed to form a barrier against any further displacement 



'"The region of the oblique belt could always be readily 

 recognised, even under poor definition, by reason of an 

 abnormal dark patch of matter which occupied the site 

 where the oblique bolt crossed over the central part ot tne 

 zone in which it was situated. This patch presented a 

 concave outline both east and west, and the oblique belt 

 passed uninterruptedly through it. A white spot (c) pre- 

 ceded it, and a fainter one was at times seen on the 

 following side. This curious patch became visible earlier 

 than did the oblique belt, and its greater durability enabled 

 it to remain in view long after the belt had ceased to be 



^' The' quicker drift of the spots A and B relatively to E, 

 F and G will be noticed in the drawings. The dark 

 natch, with its condensation D, was carried along at about 

 the same rate of velocity as A and B, and all, therefore, 

 participated in one and the same current 1 he white 

 spot C drifted at the .same rate as the spots E, b, G ana, 

 as will be noted, it was being gradually overtaken by tne 

 dark patch. Thus we observe the relative movements ot 

 two independent currents. They disclo.sed the noteworthy 

 fact that the dark patch was in reality a distended part ot 

 the current about A and B. having evidently forced its 

 way northwards across the slower current round C to tne 

 spots E F, G. The condensation D formed part of the 

 oblique belt. Whether the rest of this belt particip.atcd in 

 the quick current of A and B is not known; but if sucn 

 was really the case, we have here at least a clue as to the 

 cause of the curious trend of the belt. The material of a 

 1 Opp"iit on of Jupiter occurred on March 31, 1910. 



NO. 2134, VOL. 84] 



