VIU 



Siipplement to Nature, March 21, 1901. 



really existed. ^ It is to be regretted that before making 

 the statement (vol. i. p. 219) that a pair of gemsbok 

 horns obtained by Oswell, which measure 44 inches, are 

 the longest but one on record the author did not consult 

 some recent works on horn-measurements. Had he done 

 so, he would have found that these dimensions are ex- 

 ceeded by five examples, the "record" length being 

 475 inches. 



Of the profusion of great game in Central South Africa 

 in Oswell's time, and of its subsequent extermination, the 

 story has been told so often that its recapitulation here 

 would be superfluous. Whether Oswell and his suc- 

 cessors were altogether free from blame in regard to in- 

 discriminate slaughter is a question which it is not our 

 province to answer on this occasion. He himself wrote, 

 somewhat pathetically, in after years as follows : " I am 

 sorry now for all the fine old beasts I have killed, but I 

 was young then, there was excitement in the work ; I had 

 large numbers of men to feed ; and if these are not 

 sound excuses for slaughter, the regret is lessened by the 

 knowledge that every animal I shot, save three elephants, 

 was eaten by man, and so put to a good use." 



That Oswell was a gallant and noble-hearted gentle- 

 man, as well as a true sportsman, will, we think, be the 

 verdict of all who read a very attractive, albeit in certain 

 jrespects a somewhat saddening, book. R. L. 



THE WORKS OF C. F. GAUSS. 

 Carl Friedrich Gauss Werke. Achter Band. Pp. 458. 

 (Leipzig: Teubner, 1900.) 



THE contents of this volume consist mainly of 

 reviews, correspondence and a series of posthu- 

 mous fragments relating to various branches of pure 

 mathematics. It not unfrequently happens that a 

 mathematician's unfinished essays, only brought to light 

 after his death, are even more stimulating than the 

 finished works which he published during his life. Thisi 

 is especially true of Gauss. As he says himself in one of 

 his letters, he aimed at the utmost perfection of form ; 

 he worked, as it were, in marble, and did not give his 

 masterpieces to the world until they had been elaborated 

 to the highest degree of symmetry and polish. The 

 result is that, when we study the productions of his 

 genius, we feel a kind of awed admiration which is not 

 entirely free from a sense of chill ; his stately synthesis 

 compels our assent, but does not always attract our 

 sympathy, and rarely suggests the idea of even the 

 humblest emulation. Gauss's own countryman, Jacobi, 

 in his lectures on the theory of numbers, refers with a 

 touch of bitterness to the frozen austerity of Gauss's 

 demonstrations. 



Much, then, as we may deplore the fact that Gauss's 

 official duties, as well as his fastidious habit of com- 

 position, prevented him from working out the details of 

 so many of his great ideas, there is a certain consola- 

 tion in being able to see, in the notes which have been 

 preserved, some traces of the inception of his far-reach- 

 ing discoveries. There is no need now to emphasise the 

 extraordinary way in which he anticipated many of the 

 most important results obtained by other men. We can 



1 Since the review was written Dr. Matschie, of Berlin, has proposed to 

 split up the African elephant into several so-called species. 



NO. 1638. VOL. 63] 



understand the suspicion and incredulity with which his 

 claims were regarded by some of his contemporaries ; 

 but no one can doubt the extent and independence of 

 his researches in the theories of elliptic functions, of 

 modular functions, and of non-Euclidean geometry, not 

 to mention other things. It must have been very trying 

 to him to see so much of his work forestalled, so far 

 as priority of publication was concerned ; and on the 

 whole it may be said that he betrayed less disappoint- 

 ment than might have been expected. It is true that in 

 this volume there are letters of a rather controversial 

 kind, in which he emphatically claims priority in the dis- 

 covery of the method of least squares ; but he refers 

 appreciatively to Legendre's work on cometary orbits, 

 adding pathetically, 



" Es scheint mein Schicksal zu sein, fast in alien 

 meinen theoretischen Arbeiten mit Legendre zu con- 

 curriren." 



And in writing to Bolyai (a friend, it is true) about the 

 non-Euclidean geometry, after saying that his many 

 engagements prevent him from thinking of the subject 

 for the present, he continues : 



" Es soil mich herzlich herzlich freuen, wenn Du mir 

 zuvorkommst, und es Dir gelingt alle Hindernisse zu 

 iibersteigen. Ich wiirde dann mit der mnigsten Freude 

 alles thun, um Dein Verdienst gelten zu machen und 

 ins Licht zu stellen, so viel in meinen Kraften steht." 



The letters and notes on the foundations of geometry 

 form one of the most interesting sections of this volume. 

 It is clear that before the end of 1799 Gauss had critic- 

 ally examined the theory of parallels, and begun to doubt 

 its necessary truth : he says that he could prove the 

 whole of geometry if it could be shown that a rectilinear 

 triangle is possible, the area of which exceeds that of any 

 given surface ; but that, far from assuming this as an 

 axiom, he thinks it may be possible that, however great 

 the sides of the triangle are taken, the area is less than a 

 certain fixed quantity. He returned to the subject from 

 time to time during many years. It appears from a letter 

 of Gauss to Gerling, dated March 16, 18 19, that he had 

 then obtained some of the principal results in that 

 species of geometry in which the sum of the angles of a 

 rectilinear triangle is less than two right angles, for he 

 gives the formula 



irCC 



" Limes arete triansjuli plani = , — ; — - " 



^ ^ {log hyp (I .+ ^2)}" , 



where C is a certain constant determined by the space 

 under consideration. Gauss refers again to this constant 

 in a letter of November 8, 1824, and speaks of it as "a 

 definite (an sich bestimmte) linear magnitude existing in 

 space, although unknown to us." This is not very clear, 

 and the context does not enlighten us, in spite of its 

 criticism of the philosophers. The fact is that Gauss 

 nowhere (apparently) in these fragments expresses him- 

 self with perfect clearness about this constant. He had 

 arrived at the notion of an upper limit to the area of a 

 triangle ; but he never suggests the possibility of an 

 upper limit to the distance between two points, and, in 

 fact, assumes that there is none. Moreover, he dismisses 

 without sufficient consideration the question of how such 

 a thing as an absolute distance is conceivable, and ignores 

 the fundamental difficulties which beset the application of 

 number to measurement. It is very remarkable that so 



