816 



SCIENCE. 



[N. S. Vol. IX. No. 232. 



how can we couclude therefrom that FE = AB 

 and the angle EFC = J - ? It is not possible to 

 mend the false deduction, wherein Legendre's 

 inadvertence was so gross that, without remark- 

 ing this grave en-or, he considered his demon- 

 stration as very simple and perfectly rigorous." 



Now for a specimen of Lobachevski's philos- 

 ophizing : " Strictly we cognize in nature only 

 motion, without which sense impressions are 

 not possible. Consequently all other ideas, for 

 example, geometric, are artificial products of 

 our mind, since they are taken from the proper- 

 ties of motion ; and, therefore, space in itself, 

 for itself alone, for us does not exist. 



Accordingly it can have nothing contra- 

 dictory for our mind if we admit that some 

 forces in nature follow the one, others another 

 special geometry. 



To illustrate this thought, assume, as many 

 believe, that the attractive forces diminish 

 because their action spreads on a sphere. In 

 the ordinary geometry we find 4-j-^ as magni- 

 tude of a sphere of radius r, whence the force 

 must diminish in the squared ratio of the dis- 

 tance. 



In the imaginary (sic) geometry I have found 

 the surface of the sphere equal to 



and possibly in such a geometry the molecular 

 forces may follow, whose whole diversity would 

 depend, consequently, on the number e, always 

 very great." 



How far Lobachevski was, not only from 

 Riemann's geometry with closed finite straight 

 line, but also from the perspective point of 

 view where the straight is closed by having 

 only one point at infinity, is illustrated by the 

 following sentences of the introduction. "I 

 consider it not necessary to analyze in detail 

 other assumptions, too artificial or too arbi- 

 trary. Only one of them yet merits some atten- 

 tion — the passing over of the circle into a 

 straight line. However, the fault is here visible 

 beforehand in the violation of continuity, when 

 a curve which does not cease to be closed, how- 

 soever great it may be, transforms itself directly 

 into the infinite straight, losing in this way an 

 essential property. 



In this regard the imaginary geometry fills in 



the interval much better. In it, if we increase 

 a circle all of whose diameters come together 

 at a point, we finally attain to a line such that 

 its normals approach each other indefinitely, 

 even though they can no longer cut one an- 

 other. This property, however, does not ap- 

 pertain to the straight, but to the curve which 

 in my paper ' On the Elements of Geometry' I 

 have designated as circle-limit.''^ 



Lobachevski anticipated in 1835 all that was 

 said not long ago in the columns of Science on 

 the length of a curve. For example : "In fact, 

 however little may be the parts of a curve, they 

 do not cease to be curves ; consequently they 

 can never be measured by the aid of a straight. ' ' 



' ' Lagrange takes as foundation the assumption 

 of Archimedes that on a curve one can always 

 take two points so near that the arc between 

 them may be considered greater than its chord, 

 but smaller than the two tangents from its ex- 

 tremities. Such an assumption is actually 

 necessary, but by it is destroyed the primitive 

 idea of measuring curves with straights. Thus 

 the evaluation of the length of a curve represents 

 not at all the rectification of the curvature ; but 

 it seeks a wholly different aim — the finding of a 

 limit which the actual measure would approach 

 the more as this measure was made the more 

 exact. But measuring is considered more ex- 

 act the smaller the links of the chain employed. 

 This is why in geometry one must show that 

 the sum of tangents decreases while the sum 

 of chords increases until the two sums dif- 

 fer indefinitely little from the limit both ap- 

 proach, which geometry assumes as length of 

 the curve." 



In the splendid treatise which follows this in- 

 teresting introduction Lobachevski has given a 

 complete coherent development and exposition 

 of the non-Euclidean geometry. Until I visited 

 Maros-VAsilrhely it was not known that Bolyai 

 J4nos had actually commenced and made re- 

 markable progress in an even greater, more 

 masterful treatment of the whole matter. From 

 the mass of John's papers tumbled in a big 

 chest I singled out especially a manuscript in 

 German entitled 'Raumlehre,' and on pointing 

 out to Professor Bedohazi J&nos some of the 

 striking passages in it he promised its publica- 

 tion. 



