8U 



SCIENCE. 



[N. S. Vol. XX. No. 520. 



In this part the fundamental question is : 

 Tfhat is the nature of the axioms of geom- 

 etry? Our author's views may be seen in 

 the following quotations : 



The axioms of geometry are neither syiithetie 

 judgments h priori, nor' experimental facts. They 

 are oonventions: our clioiee among all possible 

 conventions is guided by experimental facts, but 

 it remains free and is limited only by the neces- 

 sity of avoiding all contradiction. Hence the 

 postulates can remain rigorously true even though 

 the e.xperimental laws which have determined 

 their adoption are only approximative. 



In other words, the axioms of geometry (I am 

 not speaking of those of arithmetic) are merely 

 disguised definitions. Consequently the question : 

 ' Is Euclidean geometry true ? ' has no mean- 

 ing. As well ask whether the metric system is 

 true and the old measures false, whether Cartesian 

 coordinates are true and polar coordinates false. 

 One geometry can not be more true than another, 

 it can only be more convenient. 



Euclidean geometry is and will remain the 

 most convenient : 



1. Because it is the simplest; and it is so 

 not only in consequence of our mental habits, or 

 of I know not what direct intuition we may have 

 of Euclidean space, but it is the simplest in itself, 

 just as a polynomial of the first degree is simpler 

 than one of the second. 



2. Because it accords well with the properties 

 of natural solids. 



Beings with minds and senses like ours, but 

 who had received no previous education, might 

 receive, from an external world suitably chosen, 

 impressions such that they would be led to con- 

 struct a geometi'y other than that of Euclid and 

 to localize the phenomena of that external world 

 in a non-Euclidean space, or even in a space of 

 four dimensions. 



If, on the other hand, we whose education 

 has been received in our actual world were sud- 

 denly transported into this new world, we should 

 have no difficulty in relating its phenomena to our 

 Euclidean space (pp. 66-8). 



If the geometry of Lobatseheffsky is true, the 

 parallax of a very distant star would be finite; 

 if that of Eiemann is true, it would be negative. 

 These are results which seem within the reach of 

 experiment, and there have been hopes that 

 astronomical observations might enable us to de- 

 cide between the three geometries. 



But in astronomy ' straight line ' means sim- 

 ply ' path of a luminous ray.' If, to suppose the 



impossible, negative parallaxes were found, or if 

 it were demonstrated that all parallaxes are 

 superior to a certain limit, two courses would 

 be open; either we could renounce Euclidean 

 geometry, or we could modify the laws of optics 

 and admit that light does not travel rigorously 

 in a straight line. It is useless to add, that every 

 one would regard the latter as the more ad- 

 vantageous, Euclidean geometry has nothing to 

 fear from new experiments (p. 93). "No ex:- 

 perience will ever contradict the postulate of 

 Euclid, nor will any ever contradict that of 

 Lobatseheffsky" (p. 95). 



The third part, devoted to force, consists of 

 chapters dealing with ' Classic mechanics,' 

 ' Relative movement and absolute movement ' 

 and ' Energy and thermodynamics.' 



Here, as in geometry, our author finds that 

 the fundamental principles are neither a priori 

 truths nor experimental, facts but convenient 

 definitions or conventions. 



If the principle of inertia, for example, 

 were an a priori truth, how could the Greeks 

 believe that movement ceases as soon as the 

 cause which originated it ceases to act ? How 

 could they believe that every body free from 

 constraint would move in a circle, the noblest 

 of all motions? 



Is there any more warrant to say that the 

 velocity of a body can not change without 

 cause for the change, than that it can not 

 change its position or the curvature of its 

 path except under the influence of an exterior 

 cause ? 



Have any experiments ever been made on a 

 body subject to no force, and if so how was 

 it known that no force was acting? A sphere 

 rolling on a marble table for a very long time 

 is a usual example, but has the force of 

 gravity ceased to act? 



Can the law that the acceleration of a hody 

 equals the force acting on it divided hy its 

 mass be verified experimentally? To do so 

 the acceleration, the force and the mass must 

 be measured. If we overlook the difficulties 

 connected with the measurement of time, it 

 may be granted that the acceleration can .be 

 measured, but there are inextricable difficulties 

 in the definition of mass and force. Useful 

 definitions must teach how to measure mass 

 and force, and require definition of the 



