.314 



NA TURE 



[February i, 1906 



It is time to refer more particularly to the contents 

 of the book. The volume opens with a chapter on the 

 nature of mathematical reasoning:, which is shown 

 to be contained in a power of generalisation dependent 

 on recurrent reasoning. When we have proved a 

 theorem for one number, and show that if true for 

 any number a, it is also true for a number a+i, we 

 may asserl il to be true for all numbers. This is the 

 generalisation which, according' to Poincare, li'--. at 

 the bottom of all mathematical argument, and allows 

 us to pass from the finite to the infinite. The second 

 chapter brings mathematical quantities into relation- 

 ship with experiment, and treats, among other things, 

 of incommensurable quantities and the creation of 

 the physical and mathematical continuum. YVe are 

 made to understand how, adopting a certain defini- 

 tion of a line which would satisfy most of us, 

 the diagonal and inscribed circle of a square have 

 no point in common, and we are then asked to 

 admit the possibility of a curve which has no tangent. 

 But a greater surprise is in store for us in the second 

 portion of the book, where, in the course of an admir- 

 able discussion of the geometries of Lowatchewski 

 and Riemann, we are introduced to the possibility 

 of a fourth geometry, in which a right line may be 

 at right angles to itself. This part of the book will 

 probably be the one most valued by the student of 

 experimental science, because it deals with an aspect 

 of the subject which, though foreign to his customary 

 plane of thought, must be of the highest interest if 

 he desires to dip a little below the surface. 



The conception of space and the relation of 

 geometrv to experiment are discussed briefly, but 

 with great precision and clearness. The foundations 

 of geometry are shown to be experimental. " If 

 there were no solid bodies in nature, there would be 

 no geometry." Yet, though founded on experiment, 

 the laws of Euclidian geometry can never be upset 

 bv experiment : 



" If, then, to contemplate the impossible, one were 

 to discover negative parallaxes, or to find that all 

 parallaxes lie above a certain limit, one would have 

 the choice between two conclusions : We might reject 

 the Euclidean geometry or, on the other hand, we 

 might modify the laws of optics by admitting that 

 light is not accurately propagated in straight lines. 

 It is unnecessary to add that everybody would choose 

 the latter alternative as most convenient." 



The final conclusion is that " geometry is not 

 true: it is convenient." 



More than one-third of the book having "been taken 

 up with the discussion of the fundamental notions of 

 mathematics, we are fully equipped to enter into the 

 discussions of the laws more particularly associated 

 with physics. All those who care to think of these 

 matters at all must have given some attention to the 

 nature of the so-called laws of motion. They will 

 find much in Prof. Poincare's reflections that has been 

 familiar to them, and something, perhaps, that they 

 have vaguely felt, but not been able to put into 

 definite form. One point which is brought out clearly, 

 which, speaking for myself, I had not sufficiently 

 NO. 1892, VOL. 73] 



realised, consists in the difficulty of finding inde- 

 pendent measures of both force and mass unless the 

 third law of motion is treated as an axiom. The dis- 

 cussion of the third law will be found to be full ol 

 interest. Too little importance, perhaps, is attached 

 to what the author calls " anthropomorphic 

 mechanics." This is surprising, as anthropomorphic 

 ideas are used by him so freely and convincingly in 

 his foundation of geometrical laws. It is true enough 

 that no one has yet been able to find a scientific basis 

 of mechanics in an anthropomorphic conception of 

 force, but at the same time I do not believe that 

 anyone has ever truly reasoned about force without 

 such idea forming the real moving spring of his 

 thoughts. " One could not maintain," Prof. Poincare 

 says, " that the sun is conscious of a muscular effort 

 when he attracts the earth." That is true enough, 

 bu1 we are able in our imagination to attach the idea 

 of muscular effort to every effect of force in the same 

 win as we can feel sympathetic pain for a friend 

 who lies on the operating table, though our reason 

 tells us that he himself is quite unconscious of pain. 

 In that case we project our own sensitive and con- 

 scious mind into his unsensitive body, and to some 

 extent feel the operator's knife. There are probably 

 great differences in the way different brains work, but 

 I could not myself form an idea of the mechanics of 

 the solar system without imagining myself at its 

 centre and consciously pulling the planets towards me. 

 In the same way, if I imagine myself freely placed 

 in space, I at once become conscious of a pull towards 

 the sun. That anthropomorphism has played an 

 important part in the history of mechanics is admitted 

 by the author, but he restricts the philosophy of science 

 to the discussion of the symbols which can be reduced 

 to measurement. 



Prof. Poincare's discussion of the principles of the 

 conservation of energy will be read with interest. 

 That principle has been abused by energy specialists 

 in a manner which could not fail to call forth a whole- 

 some reaction. The weak point which Prof. Poincare 

 specially exposes, seems, however, to me to touch 

 not so much the enunciation of the principle as the 

 difficulty of identifying potential and kinetic energies 

 in cases where the mechanism of the phenomena is 

 unkxiown to us. This only means that science is not 

 sufficiently advanced to specify completely the different 

 forms of energy. This most of us admit, while the 

 Energetiker deliberately uses the principle of energy for 

 the purpose of hiding his ignorance. Thermodynamics 

 is briefly dwelt upon, but we should have liked to 

 hear more of the author's views on the dissipation of 

 energy. The far-reaching consequences of the gradual 

 decay of regular motion assign to the second law of 

 thermodynamics a predominant place, and put the im- 

 portance of the first law completely into shade. Lord 

 Kelvin's principle of dissipation of energy has opened 

 out a common ground lying on the borderland between 

 physics and metaphysics which has not been cultivated 

 so much as it deserves to be. 



It is not possible to follow Prof. Poincare 1 in his 

 discussions of selected questions of modern theories 

 of physics. They will be read with interest, though 



