274 



NATURE 



[August 31, igu 



the ancestral relation, which, to avoid a vicious circle, 

 is defined so as to apply to members of an infinite 

 class. As the authors explain in a note, this section 

 is mainly based upon Frege's work, and is used 

 afterwards in deduce the properties of finite cardinals 

 and the transfinite cardinal X- Here we find the 

 Peanesque notation in all its development, and must 

 make up our minds to learn it thoroughly, or else to 

 express its formulae in an equally exact, but less 

 unfamiliar symbolism. This leads us to the few 

 critical remarks that we venture to offer on this ad- 

 mirable and elaborate work. Every communication 

 of ideas from one mind to another is made by means 

 of a conventional symbolism; no symbolism can be 

 more exact than language, because language is, in 

 the last resort, required to explain and define it. But 

 it may be more concise than language, and this is 

 tin- real virtue of the Peano notation and its deriva- 

 tives. To show how easy it is to exaggerate the 

 value of the notation as such, we may take an ex- 

 ample from p. 16 of the present work. The authors 

 say that "it is an obvious error, though one easy to 

 commit," that "No A is B" is the contradictory of 

 "ever^ A 'S B," and proceed to add that the sym- 

 bolism exposes the fallacy at once. Really it does 

 nothing of the kind; truly the symbol- {'(.v).^(.v)}, 

 the contradictory of (x).$(x), is different in j,<rm 

 from (*). ~$(x), but how can we tell from looking 

 at them that these last two symbols are not equiva- 

 lent? Again, the authors profess to give a proof of 

 the law of excluded middle; they assume it in de- 

 fining the assertion symbol, for they practically say 

 "if the proposition to which this sign is prefixed is 

 false the book is in error," tacitly assuming (here) 

 that all their propositions are significant. The lav 

 ol excluded middle is surely axiomatic for a signi- 



" proposition, the only trouble is in being quite 



that our assertions ate really significant, and in 



this it is reason that must guide us, not symbolism, 

 though a proper choice ol symbolism may conduce to 

 economy of thought. As an illustration, we nia\ 

 refine a little on the paradox of Epimenides. Sup- 

 pose a Frenchman or a German ass, its "Every 

 statement thai has ever been made b} an Englishman 

 is false." This is a significant statement, and as 

 such must be true or false. But suppose an Eng- 

 lishman says the same thing: the proposition ceases 

 to be Scant, unless he adds "except one," when 



it again becomes significant. Questions of this kind 

 are not so trivial as they appear, an, | a really philo- 

 sophical stud} ol language might do a good deal 

 towards making more definite the metaphysical basis 

 of knowledge. G. B. M. 



MOVEMENT AND ESCAPEMENT. 

 1 •' Wouvement. VIesures de Vitendue et mesures tin 

 temps. Bj Prof. J. Andrade. Pp. vi + 328. 



(Paris: I.ibrairie helix Air. in, inn.) Price I) 



1 1 : n : , 



SOME literar} effusions— for instance, the novel 

 with a purpose present to the reviewei an awk- 

 ward problem, namely, whether to concentrate his 

 on 1 1: novel as such, or on the purpose. 

 NO. 2183, VOL. 87] 



111, present work might almost be included in some 

 such category, inasmuch as it may be regarded from 

 the point of view of a mathematician pure and 

 simple, of a more or less practical mechanic, 

 or even of an astronomer, while all the time it 

 apparently claims to be a philosophical treatise, and 

 as such to appeal to what may be called the general 

 reader. In some parts of the book the philosopher 

 is much in evidence, and in many places the absence 

 of diagrams, and the assumption that the reader will 

 understand determinants, vectors, or even ordinary 

 equations of motion without explanation, would cer- 

 tainly repel the ordinary reader. The mathematician 

 will find perhaps little that is novel. The suggestions 

 of non-Euclidean space, whether that of Lobatchewsky 

 or of Riemann, are little more than suggestions, 

 and can only give those to whom such idea- are new 

 the kind of shock the earlier cyclists felt on first riding 

 a free-wheel. On the other hand, a very good 

 historical sketch, amply provided with diagrams, 

 is given of the development of scientific clock- 

 making with due respect to the great English horo- 

 logists. 



A brief sketch of the contents of the book will 

 serve to indicate the scope of the author's endeavour, 

 and it is difficult to conceive how, within the limits 

 ol such a volume, a perfectly satisfactory result could 

 have been achieved. Perhaps only a fellow-country- 

 man ot tin- great French philosophers of the past 

 would ever have attempted such a task. The first 

 part treats ol geometrical ideas of number and space, 

 the author showing a decided preference for vectorial 

 or polar coordinates, and for rotation as a means of 

 translation. The finite straight line is elaborately dis- 

 cussed, and ordinary geometrical propositions re- 

 garded from the point of view that came into vogue 

 about a quarter of a century ago, when Nixon's 

 Euclid began to oust Todhunter's in some schools. 

 Iriaiigl's and solids, plane and spherical an -, 

 volumes, velocity, vectors, the theorems ol Ampere 

 and Stokes, moments, composition of vectors and 

 vectorial quantities, bring us through trigonometry 

 and statics to non-Euclidean geometry by a somewhat 

 tortuous route. 



The second part introduces force, one chapti r being 

 devoted to the notions of astronomy and celestial 

 mechanics from Hipparchus to Newton, and another 

 to the principles of dynamics, equilibrium, and the 

 two fundamentals, which, in the author's view, are 

 clock and orientation; a third dwells on the vital im- 

 portance- ol a function of forces, on stability and con- 

 servative systems, on isolated m-mn*, Painleve's 

 theorem and Laplace's invariable plane; and these 

 are followed by simple and damped oscillations, 

 spiral movement, elastic bodies, and fluids, and the 

 bending ol springs. The third pan deals with optics 

 more especially ol the telescope, with a more special 

 devotion to different methods ol geodesy from Picard 

 to the use of invar and the Jaderin wire, and the 

 correction ol the units of the metric system. 



The fourth and last part deals with the chrono- 

 meter in general, and escapements in particular, call- 

 ing for continued experimental work on indicated 



