206 



NA TURE 



[July 19, 1906 



where everything mathematical is as popular as it is 

 unpopular in England, the philosophy of mathe- 

 matics has taken such hold of public thought that the 

 multiplication of books on the subject has in a small 

 wav resembled the multiplication of school geometries 

 with us. But a philosophical treatise stands on a very 

 different level from a mere examination text-book, and 

 if we have been somewhat severe in the past in our 

 criticisms of the work of isolated writers in France, it 

 was felt that what was wanted was something more 

 than a number of isolated writings, each, from the 

 nature of the case, presenting the views of one indi- 

 vidual without much reference to the subject and its 

 literature considered as a whole. The opening words 

 of M. Couturat's preface, " The present book has no 

 pretension to originality, and this is precisely what 

 ought to recommend it to the reader," show that the 

 author has been at great pains to fill the want. His 

 book is to a large extent based on Mr. Bertrand 

 Russell's English treatise with the same title, and is 

 intended to provide a risumi of our present knowledge 

 on the philosophy of mathematics. It is somewhat 

 remarkable that up to the middle of last century logic 

 and mathematics were regarded as essentially distinct, 

 and it was largely the result of the labours of Boole, 

 Peano, Cantor, and others that led to the gap being 

 filled and to the opening out of what has proved to be 

 one of the most fertile regions of modern thought. 

 The complete rapprochement owes its existence very 

 largely to the symbolical logic or " logistic " of Peano, 

 and leads to the conclusion that mathematics is 

 entirely and exclusively founded on the principles of 

 logic. This view is, as M. Couturat shows, dia- 

 metrically opposed to the philosophy of Kant of 

 which a summary has been given in the appendix. 

 It need hardly be said that Russell's treatment is in 

 many places closely followed, and it is the author's 

 hope that the book will induce French writers to con- 

 tribute to our knowledge of mathematical philosophy 

 in a way that has not been done hitherto. 



A perusal of Mr. Oom's pamphlet will convince any 

 reader that however much has been done elsewhere in 

 facilitating calculations by the introduction of graphical 

 methods, the observatory of Lisbon under the director- 

 ship of Vice-Admiral Campos Rodrigues has made a 

 number of very distinct advances. For the correction 

 of level and deviation error diagrams are used, as 

 also for the corrections due to precession, and a still 

 happier thought is the construction of slide rules for 

 the performance of addition operations other than the 

 addition of logarithms performed by the ordinary slide 

 rule. Thus, for example, a slide rule graduated in 

 reciprocals is used to work out relations between the 

 conjug'ate foci of a lens ; another, graduated in squares, 

 is applicable to quantities connected by the relation 

 between the sides of a right-angled triangle, and is 

 particularly useful for calculations of probable error, 

 and so on. Possibly some reader of N.ature will write 

 and say that these slide rules have been in existence 

 previously. In any case they are worthy of note, and 

 M. Rodrigues appears to have devised them " off his 

 own bat." 



The issue of a scries of " Cambridge Tracts in 

 NO. 1 916, VOL. 74] 



Mathematics and Mathematical Physics " under the 

 able editorship of Messrs. Leathem and Whittaker 

 affords evidence of the activity of the younger genera- 

 tion of Cambridge mathematicians. The idea is a 

 good one. Many people have ideas about the best 

 methods of treating some particular piece of work,, 

 which do not cover a sufficiently wide range tO' 

 form a book, and are unsuitable for publishing and 

 possibly burying in a volume of transactions. So 

 much is this the case that we should not be surprised 

 if Mr. Leathem finds himself besieged by tracts sub- 

 mitted for publication. His own contribution deals 

 with an important point. Many physicists in the solu- 

 tion of problems have had to transform volume and 

 surface integrals in a way that they either have 

 known or ought to have known was not perfectly 

 rigorous, but with the knowledge that the results 

 would be all right; this particularly applies to integrals 

 extending to infinity, and we have here in a convenient 

 form a study of these transformations in their mathe- 

 matical aspect. In discussing the application of 

 infinitesimal analysis to potential properties of bodies 

 of discontinuous structure, Mr. Leathem introduces 

 the notion of pliysical smallness. The term is, perhaps, 

 not altogether a happy one, as physics concerns itself 

 not only with bodies of finite size, but with molecules 

 which are of a higher order of smallness than the 

 elements contemplated. The important point is that 

 the applications of differential equations are based on 

 the consideration of elements which for purposes of 

 analysis may be regarded as infinitesimal, but which 

 may be regarded, on the other hand, as infinitely great 

 compared with the dimensions of molecular structure. 

 It would be better to call such elements " differential 

 elements " since they represent the dx dy dz of the 

 formulae. The careful discussion of the legitimacy of 

 the assumptions involved, as given by Mr. Leathem, is 

 important, as we often find unscientific writers 

 announcing as a great discovery the view that there is 

 no such thing as temperature, quite forgetting that 

 the notion of " temperature at a point " stands on 

 much the same footing as that of " density at a 

 point " or, indeed, many other similar concepts with- 

 out which the study of mathematical physics would 

 not have made the progress that it has made. 



The series of papers and books by C. A. Bjerknes 

 the father and V. Bjerknes the son well illustrate the 

 proper spirit of scientific inquiry as opposed to the 

 spirit of the unscientific faddist whose rejected ad- 

 dresses give so much trouble to reviewers. The discus- 

 sion of the fields produced by bodies moving in fluids, 

 if it has not given us a new theory of matter has cer- 

 tainly greatlv helped us to understand the lines on 

 which such theories should be laid down. The elder 

 Bjerknes confined his attention to solid spheres moving 

 in liquid; in the present instance, " bodies " are repre- 

 sented bv portions of fluid differing from the remainder 

 by the fact that in the latter the equations take a 

 simple form. But is not this merely the vortex atom 

 theory? It is true that on pp. 134-7 Prof. Bjerknes 

 compares his results with those of von Helmholtz and 

 Lord Kelvin, and points out the differences in his 

 method of treatment, but all these investigations are 



