Febkuarv 26, 1903] 



NA TURE 



j°7 



It is clear that the adjustment will depend in part 

 on the position of the lighthouse, especially on its 

 height above sea-level, and that a system of lenses 

 put together without any reference to this was bound 

 to be wrong. 



The results were entirely satisfactory, and the 

 Whitby light when reconstructed gave admirable re- 

 sults. 



A good deal of correspondence followed with some 

 members of the Commission as to the form of lamp 

 and the best height for the principal focus of the 

 system above the wick, and as time went on various 

 other improvements were introduced; but Mr. Chance's 

 position was now assured, and is evidenced by the 

 long list of splendid lights we owe to him. 



One of the improvements worked out in collabora- 

 tion with Mr. Thomas Stevenson is the dioptric mirror, 

 whereby the rays which leave the lamp at the back 

 are totally reflected by suitable curved prisms and 

 issue in the direction in which the light is required 

 to travel. 



The whole book, however, is most interesting, and 

 forms a striking illustration of the application of 

 science to industry. Mr. Chance realised the need of 

 this, and his success was the consequence. 



His contributions to the mathematical side of the 

 subject are summed up in two papers read before the 

 Institution of Civil Engineers in 1867 and 1879. The 

 first deals with lighthouses in general ; its value as a 

 reprint is, however, impaired by the omission of the 

 careful figures by which it was illustrated; in the 

 second, the question of the application of the electric 

 light to lighthouses is considered. Electric light was 

 employed at the South Foreland in 1872 and at the 

 Lizard lighthouse in 1878. The apparatus in the 

 latter case was designed by John Hopkinson, who 

 on Mr. Chance's retirement became scientific adviser 

 to the firm. R. T. G. 



THE INFINITIES OF MATHEMATICS. 

 Die Grundsatze und das Wesen des Unendlichen in 



der Mathematik und Philosopiiie. Von Dr. Phil. 



Kurt Geissler. Pp. viii + 417. (Leipzig: Teubner, 



1902.) Price 14 marks. 

 "E" VERY serious inquiry leads, sooner or later, to 

 - 1 — "* metaphysics, and thus to antinomies which no 

 merely logical process can reconcile. The pure mathe- 

 matician is one of the first to reach this conclusion, 

 because his methods are mainly logical, and the notions 

 with which he deals are few and abstract. Why is it, 

 then, that (as a rule) he regards the philosopher with 

 a mixture of pity and disdain, and rarely takes part in 

 any strictly metaphysical discussion? Each is vitally 

 concerned with number, space and time ; why do the 

 conclusions of the one appeal so little to those of the 

 other? Leaving the philosopher to answer for him- 

 self, we may endeavour to construct the mathema- 

 tician's apology. 



It is mainly that, while he reaches the fundamental 



paradoxes as soon as the metaphysician, his attitude 



towards them is different. As it seems to him, the 



philosopher, after an imperfect analysis, tries to save 



NO 1739, VOL. 67] 



the situation by a still more imperfect transcendental 

 synthesis. To swamp all distinctions in the Absolute, 

 while assuring us that the distinctions persist, is a 

 childishly simple course, especially when adopted by 

 someone who has a very vague conception of the distinc- 

 tions which he proposes to abolish. Surely it is reason- 

 able to examine our concepts as carefully as we can, to 

 disi over, if possible, which are simple and which are de- 

 rived or composite. Until we do this, we have no right 

 to say what are the ultimate logical inconsistencies, still 

 less how we propose to reconcile them. The presupposi- 

 tions of arithmetic and geometry have recently been 

 analysed with great care, and definite results of 

 primary importance have been obtained ; the philo- 

 sophical bearing of these conclusions is obvious, and 

 henceforth no metaphysical theory that ignores them 

 will be accepted by mathematicians. Difficulties re- 

 main, of course; some have emerged which were 

 previously unsuspected ; but at any rate the ground has 

 been cleared of many merely sophistical paradoxes, and 

 the real issues have been made clearer. 



Dr. Geissler's book is rather pathetically disappoint- 

 ing; he has evidently tried to master modern critical 

 theories, but has failed in the attempt. The whole 

 arrangement of the work is unsatisfactory, starting as 

 it does with a vague spatial intuition, and constantly 

 mixing up arithmetical difficulties with those of geo- 

 metry. In the forefront of all discussions of mathe- 

 matical infinity must be put the notion of the arithme- 

 tical continuum; this, at any rate, is precise' and 

 definite. From it we get the concept of a continuous 

 real variable, and thence can proceed to the differential 

 and integral calculus treated by the method of limits. 

 This involves the use of a fluent differential, but there 

 are no serious logical difficulties. Dr. Geissler's atti- 

 tude is anything but precise, and not always consistent ; 

 he appears to try to establish the existence of infini- 

 tesimals of different orders as actual entities, and this 

 partly by geometrical considerations. In this region 

 of thought geometrical intuition is wholly untrust- 

 worthy ; and it is doubtful whether any satisfactory 

 analytical theory can be constructed on the basis of 

 what we may call fixed infinitesimals. It is certain, 

 for instance, that in the arithmetical continuum there 

 is no natural series of orders of infinitesimals. What 

 is the precise nature of geometrical continuity, and how 

 far it can be expressed by arithmetical means, is a very 

 difficult question, upon which Dr. Geissler does not 

 help to shed any light. 



One important point the author does emphasise, 

 though sometimes with more zeal than discretion. 

 The terms infinite and infinitesimal have no precise 

 meaning except in relation to a context and to certain 

 presuppositions. Thus, in projective geometry, the 

 statement that all points at infinity lie in a plane is 

 a convenient summary of a set of facts about parallels ; 

 on the other hand, in the theory of algebraic functions, 

 we assume that in the plane of the complex variable 

 there is only a single point (not a line) at infinity. Each 

 statement is true in its context, and out of its context 

 it means nothing- at all. If, with Dr. Geissler, we set 

 off equal finite segments continually along a Euclidean 

 straight line, we may assert the possibility of any 



