NATURE 



417 



THURSDAY, SEPTEMBER 2, 1897. 



THE NECESSARY POSTULATES OF 



GEOMETRY. 



An Essay on the Foiniddtioiis of Geometry. By 



Bertrand A. W. Russell, M.A. Demy 8vo. Pp. xvi + 



201. (Cambridge: at the University Press, 1897.) * 



THE title of this essay suggests a number of distinct 

 problems. We may ask what the postulates of 

 geometry are, or we may seek the source of our know- 

 ledge of them ; in the latter inquiry, again, we may set 

 out to discover how the fundamental geometrical notions 

 grew up, or it may be our object to ascertain how we 

 can have certainty concerning them. Mr. Russell's essay 

 deals with the last of these questions. It is, on the one 

 hand, a criticism of existing theories of geometry, and, on 

 the other hand, it is constructive, and aims at formulat- 

 ing a new philosophical theory of the foundations of the 

 science. 



An abstract geometry, logically arranged, would start 

 from a small number of definitions and postulates, and 

 would proceed deductively. In the process there would 

 occur places in the argument where a choice would be 

 possible among different hypotheses, and at such places 

 ambiguity would be removed by the introduction of fresh 

 postulates. There would thus be different orders of 

 postulates, some being required in order that there might 

 be any geometry at all, and others being adapted to make 

 geometry applicable to the formulation of experience. 

 The problem of separating the postulates into such 

 classes is the problem of transcendental geometry, or, as 

 the author calls it, Metageoinetry. 



Mr. Russell gives in his first chapter an outline of the 

 history of metageometry. He shows how it began with 

 attempts to deduce Euclid's parallel axiom from the re- 

 maining axioms, and how this attempt issued in the con- 

 struction of logically consistent geometries which did not 

 adopt the axiom ; he describes how Riemann and Helm- 

 holtz attempted to classify geometrical axioms as suc- 

 cessive determinations of space considered as a particular 

 example of a more general class-conception— that of a 

 manifold or numerical aggregate, and here he does not 

 omit to summarise the extremely important results ob- 

 tained by Lie in modifying and completing Helmholtz's 

 investigation ; lastly he explains how Cayley and Klein 

 connected metageometry with projective geometry, and 

 here he incidentally gives an account of what projective 

 geometry is, and of its independence of the notion 

 of measurement, a notion which was fundamental in 

 Riemann and Helmholtz's methods. The chapter is 

 partly historical and partly critical. It contains inter 

 alia an answer to Cayley's challenge demanding that 

 philosophy should either take account of the use of 

 imaginaries in analytical geometry, or show that it has 

 a right to disregard it (pp. 41-46). 



The second chapter contains a criticism of philo- 

 sophical theories of geometry propounded by Kant, 

 Riemann, Helmholtz, Erdmann, Lotze, and others. 



In the third chapter we have a discussion of the ques- 

 tion what postulates are necessary in order that there 

 may be a geometry at all. The same result is arrived at 

 NO. 1453, VOL. 56] 



whether the subject is considered from the projective or 

 the metrical point of view ; it is that the necessary pos- 

 tulates are those of homogeneity, and continuity of space, 

 and the existence of the straight line as a unique figure 

 determined by two points. Thus the most general 

 possible geometry includes Euclidean geometry, the 

 hyperbolic geometry of Lobatschewsky, the spherical 

 geometry of Riemann, and the elliptic geometry of Klein, 

 but besides these there is no other. The postulates 

 necessary to this general geometry are declared to be 

 h priori axioms, while the parallel axiom and the axiom 

 of three dimensions are found to be empirical. 



The fourth chapter deals with some dif^culties met 

 with in the previous chapter, and traces some of the 

 philosophical consequences of the theory proposed. 



Mathematicians will turn with most interest to 

 Chapter iii., to see what Mr. Russell lays down as the 

 essential postulates of geometry, and how he establishes 

 his conclusions. The chapter is divided into two 

 sections, dealing respectively with the "Axioms of 

 Projective Geometry," and the " Axioms of Metrical 

 Geometry." In projective geometry, as the author 

 points out, the notions of the point, straight line, and 

 plane are presupposed. Technically, the subject starts 

 from these notions, and determines by the methods of 

 projection and section what figures are equivalent to a 

 given figure. Philosophically, the subject has a wider 

 aim, consisting in the determination of all figures which 

 cannot be distinguished by their internal relations when 

 quantity is excluded (p. 133). The kernel of the argu- 

 ment consists in the identification of projective equiva- 

 lence with qualitative similarity. The author attempts 

 to prove that a form of externality (a notion essential 

 to the knowledge of a world of diverse and inter-related 

 things) must possess precisely the properties attributed 

 to space in projective geometry, these properties in- 

 cluding homogeneity, and continuity, and the possibility 

 of the straight line, or in other words of a unique figure 

 determined by two points. He seeks, in fact, to deduce 

 these properties of the form from the relativity of 

 position. Without wishing to impugn the correctness 

 of the deduction, or to deny the legitimacy of the con- 

 clusion, we cannot help thinking the argument obscure. 

 This is especially the case in all that concerns the 

 notion of the point. Thus, in speaking of the infinite 

 divisibility of the form of externality (p. 138) he says : 



"The relation between any two things is infinitely 

 divisible, and may be regarded, consequently, as made 

 up of an infinite number of the would-be elements of 

 our form, or again as the sum of two relations of 

 externality." 



He finds in the notion of the point "a self-contra- 

 dictory result of hypostatizing the form of externality." 

 This difficulty he recurs to again and again. Would it 

 be presumptuous for a mere mathematician to suggest 

 that this alleged contradiction may arise from the adop- 

 tion of an antiquated mode of statement ? We are told 

 (p. 188) that the difficulty is extremely ancient. Is it 

 not safe to say that the ancient philosophers had not 

 firmly grasped and completely analysed the concept 

 of the mathematical continuum.^ Mr. Russell says 

 (p. 189): 



