;2o 



NA TURE 



[September 27, 1894 



the progress of research is sjre to be considerable. A 

 student will have before him work whose style has never 

 been surpassed, and demonstrations which are absolutely 

 rigorous. In the latter respec: Gauss' work seems to 

 have left a lasting impression upon his mind. 



1 conclude by quoting the noble words from the 

 British Association address : — 



" But in science sophistry is impossible ; science 

 knows no love of paradox ; science has no skill to make 

 the worse appear the better reason ; science visits with a 

 not long-deferred exposure all our fondness for pre- 

 conceived opinions, all our partiality for views that we 

 have ourselves maintained, and thus teaches the two 

 best lessons that can well be taught— on the one hand 

 the love of truth, and on the other sobriety and watchful- 

 ness in the use of the understanding." 



P. A. MacMahox. 



"I 



ABSTRACT GEOMETRY. 

 Crundziige der Geoinelrie von meltrcren Dimensionen tiiid 

 viihreren Arten gradliniger Einheiten in chiiuntarcr 

 Formentwickelt. \'on Guiseppe Veronese. (Leipzig: 

 Teubner, 1S94.) 



MODERN speculations on the Foundations of Geo- 

 metry have raised the question of the character 

 of Geometry as a science, and the question has been 

 answered in different ways. Some writers have held 

 that our space-intuition is an absolute guarantee of the 

 truth of geometrical axioms ; others have treated 

 Geometry as a science of observation and experience, 

 whose results accordingly are liable to the same kind and 

 degree of inexactness as any other Physical Science. 

 If either of these answers were correct, the! method of 

 Geometry would seem to require revision. The method 

 is to deduce the properties of figures by logical processes 

 from definitions and a few propositions (."Vxioms) assumed 

 m advance. But if space-intuition were a sufficient 

 guarantee for the truth of the Axioms, it would seem to 

 serve equally well for a guarantee of the truth of |many 

 of the Propositions, and there would appear to be no 

 good reason for assuming as few as possible and deduc- 

 ing the rest. If, again. Geometry is to be purely a 

 Natural Science, there would be simplicity in proving 

 its propositions by the help of well-made constructions 

 and good instruments of measurement. There seems to 

 be room for a third view of Geometry as an abstract 

 formal science to which the method always known as 

 geometrical would be proper and natural. In such a 

 view abstraction might be made of all space-intuition, 

 and there would remain a body of logical truths in which 

 the .Axioms would occupy the place of Definitions or 

 well-defined Hypotheses. The science would be 'at the 

 same time founded upon intuition and independent of 

 intuition. If its Definitions and Hypotheses are never 

 in contradiction with themselves, or with each other, or 

 with our space-intuition, then will its conclusions alw.iys 

 be verified within the limits of exactness that belong to 

 observation. It will be a formal science ready for prac- 

 tical applications. 



The theory of Abstract (ieometry in the sense just 

 described is the subject-matter of Prof. X'cronese's 

 treatise He lays down in his Preface the nature of 

 Geometrical .Axioms as the simplest truths of space- 



NO. 1300, VOL. 50] 



intuition ; he describes the character of a system of 

 -Axioms in that they must be independent, as few as 

 possible, and yet sufficient for the establishment of the 

 properties of figures without tacit assumption of other 

 axioms. No definition or axiom is satisfactory which 

 contains any notion not previously cleared up, or any- 

 thing to be afterwards deduced. Any geometrical figure 

 regarded as existing in the space of intuition may be re- 

 placed by a well-defined mental object or '• Form,' in 

 the sense of the word fully described in the Introduction. 

 The geometrical axioms are replaced by hypotheses 

 serving to discriminate among possible forms or possible 

 formal relations. Intuition is taken as a guide to the 

 choice of hypotheses. The distinction is drawn between 

 .Abstract Geometry and its practical applications, and 

 it is pointed out that there may be axioms of great im- 

 portance for the latter which are useless restrictions in 

 the former : such axioms are that the space of intuition 

 is the Euclidean space form, and that the space of 

 intuition has three dimensions. For our author all con- 

 ceivable space forms are in theory equally admissible, 

 and the number of dimensions of space is unlimited. 

 The straight line, the plane, space of three or n dimen- 

 sions, are all regarded as existing in the General Space. 

 His method is the method of Pure Geometry, and his 

 work is free from any trace of axes, coordinates, and 

 Algebraic processes. Apparently this method has noi 

 previously been applied to the discussion of space 

 more than three dimensions. 



A reader who approached Prof. X'eronese's book in the 

 hope of finding a logical development in purely Geo- 

 metrical form of the theories of the non-Euclidean Geo- 

 metry would be disappointed, for (he work is throughout 

 subordinated to the Euclidean system ; nor would the 

 reader be better satisfied if he sought merely for thf 

 logical establishment of the Euclidean system, for it is 

 throughout treated as a limit included in a more general 

 possible system. It is well known that the Euclidean 

 Geometry is the limiting form between the Hyperbolic 

 and Elliptic Geometries, and this is the case whatevei 

 more particular character we attribute to either of these 

 Geometries. Hyperbolic Geometries differ with the form 

 chosen for the " Absolute," there are two Elliptic Geo- 

 metries according as two straight lines have one or two 

 common points. All these systems have Euclid's system 

 as a limit. In the elements of an Abstract Geometryi 

 developed in an orderly way we shall be presented time' 

 after time with a choice of hypotheses. Our choice at 

 any time will determine to some extent the space form ol| 

 which we treat. Our series of hypotheses will limit US| 

 to a particular space form. If one of our hypotheses is 

 the existence of straight lines, we shall come upon the| 

 Euclidean system or a non-Euclidean system having thci 

 Euclidean as its limit. Wc may state at once that Prof | 

 X'eronese's hypotheses lead him to a system which, in anj 

 absolute sense, is the so-called Spherical Geometry, as] 

 distinguished from the Elliptic Geometry proper. Ac-'t 

 cording to this system two straight lines cut in twc 

 " opposite " points, and the length between opposite pointsi 

 Is constant. This, however, is only true In an " absolute* 

 sense," the length in question being actually infinite ini 

 comparison with any perceivable length treated as ai 

 I unit. The doctrine of the "actually infinite" is that 



