NA TURK 



493 



THURSDAY, SEPTEMBER 20, 1894. 



THE PRIXCIPLES OF PURE MA TIIE.MA TICS. 



Grtitidziige der Geometric von mehreren Diineusioiieii iitid 

 inehrcrcn Arten gradlinigcr Einheiteii in elementarer 

 Form entwickelt. Von Guiseppe Veronese, Pro- 

 fessor an der kcinigl. Universitiit zu Padua, 

 xlvi. + 7iopp. Leipzig: Teubner, 1894.) 



THE work before us is an authorised German 

 translation, by Lieutenant Adolf Schepp, of 

 Wiesbaden, of Prof. \'eronese's treatise on the Founda- 

 tions of Geometry, first published at Padua in 1S91 ; and 

 the translator tells us in his short preface that the author 

 I1.15 communicated to him the corrections and improve- 

 ments of his work that have occurred to him since its 

 publication in Italian. To give such an account of the 

 contents of a book, so important, so original, and, it may 

 be added, so controversial, as would serve to render its 

 purpose and method generally intelligible, and at the 

 same time to subject it to adequate criticism, would re- 

 ciuire a memoir rather than a review. We shall therefore 

 endeavour only to give such a description as will recom- 

 mend it to the attention of all w-ho are interested in the 

 logical basis of Pure Mathematics. 



The Preface of thirty-four pages gives a general view 



»'f the author's system ; the Introduction of 222 pages is 



'levoted to the logical establishment of the notions of 



Number and Continuous Quantity; after that the First 



i 'lit deals with tlie Straight Line, the Plane, and Space of 



I'hree Dimensions ; the Second Part is occupied with the 



theory of Space of four or « Dimensions. The treatise 



ends with an interesting historical and critical discussion 



olthe most important previous works on the same subject, 



■ul some notes elucidatory of special principles. .A. full 



ible of contents is given, and a list of authors quoted ; 



but it would have added much to the value of the book as 



a work of reference if a good alphabetical index of sub- 



:ts had been included. The remainder of ths present 



tice will be confined to the Introduction, the specially 



ometfical parts of the book being reserved for another 



occasion. 



The thorough revision to which, in this century, the 

 underlying principles of mathematical reasoning have 

 been subjected is not less remarkable than the great ad- 

 vances made in the ulterior developments of these 

 principles. Crelle has said that for those who probe the 

 depths, equally with those who build in the heights of 

 mathematical thought, there ever remain unexplored 

 mysteries. Among those who have probed the depths, 

 the investigators, namely, who have occupied themselves 

 with the notions of number and quantity, the continuum 

 of real numbers, the infinitely great and the infinitely 

 small, there has been much divergence of opinion as to 

 the logical grounding of the subject. Such discussions 

 appear to be foreign to the taste of our English writers. 

 With us Arithmetic is an atlair of sums to be done by 

 uiles ; Algebra is arrived at by noting the laws of opera- 

 lion with the numbers of Arithmetic, and giving to 

 them the power of holding generally. We are too used 

 to the process humorously described by Clifford ' — 



' " Lectures .ind Ess.lys," vol. i. p. 336* 



NO. 1299, VOL. 50] 



"In the science of number while five-sevenths of four- 

 teen has a meaning, namely, ten, five-sevenths of twelve 

 is nonsense. Let ui then treat it as if it were sense, and 

 see what comes of it." This method, whichClifford held 

 to be '• logically false and educationally mischievous,' is 

 not that adopted by continental writers, and in particular 

 it is not the method of Prof. \'eronese. For him it is 

 necessary so to define the abstract notions — number, 

 quantity, and so on — that the laws of operation with 

 them may be logically well-grounded upon the definitions. 

 Let us see how he sets about the notion of (positive 

 integral) number. 



Readers of Clifford's lecture just now quoted will 

 remember that the eriix of Arithmetical theory lies in 

 the proof of the statement that the number of things in a 

 group is independent of the order in which they are 

 counted ; and the difficulty of proving it is not 

 diminished by the facts, firstly, that everyone is firmly 

 convinced of its truth, and secondly, that the whole 

 system of Arithmetic is the work of the human mind, 

 and therefore the theorem must somehow be implicitly 

 included in the tlefinitions when these are given with 

 sufficient clearness. Prof. Veronese, like Kronecker,' 

 appears to regard the ordinal number as logically pre- 

 ceding the cardinal number ; in other words, he makes 

 the idea of a group of things arranged in an order more 

 elementary than the idea of the number obtained by 

 counting the things in the group, i'ronecker, going out 

 from this notion, rapidly arrived at the required result, 

 but our author is not satisfied with his reasoning. His 

 own process is much more leisurely. He starts from the 

 notions of unity and multiplicity (Einheit and Mehrheit) 

 and explains the operations of uniting (Vereinigung) the 

 objects of a series into a group, and of separation 

 (Zerlegung) of a group into objectsby successively taking 

 away (Wegnehmen) object after object from the group. 

 He defines an ordered group, and explains the unique 

 correspondence of elements in two such groups. It is 

 only after all this that he is prepared to define a num- 

 ber as an ordered group of units arranged to correspond 

 uniquely and in the same order to the objects in an 

 ordered group of objects. This definition is found to be 

 a sufficient ground for the definition of counting, for 

 proving the crucial proposition above referred to, and for 

 the establishment of the commutative law of addition 

 and the remaining laws of operation with positive in- 

 tegral numbers. 



To establish the notion of Quantity and the extended 

 conception of Number, with which Algebra has m.ade us 

 familiar, it is necessary, as many writers (including Du 

 Bois Reymond) have pointed out, to frame an account of 

 the Fundamental Form, or, as we may call it, the Locus 

 /// y«o of real Quantity. A numerical fraction implies a 

 something divisible into equal parts, and a part of it con- 

 taining a certain number of these parts. .-\ square root 

 implies an exact measurement of a side of a square 

 which has a given area. These examples show that we 

 do tacitly or expressly assume a someuhctl capable of 

 exact division into parts in arbitrary ways corresponding 

 to various mathematical ideas. This somew/ia/ is the 

 Fundamental Form (Grundform), and it has been fre- 

 quently figured as a geometrical straight line, as by Du 



■ ";Vebcrdcn ZahlbegrilT." C«//;. BJ. ci. icS?. 



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