494 



NA TURE 



[Septe.mi!er 2o, 1S94, 



Bois Revmond 'in his Alsemenie Functionentheorie'. 

 Accordins: to Prof. Veronese it is more proper to give an 

 independent abstract account of the Kundamental Form, 

 partly because the properties of the straight line arc 

 afterwards to be determined in accordance with abstract 

 definitions. He therefore takes as a guide the rectilinear 

 continuum of intuition, of whose properties he gives an 

 analysis, and then proceeds in an abstract manner. He 

 defines a Form as anything whose marks are part, whole, 

 order, and kind of position. Thus a line regarded as 

 consisting of segments limited by points in a certain order 

 having positions on the line is a Form, a song regarded 

 as consisting of certain words pronounced in a certain 

 order and each in a certain musical pilch is a Form. He 

 explains how one Form may be determined by other 

 Forms, and how the identity of two Forms may be 

 inferred from the identity of the Forms that severally 

 determine them. To avoid the circular reasoning that 

 must ensue, unless some Forms are known to be iden- 

 tical there arises the necessity for introducing the Funda- 

 mental Form as a standard which serves for the deter- 

 mination of all others. He describes successively a 

 system of one dimension as a form given by a series of 

 elements whose order, from a certain element, is a mark 

 of the form, a homogeneous system of one dimension, 

 and a system of one dimension identical in the positions 

 of its parts. Such a form is chosen as Fundamental 

 Form, The operations of uniting segments of the Form 

 and of separating united segments are described, and 

 shown to obey the Laws of Algebra for addition and 

 subtraction. The relations of segments as multiples or 

 factors of other segments lead to the laws of multiplica- 

 tion and division, and to the description of the Scale 

 founded upon any segment as unit. The Range of the 

 Scale (Gebiet der Scala) is the part of the Fundamental 

 Form arrived at by continual repetition of the segment 

 chosen as unit. 



The author is now prepared to introduce the concep- 

 tions of the infinitely great and the infinitely small. He 

 asiumes that there is an element of the Fundamental 

 Form which lies outside the Range of the Scale founded 

 on any segment as unit. This assumption is apparently 

 free from any contradiction. Such an element being 

 chosen, the ssgment limited by it, and any clement 

 within the range of the scale, is infinitely great in reference 

 to the unit of the scale ; had this segment been chosen 

 as unit, the original unit would have been infinitely small. 

 From the nature of the Fundamental Form, as a homo- 

 geneous system identical in the position of its parts, 

 foUotvs the necessity of assuming any number of orders 

 of infinite segments and any number of orders of in- 

 finitesimal segments. 



To every segment corresponds a numerical symbol, 

 just as m particular the natural numbers correspond to 

 the segments which are exact multiples of that one chosen 

 as unit. The ordinary Laws of Algebra holding for the 

 segments hold in like manner for the numbers thus intro- 

 duced. To the infinitely great and infinitely small seg- 

 ments of different orders correspond infinitely great and 

 infinitely sm.iU numbers of different orders. It is proved 

 that the numbers thus arrived at are not identical with 

 Cantor's " Transfinitc numbers" {Achi .Mnlhcmatiin, 

 I5d. II}. After the introduction of these numbers, and 

 NO. 1299, VOL. 50] 



the enablishment of the laws of operation with them, 

 come the hypotheses of continuity of the Fundamental 

 Form, an idea here treated in a very instructive manner, 

 the proof of the existence of Limits so elaborately dis- 

 cussed by Du Bois Reymond, the notions of conjmensur- 

 able and incommensurable segments, and the theory of 

 Proportion, the last being especially interesting. A 

 chapter is added for the sake of completeness, in which 

 the properties of real, positive and negative, rational and 

 irrational, numbers are established on the basis of 

 principles already discussed. 



It is a cardinal feature of the author's account of the 

 theory of (Tuantity to dispense with the so-called .\xiom 

 X of Archimedes, according to which it is inherent in 

 the notion of quantity that when one quantity a is greater 

 than another b, there exists a number « such th.-\i 

 ;//' is greater than a. This axiom has been found bv 

 other writers, as Stolz, extremely useful in establishing 

 the properties and relations of finite quantities, but 

 appears to involve difficulties in connection with the 

 infinitely great and the infinitely small. At the expense 

 of greater length of explanation, .i^rof. Veronese has freed 

 the theory from the axiom and the involved difficulties. 

 His own exposition is gener.illy clear, though his doctrine 

 of "commensurable numbers of the second kind'' 

 (pp. 1S2 and 213) is not without obscurity. Could not 

 an cx.nmplc have been given ? 



Enough has been said to show th.it Prof. \"eroncse - 

 book treats of a great deal besides the Foundations o; 

 Geometry — his Introduction might, in fact, well be en- 

 titled th; Foundations of Mathematics. He tell; us that 

 although some parts of it will be useful in Geometry, 

 much has been worked o it simply for its own sake. We 

 may well be grateful to him for the patience and trouble 

 that he has expended in clearing up the Logic of the 

 operations that most of us, without a thought of under- 

 lying diflicultics, cheerfully perform with confidence and 

 success. He has none of the charm of style to be found 

 in the writings of Clifford or Kronecker. Rigorous he is, 

 thoroughly common-sense, careful almost to tediousness, 

 and extremely leisurely. For the elucidation of the very 

 difficult subject he has chosen, these qualities are pei 

 haps the greatest of merits, yet we fear that they will no; 

 render his writings acceptable to readers unprepared for 

 a consid erable sacrifice of time. A. E. H. L. 



TEXT-BOOKS OX CRGAXJC CIIEMISTRV. 

 Organic Chemistry. Part I. By W. H. Perkin, jun., 



Ph.U,, F,R.S., and F. Stanley Kipping, Ph.D., D.Sc. 



(London: W, and R. Chambers, 1S94.) 

 Lessons ill Organic Chemistry. Parti. Elementary, By 



G. S. Turpin, M.A. (Camb), D.Sc, (Lond.) (London : 



Macmillan, 1894.) 



IT is not surprising that "organic" chemistry should 

 have received less attention in this country than on 

 the continent, considering that the professors in nearly 

 all the chief British universities have been notoriously 

 neglectful of this department of the science, and that the 

 highest degrees in connection with chemical science 

 have been until recent years generally attainable without 

 a practical acquaintance with the subject, an. 1 without 

 evidence of capacity for research. 



