NATURE 



[November 7, 1895 



The treatise of MM. Laisant and Lemoine is intended 

 to serve a double purpose : to provide a strictly scientific 

 introduction to the science of arithmetic, and to ex- 

 emplify a system of orthography which has been 

 adopted by the " Soci^td filologique fran^aise." This 

 is not the place to discuss the system of spelling which is 

 advocated : suffice it to say that it is one of those com- 

 promising systems which, while rebelling against tradi- 

 tion, stop short, by a long way, of a strictly phonetic 

 orthography. The consequence is a mass of inconsist- 

 ency which has not even the excuse of a historical de- 

 velopment : thus, for instance, " philosophie " becomes 

 " filosofie," while " science " retains its traditional form. 

 Again, "commun" is changed into "comun," while 

 *' irrdductible " is left alone. 



From a mathematical point of view, the book, as might 

 be expected from the reputation of its authors, is very in- 

 teresting and valuable. A treatise on elementary arith- 

 metic may be criticised in two different ways, according as 

 it is estimated in relation to pure science or to pedagogy. 

 Most text-books on arithmetic are utterly unscientific, 

 and a treatise like this, which aims at a rigorous method 

 and, on the whole, achieves it, is a welcome acquisi- 

 tion. 



The book deals with the four fundamental operations 

 as applied to whole numbers and fractions ; the metric 

 system ; elementary theory of numbers (prime factors, 

 G.C.M. and L.C.M., recurring decimals, &c.) ; incom- 

 mensurables, squares and square roots; ratio and pro- 

 portion. Most of it is quite admirable ; and the criticisms 

 which follow are offered in no captious spirit, but as a 

 kind of acknowledgment of the really scientific character 

 of the book. 



The authors begin by defining addition as an operation 

 which is independent of the order of the things added ; 

 or rather they refrain from giving a definition of addition, 

 and state that any definition must be subject to the con- 

 dition above stated. Now in arithmetic it is not things, 

 but numbers, that are added, and it is quite possible to 

 give a satisfactory definition of an arithmetical sum. 

 Thus take two groups of objects (in the most abstract 

 sense), count the first group, then the second, and thirdly 

 count the first group as before, but go on counting as you 

 pass on to the second group instead of beginning again. 

 Three numbers are thus obtained, and the third is defined 

 to be the result of adding the second to the first. The 

 commutative law follows easily enough. 



The authors' definition of a sum is equally applicable 

 to the addition of vectors, and this fact vitiates their 

 statement that " a quantity A is said to be greater than 

 another quantity B when A results from the addition of 

 a quantity C to B." 



The objection that concrete "quantities" are intro- 

 duced, whereas pure arithmetic is concerned with 

 numbers, and numbers only, applies to other parts of 

 the book, notably to the chapter on fractions. It is 

 quite true that concrete illustrations, such as those 

 afforded by a two-foot rule, are very useful, and indeed 

 indispensable for the purposes of primary instruction in 

 the subject, but the theory of fractions is independent of 

 these apphcations. This may be seen, for instance, in 

 Biermann's "Theorie der analytischen Functionen" 

 (after Weierstrass) ; and it is not difficult to see in 

 NO. 1358, VOL. 53] 



Euclid's arithmetical books some foreshadowing of this 

 way of looking at the matter. 



Then, again, the treatment of irrational numbers, 

 although greatly superior to that usually found in text- 

 books, does not seem wholly satisfactory. The authors 

 evidently intend to adopt the method of Dedekind, or 

 rather, perhaps, that of Heine, but the way in which this 

 is presented is not very clear. According to Dedekind, 

 the existence of a single definite irrational or transcen- 

 dental number is established when we are able to define 

 a "Schnitt" in the (discrete) multiplicity of rational 

 numbers ; that is to say, when we are able to find a 

 criterion which separates all rational numbers into two 

 groups, A and B, such that every number, say a, which 

 belongs to A, is greater than every number b which 

 belongs to B. Thus, for instance, if we assign a rational 

 number to A when its square exceeds 2, and to B when its 

 square is less than 2, we establish a " Schnitt " which 

 defines the irrational number ^^2. Heine's method is 

 not very different from Dedekind's ; thus his way of 

 defining ^Jt, consists in selecting from the groups A and. 

 B, as above defined, two sets of rational numbers : — 



such that 



a. 





bn 



<do < bo ■ 



. > an > 



. <bn< 



{al > 2) 



32 < 2) 



and then showing that if n, ni are taken large enough,, 

 an - bm. can be made as small as we please. MM. Laisant 

 and Lemoine, after introducing the problem of measuring 

 an incommensurable quantity, proceed : ' Jusqu' k present 

 on n'a rien trouve de mieus pour remplir ce but que 

 d'indiquer tous les nombres entiers ou fractionaires 

 mesurant les quantites plus grandes et les quantitds plus 

 petites que A." Now, even if we waive the objection, 

 already brought forward, that the measurement of quan- 

 tities is independent of pure arithmetic, and that the 

 assumption that every quantity admits of arithmetical, 

 measurement in terms of an arbitrary unit requires justi- 

 fi cation (and this can only be given after the theory of 

 irrational numbers has been established, if indeed then),, 

 the above statement is not satisfactory ; because if A is 

 really a concrete quantity, the "nombres entiers ou 

 fractionaires," &c., can only mean " quantities commen- 

 surable with an assumed unit," and it remains to be 

 proved that the choice of a unit has no influence on the 

 result ; while if A is a number, no criterion is given by 

 which we can decide whether a given rational number is 

 greater or less than A. On this point we cannot do 

 better than refer to the preface to Dedekind's invaluable 

 tract, " Was sind und was sollen die Zahlen ? " in which 

 the author expressly rejects all theories of irrational 

 numbers based on the assumption of measurable quan- 

 tities. That he is right in so doing must be admitted by 

 all who reflect on the subject with sufficient attention. 



These observations, as already remarked, are not in- 

 tended to detract from the undoubted merit of the book. 

 Arithmetic is a thorny subject, the very elements of which 

 abound in points of great difficulty and delicacy ; and any 

 serious work on the science is certain to contain passages 

 giving occasion for criticism or controversy. MM. 

 Laisant and Lemoine deserve our gratitude for having; 



