NATURE 



21 



THURSDAY DECEMBER 12, 1895. 



THE HISTORY OF MATHEMATICS. 

 GeschicJite der Mathematik ini Alterthum und Mittel- 



alter. Vorlesungen von H. G. Zeuthen. Pp. viii. -f- 344. 



(Kopenhagen : Host, 1896.) 

 A Primer of the History of Mathematics. By W. W. 



Rouse Ball. Pp. iv. + 146. (London : Macmillan and 



Co., 1895.) 



THE first of these books is a translation, with some 

 alterations and additions by the author, of a work 

 originally published in Danish in 1893. Its aim is to 

 supply students and teachers of mathematics with trust- 

 worthy information about such parts of the history of 

 the subject as are really important for them to know ; 

 and, in particular, account has been taken of the fact 

 that candidates for the Danish teachers' certificate are 

 expected to show an acquaintance with the text of 

 Euclid's Elements, as well as a general knowledge of the 

 history of mathematical science. This plan has been 

 effectively and judiciously carried out, and the result is a 

 work of permanent value and interest, admirably suited 

 for the class of readers for whom it is designed. 



Dr. Zeuthen has very wisely adopted the course of 

 tracing the lines of formal development along which the 

 science of mathematics has progressed. This does not 

 exclude the special consideration of the works of mathe- 

 maticians of first-rate importance, while at the same time 

 it brings clearly before us the different springs and rivu- 

 lets, so to speak, which have converged into the broad 

 and deep streams of modern geometry and analysis. 

 Thus we are shown how the speculative and logical 

 intellect of the Greeks built up a system, mainly geo- 

 metrical, which at last seemed to become inert from the 

 very perfection of form which it had attained ; how the 

 decimal notation of arithmetic and the elements of 

 algebra, slowly evolved by the Indians, eventually made 

 their way into Europe ; and, finally, how mathematical 

 science, preserved from extinction by the Arabs amid the 

 ignorance and bigotry of the dark ages, woke up to new 

 life in the thirteenth century, and grew slowly but steadily 

 thenceforth, until the analytical method of Descartes and 

 the invention of the infinitesimal calculus marked the 

 beginning of a new era, the splendour of which has 

 perhaps unduly obscured the achievements of all the ages 

 which went before. We are all too apt to forget what 

 we owe to our forgotten, unhonoured ancestors ; and it 

 is well that we should be now and then reminded that 

 the exact science to which we are indebted for mos 

 our " triumphs of civilisation," and which is ruly the 

 finest product of the human mind, may be traced back 

 to the dawn of history and the rude efforts of primitive 

 man in the arts of counting and measuring. 



But to return to Dr. Zeuthen's book. After a few 

 introductory pages, devoted to VorgescJiichte and the 

 Egyptians and Babylonians, we reach the longest and 

 most interesting section of the work —that which treats 

 of Greek mathematics. Here we have a clear, well- 

 proportioned outline of the progress made from Thales 

 to Diophantus, and of the scope and methods of Greek 

 geometry and arithmetic, together with a sufficient 

 NO. 1363, VOL. 53] 



analysis of the works of Archimedes, Apollonius, Dio- 

 phantus, and more particularly Euclid. An interesting 

 account is also given of the researches connected with 

 the three famous problems of antiquity — namely, the 

 trisection of an angle, the duplication of a cube, and the 

 quadrature of a circle. (Archytas's construction for the 

 second of these problems (pp. 84-5) is not very easy to 

 follow : it would be an improvement to give a figure 

 showing the circle traced out by the point Y.) 



The amount of space given to the discussion of Euclid's 

 Elements is justified by the importance of the subject. 

 Dr. Zeuthen's criticisms of the definitions and postulates, 

 and of Euclid's geometrical presuppositions, are very 

 instructive ; and his analysis of the contents of books v., 

 vii.-xiii., ought to be of great service to students of the 

 original text. There is only one point about which we 

 would venture, with all respect, to differ from the author's- 

 conclusions. Dr. Zeuthen appears to regard the arith- 

 metical books, and in particular book vii., as containing 

 a theory of commensurable quantities ; so that, for 

 instance, the theory of proportion in book vii. is merely 

 the substance of book v. adapted to commensurable 

 ratios, and was composed because the general theory 

 of proportion was still unfamiliar. This is not very 

 plausible, prima facie; -and it is, we think, disproved by 

 the facts of the case. 



The second definition of book vii. undoubtedly means 

 "A number is an assemblage (ttX^^ov, not fiiyedos) com- 

 posed of units " ; and if the word fiovds has the same 

 meaning in the first definition that it has in the second 

 (and this is almost necessarily the case), the true sense of 

 the first definition will be "Any object whatever is a 

 unit [or is regarded as a unit] when it is spoken of as 

 one." If this be so,4it is quite clear that we have to deal 

 with a purely arithmetical theory ; and there is nothing 

 whatever in the form or matter of the seventh book in- 

 consistent with this conclusion. It may be added that 

 the definition of proportion in the seventh book is 

 essentially distinct from that in 'the fifth ; and that in 

 some MSS., at any rate {rf Gow's " History of Greek 

 Mathematics," p. 74, note 3), numbers are represented 

 by dots, and not by lines. 



On the other hand, the tenth book does deal with 

 quantities, and begins with a definition of commensurable 

 quantities ; a definition which, according to Dr. Zeuthen's 

 theory, ought to have been given long before. It is 

 quite possible that Euclid perceived the analogy between 

 commensurable numbers and commensurable magni- 

 tudes ; whether he realised the ratio of two commensur- 

 able quantities as a fraction in precisely the same wa 

 as we Qo s doubtful : it is almos certain that he ha 

 no conception of surds as numerical quantities. 



The Indians did not scruple to introduce irrational as 

 well as negative numbers ; by so doing, and by their 

 invention of the decimal system of notation, they 

 immensely extended the range of analysis, probably 

 without knowing what they did. Except in the region 

 of diophantine analysis, their particular discoveries are 

 comparatively unimportant. However, Dr. Zeuthen does 

 them full justice for all their researches ; and we are 

 glad, also, to see that he has a kindly word for the 

 Arabs, who not only guarded the treasures of learning, 

 but added something of thoir own to the store. Thus 



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