June 18, 1909] 



SCIENCE 



975 



of cathartic and exercise," designed, that is, 

 to aid the beginner in the discovery of para- 

 logisms and in the discrimination of true 

 principles from the specious and false; the 

 still extant " Data " ; the book " On Divisions 

 (of Figures)," which though "lost in Greek," 

 " has been discovered in the Arabic " ; the three 

 lost books of " Porisms " ; the two works en- 

 titled " Surface-loci " and " Conies," both 

 lost ; the still extant " Phsenomena," an astro- 

 nomical work ; and the " Optics," edited by 

 Heiberg in 1895. In the two chapters (27 

 pages) devoted to Greek commentators upon 

 Euclid and especially to Proclus and his 

 Sources, Dr. Heath has given not only de- 

 tailed citations of the immense body of litera- 

 ture bearing upon these matters, but — what is 

 far more — a most luminous and valuable 

 digest of it all. Then follows the chapter of 

 18 pages dealing with the text. Here we have 

 an interesting account of the several cele- 

 brated manuscripts from which knowledge of 

 the content of the " Elements " is mainly de- 

 rived, a statement of the critical principles 

 followed by Heiberg in the comparison and 

 evaluation of the sources, and a good indica- 

 tion of the ingenuity, of the prodigioui 

 scholarship, labor and devotion that enabled 

 Heiberg to produce his monumental work on 

 Euclid. This work was published between 

 1883 and 1888. It is the "definitive text" 

 contained in it that Dr. Heath has translated 

 into English and that his scholarship has 

 enabled him to set in the light of modern re- 

 searches into the foundations of geometry. 

 In view of the empire that Euclid has exer- 

 cised in British education, it is especially in- 

 teresting to learn, in the chapter on Trans- 

 lations and Editions, page 95, that the great 

 Greek classic, or some portion of it at all 

 events, found itself in English dress as early 

 as the first half of the tenth century. In sup- 

 port of this contention. Dr. Heath cites the 

 following quaint lines from " Eara Math- 

 ematica " : 



The clerk Euelide on this wyse hit fonde 

 Thys craft of gemetry yn Egypte londe 

 Yn Egypte he tawghte hyt ful wide, 

 In dyvers londe on every syde. 

 Mony erys afterwarde y understonde 



Yer that the craft com ynto thys londe. 

 Thys craft com into England, as y yow say, 

 In tyme of good kyng Adelstone's day. 



If among the many students of recent Amer- 

 ican and English, Italian and German work 

 in the foundations of geometry, there be any 

 one who imagines that nearly all of the fine 

 things presented in such work are new, there 

 await him in the closing chapter of Dr. 

 Heath's introduction a pleasant surprise and 

 a happy emancipation. Indeed this chapter 

 of 38 large octavo pages is an exceedingly val- 

 uable contribution to the historico-critical 

 literature that pertains to the common ground 

 of logic and mathematics. In it is set forth 

 in clear and orderly fashion the best thought 

 — and how fine and penetrating much of it is ! 

 — of pre-Euclidean and post-Euclidean philos- 

 ophers and geometricians from Plato and 

 Aristotle down to Proclus — a period of nine 

 centuries — ^the best thought, I say, of the best 

 intellects of antiquity regarding such eternally 

 interesting matters as the nature of scien- 

 tific (and especially) geometric elements, of 

 the nature and significance of axioms (or com- 

 mon notions) and definitions, of hypotheses 

 and postulates, of existence assumptions and 

 existence proofs, of theorems and prohlems, 

 lemmas, cases and porisms, of methods of 

 argument and demonstration, ohjection, re- 

 duction, analysis and synthesis. Much that 

 has been recently written about these things 

 is new and much of it is but repetition — 

 repetition that is sometimes inferior in point 

 of form and doubtless often unconscious. It 

 is no small service to give the reader, as Dr. 

 Heath has here done, a lively sense of the 

 scientific atmosphere in which Euclid wrote 

 and of his indebtedness and through him that 

 of all subsequent time to his predecessors and 

 contemporaries. Aristotle's statement, quoted 

 by Heath from the "Posterior Analytics," 

 that, " other things being equal, that proof is 

 the better which proceeds from the fewer pos- 

 tulates or hypotheses or propositions," reminds 

 one of the famous dictum enunciated sixteen 

 hundred years later by the " Doctor in- 

 vincibilis," William Occam : Entia non sunt 

 multiplicanda prceter necessitatem. Nothing 

 conveys better the animating spirit of the 



