172 



NA rURE 



\yune 20, 1889 



plate following we have the magnified details of the 

 mouth-plates, the supradorsal membrane, the adambula- 

 cral plates, and other characteristic portions. 



The publication of this Report cannot fail to give a fresh 

 stimulus to the study of this hitherto rather neglected 

 group of the Echinoderms, and the best thanks of every 

 student of natural history are due to Mr. Sladen for the 

 thorough and honest manner in which he has accom- 

 plished a troublesome and arduous task. 



GREEK GEOMETRY FROM THALES 

 EUCLID. 



TO 



Greek Geometry from Thales to Euclid. By Dr. G. J. 

 Allman, F.R.S. (Dublin : Hodges, 1889.) 



THE subject-matter of this work has at different times 

 been brought under the notice of the readers of 

 Nature, for it is very little more than a collected and 

 corrected reproduction of papers which have at varying 

 intervals appeared during the last eleven years in 

 Hermathena. In all our previous notices, we believe, we 

 strongly insisted upon the desirability of Dr. Allman's 

 giving a permanent form to his labours, which should 

 render his brilliant achievements the more readily acces- 

 sible to mathematical and, we may say also, to general 

 readers. Hitherto all the original investigation in this 

 direction has been carried on by German, French, and 

 Danish writers, for Mr. Gow's " Short History of Greek 

 Mathematics," interesting though it is, is confessedly not 

 founded upon independent research, nor does Mr. Heath's 

 " Diophantus," concerned as it is with Greek algebra, form 

 exception to our statement. In the historical domain of 

 mathematics, Montucla held sway until quite recently, and 

 even the latest French work, by M. Marie, the outcome of 

 forty years' travail, holds fast by him, so that Heiberg 

 (quoted by our author) writes : •' The author [Marie] has 

 been engaged with his book for forty years : one would 

 have thought rather that the book was written forty years 

 ago.'' Far different is the case with Dr. Allman : all along 

 the line of his labours he has consulted the original 

 Greek authorities, and fought every inch of the ground 

 with such experts as Heiberg, Bretschneider, Cantor, 

 Tannery, and several other writers we could name, many 

 times adopting their results, but in nearly as many cases 

 putting forward and convincingly maintaining views of 

 his own. In evidence that the views we have all along 

 held of the importance of this contribution to our know- 

 ledge of the early Greek geometers was not a singular 

 one, we have now the confirmation of the favourable 

 reception the papers in their original form met with 

 from many competent authorities on the Continent and 

 elsewhere, the outcome of which has been the present 

 handy volume. Dr. Allman states that " it has been, 

 throughout, my aim to state clearly the facts as known 

 to* us from the original sources, and to make a distinct 

 separation between them and conjectures, however prob- 

 able the latter might be." This testimony is, we believe, 

 true : certainly the reader is put in possession of the 

 facts so far as they are at this date obtainable. 



We may just call to mind the points discussed. In an in- 

 troduction the authorities on the early history are named : 



had Eudemus's history come down to us we should possibly 

 have had a summary of the period treated of here, but 

 now we are dependent upon Proclus. Then the work 

 of Thales, of Pythagoras and his school, of Hippocrates of 

 Chios, of Democritus, and of Archytas, is clearly dis- 

 cussed in Chapters I. to IV. In Chapter V., as we showed 

 in a former notice, ample justice is done to Eudoxus, and 

 his right place in the history of science is duly assigned. 

 " In astrologia judicio doctissimorum hominum facile 

 princeps," writes Cicero ; in his " Histoire de I'Astronomie 

 ancienne" Delambre has, "rien ne prouve qu'il fut 

 geometre " ; and even De Morgan writes, " he has more 

 of it [of fame] than can be justified by any account of his 

 astronomical science now in existence." M. Marie is 

 more just ; though he devotes only two pages to the 

 account of his work, he remarks, " il n'dtait pas au reste 

 moins bon geometre que bon astronome " (cf. Delambre, 

 supra). Had Dr. Allman done no more than reinstate 

 in its proper place a name " highly estimated in antiquity," 

 this would have been a raisoti d'etre for his work. We 

 must remember, however, with regard to this tardy act of 

 justice, that " it is only within recent years that, owing to 

 the labours of some conscientious and learned men, 

 justice has been done to his memory, and his reputation 

 restored to its original lustre." In the following chapters 

 (VI. to VIII.), we have accounts of the successors of 

 Eudoxus, viz. Mena^chmus, Deinostratus, and Aristasus. 

 The concluding chapter takes up the work of Theastetus, 

 and herein we have a discussion of the part which Euclid 

 himself most probably contributed to his well-known 

 " Elements." 



All readers of this standard contribution to the early 

 history of geometry, which has placed its author in the 

 first rank of writers on the subject, and thereby brought 

 credit to the whole body of English-speaking mathe- 

 maticians, must hope that Dr. Allman will not lay his 

 armour down, but that, after a brief respite it may be, he 

 will undertake some such work again on a kindred 

 subject. We would have suggested a careful edition of 

 the text of Euchd had not labour in this direction been 

 anticipated by Dr. Heiberg in his recently completed 

 edition of the " Elements." 



A bust of Archytas, from Gronovius, forms the frontis- 

 piece, a few notes are appended at the end to bring infor- 

 mation as to books and editions up to date of issue, and a 

 full index completes the volume. 



One of the notes (p. 218) on "the theorem of the bride" 

 is very interesting to us. On pp. 633, 637, of " Clifford's 

 Mathematical Papers," we have given footnotes on the 

 term " the figure of the bride's chair," which Clifford 

 evidently used for a particular figure of Euclid i. 47. We 

 had an idea at the time of writing the notes that the term 

 ought to occur in Arabic, and so made application to Mr. 

 Spottiswoode (a fair Arabic scholar himself), and through 

 him to Oxford authorities, but no one could identify the 

 expression. Dr. Allman notes : " M. Paul Tannery (' La 

 Geom^trie Grecque,' p. 105) has found in G. Pachymeres 

 (• MSS. de la Bibl. nationale ') the expression to dfdyprjiia r^s- 

 vvfKpTjs to designate the ' theorem of Pythagoras.' " This 

 seems to point to the old Egyptian idea as handed down 

 by Plutarch (cf. Allman, pp. 29-32). 



R. T. 



II 



