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532 CBITICAL NOTICES : the word inroOeo-eis occurs we find the same interpretation adopted by Mr. Burnet. Aristotle's words are : i; yap apfrr/ KOL /xo^^pi'a T?)V "PX*? 1 ' ^ t** v ^Sfipfi rj Sf o-(i>ei, iv Se rat's Jrpdfeo-i TO ov evtxa apr/, oxr- Trep fv TOIS /J.a.Orjfj.a.TiKoi'i o.l vTTodftreis ovTf Sr) tKtl 6 Adyos StSao-KaAtKos TU>V ap^tav ovrf (VTaWa, dAA' dperJ) rj <f>v(riKrj f/ (Ourrrj rot 1 opOoBoieli' epi rrjv apxyv. Mr. Burnet comments as follows : " The second inter- pretation suggested by Prof. Stewart, though with some doubt, seems to be certainly right. An wrofeo-is in mathematics is cer- tainly the assumption of the thing to be proved or the thing to be done from which an analytical proof starts." Although, of course, Mr. Burnet cannot be assumed to assent to Mr. Stewart's reasons, it is perhaps worth while to consider these, since Mr. Burnet gives no additional reasons of his own. wrd$eo-ts in the present passage must mean either (1) MP, an ultimate dp^ of the science, or (2) SP assumed to be true for purposes of analysis. Mr. Stewart holds that (1) is not in accordance with strict Aristotelian usage, on the strength of three passages in the Post. Anal., 72 a, 14, diieVoi' 8" dpx^/s OTjAAoyioriK^s Oc&tv [lev Aeyo> f/v /xj/ tori 8eiai, 1078' dvayxjj e (lv TOV faaOrjaofnevov n rji' 8' avdyKy ii.v TOV oriovv /xa^^tro'/xevov, dfi'vifj,a. . . . @t(rew<; 8 r/ tiev mrorepovovv rSiv iiopiW T^S u7ro<av(ra>s Xafj-fldv- ovfra, olov A.eyw TO flvai TL 17 TO /j.rj elvai TI, vw6&f(ri'i, r/ 8' aveu TOVTOU opio-ynos : 92 b, 15, Tt /xer yap trrj/MilveL TO Tptytovov eAa^Sei/ 6 ye</)/xcVpi7S OTI 8 co-ri StiKvvai : 76 b, 35, ol fiev ovv opoi OVK fltrlv wroOeirfig ovSev yap flvat rj /njj eTvat Xiyovrai . . . Tors 8' opous /xdvov ^vvitcrOai. 8ei. No doubt the second of these passages seems to say that the geometer proves the existence of his subject S. But it is clear from the context that rpiywvov must be taken as an example of P, not of S (see Zabarella's Commentary). The words then merely mean that in geometry you assume the meaning of P, and then prove that it exists, i.e., that S is P. The third passage merely repeats the dis- tinction explained in the first. Thus the suggestion that the (9eWs of geometry are opttr/xot alone, not t)7ro$<Ws, seems to fall to the ground : and indeed it is obvious that no science can prove any- thing about S unless it assumes its existence, though of course the existence of S may be so obvious as not to require explicit state- ment. It would be easy to bring forward many proofs that the apai of the synthetic process in geometry are -ra-otfeWs + opwrttoi : perhaps the following words taken from the same chapter of the Post. Anal, as Mr. Stewart's third quotation will suffice : lo- 8' iota tiv Kal a A.axi/JaVeTai tlva.1, Trepi a rj 7rMrj-?;/xi7 Oetapel ra {nrap^oi'Ta KO.O avra, OLOV /xovdSas 17 apiB/j.rjTiKrj, rj 8e ye<o/xTpia <Tr)[ji,(ia KOL ypaii/tas. TavTa yap Aa/x/JaVovo-i TO elvat KOI roSi elvat (76 b, 3) : again, the words that immediately follow Mr. Stewart's third quotation, dAA' otriav ovTtav r<3 eKfiva tivai ytVcTai TO o-iiyUTrt'patrtia. ovB' o y jjfvSrj vTroriflerai . . . (76 b, 38). So much for the general question as to the meaning of vTro&. In the present passage in Eth., vii., the evidence is on the whole in favour of taking ai vjro0ms to mean the principles of mathe- matics. As Mr. Stewart admits, the words oiVe Si) eKel b Adyos