On Certain Mathematical Terms in Aristotle s Logic - Part I

7/29/2019 On Certain Mathematical Terms in Aristotle s Logic - Part I

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On Certain Mathematical Terms in Aristotle's Logic: Part IAuthor(s): Benedict EinarsonReviewed work(s):Source: The American Journal of Philology, Vol. 57, No. 1 (1936), pp. 33-54Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/289829 .

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ON CERTAIN MATHEMATICAL TERMS INARISTOTLE'S LOGIC.1

PART IMathematics in Aristotle's time was being reduced to a strictlydeductive science.2 It was, moreover, much cultivated in theAcademy, where important work was done.3 It is not surprising,then, that it should have exercised a great influence on Aristotle'smethod, ideas, and terminology, as he worked out his science of

deductive proof.As nothing but scanty fragments of pre-Aristotelian mathe-matics remain, it is often difficult to find external evidence forthe existence of a given mathematical term during or before thetime of Aristotle. The fragment of Hippocrates of Chios,4 ifthe language is Hippocrates' and not Eudemus',5 shows that theterm VrOKEtrat and the designation of parts of figures by letters,together with certain syntactical and stylistic characteristics,were in use before Plato's time; 6 but for such important

This study was begun during my tenure of a Sterling Fellowship atYale University, and completed while a Junior Fellow at Harvard. Itake this opportunity of thanking Professor W. D. Ross for his greatkindness in reading through an early draft of the manuscript and givingme the benefit of his detailed criticism.3 Cf. J. L. Heiberg, " Mathematisches zu Aristoteles," p. 4 f, in Abh. zurGesch. d. math. Wiss., XVIII (1904), and H. G. Zeuthen, "HvorledesMathematiken i Tiden fra Platon til Euklid blev rationel Videnskab," inM6moires de l'Acad6mie Royale des Sciences et des Lettres de Danemark,Section des Sciences, 8me serie, t. I, no. 5 (1917).8A list of mathematicians connected with the Academy is given byProclus (In Primum Euclidis Elementorum Librum Commentarii, p. 67,8 ff. [Friedlein] hereafter cited as Proclus, in Eucl.). Cf. P. Tannery,La g6ometrie grecque, pp. 130 ff.'Ap. Simplicius, Comm. in Aristotelis Physica [Diels], pp. 61 ff.Latest edition: F. Rudio, Der Bericht des Simplicius iuber die Qua-draturen des Antiphon und des Hippokrates, Teubner, 1907.6 Even here we cannot be sure that the expressions are Hippocrates'and not Eudemus'; eyypdietv is used for 'inscribe' in accordance withAristotle's usage and presumably Eudemus', while Plato has e'7relveLv.6For v7roKeTCra cf. p. 58-17, 64-12, Rudio; for the use of the letters withthe article only cf. 58, 14R; 62, 19R; for the construction rTo ' , Aor e0' oi A cf. 58, 6R; 60, 21R. Further, the use of eaTr (59, 5 Rudio)KelaOO (58, 9R), ri-trrE e'rl c. ace. (58, 12R) can be noted.3 33

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BENEDICT EINARSON.mathematical terms as AhyiX/a,d6wmita,aUTr)yta, rporaatg, avrtcrTpeo/w,EKOCats, nd av/ rEpaojYawe have only the testimony of post-Aris-totelian mathematicians, themselves so thoroughly influenced bythe Aristotelian terminology that they cannot be consideredreliable witnesses to a purely mathematical tradition of language.A further difficulty is the fact that Euclid, Archimedes, andApollonius had but little occasion to put into writing certainterms for general mathematical procedures, such as avlTrePpaaola,avaXAvtv, and aVTVCTpeCIELv, although these terms were current intheir times.7 In much the same way, an orator will only inci-dentally, and as it were by accident, use the technical languageof rhetoric in a public address. Much of the older terminologymay have perished completely,8 as Euclid was the standard for alllater ages,9 and his systematic and well organized terminologydispensed with much of what had been current before his time.It is obvious, for instance, that he avoids the word opos, for termof an avaAoyta, which occurs in Plato and Aristotle,l1 but onlyonce in the Elements; 1 that he prefers r-yovelEvo and E7roeiV01o(antecedent and consequent) in the discussion of progressionsto their equivalents uetgwv and EXarTTv; 12 and there are indi-

7For davaXv?L, dvaLXveLvnd avurepaaota cf. s. vv. in Heiberg's index, vol.III of his edition of Archimedes; for dvTaffpo(0f cf. Apollonius, Conica, I,p. 284, 19, Heiberg.8The words eyTeiVW,or inscription, and 7raparelvw, or the Euclidean,rapapd\XXw, occur to my knowledge only in Plato (Meno, 87a, Rep. 527a)and writers dependent on him (as Proclus, in Eucl., pp. 79-80). Theexpression e,UdCOoyevSect. Can., Euclidis Opera, viii, p. 162, 3 and 164, 8[Menge-Heiberg]), to refer to a previous theorem, is found only oncein Euclid's Elements in an interpolated passage (Elem., X, 10 [iii,p. 32, 15, Heib.]; cf. id. Prol. crit., V, lxxxi). It may very well beof considerable antiquity, as it is easily connected with zaOr?acrainthe sense of mathematics. Certain words, such as erep6otZces, 6po3POs,and rpareitov, occur but once in Euclid (cf. Heiberg, Math. zu. Ar., p. 11).I For Apollonius cf. T. L. Heath, Apollonius of Perga, p. xcv. Thedependence of Archimedes is shown, among other things, by his exactagreement with Euclid in the terminology of proportion.0 Aristotle, E. N. 1131b5 ff.; Plato, Phileb. 17cll (applied to music)." V, def. 8.12 In discussing geometric progressions Euclid often uses concomi-tantly two sets of designations to distinguish the terms: Jeitlwv andXiaTrwdv,and 77yoLfevos and e7ro6evos. As his phrase is (Elem., VII, 21,24, 30, 34; VIII, 1, 4, 8, 21) 5 re ,eiwsv Trov geilova Kal 6 eXaTwV TOVeXarTova, TOVTerOTtv ' Te 7'yovI!Levos TOP i1yovoevov Kal 6 er6oUevoS TOv eriolevov,

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HEMHEMATICAL TERMS IN ARISTOTLE'S LOGIC.cations that Xoyos (in the sense of ratio) has been substituted,most probably by some predecessor, for taaorrVVa,which does notoccur in this sense in the Elements.22aAristotle's terminology, as has been recognized,l3was in partit is likely that 4iyo6ulevos, nd Ecro/6evosare meant to explain LetL4wv andeXarrwv (which are strictly after all subject to misapprehension, as"greater " and "smaller" might be taken to refer not to the twonumbers in the same ratio, or Xo'yos,but to the actual size of numbersanywhere in the ivaXo'yia). Meliwv and e\Xdrrw,T then, as requiringexplanation, would be the older terms, and this presumption is con-firmed by the fact that 9-yo6/ievosand CTrofJevosre sometimes used alone,as being the more natural terms (IX, 12, 16, 17, 19, 36). In the theoremto which these words refer, however (VII, 20), tuei4wvand eiXcaao areused alone, as it was no doubt evident from the proof what was meant.Further in the theory of alternate and inverse ratio (Eucl. El., V,def. 12 and 13) antecedent and consequent are more convenient terms than,eitlov and `XarroY.

12a Atisr7txa and 5pos were used in the descriptions of intervals inmusic (Plato, Philebus, 17cll; Aristoxenus, Harm., 70, 25 [Marquard]).In Archytas, fr. 2 we have feto'vwv opwv 8stdai-rua (p. 335, 4 and 12[Diels]) used generally to cover all three progressions. The term forratio in Aristotle is X6yos (E. N. 1131a31). In case 6tdartz,a was theolder word (as the presence of such a term as 5pos, literally "boundarystone "-to mark off the distances-would suggest), and X6yoswas latersubstituted for it, we should have an explanation of the origin of thephrase 8trrXaatw,v6yos used by Euclid (Elem., V, def. 9; cf. def. 10;VI, 20; X, 9). Taking a geometrical progression represented by equaldistances (6taar-jutara) between points representing the terms (opot),we should find that by doubling the distance we obtained a duplicateratio, that is, a2: b2 instead of a: b. The ratio a2: b2 would be a8it7rXdcriov i&atdrTca, then, and by substitution of X6yos, a &trXaaiwv Xoyos.That with the word Xoyosalone, unaccompanied by any notion of distance,we should not obtain such a meaning for " double ratio," appears fromthe fact that S&rXdialosX6yos actually does occur in a different sense,and that it was to distinguish the two uses that the form 8t7rXaaiwvwasintroduced (cf. Sir T. L. Heath, The Thirteen Books of Euclid's Elements,II2, 133). Cf. also Paul Tannery in Bibl. Math. 3, Folge 3 (1902), pp.162 ff. It was perhaps through a rptwrXdatov 6tadrlama that Hippocratesconceived the idea of reducing the problem of duplication of the cubeto that of finding two middle proportionals between a given line andits double. Here the proportion would be a: x:: x: y:: y: 2a, and2a: a is the "triple distance" of x: a, or x3: a3. Then x3 equals 2a3,and x is the side of the cube whose volume is twice that of the cubewhose side is a.

18 Diels, Elementum, p. 20.

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BENEDICT EINARSON.developed in the Academy. But the Academy produced severalmathematicians of importance,14 and its members must haveused mathematical terminology either traditional or devisedin part by themselves.15 The difficulty then arises, for wordstraceable in a mathematical sense to the Academy and in adialectical sense to Aristotle, as to which is the originalmeaning; and it is usually extremely difficult to decide, as in thefrequent absence of external evidence the origin of the termmust be determined by its own superior applicability to the onefield or the other.The words avvdAmts,vaXvetv, and avaXvrtKa are almost certainlyof mathematical origin, as was recognized long ago.16 In E. N.1112b20 the mathematical connection of the word is clear: o yapI3ovXEv6o'eVOg E'OLKe T7TEtVKalt aaXkVELvOv efpWd/EvOV poT7rov (i. e. asdescribed ib., 15, working back from the result aimed at) owarep8LtypauFLa.17 The analysis as described here correspondsto thedescription of mathematicalanalysis given by Pappus.18 Proclusattributes the discovery of this procedure to Plato.l914Theudius of Magnesia is supposed to have written the text-book ofthe Academy (Sir T. L. Heath, op. cit., I2, p. 117), and Menaechmusis supposed to have been the first to deal with the conic sections (thefragments have been collected by M. C. P. Schmidt in Philologus,XXXII (1884), pp. 72-81).15Cf. Heiberg, op. cit., who believed that it was through Plato'sinfluence that a7/zelov was finally substituted for acrtYii (p. 8).16 Blancanus, Aristotelis loca mathematica, p. 35 f. (Bologna, 1615),suggested the mathematical derivation and is followed by Waitz(Organon, I, p. 366), and F. Solmsen, Die Entwicklung der aristotelischenLogik und Rhetorik, p. 123 and n. 2. Cf. Susemihl and Hicks, ThePolitics of Aristotle, note on 1252a18.

17 Cf. Soph. El. 175a26 ff. avouLaiveL 7roreKaOaCrepv rTOs aa'ypa/Aaowtv:Kai &yapKeL dvaXaravTes eviore c'vvOeivai7radXvdvva7rov0ey . . for themathematical connection of the word."8Pappus, Collect., VII (vol. 2,p. 634 ultsch) : &vdXvo-usotvvva-rT, 68bsarbTOU VTroVlvOV 'SdO\Xo7OVO^VOV 6&&TWV i 7) dOKOXO6OWVlr 6oJoXo0oyoLevov oav-0t&rei' jv Vv yap T?r dvaaX6oeE rTb fTro6JLevov Ws yeyovob viroOCtlevo rTb It o5 TOUTO

v,lt3alves oTKO7rovLeSAcaKatl rdXLv EKievov rTbTpo7oyfolevov, &wsav oirws draxroSlfovresKaTavT'rrawlUJev ert r WV 716S73yPVopLromlvwv lX rdtv dapXPs IX6v-Twv. Cf. F. M.Cornford's excellent discussion of this passage in " Mathematics andDialectic in the Republic VI-VII," Mind, XLI (1932), pp. 46 ff.1" Op. cit., 211, 21 (be60o5os io 8a T7iS dvaXvcrews) 'vz Kat 6 IIXcirwv Wsqbao'tvAewsdiALavrL rapaseUKwer. Whether Plato actually discovered it ornot, we may be fairly certain it was used by Leodamas.

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MATHEMATICAL TERMS IN ARISTOTLE'S LOGIC.In the game of Socratic debate as practiced in the Academy,it was the custom to set up a statement, such as "pleasure is

good," called Oemtsor 'rpo4pX',pa,2a which was attacked by oneparty, the questioner, and defended by the respondent. TheTopics is a handbookfor this kind of dialectic, and the point ofview of the dialectician is always taken into account in thePrior Analytics. It is clear that the questioner would have todiscover his arguments by a method exactly parallel to that ofanalysis in geometry: he would have to take the conclusion hewished to establish and work back from it until he reached agroup of premises he felt sure would be acceptable to his inter-locutor.Such would perhaps be the most natural explanation ofavaAXvmtnd aYvaXvrca as used of the syllogism, and no doubtthis sense was present in Aristotle's mind as he worked out hisdoctrine, but the actual usage of the word in the Analytics issomewhat different.The passage from 46b38 to 51b5 of the first book of the PriorAnalytics, where avdXvats s discussed, is introduced with the fol-lowing words: " It is clear from the foregoing what the elementsof demonstrations are and why, and further what things wemust consider in the case of each 7rpo'pXqua i. e. thesis [orconclusion] determined in quantity and quality); it remainsto say how we shall reduce (ava'opsev ==avaAvcro/Ev) reasoningsto the aforementioned figures, for this part of our investigationstill remains to be treated. For, if we should both understandthe way syllogisms come into being, and should be able to dis-

20Topics, 120b18; cf. 104b7, 104b35. The reasons for believing thatthe dialectical game was practised in the Academy are (1) Plato'sinterest in the procedure and (2) the presence of many titles dealingwith dialectics in the works of Xenocrates and Aristotle (for Xenocratescf. Diog. Laert., IV, 13: XVaLsrGv irepl robs X6yovs L'; Xvkaesa's'; Oe'aewv,tptXla Kqty'; 7rS 7repi rb BtaXeTye(OarrpayiarTelas tiXia, la, ia',a,91'; forAristotle we have in the Topics a dialectician's handbook, or a re'Xv7tLaXeKTLKt, and among the lost works such titles as bro,trjuara e7nrXeip1l-ZarIKd Y' (D. L., V, 23), epwrTjfewS Kcai roKp1lwcos a'j' (D. L., V,23), ffOaeLs ew7rXeLp77parTKaL', OecretLsprpTKai 8', eO&ES tLX\Kal8f, Oeaetswep'i 'vxj7s a' (D. L., V, 24), to go no further. The game as presentedin the Topics must have centered in the Academy, as the works ofPlato are everywhere touched upon. Cf. E. Hambruch, Logische Regelnder Platonischen Schule in der Aristotelischen Topik, Programm, Berlin,1904.



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BENEDICT EINARSON.cover them (cf. the use of EVplaKoin rhetoric), and in additionshould (be able to) reduce reasonings that have already beenmade to the aforementioned figures, our initial purpose wouldbe attained." 21

In these words Aristotle is proposing to take ovXXoyiaLUoIasfound and reduce them to the three syllogistic figures. He ishere using the word avXXoytLcraosoosely, for reasoning in general,and is probably referring especially to a method of taking anargument found in some philosophical or technical author andreducing it to a syllogism in one of the three figures. Similarly,the works entitled 7rpordcaE 22 probably contained collections ofpremises gathered from the writings of philosophers and thegenerally received opinions of mankind, for use as referencematerial in preparationfor debates or as ZvSo$a o be discusssedand reconciled in a philosophical treatise.23 Jaeger has calledattention to Aristotle's recasting a passage of the Euthydemusinto syllogistic form.24The procedure recommendedby Aristotle for the analysis ofsyllogisms is first to discover the two premises-supplying anythat are lacking if necessary-and next to find the terms, that

21As my rendering is more of a paraphrase than a translation, Iappend the Greek: KKTvwv .v oiv atlaciroSeles ylvovrJaTal 7rTwKaiels oroiaPXevrrTovKaO'KKaTTovp63Xraj,ia,qavep6vKrTvelp,fugvwvvr&s8' avdsolzevrois

v\XX\oyLrLouJosls rT 7rpoELpr7/gJvaX/zLara,XeKTOV&vet1 /ALETaaTra- XOLtc6 apeTt7OUTO TjS OKEIWEUs. el &yap rTV Te Y've'itv rTWv 'v\XX\o'yL7zTv OewpOLiALev Kal TroepiptoKEiv EXOI/Uev bva/.L, Ert & roi)s 'yeyetv,Aezvovs avaX6otLev els r& TrpoeLp77ihvaoX?r ara, rTXos &v e'XoL e' aipXs Trp6Oe?Ls. The TrpoOeaLs is very much likethat in Topics, 0lbll-13. We may compare Aristotle's yeyev?7,rU/Lvovwith the yeyevia0Ow or yeyoverwt of analytical proof in mathematics: cf.Menaechmus ap. Eutoc., vol. III, p. 78, 16, Heiberg; Archimedes, vol.I, pp. 202, 10; 184, 5; 206, 24, Heiberg.

22 Cf. D. L. ap. Rose, pp. 3-9, lines 34, 46, 47, 67; Hesychius, ib.,pp. 11-18, lines 34, 38, 44; Ptolemy, ib., pp. 19-20, lines 63, 79, 80.23The constant attention paid to 'WvoSa is shown by the frequencyof the verb SOKeE,which introduces them. (Cf. Bonitz, Index, 203a27-37.)This would make it probable that Aristotle had collections of theseopinions ready to hand. Cf. Topics, 105b12, eK\XyeLv 6e Xpi (sc. rasIrpordaaeLs) Kai eK TlV 'yeypaCziievwv kX6ywv, KTX.24Jaeger, Aristotle (English translation), p. 62: "... and since thewords are not a plain citation from the Euthydemus, but a compressionof Plato's exposition into several fairly long syllogisms. ..."

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MATHEMATICAL TERMS IN ARISTOTLE'S LOGIC.which occurs twice being the middle, and the others being easilyidentifiable from their position in the conclusion.25The analogy of this with mathematical " analysis" is clear:we have the reasoning, especially the conclusion,26 given: theproblem is to find the syllogism which produced it. In thiscase Aristotle is using avar5Xvtlof passing from the conclusionto the principle from which it is deduced, and thus connectingthe conclusion with its source, and not primarily of the processwhereby the reasoning is retraced. He is justified in thismodification by the meaning of the mathematical term itself,which is literally a "breaking-up." 'AvaAXvosts used, forexample, of breaking up figures into their elements in the DeGeneratione et Corruptione.27 This sense was transferred, by ametaphor common in mathematics,28from the figures to theproofs (both are called Staypa'/piaTa), and avaXv'Etv caypa/Aluamight have meant either to break a figure up into its parts orto divide a theorem into its premises or urTotxEa, tarting fromthe figure or the theorem as a whole.From the general meaning of reducing reasoning to syllogisticform is derived the common meaning of reducing a syllogism inone figure to a syllogism in another.29 The procedure is anal-ogous in both cases: in the one we find the syllogism behind apiece of reasoning not expressed syllogistically; in the other wefind the syllogisms in other figures behind reasoning expressedin a given figure.That Aristotle consideredthe conclusion as a starting point,30as appears from his use of avaAvatL, is confirmed by his expressionfor begging the question-ro cE apX,p or ev apxi -lrELUOatoralretv. Some confusion has been caused in the interpretation of

2a An. Pr., I, chap. 32.26 2XvXXoYtao6sfrequently occurs in the sense of cavTLre'pacrua: cf. Bonitz,Index, 712a9 f.27329a23; cf. De Caelo, 300all. Cf. Proclus, in Eucl., p. 382, 1-2

Xp' 7Troitvv lievat oTtLTrv oX~j!a evCd'-ypa//ItovPis rpiYwva avaVLerat.28 Cf. the use of Lta'ypa/C/aor theorem. (Cf. Ross on Ar., Met. B, 998a25;Plato, Phaedo, 73bl; LtayeypapaiuevwvAn. Pr. 46a8), 'ypaceov for geometri-cal proof (Topics, 158b30; Plato, Theaetetus, 147d) and /evvo'ypaqferYfor false geometrical reasoning.29 An Pr. 50b33, 51a2 f., 18, 22, 26, 32, 34, 38, 40 f., b4. Cf. also An.Pr. I, chap. 45.30 Cf. H. Maier, Die Syllogistik des Aristoteles, II, pp. 157 ff.

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BENEDICT EINRARSON.Aristotle's doctrine of petitio principii by a failure to distin-guish between the senses of apXqr.In the phrase in questionapxr is not used as a "principle of knowledge" or demonstra-tion, but in the literal sense of "beginning." The words rOevapX,or 'e apXs mean "that with which we began," that is, the7rpo6/3ryLaor question at issue,31 which is the same as the conclu-sion, and the whole phrase To ^epX alrTelOaaLmeans " topostulate or take as premise the question at issue," that is, touse the conclusion as one of the premises from which it is to bededuced. The confusion of the meanings of apX-is perhaps tobe seen in the traditional Latin translation, petitio principii,and was doubtless furthered by the occurrenceof both senses ofapx in Aristotle's discussion of the fallacy in the Prior Analytics(64b35).32 The occurrence of the fuller form T oap3X1 KEtlevov(Met. r, 1008blf.) and riT e apxis OECarweTop. 156a13 cf.159a8) shows that 7TO apXj was the Oeats,or problem at issue(cf. Top. 104b35), and is decisive in favor of this explanation,as is the fact that Xaa/d3avetvs often substituted for ateTaCOat.33Aal3advetvmeans "to take as premise" and indicates that thefallacy consists in the identity of the conclusion with a premise.The other interpretation would force one to take Xaa3avetvasreferring to the conclusion, besides involving a mistranslation ofthe common Greek phrase e apXr.3The phrase may well be of mathematical origin. In at leastone mathematically colored passage35 Aristotle refers to the

s1Cf. Waitz, I, p. 429; Bonitz, Index, 111b15 ff. That sO et a&pXjs assynonymous with 7rp6O38Xriuaan be seen from a comparison of Topics,162b34 aiTreiaat . . . TO e dpXj with 163a9 atreirat rbTrp,6PXlfia (cf. alsoAn. Pr. 43a7 and 48b34). Cf. also Top. 163a24-27: . . . rpbs yap iKeIvo(i. e. Tr avl7reppaacAa) /pXerovTes rT ev dpxy XeyoJLevairelara. ....

32 For dpXai= premises cf. An. Pr. I, 43a21.a3Cf. Bonitz, Index, lllblOff. and especially 21-27.34 Cf. Grote, Aristotle3, p. 176. Besides Grote, Prantl seems to mis-understand the phrase (Gesch. d. Logik, I, p. 311): Zunachst kommthiebei in Betracht die erschlichene Annahme des obersten Ausgangs-punktes (ro ev apxi aireloOatc), die sogenannte petitio principii . .Here "Ausgangspunkt" seems to represent acpx-jas premise; and al-though the idea read into the passage is correct, it is arrived at by amisinterpretation of the terminology.35Phys. 242b20. The theorem to which it refers is on pp. 242a15-242b19(interrupted by a long parenthetical statement from 242a31-b8), andthe enunciation is found p. 242a15-20. Cf. also Topics, 163a10-13.

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MATHEMATICAL TERMS IN ARISTOTLE'S LOGIC.initial statement of what is to be proved, or enunciation, withthese words, and similar expressions are also found in latemathematicians.36 The absence of the phrase in early mathe-matical writings can to some extent be compensated for bypointing to the mathematical associations of the words alreLarOatand XacupavE?Lvith which it is usually accompanied. Further,the expressionpresents a somewhatunphilosophical appearance,37as apXy in the theory of demonstration would naturally be inthe language of Plato and Aristotle the highest principle, orthe premises. That the Academy was careful in its choice oftechnical terms can be seen from the substitution of 0Ewp-quaorthe older Slaypa/t%/an the sense of theorem,38 and from theobjections to the term yewuerpla in the Epinomis.89 Again,"begging the question" was an extremely easy mistake to fallinto when using the "analytical" method of proof; and thatmathematicians had classified the usual errors, and consequentlythis one, would seem plausible from the fact that Euclid in hisq?ev8dpLadiscussed the "ways " of committing geometrical para-logisms.40Aristotle uses the word oroLXCovn a mathematical sense, andin a logical or dialectical sense analogous to it.41 Mathe-matically, it is an elementary theorem; dialectically, an elemen-tary principle entering into many arguments.

3aPappus, Collectio, vol. I, p. 246, 18, Hultsch: To706ov Y&p 5vTroS7b7rpoKelucevov e dpp7is6elKvvTra . . ; Eutocius, Comm. in lib. de sphaeraet cylindro (vol. III of Heiberg's Archimedes, p. 147, 7) Tb et dpXjsrpopX71fia;f. Eucl., Fragm., vol. VIII, p. 280, 17, Heiberg.87That it was not coined by Aristotle appears from Top. 163a26 f... 7rpos'yap EKEvo (i. e. rb av/wripaalca) PX\'rovres rT ev adpx XyoLeValrelcOat, where X9yot/Ae points to a current usage of the phrase.88 Cf. Proclus, in Eucl., p. 77, 16 f., Friedlein: ij08 rQTv raXaiQv ot,tL' rda7vTaOewp'/uara KaXe?Yi0wJo'aQ,wt ol 7repiZ7re6a'rrrov Kal 'A4tl5voutov,i)yo&-/hevoL Tras Oewp-qrLKaiZS7rLT( /aLS OiKELOTrpayl ei[vaL r1Yv TWv OeOwpldrwv Tapor'yo-piarvX 77VTrV 7rpoXqtcIuadTrv.To be sure, from this notice the term Oecd&pwuaappears as opposed to rrp6lX?7lua,ot 8ta-ypa./a, but i8typauf/Ladisappearsfrom mathematical usage from this time on, being replaced by 60ejpr71ua(clearly connected with the Platonic Oewpeiv nd iteiv of the Ideas) and7rp61X\7,Ca.

89 990d2. Cf. Plato, Rep. 527a.40 Proclus, in Eucl., p. 70, 11, Friedlein (Eucl., Frag. 3, Heiberg).41For the dialectical use cf. Top. 121bll, 123a27, 128a22, 143a13,148a22, 151bl8; for the mathematical Met. B 998a26, Top. 158b35, 163b24,Diel's Elementum, pp. 26-28.

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BENEDICT EINARSON.Proclus, probably on the authority of Eudemus, says that

Hippocrates of Chios and Leon wrote :rotxda or Elements.42Diels, I think rightly, argues that these titles cannot be used todate the mathematical use of UTotxea.43 More valuable is theevidence that Menaechmus, a member of the Academy and acontemporary of Aristotle, knew of the mathematical use ofrTOtXELov.44The natural source of the dialectical use, where the word is

equivalent to 7ro%,45would also be the Academy, and as thepresence of the word KaXovtLv (1014b2) in the description ofthe dialectical use in Metaphysics A would indicate that the usein question was generally current, and therefore not speciallyAristotle's own, it may be said that the evidence points to theAcademy as the source of both meanings.Zeuthen 46 connects the mathematical use of rTOtxelOVith theprocedure of analysis, by which the parts of a mathematicalproof are revealed, each of these parts being an element.47 Aselement then is best explained semantically by its connectionwith analysis, it is reasonable to suppose that the mathematicaluse is the source of the analogous use in logic. Further, a r'0rosis a source for arguments, which the disputant obtains(-roplEarOat) from it. A parallel usage is found in mathematics:just as we obtain (7roptEo(raL) an argument from a dialecticalTr0rOsor (TroLXELo48 so do we get a ropwtoraor corollary from amathematical LTTOtXELOV.The word atlrlv was seen to be connected with the point ofview of analysis in the phrase ro5dpXc jlpra v. A considerationof the derivation of a'rTua in the sense of postulate throws somelight on the phrase, and an examination of datofa on the useof a'TrrLua.

42 Proclus, op. cit., p. 66 f. 44 Proclus, op. cit., p. 72, 23 ff.43 Elementum, p. 27. 46 Cf. Aristotle, Rhet. 1396b22.46 Op. cit., p. 225: Den ved Analysen fundne Gruppe af enkelte

Saetninger, eller snarere Beviserne for disse, udg0r de Elementer(aroXceia), hvoraf den forelagte Saetning, eller snarere Beviset fordenne, er sammensat, etc. Translation: "The group of simple proposi-tions found by the analysis, or rather the proofs of these, constitutesthe elements (aorotXea) of which the proposition under consideration,or rather its proof, is composed."47 Cf. Met. A 1014a35.48 Cf. Top. 158b6, 22, 164bl9, Rhet. I, 1356al, III 1403b13.

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MATHEMATICAL TERMS IN ARISTOTLE'S LOGIC.Three uses of at$wua and aetL can be distinguished in Aris-

totle,49 the dialectical, the mathematical, and the metaphysical.The mathematical usage, as certain passages indicate,50was notoriginal with him, but taken from the mathematicians of his day.In this sense awLttzaTa re the Kotvd or KOtVa apXat51 in mathe-matics, so called becausethey are common to the various branchesof mathematics, as geometry and arithmetic, and are not con-fined to a single branch, as Euclid's ati'T/xara are to geometry.52Aristotle's use of the term in the metaphysical or epistem-ological sense is derived from this usage. 'ALw,/La s defined(An. Pr. 72al3-18) as: ;v (sc. ae/crov apXrVvOVUAAoytaTTtKV)8' avayK-q cXelt TOVTrowv UaOro'odcEvov (sc. XAyw) aw/ola' Ecart yapfvta TOetavra'rTOVT yap /aXaAiT'E7rrTO70OLOVTOtSoEltWOaleEvvoLaXeyetv." I call that immediate syllogistic principle which a man who isto learn anything at all must have (i. e. know previously) anaxiom-for there are a few such principles-, that being the termwe are most in the habit of using for such things." He has in

?9A useful discussion of this word is to be found in Maier, DieSyllogistik des Aristoteles-hereafter referred to as Syllogistik-, II,1, p. 4, n. 2.5oMet. r 1005a20: ra ev TroIs aJaO7,oaaltaXol6ueva dEttcJara. In Met. N1090a36 (cf. 1090a13) dttiL'ara is loosely used for mathematical the-orems in general, arithmetical and geometrical. Plato uses dcttc in twopassages where he has mathematical principles in mind and may beplaying on the word: cf. Rep. 510c2-dl: . . . -roracf,Levot broOearaesavr&ode'va X6yov ... er&dc(ovat 7reptavrTWv 8t86vaL &s xravTri bavepv . . ., 526a2 f.:

. . .v ols Trb e oov UeiS dtL0ore' eC0rv'.61Cf. An. Post. I, chap. 10 and especially 76a41, b20.2For the equationr of dtc,Laara and KOLvdcf. An. Post. 76b14: 7rKOtia Xery6oLevatw&Jaraand H. Maier, Syllogistik, II, 1, p. 400, n. 1.Heiberg (Mathematisches zu Aristoteles, p. 5) defends the authenticityof Euclid's term KOVtpalvvotat (= axioms) by pointing to Aristotle'suse of KOLVa. he distinction between Euclid's alr/i.jara and KOtpalvvotatrests on this traditional mathematical distinction of "common" and"peculiar " principles. The KOtvat 'votat use the general neuter of theobjects to which they apply, while the postulates apply without excep-tion to the peculiar matter of geometry: points, lines, angles, andfigures. This circumstance no doubt explains the inclusion of theassumption (KOIVt'evvota,7) that things congruent are equal among theKOval evopotat, lthough it is pretty clear that it can only apply to thematerial of geometry: it does not mention explicitly points, lines,angles, or figures but is expressed with the neuter in the generallanguage of the other axioms.

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BENEDICT EINARSON.mind here the law of contradiction (implicitly called ati&wpanMet. r, 1005b33), which a man must have who knows anything:(Met. r, 1005b15 if.) : v yap (sc. apX-v) avadyKc)lEXEV Trv OTLOV1)LEvrTa TWV OYTWVr TOVTO oVX VrOea?LS. O Of yvOpiECLv avayKalOV T)

OTtO0V yvWpt0ovrt, Kal ?tKEV EXOVra avayKatov. With the law ofcontradiction in ,rTLyap EvLa ToLavra Aristotle no doubt includesthe law of the excluded middle. The analogy of the philosopherand the mathematician is touched upon in Met. r, 1004a6;53both deal with sciences that have branches arranged in order ofpriority, and it is very probablyin accordancewith this analogythat Aristotle applies atlo/a to his laws of contradiction andthe excluded middle; for just as the axioms of mathematicsapply to all its branches, so these laws apply to all the branchesof ovao-a.54 We may interpret his saying that he chooses the termatlw/ua "because it is the term we are most in the habit ofapplying to such things" as a defense of what is, to a degree,an innovation. Similarly, KaXovpcva in Ta eV TOL /pa0LjGmacLKaXov,uEvaadLtwuaTaMet. r, 1005a20) points to the fact thatthese mathematical axioms correspondless to his definition thando the laws of contradiction and the excluded middle, which areaxioms in the full and highest sense. Again in the phrase ofthe Posterior Analytics (76b14), 7a KcoLaXcyo0',va aLctuawa,Xeyo/LEvao doubt is taken especially with Kcovaand emphasizesthe point made explicit in the context, that these "commonaxioms " are not common absolutely, but Kar'avaXoylav, that is,when formulated so as to apply only to a particular subjectmatter, have an analogous relation to these matters, the axiom"when equal numbers are subtracted from equal numbers, theremaining numbers are equal" having a relation to arithmeticanalogous to that which the axiom "when equal magnitudes are

6s Cf. Met. r 1004a2-9: Kal rooavOa pUrp7rXoaoQolas earlvaaicrep at oaioar'w,o-redvayKaLov eivat rtva 'rpc7riYv Kal 1XogdvPv aCrt/v. IVTdpxet yap evfOs yyv/tXov rb a6 [Kat r6 Ev]' 8t6 Kat atdrLo-racGL dKoXou)o0rct 0'TOSro. f'rypt 7&p o6q\X6aocposwarep 6 tauOzjarT7Kbsey6luevos' K ar e fip,alavr Erpr$T?1 TrLS al6evurpa 0rtv 7rTT5/oLr 7 KCa &\XXaLt PeTjs dvoe To.zaO A.aosLv.64 Cf. Theophrastus' definition ap. Themistius, Analyticorum Para-phrasis, ed. Wallies, p. 7, 3-6: 6 yap Oe60paoro,s Orwr Lptlera r6 datwa.alwtudar eL 6t 56a TrS 4 u1evbv rOZS oyEevtLv, &av toea c&rb ov, i] aTrXus dv&ioraatv, otov 7Tv KardIpcaatv e r7Y &dr6(pao'tv' raTOa y&p KaOdTrep or6/juvTa KalKoLva 7TraLt.

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MATHEMATICAL TERMS IN ARISTOTLE'S LOGIC.subtracted from equal magnitudes, the remaining magnitudesare equal " has to geometry.55The third use, which is the source of the preceding two, isthe dialectical. 'Atzwua is used of a premise 56 and of a demandfor assent from the dialectical correspondent and is applied tothe reductio ad absurdum, where it indicates the demand of thequestioner that the respondent shall grant the contradictory ofthe conclusion he denies.57The source of the dialectical meaning, and through it of theothers, is the common Greek use of a$LoVYn the sense of "deemfitting" or "right" that a thing should be so.68 The word as

65 An. Post. 76a37-76bl: r^T 8' 4& XpWvraL MVrats acroSeKTrKas 7TL-r7/atsra t,vPf5a csKdTrxS &L^TrfLVs T& Kod,&OL KOLV&5 Kar' dvaXoylav, wrelXph6poi-6-y Soov iv rq b7bhTO eT(tr^ox?p yet. tSLa /v otoUv pa&L/^qvivat TOavsl, Kat rbe66, KOLvd8U otov 7r tora airb towvav &\y, r tba T& XoLvd. LKavbv 8' icKaaTrov7ro0rTW 6TOP v T7j y6vel' TraTb yt&p rot7rcet, K&VALi aTr 7rdfVTCrw\df- &\XX' Trl/Aeye0wv i6vov,rT 5' dpapL0T777tKW' apiOAwv. "Of the principles used in thedemonstrative sciences are (1) those peculiar to each separate science,and (2) those which are common to several-I mean 'common' byanalogy, since only that part of them which is in the subject matterof the science is used. Peculiar to a particular science are suchprinciples as 'a line is of such a sort (i. e. the definition of line) ' and'straight (i. e. the definition of straight),' while common to severalare 'if equals are subtracted from equals, the remainders are equal.'That portion of these principles which is in the subject matter in ques-tion is sufficient for the needs of the science dealing with that subjectmatter; for its usefulness will be the same if it be formulated to apply notto everything, but to magnitudes alone (as in the case of the geometer)or of numbers alone, in the case of the arithmeticians." Aristotle'spurpose in this passage is to show that the mathematician, unlike thephilosopher, does not treat the KOLtadas cotYa. Cf. Met. r, 1005a25.

63 Aristotle, Topics, An. Pr., and An. Post. passim, e. g. Cf. H. Maier,Syllogistik, II, 1, p. 4, n. 2.67Here dalwtja is nearly equivalent to dClwals. For examples of thisapplication cf. An. Pr. 62a13, 16 f. (adilo6v). For other applications ofdtatZ cf. Top. 157a37, 157b32, 37.8 Cf. the Stoic explanation of their term dtlwta (= judgment) inDiogenes Laertius VII, 65: cwvo6acrraL 8e rb dSiwtLadarbroi datoiYOaatdOereiFOa'* 6 /yap Xeywv tjzeLpao-Trr,datouv SOKceL b i7/pap elpva (herediloiv has acquired the connotation of applying to an affirmative proposi-tion); Ammonius in An. Pr. (Wallies), p. 26, 36-27, 2; Schol. inEuclid, vol. V, p. 112, 6 (Heib.); ib., p. 113, 19. Themistius gives anerroneous derivation (An. Post. Paraphr. [Wallies], p. 7, 6-8 after thedefinition given supra, p. 44, n. 54): 8 yap a&rXcTs Kal e'rt 7rapTwvi 6\Xw

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BENEDICT EINARSON.used in dialectics could then be interpreted as "a deemingproper" by one of the parties that a subject should have acertain predicate, or that a certain proposition should be true,or finally, that the other party should grant a proposedstatement.The latter interpretation is the most probable,and the word usedin this connection is the source of the dialectical sense, as certaindialectical passages in Plato show.59In dialectics dat is used only of the premises, the conclusionbeing proved, and not asked as a concession from the respon-dent.60 In the mathematical application, then, aojuya wouldlikewise have reference to the premises. But all the premisesof a given theorem are the conclusions of previous theorems,and therefore proved, with the exception of the first principles,which are the premises par excellence, as they are not alsoconclusions, and thus the word would naturally come to applyto the first principles par excellence. Of these, some weredefinitions, some aTir7'ara, and altwiaTra would again designateespecially those principles for which it was the only name, andthus acquire, from the dialectical usage, the mathematical senseof " axiom."

Although the evidence is slight, the following arguments tendto show that altrTqa and airT, as used in Aristotle's logic, are ofmathematical origin.AITZ is often associated with &Si/6u,61and the use of 8towlitinerL TWV6oioyevip TtOeTaL, rTavrTv dLOi/Lev Ka' rTTi wvPe. Aristotle, EthicaEudemia A, 1218a28-30: 8et . ...i dwovvlo)e v dciXyws, & KaI /ierT&X6oov 7rcrTevaat ot pit3ov (where datovv implies that there is no argumentpresented). Cf. Phys. 252a24.6 Cf. Plato, Meno 93a2, where Anytus (the respondent) says: KalTrorovs (i. e. 7TOv KaXoivs KadyaOovs ) adtw 7rapa TWrSVTporTepwv iaOev. ....E. S. Thompson in his note on this passage cites as examples of dai(5with a proposition as object Gorg. 450cl, Phaedo 86d, Apol. 18d, Rep.610a, Polit. 262e. He translates ditc as "I expect you to grant" andcompares the "mainly transatlantic" use of "I claim."60So the adSifJia of the reductio ad absurdum is not proved-as itmight have been with the help of the law of contradiction and the ex-cluded middle-but asked as an Ev6ooov. It would have taken thedialectician too far afield to have brought in the law: how far can bejudged from the analysis of the reductio ad absurdum given in Maier,Syllogistik, II, 2, pp. 125 f.61Cf. Hdt., IX, 109: w6aELsuOLrb d&ve alr'jarw; I, 90: alreo 86ao-v rjTLrvaBo6Xeat To0 yevrlOai; VIII, 112: el ,z bSoovoL ra alre6LevPov; Xen., Cyr., V,

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MATHEMATICAL TERMS IN ARISTOTLE'S LOGIC.mathematics is well established.62 Further, atlr and alro3ivat arewith few exceptions in Aristotle confined to the expression rose ~ - I - 63apXS3 aiTetLv.The procedureindicated by this phrase is applicable to mathe-matics, and in his discussion of it Aristotle uses mathematicalillustrations.64 The practical restriction of the word to thisexpression would indicate that it was traditional; and the onlysource for which there is any evidence is mathematics.65 Again,Euclid's sense of a'lrla is not Aristotle's, an indication that therewas an independent mathematical tradition for the word. InEuclid the KoLvalEvvoLaLre principles common to arithmetic aswell as geometry, and are synonymous with adtloua as used by themathematicians of Aristotle's time,66 while the aIrT,IaTa dealwith points, lines, and angles, the peculiar province of geometry.The distinction between the KoLvat Evvotat and alrt'Tara of Euclidis then the same as Aristotle's distinction between Kotvat andZtLat apxa.67 As Aristotle's definition of awl'TLuadoes not agree5, 21: Tjrrldo?e&ovail oL; Hell, V, 4, 11: 86ooav& Trovv; Cyr., VIII, 7, 3:altrojUaL8 viUt,s8osvat; Hipp., 1, 1: X%plrTeatLOeot s . . 6.8a86va . . .;Symp., 4, 47: alTrovrTa TOrBS 0eos . .. T&ya& .. .86v6a; cf. ib., 8, 15:Anab., II, 3, 18: alrao-acrOatoval Aotarroaoaa vass el rTiy'EXXdSa.Plato,Theat., 146d3: gv ait-r7iels roXX&8l6s.

62 Cf. Sir T. L. Heath, op. cit., I2, pp. 132 f.63For TO e adpXjs lreiv cf. pp. 40-41 supra. Tdvavria alretv occursin Top. 163a14-28 but is replaced by rdCi'rKeit6eva Xa3vSayeeyn An. Pr.,II, chap. 15. Airoiiza occurs independently of these phrases in An.Post. 73a13, Top. 162a27 and 31. The use of an'r7la has been generalized,however, and the word is currently used as a synonym of 7rporacas orpremise, together with &dciarr7a, 6roaeiats,dpXi, XAAa, epwrf?fLa nddllwAa: cf. H. Maier, op. cit., II, 1, p. 4, n. 2.84 Cf. An. Pr. 65a4-9, Top. 163a11-13.65 The evidence is the use of al'TXnLa and TrnaOw in Euclid, Elem.,Postulata I (vol. I, p. 8, 6 and 7, Heiberg).66 Cf. p. 43 supra.67 The seventh Common Notion (Euclid [ed. Heiberg], vol. I, p. 10, 10)

Kalr TraefapL6tora eta' a\XX7Xa L'aadkXX?XolsaTr would seem to be peculiarto geometry. It was probably included among the common notionsbecause it contains no explicit mention of the subject matter peculiarto geometry: points, lines, and angles. It will be noticed that everypostulate contains a mention of these, while every common notion(with the exception of the ninth, an interpolation) is expressed bymeans of the general neuter, as ra i'aa, ra iptilav),ra e,pap4uotora.



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BENEDICT EINARSON.with the Euclidean usage, it is probablethat Aristotle took theterm from mathematics, and not the mathematicians from him,especially as datlwa, the correlative of altT7,ra, is certainly a mathe-matical borrowing.We may supposethen that the mathematicianscontemporaneous with Aristotle used atlo/ma and atT?qa for KoLvaland 8cUapXag, nd that when Aristotle took atwfa in the senseof a principle a pupil had to accept to know or learn anything,he also took its correlative acrTlTa in the corresponding sense ofa principle that a pupil did not, or even refused to, accept,68and was perhaps confirmedin his choice by the current usage ofthe language, in which aLuo implies that a request is thoughtfair and reasonable,while alT( lacks such implication.The origin of a'lrrVuas to be sought in its connection with&8'8utL:t is a request for the granting of certain data for use indemonstration. Actually the first three airq,aura in Euclid pos-tulate the possibility of certain constructions-of the connectingof two points by a straight line, of the production of a finitestraight line continuously in a straight line, and of the con-struction of a circle with any point as center and any radius.In the theorems of the Elements these postulates are referredto in sentences where the infinitive of the verb of the postulatebecomes a passive imperative. Thus the second posulate runs:(flT7arOW) Kal 7rE7repao,LEv-v evEOcav KcaLTaTO oVveXeS 7r' eVOeas eKaXeAEv.In the Elements69 we find such phrases as EK/,e/jXAe(ro7 AB(eV6ea). Similarly the Greekof the third postulate is (jr 6Oow)Kat 7ravTr KevTp Kal &tacTyLaTL KvKAov ypadeaoOat. In the Elementsthe phrase is KEVTrp /iEV TroA, s&aaOrT/rjaTLET AB KVKXo yeypda0co6 BrA.70 The verb ayayetv of the first postulate, however, isuniformly representedby icreCev'Xw,ut the two words are nearlysynonymous,71 and the slight variation need cause no difficulty.

8 An. Post. 76b30 f.9 Eucl., Elem., I, 12, 26 eKgepfi7ao0aav r' evOelas rats AA, AB e60eial atAE, BZ. Cf. I, 20, 7, I, 42, 9, I, 44, 14 (Heiberg).70Eucl., Elem., I, 10, 19; 14, 1; 14, 23.71The postulate is formulated as follows: r'7aOwodro7rravrcbsjrJielovearl 7rav aorjueoveiOeiav ypa,utJrvadyayeiv,while in the Elements the pro-cedure is expressed by such phrases as erreevuXoaw7rbrov A ar7tJelouvrrb B a-rteiov ev6Oea AB (cf. Elem. vol. I, 12, 24; 12, 3; 20, 13; 24, 24[Heiberg]). 'Hx0w, which we should have expected, is used chiefly ofdrawing a line parallel (Elem. vol. I, 88, 20; 90, 17; 102, 19) orperpendicular (vol. I, 36, 10; IV, 24, 17; 32, 4 and 10) to another,

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MATHEMATICAL TERMS IN ARISTOTLE'S LOGIC.All three postulates then can be said to point to an origin ofalrrjla in the use of the imperative in mathematics.72An obvious objection to this explanation is that the fourthand fifth postulates are of a different kind, as was noticed byGeminus.73 The fourth postulates that all right angles areequal; the fifth that two straight lines extending outward onthe same side of a straight line, and forming interior angles withit of less than two right angles, will meet on that side. Clearly,if two right angles are unequal, the sum of two right angles isnot a constant, and it would follow that the postulate wouldnot always be true. The fourth postulate is thus bound up withthe fifth, and like the latter, is perhaps a post-Aristotelianaddition.74 However that may be, the fourth and fifth aredifferent enough in kind from the other three to justify thesupposition that they are later additions to the tradition, perhapsdating from the time of Aristotle, and need not therefore standin the way of interpreting atTrrua as originally indicating con-structions, or hypotheses expressed with the imperative, such asyeypadoOw, Er-TO),r KEtic), a meaning which the word has lost inboth Aristotle and Euclid through its opposition to &alo/la.The same use of alTrrlcais seen in ro e' &pXplre'v, if a mathe-matical origin is accepted. In the "problems" in Euclid theinfinitive is alwaysused-as e. g. crovTrrracrOa,ayayev--dependingoneT understood.75 The "begging" of the e apX,c is thenthat is, in such a way that it is not primarily drawn to connect twogiven points, but for some other purpose (cf. Elem., vol. I, 146, 27;Aristotle, Meteor. 363b6). Perhaps dcyayeiv was the older and moregeneral expression (cf. Aristotle, Top. 101al6)-though the same dis-tinction between X0owand eirerevx0w is observed by Aristotle (cf. forg7ree?x6OwMeteor. 373a10, 376a17, with 375b23, 376b23; for X8Ow eteor.373al1, 363b6)-or perhaps expressions such as TrraffOwariO ravrbstfo7uelov 7rt rra,vo'rtei4ov erievyvvrvoOat(cf. ypadceaOatn postulate 3) orermt~eviat (cf. eKgaXelv in postulate 2) were avoided because it is reallythe two points that are "joined," and not the straight line, which isproperly the result of the joining.

72 For alrt connected with an imperative cf. Xen., Cyr., VIII, 7, 26:alrov/uat . . . /A7q8eis 8erw; Plato, Theaet. 146d3: evalr77Oe4sro\XX&86ws(referring to e7re-, c3); Aristophanes, Vespae, 556 and Acharn., 476.73 Proclus, in Eucl., pp. 184, 6; 188, 5; 192, 5; cf. 182, 1-6 (Friedlein).74 T. L. Heath, The Thirteen Books of Euclid, I2, p. 102.75 Cf. Euclid, Elem. i 1, 3; 9, 12 etc.4

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BENEDICT EINARSON.originally the assumption (by use of the appropriate verb inthe imperative) of the actual construction the problem isto establish, or of a construction only possible if the former isvalid. The use in the general sense of begging the questionof a phrase indicating construction is no doubt due to thefamiliar transference of the language of construction to actualdemonstration.76

Aa,/a3avo occurs in the arguments of Plato in a variety ofapplications, meaning to take a consideration in hand, insteadof dismissing it,77 to lay hold of a property of a subject ofinvestigation or problem, by enquiry, or discover its formu-lation,78to acquire a piece of knowledge or opinion concerninga thing, or simply acquire knowledge,79and to find and formulatea thing and bring it into the discussion.80 All these senses reston the comparison of a mental process to a "taking" or" grasping."That Aristotle had a mathematical meaning in mind in usingha/pa3dvo of premises is clear from his frequent use of cdXqow 81-this form being extremely common in mathematics--82and con-sequent use of the perfects eXAqirrTaL3 and elXtAriEva. In Euclid

7a Cf. n. 28 supra and Plato, Rep. 427a.77 Plato, Polit. 282a6 .. avYKPLTLKS . .6ptoKcpTwKev . .. lo a 8T7S 6CaKpctItrIS iv aClVTr6t, eOCLAyev il4Turavra.

78 Plato, Phaedrus 263b6 OVKOV rObv leXXovra re-XvT pr77ToplKiv iertLIeva7rpicrov Iuev er TrLava O6SC8L0p7aOaat, Kal el\)ervac rtva& XapaKTWrpa roV e['tovs,eT Ce darVdyKrTo rXr0os 7rXavaial Kala v J tLi. Plato, Theaet. 208d6 . . .&s ap'ac v lta(oopbv eiKcarov av XalpaCdvrs , r Cv aXXwov lacpe, X6yov . . .X77V?.... .Plato, Polit. 297d3 rot6voe rt 8ei ye 77rTeLv,oL 7radvv Ovvj0es ov5&

6tovi1el6 'v oiwsL'Sy retpWJeOa6 Xa3eliv aVro. Plato, Phaedrus 246d . . .r7v oe alriav . . . XdtwtLev. Plato, Polit. 308b f., Phaedrus 265c.

79 Plato, Soph. 238b7 . . . Kai T) rcavola r5 7rapd7rav Xd,Bo rTa u6vra .. w. pis dptOJov; Plato, Philebus 34d 7rpoTepov e'rfalverat XqprreovetrrOv.liav elpva ri Tro' eaTt IcaL irov -ylyveTat. Plato, Polit. 297b8 TrvTroavrT7vXapdbvei7rtarLt7v. Ib., 300e5 and 8 Xa3elv rTeXv7v (cf. ib., 302abd'yvot.av el\Xr(fl6rwv . .e. ir-T/L71Xv elXrfe'vat ).

s Plato, Soph. 233d3 Xdci3w/ev roivvv aaobe'rep6v rt 7rapde'tytca replTOUTOwV.s1 An. Pr. 26b8, 33a40, 35a16, 37b24, 26b12, and passim.82Euclid, Elem., I, 34, 10; 168, 25; 172, 15; 186, 3; 194, 5; 200, 1;202, 10; 184, 10; II, 12, 10; 128, 7; 194,20 etc. (Heiberg). Cf. Aristotle,Physics 233a35, 235a18 f., 238a6, 242b10, where el\X/0w occurs intheorems.83 Cf. An. Pr. 47b9, 47a18 (e'Xw,rrat) with Euclid, Elem., I, 200, 16;II, 180, 13.

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MATHEMATICAL TERMS IN ARISTOTLE'S LOGIC.two senses of edX74Owan be distinguished; that of choosing anarbitrary figure or element of a figure, or number, and that ofobtaining a figure or element or number by processes alreadyestablished, the principal examples being, for the first sense, thechoosing of a point, for the second, the taking of the center ofa circle.84 Of these senses neither is parallel to Aristotle's senseof taking a premise of a certain quality or quantity, though thefirst is nearer it than the second.85 The identity of form,however, shows that he took the mathematical and logical sensesof the word to be similar.The source of the technical use of Xap,3fdvwor the "taking"of a premise is, however,not to be sought in mathematics, as thedevelopment of the usage of the word in this direction can betraced in Plato.The meaning " grasp " allows as object a proposition, as oftenin Plato,86 and thus the mathematical use of the word forassumption" 87 and " lemma " 88 and the logical use for "takeas premise" are made possible.

84 For the choosing of a point cf. Euclid, Elem., I, 34, 10; 56, 7;76, 1; 178, 25; 184, 10; 190, 12; 200, 16 etc. (Heiberg); for the takingof the center cf. I, 168, 25; 172, 3; 174, 15; 186, 3; 194, 25; 200, 1;202, 10 etc. This use occurs only after the method for finding thecenter has been proved (I, 166, 14ff., bk. III, 1). In the fifth bookthe taking of any equal multiples whatever (cf. Heath, op. cit., II2, pp.143 f.) is, like the choosing a point, arbitrary (examples: II, 12, 2and 10; 14, 22; 22, 14; 180, 13), while the taking of a third proportional(II, 128, 7) or of the greatest common measure (II, 194, 20; 196, 19)or of the smallest numbers in a given ratio (II, 258, 16) are all likethe taking of the center of a circle in that they proceed after a methodthat has been previously demonstrated as valid.The word Xaczavovwn these senses was no doubt originally opposedto 8/1dltL. In the one case the number or point was primitively thoughtof as given or assigned by the interlocutor, in the other as either ac-quired by a legitimate procedure or taken arbitrarily by the demonstrator.

85 Not all the cases of elX'rp0w refer to the taking of a premise. Somerefer to terms: cf. An. Pr. 26b8, 12 and passim. Here there is a greatsimiliarity with the "taking" of arbitrary points.

86 Theaet. 145e ov 36valat Xagelv IKav6os 7rap'eiLavrT, erTrT7/uL7yrL IroTrTrvyYXcveL ov. Ib., 200d2, Phil. 34d . . . 7rp6repopverTL alveraL X?77?rreoertOvculav elva&l 71 rOTr'?r Kal 7rou y&lyverat.87 As in Archimedes vol. II, 12, 6 f.; 262, 18 f.; 262, 8 and 22 (Heiberg).88These two meanings are to be carefully distinguished: cf. Sir T. L.Heath, The Thirteen Books of Euclid, I2, p. 133, n. 2.

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BENEDICT EINARSON.Aristotle and Plato use Xaa,/cdvwor the gaining and formu-

lating of a point by investigation.89 In the game of questionand answer, which is by its syllogistic procedure akin to phil-osophical or scientific enquiry, the " gains " or " acquisitions "are the propositions admitted by the respondent, in other wordsthe premises,90and it was natural that as Aa/38avYcos used ofthe philosophical and scientific procedure, it should also beapplied to the point gained in the dialectical game. In thisacceptation Xaa/lfavw ecame tinged with the connotation of" taking " or " receiving " a propositionfrom the respondent,anda X/Al~jaor " acquisition " was a proposition so taken, that is, apremise. Aa/alovwthus arrived at its technical sense of "takeas premise." 91Why then are not Xafdl3av and AX//uaused technically of theconclusion? For the conclusion is as much of an acquisitionas the premises. The reason is that in both the Aristotelianand Platonic passages for the non-dialectical meaning, the ex-

pressions Xr7rrTEovnd 8e XaL,i3cdveLvith a proposition as object-clause are used of a point that must be elucidated before theinvestigation in hand can proceed.92 Aaa3aivw, then, is hereassociated not with the simple gaining or acquisition or formu-lation of a point, but rather with the formulation of a point89For Plato cf. n. 78 supra; for Aristotle cf. De Anima 415a14f.:

dvacyKaLov e TOrv i/EXXovTa ,repi TOVUTWV Keslt 7roteiaOaL XaLtelLv eKaaTrovavrTv Tr1 eOTt, ed0' OVTwS 7repi . . T. r a\XXwv e7rTLreiv; 403a5; Phys.213b30, 219a2; Meteor. 371bl.90 For the reason why Xalupcvw is not technically used of the con-clusion cf. infra, and p. 46.91 For the connotation of taking from the respondent, cf. An. Post.71a7 d/ciAorepo 7yap (sc. ol XO6yoLireE a \\o7vXXo'yit KaCol irepZecra'ywys)a T7rrpo'yLvcoWKOfILvwv 7rotovvTaL T7V 6LtoaaKaXiav, ol Iuev XaAuL3divovre sssapa

vvLevrTV, oole eLKVVVTSreSTO KaOOXOv3at TOb \Xov elvat TO KaO' eKaaTov. Top.154a25: XaP3eiv 7rap& Trcv epWTWiLewv TCLs TroavTas 7rporarets. . . . Thatthis connotation is merely adventitious appears from An. Pr. 24a24:ov -yap epwrc daXXa XaLp3afvee o da7rooecIKvwv.

2 Cf. Plato, Phaedrus 263b: OVKOVV TOrP /\Xovra TeXr%vXly 7TopLK*,TLEC'Tvat 7rproYOv eEyv e rarTa oo5 tpoCjaOa, Kai eiX\,qpevaL rTva XapaKTjpaeKaTepov TV eTi'ovs . . . ; Philebus 34d, 61a4: TO Tol'vvv rotOUrTOV dyaOb

I oOO aabCos X Kai T'ra Tv'7rov a'LrTOVX*rTeov, L'v', oirep eXe'yogev, evrTepedaotro, o6'o-joev 'eXWlev'; Aristotle, De Part. Animal. 661b28: KaO6XovO XpecvTt Xaf3eiv, 6 Kail 7rt rov7TWV Kai eiri7roXX\v Trvv VuT'epov XeXOOo/,ev6WvP acaX%piacuovand the passages adduced in n. 89 supra.

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MATHEMATICAL TERMS IN ARISTOTLE'S LOGIC.that is to be used for illuminating a given question. To discussanything we must rely on certain knowledge previously acquiredor possessed; and with the subject of discussion in mind, we canoften determine what that knowledge must be. There is a hypo-thetical necessity then to such knowledge, which is expressedby the forms Ar]7rrEovand 8e Aalaxiavetv. This intermediaryknowledge is often in the form of conclusions of syllogisms, andthus has the same form as the final proof we are seeking; butthere is a profound difference in the attitude with which it isregarded by the demonstrator: it is a means, not an end, andmust be got before he can prove his point.In this way, perhaps, kau,3advwnd XA/jua s used of apodeicticpremises can best be explained; for in the syllogism, the knowl-edge used in proving the conclusion is contained in thepremises.93The use of XAj,uLan mathematics can be similarly explained:it is used of a theorem that must be known before the theoremin hand can be proved,which, unlike the theorems which precedeit, either is of little interest, or breaks the continuity of thework: in other words, it is a theorem whose purpose is merelyto provide a premise for use in another demonstration. IfXalJL3avwtarted from the meaning of "taking" a propositionfrom the respondent for use in a proof, or meant acquisition ofa point, with no further connotation, it would be equallyapplicable to all premises, that is, to all the preceding theorems:as it is, its peculiar implication of hypothetical necessity for theinvestigation in hand renders it eminently suitable as a desig-nation for lemma.Besides this application to premises and lemmas, is foundanother use for unproved assumptions in general.94 Here thewhole body of proof is contrasted with the Xaul3avo',eva, and not,

93 The language of the Epinomis approaches very closely to that ofAristotle, although a conscious use of the syllogism cannot be shown.Cf. 980e3: XdfpwIoLev 7TroVO6e, Bs tpvvx%rpeaiPvrep6o eaftL a&uaros. Thelanguage of this whole passage is that of proof: cf. dapX? (981a3), Ow,uev(a2), v7rvp'yuevov(a2), and e6rqpalvetv(a4), which appears to be anecho of the Ieri3aansof the Republic (511b) and refers to the hypothesesmade. Aalpaivw is used for "assume" also in 982el, 987d9.94 Cf. Archimedes I, 4, 22 Heib.: ypaqovrat 7TpWrOV Ta 7e dtjaU1aT7a Kairt& Xkaufiav6fieva els Tas ai7roielites airTvP (i. e. rTWvOewpTuaciTwv cf. p. 2, 7);Aristotle, An. Post. 76b3, 6, 7, 27.

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BENEDICT EINARSON.as in the case of AXirua,only the theorem to which the lemmais attached.The explanation given above connects the term closely withthe analytical procedure; for having the conclusion in mind, wedetermine what premises are necessary for its establishment.Whether the term was actually taken from mathematicalanalysis must remain doubtful, in the present lack of evidence,though highly probable. If the connection with analysis isaccepted, the term belongs to the group comprising vadAvavm,o~ apX,jS alrTEv,TarotLeov, and o $aE8elat.The phrase 0o7repZSt 8eatL, all but universal in Euclid'sElements, occurs in Aristotle in the form o MSet8eLai or

vAAoytraaa0at to indicate the question at issue.95 Even ifHeiberg 96 is right in suggesting that Euclid was the first tointroduce the phrase, its occurrence in Aristotle would showthat orally, at least, the phrase was older. It obviously dependson the analytic point of view.AL8w&qts a natural correlative of Xa4advw, and it is naturalto suppose that it too was derived from mathematics.97 Butthe dialectical use appearsin Plato (Theaet. 166b6, Phaedo 88a2and 8,100b7,cl) and is so natural in the sense of "I grantyou " 98 in Greek that there is no need to look to mathematicsfor its origin, especially as the mathematical use had becomespecialized.99

BENEDICT EINARSON.HARVARDUNIVERSITY.

(PART II TO FOLLOW.)

89An. Pr. 46b19, 46a33 (8 9et 8ed5a) ; 46b12, 32 (8 68ei vXXoyl`aaaca).Compare also 46a33, 46b21, 57b21 and 25, 66a38, An. Post. 84b31, 86a19,85a7, 87a7, Top. 162b35, 163a4, 7.6 Litterargeschichtliche Studien iiber Euklid, p. 36.87 For Aristotle's use of aiow/tL in dialectics cf. Bonitz, Index, 194a30-35.

98 Cf. Lysias, IV, 5 \XX' 7', el osvXEerat, EX^OpO'Lbwut yAp avTr TOVrTOoSeYvyap 5aoqe'pe.89 For the term eboisevov cf. Heath, op. cit., I3, p. 132. Of thedefinitions given in Marinus' commentary on the Data (vol. VI, pp.234-256 of Heiberg-Menge's edition of Euclid) one probably gives theoriginal meaning (p. 236, 1-3): Kat 7O eYvvroOef'el &e rraparo 7rpopa3XXovrosEKTIr0efzevov eaoluevov eLval rTves V7relXqbaa&v.

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On Certain Mathematical Terms in Aristotle s Logic - Part I

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