GreekG EometricaAl Nafysis - Norman Gulley

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Greek Geometrical Analysis

Author(s): Norman GulleySource: Phronesis, Vol. 3, No. 1 (1958), pp. 1-14Published by: BRILLStable URL: http://www.jstor.org/stable/4181623.

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GreekGeometricalAnafysisNORMAN GULLEY

THIS article offers an interpretation of the passage in Pappus de-scribing geometrical analysis (Collection, vii., Preface, pp. 634-6in Hultsch; full text also in Thomas, Greek Mathematical Works,II, pp. 596-9, Loeb). The principal difficulty is that Pappus appears togive two different accounts of the direction of the analysis. (i) as anupward movement to prior assumptions from which an initial assumptionfollows (ev IL'V y&'p -T 0oV0F.XUxcCT4. xCXX0UtV (FVOAsLv). (ii) as adownward movement of deduction from an initial assumption (8vTv.npo6fX-%toc).wo interpretations have been offered. The first accepts (ii)as the proper formulation of what the Greeks called geometrical analysis,and explains (i) as merely an alternative way of describing (ii). Thesecond accepts (i) as the proper formulation, and explains (II) as merelyan alternative way of describing (i). The first is the commonly acceptedinterpretation. Lucid and detailed accouats of the method on thisinterpretation are given by Heath (The Thirteen Books of Euclid's Elements,I, pp. I37-142), Robinson (Mind, N.S. XLV, I936, pp. 464-473), andCherniss (Review of Metaphysics, IV, i9gi, pp. 4I4-4I9). The method isone of assuming to be true a geometrical proposition which it is requiredto prove, or assuming a geometrical problem to be solved, and, byanalysis, deducing con-sequences until one reaches either a propositionknown independently to be true, or a construction which it is possibleto satisfy, or a proposition known to be false, or a construction which itis impossible to satisfy. In the first case, the last step in the analysisbecomes the first in the synthesis, which repeats the steps of the analysisin reverse order until the original assumption is reached and so proved.In the second case one may conclude, without resort to the synthesis,that the original assumption is false, or the solution impossible. Arequirement of the method is that the implications at each step arereciprocal. The second part (ii) of Pappus' account seems, quite clearly,to be describing this method. So also do the definitions of analysis andsynthesis interpolated in Euclid XIII (see Heath, op. cit. p. I 38). More-over, as Robinson emphasizes, there are excellent examples of its use inArchimedes, in Pappus, in the alternative proofs of XIII, i-g, interpolatedin Euclid, and elsewhere (loc. cit. pp. 469-472; see also Heath, op. cit.,I, pp. I4I-142). That this, and no other method is what the Greekgeometers called analysis is, says Robinson (Plato's Earlier Dialectic,

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ed. 2., p. i66), "really as certain as anything in the history of thought".The second interpretation contends that this is not only not certain, butfalse. The only detailed argument for this other view is that of Cornford,in Mind, N.S., XLI, 1932, pp. 43-50, but severalother scholarsappearto share his view.1 Cornford argued that the method of analysis is notone of deduction; the procedure is not to see what follows from theoriginal assumption, but to see from what the original assumptionfollows, and, having discovered that, to proceed backwards until aproposition is reached independently known to be true. Then, bysynthesis, the original assumption is deduced and so proved. It is onlythe synthesis which is deductive, and not also the analysis. The first part(i) of Pappus' account ('V ,uevyap -U vcLVXUCL... XX05XOV OVOeLav)seems, quite clearly, to be describing this method. So also do thedefinitions of analysis and synthesis given by the Greek commentators onAristotle and by Proclus. Moreover there are, as we shall see, illus-trations of this method of analysis in Aristotle, in the Greek commentarieson Aristotle, and in Proclus.An assumption common to both these interpretations is that Pappus,throughout his account, is describing, with perfect consistency, a methodwhich, exclusively of all other forms of analysis, was called by theGreeks geometrical analysis. It is this assumption which seems open toquestion; it is in the attempt to make one description of analysisconsistent with the other that the weaknesses of each interpretationbecome apparent. Thus Cornford arguedthat Pappushasbeen "lamentablymisunderstood" because his description of analysis as proceeding 8r.&rCveiJq &xo?CoiU$Ocvas been interpretedas if Ta riKq &xOXou6%eant TXauxLPavov-n, logical consequences. It should, he argued, be taken tomean "the succession of sequent steps", without any implication that thesequence is deductive (loc. cit. p. 47, n. i). He considered that if itmeant "logical consequences", it introduced a logical impossibility intoPappus' account, since he considered it to be logically impossible tohave the same series of steps giving logical consequencesn both directions.But he overlooked the point that when each of the propositions in theseries is convertible, then there is no logical impossibility of this kind.And Robinson rightly made this a principal criticism of Cornford's view1 See H. D. P. Lee in C.Q: XXLX, pp. II8-I24; A. S. L. Farquharson in C.Q. XVII,p. 21 ; B. Einarson, in a valuable discussion of mathematical terminology in Aristotle(A J.P. LVII, pp. 3 3- 4, IS 1-172), commends Cornford's account of the Pappus passageas "an excellent discussion' (p. 36 n. i8), but his later remarks on analysis (p. I 3)seem to imply that he would not accept Cornford's conclusions.2


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(Mind, N.S., XLV, pp. 468-9). The interpretation which takes rOC'&ix6?'ouO%to mean logical consequences has, however, to explain, con-sistently with its own thesis, Pappus' description of analysisasa resolutionbackwards from an original assumption to prior propositions from whichthat assumption follows. Robinson says that this description is "notincorrect on the usual view of analysis; it is merely unexpected". "Sinceon the usual view the implication goes both ways, Pappus would becorrect whichever way he said it wvent".He suggests that "the reasonwhy he expresses himself here in this unexpected way is that he islooking at analysis as existing for the sake of synthesis; this makes himdescribe the steps of the analysis, not as they appear while you are doingthe analysis, but as they appear in the subsequent synthesis" (ibid. p. 473).In other words, Robinson says that, since the implications are reciprocaland the conclusion of the analysis becomes the premiss of the synthesis,the analysis can correctly be drescribed as a search for premisses or forantecedent propositions, since propositions which in the analysis werelogical consequences become, in the synthesis, propositions from whichthe original assumption follows as conclusion.' Thus (i) becomesconsistent with (uI), as an alternativeformulationof (II). This inter-pretation bases itself on no external evidence for (i) being an alternativeformulation of (II). And it is clear that if there is reliable cxternalevidenceto provide a d fferent explanation for Pappus' description (i), thatexplanation should be preferred to one which, from an assumption of theinternal consistencyof the passageas a whole, uses (ii) to interpret (i).I shall argue that such external evidence is available. There are, in fact,several accounts of geometrical analysis, both earlier and later than thatof Pappus, which describe it in almost precisely the same way as (i), andit is, as we shall see, possible to supplement Cornford's evidence on thispoint. And in the light of the evidence for both (i) and ({I) as recognisedforms of geometrical analysis, a third possible interpretation of thePappus passage suggests itself. It is that Pappus, though apparentlypresenting a single method with a single set of rules, is really repeatingtwo different accounts of geometrical analysis, corresponding to twodifferent forms of the method, and is assuming the equivalence of (i) and(ii) for all cases of analysis, overlooking the inconsistencies involved inthis assumption. If we assume that at least one account correctly de-1 Heath too (op. cit. pp. 139, I40) appearsto assumethat the reason why Pappusde-scribes analysis in this way must be that he is assuming the convertibility of each step inthe argument, and thus feels himself free to vary his description of the direction of theanalysis.

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scribes the method, then either (a) one account is the correct one, andthe other is based on a tradition which confuses geometrical analysiswith some other form of analysis, or (b) both describe metho(dsrecognisedby the Greeks as forms of geometrical analysis. I shall argue for (b).I shall accept the view that the Greeks recognised a form of geometricalanalysis in which both analysis and synthesis were strictly deductive.Certainly there is no mention in any description of the method inantiquity of the essential condition of the successful practice of themethod - that the implications at each step should be reciprocal. Butthere is at least no doubt that Greek geometers were aware that a largenumber of geometrical propositions were convertible (see Aristotle,An. Post. 78a, IO-13; Proclus, In Eucl., Friedlein, pp. 72, 26ff., 2S2,Sff.), that they practised a method of analysis where the steps were infact convertible, and that before the time of Pappusa formulation of thismethod had been made representing the analysis as deductive. Thus todecide between (a) and (b) is to decide whether the tradition on whichPappus' account (i) is based is a sound one or not.There are many references to analysis in antiquity to illustrate thegeneral meaning of the term, its meaning in logic, and its meaning asdescriptive of a geometrical method. The Greek commentatorsdescribe at some length, principally when explaining the title ofAristotle's Analytics, the application of the term in many fields - byypotc'noelLXoL, puaLoyoL, cpoL?OQOL, and y wttpxO.1 It is a process ofresolution, of a whole into its parts, of a compound into its elements, ofthe complex into the simple. In its logical sense it is contrasted as anupward movement with the downward movement of a'vOeatc, &O8&LiL4,8LodpXsat; it is an ascent to what is npo6repov,o principles (&pyoxL) rcauses (ctVruo), rom which the truth of the proposition which was thestarting point of the analysis can be demonstrated; as a logical methodit is classified as one of the four divisions of dialectic - aCG,Waet8tq, aLuLtpeaL, and OpLa:k4, a classification of which the Greekcommentators find the basis in Plato, though it does, of course, owemuch to Aristotle. 2 Two principal forms of analysis are recognised:1 Alex. In An. Pr. I, 7, 12ff. (Wallies); Ammonius, In An. Pr. I, 5, ioff. (Wallies);Philoponus, In An. Pr. I, E, i6 ff. (Wallies); Eustratius, In An. PostII, 3, I Off (Hayduck);David In Porph. Is. 9, i iff., 103, 23ff. (Busse).2 Amm. In Porph. Is. 34, 20ff.; In Anal. Pr. I, 5, 22ff.; Philoponus, In Anal. Pr. I, 5,i 8 ff.; In Anal. PostII, 334, 24 ff.; Eustratius In Anal. Post II, 3, 13 ff.; David In Porph.Is.88, 6ff. See also Albinus, DidaskalikosV (1i6-7 in Hermann's Plato, vi), and Proclus,In Eucl. (Friedlein), 42, I 2 ff.4


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(a) a "reduction" of sensible particulars to a single form, for the purposeof defining a term (thus auvy y is a form of analysis): (b) an ascent toprior propositions or premisses. Though Plato was the first to formulatethese methods as methods of philosophical inquiry, he does not himselfuse the term (XvOCauaLo describe them. Its first occurrence as a logicalterm is in Aristotle. There it is used of the method of "breakingup" anargument into its premisses, and these into their terms (see An. Pr. 46,4o ff.). It is essentially a method of working back from a given conclusionto the premisses from which that conclusion is deduced. And the de-scriptions of logical analysis in the Greek commentaries naturally followthe terminology of Aristotle's discussion of syllogistic analysis, just asthe terminology used to describe the propositions which constitute thelimit of the analysis reflects Aristotle's terminology in specifying theconditions for the premisses of o au)oytaot6e La-T-1J OVLX64 (An. Post.7ib, I6-22) - 7p()o, 05y4a, 7pOTEppOA,ona, yvxpc p.In the formulations which the commentators give of geometricalanalysis, this same terminology is used. There is the same generalemphasis on analysisas a discovery of proof, and it appears to be assumedthat geometrical analysis is simply the application to a particular subjectof a method which has other and wider applications. The descriptions arevery closely parallel to Pappus' description in the first part of his account.Alexander (In An. Pr. I, 7, I i- I 8), described it as a method which takesthe auv=?pOC (conclusion) as starting-point, and proceeds upwardsprogressively, through the assumptions necessary for the demonstrationof the conclusion, ?7:L Tag4 &pXq. Proclus, in commenting on themethod &tocycoyn rs &UvOTov (In Eucl. p. 25S, I2ff., Friedlein), saysthat in general all mathematical proofs are eitherfrom Cp;o'Lr to p cLand the latter he divides into those OerLxx'L tiCiv &pyfv, which areanalyses as opposed to syntheses, and those &VoLprTLxOx i.e. proofs byaocyCy4 ?La,&&iVXTOV.Similarly (ibid. p. 69, I 6- I9) he contrasts &oTr6eLr,as used in the passage from first principles to the things sought, with3ovXuag,as used in the opposite process from the things sought to thefirst principles (cf. 8, g-8; 5, I6-I8; 2II, 19-21). Ammonius (In An.Pr. 1,5,27-3i) quotes Geminus' definition of geometrical analysis asMro8sLrCo 'UP?CL4,and describes it, with the complementary synthesis,in much the same terms as Alexander. A fuller account by Philoponusdescribes it as the discovery of the premisses from which the truth of theconclusion is inferred, the procedure being to assume the conclusion astrue, and to analyse it until one reaches TLV%OPO)OYO\)tLVM XOL T&q &PX&4Tiq yecoeU'pX'Caq (In An. Post. 1,I62, 23 ff.). He gives a detailed illus-

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tration of this (In Phys. 333, 3 if., Vitelli) by taking the proposition thatthe three angles of a triangle are equal to two right-angles as the starting-point, and analysing the prior conditions for the establishing of its truthuntil he reaches, as the limit of the analysis, the definition of straightline (see Euclid, I, io); he then adds the complementary synthesis (cf.his remarks, In An. Post 1, 3 19, I S ff., whcrl he says that in syllogism thelimits of analysis are O'pLapoL or &awaoL porOasCL, adding that in analysingthe proposition that the three angles of a triangle are equal to two rightangles, the limit is the definition of straight line and angle. SimilarlyEustratius, In An. Post., p. 3g, 11.23-6 (Hayduck). Proclus (in Eucl.pp. 382,422) speaks of the analysis of rectilinear figures into triangles,the ultimate figures into which they can be analysed).It is clear that this method, as the commentators describe it, is closelyakin to what they describe as philosophical analysis and syllogisticanalysis. In each of the three forns of analysis there is a similar processof upward resolution into principles, and in the formulation of eachmuch the same terminology appears. A striking instance of resemblancebetween formulations of philosophical and of geometrical analysis is theresemblance between the formulation of geometrical analysis in thecommentaries on Aristotle and the formulation in Albinus of whatAlbinus classifies as the second of three forms of philosophical analysis inPlato.' Cherniss (IOc. cit. p. 41 9) says of this latter that it is "areductionof Phaedrus 245 c-e to a scheme which is appareintly influence(d bygeometrical analysis". 2 Equally well, Albinus' formulation may be takenas an indication of the influence of philosophical formulations of analysison the formulation of geometrical analysis. Certainly the way in whichgeometrical analysis is formulated in Proclus and in the commentaries onAristotle suggests that to its formulation the development of philo-sophical analysis by Plato and of syllogistic analysis by Aristotle con-tributed a great deal. In Plato's case this is no doubt the explanation ofthe close association in antiquity of Plato's name with geometrical1 It is introduced as 1)&OCCi)v8LXVUIL6V&)Val U7O8ELXxVu[LkvcaV&vokq rTdroe; &vaTro8ae(x'roug xaxl &?Lkaouq 7po-r&aeL;, and then described more fully as follows (Didaskalikos V.p. I57, 11. 19-23, in Hermann, vi.): - eaOrTOw nt 8ii 6 o4,Tou.VOV x>ol oEcapeLv,rtLvo&rl -=p6repo ocivtoi3,xoL Ta5-rua&0o8etXV6eLV&Tr6mvUvripov &nd & 7rp6TEpot&vt6vtr,gjq &V IXOWoEV &17l TO6%pCOYV xKo 6[LO),OYOU?4LVOV, OC76 To'uTou 8g CaP4LCVOL 7id T6To,UgCVoVareXeua6FLaO auvOvtLX7I tp6nc-. The other two forms of analysis are(i) the 'ascent' from sensible particulars to Forms (illustrated from the Symposium)(II) the upward path of the Phaedoand Republic.2 The analysis of the proposition that soul is immortal is given also by Ammonius,In An. Pr. I, g, 34ff., as an illustration of syllogistic analysis.6


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analysis; in both Diogenes Laertius (III, 24) and Proclus (In Eucl. p. 2 I I,I8-23) we find the statement that Plato explained or communicated themethod of inquiry by analysis to Leodamas. Other evidence suggests,however, that the method was not initially a philosophical formulation.It suggests rather that the work of Plato and Aristotle led to the moreprecise formulation of a method of geometrical analysis already practisedand already formulated, although this latter formulation doubtlesslacked the more precise mathematical terminology which belongs to thesecond, and, almost certainly, later formulation (Pappus' description(II)). Thus when Plato first introduces in the Meno he method of analysisf' o60e'acwq,he likens it to a method of analysis in geometry which,he says, geometers frequently practise, and which he represents as amethod of analysing antecedent conditions for the possibility of solvinga problem.' Much more important evidence is that of Aristotle. Inseveral passages he indicates clearly the method recognised as geo-metrical analysis in his time; the implication is that this method wasinitially formulated by geometers. At the same time these passagesconfirm the view that the tradition which Pappus follows in his de-scription (i) of the methodi s a reliable one. There is, in the first place,the familiar passage in the Ethics(E.N. I I I2 b, 28ff.) in which the processof deliberation in the sphere of action is compared with the analysis of afigure in geometry (86x poq4tc). It is a method which, as a startingpoint, assumes the end to be achieved, or assumes a problem solved, andworks backwards through the prior conditions for the achievement ofthe end or the solution of the problem, until it reaches a conditionwhich can be satisfied in action (deliberation) or in construction(geometry). In E.E. I 227b, I 2 ff., the proposition that the angles of atriangle are equal to two right angles is given as an example of a propo-sition which can be analysed by this method (cf. the remarks of theanonymous commentator on the former passage (p. Igi, Heylbut),which describe geometrical analysis as a process of assuming the con-clusion to be true, working backwards through prior assumptions 'erLrrv apyjPv, hen proving the conclusion by synthesis). In the account of"production" (y&veaLq) in Met. io32b,6-29 Aristotle calls the activityin the ascent from starting-point to conditions v6onLq,as he does later inIO I a, 2I-33 in describing how geometrical relations are discovered by"actualisation". In vAYsc we perceive the divisions which exist po-1 Meno. 86e-87b. The example given is of a 8LopLtqL6q, the determination of theconditions for the possibility of the solution of a problem. See Thomas, GreekMathematicalWorks, , pp. 394-7.

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tentially in a given figure, and "actualise" them by construction. Thisanalysis is illustrated' by the process of discovery (vpreatq, 11. 23, 30)of the proof that the angles of a triangle are equal to two right angles 2,and that the angle in a semi-circle is a right angle (see Euclid, III,3 ;the premiss in Aristotle's passage appears to be Euclid, I, 32). Since8aLypoctqLs used both for geometrical figures and geometrical proofs,aVOCS)V 8aLypLa[L o might mean, as Einarson points out 3, "either tobreak a figure up into its parts or to divide a theorem iilto its premissesor aotLeXZ". Aristotle makes clear elsewhere what already seems to beimplied by these examples and by the formulation itself of the method -that the implications are not reciprocal and that it is only the synthesis orproof which is deductive. In Physics 2ooa, Aristotle at once compares andcontrasts logical necessity in mathematics and "hypothetical" or "tele-ological" necessity in natural processes. His primary purpose is to notethe analogies between deduction in mathematics and the analysis of thepreconditions for the achievement of an end in nature. Thus if wecompare the relation of necessity between premisses (i) and conclusion(II) with that between end, as consequent, (a), an(dpreconditions (b),we find that while (i) is false if (ii) is false, and (a) is not achieved if (b)is absent, yet (Ii) does not entail (i), nor does the presence of (b), whichis a necessary but not also a sufficient condition of (a), entail theachievement of (a). In each case the relation is irreversible, for in eachcase there is"the necessitation of a aUt6pxa:obyan'pp',unaccompaniedby a necessitation of the ip-n by the av=_'pxaua " (Ross, Aristotle's Physics,p. 532). So in mathematics, "since the straight line is what it is, it isnecessary that the angles of a triangle should equal two right angles", butnot conversely; the premiss (i) is logically prior to the conclusion (iX) inthe sense that (i) can be assumed without assuming (iJ), but (ii) cannotbe assumed without assuming (I). The second example given in Met.I o I a - that the angle in a semi-circle is a right angle - is found again inAn. Post. 94a28-35, and there the way in which Aristotle desribes theanalysis suggests that in this example too he is assuming that the relationbetween premiss and conclusion is non-reciprocal. For he asks from whatassumption (-0o5 O,v-roq;) he proposition follows, and finds the premissin the definition of a right angle as the half of two right angles.We have then, in Aristotle, the recognition of a method of analysis ingeometry, corresponding to Pappus' description (i), and illustratedl byI See Cornford, loc. cit. pp. 44-45, and Ross, Aristotle's Metaphysics, i, pp. ?68-73.2 This favourite example is used again in reference to analysis in An. Pr. 48 a, 2 9.8 Ioc. cit. p. 39. cf. Cornford. loc. cit. p. 44.8


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examples where the relation between PCnpZ-nd au,7repxastx s recognisedto be irreversible. Moreover, what seems to be assumed in Aristotle, inProclus, and in the commentaries on Aristotle, is that geometricalpropositions can be ordered in a hierarchy, and it is principally in its usein furthering such an ordered science of geometry that the importanceof a method of non-deductive analysis would appear to lie; its task wouldbe that of systematising geometrical knowledge and co-ordinating resultsby leading propositions back to first principles - to axioms or definitionsor something already demonstrated. It is indeed possible, in view of thelarge part played by geometrical ideas in the Analytics, and especially inthe account there of first principles, that this conception of an orderedscience of geometry was taken by Aristotle from the geometers of histime asa model for his own conception of scientific knowledge generally.lWhat at least is a reasonable assumption is that it was primarily inrelation to the ideal of reducing geometry to a system that the geometersviewed the fLnction of analysis, and that their formulation of the method,as it was known to Aristotle, reflected principally its function-of reducinggeometrical propositions to first principles. In their descriptions of themethod it is this function which the Greek commentators stress. Andthese descriptions, together with those of Aristotle and of Proclus,afford sufficient evidence that it is not because he happens to be lookingat deductive analysis upside-down that Pappus describes analysis as asearch for prior propositions; it is because he is repeating a recognisedformulation of geometrical analysis, distinct from that described in hissecond account.What is absent from these sources is any account of analysis corre-sponding to Pappus' account of it as deductive. There is one passage inAristotle the elucidation of which would appear to make such an accountfrom the commentators especially relevant. It is the passage in thePosterior Analytics (78a) where Aristotle says that mathematical propo-sitions are more commonly convertible than dialectical propositions, andthat this makes analysis in mathematics easier. The commentatorsexplain analysis here as a method in geometry of discovering thepremisses from which a proposition assumed true as conclusion can bedemonstrated, (Themistius, In An. Post. 1,26, 11. 2 2-4; Philoponus, InAn. Post. 1, I62, 11.16-28). Thus they assume that what Aristotle isreferring to is geometrical analysis, and the account which they givecorresponds with the account in the first part of the Pappus passage,showing no awareness of any other formulation of the method relevant toI See H. D. P. Lee, C.Q, XXIX, pp. I1 3-1 24.

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cases where the implications are reciprocal. Before discussing theircomments further, it is important to consider the precise relevance ofthe passage to geometrical analysis. Aristotle says that if false premissescould never give true conclusions analysis would be easy, for premissesand conclusion would in that case inevitably reciprocate. After illus-trating this he says that reciprocation of premisses an(l conclusion is morefrequent in mathematics, because mathematics takes definitions, butnever an accident, for its premisses. The main point to be considered iswhether Aristotle's illustration is an illustration of the method ofgeometrical analysis, presupposing perhaps a known formulation of it asa deductive procedure. The illustration is as follows: -3atc yap TO A ov- -ro'Aou 8' "VTOg 'aN ?a', Xt%

O6- a-GrLV, lov TrOB. Ex Tourov apoaei 6-nt C`xLv zXZvo.Ross 1 says that &vocvxeLv n the present passage means "the analysis of aproblem, i.e. the discovery of the premisses which will establish thetruth of a conclusion which it is desired to prove". He takes A as aproposition assumed true as conclusion, and the movement from A to Bas a movement from proposition to premisses. So does Philoponus.Taken in this way the illustration is recognisably an illustration of ageometrical analysis which deduces consequences till something knownto be true is reached, and this last becomes the premiss of thesynthesis which demonstrates the truth of the original assumption. Thisinterpretation is, I think, wrong. The point which Aristotle wishes toillustrate is that where premisses and conclusion reciprocate, then, iffrom certain premisses you reach a conclusion known to be true, it ispossible to reason in reverse to the truth of the premisses, but thatwhere there is no such reciprocation, then the fact that from certainpremisses you reach a conclusion known to be true does not allow youto infer the truth of the premisses; for if A entails B and B is true, it doesnot follow that A is true, since true conclusions can follow from falsepremisses.2 A is premiss, B is conclusion. For if B is premiss and Aconclusion, then if B is known to be true, and B implies A, then A mustbe true, whether the relation of implication between A and B is reciprocalor not. Thus Aristotle's illustration: (a) is a perfectly good illustrationof the point that it is only if the relation between A and B is reciprocalthat you can be sure that your premisses are true: (b) is relevant to1 Aristotle's Prior and PosteriorAnalytics, p. 549.2 This is precisely how Ross explains Aristotle's point (ibid.). Yet he has just arguedthat A is conclusion, B premisses.I 0


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analysis merely in that if, in an analysis of B, you see that A implies B,which is true, then you can be sure that your premisses are true if therelation between A and B is a reciprocal one. Both Themistius andPhiloponus seem to realise that the illustration is intended to illustrate(a), but at the same time try to make it an illustration of geometricalanalysis and synthesis. And it cannot logically be made to perform thisdouble service. Thus Themistius explains the comparative ease ofanalysis in mathematics by saying that whereas in dialectical arguments(&'XaoyoL), where gvaoRa may form the premisses, the search for thepremisses of a given conclusion is 'v &brELpoL, the mathematician,taking only definitions and properties, which are few in number, as hispremisses, searches within a limited and clearly defined field, and neverin fact takes anything that is not true for premisses. In dialecticalanalysis the difficulty is that, since true conclusions may follow fromfalse premisses, one can never be sure that one's premisses are not false,being as they are indefinite in number and lending themselves to variousambiguities (2 6, 11.24-3 3). Similarly Philoponus (I 6 2, 1. 28- I 6 3, 1. I 3).In each case the remarks are prefaced by a description of geometricalanalysis as a discovery of premisses. It is not clear at this point whetherthe analysis is taken to be an analysis of A or B. What is to be noted isthat the distinction made between mathematical and dialectical analysisis not one of direction,but between simplicity and complexity in the fieldof inquiry. Philoponus now goes on to deal with Aristotle's remarksabout reciprocation as a separate point. He says (i 63, 16-2 I) that if A isconclusion and if B stands for the premisses, which are true, then Anecessarily follows from B, and, reciprocally, B necessarily followsfrom A; and tlhus the finding of the premisses is easy because theynecessarily follow the conclusion. This last remark suggests that he ismaking the point that, where the implications are reciprocal, theanalysis which gives you the premisses is simply a process of deduction.But the form of his argument makes it clear that he is not concerned witha method of proving a given conclusion by a deductive analysis followedby a deductive synthesis. He is still elucidating the point that, whereas indialectical arguments the same true conclusion can follow from manydifferent premisses or from false premisses, in mathematics recipro-cation allows you to see not only that your premisses are true but thatthey are a necessary and sufficient condition of the truth of the con-clusion. And since his point is that what the reciprocal inference enablesyou to establish easily and certainly is the truth of the premisses, heought, to be consistent, to have taken A as premiss, B as conclusion;


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otherwise the premisses are known to be true before the reciprocalinference is made.These comments of Themistius and Philoponus, though offering noproper Cluci(lationof Aristotle's point, do at least make clear that neitherof them knows of a formulation of geometrical analysis representingboth analysis an(l synthesis as dedLctive. A further possible inferencefrom their remarks is that the description of geometrical analysis as amethod of fin-ding premisses or prior propositions from which a givenconclusion will follow was accepted as a correct formulation irrespectiveof whether the implications were non-reciprocal or reciprocal. Theconfusion in the remarks makes this unlikely, perhaps, but it doesdeserve note that in commentaries both on a passage (Phys. 200a) wherea case of non-reciprocal implication is discussed and on a passage (An.Post. 78a) where a case of reciprocal implication is discussed, preciselythe same formulation of geometrical analysis is presented. The con-clusions to be drawn from Aristotle's own remarks in the latter passageare (i) th-atthey recognise the practice of a form of mathematical analysisin which the implications are reciprocal, and recognise that very manymathematical propositions are convertible; they imply, at the sametime, that some are not convertible: (II) they do not themselves presenta formulation of a method of geometrical analysis in which the analysis isdeductive. There is indeed, no reliable evidence at all for dating, evenapproximately, the first formulation of that method. Heath (op. cit.p. 137 n. 4) is inclined to accept Heiberg's view, which traces the originof the definitions interpolated in Euclid XIII to Heron of Alexandria,possibly the Ist century A.D. This view is based on a comparison of thosedefinitions with definitions quoted from Heron by an-Nairizi in hiscommentary on Euclid (Curtze p. 89). What the Latin translation termsdissolutio and compositio do appear to be equivalent to the Greek &vaXuaLqand uv0act4 as geometrical terms 1, but the definitions of them are farfrom precise. In so far as they are comparable with other accounts ofanalysis, their resemblance to the accounts given by the Greek commen-taries on Aristotle is closer than it is to the account of deductive analysisin the second part of the Pappus passage. They constitute no argumentfor Heron's authorship of the definitions interpolated in Euclid. All thatcan safely be said about the formulation of a method of geometricalanalysis as deductive is that it was earlier than Pappus, and that Pappus1 an-Nairizi, in his commentary on IV, s if, himself adds an analysis of the demonstration(see Curtze, pp. i4 ff.), in Latin the solutio, as opposed to the synthetic demonstrationfound in Euclid, the compositio.1 2


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knew of it. It is extremely unlikely that the account of it as deductive inPappus represents its first formulation.The evidence we have considered strongly suggests that Pappus, for hisaccount of geometrical analysis, is repeating two different formulationsof the method, one describing it as an upward movement to priorassumptions from wlhich an initial assumption follows (i), the other as adownward movement of deduction from an initial assumption (nI). And(i) is not equivalent o (ii) for all cases of analysis,although t is possiblethat (i) was recognised as a correct formulation for the method of which(II) is the formulation. It follows, as we have already seen, that Pappus'description (i) of anialysis s not (lue to the accident that he is looking atdeductive analysis upside down. And since he appears to be assumingthat (i) is equivalent to (In) for all cases of analysis, it follows that theapparent inconsistencies in his account are due simply to the inconsistencyof this assumption. Pappus overlooks the point that (i) is a correctformulation for cases for which (II) is an incorrect formulation. Theform itself of his account suggests that it is made up of two separatedescriptions of the method. An initial statement that analysis is a passagethrough the successive consequences of an assumption is followed by twodescriptions of analysis, the first a more general one, the second dividingit into theoretical and problematical analysis. Each description isperfectly self-consistent. According to the first description, analysis is aresolution upwards through propositions antecedent to an initialassumption until something already known or ranking as a first principle(nr&RLv ppXjq "ZXov)s reached. The synthesisis then carefully describedas a process of deduction from the last step in the analysis, arranging intheir natural order (xour&puatv) as consequents what were formerlyantecedents (ta 7ZpoInYoU4svLa).ccording to the second description theanalysis is deductive; it is then said simply that the proof (&7r6CLL)will be the reverse of the analysis. But whereas the first account says ofthe end of the method only that, having reached in your analysis a firstprinciple or something already known, you proceed deductivel yuntilyou finally arrive at the construction of what was sought, the secondaccount describes as results of the method what would not alwayslogically result from the practice of method (I). It says that if in analysisyou reach something admitted to be false or impossible, then that whichis sought will be false or the problem will be insoluble. But this conse-quence is not, of course, necessarily rue for cases where the implicationsare not reciprocal, since false premisses may give true conclusions.Robinson uses this point as an argument for the view that reciprocation is

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GreekG EometricaAl Nafysis - Norman Gulley

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