Philoponus: On Aristotle Posterior Analytics 1.9–18 9781472552051, 9780715640890

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Conventions [] Square brackets enclose words or phrases that have been added to the translation or the lemmata for purposes of clarity, as well as those portions of the lemmata which are not quoted by Philoponus. Angle brackets enclose conjectures relating to the Greek text, i.e. additions to the transmitted text deriving from parallel sources and editorial conjecture, and transposition of words or phrases. Accompanying notes provide further details. () Round brackets, besides being used for ordinary parentheses, contain transliterated Greek words and Bekker pages references to the Aristotelian text.

Introduction1 Philoponus’ work, the earliest surviving commentary on the Posterior Analytics, was not the first. Theophrastus wrote a commentary in seven books,2 while Galen boasts of having composed six commentaries on the first book of the Posterior Analytics and five on the second,3 and Philoponus himself refers to Alexander’s commentary.4 We do not know the scale of these earlier works, but Philoponus’ commentary (333 CAG pages on book I alone) can claim to be among the longest ever written. Maximilian Wallies’ edition, published in 1909, contains in addition to Philoponus’ commentary on book I, the commentary on book II that is (by almost universal agreement5 falsely) attributed to Philoponus, as well as an anonymous commentary on book II. The present volume is the third of a projected four which will present the first complete translation of Philoponus’ commentary on book I and pseudo-Philoponus’ on book II into any modern language.6 Interest in this commentary has waned since the Renaissance, when it was the first of Philoponus’ major philosophical works to be edited.7 It was twice translated into Latin in the sixteenth century, with the translations being reprinted frequently from 1534 to 1569.8 In it Philoponus sets himself the task of expounding the meaning of the Aristotelian text rather than raising objections and presenting alternative theories in the ways that make some of his later commentaries (notably the Physics commentary) important philosophical works in their own right. Here, he is in the main content to follow the run of the text, clarifying difficulties and explaining Aristotle’s statements in the light of other passages in the Posterior Analytics or of other works of Aristotle. Among these, he tends to limit his references to the other logical works (the Categories, De Interpretatione, Prior Analytics, and Topics), and to the Physics, Metaphysics, and De Anima. His practice of explaining Aristotle through Aristotle has been standard ever since. The opening words of the work, ‘John of Alexandria’s lecture notes from the meetings of Ammonius, son of Hermeias, on the first book of Aristotle’s Posterior Analytics, together with some observations of his own’, indicate that it contains some original contributions of Philoponus, but that it is largely a record of Ammonius’ views. The absence of other direct testimony about Ammonius’ lectures on the

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Posterior Analytics makes it impossible to determine how much is due to Philoponus. That Philoponus does not follow his teacher slavishly, however, is shown by two passages in the section translated in the present volume (157,20-158,13 and 177,19-178,13). In the former passage Philoponus rejects Ammonius’ explanation of 77b34-5 because it ‘does not accord with the passages in Aristotle’. In the latter he invokes Themistius’ interpretation of 78b28-9 to show that Ammonius’ interpretation is incorrect. Criticisms of Alexander (122,10-20, 126,3-10, 159,17-160,13) and Proclus (112,25-36) may also display his originality, but they could also be due to Ammonius (as is the criticism of Proclus at 112,20-36) or to others in the tradition (as is the criticism of Alexander at 111,31-112,8). In any case it is clear that Philoponus engages vigorously with the earlier commentators on the Posterior Analytics. Philoponus presumably decided to offer this sort of exposition of Aristotle’s work because he thought that the Posterior Analytics is difficult (and so in need of a detailed guide and explanation), and also because he thought that it is important. Whatever our view of its importance, at least we may agree that it is difficult and in need of clarification. It is neither a systematic exposition of its subject (the theory of demonstration) nor does it proceed in the main by examining earlier opinions on the subject, as is Aristotle’s frequent practice in other works. Its train of thought from chapter to chapter, from paragraph to paragraph and even from sentence to sentence is frequently obscure. The expression is also more condensed and opaque than is usual for Aristotle, and there is relatively little argument, much of the burden of the discussion being carried by examples (a practice which Philoponus continues, adding additional examples of his own to the many already in Aristotle). In the commentary on chapters 9-18, we note a continuation of the tendency (already noticeable in the commentary on the opening chapters) to supplement Aristotle’s discussion, sometimes by working out arguments and examples in greater detail than Aristotle does, sometimes by showing how Aristotle’s claims apply to cases that he does not discuss, sometimes by drawing distinctions not found in the Aristotelian text and exploring their implications for the theory of demonstration, and sometimes by discussing puzzles not posed by Aristotle. The first of these features is found in the treatment of the demonstration that walls do not breathe, where Philoponus investigates why conclusions of this sort cannot be inferred in the first figure (176,16-177,4); also in discussing the thesis of chapter 14, that the first syllogistic figure is the most scientific, where he treats Aristotle’s arguments in favor of this thesis, justifying some of Aristotle’s brief remarks (184,13-24 and 185,9-26) and supplying a further argument of his own (184,24-7). This feature is particularly noticeable in the detailed discussion of Aristotle’s treat-

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ment of errors in chapters 16 and 17, in which Philoponus systematically goes through all the possible cases in the relevant syllogistic figures.9 The second feature is exemplified in Philoponus’ discussions of cases where the cause occurs and the effect does not (172,17-173,20), in his account of why circular wounds heal slowly (182,13-27) and his claim that medicine is subordinate not only to geometry but to astrology as well (182,27-183,3). Instances of the third feature are found in Philoponus’ distinction between mistakes ‘in respect of the matter’ and ‘in respect of the form’ (152,15-22) as well as in the observation that the requirement that demonstrations have immediate premises entails that previously proved conclusions cannot be used as demonstrative premises – an exceedingly strict interpretation which, Philoponus notes, has the untoward consequence that strictly speaking only the first two proofs in Euclid’s Elements qualify as demonstrations (120,3-14 with note). The last-mentioned feature is found at 146,14-147,16, where Philoponus identifies a puzzle about subalternate pairs of sciences, where the superior science demonstrates the principles of the subordinate one and the expert in the subordinate science knows the facts of the superior science: does this mean that the subordinate science contains arguments for its own principles? Also noteworthy is the prominence of mathematical and astronomical examples,10 including the use of a theorem in Euclid’s Optics to show how geometry explains results in optics (178,20-179,10). In this practice Philoponus is following Aristotle’s lead, but he employs such examples even more frequently than Aristotle does, and on some occasions his comments are far more extensive than are needed to explicate the text. The long treatment of Bryson’s quadrature of the circle (111,17-114,17) is the most extreme case found in the present volume. With the exception of this last-mentioned case, the geometrical examples mostly come from Euclid’s Elements and are of an elementary character. Some of the definitions of ratios of numbers come from Nicomachus’ Introductio Arithmetica, on which Philoponus wrote a commentary.11 It is interesting that Philoponus is guilty of a number of mathematical errors (see notes 4, 14, 19, 26, 52, 274, 300 on the translation), which may indicate that although he was interested in mathematics he had little mathematical comprehension or talent. In this connection it is interesting to note that he took Alexander to task for a mathematical mistake, saying that ‘he did not have much of an aptitude for mathematics’ (159,18-19). A similar verdict may perhaps be rendered on Philoponus’ abilities as a logician (see notes 167, 176, 202, 263, 270, 294, 299, 304, 342, 369, 374, 385, 386, 388, 389, 391, 392, 403, 409, 410, 438, 441, 446, 449, 450, 464, 471 to the translation). Beginning with chapter 11, Philoponus prefaces his commentary

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to each chapter (and sometimes to more than one section of a single chapter) with a long introduction and summary (chapter 11: 135,3138,25; chapter 12: 145,13-148,5, 150,11-152,3, 154,13-155,20; chapter 13: 166,15-169,27; chapter 14: 183,6-184,27; chapter 15: 185,29-186,21; chapter 16: 191,3-193,29, 198,7-25; chapter 17: 207,4208,11; chapter 18: 213,17-215,5). These sections are much longer than in the preceding chapters and represent a change in Philoponus’ style of commenting. Another point of interest is the emphasis he gives to first philosophy and dialectic. Philoponus notes that they have some common points: whereas ‘each science is concerned with a single genus  only first philosophy and dialectic have all things that are as their subject’ (142,25-143,1) and they both ‘employ the principle of non-contradiction as common and pervading all things that are and are not’ (140,23-5). In a neat (though not most clearly set out) taxonomy, Philoponus maintains that first philosophy, dialectic and all the sciences employ the common axioms, that as opposed to the sciences, first philosophy and dialectic are not restricted to a single genus, and as opposed to first philosophy and the sciences, dialectic gets its premises by asking questions rather than obtaining them ‘from the very nature of things’ and aims at refuting the interlocutor rather than establishing the truth (143,6-29). As to first philosophy in particular, he maintains that it ‘discovers and demonstrates the principles of every science’ (118,23-4, 119,8-21) and in particular that ‘in the third book of the Metaphysics Aristotle demonstrates the very axiom of contradiction’ (141,8-9). Philoponus says even more about dialectic despite the fact that Aristotle mentions it only three times in the chapters translated in this volume (77a29, a31, b39) and mentions dialectical conversations (if this is the correct translation of dialogoi) only once (78a12). Many of the points he makes about dialectic are found in comments on passages that have no direct connection with that topic. This is especially noticeable in the case of the references to dialectic in the discussion of errors in deductions (154,15-155,20, 156,4-5, 158,16-28, 162,32-163,13, 163,25-164,3), where Philoponus is on the verge of identifying dialectical reasoning with fallacious reasoning. The final feature I will take up is Philoponus’ idiosyncratic treatment of Aristotle’s views on scientific principles. I will here simply sketch out some of the most salient features of his interpretation, since a thorough treatment is beyond the scope of this introduction.12 Philoponus begins by dividing ‘the things in every demonstration’ in a way not found in the Posterior Analytics (7,18-10,20): ‘the problem proposed for demonstration, and the premises through which the problem is established’. Further, ‘in every problem the following two things are observed: the given and the sought’. He goes on to identify the given with the subject term in the conclusion of the demonstra-

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tion and the sought with the predicate. He then proceeds to bring this result to bear on Aristotle’s distinction between two kinds of prior knowledge: knowledge ‘that it is’ and knowledge ‘what it is’: with respect to the given it is necessary to have both kinds of prior knowledge, while with respect to the sought it is necessary to have only the second kind and impossible to have the first kind. In effect the demonstration proves of the sought ‘that it is’. Axioms are the major premises of demonstrations (8,7-8). Some axioms are common to all sciences, some are common to more than one science but not to all, and some apply only to a single science (10,27-11,3). Scientific principles are divided into axioms and theses, and theses are subdivided into definitions and hypotheses (thus far Philoponus follows Aristotle (72a14-21)). The difference between axioms and theses is that the former are self-guaranteeing and we know them from within ourselves, while the latter are self-guaranteeing too, but require some attention in order to be understood because they are not selfevident without qualification, and we take them from the teacher of a given science. Hypotheses differ from definitions in that hypotheses predicate one thing of another while definitions do not, but are identity statements. Further, there are two species of hypotheses, known respectively as hypotheses (homonymously with the genus) and postulates. Both kinds are taken from the teacher without demonstration, but hypotheses are true, appear true to the learner, and require little attention for their truth to be seen (127,31-3, 129,3-5), while postulates do not appear true (129,5-6). Finally, there are two subspecies of hypotheses (homonymously with the genus): hypotheses in the strict sense and hypotheses relative to the learner (127,25-33, 128,13-15). Philoponus’ interpretation has not been followed by other commentators, with good reason. A catalogue of some of its characteristics may provide the groundwork for a study of Philoponus’ methods. First, he employs his normal practice of explaining Aristotle through Aristotle, bringing in material (sometimes uncritically) from elsewhere in the Posterior Analytics. This accounts for some weaknesses in his treatment of hypotheses and postulates. (Notably it depends heavily on the discussion at 76b23-34 of hypotheses that are different from the hypotheses identified at 72a18-20 as a kind of scientific principles. The hypotheses in question, and the postulates from which they are distinguished, are identified as being demonstrable, that is to say, not scientific principles, which has catastrophic consequences when this passage becomes a basis for an interpretation of the doctrine of scientific principles.13 One of these consequences is that Philoponus treats hypotheses as principles.) Second, Philoponus places greater weight than modern commentators on Aristotle’s references to the psychological attitude appropriately held towards principles. For example, he distinguishes

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axioms from other principles on the grounds that we know them ‘from within ourselves’ (autothen) and without demonstration (34,1011, 34,20-2) – features which surely apply to all indemonstrable principles! If we have our knowledge of the principles ‘from within ourselves’ and this knowledge is ‘natural to all men in common’ (127,21-4), then it requires some work (which Philoponus does not provide) to square this view with the important Aristotelian distinction, fundamental in the Posterior Analytics, between what is better known in nature and what is better known to us. (Here too some of the difficulty is due to the account of hypotheses and postulates in 76b23-34, where the principal difference between hypotheses and postulates is that the former the learner is ready to accept, while the latter he either disbelieves or has no opinion on them.) Third, he employs authorities other than Aristotle, sometimes without sufficient care. This is clear in his use of Euclid as an authority for the existence of axioms that hold in only one science – a doctrine that goes squarely against Aristotle’s account. There is also good reason to think that he bases some of his treatment of scientific principles on Proclus’ discussion in the introduction to his Commentary on the First Book of Euclid’s Elements, again not always with fortunate consequences. The picture that emerges from a study of the relevant passages is that Philoponus utilized Euclid and Proclus to explicate Aristotle, but that neither of these authors can be used uncritically as a reliable guide to Aristotle’s account of scientific principles. There is an obvious connection between Aristotle’s division of three kinds of principles and Euclid’s: both have definitions; one of Aristotle’s standard examples of common principles, which he sometimes calls axioms, that if equals are subtracted from equals the remainders are equal, is identical with one of Euclid’s ‘common notions’, and Aristotle’s hypotheses correspond to some extent with Euclid’s construction postulates. In fact, Aristotle seems to have based his doctrine of the types of principles on mathematical practice. But he did not follow the mathematicians slavishly; his need to produce a doctrine applicable to all sciences, not just to geometry or to mathematical subjects in general, forced him to generalize. Thus, his hypotheses posit the existence of primitive entities whereas Euclid’s corresponding postulates posit constructibility. But there are other mismatches: two of Euclid’s five postulates have nothing to do with constructing basic entities, one of his common notions (that two lines do not enclose an area) applies specifically to geometry, and he does not include two of Aristotle’s examples of axioms (the principle of non-contradiction and the law of the excluded middle) anywhere among his principles. Some of Euclid’s definitions present problems as well. There are reasons to think that Euclid based his division of principles on Aristotle’s discussion in the Posterior Ana-

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lytics, and when he needed principles that are neither definitions nor construction postulates nor principles that hold in other branches of mathematics as well as geometry, he felt strongly enough bound by Aristotle’s doctrine of three kinds of principles that he simply fitted them into whichever type of principle seemed most reasonable.14 Proclus goes to great efforts to make sense of Euclid’s division of principles. The relevant passages of his commentary are 75,27-77,6, 178,1-179,12, 181,1-15, 181,24-183,13, 183,20-184,10, 184,12-29, 193,15-194,9, 195,23-196,14.15 He discusses several previous accounts, frequently mentioning Aristotle in this connection. He correctly shows that each of these accounts fails to account for Euclid’s principles as they stand, and he ends up apparently approving incompatible accounts (184,12-22, 194,4-8, 195,23-5). It is easy to see references to Proclus’ discussion in Philoponus. Since Philoponus states that his commentary on the Posterior Analytics is based on Proclus’ student Ammonius, the likelihood that Philoponus actually made use of Proclus’ commentary on Euclid is large and accounts for many of the peculiarities of Philoponus’ account of Aristotle’s principles. The goal of this translation is to render Philoponus’ text faithfully into acceptable English, while striving for consistency in the translation of important terms. The Greek-English Index and the EnglishGreek Glossary provide complete information about how individual words have been translated. Philoponus provides lemmata which indicate the stretch of text he is discussing. Since he does not discuss every sentence there are gaps between the lemmata. In conformity with the practice of this series, I have translated the entire text of Posterior Analytics 1, chapters 9-18. Text that is not included in the lemmata is placed in square brackets. The translation of the Aristotelian text differs from other translations in two principal ways. First, as we can tell from the commentary, Philoponus’ text of Aristotle differed in places from modern texts. Accordingly, I have translated Philoponus’ text, recording the differences from Ross’ edition in the notes (collected in an Appendix). Second, since Philoponus quotes and paraphrases the Aristotelian text throughout his commentary, I have produced a translation that is more ‘literal’ and less idiomatic in English than is perhaps desirable, in order to preserve the Aristotelian phraseology in translating Philoponus’ text. In fact, Philoponus quotes Aristotle more frequently than Wallies’ text indicates, and the present translation makes an effort to reflect this fact by putting all the quoted material in inverted commas. However, the flexibility of Greek word-order raises a problem in this connection. As an example, consider 144,21-4: epistêmonikai protaseis, ex hôn ho sullogismos, hôrismenai eisi kath’ hekastên epistêmên, touto an eiê erôtêma epistêmonikon ho an êi ek protaseôs

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tinos ex hôn ho kath’ hekastên epistêmên sullogismos ginetai, which refers to 77a37-40: protaseis de kath’ hekastên epistêmên ex hôn ho sullogismos ho kath’ hekastên, eiê an ti erôtêma epistêmonikon, ex hôn ho kath’ hekastên oikeios ginetai sullogismos, ‘premises in each science are the things from which the deduction in it [is formed], it follows that there is such a thing as a scientific question, [namely, one formed] from premises on the basis of which a proper deduction in any [science] is formed’. Philoponus characterizes the protaseis as epistêmonikai, and switches the order of ex hôn ho sullogismos and kath’ hekastên epistêmên, also adding a verb (hôrismenai eisi) which changes the syntax of the sentence. He replaces Aristotle’s eiê an ti (‘it follows that there is such a thing as’) with touto an eiê ho an êi ek protaseôs tinos (‘will be any [question] formed from any premise’), he adds epistêmên to go with kath’ hekastên, he omits oikeios and reverses the order of Aristotle’s ginetai sullogismos. The resulting translation is ‘scientific “premises from which the deduction” [is formed] are determinate “in each science(e)”, a “scientific(e) question” will be any [question] formed from any premise “on the basis of which” a “deduction” “is formed in any” science(e)’. Paraphrases are not put into inverted commas, even when the paraphrase in Greek is so close that the English translation of Philoponus is identical with that of Aristotle. This practice, although typographically inelegant, displays how closely Philoponus was working with the text of the Posterior Analytics, and helps us see how he went about his task as a commentator. I should point out, however, that there are borderline cases, where it is not clear whether Philoponus is deliberately quoting Aristotle or simply using the word he would ordinarily use, and in such cases one reader’s judgment may differ from another’s. I use square brackets to indicate words that are not in the Greek but that need to be supplied in English. And since epistasthai, ginôskein, and eidenai, can all be translated as ‘know’, I distinguish them (and associated words such as epistêmê and gnôsis) as follows: epistasthai is ‘know(e)’ or ‘have scientific knowledge(e)’, ginôskein is ‘know(g)’, and eidenai is ‘know(o)’, and similarly epistêmê is ‘scientific knowledge(e)’ or ‘science(e)’, epistêmôn is ‘expert(e), while gnôsis is ‘knowledge(g)’ and translations of gnôrizein and gnôrimos are signalled in the same way. In cases of doubt the English-Greek Glossary can be used as a guide. My greatest debt, as with my previous volume in this series, is to Richard Sorabji, who invited me to contribute this volume. The six readers (Owen Goldin and five who have remained anonymous) took the trouble to read the translation and offer suggestions for improvement, and I am grateful to them for their help. I also want to thank Voula Tsouna for her help in matters of translation and Hellenistic terminology, and Eleni McKirahan, who did me the favour of drawing the diagrams for pp. 17, 18, 20, 64 and 76 with AutoCAD. The

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translation benefited from the Greek-English Indices of previously published volumes in this series. In order to achieve accuracy and completeness in the Greek-English Index of this volume and also to make the task of composing it less burdensome, I wrote a number of computer programs to manipulate the text of Philoponus given in the Thesaurus Linguae Graecae. Much of the work on this volume was completed on a sabbatical granted to me for this purpose by Pomona College, during which I had the privilege of working at the American School of Classical Studies at Athens. I want to express my sincere thanks to these institutions for their support and assistance. I dedicate this volume (as I did my volume on the first eight chapters of Philoponus’ commentary) to my lovely wife Voula and to our equally lovely daughter Eleni. Notes 1. Much of this introduction is taken from the introduction to McKirahan (2008). 2. D.L. 5.42. 3. Galen De suis libris 14. 4. The surviving material from this work is collected in Moraux (1979). 5. See Wallies (1909), Praef. v and Goldin (2009), 1-4. Sorabji, however, has recently raised a consideration in favor of authenticity, in Goldin (2009), vii-ix. 6. The final volume, on book I chapters 19-34 is presently being prepared by O. Goldin and M. Martijn. 7. The Aldine editio princeps of 1504 was followed by a second edition in 1534 and a third in 1558 (Wallies, Praef. xxiii with n.). 8. The first translation, by Theodosius, was first published in 1539 and was reprinted 9 times during this period. The second, by Rota, was published in 1559 and reprinted in 1560. (For references, see Schmitt (1987), 216, 228.) 9. However at 205,10-14 he comments on only one of two cases mentioned at 80b2-5. 10. See Subject Index for references. 11. This commentary has recently been published with Italian translation and commentary (Giardina (1999); see also D’Ooge (1926)). 12. For a full discussion, see McKirahan (2009). 13. See also 127,18-129,25. 14. I discuss the material in this paragraph at length in McKirahan (2000). 15. References are to Friedlein (1873).

Textual Information Textual emendations The text translated here is that printed in Ioannis Philoponi in Aristotelis Analytica Posteriora, ed. M. Wallies, CAG 13.3 (Berlin: Reimer, 1909), with the following emendations: 119,2 Punctuating with a comma; Wallies has a semicolon. 141,12 Keeping the MS reading deon; Wallies prints his own conjecture dein. 145,25 Keeping the MS reading hôs which Wallies deletes. 147,11 Accepting the suggestion of an anonymous referee . 176,31 Keeping the MS text di’ heautou; Wallies in his apparatus criticus prefers di’ autou or di’ ekeinou. Notes on the text of Aristotle’s Posterior Analytics Discrepancies betwen Philoponus’ text of the Posterior Analytics and the text as given in Ross’ edition: 76a12 The commentary indicates that P.’s text had to de dihoti heteras, tês anô, where Aristotle and the lemma have to de dihoti tês anô. 76a32 P. reads ho ti where the manuscripts and modern editions have ti. 76a35 P. adds ti before trigônon. 76b18 P. has arithmós esti where Aristotle has arithmòs ésti. 76b35-36 P. has oude  legontai where Aristotle has ouden  legetai. 77a17 P. adds pan before mê zôion. 77b1P. adds ê before deiknutai. 77b1 P. omits ha after ê. 77b28 P. has ditton aei where Aristotle has aei to ditton. 77b41 P. has ta hepomena amphoterois where Aristotle has amphoterois ta hepomena 78a2 P. has houtôs where Aristotle has houtô. 78a3 P. has takhistê where Aristotle has takhistêi. 78a4 P. has poluplasiôn where Aristotle has pollaplasios.

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78a7 P. has pseudôn where Aristotle has pseudous. 78a16 P. has touto where Aristotle has tout’. 78a20 P. omits ho before artios. 78a26 P. omits ei before di’ amesôn. 78b29 P. has de esti where Aristotle has d’ ésti. 78b35 P. has tôi where Aristotle has to. 79a25 P. adds tou skhêmatos after touto. 80a35 P. has huparkhei where Aristotle has huparkhoi. 80b17 After mê atomôs huparkhousin, P., in agreement with MS d of the Posterior Analytics (Laurentianus 72, 5) has ê mê huparkhousin; Ross, following the other principal MSS, omits these words. 80b35 P. has melloi where Aristotle has mellei. 81a27 P. has hupallêla where Aristotle has hup’ allêla. 81b3-4 P. (agreeing substantially with MS n of APo) has estai di’ epagôgês gnôrima, ean tis boulêtai gnôrima poiein where Ross, following the other MSS, has estai di’ epagôgês gnôrima poiein.

PHILOPONUS On Aristotle Posterior Analytics 1.9-18 Translation

John of Alexandria’s lecture notes from the meetings of Ammonius, son of Hermeias, on the first [book] of Aristotle’s Posterior Analytics, together with some observations of his own Chapter 9 75b37-41 Since it is obvious that it is not possible to demonstrate each thing except from its own principles, if what is being proved belongs to it qua that very thing, it is not possible to know(e) ‘this’ [if it is proved from premises that are true, indemonstrable and immediate. For if that is so, it is possible to construct proofs the way Bryson proved the squaring [of the circle]. In addition to what he has already proved about scientific knowledge(e), he also adds that taking premises that are true and immediate is not enough to make a demonstration; they also need to be appropriate to the subject of demonstration. For if I say that every stone is coloured, and every coloured thing is a body, therefore every stone is a body, I have taken premises that are true and also immediate (for I need no middle term to demonstrate either that a stone is coloured or that what is coloured is a body1), but the middle term is not appropriate to the subject, since being coloured belongs to many other things too.2 But as has been said many times, the demonstration must be based on principles that are appropriate to each thing. This is in order that the middle term be appropriate to the extremes and common to nothing else.3 And so, he says, the premises must be taken from things that are not only true and immediate, but also appropriate to the conclusion. Since that way, he says, it is even possible to prove Bryson’s quadrature on the basis of things that are common and are not principles appropriate to the claim in question. Aristotle says only so much about Bryson’s quadrature, but Alexander says that Bryson attempted to square the circle as follows. A circle is larger than any rectilinear figure inscribed in it and smaller than [any rectilinear figure] circumscribed [about it]. (A rectilinear figure drawn inside a circle is said to be inscribed in it, and one [drawn] outside [is said] to be circumscribed.4) Also a rectilinear figure drawn between the inscribed and the circumscribed rectilinear figures is smaller than the circumscribed one and larger than the inscribed one. But things that are larger and smaller than the same things are equal to one another. Therefore the circle is equal to the

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rectilinear figure drawn between the inscribed and the circumscribed [figures]. But we can construct a square equal to any given rectilinear figure. Therefore it is possible to make a square equal to the circle. Thus Alexander. But the philosopher Proclus said that his teacher5 objected to Alexander’s explanation because if that is how Bryson squared the circle, he concurred with Antiphon’s quadrature. For Antiphon too made the figure drawn between the inscribed and the circumscribed rectilinear figure fit with the circumference of the circle to the point where he made (as he claimed) a straight line coincide with a circumference, which is impossible. This was discussed in the Physics.6 Therefore Aristotle would not contrast Bryson’s quadrature as different from Antiphon’s7 if this is how Bryson squared [the circle]. But, says Proclus, I declare the axiom to be false. For it is not true that things larger and smaller than the same things are equal to one another. Ten is larger than eight and smaller than twelve, but likewise nine is smaller than twelve and larger than eight, but of course it is not the case that ten and nine are equal just because they are both larger and smaller than the same things, namely twelve and eight.8 Therefore it is not the case that even if the circle and the rectilinear figure drawn between an inscribed [figure] and a circumscribed [figure] are larger and smaller than the same things, namely, the inscribed and the circumscribed [figures], they are for that reason automatically equal to one another too – unless as was already stated, someone claims that the rectilinear figure drawn between the inscribed and the circumscribed [figures] à la Antiphon, coincides with the circle, which is impossible. For a straight line never coincides with a circumference. Against Proclus’ [account] it can be said that if this is how Bryson constructed the quadrature of the circle, he did not construct it at all, but begged the question. For those who attempted to square the circle did not investigate whether it is possible for there to be a square equal to a circle, but supposing that it is possible for there to be one, they attempted to generate the square equal to the circle. But what was just said by Proclus, as our teacher9 said, proves that it is possible10 for there to be a square equal to the circle, even if this is in fact granted. But he still did not draw a square equal to the circle nor did he teach how this might come to be – which is what those who attempted to square the circle wanted to do. And Aristotle spoke as if the circle had been squared by Bryson, even though it was not done geometrically. And so Proclus’ explanation does not appear to be naturally fitting. Therefore11 Proclus said that Bryson squared the circle as follows. A circle, he declares, is larger than any inscribed rectilinear figure and smaller than any circumscribed one. When there are things that are larger and smaller than something, there is also something equal

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to it. But there are rectilinear figures larger and smaller than the circle. Therefore there is also one equal to it. But even if someone were to grant that this is how Bryson constructed [his quadrature], it is possible to claim against it that the account is true for things of the same kind – that when there are things that are larger and smaller than something, there is also something equal to it – but this is not true for things not of the same kind. To be sure it is proved by the Geometer that straight line AC drawn at right angles at the end of CB, a diameter of semicircle CDB [Diagram 1] lies entirely outside the circle, and that of the two angles

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formed by the circumference and the diameter and by the [line] drawn at right angles and the circumference – I mean the exterior angle ACD and the interior angle DCB12 – the exterior angle is smaller than any acute rectilinear angle and the interior angle is larger than any acute rectilinear angle.13 And notice that here although we have proved that it is larger and smaller than the same acute rectilinear angle,14 we still cannot find [an angle] equal [to it] because the magnitudes are not of the same kind. For the angles in question are composed of a straight line and a circumference, and we call them horn angles. And the surprising thing is that even though the exterior angle can be increased and the interior angle decreased ad infinitum, and – vice versa – the interior angle can be increased and the exterior angle decreased ad infinitum,15 the exterior angle as it is being increased ad infinitum will never become equal to any rectilinear acute angle, but will always be smaller than any [given rectilinear acute angle] nor will the interior angle as it is being increased ad infinitum ever become equal to a right angle. We will increase the exterior angle by drawing smaller circles [Diagram 2].

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For example if I cut diameter CB at point E and [if I cut] straight line CE in two at point F, and with centre F and radius FC draw a circle whose semicircle is CGE, the exterior angle ACG has been increased,16 none the less it is still smaller than any acute angle for the reason that has been stated. For the theorem has been proved by the Geometer for every circle.17 In the same way, if I again cut the

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diameter of the inner circle and inscribe a smaller circle, and do this ad infinitum,18 I will always increase the exterior angle and decrease the interior angle, but the exterior angle will never become equal to an acute rectilinear angle nor will the interior angle,19 but the exterior angle will always be smaller and the interior angle always larger.20 This is how I will increase the exterior angle and decrease the interior angle. And, vice versa, I will increase the interior angle and decrease the exterior angle by circumscribing larger circles as follows [Diagram 3]. I produce diameter CB in a straight line to E and with centre B

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and radius BC draw a circle whose semicircle is CFE. And it is clear that semicircle CFE will fall inside straight line AC because it has been proved that a line drawn at right angles at the end of a diameter lies entirely outside the circle.21 Further, it is clear from the following that no part of the outer semicircle CFE coincides with any part of the inner semicircle CDB. For if it coincides, let straight lines GB and GH be drawn from G, the point at which they coincide, to B and H, the centres of the circles. Now since point H is the centre of the inner semicircle, HG is equal to HC, and since B is the centre of the outer semicircle CFE, BG is equal to BC. But BH and CH are equal to HG. Therefore GB is equal to BH and HG. Therefore the two sides GH and BH of triangle GHB are equal to one side, GB, which is impossible. Therefore it is not the case that any part of the outer circle coincides with any part of the inner [circle]. Therefore the outer circle cuts angle ACG. And in the same way by drawing outer circles ad infinitum22 I will decrease the exterior angle and increase the interior angle ad infinitum,23 and the interior angle being increased will never become equal to a right angle but always becomes larger than any acute rectilinear angle.24 Now if it has been proved that something can be larger and smaller than the same thing without being equal25 because of the dissimilarity of the magnitudes, Bryson was wrong to assume that if the circumscribed rectilinear figure is larger than the circle and the inscribed [rectilinear figure] is smaller, it follows that what is between the inscribed and the circumscribed [figures] is equal. For here too the magnitudes are dissimilar – I mean the rectilinear figure [is dissimilar to] the circle – and so they will not be equal either.

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75b41-76a1 For such arguments prove [their conclusion] in virtue of some common property that belongs to something else as well. [This is why the arguments apply also to other things that do not belong in the same genus.] That when there are things that are larger and smaller than something, there is also something equal to it – the principle from which Bryson thought he proved the quadrature of the circle26 – is not proper to geometry, but is common to very many other things as well.27 It is especially characteristic of dialectic, not geometry, to make use of such [claims], because they do not prove the claim in question on the basis of geometrical principles.

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76a1-3 Therefore, he does not know(e) in virtue of that very subject, but [only] accidentally, for the demonstration would not apply to another genus as well. If, he says, he proved the quadrature on the basis not of appropriate principles, but of some [principles] that are more general, it follows that he proved it on the basis not of per se attributes, but of accidental ones. What it is to prove per se and not accidentally he goes on [to define] next: when we know(g) something from its appropriate principles and not from some more general ones that can apply to other things as well . And this is why he said above that there is no scientific knowledge(e) or demonstration of perishables except as if accidentally, meaning by ‘accidentally’ constructing the demonstration from some more general [principles].28

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76a4-7 [We know(e) each thing not accidentally when we know(g) it in virtue of the very thing in virtue of which it belongs, from the principles of that very thing qua that very thing.] For example, the property of having [angles] equal to two right angles – [we know it not incidentally when we know] that it belongs to that to which the property belongs per se [and know this] from the principles of that very thing.

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‘The [property of] having [angles] equal to two right angles’ belongs per se to triangle and he demonstrates this not from some more general [principles] but ‘from’ ‘principles’ appropriate to the subject of scientific knowledge(e). For example, he proves that the three angles of a triangle are equal to two right angles,29 by producing one of the sides [Diagram 4] and proving that the two right angles – [the sum of] the interior angle and the adjacent exterior angle – are equal to the three interior angles, so that the following deduction results: when one of the sides is produced, the three angles of the triangle are equal to two adjacent angles; two adjacent angles are equal to two

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Diagram 4

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right angles; therefore the three angles of a triangle are equal to two right angles.30 That the adjacent angles are equal to two right angles is proved from the fact that two adjacent angles are either equal to two right angles or are 31 right angles. From where [did he prove] that two adjacent angles are either equal to two right angles or are two right angles? We know(g) from the definition of right angles that if ‘a straight line standing on a straight line makes the adjacent angles equal to one another, each of the equal angles is a right angle’.32 So after arriving at the definitions and principles of geometry we do not investigate further, but we have the triangle demonstrated on the basis of the principles of geometry. 76a8-9 And so if that too belongs per se to that to which it belongs, the middle term must be in the same genus. ‘If’, he says, the [term] predicated in the problem33 ‘belongs per se’ to the subject (for by ‘that too’ he means that about which he just spoke, namely that a triangle has [angles] equal to two right angles),34 ‘the middle’ through which this is proved ‘must’ ‘be in the same genus’ as the extremes. The middle term is that the two adjacent angles of the triangle are equal to its three interior angles.35 For it holds per se of the three [angles] of a triangle that the two adjacent [angles] are equal to them, and in turn it holds per se of the two adjacent angles that they are equal to two right angles; rather that they are two right angles, as the definition of right angles proves. Therefore he did well to say that if the predicate [term] in the problem belongs per se to the subject, the middle term too will necessarily be ‘in the same genus’ as the extremes. This was previously demonstrated independently.36 76a9-10 Otherwise, at least [if we know it] the way facts in harmonics [are proved] through arithmetic.

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If, he says, the demonstration is based not on principles appropriate to the subject but on the principles of the proximate science(e)37 that includes the one in question, the demonstration must be based on this [science], if it is truly a demonstration – for example if we prove ‘facts in harmonics’ through the principles of ‘arithmetic’. For the ‘musician’ theory says that the interval of a fourth, for example, is concordant because such an interval has the ratio of four to three,

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and the ratio of four to three is concordant. But that such a ratio is concordant the harmonician could not prove; for it is the job of the arithmetician to discuss concordant ratios, for example that the number eight has a ratio of four to three with regard to the number six, and is concordant because both are measured by the common measure.38 For the number two three times measures the number six and four times [measures] the number eight. But it has been said39 that in the case of subalternate sciences(e) nothing prevents the demonstrations from being transferred. 76a10-15 Such things are proved in the same way, but there is a difference. For the ‘that’ is the concern of one science(e) (since the subject genus is different), but the ‘why’ is the concern of the superior [science], whose per se affections they are. [And so from this too it is obvious that it is not possible to prove each thing without qualification except from its own principles.] By ‘such things’ he means the things proved through the principles of the more general science(e). These, he says, are proved similarly to the other things proved through the appropriate principles, i.e., the demonstration of these is scientific(e). ‘But there is a difference’, because in that case both the ‘that’ and the ‘why’ come from the same science(e); for it is a matter of geometry that the three angles are equal to two right angles, and so are the principles from which this is proved. However in the case of music, in which the things demonstrated are demonstrated through the principles of arithmetic, the fact of the things demonstrated, for example, that the interval of a fourth is concordant, is the concern of music, but the ‘why’, that is, why it is concordant and what ratios are concordant,40 is the concern of arithmetic. Now this is what he means to say. But since for brevity’s sake he said ‘for the “that” is the concern of one science(e)’ without saying which, he then went on and added ‘but the “why” is the concern of’’ another,41 ‘the superior’ – indicating that the ‘that’ is the concern of the lower one. For if the ‘that’ is the concern of the one and the ‘why’ is the concern of the other, and the ‘why’ is the concern of the superior one, it remains that the ‘that’ is the concern of the subordinate one. [He says] ‘whose per se affections they are’ because in the case of subalternate sciences(e) the per se attributes of both [sciences] are primarily attributes of the more general one.42 76a15 But the principles of these [sciences] have a common feature. [He says this] instead of ‘the principles of these [sciences], both the subordinate and the superior, are referred to “what is common”’ to every [science], namely first philosophy. For this is what discovers

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and demonstrates the principles of every science(e), which is why it is called the art of arts and the science(e) of sciences(e).43 It seems to me simpler to understand the claim in question in this way. For since he said that it is possible, in fact, to demonstrate the [propositions] of a subordinate science(e) from the principles of the superior one and then also added the difference by which things demonstrated in this way differ from things demonstrated through the appropriate principles, lest anyone suppose that the things demonstrated through the principles of the superior [science]44 are not demonstrated because they are not [proved] through appropriate principles, he says that ‘the principles of these [sciences] have a common feature’, that is, that the principles of subalternate sciences(e) are common, as holds with arithmetic and music. For the demonstration takes place in both [sciences] by means of common principles. For if the things through which music forms its demonstrations seem to belong more to arithmetic, nevertheless arithmetic itself is a principle of harmonics,45 and so the principles of arithmetic and harmony are the same too. For the things predicated of the superior things are also predicated of the subordinate things. Likewise also for natural philosophy and medicine and for geometry and mechanics.46 76a16-22 But if this is obvious, it is also obvious that it is not possible to demonstrate the proper principles of each thing. [For they [the principles on which such proofs would be based] will be the principles of all things, and the science(e) of them will have authority over all things. For indeed the person who knows(o) from the higher causes knows(e) in a higher degree, since he knows(o) from prior [causes] when he knows(o) from uncaused causes. And so, if he knows(o) in a higher and in fact the highest degree, that science [or, scientific knowledge(e)] will also be higher and the highest.]

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On the basis of what has been said he infers a corollary concerning the geometers. For, he says, if it is not possible to demonstrate anything except on the basis of its own proper principles, it will be evident that it is not possible for any science(e) to demonstrate its proper principles, if in fact a demonstration must be based on proper principles, but there can be no principles of principles. But if, he says, there is a science(e) that demonstrates ‘the’ ‘principles of each thing’, that will be a genuine science(e) and principle47 from which they demonstrate the principles of the other sciences(e), a common principle of everything, and that will be the art of arts and the science(e) of sciences(e),48 and this is first philosophy, which is discussed in the Metaphysics.49 For if ‘the person who knows(g) from the’ primary principles ‘knows(e) in a higher degree’, and this [science] knows(g) from principles that are common and [are principles] of every sci-

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ence(e), this will be a science(e) and a principle. Since if knowledge(g) is not through these, it belongs through other things that are effects.50 But this is what we saw to be scientific knowledge(e) in the strict sense – the kind that knows(g) things on the basis of the things that are the absolutely primary principles and are only causes and in no way effects.51

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76a22-5 But demonstration does not apply to another genus, except in the way that it was said that [demonstrations] from geometry [apply to demonstrations] in mechanics or optics [and [demonstrations in] arithmetic [apply to demonstrations] in harmonics.] Because it is impossible, he says, for demonstrations to cross over so that they apply in different sciences(e) unless they are subalternate, with the result that the lower-ranking [science] employs the demonstrations of the higher [science], so that mechanics or optics employs the [demonstrations] of geometry and harmonics [employs] the [demonstrations] of arithmetic. 76a26-30 It is difficult for anyone to know(g) whether he knows(o) or not. For it is difficult to know(g) whether or not we know(o) from the principles of each thing [– and this is what it is to know(o). But we suppose that we know(e) if we have a deduction that has some true and primary premises. But it is not the case, but [in addition the principles] have to belong to the same genus as the first ones.] ‘It is difficult’, he says, ‘to know(g)’ whether or not the deduction proceeds demonstratively. For if someone takes premises that are true and immediate, ‘we suppose’ that he has taken ones that are demonstrative too. ‘But it is not the case’, since it has been proven that the premises must be not only true but also taken from the proper principles. This is why the first and second theorems of geometry are demonstrated in the strict sense, while the following ones are deduced truly but not demonstratively. For they are not proved from premises that are proper or immediate. For the following ones are always proved making use of things that have been proved.52 But even so the deduction that occurs in these cases is also called a demonstration, rather carelessly, unless someone says that these things too are demonstrated by ascending necessarily through prior [conclusions] to the principles of geometry through which the primary things are proved. He says this himself further down.53 So since it is difficult to know(o) the nature of things and the per se attributes of each kind of thing, this is why it is also difficult to know(o) whether or not a deduction is demonstrative.

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Translation Chapter 10 76a31-2 I call principles in each genus those [regarding] which it is not possible to prove that they are.

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The principles in each science(e), he says, are the things that it is not possible to demonstrate, but are granted. For example point, line, and so forth are principles of geometry.54 It is not geometry’s task to prove that points are without parts or that lines are one-dimensional, but it assumes these as being outside the realm of demonstration. So in this way in the remaining cases too the primary indemonstrable things are principles of science(e). 76a32-3 Now whatever55 both the primary things and the things that depend on them signify is assumed. Since he said that the ‘principles in each genus’ are ‘those which it is impossible to prove that’ (76a31-2) they are, starting at this point he teaches which things that are employed in demonstrations it is necessary to prove that they are and which it is not necessary. He says the same things that he said in a riddling way at the beginning.56 But we will make them clear, taking our starting point from this. He says that all the things employed in demonstrations are three in number.57 More completely, the things in which demonstration consists are divided into two: premises and the conclusion drawn from them. Since the conclusion contains two terms, the predicate and the subject, of which one, the subject, is given and the other, the predicate, is sought, all the things [in which demonstration consists] are divided into three: the premises (or rather among the premises the axioms, which entirely fill the place of the major premises in demonstrations, for the reason we stated at the beginning58), the given, and the sought. Now in all these cases he says in general that what each signifies is assumed in advance. But in the case of the axioms and the givens it is necessary to assume separately also that they are. For the premises consist of these, and because the premises are the case it is necessary that the conclusion is the case too.59 But with regard to the sought alone do we assume what it signifies, but not that it is, since [if we did] it would no longer be sought. For the demonstration is about this. ‘Now what both the primary things and the things that depend on them signify is assumed’, that is, in the case of both the premises and the conclusions that are based on them what each of them signifies must be assumed in advance.60 For the text has ‘whatever’ instead of ‘what’, adding the omicron.61

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76a33-4 But it is necessary to assume that the principles are and to prove that the others are. Again he calls premises ‘principles’62 with reference to which it is necessary to know previously(pg) not only what they signify but also that these things that are said to be actually are. For if it is not given that the premises are the case, it is not possible for the conclusion to be drawn.

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76a34-5 For example, [it is necessary to assume] what unit [signifies], and what straight and what63 triangle do. He takes the unit as an example of givens, or simply of principles of demonstrations for which we should assume in advance both what they signify and that they are; however, ‘straight’ and ‘triangle’ are among the things sought. For we said earlier64 that sometimes both straight and triangle become the sought. But we likewise said that they can be given too.65 So it is a common feature of these cases that it is necessary to assume what each of these things is – that is, to set out their definitions in advance.

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76a35-6 And that the unit and magnitude are, but it is necessary to prove the rest. Having said that in all cases the ‘what it signifies’ is assumed (76a32-3), he now proves that in some cases the ‘that it is’ is proved, while in some others it is assumed. Alexander here understands ‘magnitude’ to mean ‘straight’. But this is not right; he [Aristotle] gives unit and magnitude as examples of givens for which ‘that it is’ is also assumed. For neither unit nor magnitude is ever assumed as a thing that is sought, for in arithmetic we do not have any theorem that proves that this is a unit,66 nor in geometry [do we have any theorem that proves] that this is a magnitude, but these are always assumed as things that are. So in these cases, he says, the ‘that it is’ is assumed. But it is proved for the rest, namely for straight and triangle. This is why he previously mentioned triangles as well, ‘for example what unit [signifies], and what straight and what triangle do’ (76a34-5), giving simultaneously an illustration of both things that are assumed and things that are proved. Next, wanting to make a distinction between what things are assumed and what are sought, he says that unit and magnitude are things assumed, but not so for triangle. But we have said that sometimes triangle too is assumed,67 but unit and magnitude are never sought.

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Translation 76a37 Of the things they use in demonstrative sciences 

He now expounds and presents more precisely what he said a little above without making the proper distinctions. He previously said that it is possible to employ the same axioms in different sciences(e),68 for example that things that are equal to the same thing are also equal to one another. For both the geometer and the arithmetician will employ it. But now he says that not even the different sciences(e) [employ] the same axioms except homonymously.69 For, he says, when the geometer says that things that are equal to the same thing are also equal to one another, he applies the axiom as if [it said] ‘in the case of magnitudes’.70 Never mind if the account were true only for magnitudes; in that case the geometer would still employ it none the less. For he does not apply the axiom as if the account were true for all sciences(e), but as if [it were true] for magnitudes alone. And likewise for all the other axioms. The arithmetician [does] similarly, using the same axiom as if it applies to only his own subject matter. And so the axioms are homonymous and not the same. 76a38  some are proper to each science(e) and some are common  We said previously71 that some of the axioms that we use in demonstrations are common to all [sciences] or to some, and ‘some are proper to each science(e)’72 – proper to geometry, for example [the principle] that things that coincide with one another are equal to one another; common to several [sciences], for example that things that are equal to the same thing are also equal to one another. 76a38-40  but common in virtue of a proportion, [since a thing is useful insofar as it is in the genus that falls under the science.] He says, I say that they are ‘common’ not in the strict sense but ‘in virtue of a proportion’, because just as the statement is true for magnitudes it is also [true] for numbers. And so the commonality is not in virtue of the subject but only in virtue of the name, just as in fact we speak of the feet of a couch or the head of a mountain, and similarly of an animal too. So we take the homonymy from a proportion. But the subject is admittedly different.73 76a40-b2 Examples of proper ones are that a line is such and such, and that the straight is such and such. [An example of a common one is that if equals are subtracted from equals, the remainders are equal. Each of these is sufficient insofar as it is in the genus. For it will do the same even if [the geometer]

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assumes it [as holding] not for all things, but only for magnitudes, and for the arithmetician [it is assumed as holding] for numbers.] That a line, for example, is the flow of a point or a one-dimensional magnitude,74 and a straight line is one that lies evenly with the points on itself, or one whose middle [points] are in front of its ends.75 These are proper to geometry. 76b3-5 Proper things are (1) those things that are assumed to exist, concerning which the science (124,5) investigates the attributes that belong to them per se. [For example, arithmetic [assumes] units, geometry points and lines.] Having said that some axioms76 are common and some are proper, he now says which givens77 are proper to each science(e). The givens that are proper to each science(e), he says, are those which we always assume as existing,78 and investigate what things belong to them per se. ‘For example arithmetic [assumes] units, geometry points and lines’. But they investigate the per se attributes of these things.

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76b5-8 For they assume that these exist and that they are this. [(2) [Concerning] the per se attributes of these they assume what each signifies – for example, arithmetic assumes what odd, even, square, and cube signify ] That is,79 [they assume] both that each of these is and what each signifies. But as to the affections that are their per se attributes, only what each of the affections signifies is assumed by the sciences(e), and whether it is or is not is investigated. 76b9 (124,15)  and geometry [assumes] what irrational or to inflect or verge [signify]. Clearly [he is talking] about irrational magnitudes. For geometry deals with these, assuming what irrational magnitude signifies – that it is incommensurable and has no common measure in relation to the other,80 as is the case with the diagonal and side of a square. However, whether or not the diagonal is irrational81 it does not assume, but proves.82 But if it is ‘proportional’ (for it is written this way in the manuscripts too)83 it is because [geometry] also deals with proportional magnitudes. This has a common [point] with arithmetic too.84 But being inflected is not the same as verging; for being inflected holds for a single line; a line not all of whose parts are on a straight line with one another, but however they may be. On the other hand, straight lines are said to verge if when produced in one

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direction they coincide at a single point as the diagonal does in relation to the side. 76b9-11 But that they are, they prove through the common principles and from the things that have been proved. And astronomy likewise. 125,1

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The sciences(e) prove the per se attributes of all the things that have been mentioned from certain common axioms and the theorems that have already been demonstrated previously. Not all things are proved proximately through the axioms: the third [theorem] is proved through the second and the second through the first.85 He calls astronomy astrology.86 For in this [science] too, for some things only ‘what it signifies’ is assumed, and for others we also investigate ‘that it is’.87 76b11-12 For every demonstrative science is concerned with three things  Notice that we made our division of the things in which every demonstration consists taking our starting point from this claim.88 76b12-14  all the things posited to exist (these are the genus [whose per se attributes the science investigates); the common principles, called axioms, from which as primary [the science] forms its demonstrations,]

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Namely, the givens. He is clearly speaking of the subject genus for each science(e), from which every term is assumed as given,89 as we said in our earlier remarks too.90 Second he speaks of the axioms from which every demonstration [proceeds]. 76b15  and third the attributes, [concerning which [the science] assumes what each signifies.]

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Namely, the soughts, which belong per se to the subjects. They are assumed only in respect of what each signifies.91 76b16-18 However, nothing keeps some sciences from overlooking some of these [(for example, from hypothesizing that the genus exists), if it is obvious that they exist ]

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Since he said that – there being three things of which every demonstration consists92 – for all of them what each signifies is assumed in advance by the sciences(e),93 but for some of them ‘that it is’ [is assumed in advance] too, but the sciences(e) do not always assume the

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axioms in advance or hypothesize the subject genus as existing – this is why he says that in cases where one of the three stated kinds is clear because it is evident, it is not necessary for it to be assumed in advance by the sciences(e). For the natural philosopher does not define what hot or cold signify, and similarly the geometer does not define what magnitude is, because these things are more evident from perception itself.94 76b18-19 (for it is not clear in the same way that number is95 and that cold and hot are) 96 This is what the Philosopher says. But Alexander97 says that number is more clear and hot and cold are less [clear]. But the truth is not like that. For cold and hot and every other natural thing are clearer than number. This is why the natural philosopher does not define what cold and hot are: these are clear because they are evident. However, the arithmetician defines what number and unit are – that a number is a plurality composed of units, and a unit is that in virtue of which each entity is called one.98

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76b19-21  and not assuming what the attributes signify if they are clear, just as in the case of the common [principles] one does not assume what taking equals from equals signifies because it is familiar. He says that the expert(e) does not define either the per se attributes, ‘if they are clear’,99 or the axioms. For he does not say what ‘things that coincide with each other are equal to one another’ signifies, nor what ‘if equals are subtracted from equals the remainders are equal’ is, and similarly in the case of some affections. For the geometer does not define, for example, what it is to have its three angles equal to two right angles, or what adjacent angles are. And likewise in the other cases.

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76b21-2 But nonetheless, these things are by nature three: that concerning which [the science] forms its demonstrations, what it demonstrates, and the things from which [it demonstrates]. Whether these are all assumed in advance by the sciences(e) or some are overlooked because they are evident, none the less, he says, in every science(e) there are three things from which every demonstration is constituted:100 the genus concerning which the science(e) demonstrates the things that belong per se – which is the given;101 the very affections that belong to it per se102 – and this is the sought; and third, the axioms from which the affections are proved as belonging to the subjects.

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Translation 76b23-4 What must be because of itself and must seem [to be because of itself] is not a hypothesis or a postulate.

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Having said103 that there are three things of which every demonstration [consists] and that the sciences(e) assume these in advance – all of them in respect of ‘what it signifies’ and some in respect of ‘that it is’,104 and that in some sciences(e) it happens sometimes that some of these are not assumed on the grounds that they are clear because they are evident, in fact just as those who are experts(e) in a particular field do not assume in advance the meanings of the axioms since they are clear because they are evident – for example what ‘if equals are subtracted from equals’ signifies and that [both members of] a contradiction cannot simultaneously be predicated truly – since he mentioned the axiom generally, he wants to present its characteristic feature to us and distinguish it from the things called hypotheses and postulates, and further [to distinguish] these things from one another and from definitions. Now this distinction was already presented at the beginning.105 But since he mentions these things now too, it is reasonable for us to set out his entire thought on the claims in question, summarizing the entire division from above in its complete form, which is as follows. Some immediate premises are self-guaranteeing and are natural to all men in common, and some are not [natural] to all.106 Those that are natural to all in common because we have conviction in them from within ourselves, are called axioms and common notions.107 Those that are not natural to all but to some are called theses, and of theses those that declare what each thing is are called definitions, while those that predicate one thing of another108 are called by the common term hypotheses.109 Some hypotheses are true and appear true to the learner and are taken [by the learner] from the expert(e), while others [are taken by the learner from the expert] although they do not appear true [to the learner]. Those that are taken and appear true to the learners are called hypotheses in a special sense of the term, homonymously with the genus,110 and those [that are taken] although they do not appear true [to the learners are called] postulates.111 All hypotheses that [are taken] in circumstances where they both appear true to the interlocutor and are true, and do not require much attention for their truth to be observed, are called hypotheses in the strict sense.112 For example if someone wants to establish that the soul is immortal, he will ask his interlocutor if the soul appears to him to be self-moving, or if contraries come to be from one another. If [the interlocutor] agrees to the question, he will say that it is so. For even if these things require some attention, still they are true and it immediately becomes clear that they are so to a person who thinks about them for even a little. Now these are called hypotheses in the

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strict sense. Definitions whose truth is evident are of this kind as well.113 For example, a triangle is a figure enclosed by three straight lines, and a circle is a figure enclosed by one line such that all the straight lines falling on it from one point inside the circle are equal to one another.114 For that the circle is enclosed by one line and that the [lines] from the centre are equal is evident to anyone who only understands that the distance from the centre, where one end of the compass is placed, to the other end is a straight line which generates the circle by being turned completely around.115 And in similar cases it likewise requires but little attention. But all that appear true to the interlocutor and are true but require demonstration and further elaboration are hypotheses relative to the learner but not in the strict sense. Once again definitions are an example of this. For all of them that have their conviction from within ourselves and also appear true to the learner are called hypotheses in the strict sense as we have already said. But all those that do not have their conviction from within ourselves but require demonstration116 – for example the definitions of point, line, and surface, and also that the sun is larger than the earth and that the earth is in the middle [of the cosmos] or has the relation of a centre117 – or whatever appears true to the interlocutor and is taken [to be true], since these require demonstration but are taken without being demonstrated, they are called hypotheses relative to the learner. And clearly a definition is an hypothesis that it [the definiens118] is the same as the subject, but it is different in the relation119 as are affirmation and negation,120 and premise, problem121 and conclusion.122 For when the definition is taken as a part of a premise, it is an hypothesis and not a definition, as when, for example, we employ the definition of circle in a premise, saying in [the course of proving] a theorem ‘the lines falling from the centre on the circumference are equal to one another’ or ‘when a straight line standing on a straight line makes the adjacent angles equal to two right angles’ and the like. But when it is not employed as a premise but as reporting the nature of the thing as that very thing, then it is a definition, not an hypothesis.123 However, it is not the case that a hypothesis is in all cases also a definition. For example, that contraries are generated from one another is an hypothesis but not a definition, and that the moon is illuminated by the sun is an hypothesis but not a definition, if they are taken and appear true to the interlocutor. Now all hypotheses that appear true to the learner or the interlocutor are more properly called hypotheses as has been said, homonymously with the genus. But all that [are taken] though they do not appear true to the interlocutor, but are true and require but little explanation, are called postulates in the strict sense. For example, to draw a circle with any centre and radius and to draw a straight line from any point to any other point124 and the like. But if they [are

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taken] without appearing true, if they are false and contrary to the opinion of the interlocutor, or even if they are true and require more demonstration, they are called postulates relative to the learner.125 Examples of false ones: if one of the followers of Democritus or of Anaxagoras should ask that it be granted to him that atoms or homoeomeries are the principles of things.126 An example of ones that are true but require more explanation: the one that says that [lines] extended from less than two right angles meet.127 For the geometer assumes this without demonstration as a postulate, although Ptolemy and Proclus devoted entire books to it.128 Often the postulates are in fact contrary to the opinion of the interlocutor but the interlocutor grants it in order for some conclusion to be reached. For example if someone who believes that there is no void should grant that there is void in order to know(o) to what purpose the hypothesis will be useful to the one who is postulating it. Often neither of the contradictory pair appears true to him, for example, that the stars are even or odd [in number], but again he grants the person doing the deduction what he wants to assume in order to see what follows. So necessarily both axioms and hypotheses are true in all cases, but postulates are not true in all cases, but are granted by the person at whom the reasoning is directed. Now this is the entire division. But the things that have been discussed are different from one another. Axioms differ from natural129 hypotheses because, as we distinguished them in the beginning, axioms are natural to all130 and each person gets them from within himself and projects131 them even if the teacher does not state them.132 But even if some hypotheses are apparent and so in some way are self-guaranteeing like axioms, still not everyone can apply them from within himself, but [some people] need to hear something about them from their teacher, even though when they have heard they do not always need an argument to prove them since their truth is evident – as it is established that what is enclosed by three sides is a triangle and what [is enclosed] by four is a quadrilateral, and what has a right angle is rectangular, while what has an acute angle is an acute-angled [figure], and all such cases. This is how axioms differ from hypotheses in the strict sense. Axioms and hypotheses differ from postulates in the strict sense: hypotheses because they are taken from the teacher and appear true to the learner, while postulates which the teacher asks to be granted do not in all cases appear true. For example, if he were to say ‘let it be granted to me to draw a straight line from any point to any point and to draw a circle with any centre and radius’.133 [And postulates differ] from both [axioms and hypotheses] because axioms and hypotheses have conviction, while postulates require more or less explanation. He presented the distinction among these at the beginning,134 and now he distinguishes axioms from both hypotheses relative to the

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learner and postulates, and [distinguishes] these from each other, on the grounds that postulates differ from hypotheses in that hypotheses appear true to the learner while postulates are granted even though they do not yet135 appear true, and axioms [differ] from both in that they are common since they appear true to everyone and they have their truth from within ourselves,136 while hypotheses do not appear true to everyone and neither do postulates. All of them differ from definitions in that they are all premises predicating one thing of another either universally or individually, and either affirmatively or negatively,137 while definitions predicate one thing of another neither universally nor individually, and neither affirmatively nor negatively, but only indicate the being of each thing in accordance with what it is. But if definitions are ever employed in premises they no longer remain definitions, but become premises which have the definiendum as subject and the definition as predicate138 and then these too become hypotheses.139 ‘What must be because of itself and must seem [to be because of itself] is not a hypothesis or a postulate’. He says in what way axioms differ from both hypotheses and postulates: an axiom is that which necessarily is because of itself and appears to be so by necessity, which holds for neither hypotheses nor postulates. For an hypothesis does not necessarily appear to be so since it does not appear true to all, while a postulate does not at all appear true, as has been said. 76b24-7 For the demonstration [is] not [addressed] to external discourse but to the discourse in the soul, [since deduction is not [so addressed] either. For it is always possible to object to external discourse, but not always to the internal discourse.] He said that axioms must be [true] through themselves and must necessarily appear to be true. But in case anyone says that axioms need not always appear to be true (for what happens if someone does not grant that in every case either the affirmation [holds] or the negation [does] or that if equals are subtracted from equals the remainders are equal?), he says in response to such a person that demonstrations are ‘not’ directed ‘to external discourse’, namely what is said out loud (for if someone is contentious and battles against things that are evident, it does not follow from that that the demonstrator will not be demonstrating), but at the very nature of things and at the internal discourse. For even if someone says that it is not true that in every case either the affirmation [holds] or the negation [does], it is not the case that he necessarily thinks that way. So this is why he says that we say that axioms necessarily are [true] and necessarily appear true – in that the interlocutors admit them not in their external discourse but in the internal disposition of their soul. If, he says, demonstrations were made with external discourse

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in mind, not only can there be no demonstration but neither [can there be] a deduction at all. (The interlocutor could refuse to grant any affirmative proposition, for example, that it is not true that every animal is a substance or that every body is three-dimensional, or any other affirmative proposition; but unless an affirmative proposition is granted, there is no deduction.) But even if such a person does not grant it in his external discourse, still in his internal discourse he will necessarily grasp it in that way. And so demonstrations are [directed] ‘to the discourse in the soul’, not to external [discourse]. 76b27-34 Now whatever provable things a person assumes without having proved them himself, if he is assuming things that seem true to the learner, he is hypothesizing them – and it is not a hypothesis without qualification, but only in relation to him [the learner] [– and if he is assuming the same thing when there is either no opinion or even the contrary opinion, he is postulating. And a hypothesis and a postulate differ in this, for what is contrary to the learner’s opinion is a postulate – anything that can be demonstrated and that a person assumes and employs without having proved.] In these words he distinguishes how a hypothesis relative to the learner is different from a postulate. At the beginning he distinguished hypotheses without qualification from postulates without qualification.140 And so he says that all things that are assumed that need demonstration and appear true to the learner are called hypotheses – not hypotheses without qualification but hypotheses relative to the learner. But if they do not appear true or in fact are contrary to the things that appear true to the learner, they are called postulates. We have provided examples of these.141 76b35-7 Definitions are not hypotheses for they are not said to be or not be,142 [whereas hypotheses are among the premises, while it is necessary only to understand the definitions.]

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Here he distinguishes definitions from hypotheses in that hypotheses affirm or deny one thing of another, while definitions neither affirm nor deny anything of anything, but only state what the thing in question is.143 And so hypotheses are necessarily ‘among’ the ‘premises’, but definitions are not among the premises, but are only like utterances, allowing [us] to understand what the thing said is. 76b37-8 But this is not a hypothesis (unless someone is going to say that even hearing is a hypothesis). He says that to say that a definition is an hypothesis is like saying

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that simply hearing a verbal expression and understanding what is indicated by the verbal expression is an hypothesis. For example, man, dog, and such like. This is what he means by ‘hearing’. But an hypothesis is called an hypothesis because it hypothesizes that one thing belongs to another, for example that the soul is immortal or mortal, or that the cosmos is generated or eternal. It is not an hypothesis simply to say that the soul is immortal or the cosmos eternal, since by the account that he himself stated, every phrase will be an hypothesis. But just as it is not the case that every phrase that signifies something is an hypothesis, so also it is not the case that definitions are hypotheses. For they indicate what the definiendum signifies, but not that it belongs or does not belong to something. Therefore definitions are not hypotheses.

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76b38-9 Hypotheses are, rather, all those things that are [true], such that by their being [true] the conclusion follows. Because the premises are true the conclusion must be true as well. Therefore hypotheses are certain premises. But it is not true without qualification that every premise is an hypothesis. It is not an hypothesis that man is an animal.144 Rather, hypotheses are premises of a certain kind, and how they differ from postulates and axioms we have gone through precisely. 76b39-77a3 Nor does the geometer hypothesize falsehoods, as some declare, [saying that we must not make use of what is false and that the geometer speaks falsely when he says that a line he has drawn is a foot long even though it is not a foot long or straight when it is not straight. But the geometer does not base any conclusion on the grounds that the line that he himself has spoken of is this one [which he drew], but on the things indicated by these [figures].] Since he said that demonstrations are ‘not’ directed ‘to the external discourse’ ‘but towards the discourse in the soul’ (76b24-7), he confirms this from the sciences(e). Some people145 who are not aware of this have supposed that the geometer does not demonstrate things that are true, because he assumes false hypotheses, as they thought. For the geometer assumes, for example, that this is a circle although it is not a circle, or that this is a straight line although it is not a straight line and then infers the consequences. So he says that since the reply is not to the external discourse but to the [discourse] in the soul, this is why the geometer constructs his demonstrations with reference not to the [line] he assumes as straight but to the [line] he conceives mentally. Even if he takes [a line drawn] on the board as ‘a foot long even though it is’ not ‘a foot long’, but, for example a finger

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long, he constructs his demonstrations not about [a line] that is not a foot long, but in respect of his own conception, with reference to which he assumed [that] the line [on the board is a foot long]. That is how it works in all cases. 77a3-4 Further, every postulate and hypothesis is either universal or particular, whereas definitions are neither of these.

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This is the second way of distinguishing definitions from postulates and hypotheses, because both postulates and hypotheses are either particular or universal, since in fact every premise is either universal or particular, whereas definitions predicate nothing either universal or particular. For we do not say that ‘rational mortal animal capable of receiving intelligence and scientific knowledge(e)’ belongs to man, but that man himself is a rational mortal animal capable of receiving intelligence and scientific knowledge(e). Chapter 11

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77a5-6 There is no need for there to be Forms or some one in addition to the many in order for there to be demonstration. Since he has said numerous times that demonstrative premises must hold universally of the primary subject, someone might think that in this way he is introducing the Ideas,146 since according to the people who hypothesized them, they are absolutely primary. This is why he eliminates this suggestion, positing that introducing the Ideas is not an inevitable consequence of demonstration. The Ideas are abstracted from the many and are not predicated of them either in every case or in any case,147 whereas demonstration hypothesizes that which [is predicated] primarily of every one, and this is what he calls the universal.148 This I declare is the form that is coextensive with the many. And so demonstration does not introduce the Ideas. ‘There is no need for there to be Forms or some one in addition to the many in order for there to be demonstration’. That is, there is no need on account of what has been said in the account of demonstration for the Ideas to be introduced on the grounds that each [Idea] is one thing. For even if we say that there must be a universal in demonstrations, we are not claiming that this is the kind of thing that those who introduce the Ideas declare is the Form of the many animals, for example.149 Indeed consider how displeased he is at the theory of Ideas, so that even if anyone were to suppose that he says something that introduces the notion of the Ideas, he himself anticipates this and rebuts it. In fact he next goes on to ridicule the Ideas, calling them nonsense.150

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77a6-7 But it must be true [to predicate] one thing of many, for if this is not the case there will not be a universal. If there is demonstration, he says, there is no need for the Ideas to be, ‘but’ there ‘must’ be ‘one thing’ [predicated] ‘of many’. For if, as we have proved, universals must be employed in demonstrations, and universals belong to every instance of the subject, since [they hold] always and per se of it as the primary [subject],151 if there is demonstration there is every need for there to ‘be’ some ‘one thing’ that belongs to ‘many’. If this is not the case, neither ‘will there be’ ‘a universal’, he declares.

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77a7-8 And if there is no universal, there will not be a middle term, and so neither will there be a demonstration. If the major term is not predicated universally of the middle term, the major term does not belong primarily or qua itself to the middle term. For this is the kind of thing the universal has been proved to be. But if the major term does not [belong] primarily to the middle term, it will not be a middle term in the strict sense, but it needs another middle term to which the major term belongs primarily. For example, if having its three angles equal to two right angles [does] not [belong] universally to the isosceles, isosceles will not be a middle term, since it will need another middle term, triangle, to which it belongs primarily to have its three angles equal to two right angles. And so if [the property of] being universal is eliminated, [the property of] being a middle term in the strict sense is eliminated. But if [the property of] being a middle term in the strict sense is eliminated, the demonstration is eliminated, since demonstrations [proceed] through immediate premises. And so, if the middle term is not a middle term in the strict sense, the premise is not immediate either, and if it is not, ‘neither’ will there be ‘a demonstration’. 77a9 Therefore it is necessary for there to be one and the same thing over many and not homonymously. He did well to add ‘not homonymously’. For the single word ‘crab’ is applied homonymously to several things: the constellation, the animal and the tool.152 Similarly ‘dog’ is applied to the terrestrial animal, the sea-animal, the star, and the philosopher.153 Not even dialectic (to say nothing of demonstration) employs homonymous things without specifying the distinctions.154

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Translation 77a10-11 No demonstration assumes that it is impossible to assert and deny simultaneously unless it is necessary to prove [the conclusion too in that way.]

He155 just said that sciences(e) make use of axioms and assume them in their premises, (for example, since A and B are equal to C, and things that are equal to the same thing are also equal to one another, therefore A is also equal to B156). But the axiom of contradiction divides the true and the false for all things that either are or are not,157 although in some cases it seems not to be so and it seems that both members of a contradiction can be true simultaneously. (For example, we say of the soul that it is both immortal and not immortal – clearly in different respects; it is immortal and eternal in its essence but not in its activities.158 Likewise the elements are both ungenerated and generated – ungenerated in their totality but generated in their parts. Likewise too the heaven moves and does not move; it does not move as a whole so as to change its entire place, but it moves in respect of its parts. The same holds for the cosmos as well – the cosmos seems to be eternal .159 And the same holds for very many other things). This is why in some cases when we are inferring an affirmative conclusion we are compelled to construct the denial of the negation, lest in the case in question there seem to be room for the negation to hold in some way. Take nous for example. If we deduce that it is immortal or eternal, lest it seem to be immortal in the same way as the soul, so that the negation can in some way be true of it too, we add in the conclusion that nous is therefore immortal and not non-immortal, and eternal and not non-eternal.160 And likewise in similar cases. In the present passage, therefore, in respect of conclusions of this sort he investigates in which term of the deduction it is necessary to employ the axiom of contradiction, in order that by employing it we may prove in the conclusion that only one member of the contradiction is true and the other one [is true] in no way. Clearly it cannot be inferred in the conclusion unless it is employed in the premises. Now since there are three terms in a deduction, in which of them must we employ the axiom of contradiction? Clearly the [axiom of] contradiction is in the premises; however it is possible to consider it in the case of each term, as when we say ‘man’ and ‘not-man’, ‘stone’ and ‘not-stone’, which are called indefinite in De Interpretatione.161 But since things predicated162 in premises are either affirmed or denied of something and since the negative particles that cause a negation are taken with one term only, this is why the axiom of contradiction is considered as applying to terms – but [in such cases] we must also understand the remaining term which completes the premise. And to speak simply, since, as I said, the things found in the conclusion are necessarily found in the premises as well and there cannot be

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anything in the conclusion that has not been supplied in the premises – but as I said, in some conclusions we make the inference stating the entire axiom of contradiction, for example, that this is an animal and not a not-animal – he investigates where in the premises this must be employed. Since there are three terms in the premises, which one must be brought forward in the form: ‘this and not-this’? He says that [it must be brought forward] only in the major term and in neither of the others. For if we employ either of the other [terms] in this way, it will be useless for the conclusion. What do I mean [by saying] that it will be useless to us for the conclusion? It is entirely impossible to use any axiom of contradiction163 in the minor or middle terms. For the contrary conclusion will be drawn, and so it will be absolutely necessary for both members of the contradiction to be true simultaneously of the same things unless the terms are coextensive. For example,164 in the deduction which [Aristotle] himself gives: Callias is a man, a man is an animal and not a not-animal, therefore Callias is an animal and not a not-animal. But it is impossible to use the denial of the negation for the minor term,165 since the contrary of the affirmation will be inferred and the negation too will be true in the strict sense, but the negation is not in the strict sense simultaneously true with the affirmation. For not only is Callias a man but also not-Callias [is a man], unless everything that is not Callias [is not a man]. And so it is not possible to say ‘Callias and not not-Callias is a man’, since not-Callias (for example Socrates and Plato) is a man too, even if not everything [that is not Callias is a man]. This is why the axiom of contradiction cannot be used in the middle term either. For not only man but also not-man is an animal, even if not everything [that is not a man is an animal]. Clearly if we say that both members of a contradiction are simultaneously true for the minor or the middle term we are not assuming a contradiction in the strict sense, but the positing and denial of one term only, which expression we called in the De Interpretatione an indefinite subject.166 For the negation is a denial that the predicated term holds of the subject. Now we say that in these cases it is terms, not propositions, that contradict one another. Contradiction in the strict sense is found in propositions. But the contradictions in terms, if the terms are taken as subjects, can be simultaneously true when one and the same thing is predicated of both, as has been said. This is why it is not possible to employ the axiom of contradiction in either the minor or the middle term, as we said. Therefore only in the major term can we use this axiom, saying that man is an animal and not a not-animal; for in this way we can conclude ‘Therefore Callias is an animal and not a notanimal’. But Aristotle took the minor and the middle terms as subject terms and in that way proved that it is not possible to employ the axiom of contradiction in those cases.

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But since the middle term is not only the subject for the major term but is also predicated of the minor term, if we take it as predicate nothing prevents us from using the axiom of contradiction for it. I say that Callias is a man and not a not-man. Indeed it is always possible to use this [claim] in this way, but it will not help us towards the conclusion. What was sought was this: in deductions where we employ the axiom of contradiction in the conclusion, in which term do we employ it as useful to us in drawing this kind of conclusion? So only in the major term should this axiom be employed in cases where we want to infer a conclusion of this kind. Either the goal of the present discussion should be determined in this way, or it should be said that the present purpose is to prove how we employ the axiom of contradiction in deductions – on the one hand how [we employ it] in direct [deductions] and how in [deductions] per impossibile, and what is the difference between them, because in direct [deductions] they are not always found – only in cases where we want the conclusion to be of this form – whereas they are always found in [deductions] per impossibile. In direct [deductions they are found] in the major premise or simply at the end, for they are in the conclusion, but in [deductions] per impossibile [they are found] in the minor or simply at the beginning. For every proof per impossibile employs the axiom of contradiction right at the beginning.167 The geometer posits that the side is either commensurable or incommensurable, and by assuming one member of the contradiction – the false one, that it is commensurable – and proving that something impossible follows from this – that the same things are even and odd168 – he thus infers the opposite. But Themistius169 gives the goal of the present discussion as related to what was said previously. For, he says, since [Aristotle] said in the previous discussion that when the things on which demonstrations are based are clear, sciences(e) do not always employ them, he now wants to prove this very point. At any rate notice that just about no science(e) employs the axiom of contradiction in its deductions, but because it is obvious and evident, it overlooks it as being clear to all, even if the person doing the deduction does not employ it in a premise. Next, since in some deductions it is necessary to employ it as a premise, he says in which deductions we employ it – namely, in those where we want to infer a conclusion of the same kind – and how it is assumed in advance in the premises. ‘No demonstration assumes that it is impossible to assert and deny simultaneously unless it is necessary to prove the conclusion too in that way’. No demonstration – direct demonstration – employs the axiom of contradiction, he says, unless it is necessary to infer ‘the conclusion too in that way’, namely, unless the conclusion too must employ the axiom of contradiction. In that case the deduction will employ it in this way. We have said170 when there is need to infer the

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conclusion in that way, namely, where it is not possible for the [term] predicated of the subject to belong in one respect and to not belong in another. For it is in order to prove that the two members of a contradiction are never simultaneously true in any way that we need to infer such a conclusion.

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77a12-13 It is proved by assuming that the first is true of the middle and not true to deny [of it]. This kind of conclusion – namely, the kind that employs the axiom of contradiction – is proved if we assume that the first term – namely, the major term – is truly asserted of the middle term and not truly denied – namely, if we employ the axiom of contradiction with respect to the major term [claiming] that if the affirmation is true of it, the negation is in no way true too.171

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77a13-15 It makes no difference to assume that the middle is and is not, and likewise for the third. That is, nothing prevents us from taking the middle term and likewise the minor term as well both affirmatively and negatively. And so if nothing prevents them from being taken simultaneously both affirmatively and negatively,172 it is impossible to employ the axiom of contradiction in a case where in fact they are simultaneously true. But according to Alexander it means the following: if it is true to predicate the major term both of the middle term itself and of its negation, it makes [no] [difference] to the deduction and the conclusion in question if we take either the middle term or the last [term] in this way. 77a15-20 For even if it had been given that anything of which it is true to say [that it is a] man (even if it is also true to say [that it is a] not-man), but if only [it is true] that man is an animal but not every173 not-animal, [it will be true to say that Callias (even if not-Callias) is nevertheless an animal and not every174 not-animal. The reason for this is that the first is predicated not only of the middle but also of another, since it applies to many things.] ‘If it had been given that anything of which it is true to say man’ is equivalent to ‘animal [is said] of every man’.175 For a person who says ‘anything of all of which man [is said]’ is saying that animal [is said] of every man. Now if a premise like ‘animal [is said] of every man, but not-animal is not’ is given and taken, this entire thing, if it is taken thus, is sufficient to yield the conclusion that Callias is an animal and not a not-animal, if this is what was proposed to prove. For even if it

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is true, he says, that animal [is said] of man and also of not-man and that man [is said] of Callias and of not-Callias, nevertheless if taken they are redundant and contribute nothing to the proposed conclusion. For if ‘animal [is said] of every man and not-animal is not’ is taken, it is sufficient by itself. For the major premise is the only one that when taken in this way is useful for a conclusion of this form. 77a20-1 And so even if the middle is both it and not it, it makes no difference to the conclusion. The middle term will not help us at all towards the conclusion not only because it is impossible to use the axiom of contradiction with respect to it but also because it will not help us towards the conclusion even if both members of the contradiction are simultaneously true of it in the conclusion. Nor will there be greater conviction in the conclusion whether or not they are simultaneously true, as it helps if both members of the contradiction are not simultaneously true with respect to the major term but are simultaneously true with respect to the minor term. 77a22-5 Reductio ad impossibile demonstration assumes that everything must be asserted or denied, and [these [i.e., the law of the excluded middle and the principle of non-contradiction] are not always [assumed] universally, but so far as is sufficient, sufficient for the genus. By ‘for the genus’ I mean the genus concerning which [the science] brings its proofs, as was stated previously.] After saying how and when we use the axiom of contradiction in direct proofs, he says about [proof] per impossibile as well that every [proof] per impossibile uses it176 – not universally to be sure (that is, not on the grounds that it pervades all things that are and are not), but to the extent that each science(e) employs it as far as it needs it. It needs it [as applying] to certain subject matter, as geometry [employs it] as applying to magnitudes, arithmetic as applying to numbers, and the others as applying to their proper subjects. So when the geometer says that something is either commensurable or not-commensurable, he does not take not-commensurable as able to signify white, for example, and animal and what is not, but only what is contrasted with what is commensurable in magnitude. For even if it did not divide the true and the false in all things that are and are not, but [did so] only in respect of magnitude, he would have used it all the same. Likewise the arithmetician and each of the experts(e) in a particular field. Only first philosophers and dialecticians will employ it as common and pervading all things that are and are not.

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77a26 All sciences(e) share with one another in the common principles. He said earlier too that some axioms are entirely general, applying to every science(e), some [apply] to several, and some [apply] to one.177 So, ‘all sciences(e)’ ‘share’ in the most general axioms. For no science(e) shares with another in its givens and soughts,178 except for subalternate [sciences(e)].179 However, they all share in the most general axioms, for example in the [principle] that both members of a contradiction cannot be simultaneously true. Every art and every science(e) is bound to employ this one. He said this previously too.180 But perhaps he repeats his account on this matter at the present point in order to add that dialectic employs this axiom, and also first philosophy itself, which in the Metaphysics he calls wisdom,181 likewise uses it even though it demonstrates the principles of all things and in particular this axiom of contradiction. At any rate, although in the third book182 of the Metaphysics Aristotle demonstrates the very axiom of contradiction,183 he still employs it in his arguments against those who introduce the views that apprehension184 is impossible and everything is indeterminate. For, he says, you either answer us or you don’t answer185 – which is a contradiction on the grounds that one must186 necessarily admit one part or the other and cannot refuse to do both. For if you do not answer us, he says, we don’t have the time to go to study with you and really discuss. Anyway, how are you trying to persuade us that apprehension is impossible without giving any answer to the question we posed? If you answer and teach us, you will necessarily be using words for your teaching, and words that are not without significance but that signify certain things. Therefore you understand them and consequently even for you there is apprehension, which is what you wish to eliminate – and this very [result is produced] through a demonstration.187 So if each science(e) that uses the axiom of contradiction does not use it as applying universally but only specifically and as far as it is useful for its own purposes – namely as applying only to its subject genus188 – how, then, does he here say again that the sciences(e) share ‘with one another’ ‘in the common [principles]’? For if they do not employ them as applying generally but as proper [principles], they should not be said to share [them] with one another. I say that even if the general axioms contribute to sciences(e) and this is how each [science] makes the common [principles] proper to itself, still the very signification of the [principle of] contradiction is common [to them all] and it makes no difference whether it is assumed for magnitudes or numbers or anything else. For a contradiction is necessarily a contradiction of something. So the [principle of] contradiction without qualification is common to every science(e), but it becomes proper insofar as it concerns magnitudes, numbers, or the attributes of these things. The same holds for the rest.189

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Translation 77a27-8 I call common the principles that people use as a basis when they are demonstrating, [not the things about which they prove or what they prove.]

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Namely, the axioms, which every science(e) uses as principles to demonstrate the claims in question. These [the claims in question] are not the same for the [different] sciences(e), but are proper to each one.190 77a29 Dialectic too is [common] to all [sciences]191

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It uses the axioms192 of all [sciences] because it makes deductions about everything. But each science(e) discusses things on the basis of what belongs per se to them, whereas dialectic [does so] on the basis of reputable opinions.193 For dialecticians will attempt to prove that two sides of a triangle are larger than the remaining one, but not on the basis of what belongs per se to triangles as geometers do, but on the basis of some reputable opinions – for example that they are larger because if some fodder is placed at one angle or another, and you set loose an irrational animal or a human to go towards it from the other end of a straight line – for example if the thing is placed at A and the thing set loose is at B, clearly it will go to the thing that has been placed via the unique straight line AB and not via two. Thus it is evident that the two sides are larger than the remaining one, for it will go the shortest way. 77a29-31  as would be an attempt to give a universal proof of the common principles, [such as ‘everything must be either asserted or denied’ or ‘equals from equals’ or others of this kind.]

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Namely, first philosophy, which does attempt to demonstrate all the axioms.194 But even this employs these and above all the most general axiom of all, [the axiom of] contradiction, as we said that Aristotle too [pointed out] in his arguments in the Metaphysics against those who wish to eliminate apprehension.195 It is evident that the arguments against these people are not those of an expert(e) in some special area, but [of an expert(e)] in first philosophy. For each science(e) is concerned with a single genus, as has been said many times, and only first philosophy and dialectic have all things that are as their subject. So discussing about all things that are, whether they have a definite nature or not,196 is the province of first philosophy. 77a31-3 But dialectic does not have any definite [subjects] in this way, nor any single genus. For otherwise [the dialectician] would not be asking questions197

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After saying that all sciences(e) make use of the common [principles] – and in fact dialectic and first philosophy do so above all198 – he next distinguishes dialectic from all the rest and from first philosophy. [It differs] from all the rest in that each of the others is concerned with a single genus while it [is concerned] with everything. It differs both from the rest and also from first philosophy in that dialectic asks questions of the interlocutor and after taking premises from him establishes the proposed claim on the basis of the premises granted by him, and establishes the same result from both members of a contradiction. For example, [the dialectician] will establish the immortality of the soul by asking the interlocutor whether the soul seems to him to be self-moving or not. And no matter which member of the contradiction the interlocutor may reply, he will take this and attempt to prove its immortality, clearly on the basis of reputable and plausible opinions, though not on the basis of necessary [attributes] or of [attributes] that belong per se to the thing.199 But sciences(e) do not [proceed] in this way. They do not obtain their premises by asking questions but get them from the very nature of the things, even if they do not seem true to anyone,200 and on that basis infer what they want, but do not [infer] the same conclusion from both members of a contradiction. For it was proved in the second book of the Prior Analytics that it is impossible to draw the same conclusion demonstratively from contraries, namely, from both the affirmation and the negation.201 So this is why the sciences(e) do not obtain their premises from asking questions. For it will not be possible to infer the claim in question scientifically(e) if the interlocutor decides to grant not the true premise but the false one. This is why experts(e) obtain their premises on their own from the nature of the things. Sometimes indeed the sciences(e) draw the same conclusion from contraries, [proving it] from one member of the contradiction by means of a direct [proof] and from the other by a [proof] per impossibile.202 For instance, that the diagonal is incommensurable with the side. If someone were to say that it is commensurable with the side, he will hypothesize this and prove that something impossible follows, and will eliminate it in this way. For since in every case either the affirmation or the negation [holds], when one member is eliminated the remaining one must be true. He does not say here that it is impossible for the same conclusion to be drawn from both members (from the one [in a] direct [proof] and from the other [in a proof] per impossibile), but only that in the same manner of proof it is impossible for the same conclusion to be drawn from contraries. And this is what the dialectician does. He will prove that there is foreknowledge both from the claim that the soul is immortal and from the claim that it is not immortal. And what is surprising when the dialectician also establishes both members of a contradiction? It is not that he thinks that it is possible for both members of a contradiction to be simulta-

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neously true, but that he compels his interlocutor to this very absurdity – whichever member of the contradiction [his interlocutor] admits, he proves it false and its contrary true. 77a33-5  for it is not possible for a person who is demonstrating to ask questions, [because he cannot demonstrate the same thing if either opposite holds. This was proven in my treatment of the syllogism.]

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But what’s this? Since first philosophy too in its demonstrations does not get its premises by asking questions, but assumes them, does it follow that it is concerned with a single genus and does not have all things that are as its subjects? Now ‘it is not possible for a person who is demonstrating to ask questions’ was perhaps not said with an eye to demonstrating that it is not concerned with a single genus (for it is possible to get premises not demonstratively but by asking questions even if it is concerned with a single genus) but with an eye on the claim ‘it does not have any definite [subjects]’ (71a31-2), so that ‘not having any definite [subjects]’ and ‘not having any single genus’ do not both signify the same thing, but that ‘not having any definite [subjects]’ signifies that it is not restricted to particular members of contradictions, as is the case with each science(e).203 For geometry is restricted to proving that the diagonal is incommensurable with the side, and not the contrary, and that two sides of a triangle are larger than the other [side], and the contrary not at all. And in each theorem it is restricted to only one member of the contradiction, and to the other in no way. And likewise for the other sciences(e). But dialectic is not restricted to either member of a contradiction, but as I said, it establishes them both, leading the interlocutor around to a contradiction and aiming to prove him ignorant. And in order to establish ‘that it does not have any definite [subjects]’ he says ‘for otherwise [the dialectician] would not be asking questions (71a31-3). For it is not possible for a person who is demonstrating to ask questions’. For he asks questions not in order to demonstrate but in order to establish the contrary of what is granted, that is, in order to establish the claim in question on the basis of what the other person grants, no matter what it is. He did not establish ‘nor [does it have] any single genus’,204 because it is clear. For it is plain to everyone that dialectic is not concerned with any single subject. Chapter 12

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77a36-41 But if a deductive question is the same thing as a premise of a contradiction [and premises in each science are the things from which the deduction in it [is formed], it follows that there is such a thing as a scientific question, [namely, one

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formed] from premises on the basis of which a proper deduction in any [science] is formed. Therefore it is clear that not every question can be geometrical or medical, and similarly for the other [sciences].] In this he wants to prove (a) that each expert(e) in a particular field will not ask every possible question, but only ones that are appropriate to his particular science(e) – geometers [will ask] geometrical [questions] and arithmeticians arithmetical [ones] – and (b) that they do not have to answer every possible question, but only those that are asked on the basis of the appropriate science(e). He proves this very clearly and concisely. If, he says, a ‘deductive’ ‘question is the same thing’ as a premise in respect of each member of a contradiction, that is, if a person asking a question in a way suitable for deduction is asking for a premise (for example, whether the soul is immortal or not, or whether the cosmos is eternal or not), and scientific(e) ‘premises from which the deduction’ [is formed] are determinate ‘in each science(e)’, a ‘scientific(e) question’ will be any [question] formed from any premise ‘on the basis of which a’ ‘deduction is formed in any’ science(e); if it is not formed from these, the question will not be scientific(e). For if he asks a scientific(e) question as205 being also a scientific(e) premise, whether negative or affirmative, and the premises in every science (e) are determinate (for the [premises] of ‘musicians’ are not useful to geometers nor are those of arithmeticians [useful] to astronomers), geometers and each expert(e) in a particular field should not ask every premise, but only those having to do with the subject genus [of their own science]. This is why an expert(e) should not answer every question either (for if anyone asks a geometer which is the most beautiful line – the straight line or the circle,206 he should not answer this kind of question), but only those that belong to his own science(e) or to the one immediately superior to it.207 For the expert(e) must also know(o) the facts of the superior [science], as a ‘musician’ 208 [must know] the facts of arithmetic (since he employs the theorems of that [science]) and the ‘mechanic’ – and the ‘optician’ likewise – [must know] the facts of geometry, since the [theorems] of the subordinate [science] are proved on the basis of the theorems of the superior [science]. 209 So if a ‘musician’ is going to be an expert (e) he should know (o) the things on the basis of which he demonstrates his own appropriate facts, namely the facts of arithmetic. Likewise the ‘optician’ or ‘mechanic’ [should know] the facts of geometry. However, no science (e) will render an account of its own principles. For example a geometer ‘qua geometer’ will not have to render ‘an account about’ his own proper ‘principles’ (77b5-6) nor will any of the others. It is worthwhile to pose a puzzle. If each science(e) must know(o) the

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facts of the [science] immediately superior, and the superior [science] demonstrates the principles of each [science immediately subordinate to itself], it follows that each science(e) knows(o) the arguments for its own principles. For example, the four elements are a principle of medicine, and natural philosophers discuss them. So if one who is really a doctor must know(o) the facts [studied by] natural philosophers, he will also know(o) the arguments concerning the four elements. But if he knows(o) he will also give an account to anyone who asks about them. How then does he say that ‘he should not have to render’ ‘an account’ ‘of his principles’ (77b5-6)? For if when asked he must give answers as well about the facts of the immediately superior science(e), from which he establishes his own appropriate facts – for example, a doctor [must give answers] about facts in natural philosophy and an ‘optician’ about geometrical facts – but the [doctrine] of the four elements, which are the doctor’s principles, is a theorem in natural philosophy210 and therefore the doctor will answer questions about them. And likewise in the other cases. To this puzzle the Philosopher211 said that while it is necessary for the subordinate [science] to know(o) the facts of the superior science(e) since through them he demonstrates his own appropriate facts, still, he need not know(o) them demonstratively, but only [know(o)] the ‘that’ without the ‘why’. Therefore he will know(o) the theorems of the superior [science] as his own proper principles and when asked about facts of the superior [science] he will give only the ‘that’ but not the ‘why’, as doctors [will answer] that bodies have only four elements and neither more nor less. And similarly for the rest. We have given the origin of this puzzle in saying that the expert(e) must know(o) the theorems of the science(e) immediately superior to his own, if in fact it is necessary for him to be really accomplished, as ‘opticians’ or ‘mechanics’ [must know] the facts of geometry and the doctor [must know] the facts of natural philosophy. And you should not find it at all surprising if a person in this condition were to demonstrate his own proper principles. But Aristotle does not say this, but that geometers should be asked questions either about geometrical facts or about [the theorems] that are demonstrated on the basis of the principles of geometry – about optical [theorems], for example.212 And so contrariwise he wants to ask the superior [science] about the facts of the subordinate one. And clearly the superior one should not be asked about all the facts of the subordinate one, for about ophthalmia, for instance, it is not [natural philosophy] that should be asked, ,213 which depends on the principles of natural philosophy and is concerned with the observation of nature, as for example how nourishment is distributed to every part, how it alters and how it is nourished, and about growth and decline and similar matters. And likewise in the case of the other [sciences]. But if it is for the superior one to state the theorems of the subordinate one, the

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puzzle we stated is no longer relevant, since no science(e) will be demonstrating its own proper principles. ‘But if a deductive question is the same thing as a premise of a contradiction’. In De Interpretatione he called a dialectical question one that is a member of a contradiction and that potentially contains in itself the other [member] as well, and which must be answered only ‘yes’ or ‘no’.214 (In this a question is different from an inquiry, since inquiries require a longer statement.215) What he there called dialectical he now calls deductive since he has just spoken about dialectic and the difference between it and the sciences(e)216 – in order that he may appear to be speaking of this kind of question only as applying to those kinds of [namely, scientific] discourse. This is why he calls it deductive. In fact there too dialectical [questions] were understood as being suitable for deductions. ‘Premise of a contradiction’ means either member of a contradiction, which becomes a premise when it is taken in a deduction. It is clear that such questions are premises, since such questions can be found in no statements other than declarative ones. People making wishes and giving orders cannot express their statements in such a way that the person to whom the statement [is addressed] says only ‘yes’ or ‘no’ in reply. This occurs only in declarative statements. And declarative statements are either affirmations or negations. Therefore a deductive question is the same thing as either an affirmation or a negation.

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77a41-b3 But [geometrical questions] must be based on the [principles] from which something either217 is proved concerning the subject matter of geometry or218 is proved from the same principles as geometry, such as optical [questions]. [And similarly for the others.] I call geometrical, he says, the theorems that depend on the principles of geometry from which geometrical theorems are proved. Even non-geometrical ones belonging to another art will be no less geometrical, where that [art] employs the same principles as geometry – ‘optical’ [theorems], for example, since these too are proved by means of the principles of geometry. And so these too will be geometrical and geometers can be asked questions about them no less than about ones that come directly from geometry. And likewise for the other subalternate [pairs of sciences].219 77b3-6 On these [questions] an account must be given from geometrical principles and conclusions, [although a geometer qua geometer should not have to render an account of his principles, and similarly in the other sciences.] Because everything that is proved in sciences(e) is proved either from

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the principles of the sciences(e) or from what has been proved from the principles. For example the first theorem of the Geometer220 is proved from the principles, and the second from the first. For the second is proved via the conclusion of the first theorem, which was inferred from the principles. And so for them all. 77b6-9 Therefore each expert(e) should not be asked every question, [nor should he have to answer every question on every subject, but only the ones determined by his own science.]

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When a geometer is asked whether a circle contains the greatest area of all isoperimetric figures or whether it has [lines] from the centre that are equal, he should give an answer; these questions are geometrical. But if someone were to ask him whether the circle is the most beautiful line, he should not answer.221 Nor ‘should he have to answer’ every [question] (since geometers do not answer questions about a topic in music), or every [question] ‘on every subject’. 77b9-11 But if someone converses with a geometer qua geometer in this way, it is obvious that [he does so] well [if he proves something on the basis of these [principles]. Otherwise, [he does] not [do so] well.] If a questioner or simply a person constructing arguments against a geometer discusses with the geometer in this way – on the basis of the principles of geometry – he will be discussing ‘well’. But if he is going to discuss with the geometer not on the basis of the principles of geometry, [he will be discussing] ‘not well’, as when the squaring of the circle is investigated by geometers. Antiphon and Bryson thought they had discovered this, wrongly postulating that they be granted things to which geometers would not assent. The former, that a straight line coincides with the circle, and the latter that if there is something of a different kind that is larger and smaller than something, there is also something equal to it.222 Now these men discuss badly with geometers, but in squaring lunules Hippocrates did not [discuss] badly or outside of the principles of geometry. His mistake was that he postulated that what holds for a part holds for the whole as well. 223 77b11-12 But it is clear that he does not refute the geometer either, except accidentally.

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For if anyone asks a geometer something that is not geometrical, for example whether a straight line or a circle is the most beautiful line, and then thinks that he is refuting him when he answers naively that the straight line is, he has not refuted ‘the geometer’, since he did not

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refute him about anything that pertains to geometry except ‘accidentally’, since it happened accidentally that the person refuted is a geometer. For he did not refute him qua geometer any more than a doctor is refuted qua doctor if he is refuted in regard to theorems in musical theory. 77b12-15 And so one should not discuss geometry with people who do not know geometry, for he will not notice that he is discussing defectively. [And likewise for the other sciences as well.]

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Just as each person should be asked questions about appropriate topics, so also geometry should be discussed ‘with’ geometers and not with people who are ignorant of geometry, since non-experts(e) do not know(o) how to judge matters that fall under the sciences(e). This is why uneducated people will frequently appear superior to experts(e).224 77b16-17 Since there are geometrical questions, are there also ungeometrical ones? Since he said that geometers, doctors and other experts(e) should not be asked every question but only the ones based on their own proper science(e), and that an expert(e) should not answer every question but only ones appropriate to him (for example a geometer [should answer only] geometrical [questions] and doctors [should answer only] medical [questions]), he here investigates whether ‘there are also ungeometrical’ questions just as ‘there are geometrical questions’, and likewise for the other sciences(e). What kinds [of questions] are these and in respect of what kind of ignorance will they count as ungeometrical? To make the account clear, let us expound the present statement precisely taking material from what he is going to say at greater length in what follows. He says that there are two kinds of ignorance:225 [ignorance] in respect of negation and [ignorance] in respect of disposition. [Ignorance] in respect of negation is where a person has no opinion on the matter, while [ignorance] in respect of disposition is where a person has an opinion on the matter but not one that is secure and exact. For example a person who does not know(o) at all whether or not parallel [lines] are straight, and if they are, whether or not they intersect, has ignorance in respect of negation, but if anyone were to suppose that parallel [lines] intersect, he has ignorance in respect of disposition. For he is badly disposed concerning the matter [and has] what Plato calls twofold ignorance.226 Now in respect of which kind of ignorance are questions geometrical or musical? Ignorance in respect of disposition in turn is divided into

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two kinds: in respect of the matter and in respect of the form of the deductions.227 For a person who by reasoning fallaciously misleads himself that parallel [lines] intersect draws the false conclusion either because he has assumed false premises and combined them deductively or because he assumed true premises but combined them invalidly – for example [by reasoning] in the second figure from two affirmative [premises]. Now which of these is more of a geometrical mistake – a deduction that is mistaken in respect of the matter or one [that is mistaken] in respect of the form? I say that just as there are two ways of being unrhythmical – one case is where something is not at all of a nature to have rhythm (as if someone were to call a point unrhythmical because it is not of a nature to have rhythm), while the other is a thing that has [rhythm] but has it badly, (as we call a line of poetry whose rhythm is bad unrhythmical) – so what is ungeometrical is either what is entirely in negation of geometry (this includes things that fall under other sciences(e), for example if someone were to ask the geometer about a theorem in musical theory or medicine; this is a matter of ignorance in respect of negation and it is ungeometrical in the way that a point is unrhythmical; [the person who makes this mistake] does not even know(o) whether this [question] belongs to geometry), or it is geometrical but distorted, as if one were to say that parallel [lines] intersect. In a way this is geometrical, in that parallel [lines] and intersecting are entirely geometrical, but it is ungeometrical in that he is wrong to say that parallel [lines] intersect, in the same way as we say that a line of poetry whose rhythm is bad is unrhythmical and that a performer of tragedies who has a bad voice does not have a voice. This is a case of ignorance in respect of disposition. And since this has two kinds (since as we said, false conclusions are drawn either from a mistake about the matter or through an invalid combination of premises) it is clear that in geometry fallacies always occur because of a mistake about the matter, when the geometer assumes premises that are false and so draws a false conclusion. In geometry you will not find cases where the matter is true and the conclusion is invalid, but only cases where the matter is invalid.228 Aristotle labels both cases mistakes, as a term common [to both] but [he labels] a mistake due to the form a fallacious argument, as a term for it alone, since it is not based validly on deductive reasoning, whereas [a mistake] in the matter [he labels] simply a mistake.229 77b17-18 And in each science, in respect of what kind of ignorance are [questions] geometrical? Since there are two kinds of ignorance, as we said that he too will say at greater length in what follows – [ignorance] in respect of negation and [ignorance] in respect of disposition – in respect of which kind of

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ignorance in each science(e) do we call questions ungeometrical, unmedical, and similarly for the rest? After saying ‘in each science(e)’ he says ‘are geometrical’, with reference to one [science] as an example. He does not say ‘ungeometrical’ but ‘geometrical’, so we need to take the whole [expression] together and understand ‘are geometrical in respect of ignorance’, which is equivalent to ‘ungeometrical’.

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77b18-20 And is a deduction due to ignorance a deduction from the opposites [of the principles] or is it a fallacious argument 230 Before distinguishing in how many ways ignorance is said, he first distinguishes the ways in which mistakes about truth occur in each case of ignorance: in respect of the matter and in respect of the form.231 ‘Ungeometrical’ must be understood as applying in common to both, in the way just stated.232 Now, he says, is a question ungeometrical ‘a deduction due to ignorance’, one ‘from the opposites’, namely a deduction due to ignorance in respect of disposition that is based on false premises (for these are contrary to the true ones), ‘or is it a fallacious argument’, namely one involving a mistake in the form (which he does not even call a deduction because the form is invalid)? ‘Due to ignorance’ [applies] in common: [the second alternative is] ‘or a fallacious argument due to ignorance’.

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77b20-2  in geometry, or one from another art [as a musical question is ungeometrical (if the subject of discussion is geometry)] 233 Here he distinguishes ignorance in respect of negation and [ignorance] in respect of disposition. Questions that come ‘from another art’ are [ignorance] in respect of negation,234 while the false ones that come from [the science] itself are [ignorance] in respect of disposition. So in respect of which of them are [questions] ungeometrical? He next gives examples of them.

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77b22-4  and is it in one sense geometrical and in another sense ungeometrical to suppose that parallel [lines] intersect? Either he is saying ‘to suppose that parallel [lines] intersect’ is ‘geometrical’ in a way because it contains terms from geometry, but is ‘ungeometrical’ because it assumes things that are false, or, which seems rather better [as an interpretation], he is contrasting the one manner of ignorance in respect of negation with the other, which he has already mentioned.235 In fact, what follows depends on this notion.

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Translation 77b24-6 For this has two meanings just as ‘unrhythmical’ does: [one thing is ungeometrical by not possessing [geometry] and the other is by possessing it but defectively.]

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‘Ungeometrical’ clearly ‘has two meanings’ ‘just as “unrhythmical” does’ too: either what does not possess it at all, the way a musical question is ungeometrical just as a point is unrhythmical, or what possesses it ‘but defectively’ – fallacies236 in geometry are ungeometrical the way a line of poetry with bad rhythm is unrhythmical, since they are concluded from false premises, though not invalidly. The way he infers his result indicates that he is talking about this kind [of ungeometrical proofs]. 77b26-7 And this ignorance and the ignorance from this kind of principle is contrary.

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This is the [kind of ignorance] mentioned above, [the kind that consists in] possessing it defectively, where ‘defectively’ is a matter of depending on principles (i.e., premises) that are not true. This [kind of ignorance] is in fact contrary to the truth. For false premises are contrary to true ones. This is how he earlier called a deduction from a false [premise arising from ignorance] of the matter: ‘And is a deduction due to ignorance a deduction from the opposites or is it a fallacious argument?’237 After saying that there are two kinds of ungeometrical questions just as there are two [ways of being] unrhythmical (not possessing it at all or possessing it defectively), and then saying what possessing it defectively means ([having it] from principles that are opposites of the true ones), he entirely separates the other kind, that in respect of a mistake in the form, from geometry because, as I said,238 no fallacy is inferred in geometry through a mistake in the form. 77b27-8 But fallacious argument does not occur in the same way in mathematics, because the middle [term] is always used twice.239

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After saying what questions are ungeometrical and likewise [what questions] in each science(e) are not appropriate to the science(e) – namely, those due to ignorance either in respect of negation or in respect of disposition – he here wants to say that fallacious arguments do not occur in sciences(e) in the same way as in dialectical conversations. Falsehood is less troublesome in sciences(e) than in dialectical procedures. The cause of this240 is that the middle term is employed twice. For it belongs to all of the minor term and the major term [belongs] to all of it. Now since it is taken twice and since many terms are homonymous, if it is taken in one meaning when predi-

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cated of the subject (I mean the minor term) and in another when the major term is predicated of it, there is a fallacious argument. This kind of thing happens frequently in dialectical conversations. For example: wise people learn; those who learn do not know(o), therefore the wise do not know(o). For learning is homonymous, being used both of understanding and of being taught. So the error escapes notice because of the homonymy. Man is an animal, animal (zôiön) has two syllables, therefore man has two syllables. For ‘man’ has two meanings, applying to both the substance and the word, since both the thing and the word are called man. Cabbages are above the earth, what is above is larger than what it is above, therefore cabbages are larger than the earth.241 For being above is either being above in respect of magnitude or being above the surface. Now since dialecticians are not concerned with certain definite things or definite middle terms, they have plenty of opportunities for error.242 But not so in the sciences(e). For the things in each science(e) are defined and error due to homonymy is never found in them.243 For example, as he says, if both epics and a geometrical figure are called circles,244 if anyone were to ask a geometer if a circle is a geometrical figure, he immediately fixes his imagination on a drawn circle, and he is not uncertain as to what kind of circle the discussion [is about] or in how many ways circle is said. For geometers see no other circle than the geometrical figure enclosed by one line. So since the geometer’s subject matter is defined and there is no homonymy in sciences(e), it is less possible to reason fallaciously in them. For the meaning of each term is defined, for example what is a circle, what is a line, what is a plane, what it is to intersect, and so for each of the rest. Geometrical figures are like things stamped in the soul. So as soon as [a geometer] hears ‘circle’ he immediately conceives the impression in his soul and is not carried off into uncertainty. For if he were asked, he will not grant that epics are a circle, since he knows(o) only what he has defined. But it is not like that in dialectical conversations since the subject matter of dialectic is not defined. So [dialectic] will reason fallaciously using homonymous words by shifting things said with one meaning [of a word] to another [meaning].

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77b29-30 [Something is said of all of it, and it in turn is said of all of something else] (but the predicate is not universally quantified). Since he said that the middle term [is predicated] of all the minor and the major term of all the middle term, he reminds us of what he said in De Interpretatione:245 the specification is not taken with the predicate (for we do not say ‘man is every animal’) but only with the subjects.

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Translation 77b30-1 It is possible to ‘see’ these with the intelligence 246

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‘These’ means the terms. ‘To “see”’ – clearly [he means] in mathematics, since his discussion is about that. In mathematics, he says, fallacious argument does not occur in the same way [as in dialectic] – because the middle term is used twice. For each term in mathematics is like a proof because it is defined, and as soon as he hears ‘circle’ or anything like that the expert(e) immediately sees the thing that has been spoken of inscribed in himself, and he is not led to think of any other meaning but only the thing whose definition he possesses in himself. 77b31  but it escapes notice in conversations.

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Namely, in dialectical conversations the words ‘escape notice’ because they are not defined. 77b31-4 Is every circle a figure? If he draws it, it’s clear. [What then? Are epics a circle? Obviously they are not.]

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If someone were to ask a geometer if ‘every circle is a figure’, he will clearly answer ‘yes’, drawing the circle – drawing it either in his imagination or on the board. If someone asks him further ‘What then? Are epics a circle?’ in order to infer ‘therefore epics are a figure’, he [the geometer] will no longer grant that epics are a circle, since they do not coincide with the definition of a circle which [geometers] have in their soul. He247 says that epics or epigrams are a circle when they are composed in such a way that the second verse does not necessarily succeed the first and the third the second, and so on, but in such a way that the same verse can be made both the beginning and end,248 as in the following. I am virgin bronze, I lie on Midas’ tomb As long as water flows and tall trees flourish and the rising sun shines and the moon is bright remaining here in this place on the much-lamented grave I will announce to passers-by that Midas is buried here.249 Notice that just as in a circle it is possible to begin from just about any verse one wishes:250 ‘remaining here in this place on the muchlamented grave’ and then ‘I am virgin bronze’ and so on; or ‘I am virgin bronze’ and then ‘remaining here in this place’ then ‘as long as water flows’ and so on. Herodotus in his life of Homer251 says that Homer was the author of the epigram to Midas, king of the Phrygians. Now he is either calling epigrams of this sort a circle or he is calling general education252 a circle. It is called this either because it

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somehow includes all subjects, or on the grounds that all things are within its embrace (for not every person deals with the other subjects such as medicine, rhetoric or others; but practically everyone, including people who are concerned with the other rational sciences(e), [deals] with this) or, as I think, because all the poets have embraced these same subjects. They253 will think that comedies are separate from general education. But I say that they are not remote from it, particularly ancient comedy, but its subjects are contained in them in many places. Besides, the former subjects will be general in the strict sense, while these too254 [will be general], if you take a particular as standing for the whole.255 Now concerning the [epic] circle some have written listing how many poets there have been, what each of them wrote, how many verses each poem contains, and their order and which ones a person needs to learn first, which second, and so on. But after Pisander composed a treatise of the same kind, collecting and arranging in order a great deal of information and also striving for beauty of language, they say that the writings of the poets before his time fell out of favor, and this is why the poems listed in the [epic] circles are not even found.256

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77b34-9 One must not bring an objection against it if its premise is inductive. [For just as there is no premise that is not true of more than one thing (for it is not true of all, whereas the deduction is formed from universal [premises]), clearly there is no objection [that is not true of more than one thing] either. For the same propositions are premises and objections. For the objection [someone] brings may become either a demonstrative premise or a dialectical one.] What he is saying here is that those who bring objections257 against such fallacious arguments as these258 (like the one in question that says, ‘epics are a circle, a circle is a geometrical figure, therefore epics are a geometrical figure’) should not bring them through induction,259 for example producing as evidence various epics and claiming that they are not circles. Rather, they must object generally that no epic is a circle. ‘For just as there is no’ demonstrative ‘premise’ that is not universal because ‘the deduction is formed from universal [premises]’,260 so there is no scientific(e) objection that is not universal, for the objection itself, he says, becomes part of the deduction.261 For if anyone objects to the conclusion, the objection becomes a starting point for a deduction. And [if anyone objects] to a premise, the objection becomes a conclusion.262 For example if anyone objects to the premise ‘epics are a circle’ saying ‘no epic is a circle’, the objection becomes the conclusion of a deduction, when we say as follows: ‘no epic is a geometrical figure, but every circle is a geometrical figure, therefore no epic is a circle’.263 But if against the conclusion stating

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that epics are a geometrical figure is brought the objection stating that no epic is a geometrical figure, the objection becomes a premise when we say, ‘no epic is a geometrical figure, every circle is a geometrical figure, no epic is a circle’. This is how the Philosopher264 explained the passage. But I think that this is not well stated (since an objection must always be the conclusion of a deduction, for this is clearly what we need to demonstrate in order to prove that its opposite is not right)265 and that [Ammonius’ explanation] does not accord with the passages in Aristotle. He [Aristotle] says, ‘The objection [someone] brings may become either a demonstrative premise or a dialectical one ’. This is why I am deferring the precise explanation of the passage until I meet with the interpreter.266 77b40-1 It happens that some people speak undeductively as a result of assuming things that follow both [terms] 267 After saying that fallacious argument occurs less in sciences(e) than in dialectical procedures because in dialectical procedures the middle term is homonymous while not so in sciences(e),268 he now wants to discuss deductions that are mistaken on account of their form.269 Fallacious arguments due to homonymy in the middle term will not be mistaken on account of their form because in these there is not even deduction in the strict sense and in truth, but rather two premises severed apart from one another, while it is because of homonymy that the middle term seems to be connected with the extremes and to be connected deductively. So after saying that because the middle term is homonymous it often happens in dialectical conversations that many fallacious arguments occur, which does not happen in the sciences(e), he now wants to display in addition the fallacious arguments in dialectical conversations that are due to their form, where again the sciences(e) are different from dialectical conversations. Fallacious arguments, he says, occur on account of their form as well, frequently as a result of [taking] the same middle term to be predicated of the two extremes, that is, by taking two affirmative [premises] in the second figure. For they suppose that if the same thing is predicated of them both, they are predicated of one another as well. But this is not so unless they belong to the same genus.270 Substance is predicated of both stone and man, but neither is predicated of the other, neither man of stone nor vice versa, since they do not belong to the same genus. This is what he says the sophist ‘Caeneus produced’ (77b41); he deduced ‘that fire’ grows ‘in a multiple proportion’ in this way: ‘fire grows most quickly, things that grow in a multiple proportion grow most quickly, therefore fire grows in a multiple proportion’.

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And yet, he says, sometimes in such cases it is possible to make an argument or deduction that is valid, but because they do not notice it, they combine the premises invalidly. For when the terms are coextensive so that it is possible to convert the predicate with the subject, it is possible to make the form valid. For example, receptive of scientific knowledge(e) [is predicated] of every man, and of everything that is capable of laughing, therefore man [is predicated] of everything that is capable of laughing. Now if you convert man and receptive of scientific knowledge(e), saying that man [is predicated] of everything that is receptive of intelligence and scientific knowledge(e), you will make the form valid, as in the present case. For if instead of saying ‘that which grows in a multiple proportion grows most quickly’ we say that the thing that grows most quickly grows in a multiple proportion, we make the figure valid and draw the same conclusion. I don’t know(o) what went wrong with Alexander (probably because he did not have much of an aptitude for mathematics), but he declares with respect to ‘things that grow in a multiple proportion’ that he [Aristotle] means that things that grow in a multiple ratio grow most quickly, as if we were to say that 200 is to 300 as 2 is to 3. He did not know that this is not a multiple ratio but a superparticular ratio. For a ratio that is the same whether you take it twice or many times is called multiple, for example 4 is to 8 as 2 is to 4 and 400 is to 800 as 200 is to 400. Likewise 200 is to 800 as 2 is to 8. But a superparticular [ratio] is one that contains the whole and a part of it, for example 3 is superparticular to 2, for it contains the thing [2] and its half [1].271 Likewise 5 is superparticular to 3 since it contains the thing itself [3] and its two parts. And double, triple, and so on are multiple [ratios], while the ratio of three to two, for example, or four to three or five to four, etc. are superparticular ones. These are the numbers mentioned by Alexander, 3 has a three-halves ratio to 2, as 300 has to 200.272 Moreover he [Aristotle] did not take the terms as growing continuously and the increase as happening most quickly in this way. For it is not possible for superparticular numbers to grow continuously most quickly. So perhaps he thought that the multiple proportion is the one said to have the same ratio. Thus Alexander. According to the Philosopher,273 Proclus explained the passage as follows. He said that several numbers must be set out in succession, exceeding one another by the same double ratio; for example 1, 2, 4, 8, 16, 32, and always [taking] the doubles in succession. And we must say that as 1 is to 2 so also 2 is to 4 and 4 to 8 and 8 to 16 and 16 to 32 and so on in succession. For this kind of proportion with multiple ratios grows most quickly;274 but this is not so for the superparticular [ratios]. For it is not possible to find successive numbers such that as 2 is to 3 so 3 is to some other [number] and that [number] to another, but here we need a procedure – how it is possible using a single

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procedure to find, for example, numbers in the ratio of four to three, or in the ratio of three to two which Nicomachus presents.275 161,1

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77b41-78a1  for example, the one that Caeneus produces, that fire [grows] in a multiple proportion. I said that this Caeneus was the sophist276 who said that ‘fire’ is generated or grows ‘in a multiple proportion’. He established this [result] in the second figure from two affirmative [premises]. 78a1-2 For fire is generated quickly, [as he says] 277 This is the minor premise. 78a2  and so is this proportion. Clearly this multiple [proportion] ‘is generated quickly’ (78a2).

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78a2-3 (161,10) But in this way278 there is no deduction. For the two [premises] are affirmative in the second figure. 78a3-4 But [there is a deduction] if multiple279 proportion280 is predicated of the quickest [proportion]281

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That is, if we convert the terms in the major premise and say that ‘multiple proportion is predicated of’ ‘the quickest’ growth or generation. Then there turn out to be two affirmative [premises] in the first figure. So we must place a comma after ‘quickest’.282 78a4-5  and the proportion that is quickest in movement is an attribute of fire.

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This is because fire moves quickest,283 which is the minor premise; but what moves quickest moves in the quickest proportion; therefore fire moves in the quickest proportion. But it makes no difference to me if this is said about the generation of fire or about its growth or movement. 78a5-6 Now sometimes it is not possible to form a deduction from what has been assumed, and sometimes it is, but it is not noticed. We said that sometimes it is possible to transform two affirmative [premises] in the second figure into a deductive form – when the terms are coextensive, so that it is possible to convert the major

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premise in order to reduce it to the first figure.284 But when the terms are not coextensive, it is not [possible to do so]. ‘But it is not noticed’ means that even if in some such cases it is possible to infer the conclusion necessarily through the premises, as I said [holds] for things that are coextensive by means of conversion of the major premise, still the necessity ‘is not noticed’ because of the invalid combination of the premises. The cause of this is not the combination but the fact that the terms are coextensive so that they can also be combined validly.

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78a6-7 If it were impossible to prove something true from things that are false,285 it would be easy to perform analysis. He has moved on to another topic in which he again distinguishes demonstrative deductions from dialectical ones, this time on the ground that analysis is easier in demonstrations than in dialectical [deductions].286 Geometers call analysis the discovery of premises through which a true conclusion is inferred. For example if a true conclusion is proposed to us – that this triangle is equilateral – they call the procedure by which we discover the premises through which this is inferred analysis.287 If it is now proposed [to discover] how an equilateral triangle may be constructed, we will discover the conclusion through premises, beginning from some things that are granted and ending up at the sought, and this is called synthesis.288 Analysis is the opposite of synthesis: taking what was previously the sought as granted – that this is an equilateral triangle, we investigate what would be the premises through which this is established, in order that we may analyze from that and discover others, until we arrive at some things that are granted and are principles of geometry.289 And this is a mark of the highest state290 – being able in this way to comprehend from which premises each conclusion is inferred. He says that analysis would be easy if true conclusions were inferred from true premises alone, since in this way the premises that can imply the conclusion are definite. But as it is, their discovery is difficult because a true conclusion can also be inferred from false [premises]291 and these are indeterminate.292 This is another way in which demonstrative deductions are different from dialectical ones. Now precisely because they are based on [premises] that are reputable and on accidental attributes of the subject, dialecticians have an indeterminate number of premises that yield the same conclusion.293 For it can be inferred that man is an animal from the fact that he moves or walks or engages in conversation or from countless other [attributes]. And as I said, a true conclusion can be inferred from false premises as well. So in these cases if anyone wanted to discover all the premises that can imply the conclusion, he will not easily succeed because they are indeterminate. But it is not like this in the

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case of the sciences(e); scientific(e) conclusions are not inferred from false premises or from attributes that belong accidentally to their subjects, but from those that [belong] primarily and per se, and these are definite. And analysis is easier with things that are definite than with things that are indeterminate. This is why analysis is easier in the sciences(e) than in discussions. 78a8-10 For it would necessarily convert. [For let A be something that is the case, and if this [A] is the case, these things are the case and I know that they are the case, for example B. Therefore from these I will prove that [A] is the case.]

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If true [conclusions] were inferred only from true [premises], analysis would be easy because the premises would convert with the conclusion. If the conclusion is A and the premises that imply this – B – are true or indeed per se (he takes B for the two premises), then if B is the case A is necessarily the case and vice versa if A is the case then B is necessarily the case. And so the discovery of the premises is easy because they are necessary consequences of the conclusion. 78a10-13 Conversion occurs especially in mathematics, because they assume nothing that is accidental (in this too they are different from the [deductions] in dialectical conversations), but [they assume] definitions.

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He says that conclusions convert with the premises and premises with the conclusions ‘especially in’ mathematical deductions. This is in fact why it is easy to discover mathematical premises and the middle term through which the conclusion is implied proximately.294 ‘But definitions’ must be taken with ‘they assume’ so that the connection is this: ‘conversion occurs especially in mathematics because they assume nothing that is accidental’, ‘but [they assume] definitions’. And then through the middle term ‘in this too they’ (clearly, the sciences(e))295 ‘are different from the [deductions] in dialectical conversations’ because dialectical [premises] assume many things that are accidental, while sciences(e) [assume] the definitions and per se attributes of things – and these are definite.296 78a14-15 It increases not through middle terms but by adding, [for example A [is predicated] of B and B of C, and again C of D] 297 He moves on to another point by which he shows once more that sciences(e) are different from [dialectical] discussions. This is consistent with the preceding points.298 Analysis is easy in the sciences(e) for this reason too, since synthesis too is simpler and so is the increase

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of arguments. For in sciences(e) when the deductions are being increased, each time we want to adjoin conclusions to conclusions, we always add terms outside and never in between. For example, if C has been proved through A and B,299 the addition of terms always takes place outside. D is placed after C and never between A and B or between B and C. For the first theorem is proved through the definitions and axioms, the second through the first, and the third through the second,300 and this is how the increase takes place in sequence. A middle term is never inserted but each time is adjoined to the first ones. The reason for this is that demonstrations are based on attributes that belong per se and primarily, and it is not possible for anything to be demonstrated through a non-immediate premise. For the theorems in sciences(e) are proved on the basis either of the axioms or of things previously demonstrated, which themselves have been proved on the basis of the axioms. But it is not so in dialectical conversations; deductions are increased in two ways: dialectical deductions are increased sometimes when middle terms are inserted and sometimes when terms are added outside. This is because dialectical premises are not immediate but are reputable301 or depend on accidents. For example, if I say ‘man is receptive of intelligence and science(e), what is receptive of intelligence and science(e) is an animal, therefore man is an animal’, this deduction cannot be increased by inserting terms in between. But if I say ‘man talks, what talks is an animal, therefore man is an animal’, I can increase this deduction through more than one middle term because the premises are not immediate.302 For example: man talks, what talks moves, what moves is an animal, therefore man is an animal. Again, from where [do we get] that what moves is an animal? Because what moves but not because of something else moves itself, what moves itself is an animal, therefore what moves303 is an animal. And in this way it is always304 possible to increase deductions by inserting terms in between. That [it is possible to increase deductions by adding terms] outside as well is evident. And so demonstrative deductions are different from dialectical ones in this way too – in that the former insert no term in between but [add them] only outside, while dialectical ones [add them] both in between and outside.

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78a16  and this305 too ad infinitum. Because in sciences(e) it is always possible to adjoin additional [theorems] outside to theorems that have been discovered, thus discovering different [theorems that result] from the ones already discovered. ‘This too ad infinitum’ – clearly in relation to us, because we have not understood the sciences(e) as being such that no further theorem can be discovered, but [in fact we suppose] that it is always possible to

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discover new ones through what has been previously demonstrated.306 78a16-17 Also slantwise, as A [is predicated] of both C and E. 15

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[He says] that even if the increase of deductions occurs in both directions, nevertheless in sciences(e) terms are never inserted in between. This is how it occurs in both directions, according to the examples he gives. He takes A to be number, with B and D on either side of it, and he calls B odd number and D even [number]. Likewise under B [he takes] C, a definite odd number, with B being indeterminate odd [number], and similarly under D [he takes] E, which he calls a definite even number, with D being indeterminate even [number]. Now if A, number – I mean [number] without qualification, whether infinite or finite (for it makes no difference to the present point) – belongs both to B and to D, namely to odd and even, and B, odd number, belongs to C, some definite odd number, and likewise also D, indeterminate even [number ] belongs to E, some definite even [number], in this way the increase has occurred in both directions [Diagram 5].

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‘Also slantwise, as A [is predicated] of both C and E’. He should have said ‘of D’307 but he says ‘of E’. In any case in what follows intending to expound [his interpretation] precisely, he predicates A of D and this of E and so A of E. 78a17-18 For example, A is definite number (or both definite and infinite [number]) 308

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78a19-20  and C is an odd number. [Therefore A belongs to C.] A definite odd [number], for example 3, 5, and similar ones. 78a20 Also D is even311 definite number 312

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the ‘that’ and [knowing] the ‘why’ are differ-

The purpose proposed in these words is to show how deductions that establish the ‘that’ are different from those that establish the ‘why’. He says that one way they differ is when the [deductions] that establish the ‘that’ belong to one science(e) while those [that establish] the ‘why’, that is, the cause of the thing, belong to a different one. How this is the case he states in what follows.314 [They differ] in another way too in that it is also possible in the same science(e) to discover both the [deduction] that establishes the ‘that’ and the one [that establishes] the ‘why’. These – namely the [cases that hold] in the same science(e) – differ in two ways. One way315 is that [deductions] that establish the ‘why’ reach their conclusion through immediate premises (since the cause of the thing ought to belong per se and proximately), while [deductions] through non-immediate premises are deductions of the ‘that’, not the ‘why’. For example, southern stars set sooner316 than northern ones. If we say as follows: ‘southern stars are farther from the north pole, what is farther from the north pole sets sooner than the northern ones, therefore southern stars set sooner than northern ones’, this deduction establishes a true [conclusion] from true [premises]. For the farther stars are from the north pole the sooner they set. This is why Aratus says: ‘But Taurus is always ahead of Auriga in setting in the west, even though it has risen at the same time’.317 However, the cause of their setting quickly is not that they are far from the north pole. This is why this deduction is of the ‘that’ and not of the ‘why’. But if I say that southern ones have a smaller arc above the earth than below the earth, things that have a smaller arc above

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the earth set sooner, therefore southern ones set sooner, this will be the principal cause of their setting sooner and this is why it is a deduction of the ‘why’. Again, the [deduction] that demonstrates that the soul is immortal because it is a self-mover does not demonstrate it through immediates. However, the one [that demonstrates this conclusion] on the basis of its being a principle of life [proceeds] through immediates. How did we say earlier318 that it is possible to prove the ‘that’ through both immediates and non-immediates, and likewise the ‘why’ as well? We there called deductions of the ‘why’ simply the ones that establish why a particular thing occurs, even if the cause given is not proximate but non-immediate. We said that the fact that the moon is eclipsed because it is diametrically opposite the sun is a deduction of the ‘why’, because he says that the cause of an eclipse is its diametrically opposite distance from its source of illumination. However, this cause is not proximate but is non-immediate. The proximate [cause] is the occultation by the earth. On the other hand, the [deduction] that states not why the moon is eclipsed but simply that it is eclipsed is [a deduction] of the ‘that’, even if its premises are self-guaranteeing. For example, the moon does not cast a shadow at the time of full moon; if it does not cast a shadow at the time of full moon, it is eclipsed; therefore the moon is eclipsed. Notice that the premises in this are immediate, for they require no [further] middle term to prove that it is eclipsed. But even so the deduction is not of the ‘why’ since it does not contain the cause of the eclipse. Accordingly he here says that every deduction that does not take the proximate cause of a thing is [a deduction] of the ‘that’. And so he calls the one that establishes that the moon is eclipsed through its being diametrically opposite to the sun [a deduction] of the ‘that’. And reasonably so, since the cause is always placed in the middle term and the middle term is the one that is the cause of the conclusion. So when the middle term does not contain the most proximate cause, we cannot strictly say that such [a deduction] is a deduction of the ‘why’, since it does not contain the cause. If I say that southern stars are a greater distance from the north pole than northern ones are, and those that are a greater distance from the north pole set sooner, therefore southern ones set sooner, this will not be a deduction of the cause, since this is not the cause of their setting sooner; it is a consequence of it [the cause], but it is not the cause of their setting sooner. And so we have deduced from this only the ‘that’ – [namely,] that if the one is the case, the other is a consequence, but not that the one is the cause of the other. And even if the premises are self-guaranteeing and need no middle term to be convincing, if they do not contain the cause, the deduction based on them will be of the ‘that’ all the same, as is the case for the person who establishes the eclipse of the moon from its being the time of full moon and [the moon’s] not

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casting a shadow. And so deductions of the ‘why’ necessarily [proceed] through immediates. Still, those that [proceed] through immediates are not necessarily [deductions] of the ‘why’. Now this is one kind of difference between the ‘that’ and the ‘why’ in the same science(e). The other319 is when both [deductions] proceed through immediate premises – both the deduction of the ‘that’ and the one of the ‘why’ – but the [deduction] of the ‘why’ deduces the effect from the cause while the [deduction] of the ‘that’ [deduces] the cause from the effect. Proofs that are called [proofs] from signs are of this kind.320 I must say this repeating from above. Some causes and effects reciprocate and others do not. For example, if there is fire there must be ash as well, and if there is ash, there must be fire as well. Also in the case of the phases of the moon; if it is illuminated in the way it appears, it must be spherical too, and if it is spherical it must be illuminated in that way. Now in cases where cause and effect reciprocate with one another, we frequently establish the cause on the basis of the effect because the effect is better known(g) than the cause. For example, in proving that the moon is spherical on the basis of its phases – although the phases are not the cause of its being spherical, but rather that fact [that it is spherical] is [the cause] of them [the phases]. So this is called a deduction of the ‘that’, since the sphericity of the moon is deduced from its phases. Likewise ‘ash is here, where there is ash there was fire, therefore there was fire here’. But if we were to say ‘fire is here, where there is fire there must be ash as well, therefore there is ash here’, the deduction is of the ‘why’. For the effect is deduced from the cause. The former is [a deduction] of the ‘that’, since the cause is deduced from the effect. And since such things are irrefutable, they are called signs and this is a secondary kind of demonstration, or as a whole it is a demonstration from a sign. This holds for things that reciprocate. But it often happens that when the cause occurs it is necessary for the effect to occur too, but not that when the effect occurs the cause must occur as well, and vice versa, that when the effect occurs the cause must too, but not [that when] the cause [occurs] the effect must too.321 An example of the former: if a women has given birth she must be pale, but the fact of having given birth is not always a consequence of being pale, for there can be several causes of the same thing: fear, illness and other things. Likewise if someone has just walked a lot, he is tired, but it is not the case that if someone is tired he has also just walked a lot. For it is possible to be tired from doing a lot of work. And clearly in these cases the deduction will be of the ‘why’ and not just of the ‘that’, since the deduction always proceeds through the causes establishing the effect on the basis of them. But it is not possible to prove the cause from the effect since they do not reciprocate. And so in these cases the deduction is of the ‘that’ and not at all of the ‘why’. An example of the latter, namely where the cause is a consequence of the effect but the effect is not necessarily [a con-

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sequence] of the cause as well: if a woman has given birth, she has had intercourse with a man. But this cannot reciprocate – that if a woman has had intercourse with a man she has also given birth. Also, if there are fruits, there must have been rain, but if there has been rain there will not necessarily be fruits as well. In these cases the deduction is only of the ‘that’ and never of the ‘why’. 78a22-3  first, [they are different] in the same science, in two ways. After saying ‘first in the same science(e)’, he does not go on to say that [this is found] in different ones as well, but after first saying how [they are different] in the same science(e) – in two ways – and after going through the two ways he then says after much [discussion] how they are different in different [sciences] as well.322 78a23-6 First, if the deduction does not take place through immediates [(since the primary cause is not grasped, but knowledge(e) of the ‘why’ occurs in virtue of the primary cause).]

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Namely, if the premises are not immediate. For in that case the deduction will be of the ‘that’. But the deduction of the ‘why’ wants to have the primary cause in the middle term. 78a26-30 Second, [if the deduction takes place] through immediates,323 but is not through the cause, but through the better known(g) of reciprocating terms. [For nothing prevents what is not the cause [of the other] from sometimes being the better known(g) of the reciprocating terms, so that the demonstration proceeds through this.]

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The second way in which the ‘that’ is different from the ‘why’ is where both premises are immediate but where cause and effect reciprocate. When the effect is better known(g) than the cause and someone deduces from the better known(g), that is when he produces a deduction of the ‘that’, just as when [we deduce] the effect from the cause [he produces a deduction] of the ‘why’. It is clear from the examples that in these cases the premises of the ‘that’ are immediate too. That if there is ash there must be fire too is immediate, since we need no other middle term to be convinced. 78a30-3 For example, [the deduction] that the planets are near through their not twinkling. [Let C be the planets, B not twinkling, A being near. It is true to say B of C, since the planets do not twinkle.]

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He first gives examples of the second way in which there are deductions of the ‘that’ and the ‘why’ in the same science(e). If we say thus: ‘the planets do not twinkle, things that are non-twinkling are near, therefore the planets are near’, this is a deduction of the ‘that’ since the middle term is not the cause of the thing. Non-twinkling is not the cause but rather the effect of being near, since it is because they are near that they do not twinkle. Therefore the proof is from a sign and the deduction is of the ‘that’. It is a first figure deduction with two affirmative [premises]. The middle term, namely non-twinkling, is indefinite, the minor premise is CB and the major premise is BA.

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78a33-4 And [it is] also true [to say] A of B, [since what does not twinkle is near.] First he makes his enumeration of terms beginning from the minor term, C,324 and afterwards325 he begins again from the major term, A, perhaps making manifest that which is ‘as in a whole’ and ‘in every case’ as being the same, since it is the same, differing only in relation.326

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78a34-8 Let this be grasped through induction and perception. [Therefore it is necessary for A to belong to C, and so it is demonstrated that the planets are near. Now this deduction is not of the ‘why’ but of the ‘that’, since it is not because they do not twinkle that they are near, but because they are near that they do not twinkle.] That things that are near do not twinkle, he says, ‘let it be grasped on the basis of ‘induction’ and ‘perception’. For ‘ê’ here is not disjunctive327 but is used for ‘and’. For example if you place silver nearby you will not see it twinkling, but if you put it at a distance, you will see it twinkling. Likewise lamplight seen from a distance seems to twinkle and almost to sparkle, but if it is near it manifests nothing of the sort. This is why the fixed stars seem to twinkle and the planets do not. And the more remote planets twinkle more. And indeed Venus, although it appears largest to us, does not twinkle because it is nearest. The cause of this is that the visual rays become weaker [the farther] they go forth,328 and this is why, when they are affected more by the glittering of the light, they distort the image. But when the bright bodies are nearer, the visual rays are stronger when they strike them and are less affected by them. This is why people with weaker visual rays think that bright bodies, such as a lamplight or things like that, twinkle even when close by. But if the visual rays are very weak, they cannot even look at lamplight when a lot of glittering is striking them. And so things that are near do not twinkle.329

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Translation 78a39-b4 The latter can also be proved through the former, and the demonstration will be of the ‘why’. [Let C be the planets, B being near, A not twinkling. Then B belongs to C and A to B, and so A [belongs to] C. And the deduction is of the ‘why’. For the first cause has been taken.]

Before this he proved that they are near through the fact that they do not twinkle, the middle term being that they do not twinkle, and the major term that they are near. If we convert their order and put the major term (that they are near) as the middle term and the middle term (that they do not twinkle) as the major term, we make a deduction of the ‘why’. For what is strictly the cause comes to be in the middle term. For example, the planets are near, what is near does not twinkle, therefore the planets do not twinkle. This [is a deduction] of the ‘why’ because it proves the effect from the cause – namely [it proves] that they do not twinkle from their being near. 78b4-11 Again, [consider] how they prove that the moon is spherical [through its waxing. For if what waxes this way is spherical and the moon waxes [this way], it is obvious that it is spherical. Now in this way the deduction has become [a deduction] of the ‘that’. But if the middle is reversed [it becomes a deduction] of the ‘why’. For it is not because of its waxings that it is spherical, but it has waxings of that kind because it is spherical. [Let] the moon [be] C, spherical B, and waxing A.] Another example of the same thing. We have said well enough330 that if we deduce the phases of the moon from its shape the deduction will be of the ‘why’, inferring the effect from the cause. And if it is backwards,331 [it will be a deduction] of the ‘that’.

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78b11-13 But in cases where the middles do not reciprocate and the one that is not the cause is better known(g), [the ‘that’ is proved and the ‘why’ is not.] After discussing causes and effects that reciprocate, he also talks about ones that do not reciprocate. In cases where, he says, the middle term does not reciprocate with the major term, that is, the cause with the effect, when the effect is ‘better known(g)’ – (He calls the effect ‘not the cause’; this is better known(g) because if the effect occurs the cause must occur, as in the deduction that says ‘a woman has given birth, one who has given birth has had intercourse with a man’; for it is better known(g) that she has given birth; it follows from this that she has had intercourse, but this [that she has given birth] does not follow from her having had intercourse. Now this is a deduction of the ‘that’ but not of the ‘why’, because it infers the cause

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(having had intercourse with a man) from the effect (having given birth) since the cause must follow the effect; for if she in fact gave birth, she must have had intercourse with a man too) – in such cases it is not possible to deduce the ‘why’ because the result does not always follow the cause. He did not mention the other way, namely the ‘that’, where if the cause occurs the effect does too, but when the effect occurs the cause does not, because having given birth is not strictly and unqualifiedly the cause of pallor, but is so accidentally, since not every instance of pallor is the result of having given birth.332 But if we take a particular kind of pallor as occurring as a result not of giving birth but of something else, we do not do well to say in this case that when the effect occurs the cause does not. For this kind of pallor is not the effect of having given birth, but of something else. In general it is impossible for any effect to occur if the cause does not. For if everything that occurs does so as the result of a cause, it is entirely necessary for the cause to occur if the effect does. The cause of the error is homonymy and the fact that the same kind or affection can be due to more than one cause.333 This is why he only takes the possibility that when the cause occurs the effect does not always occur, but not the opposite case, where the effect occurs but the cause does not. But if cause and effect are relatives, how do we say that they do not reciprocate334 and that it is possible for the effect not to occur when the cause does? Clearly [this is the case] if we say that the cause [exists] potentially and not actually and not as a cause but as a thing. For example, a house-builder exists when a house does not, since he has this particular state.335 But if he is taken as the cause, the effect must always exist too. For these are relatives and as the one is so is the other: if the cause is potentially the effect is potentially too, and if the cause is actually, the effect must be actually too. 78b13-23 Also [it is similar] in cases where the middle is placed outside. For in these cases too the demonstration is of the ‘that’ and not of the ‘why’, [for the cause is not mentioned. For example, why does a wall not breathe? Because it is not an animal. If this were the cause of not breathing, being an animal must be the cause of breathing just as if the negation is the cause of not belonging, the assertion [must be the cause] of belonging – for example, if [the cause] of not being healthy is that the hot and the cold are out of proportion, their being in proportion [must be the cause] of being healthy. Likewise too, if the assertion [is the cause] of belonging, the negation [must be the cause] of not belonging. But for things exhibited in this way, the stated condition does not obtain. For not every animal breathes.]

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After saying that the deduction of the ‘that’ and the ‘why’ occur in the same science(e) in two ways – one where the [deduction] of the ‘why’ proceeds only through immediate premises while the [deduction] of the ‘that’ [proceeds] through non-immediate ones as well, and another, where the [deduction] of the ‘that’ deduces the cause from the effect while the [deduction] of the ‘why’ [deduces] the effect from the cause – he gave examples of the second way that have to do with the stars and the moon, but he gave no example of the first. This is why he now repeats his account of the first way, in order to confirm it too through an example. For he does not repeat this as if he were setting out another manner of difference between the ‘that’ and the ‘why’ in the same science(e) (for it is not possible to find another one) but, as I said, in order to give an example of it as well. What does ‘in cases where the middle is placed outside’ [mean]? To begin with, Alexander says that by this he signifies the second figure. And in fact the examples he gives are in the second figure. This is why someone might suppose that this is what the middle term being placed outside signifies for him [Aristotle]. But the middle term being placed outside does not signify this for him but, as Alexander says further down, he says that the middle term is placed outside instead of [saying that it is placed] further from the proximate cause. [He says] this to avoid the middle term containing the most principal and proximate cause and the premises being non-immediate. For such was the first manner of difference between the ‘that’ and the ‘why’. For he said that the deduction of the ‘that’ does not have premises that are immediate, but ones that are non-immediate.336 This is also clear from what Aristotle says. For the deductions he gives in the middle figure, namely the second [figure], he gives in what follows, where he says ‘the deduction of this kind of cause takes place in the middle figure’337 as if he had said nothing about it. But that the middle term’s being placed at a distance signifies nothing else for him than that it is remote from the proximate cause he indicates in what he says again a little below: ‘these kinds of causes resemble things said extravagantly; that is to say that the middle term is more remote’,338 namely, remote from the cause. So much for this. He gives the following example of a non-immediate premise: if we want, he says, to prove that ‘a wall does not breathe’ through the fact that it is not an animal (clearly, animal being the middle term) the deduction will be of the ‘that’ since it has not taken the proximate cause of the thing. For it is not because of this – that it is not an animal – that it does not breathe. For if this, he says, – its not being an animal – were the cause of the wall’s not breathing, then it would be necessary that every animal breathe. But as it is, many animals do not breathe – for example, all the insects and fishes.339 For it is true that anything that is not an animal does not breathe, and the consequent follows the antecedent. But this is

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not the cause of not breathing. Instead, a person who deduces that the wall does not breathe through its not having a lung is stating the deduction of the ‘why’. For it [his deduction] has the cause of the thing in its middle term. But the person [who deduces that the wall does not breathe] through its not being an animal [is stating a deduction] of the ‘that’ because the middle term is remote from the cause. At this point someone might reasonably be puzzled about what Aristotle means when he declares ‘for if’ not being an animal is ‘the cause of not breathing, being an animal must be the cause of breathing’. For in making a conversion from the antecedent he seems to be confusing conversion with opposition. If, he says, not-animal is the cause of not breathing, animal is the cause of breathing. For not-animal is the antecedent and not breathing is the consequent. What do we say to this? That if he proceeded to make the conversion simply as applying to the premises, saying that if what is a non-animal does not breathe then it is necessary for every animal to breathe, he really did confuse conversion with opposition. But as it is, that is not how he stated it. For he did not set out the argument simply as applying to the premises, but as applying to the cause and the effect. This is why he did not say ‘if not belonging follows the negation, belonging will follow the affirmation’ – clearly, breathing – but ‘if the negation is the cause of not belonging, the affirmation [must be the cause] of belonging’, in order, as I said, to take the cause and the effect. In these cases, as I said, it is true to make the conversion from the antecedent. For example if women’s having intercourse with men is the cause of their giving birth, not having intercourse with men is the cause of their not giving birth. And if not sailing is the cause of not being shipwrecked, sailing will be the cause of being shipwrecked. And if not raining is the cause of there not being fruit, raining will be the case of there being [fruit]. And likewise for all cases. And so he did not do badly in making the conversion from the antecedent, since he was taking cause and effect. ‘Also in cases where the middle term is placed outside. For in these cases too the demonstration is of the “that” and not of the “why”’. It is clear from these very words that by ‘placed outside’ he does not mean being in the second figure. In these cases, he says, ‘the demonstration is’ only ‘of the “that” and not of the “why’’’. And so he would be saying that deductions of the ‘why’ never occur in the second figure. But this is false. For notice – I say this: in the region beneath the celestial equator340 the orbit on which the sun moves has a semicircle above the earth equal to that beneath the earth;341 in regions where there are longer and shorter days, the part of the solar orbit above the earth is not equal to the [part] beneath the earth; therefore in the region beneath the celestial equator there are not longer and shorter days. This is a deduction of the ‘why’ since the proximate cause is in the middle term. And as he himself says, ‘the

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[argument] of Anacharsis’ (78b29-31), ‘among the Scythians there are no vines, where there is drunkenness there are vines, therefore among the Scythians there is no drunkenness’– it is clear that this deduction too is of the ‘why’; for the fact that there are no vines is the proximate cause of there being no drunkenness,342 and it is a second figure. And the same thing occurs in many other cases. 78b23-7 The deduction of this kind of cause takes place in the middle figure. [For example, let A be animal, B breathing, C wall. Now A belongs to all B (for every breathing thing is an animal), and to no C, and so neither does B belong to any C. Therefore a wall does not breathe.]

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Namely, where the middle term is placed outside and the conclusion is negative.343 For it is clear that if it344 is affirmative it is proved in the first figure, and if it is negative [it is proved] in the second, as in the claims in question: no wall is an animal, everything that breathes is an animal, therefore no wall breathes; there are no vines among the Scythians, where there are flute-girls there are vines,345 therefore there are no flute-girls among the Scythians. It is possible to make a deduction in the first figure as well, converting the negative premise and the conclusion. For if no wall is an animal, also no animal is a wall. Moreover, everything that breathes is an animal, and what is inferred is not ‘no wall breathes’ but ‘nothing that breathes is a wall’. And then when the conclusion is converted it becomes ‘no wall breathes’. But since it was inferred more simply in the second figure, this is why he puts it the way he does. We must investigate the reason why in the present case a deduction does not occur in the first figure. For if it is possible to infer a negative conclusion in the first figure as well as when I say ‘every man is an animal, no animal is a stone, therefore no man is a stone’, then why is it not so in the present case as well? I declare that when I say ‘every man is an animal, no animal is a stone, therefore no man is a stone’, I did not take the middle term as the cause of the fact that the major term is not predicated of the minor term. For just because man is an animal, that is not why he is not a stone. But in the case of causes it is necessary to deny the middle term of the minor term and for it to be the cause of the fact that the major term too is denied of the minor term through itself. For as in affirmative deductions the major term is predicated of the minor term on account of the fact that the middle term is too, and it is through the middle term that it too is predicated of the minor term, clearly this is how it must happen in negative [deductions] too, if indeed the middle term must be the cause of the major term’s not belonging to the minor term, and the fact that the middle term is denied of the minor term is also [the cause of the fact] that the major term is denied of it [the minor]

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through itself.346 Now since the middle term is going to be denied347 of the minor term and since it is not possible in the first figure for the minor premise to be negative, that is why it is not possible for there to be a negative deduction of the cause in the first figure unless, as I said, we make two conversions – of both the negative premise and the conclusion. But as I said, when I say ‘every man is an animal, no animal is a stone, therefore no man is a stone’, the cause of the conclusion is not the middle term but the very nature of the terms. For the fact that no animal is a stone is as if one were to say that it is not a stick or anything else but an animal. But saying ‘every man is an animal’ is not the cause of the fact that a man is not a stone, since [a man] could fail to be a stone even if it were not an animal. However, when I say ‘no man is soulless, every stone is soulless, therefore no man is a stone’, the fact that man is not soulless is the cause of his not being a stone. For since soulless is a genus of stone, and man does not share in the genus, this is why [man] will not share in any of the species of the genus either – which include stone.348 Therefore only in the second figure is there a deduction of this kind of cause.

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78b28-34 These kinds of causes resemble things said extravagantly; this is349 to state something more remote as the middle term, [like the argument of Anacharsis that there are no flute girls among the Scythians because there are no vines there either. These are the differences between the deduction of the ‘that’ and the deduction of the ‘why’ in the same science and in relation to the position of the middle terms.] The Philosopher350 says that ‘things said extravagantly’ is said by Aristotle for ‘things that are very remote’ and that what follows explains it. For, he says, ‘I say that this “extravagantly” means the middle term is remote from the proximate cause, that is, that the middle term is a non-proximate but more distant cause’. But the passage does not seem to me to signify this. For at the beginning of his account of this topic he said this very thing: ‘in cases where’ ‘the middle is placed outside’ (78b13), namely where they have a position distant from the proximate cause. So in saying ‘distant causes are like causes from a distance’ the lack of logical sequence is very clear. But as Themistius too says, this is equivalent to ‘things taken from what is excessive and excessively’.351 For to prove that the wall does not breathe it suffices to say that it does not have a lung, and the fact that it is not an animal is taken extravagantly and excessively. Similarly that southern [stars] set sooner because they have a smaller arc above the earth suffices for a demonstration. But [arguing] from the fact that they are farther from the north pole is [arguing] from the more [remote cause]. Then, when he [Aristotle]

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explains what ‘extravagantly’ [means] he goes on to say that it is extravagance ‘to state’ the account in such a way that the middle term is ‘more remote’, namely, more distant than what is strictly the cause. For if the deduction is through the middle term and this [term] is the cause of the conclusion, a person who proves [the conclusion] from such terms proves it from what is excessive. For if the conclusion is inferred through a middle term that is distantly removed from the cause, it will be proved much more through the cause itself. Anacharsis the Scythian, he says, gave this kind of cause of the fact that there are no flute-girls among the Scythians. Asked if there are no flute-girls among the Scythians he answered ‘not at all, since there are no vines either’. He too gave [a cause] that is distant. For the proximate cause of there being no flute-girls is that the Thracians do not get drunk, and [the proximate cause] of their not getting drunk is that there are no vines.352 This is mentioned in ancient records. 78b34-7 But the ‘why’ and the ‘that’ differ in another way – considering353 each of them in a different science(e). [These are the ones that are related to one another in such a way that one is subordinate to the other, as optics is to geometry]354 After first saying that having scientific knowledge(e) of the ‘that’ and [having scientific knowledge(e) of] the ‘why’ are different in the same science(e), he here makes the coordinate claim, saying ‘in another way considering each of them in a different science(e)’. Now all sciences(e) that are subalternate one to another are related in this way; the deductions of the subordinate science(e) are of the ‘that’ while those of the superior [science(e)] are of the ‘why’. For example the ‘optician’ says why things seen from a distance appear smaller and those [seen] from nearer [appear] larger, and he gives the cause of this, saying that things that are seen under a larger angle appear larger, while the same things seen from nearer are seen under a larger angle than when they were distant [Diagram 6]. For example, let CD be the

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magnitude that is seen, let the eye be at point A, and let AC and AD be the rays proceeding from the eye and striking the thing seen, along

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which our apprehension [of the thing] occurs. There is produced a triangle with the seen magnitude CD its base, and the rays from the eye AD and AC its sides, and the eye itself, A, its apex. Now if the eye moves and comes to be nearer to the thing seen, for example at point B, other straight lines, BD and BC, strike the thing seen, producing another triangle, BCD whose apex is still B, whose base is the seen magnitude CD and whose sides are BC and BD, the rays from the eye that strike the thing seen. Therefore there are two triangles with the same base and different apexes and sides. The ‘optician’ says that things seen from nearer are seen under a larger angle. But it is no longer the ‘optician’ but the geometer who proves why angle CBD is larger than CAD: because if on one of the sides of a triangle two straight lines are constructed inside [the triangle] from the ends [of the side in question], the constructed [lines] will be shorter than the other sides of the triangle and they will contain a larger angle.355 The things we have found puzzling in the theorem – I mean the optical [theorem] – are listed among the miscellaneous theorems which it is now superfluous to mention, or else my account will stray from the present discussion.

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78b37-8  and mechanics to stereometry [and harmonics to arithmetic]356 Stereometry is different from geometry in that the one is concerned with planes, the other with solids. Mechanics falls under stereometry because it employs as principles and causes the things proved by the latter.357

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78b38-9  and ‘phenomena’ to astronomy. By ‘phenomena’ he means the observation of the phenomena that is done by sailors,358 for example that on this day that particular star rises, and everything else that they know(o) from observation. This – I mean the observation of the sailors – knows(o) only the ‘that’, while astronomy also gives the causes of these things.

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78b39-79a1 Some of these sciences are almost synonymous [– like mathematical and nautical astronomy]359 He did well to say ‘almost’ for it is not true that they ‘are’ entirely ‘synonymous’, since then they would be the same. Nor are they entirely homonymous, since then they would have nothing in common with regard to their objects. But as it is, they are concerned with the same things. But even if we give accounts of them rather roughly we will see that the definitions are common. For example if someone were to say that astronomy is knowledge(g) of the things that occur in

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the heaven, he will give it as a common definition of both nautical and mathematical astronomy. 5

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79a1-4  and mathematical harmonics and harmonics that has to do with hearing. [Here it is the perceptual [sciences] that know the ‘that’ and the mathematical ones [that know] the ‘why’, for these latter people [those who know mathematical harmonics] have demonstrations of the causes]360 Pythagorean musical theory distinguishes tunings by ratio,361 not perception. But music in the popular sense distinguishes what is untuned and what is tuned by perception alone. This is why it knows(o) only the ‘that’, for example, that a tuning is concordant, whereas mathematical [musical theory] also [knows] the account [that explains] why it is concordant. That is why people do well to make fun of them by saying that their ears are ahead of their mind. 79a4-6 And often they do not know(o) the ‘that’, just as people who investigate the universal frequently do not know(o) some of the particulars through lack of observation.

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Just a person who knows(o) that no mule is pregnant but has often seen a mule with a swollen belly might suppose that it is pregnant if he has not thought of the universal formula that no mule is pregnant or that the present [animal] is a mule – so sciences(e) of the ‘why’ often do not know a particular through failure to pay attention, although they know(o) the general account. For example a ‘musician’ who has often heard a concordant tuning does not know(o) if it is concordant, even though he knows(o) universally which are the concordant ratios. Likewise too the astronomer who knows(o) that northern stars that rise together with southern ones set later than the southern ones, and has often seen Taurus setting before Auriga, will think that it also rises before Auriga and that this is why it sets before it – through failure to pay attention to the particular and because the more southerly [stars] set sooner. 79a6-9 These [the objects of the mathematical sciences] are things that make use of forms [although they are something else in their substance. For mathematical [sciences] are concerned with forms, since [their objects] do not belong to any subject. For even if geometrical objects are said of some subject, they [are] not [studied by geometry] qua said of a subject.] Since he said that the perceptual [sciences are concerned with] the ‘that’ while the mathematical ones [are concerned with] the ‘why’, calling harmonics ‘that has to do with hearing’ (79a1-2) and nautical

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astronomy perceptual, he here wants to say what the mathematical ones are362 – namely, the sciences(e) called mathematical are those that [have a subject that is] something else in substance but do not treat that, but instead strip off only the form from the matter itself. For nautical astronomy and music ‘that has to do with hearing’ (79a2) do not employ ‘forms’ but the very substance of their objects, whereas the mathematical ones [employ] ‘forms’ alone, in no way taking into account ‘the substance’ or making use of perception.

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79a10-13 There is another science that has the same relation to optics as optics has to geometry [– the study of the rainbow. For it is for the natural philosopher to know(o) the ‘that’, but it is for the ‘optician’ [to know] the ‘why’, either [the one who studies optical phenomena] without qualification or [the one who studies them] mathematically.] The relation that optics has to geometry, he says, some other science(e) has to optics. Alexander says that the [science] that considers the theorem ‘about the rainbow’ is related to optics as optics is related to geometry. For as geometry employs only the forms without the substrates, the expert in optics employs the straight line in the air, which is a natural body that has the visual rays as a substrate, and as optics grasps only the ‘that’, but geometry also proves the ‘why’, the [science] that examines the theorem of the rainbow is related in this way to optics itself. However, what this science(e) is and under which science(e) it belongs, he does not say. However Proclus said he means catoptrics. The theorem about the rainbow belongs to catoptrics, since catoptrics constructs arguments about reflection, whereas ‘opticians’ say what kinds of attributes belong to visual rays. And the catoptrician employs the [theorems] of the ‘optician’; for they simply assume the attributes of the rainbow – that it has three colours, that there are never more than two [rainbows], that it is never larger than a semicircle,363 and the like – which ‘is for the natural philosopher to know(o)’. But why such things occur is the province of the ‘optician’. The ‘catoptrician’, then, is related to the ‘optician’ as the ‘optician’ is to the geometer. That Aristotle means the ‘catoptrician’ in these words Proclus confirms from what Aristotle cites. ‘For’, he says, ‘it is for the natural philosopher to know(o) the “that”, but it is for the ‘optician’ to know the “why” – either [the one who studies optical phenomena] without qualification or [the one who studies them] mathematically’. In these words he hints at the ‘catoptrician’. For the ‘catoptrician’ is not an ‘optician’ without qualification but in respect of the study of the rainbow. And the study of the rainbow discusses the reflection of visual rays as being reflected by raindrops as if they were mirrors. Such is the ‘catoptri-

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cian’ who discusses the attributes of mirrors in the case of the reflection of visual rays. 79a13-16 There are many sciences related in this way that are not subalternate, as medicine is to geometry. [It is for a doctor to know(o) that circular wounds heal more slowly, but for the geometer [to know] the ‘why’.] 10

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He said that the deduction of the ‘that’ is different from that of the ‘why’ in different sciences(e), and he showed how [this happens] in the case of subalternate sciences(e). Now he shows that this difference appears also in the case of some [sciences] that are not subalternate. For example, he says, doctors say that ‘circular’ wounds form scabs more slowly than oblong ones, while geometers state the cause of this – that circles contain the greatest area of any isoperimetric figures. For among isoperimetric [figures] those that have more sides always contain a greater area. And since the circle is the limit of the polygons, it contains the greatest area of all figures.364 Now this cause does not seem to have been stated soundly. For if we take two wounds that are not isoperimetric but the larger one has straight sides and the smaller one is round and has a smaller perimeter and area, nonetheless the circular one will form a scab more slowly. The cause of this is that the healthy parts are not near one another but are far apart and for this reason nature has difficulty in bringing them together. This is why doctors cut such wounds and make angles, in order to obliterate the shape. For where there are angles, since the distance between the healthy parts is small, nature is able to bring the healthy parts together and form a scab since they are not far apart. It is possible to use a different example for the claim in question if we take medicine and astrology; the doctor says that seventh days are critical, but he does not know(o) why they are critical, whereas the astrologer states the causes – namely the quartile shapes of the moon365 and such things as that. Chapter 14

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79a17-22 The first is the most scientific of the figures. [For the mathematical sciences construct their demonstrations through this figure – for example arithmetic, geometry, optics, and practically all [sciences] that investigate the ‘why’. For the deduction of the ‘why’ either entirely or for the most part and in most cases occurs through this figure.] It is here proposed to prove that the first figure is most appropriate for scientific(e) matters and he proves this in many ways. First from the fact that all sciences(e) that concern the ‘why’ deduce their proper

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theorems in this figure above all, and in the second figure but rarely. In the first place because causes are usually affirmative, and no affirmative conclusion is inferred in the second figure, and though they [affirmative conclusions] are inferred in the third [figure], all [conclusions of third figure deductions] are particular, whereas causes are universal. Sciences(e) rarely infer negative conclusions. For example, when a geometer says ‘two straight [lines] equal respectively to two straight [lines] cannot be constructed on the same straight line such that they have different points of intersection on the same side [of the line] when they have the same endpoints [on the line] as the initial straight lines’. This theorem is negative. This is the first [argument] by which he will demonstrate that the first figure is appropriate. The second366 is that demonstrations occur for the most part through definitions, but ‘only through’ the first figure ‘is it possible to hunt for scientific knowledge(e)’ of definitions367 since every [term] employed in a definition is employed in an affirmation, and affirmatives are not proved in the second [figure]. Again, universals are not proved in the third [figure], whereas definitions are universal. Therefore scientific knowledge(e) of definitions [occurs] only through the first figure, when we successively discover the attributes of a subject and combine them with the genus as differentiae until we make the definition coextensive with the definiendum.368 For example if we were to want to define man, we first investigate what is the genus of man, and then, deducing that it is an animal, we again deduce each of the differentiae369 and combine them with the genus until the assemblage converts with the thing in question, and they say that this is the definition of man. But we do not grasp that this is the definition of that by demonstration, since there is no demonstration of definitions as he will demonstrate in the eighth book of the Topics,370 but definitions must be familiar simply because they are evident, since there is no definition of definition unless it is some kind of general description. Rather, demonstrations depend on definitions. The following might be a general description of a definition: a definition is a formula that holds uniquely of something and is comprised of essential attributes that hold more widely [than the definiendum].371 From this general description it is possible to deduce conclusions about definitions. For example, ‘mortal rational animal’ is a formula [that holds] uniquely of man and is comprised of essential attributes that hold more widely. But what is comprised of things [that hold] more widely and holds uniquely of something372 is a definition of that to which it holds uniquely. Therefore ‘mortal rational animal’ is a definition of man. He also says that the first figure is especially appropriate to science(e) in a third argument,373 that this [figure] ‘has no need’ of the other figures, whereas the others need it (77a30). For every demon-

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stration takes place through immediate premises. So if non-immediate premises are taken and we need to demonstrate each of them by taking middle terms until we reach immediate premises, this cannot be done through any figure but the first. What do I mean? Suppose that something has been proved in the second figure and that it is necessary to demonstrate each of the premises. It is clear that the affirmative one can be proved only through the first figure, for the second [figure] infers no affirmative conclusion, while the third [infers] nothing universal. Therefore the affirmative [premise] can be proved only in the first figure. Likewise the negative one. For it is not proved in the third [figure], because the conclusions of the third [figure] are particular. And it cannot [be proved] in the second [figure], because an affirmative premise is needed to prove a negative [conclusion] (since nothing is proved from two negative [premises]), and the affirmative [premise] in turn cannot be proved except in the first figure. But unless that [premise] is proved, the negative [premise] is not proved either, unless the affirmative premise is immediate and self-guaranteeing.374 Yet another argument. It is proved in the Prior Analytics that deductions in the second and third figure do not have their necessity evident,375 and are completed by reduction to the first figure.376 Therefore for this reason too the first figure is appropriate to science(e). 79a22-4 And so for this reason too it will be the most scientific. For investigating the ‘why’ is the chief thing for knowing(o).

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If demonstration is rather a deduction of the cause, and sciences(e) that deduce causes employ this figure above all and the second [figure] but rarely, this above all will be the figure that is ‘chief’ and appropriate to science(e). 79a24-9 Next, only through this figure377 is it possible to hunt for scientific knowledge(e) of what it is. [For in the middle figure affirmative deductions do not occur, but scientific knowledge(e) of what it is is [knowledge] of an affirmation. And in the last [figure affirmative deductions] do occur, but they are not universal, whereas the what-it-is is universal. For man is not a two-footed animal [merely] in a way.] Namely,378 the [knowledge(e)] of definitions, since definitions are both affirmative and universal. For no one in defining man says that man is somehow two-footed, but universally ‘every man is a two-footed rational animal’.379 79a29-32 Further, this has no need of the others at all, but they

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are filled in and grow through this, until they reach the immediates. [So it is obvious that the first figure is the most important thing for knowing(e).] That is, if the premises are not immediate, the first [figure] is sufficient by itself for inserting middle terms successively until we arrive at the immediate premises. For if we want to prove a negative [premise] there is no need for the second figure. For the negative [premise] will be proved either directly or by conversion. For example, if we want to prove that A belongs to no B and take middle term C, it is of course necessary for C to be predicated either of A or of B.380 Now if C can be predicated of every B, we will prove the extremes directly: A belongs to no C, C belongs to every B, therefore A belongs to no B. But if C is not predicated of B, but [is predicated] of A, it is possible to deny the extremes of one another in the second figure. For if C [is predicated] of every A but of no B, A [is predicated] of no B.381 But it will have ‘no’ need of the second figure, since it can [be proved] through the first too. For C [is predicated] of every A and of no B. But then B [is predicated] of no C. And you will infer [that] B [is predicated] of no A, and if you convert the conclusion you will find that A [is predicated] of no B. But in the case of the third figure it is not possible, as we proved. For the third [figure] is entirely useless for science(e) since sciences(e) infer no conclusion that is particular.

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Chapter 15 79a33-4 Just as A can belong atomically to B, it can also not-belong atomically. Since he said (79a29-32) that the first figure has no need of the others, ‘but they’ ‘are filled in and grow’ through the first figure ‘until’ it reaches immediate382 premises by continually inserting middle terms through which the premises are proved,383 he now wants to prove here that immediate premises are not only affirmative but also negative. For it is possible to deny one thing immediately of another. He here presents us with a rule as to how it is necessary to know(g) terms that are immediately denied of one another, since this is not easy to notice. Anyone can know(o)384 which affirmations are immediate: whichever ones have a predicate that belongs primarily to the subject and to nothing else prior to it, as substance is predicated immediately of body, and this [body] [is predicated immediately] of ensouled, and similarly in other cases. But in the case of negations the distinction is not easy. So what does he say? That when terms that are denied of one another are such that neither of them has any [term] affirmed of it because of the fact that it is most general and can be predicated of certain genera,385 they are said to be immediately denied of one

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another. For we call premises immediate when it is not possible to insert any term between [them] and form a deduction. But if either of the terms has some [term] predicated of it, this can be inserted between them and form a deduction, being predicated of one [of the terms] and denied of the other, or having the other denied of it, in order in the first case to form a second figure [deduction], and in the second case a first [figure deduction].386 And so only negations that have both terms from most general genera will be immediate. Since if in fact either or both of them are subordinate, such a negation is not immediate. 79a34-5 By belonging or not belonging atomically I mean that there is no middle term between them.

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He indicates clearly that ‘atomically’ means the same as ‘immediately’. And he does well to call ‘immediately’ ‘atomically’, because it is impossible for immediate premises to be split387 by the insertion of a middle term and so make one premise into two. 79a35-6 For in this way it will no longer be possible to belong or not belong in virtue of something else.

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For just as if I say ‘man is a substance’, I am not predicating substance of man primarily but ‘in virtue of something else’, and this is why the premise is not immediate (for [it is proved through the middle term] ‘rational’ and this [is proved through the middle term] ‘animal’ and this [is proved through the middle term] ‘body’) and substance is predicated immediately only of body, and rational [is predicated immediately] of man because between them no other term can be inserted to which the predicate will belong primarily – so too there will be immediate negations in which the term cannot be denied of another subject.388 For then the negation is immediate and not ‘in virtue of something else’. 79a36-8 Now when either A or B or even both A and B are in something as in a whole, it is not possible for A primarily not to belong to B.

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First he takes the subject as not being in anything else but as the most general genus,391 and the predicate as being in C. Now if C belongs to all A and to no B, A too will belong to no B. And so premise AB is not immediate but A is denied of B through the middle term C. For since C [belongs] to no B, this is why neither does A, which is a part of C. Let A stand for continuous, B for substance and C for quantity. Now if quantity [belongs to] everything continuous but to no substance, neither will ‘continuous’ belong to any substance: that is, no substance is continuous, since it is not a quantity. Notice that although the one term, ‘substance’ is immediate, since the other is not immediate392 the negation has proved to be not immediate.

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79a39-b1  (for it is possible that A is in something as in a whole but that B is not in this) [there will be a deduction of A’s not belonging to B. For if C belongs to all A and to no B, A belongs to no B.] Since he said that A is ‘in C as in a whole’ and B is in no C, and since contrasted species are both denied of one another and are included under the same species (as rational belongs to nothing irrational, nor does irrational to anything rational, and nevertheless both are in animal as in a whole), to keep anyone from thinking that all things that are denied of one another are such that when one is included under something more general the other is too, he adds that ‘it is possible’ that one of the things denied ‘is in something as in a whole’ while the other ‘is not in this’ very thing, as for instance in the examples we gave ‘continuous’ is in ‘quantity’ as in a whole and ‘substance’ is in nothing. After saying that it is possible, he does not say here how it is possible, but he will add that point in what follows.393 It should be noted that he forms a deduction in the second figure. He says that C [belongs] to all A but to no B. For when the predicate term of the negative problem is in something else that is more general, the deduction takes place in the second figure. However it can also [take place] in the first – only A is no longer inferred [to belong] to no B, but B [is inferred to belong] to no A. But when the conclusion is converted neither will A belong to any B. I needed to note this because he declared that premises are filled in through the first figure only.394 But it is possible to prove the negative premise through the second figure as well if the affirmative [premise] is immediate, as I also said earlier395 and have now confirmed by illustrations. For if we want to prove the negative premise ‘no substance is continuous’, when ‘quantity’ is put in between [the given terms], the deduction takes place in the second figure with an immediate affirmative premise. For quantity [belongs] to everything continuous but to no substance, and so neither does continuous. Now

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if the predicate term in the negation is in some whole, the deduction takes place in the second figure, as we have shown. But if the subject is something that belongs in something more general, then the deduction takes place in the first figure, as he posits in what follows. 25

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79b1-4 Similarly if B is in something as in a whole, for example D. [For D belongs to all B, and A to no D, and so A will belong to no B through a deduction.] He now takes the converse [case]: the subject is in something else and the predicate is not. Let A stand for substance, B for continuous, and D for quantity. Now suppose that quantity belongs to everything continuous and substance [belongs] to no quantity, so that substance does not [belong] to anything continuous either. Therefore the premise ‘nothing continuous is a substance’ is not immediate. 79b4-5 It will be proved in the same way even if both are in something as in a whole. For example let A stand for body and B for continuous. It is clear that A is in something as in a whole, namely substance, and B is in quantity. Therefore A will not be denied immediately of B – that is, body [will not be denied immediately] of anything continuous, but there turn out to be two prior deductions. 79b5-11 But that it is possible for B not to be in something that A is in as in a whole, or for A [not to be] in something B [is] in, is obvious from non-overlapping columns. [For if nothing in column ACD is predicated of anything in column BEF, but A is in H as in a whole and H is in the same column [as A], it is obvious that B will not be in H, for then the columns will overlap.]

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After saying ‘for it is possible that A is in something as in a whole but that B is not in this’ (79a39-b1), he now wants to prove this. He calls the series descending from each genus a column – for example substance-body-ensouled-animal-rational, or substance-bodysoulless-heavy.396 These combinations overlap, since genera common [to them] are predicated of them. Now in the case of ones that do not overlap, he says, it is clear that it is possible for one term to be in something and the other not to be in the same thing. For if we take one middle term and deny it of one term in the other column – for example [if we deny] animal of everything continuous, animal will be included under body or substance, but continuous will not. For if continuous too is included under the same thing as animal, the columns will overlap; but they do not overlap.

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79b11-12 Similarly too if B is in something as in a whole. Namely, it is also possible for A not to be in the [same thing] as B. 79b12-15 But if neither is in anything as in a whole and A does not belong to B, it is necessary for it to atomically not-belong [to B]. [For if there is going to be a middle [term], it is necessary for one or the other of them to be in something as in a whole.] That is, if neither A nor B belongs in anything more general but they are themselves the most general things, and one of them is denied of the other, it is necessary that it is denied immediately. 79b15-18 For the deduction will be in either the first figure or the middle. [Now if it is in the first, B will be in something as in a whole (for the premise with this as a subject must be affirmative).] If, he says, there is a middle term between A and B, deductions occur both in the first figure and in the second because ‘B’ ‘will be’ ‘in something as in a whole’ and will not be most general. It is clear that this is true. For the minor premise must be affirmative, and B is the minor term. And so if there is a middle term C, it will belong to all B but A will belong to no C. And so it proves to be a first figure [deduction]. We have already spoken about these matters. 79b18-19 But if it is in the middle, one or the other of them [will be in something as in a whole], for there is a deduction when a privative is taken as attached to either of them as a subject. If when a middle term is posited a second figure [deduction] is formed, the middle term can be denied of either of the extremes. The cause of this is the fact that in the second figure it makes no difference in which premise the negative [predication] occurs. For regardless of whether C belongs to all A and no B or vice versa, either way it is possible to infer that A [belongs] to no B. And so AB will not be an immediate negation. For regardless of whether the middle term belongs to all A as in the second figure397 or to B as in the first and the second,398 the negation will not be immediate, that is, denying something that is most general of something that is most general. 79b20-2 But if both are negative there will be no [deduction]. [Therefore it is obvious that one thing can not-belong to another atomically, and we have stated when this can happen and how.] To keep anyone from saying that a middle term can be taken that

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belongs to neither extreme and that in this way A is denied of B through a deduction when both are most general, he adds that if a middle term is taken such that it is neither in A nor in B, it will make both premises negative and in this way the combination will be invalid.399 Chapter 16

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79b23-4 Ignorance not due to negation but due to disposition is on the one hand error that occurs through deduction  An expert(e) should not only present what belongs to the teaching of his science(e) but also expose errors that attend the science(e). So after Aristotle says what science(e) is and on what it depends, what the difference is between deductions of the ‘why’ and [deductions] of the ‘that’, and that there are also immediate negations just as there are immediate affirmations400 – since the discussion of immediate negations contributes to his doctrine on error he now proposes to expound in how many ways error occurs in sciences(e). But in order to find in how many ways error occurs in sciences(e) he employs the following division: ignorance, he says, is either ‘due to negation’ or ‘due to disposition’. He does not mention ignorance due to negation in the present passage since it does not at all pretend to be scientific knowledge(e). What is known as simple [ignorance] is ignorance due to negation: for example a lay person does not know(o) whether or not triangles have their three angles equal to two right angles. It is simple [ignorance] because he is ignorant and knows(o) this very thing – that he is ignorant. And so it [this kind of error] is not going to bother experts(e) and does not pretend to be scientific knowledge(e). That is why he does not even mention it. He makes a division of ignorance due to disposition. Ignorance due to disposition is said to be the opposite of true knowledge(g), which is false. For example, if someone were to say that the two sides of a triangle are smaller than the remaining one. This is [ignorance] due to disposition since a person who has it is disposed in a certain way because of it and erroneously supposes that he knows(o) it. This is why this kind of ignorance is called twofold.401 For he does not know(o) that he does not know(o), but though he does not know(o), he is ignorant of this very thing, that he is ignorant. Now this ignorance due to disposition occurs with both immediate propositions and non-immediate ones.402 For just as it is possible to truly affirm or deny one thing of another immediately, it is also possible to falsely affirm or deny immediately. For example, if it is true that substance belongs to no quality, which is an immediate negation, the [proposition] stating ‘substance belongs to every quality’ is a false immediate affirmation. Similarly too if the immediate

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affirmation stating ‘substance belongs to every body’ is true, then the negation stating ‘substance belongs to no body’, which is immediate, is false.403 Likewise for non-immediates too. So the error – that is, the false belief – occurs in the case either of immediate propositions or of non-immediate ones. And in both cases it arises either when there is a deduction of it or when there is not. For it is possible to have the erroneous opinion that the three angles of a triangle are less than two right [angles] even without a deduction, just as in perception we may make mistakes in hearing or seeing. But it is also possible to be led astray by a deduction, or rather by a fallacious argument. The error that arises without a deduction is simpler – as if as a result of [hearing] a presentation someone were to believe that natural bodies are composed of atoms or that there are two principles of things that are,404 without employing any reasoning. For what kind of division [into different kinds] could occur? But error through deductions is more varied and complex. For deductions, or rather fallacious arguments through which we may be deceived, are more varied. For the deduction through which the error [occurs] will be either affirmative or negative.405 Now if the fallacious argument is affirmative it must take place in the first figure, for since the truth is a universal negative, the falsehood that is its opposite will be a universal affirmative. For since we are considering the error that pretends to be scientific knowledge(e) and since scientific knowledge(e) is of universals, the error must be inferred fallaciously through universal [premises]. For it cannot [be inferred] through contradictory opposites but through contraries (these are the universals), in order that it may have the appearance of scientific knowledge(e). Now if the fallacious argument is affirmative and universal, it must be reached through the first figure. For [it can be reached] neither through the second [figure], since it [the second figure] infers nothing affirmative, nor through the third [figure], since it infers nothing universal. Now if the fallacious argument is affirmative and immediate,406 it will have either two false premises or only one. If it has two false ones, either both will be universally false or one will be universally [false] and the other partially [false]. Both are universally false when the middle term that is taken to form the deduction is entirely unrelated to the extremes. For example since the true negation stating that substance belongs to no qualification is immediate, while the contrary error states that substance belongs to every qualification, if the deduction through which the error has arisen proceeded through two false premises, both [of them false] in virtue of every kind of falsehood,407 the middle term that is taken will be included under neither of the extremes, neither under substance nor under quality, but it will be entirely unrelated to them – as for example if it were quantity. For the fallacious argument will be ‘substance

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[belongs] to every quantity, quantity [belongs] to every qualification, therefore substance [belongs] to every quality, that is, to every qualification’. But as I said, it is possible for one of the premises to be false not universally, but false in some case, and this is none other than the minor premise. This occurs when the middle term is taken from the things that fall under the minor term, for example whiteness, which falls under quality. For then we say ‘substance [belongs] to every whiteness, whiteness [belongs] to every qualification’, and the latter premise is false not universally but in some case. He expounded precisely what is a universally false premise and what is [a premise that is false] in some case in the second book of the Prior Analytics.408 For he says that a universally false premise is one that introduces the contrary of what is universally true. For example the one that says ‘no man is a stone’ is true, and the contrary of this, stating ‘every [man] is a stone’ is false, since ‘every’ is contrary to ‘no’. Now what entirely eliminates what is true is universally false. And what is false in some case is the contradictory opposite. Since [the premise] ‘rational does not [belong] to every animal’ is true, the one that states ‘rational [belongs] to every animal’ is false in some case. This is how it is if both [premises] from which the falsehood is inferred are false. But if one of them is true, none other than the major premise can be true; it can never be the minor premise. For since both the minor term and the major term have been taken as most general genera409, it is clear that the middle term cannot be truly asserted of the minor term, since under these circumstances it must extend more widely. But the minor term is posited to be a most general genus. And clearly the middle term cannot be coextensive with the minor term, for nothing is coextensive with the most general genera. Therefore the minor premise can never be true. Therefore only the major premise will be true, for there can be a middle term among those falling under the major term. And thus the major term will be predicated of it truly, and this one will be [predicated] of the minor term falsely. For example, if I take body: substance belongs to every body (which is true) and body [belongs] to every quality (which is false) – this is how ‘substance belongs to every qualification’410 will be inferred. Now this is in how many ways fallacious arguments occur when the error is affirmative and immediate.411 How [they do] when the error is negative and how when it is non-immediate and both affirmative and negative,412 we will indicate further down. But first we will examine the passage in which he sets out the first way [of there being error in science], which we have stated. 79b25-7  and in the first place, in cases of things that belong or do not belong primarily this occurs in two ways. [For [it

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occurs] either when he just thinks it belongs or does not belong, or when he gets the belief through a deduction.] After saying that ‘ignorance’ ‘due to disposition’ ‘is on the one hand’ ‘error that occurs through deduction’ (79b23-4), even though he ought to go on to say that [ignorance due to disposition] also is found without deduction (since this is the following topic), he resumes his account and says, ‘in the first place, in cases of things that belong or do not belong primarily this occurs in two ways’. By ‘this’ he means not the kind ‘that occurs through deduction’ but the kind that occurs ‘due to disposition’. For this is what he divides into the kind [that occurs] through deduction and the kind [that occurs] without deduction. ‘In cases of things that belong or do not belong primarily’ means among the immediate premises or propositions whether affirmative or negative. For in each of these it is possible to believe the contrary error both without deduction and also with deduction. Perhaps it is because he wanted to make the division of error with deduction right at the beginning before mentioning the kind without deduction that he said ‘is on the one hand error that occurs through deduction’ (79b24). And then, thinking it good to say first in how many ways error due to disposition occurs, he took up his account and said ‘in cases of things that belong primarily’ etc. 79b28-9 In the case of simply thinking there is just one kind of error, [but there are several in the case of error due to deduction.]

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For if someone simply makes an error without deduction, there is ‘just one kind of’ error’ involved with this kind of false belief. For since it occurs without any deduction, there is no variety in it or [the possibility of] division as there is in the case of the kind that occurs through deduction. 79b29-33 For let A atomically not belong to any B. [So if he deduces that A belongs to B by taking middle term C, he will be in error through a deduction. Now it is possible for both premises to be false, but it is also possible for only one to be false.]

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If a simple negation is immediate, then if the affirmation that is opposite to it occurs through a deduction, it will take a middle term, C, and it will be deduced erroneously through the middle term C that A belongs to every B. How many ways fallacious arguments occur in this case he goes on to say in what follows. 79b33-5 For if it is not the case that A belongs to any C or that

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If, he says, the middle term taken is so related to the extremes that the major term is denied of it and it [is denied] of the minor term, but the reverse is taken in the fallacious argument – namely, that the major term belongs to the middle term and this [belongs] to the minor term, the fallacious argument is clearly based on [premises that are] both false. 79b35-7 But it is possible for C to be related to A and B in such a way that it neither falls under A nor belongs universally to B. Since he said that both premises can be false, he here adds how this can be: when, he says, the middle term is such that it does not fall under the major term and is not predicated of the minor term. It is like this when it is unrelated to both. 79b37-8 For it is impossible for B to be in anything as in a whole [(for it was said that A not-belongs to it primarily)]413

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He demonstrates both of his claims. It is clear that the middle term can be predicated of B non-universally – rather it is necessary for it to be predicated of B non-universally. For since A was denied primarily and immediately of B, B cannot fall under anything else, for then A would not be predicated of B immediately but through the middle term under which B is included too.414 79b39-40  but there is no necessity that A belong universally to all things that are, [and so both [premises may be] false].

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Because it is possible that A too belongs to no C. For if there were some most general genus of all things that are and we took this in addition to A, it would be impossible for that [the most general genus] not to have to be employed in an affirmation. For whatever middle term could have been taken would have to be in that and, that is, as we have seen,415 predicated of all of it. But since there is no such thing, it is possible for a middle term to be taken that is unrelated, and so the major term will not be predicated of it. 79b40-80a3 [But it is possible to take one of the two [premises] as true – not, however, whichever one you like, but only AC. For premise CB will always be false because B is in nothing,] whereas AC may [be true], for example if A belongs atomically to both C and B.

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That the major premise AC can be true is confirmed here through these considerations. It is possible, he says, that ‘A belongs atomically to both C and B’. But how he here means that A belongs immediately to both C and B became a major question for Alexander.416 And after much [discussion] he states the true sense of the passage: ‘belonging’ here means not ‘being affirmed’ but simply ‘being predicated’, whether affirmatively or negatively. So, he says, A can be predicated immediately of both B and C – affirmatively of C and negatively of B. For if C falls immediately under A and not after several others – for example if A is substance, if C is something that falls immediately under substance, such as body, substance will be predicated of it immediately; but it [substance] will be denied immediately of B, namely, quality. But if it can be predicated of C immediately, the major premise, AC, can be true.417 80a3-4 (For when the same thing is predicated primarily of more than one thing, neither will be in the other.) Since he said that premise CB, namely, the minor premise, is always false but the major premise, AC, can be true, he proves that the major premise can be true, saying ‘for example if A belongs atomically to both C and B’ (80a2-3). That the minor premise is always false he proves here, employing a general argument. He says generally that ‘when’ anything that is one and ‘the same thing’ ‘is predicated’418 ‘of more than one thing’ in whatever way, whether affirmatively of all of them or negatively of all, or affirmatively of some and negatively of others, none of the rest belongs in any of the others. Let us see the truth of the argument going through the details. Let A be immediately denied of both C and B. I say that B will not belong to any C either. For if this is false, B will belong to every C.419 Now if A [belongs] to no B and B [belongs] to every C, A belongs to no C through B as a middle term. But it was posited that it not-belongs immediately. Therefore B belongs to no C. Then for the same [reasons] C too will belong to no B. But then, let A be affirmed immediately of both – both B and C.420 Then again B will not belong to C. For if it belongs, since A [belongs] to every B and B [belongs] to every C, therefore A too [belongs] to every C through B as a middle term. But it was posited that it belongs immediately. Therefore C likewise will belong to no B. But then let it be immediately affirmed of one of them and [immediately] denied of the other. For example let A belong immediately to every B, and to no C. I say that B too will belong to no C. For if [it belongs] to every [C], since also A [belongs] to every B and B [belongs] to every C, A too will belong to every C. But it was posited [that A belongs] to no [C]. Likewise neither does C [belong] to any B. For if [C belongs] to every [B], since A [belongs] to no C and C [belongs] to

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every B, A will belong to no B. But it was posited [that A belongs] to every [B]. Thus for the same [reasons] if there are more than one thing of which A is predicated immediately, it is not possible for one of them to belong to any of the others. Now if, when the same thing is predicated immediately of some things it is impossible for those things to belong to one another, and A belongs [immediately] to both C and B, it is impossible for these to be affirmed of one another. And so the fallacious argument affirming that C [is predicated] of B must make the minor premise false. 80a4-5 But it makes no difference if it belongs non-atomically.

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‘For when the same thing is predicated primarily of more than one thing’ (80a3-4) is not directed at what has just been said, but at the preceding point: ‘but it is also possible to take one of the two [premises] as true – not, however, whichever one you like, but AC’ (79b4080a1). Then [he says] next, ‘but it makes no difference even if it belongs non-atomically’, that is, it makes no difference as to whether the major premise (AC) turns out to be true whether A (the major term) is affirmed of C (the middle term) immediately or non-immediately but through several [terms], that is, if the middle term proves to be ordered not immediately under the major term but after many. For if A is substance, then if C is body, A is predicated of it immediately; but if it is something lower down such as animal, rational, or something else like that, A [does] not [belong] to C immediately, but it does belong truly. So with regard to the major premise turning out to be true it makes no difference whether the middle term is posited [to belong] to the major term proximately or after many. 80a6-9 [Error about [something’s] belonging [to another] occurs in these ways and only [in them] (for deductions of belonging are not found in any other figure),] but error about [something’s] not belonging [to another] [occurs] in the first and middle figures.

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We said that his present purpose is to deal with the error that is contrary to scientific knowledge(e) and that error due to disposition occurs in relation to either immediate premises or non-immediate ones, and that in both it is found either in connection with a deduction or without a deduction,421 and that if a scientific(e) proposition or premise is affirmative, the contrary error will be [the error] of the negation that is contrary, not of the one that is the contradictory opposite,422 because it pretends to be scientific knowledge(e) and wants to draw universal conclusions, but if the scientific(e) premise is negative, the error will be an affirmative [premise].423

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Now since he said how error concerning immediate affirmative premises occurs and is affirmative because [it occurs] only in the first figure with either one or both premises being false,424 he now proves how the negative error that is the opposite of the immediate affirmative premises425 occurs. It occurs, he says, in both the first and the second figures because in both a universal negative [conclusion] is inferred, either through two false premises or through only one false premise and one true one. How each occurs we will know(o) by going through the passage in detail. To begin with he shows how it is proved in the first figure in cases where both premises are false.

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80a9-14 [So let us first say in how many ways it occurs in the first, and how the premises are related. It can happen when both premises are false,] for example if A belongs atomically both to C and to B. For if A is taken [as belonging] to no C and C [is taken as belonging] to all B, the premises are false. Again he takes A as the major term and B as the minor term, and again he affirms A of B immediately. For example, let A be animal and B be terrestrial; animal belongs immediately to every terrestrial thing. Now the error stating that A belongs to no B could infer this in the first figure from two false premises if it took a middle term such that the major term is predicated affirmatively and immediately both of it and of the minor term – namely, if from the start it takes things falling under the major term that are coordinate members of a division with one another. For example, if the major term is substance and the minor term is body, the species coordinate with body (I mean bodiless)426 must be taken as a middle term.427 Likewise if we had taken animal as the major term and terrestrial as the minor term, we will take winged, which is coordinate with terrestrial, as the middle term. For animal clearly belongs immediately to both winged and terrestrial. So given that the middle term is of this kind, if he assumes that A belongs to no C and C [belongs] to every B, both will clearly be false. For we proved generally that when one thing is predicated immediately of two things, whether it is predicated affirmatively, or negatively, or affirmatively of one and negatively of the other, neither of those things belongs to the other.428 And so if he assumes [that] A [is predicated] of no C when A belongs to all of it [C], and that C [belongs] to every B when it belongs to none, he will be assuming two false [premises]. For as we said, let the major term, A, be animal, the middle term, C, winged and the minor term, B, terrestrial; animal belongs immediately to both. So if anyone assumes that animal [belongs] to no winged thing and winged [belongs] to every terrestrial thing, he will be assuming two false [premises] and will infer that animal [belongs] to no terrestrial thing. But it is possible to assume two [premises] that are not universally

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false. For example, animal belongs to every rational thing. It will be inferred not to belong if mortal is taken as the middle term. Now if I say ‘animal [belongs] to no mortal thing, mortal [belongs] to every rational thing, therefore animal [belongs] to no rational thing’, the major premise will be universally false while the affirmative minor premise is false in some case.429 Therefore the middle term must be taken as belonging to some instance of the minor term and not to another. For if it is taken to belong to every instance, it will be false in some case. But the major term cannot be taken as false in some case. For since A is the most general [term], if some term is taken that falls under it, A will belong to all of it truly and to none of it falsely. But if the [term] taken is unrelated to A, it will belong to no instance of it truly but to all of it falsely.430 Therefore the major premise cannot be false in any instance. 80a14-15 It can also occur when one of them is false, no matter which. In the first place he takes a true negative major premise and a false minor premise. This occurs when the middle term taken is unrelated to both. For example, stone, animal, rational: animal belongs to no stone, stone [belongs] to every rational thing, therefore animal will belong to no rational thing. And only the negative [premise] is true. In this way this argument is clear through the terms,431 but he confirms it also through some more general arguments. 80a15-17 [For it is possible for AC to be true, but CB false] – AC to be true because A does not belong to all things that are432

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He proves concisely that the negative [premise] – I mean the major premise – can be taken as true because it is not possible to take any term such that it is predicated of all things that are. For if there were some common genus of all things that are, and if this had been taken in the major term, it would not be possible to deny this truly, but it would be affirmed truly of all things that are, just as if we were to take substance and there were nothing in the universe apart from substance, it [substance] would have to be affirmed of anything that is taken. Now since there is no common genus of things that are, but the categories are divided, it is always possible to take a term from another genus of which it is possible to deny the major term truly. But it is clear that if a true major premise is taken it is impossible for a true minor premise to be taken, but it will have to be false.433 For if the major term belongs to all [instances of] the minor term but to none [of the instances of] the middle term because it is unrelated to it, clearly neither will the middle term belong to any [instance] of the minor term. For if it is unrelated to the most general [term],

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clearly [it is] also [unrelated] to everything that falls under the most general [term]. And so if the middle term is taken to belong to all [instances] of the minor term, the premise will have to be false. Let it be proved by us in this way that if the major premise is negative and true, the affirmative minor premise cannot be true. Aristotle proves this plainly through the figures in two ways, of which the first is the following.

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80a17-19  and [it is possible for] CB [to be] false because it is impossible for C to belong to B when A belongs to no C. For premise AC is no longer true. It is possible, he says, or rather necessary for the minor premise ‘CB’ to be ‘false’, since it is impossible, if in fact A is truly denied of C, to predicate C truly of B. For if C is truly affirmed of B, A is no longer truly denied of C. For if C belongs to every B but A too clearly belongs to every B, it will be inferred in the third figure that A belongs to some C. But it was posited that A belongs to no C. And so if the major premise is negative and true, the minor premise cannot be affirmative and true. 80a19-20 At the same time, if in fact both [premises] are true, the conclusion will be true also. Because in another way too it is impossible, given that the major premise is true, for the minor premise not to have to be false. For if it is true too, ‘the conclusion also’ ‘will be’ ‘true’, which is impossible. For example if A truly [belongs] to no C but C truly [belongs] to every B, A will truly belong to no B either. But it was posited that it belongs to all. Therefore it is not possible, given that the major premise is true, for the minor premise not to have to be false.

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80a21-6 It is possible for [premise] CB to be true if the other [premise] is false, [for example if B is in both C and A. For it is necessary for one of the two to fall under the other, so that if he takes A as belonging to no C the premise will be false. So it is obvious that both when one premise is false and when both are false the deduction may be false.] He posits the opposite case, where the minor premise is true and the major premise is false.434 This occurs, he says, if a middle term is taken such that the minor term falls under it and it [the middle term] is included in the major term.435 For if the middle term is taken such that the minor term is included in both it and the major term, ‘it is necessary’, he says, ‘for one of the two to fall under the other’, that is, the middle term under the major term and the major term under the

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middle term.436 For if A belongs immediately to B, and C belongs to B,437 there is every necessity for A and C to be coextensive, so that both A is included in C and C is included in A. For things of that sort are coextensive. For example, let A be capable of laughing, B man, and let walking erect be the middle term. Now it is clear that man is included in walking erect and that walking erect is included in capable of laughing and that capable of laughing is included in it. And to speak concisely three terms must be taken that are coextensive, for if they are, then the major term is falsely denied of the middle term, and the middle term is truly affirmed of the minor term.438 80a27-8 In the middle figure it is not possible for both premises to be false as wholes. After saying in how many ways falsehood439 can occur in the first figure in the case of immediate negative premises,440 he now passes on to the second figure and says that it is not possible to employ premises that are universally false. And this is what he proposes to prove to begin with. Let A belong to every B441 and suppose someone says [that A belongs] to no [B], and attempts to infer this in the second figure. I say that it is not possible to take either premise442 as wholly false. For if in the middle figure the middle term, C, must be affirmed of one [extreme term] and denied of the other, clearly the middle term must be such that it is both falsely denied of that of which it is denied and falsely affirmed of that of which it is affirmed – not falsely in some case and truly in some case, but entirely falsely. He says that this is impossible. How it is impossible we will know(o) through the very passages. 80a28-30 For when A belongs to every B it will not be possible to take anything that will belong to all of one of the two but to none of the other.

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That is, if ‘A belongs to every B’ it is not possible to think of a term such that it belongs to all of one of them and to none of the other. For if it belongs to every A it will belong to B too. For B falls under A. Or if it belongs to every B it will have to belong to some A as well, since B falls under A. For example, let animal belong to all rational things. Now if I take something that belongs to all A, for example substance, it will have to belong to rational too. Likewise let something be taken as belonging to every rational thing – using reason, for example. This will have to belong to some animal as well. Likewise if something is denied of animal – soulless, for example – this must be denied of rational too. And likewise too, if something is denied of rational – non-rational, for example, it must also be denied of some animal. Therefore it is not possible to take anything that can be affirmed of

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all of one and denied of all of the other. For if it is affirmed of either it must [be affirmed] of the other as well, either universally or partially.443 For if it is affirmed universally of the major term it will have to be affirmed universally of the minor term too, since the minor term is part of the major term. But if it is affirmed universally of the minor term it will be affirmed partially of the major term. Likewise if it is denied of all of either term, it will have to be denied of the other too, either universally or partially just as when it was affirmed.444 Now since this is the case, if we take the middle term as falsely denied universally of one and truly affirmed universally of the other,445 of course it is clear that it will be truly predicated universally of that of which it is falsely denied universally and it will truly be denied universally of that of which it is falsely predicated universally.446 But we proved this impossible. Therefore it is not possible in the second figure to take two universally false premises. 80a30-1 But it is necessary to take the premises in such a way that [the middle] belongs to one [extreme term] and does not belong to the other, if there is to be a deduction. For in the second figure it is entirely necessary for the premises to be of dissimilar forms447, given that the figure is to be deductive. But also the deduction of error that pretends to be scientific knowledge(e) will employ deductions that are valid in their form and get its falsehood only in connection with their matter, as we said earlier.448 For it is sophistical to construct formally fallacious arguments.

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80a32-3 So if they are false when taken this way, clearly if taken in the contrary way they will be the other way round. But this is impossible. Since he proved that it is impossible to find a term such that it is truly affirmed of all of one and truly denied of [all of] the other, from this he proves that it is impossible to take both as entirely false. For if, he says, it is possible to affirm C falsely of all of one and to deny it falsely of all of the other, clearly the [premises] contrary to these will be true. For if C [belongs] falsely to no A, it will belong truly to every [A]. Likewise if it belongs falsely to every B, it will belong truly to none. And so it will be predicated truly of all of one and will be denied of [all of] the other. But this has been proved impossible. For the true opposites of universally false premises are contraries, that is, universal. But the true opposites of [premises] that are false in some case are contradictories. For if ‘stone [belongs] to every man’ is false, the true opposite of this is not the contradictory ‘not to every’ but ‘to none’. Likewise if ‘animal [belongs] to no man’ is false, the true opposite will be not that animal [belongs] to some man but that [it

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belongs] to every [man]. And so the true opposites of universally false [premises] are universal. And so if C [belongs] falsely to no A and falsely to all B, it will be possible to say that both are universally true – that C [belongs] to all A and to no B. But this has been proved impossible.449 Alternate proof. If [C belongs] truly to all A and to no B, also A will belong to no B. But it was posited [that A belongs] to all [B]. The same thing will happen if [C belongs] falsely to no B and [falsely] to all A. For in the second figure the privative makes no difference. ‘If they are false when taken in this way’ means that if in the second figure it is necessary to take one negative [premise] and one affirmative [premise] in order to deduce the negative conclusion that is false, and these [premises] must be entirely false, ‘clearly’, he says ‘they will be’ ‘the other way round’ – that is, the negative [premise] will be true when it becomes affirmative and likewise the affirmative [will be true when it becomes] negative. But this has been proved impossible.450 And so it is impossible to take both as universally false. 80a33-8 But nothing prevents each of them from being false in some case, for example if C belongs451 both to [some] A and to some B. [For if it is taken as belonging to all A and no B, both premises are false – not wholly, but in some case. And likewise if the privative is placed the other way around.]

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After proving that it is not possible to take [premises] that are both entirely false, he passes on [to consider whether it is possible] to take [premises] that are both false ‘in some case’. For when the middle term is so related to both extremes that it is predicated of each of them partially, it is clear that if it is taken to belong to none of the one and all of the other, both [premises] will be false of something. For example, let A be animal, B rational, and C, the middle term, mortal. Now clearly mortal belongs to some animal and to some rational thing. So if it is taken [as belonging] to no animal and to every rational thing, both [premises] will be partially false. And likewise if [it is taken as belonging] to no rational thing and to every animal.452 But is it possible also for one [premise] to be universally false and the other [false] in some case? I say that if the major premise that is taken is universally false, it is impossible for the minor premise that is taken to be false in some case; but it must be [universally] true. But if the minor premise is universally false, the major premise can be universally true too but it can also be false in some case. For in the first place let it be posited that the major premise is universally false. I say that the minor [premise] cannot be false in some case, but must be [universally] true. For if C belongs truly to all A, and is taken as not belonging falsely, clearly it will be true to say that C belongs to

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all B. For what belongs to all A also [belongs] to B. Again, if C belongs to no A, clearly neither [does it belong] to any B. So if it is taken [as belonging] to all A and no B, the affirmative major premise will be false and the negative [premise] will be true. Terms for these are: in the first, major term: animal, minor term: rational, middle term: substance; in the second, major term and minor term: the same, middle term: soulless. But then let the minor premise be posited [as being] universally false. Now if the middle term is unrelated to both the extremes, the major premise will be universally true. I put it like this: ‘soulless [belongs] to every rational thing’ – universally false, and the major premise turns out to be universally true: ‘soulless [belongs] to no animal’. But if the middle term is appropriate to the minor term so that it can belong to all of it, the major premise will be false in some case. For example, if the major term is rational, the minor term is man, and the middle term is mortal, then if I say ‘mortal [belongs] to no man, mortal [belongs] to every rational thing’, the major premise will clearly be false. 80a38-9 It is possible for one to be false, in fact either one. For if one [premise] is true the other can be false, and this is sometimes the major premise and sometimes the minor premise. When the minor premise is negative, if the major premise is true, the minor premise will be universally false, but if the minor premise is true, the major premise can sometimes be universally false and sometimes [false] in some case. Let the major premise be affirmative and true. If C belongs to all A, of necessity it will belong to all B too, since B is a portion of A. But also when C belongs to all A and A [belongs] to all B (for this is posited) it will be inferred in the first figure that C [belongs] to all B. And so if this is the case and C is taken as belonging to no B but to all A, the major premise will be true and the minor universally false. But it is not possible for the minor premise, being negative, to be false in some case, because what belongs to all A also belongs to all B. So if it is taken [as belonging] to none, it will be universally false. For example, let the major term be animal, the minor term rational, and the middle term ensouled. Then if ensouled is said to belong to every animal but to no rational thing, the minor premise will clearly be universally false. But if we take the minor premise, which is negative, as true, it is necessary to take the middle term either as entirely unrelated to the major term (for in this way it will also be unrelated to the minor term) or as unrelated to the minor term but not to the major term. But if it is unrelated to the major term, it will falsely be affirmed of every A and truly denied of B; for example, if the middle term is soulless, since soulless belongs truly to no rational

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thing, but falsely to every animal universally. But if the middle term is unrelated to the minor term – irrational, for example – irrational [belongs] truly to no rational thing but partially falsely to every animal. This is how it is when the minor premise is negative. 80a39-40 For what belongs to all A also belongs to B.

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For if C [belongs] ‘to all A’ and A [belongs] to all B, C ‘also’ [belongs] ‘to’ all ‘B’. 80a40-b2 So if C is taken as belonging to A as a whole, [and as not-belonging to B as a whole, CA will be true and CB false.] He here posits that the major premise453 is true and the minor premise454 is false. 80b2-5 Again, what belongs to no B will not belong to every A either. For if [it belongs] to A, [it belongs] to B too. [But it does not belong [to B]. So if C is taken to belong to A as a whole and to no B, the premise CB is true, but the other premise is false.]

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He wants, then, to take the minor premise as true. Now if the middle term, he says, is going to belong to no B, since B is a part of A, C will not belong to all A either. For if C [belonged] to all A, since A too [belongs] to all B, C too will belong to all B in the first figure. But it is posited [that it belongs] to none. And so C will not belong to all A.455

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80b6-16 Similarly if the privative is transposed. For what belongs to no A will belong to no B either. [Therefore if C is taken to not belong to A as a whole but to belong to B as a whole, premise CA will be true and the other premise false. And again, what belongs to all B is false to take as belonging also to no A. For if it belongs to all B, it must also belong to some A. Therefore if C is taken to belong to all B and to no A, CB will be true and CA false. So it is obvious that there can be a deduction of error among atomic propositions if both premises are false and also if only one of them is.]

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If the major premise is negative, supposing it is true, then the minor premise will be universally false. For if C [belongs] to ‘no’ ‘A’ but A [belongs] to all B, C ‘will belong to’ ‘no’ ‘B’ in the first figure. Now if C is taken [as belonging] to all B, it will be universally false. For example, soulless [belongs] to no animal but to every rational thing. But if the minor premise is true, the major premise will be false only in some case. For what belongs to all B will belong to some A as well. For if C [belongs] to all B and A too [belongs] to all B, C [belongs] to

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some A in the third figure. Now if it is taken [as belonging to] none, it will be false in some case. The terms are animal, terrestrial and winged; terrestrial [belongs] to no winged thing but to some animal. So if we take terrestrial [as belonging] to no animal, it will be false in some case.

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Chapter 17 80b17-20 In the case of things that belong or not belong456 non-atomically, when the deduction of a false [conclusion] takes place through the appropriate middle term, it is not possible for both premises to be false, [but only the one with the major extreme term.] He proposed to present the deductions of error in all the ways they occur.457 Since he has said that they occur either with immediate premises or with non-immediate ones,458 he now passes on to expounding how they occur with non-immediate premises. In the case of immediate [premises] he first presented the affirmative deductions of error that say that what belongs to none [of one thing] belongs to all [of something else].459 Now he presents first the negative [deductions] that say that what belongs to all [of one thing] belongs to none [of something else], and [of these] first the ones in the first figure.460 He says that when the deduction of error infers a falsehood through the same middle term through which a true conclusion is inferred too, the minor premise will have to be true and the major premise will have to be false and universally false. For if it is truly inferred that A [belongs] to all B through middle term C, A belonging to all C and C to all B, and the deduction of error wants to conclude that A [belongs] to no B through the same middle term, clearly it will always keep the minor premise affirmative, as it is by nature, in order for the form to remain deductive,461 and it will transform the major premise, which is affirmative as well, into a negative one. Now if, as we saw, it is true that A belongs to all C, and he assumes that [it belongs] to none, it will be wholly false. Let the terms be substance, animal, man. For if we want to prove that substance belongs to every man through the middle term animal, the [deduction] that produces error will employ the same middle term to prove that substance belongs to no man, and will say that substance belongs to no animal (which is wholly false) and animal [belongs] to every man (which is wholly true). Now just as with demonstrative deductions the contrary error that uses the same middle term always keeps the same minor premise because it is affirmative, and makes the major premise false by transforming the major premise into a negative one, so too even if a deduction is dialectical and true,462 the [deduction that] infers the

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contrary of this will infer it through the same middle term. For he says that it will always have a true minor premise and a major [premise] that is universally false for the same reasons. The middle terms through which demonstrative [deductions] are inferred are those taken from the essential attributes. For example, to prove that man is an animal the middle terms might be ensouled, rational, two-footed, and similar ones which he designates as being from the same column.463 But those who infer the same thing dialectically take engaging in dialectic (for example) as a middle term, or being moved by itself, or being awake or sleeping or things of that sort. For if you employ each of these as a middle term you will infer that man is an animal. These [terms] too he designates as being from the same column because the former ones all infer their conclusion demonstratively, while the latter ones do so dialectically.464 80b20-4 (I call ‘appropriate’ a middle through which the deduction of the contradiction takes place.) [For let A belong to B through middle term C. Now since it is necessary to take CB as an affirmative if a deduction is to occur] 465

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By ‘contradiction’ he here does not mean contradiction in the strict sense but the universally true proposition that is the opposite of the false one. He says that I call ‘appropriate’ that ‘middle through which’ the true conclusion that is the opposite of the false one is inferred. 80b24-6  it is clear that this will always be true, for it does not convert. [But premise AC is false, for when this is converted the deduction becomes contrary.]

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That is, it is not transformed into a negative [premise] by the error.466 For in the first figure the minor premise cannot be negative. But it is necessary to convert467 the major premise and make it negative instead of affirmative in order for the negative falsehood to be inferred. Now if the [premise] that is affirmative is true, clearly the negative one that results468 will be false. 80b26-31 Similarly also if the middle term (e.g., D) is taken from a different column, [if it is in A as in a whole and is predicated of every B. For the premise DB must remain and the other must be converted, and so the former is always true, but the latter is always false.]

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the [deduction] of error is going to use the same middle term as the dialectical deduction used, the same [consequences] will follow.470 80b31-2 In fact this kind of error is almost [the same as that which occurs through the appropriate middle term.] The ‘error’ that has to do with demonstrative deductions is ‘the same’471 as the [error] having to do with dialectical ones when both the [deduction] of error and the one that infers the truth use the same middle term. For in both cases the major premise is always false and the minor premise is always true. They differ in the fact that their middle terms are in different columns, as has been said. 80b32-4 If the deduction does not take place through an appropriate middle term, [when the middle term is under A and belongs to no B, both premises must be false.] If, he says, the deduction of error does not employ the same middle term as the [deduction] of truth uses, but a different one,472 it is possible to take [premises] that are both universally false. For example, if the middle term is taken such that it falls under the major term and is a coordinate member of a division with the minor term, as [occurs] if animal belongs to every man, and, wanting to prove that [it belongs] to no [man] he takes horse, for example, as a middle term: animal [belongs] to no horse and horse [belongs] to every man – and both are universally false. But if a middle term is taken that is more specific than the minor term, the major premise will be wholly false and the minor premise will be false in some case. For example, if animal belongs to every irrational thing473, and we take able to whinny as the middle term: animal [belongs] to nothing that is able to whinny, and this is wholly false. But that able to whinny [belongs] to every irrational thing is also false in some case. It is also possible to take a major premise that is false in some case and a minor premise that is universally false, if we take a middle term that is unrelated to the minor term and belongs to some of the major term and does not belong to some; for example rational, mortal, angel474: rational [belongs] to no mortal thing (and this is false in some case), but mortal [belongs] to every angel, and this is universally false. In this way both are false. It is also possible for one [premise] to be true and one false.475 Now if a middle term is taken that is unrelated to both extremes, the major premise will be true and the minor premise universally false. For example, animal, stone, man476: animal [belongs] to no stone, and stone [belongs] to every man. Since the middle term is unrelated to both [extremes], the major premise is always true since it is negative,

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and the minor premise is always false, and so the major term is truly denied of it [the middle term] and it [the middle term] is falsely predicated of the minor term. But it is not possible to take a true minor premise and a false major premise unless the middle term is appropriate. For477 if C [belongs] to all B, it is possible both that A [belongs] to all B, and that A [belongs] to some C.478 But in fact it is posited [that A belongs] to no [C].479 Alternative proof. If B falls under both C and A, one of these will fall under the other too,480 so that A and C cannot be denied of one another. But neither is it possible to take the major premise as true and the minor premise as false in some case. For example if A [belongs] to no C and C [belongs] truly to some B. For if C [belongs] to all B falsely in some case, it therefore [belongs] truly to some [B]. And so A too does not [belong] to all B. But it is posited that in fact [A belongs] to all [B], which is impossible. 80b35-9 For the premises must be taken contrarily to how they are, if there is going to be481 a deduction. [But if they are taken that way, both become false: for example, if A belongs to D as a whole and D to none of the B’s. For if these are converted there will be a deduction and both premises will be false.]

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‘Contrarily’, clearly ‘to how they are’ by nature. For if C482 falls under A it is clear to everyone that A belongs to C. But C belongs to no B. So if it is necessary that the form turn out to be deductive, let them be taken ‘contrarily’ ‘to how they are’,483 with the result that the major premise becomes negative and the minor premise [becomes] affirmative. If they are taken in this way both will be false. 80b40-81a4 When the middle [(for example, D)] is not under A, [AD will be true and DB false. AD is true because D was not in A, but DB is false, because if it were true, the conclusion would be true too, whereas it was [given as] false.]

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He says how one [premise] turns out to be false and the other true: when, he says, the middle term is unrelated to the major term.484 For it is of necessity truly denied of it. If A belongs truly to no C485, C too will belong to no B. If it is taken [as belonging] to all [B] it will be false. But if anyone were to say that A can belong to no C but C [can belong] to all B, clearly A will also belong to no B. But it was posited [as belonging] to all [B]. 81a5-14 When the error occurs through the middle figure, it is not possible for both premises to be entirely false [(for when B

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is under A, nothing can belong to all of one but none of the other, as was said earlier too), but one can be [entirely false], in fact either one. For if C belongs to both A and B, if it is taken to belong to A but not to B, CA will be true and the other premise false. Again, if C is taken as belonging to B but to no A, CB will be true and the other premise false.] After saying how in the first figure a deduction of non-immediate error occurs that is negative and opposite to the true affirmation,486 he now says in addition how it occurs in the case of the second figure – first, as he also says [occurs] in the case of immediates,487 because in this case too it is not possible to take [premises] that are both universally false. The demonstrations are the same. However it is possible to take [premises] that are both false in some case.488 And to state it briefly, the things we proved to happen in the case of immediate premises will also result in the present case. 81a15-24 [Now if the deduction of error is privative, it has been said when there will be error and through what premises.] But if it is affirmative, then when it takes place through the appropriate middle term, it is impossible for both [premises] to be false. [For if there is going to be a deduction, it is necessary for [premise] CB to remain, as was said previously too. And so AC will always be false, for this is the one that converts. Similarly too if the middle is taken from a different column, as was said in the case of privative errors. For DB must remain and AD must convert, and the error as the same as above.] After speaking about non-immediate negative error, he wants to talk about affirmative [error] as well. Again, this too [occurs] only in the first figure. For the second [figure] draws only negative conclusions, and the third [draws only] particular ones. In this case, when the deduction of error takes place through the same middle term as the true [deduction] (no matter whether it [the true deduction] is demonstrative or dialectical), it is wholly necessary for the minor premise always to be true (since it is the same as the [minor premise] of the true deduction), and for the major premise always [to be] false. For since the major premise in the true deduction is negative, in the [deduction] of error it will be transformed into an affirmative [premise] in order for the conclusion to become affirmative. Now if the negative [premise] is true, the affirmative one will be false. Only the major premise will be universally false, since if the minor premise is converted it makes the form invalid. And so the minor premise is always true and the major premise false.

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81a24-6 But when [the deduction takes place] not through the appropriate [middle], if D is under A, this will be true, but the other [premise will be] false. If an affirmative deduction of error does not proceed through the same middle term as the true one, both premises can be universally false, as when the middle term is unrelated to both extremes. For example, let it be posited that animal belongs to no stone; if anyone were to say that [it belongs] to all [stones] and were to take as a middle term something unrelated to both, for example stick: animal [belongs] to every stick, stick [belongs] to every stone, and both [premises] that he has taken are universally false. But it is impossible to take [premises] that are false in some case. However, it is possible to take a major premise that is universally false and a minor premise [that is false] in some case, if you take a middle term that is more specific than the minor term. For example, fulgurite stone; animal [belongs] to every fulgurite and fulgurite [belongs] to every stone, and it is false in some case. For stone contains fulgurite stone. Conversely, you will make the major premise false in some case and the minor premise universally false if you take a middle term that is more general than the major term, for example, animal, ensouled, stone. In this way both [premises] are false. But it is also possible for one [premise] to be false and the other one true. For if the middle term falls under the major term, the major premise will be true and the minor premise false; for example, animal [belongs] to every man and man [belongs] to every stone. But if the middle term contains the minor term, the major premise is false and the minor premise is true; for example, animal [belongs] to every soulless thing, and soulless [belongs] to every stone. But it is also possible for one to be true and the other false in some case. For example, if I take a middle term that is more general than the major term and that can also belong to the minor term, the major premise will be false in some case and the minor premise will be true; for example, animal [belongs] to every body and body [belongs] to every stone. But if the middle term is more specific than both the major and the minor terms, the major premise is true and the minor premise is false in some case. For example, body [belongs] to every stone, and stone [belongs] to every substance. But it is necessary to note that here the conclusion too is false in some case. If the conclusion is universally false, it is impossible for the major premise to be true and the minor premise to be false in some case. For if A [belongs] truly to all C and C [belongs] truly to some B (for what is false in some case is true in respect of something), it follows that A belongs truly to some B. But it was posited [that it belongs] to no [B]. Therefore it is not possible for the major premise to be universally true and the minor premise to be false in some case.

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81a26-8 For it is possible for A to belong to several things which are not under one other.489 [But if D is not under A, this will clearly always be false (for it is taken as an affirmative)] 490 Here ‘belong’ is taken in the sense of being predicated either affirmatively or negatively.491 For since it is posited that A belongs to no B and he also posited that A [belongs] to all D, clearly in order to infer that D [belongs] to no B and B to no D, he says that this is not at all impossible, given that A belongs to more than one thing either affirmatively or negatively or affirmatively to one and negatively to another, and they are not one under the other. For example let A belong to all D and to no B. I say that B and D do not belong to one another either in the second figure. Terms for this are animal, man, stone. But let something be predicated affirmatively of both, for example, animal [is predicated affirmatively] of both man and horse, and [let something be predicated] negatively of both, for example animal [is predicated negatively] of stone and stick. And clearly neither man and stone nor man and horse belong to one another, nor do stone and stick.

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81a29-31  but DB can be both true and false. [For nothing prevents A from belonging to no D, but D to all B, as animal [belongs] to scientific knowledge(e) and scientific knowledge(e) to music] 492 If, he says, the middle term does not fall under the major term, the major premise will of necessity be false if in fact the major term is affirmed of an unrelated [term], and the minor premise can be sometimes false and sometimes true. For if the middle term is also unrelated to the minor term, the minor premise too is false; we have given examples. But if it contains the minor term, the minor premise will be true. For example, animal, scientific knowledge(e) and music. For animal [belongs] falsely to all scientific knowledge(e) and scientific knowledge(e) belongs truly to all music. 81a31-7  or A from [belonging] to none of the D’s and D [from belonging] to none of the B’s.493 [Therefore it is obvious in how many ways and through what [premises] errors in deduction can occur both in the case of immediates and in the case of what is [proved] through demonstration.] Because when the middle term is unrelated to both extremes, both premises will be false.494

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81a38-9 It is also obvious that if any sense is absent, some scientific knowledge(e) must be absent too, [which it is impossible to get] 495 The purpose proposed is to prove that ‘if any sense is absent’, the scientific knowledge(e) concerned with it is absent too. For example, harmonics is a science(e) whose objects are things that are heard, geometry and astronomy [are sciences(e)] whose objects are things seen, and arithmetic is acquired from just about all the senses, since number is something that sensibles have in common.496 Now if sciences(e) are concerned with the per se attributes of sensibles, it is clear that if any of our senses is absent, supposing that there are some other sensibles in addition to the ones we now know(g) but no science-producing(g) sensation of these,497 it is entirely necessary for some scientific knowledge(e) to be absent. For example, if we did not have the sense of hearing, we would entirely lack the science(e) of harmonics, which we now in fact possess. Similarly, if there were no sense of touch, doctors would lack scientific knowledge(e) of the pulse. If there were no taste, again the scientific knowledge(e) that distinguishes the power of flavors would be absent. And likewise for the rest. This is the present topic. He proves this in the following way. If it is impossible to learn anything in the sciences(e) except through universals (for example that every body has three dimensions or that two sides of any triangle are larger than the remaining one, or that everything that is a boundary of something has one dimension fewer than the thing whose boundary it is, or that all things that are equal to the same thing are also equal to one another),498 and further if demonstrations in sciences(e) proceed through universals and it is impossible for a person who is demonstrating to know(o) or grasp universals except through induction (for when you ask a student to grant you to draw a straight line from any point to any point,499 or any of the other postulates or axioms, you bring confirmation of these through induction)500 – now if we acquire universals through induction and a person without sensation cannot employ induction (for induction has to do with particulars and particulars are sensible), it is clear that if any sense is absent the scientific knowledge(e) is also absent that judges the attributes of the sensibles of which the sense is absent. A person blind from birth will not have scientific knowledge(e) of colours or shapes. Nor can a person deaf from birth have scientific knowledge(e) of harmonics. And of course we do not say this because the senses are the causes of scientific knowledge(e) (for it is not right

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for what is inferior to be the cause of what is superior), but because our souls are possessed by the drowsiness or sleep of birth, as it were, and need the senses to waken them and rouse up the spark of knowledge(g) hidden within us. And just as a person who has deeply forgotten one of his friends might perhaps never come to think of that person without some external prompting, but when someone shows him a coat or something else that belongs to his friend he is immediately moved to think of his acquaintance and he rouses up the impressions in himself,501 so too sensibles stimulate the soul to project the concepts of the paradigms which [concepts] are joined essentially with it.502 For example, when we gaze upward into the heaven and see the order of the things in it we come to think of the one that put it in order, and we proceed backwards in thought from bodies to incorporeal power. Also it is through hearing that we come [to think of] the judgment of truth and from sensible harmony [that we come to think of] the universal formulas of harmony. This is why Plato says that the Demiurge has given us ears and eyes through which we invented philosophy.503 But that the soul does not acquire knowledge(g) of things from sensibles has been sufficiently proved in the commentary on the Phaedo.504

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81a39-b2  if in fact we learn either by induction or by demonstration. [But demonstration is based on universal premises, whereas induction is based on particulars, whereas it is impossible to investigate universals except through induction.] For it is by induction that we learn the axioms and postulates, which are universal premises.505 And through these [we learn] the demonstrables. For demonstrations [proceed] through universals. Now if without induction it is impossible to observe the universals on which demonstrations [depend], it is also impossible without sensation to have scientific knowledge(e) of the universals (I mean the axioms on which demonstrations [depend]506).

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81b2-4 Since even the things called abstract can be known(g) through induction (if anyone wants to make them known(g))507 that some [attributes] belong to each kind 508 That ‘even the things called abstract’ (even they seem to come to be known(g) by thought and not to require sensation) – nevertheless these too become ‘known(g)’ ‘through induction’ and sensation. For a person who has recently approached geometry, upon hearing that things equal to the same thing are also equal to one another, even without understanding very well what is being said, will recognize it by induction, when we say, for example, that if there should be two magnitudes that are each two cubits long, and also a third which is

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equal to one of the ones mentioned, it will of course have to be equal to the other one too, and similarly for the claim that if equals are subtracted from equals the remainders are equal: he will recognize what is said through induction of particulars.509 Likewise too in all other cases. 25

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81b4-5  (even if they are not separate), in that each is the sort of thing it is. That is, even if they do not exist on their own in the way each is said to be – for example equal or surface or anything else of that kind510 – these things can be confirmed through induction in the same way, for example that a surface has only length and breadth (for we are more convinced of this when we have performed an induction through sensible instances),511 and that things equal to the same thing are also equal to one another. Now he may be saying here that we know(g) abstract things too by induction, in the same way [that we know] that every human moves the lower512 jaw or is two-footed or walks erect. For these cannot be separated from their subjects even in thought, the way circles, triangles, and so forth can be. So, he says, it is by induction that we know(g) ‘each’ of these ‘in that’ it is ‘the sort of thing it is’. 81b5-9 But those who lack sensation cannot perform inductions. [For sensation is of particulars, since it is not possible to acquire scientific knowledge(e) of them either from universals without induction, or through induction without sensation.]

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This is the result of his summary of what he said above. For after saying ‘it is impossible to investigate universals except through induction’ (81b1-2) and then establishing this in what follows, he here gives what is missing for his summary, saying ‘those who lack sensation cannot perform inductions’. For if induction is knowledge(g) of particulars and particulars are sensibles, and we know(g) sensibles by sensation, then a person who lacks one of the senses cannot know(g) through induction the sensibles that are subject to that sense. Then, lest anyone say ‘but even though it is impossible to know(g) these things through induction because we do not have apprehensive sensation513 of them, even so it is possible to know(g) them by reason’ – in refuting this very claim he goes on to say ‘since it is not possible to acquire scientific knowledge(e) of them’. For [scientific knowledge(e) comes] neither from universals without induction nor through induction without sensation. For if reason and scientific knowledge(e) are of universals, not particulars, and if the knowledge(g) of universals arises in us through induction, while induction [arises in us] through

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sensation, a person who lacks sensation therefore cannot perform inductions, and a person who cannot perform inductions does not have knowledge(e) of the universals either. Therefore it is completely necessary that if a sense is absent, scientific knowledge(e) of the sensibles subject to that sense is absent as well.

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Notes 1. P. apparently considers the truth of these premises as obvious, but fails to recall that things that are more familiar to us are not necessarily principles. One can imagine that the second premise (‘every coloured thing is a body’) could be demonstrated from more basic premises involving the fact that bodies have surfaces and colour is a property of surfaces. 2. That is, rocks are not the only coloured things. 3. P. makes much use of the word ‘appropriate’ (oikeios), which is not found in the Aristotelian passage. The passage is problematic, since at least superficially it appears to contradict the important fact (discussed in the next chapter of APo) that common principles apply in different sciences, and therefore to things that are not ‘in the same genus’ (homogenês) (76a1). Doubtless with ‘universal’ (in the sense specified at 73b26-74a32) demonstrations in mind, P. interprets the passage as requiring that the terms in all premises must convert. Here and elsewhere (see n. 385) he does not recognize that in a fully worked out genus/species tree schema, the connection between a genus and the species falling immediately under it is immediate, and so provides a suitable (albeit non-convertible) premise for a demonstration. 4. The definition of inscribed figures neglects to specify that the vertices are points on the circle, and the definition of circumscribed figures neglects to specify that each side of the figure is tangent to the circle. 5. Syrianus. 6. P. refers to Physics 185a16-17, which does not specify exactly what Antiphon’s error was. In his commentary ad loc. (in Phys. 31,9-32,3), P. gives a fuller account of Antiphon’s argument which proceeds by inscribing and circumscribing sequences of polygons with smaller and smaller sides, to the point that ‘because they are very small they coincide with the circle’. 7. Aristotle contrasts the two attempted quadratures at SE 172a3-8. 8. Proclus’ objection is based on an unnecessarily ungenerous interpretation of the ‘axiom’ and misses the point, which is that if two things are such that whenever either of them is larger (smaller) than something else the other is also larger (smaller) than it, the two things are equal, not that given two things (9 and 10 in the example) if there is something smaller than both (8 in the example) and something the larger than both (12 in the example), then the two things are equal. 9. Ammonius. 10. The argument concludes that there is a square equal to the circle, not that it is possible for there to be such a square. 11. Proclus’ account of Bryson’s quadrature does not follow from what he has said. This suggests that his account is sheer conjecture. 12. As P. makes clear below (113,13-14), these angles have one straight side (AC and CB, respectively) and one curved side (CD in both cases). He calls ACD the exterior angle and DCB the interior angle. 13. That is, if we take a point Z on semicircle CDB and draw straight line

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CD, as Z approaches C, angle ACZ approaches zero and angle ZCB approaches a right angle. See n. 15. 14. P. may have in mind the property that for any point Z on semicircle CDB there exist points X and Y on CDB such that X is between C and Z, and Y is between Z and B. Angle ACX is less than angle ACZ, and angle ACY is greater than angle ACZ. But this is irrelevant to the question at hand. In fact P.’s claim is not true: he has not proved that either the interior or the exterior angle is larger and smaller than the same acute rectilinear angle; he has proved that the one is larger than any such angle and the other is smaller. 15. ‘Ad infinitum’ (ep’ apeiron) is an odd way to state that the two angles can be increased as close as you like to a right angle and decreased as close as you like to zero, and as the one approaches a right angle the other approaches zero. P. may be thinking of a procedure for determining the sequence of angles that involves the continued bisection (ad infinitum) of the original (rectilinear) angle (ACD). 16. Exterior angle ACG is greater than exterior angle ACD. 17. Euclid, Elements book 3, proposition 16. See Heath (1925) vol. 2, 39-43. 18. Namely, if we inscribe circles whose diameter approaches zero. 19. P. should have said ‘nor will the interior angle become zero’ (or in view of the difficulty of expressing the notion of a angle of zero size, perhaps ‘nor will the two lines that constitute the interior angle ever coincide’.) I conjecture that something has fallen out of the text. 20. That is, smaller than a right angle and larger than zero. 21. Euclid, Elements book 3, proposition 16. 22. P. speaks of constructing a sequence of circles (more precisely, semicircles), presumably each time doubling the radius in the same way as in the construction of semicircle CFE. 23. See n. 15. 24. ‘Always’ perhaps in the sense that no matter how large an acute rectilinear angle we take (even if we take a sequence of such angles that approaches a right angle as a limit), the exterior angle will prove to be larger than every single rectilinear angle we take. 25. The case of the horn angles shows that the interior angle (DCB) is not a right angle but like the right angle ACB it is larger than every acute angle. We may add the obvious fact that both the interior angle and the right angle are smaller than every obtuse angle. Likewise exterior angle ACD is not identical with the straight line AC (whose angular size is zero) but both ACD and AC are smaller than every acute angle. 26. P. surprisingly reverts to Proclus’ interpretation of Bryson’s proof, which he rejected at 112,25-36. 27. Again P. fails to notice that this objection applies to the common principles which are essential to demonstrative sciences (see n. 3). Compare Heath (1949) p.50: ‘Aristotle himself seems to object, not to Bryson’s principle in itself (whatever it was) but only to its use in the particular case because it is not specifically geometrical but is of wider application. But it is difficult to see why it is less legitimate to use it (if true) in a particular application than it is to use, in like circumstances, the axiom that ‘things which are equal to the same thing are equal to one another’, the ‘common axiom’ which Aristotle is constantly citing. It seems, on the whole, that we are not really in possession of sufficient information about Bryson’s procedure to enable us to judge how far Aristotle’s objection was sound’.

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28. This is an odd use of the expression ‘accidentally’ (kata sumbebêkos). See n. 27. One way to solve the problem is to take Aristotle to be saying that Bryson used a principle that does not in fact apply per se to the figure he constructed, but thought that it did. For example (to stay with P.’s discussion), he may have supposed that a principle that applies per se only to rectilinear angles applies per se to horn angles as well and may have based his proof on an application of it to horn angles. In that case even if the principle happens to hold for horn angles (or for the horn angles in the proof), so that the argument yields a true conclusion, it is not a demonstration of the conclusion (and so Bryson does not know(e) it). In this case the conclusion holds but not ‘qua that very thing’. The principle in question is ‘common’ in the sense that it holds per se of straight lines and accidentally of the horn angles in question. And so it is used in arguments that ‘apply to other things that are not in the same genus’ of angles as horn angles – that is, arguments that apply to straight lines. In the case in point, assuming that the principle does hold (albeit accidentally) of the horn angles, we do not know(e) the conclusion but know(g) it accidentally. I put this interpretation forward because as far as I know it has not been suggested previously, but I admit that in some places it requires taking the Greek in not the most natural way. For example the two occurrences of kai in 75b42 need to be taken to mean ‘in fact’ rather than ‘also’ (as I have translated it in the lemma). And genos (76a3) does not refer to the subject genus of geometry but to a class of angles (similar sungenôn - 76a1). 29. This theorem is Euclid, Elements, book 1, proposition 32. 30. That is, LBAC + LACB + LCBA = LACB + LBCD. 31. The word ‘two’ is added by Wallies. 32. Euclid, Elements, book 1, definition 10. The general claim needed to prove the theorem in question, that all adjacent angles (not just the case where adjacent angles are equal) are equal to two right angles (Euclid, Elements, book 1, proposition 13) follows trivially from this claim. 33. ‘The problem’ is Wallies’ addition. Without it, the text reads: If the predicated belongs per se in the subject. 34. This is not a term, but a proposition (in fact, the conclusion of the demonstration). Rather, Aristotle is referring to the predicate: ‘has angles equal to two right angles’. 35. This too is a proposition (in fact, the minor premise), not a term. 36. APo 1.7, 75b8-12. 37. That is, the immediately superior science. ‘Proximate’ shows that P. is thinking of the cases mentioned at 79a1-2, a10-13, in which the subordination relation holds for more than two sciences: optics is immediately superior to the study of the rainbow, and geometry is immediately superior to optics. 38. P.’s definition of concordant implies that any pair of numbers that are not relatively prime to one another is concordant. The proemium of the pseudoEuclidean Sectio Canonis defines as concordant any pair of numbers in which one is a multiple of the other or has to the other the ratio of (n+1) : n. Cf. pseudo-Aristotle Problems 921b1-13. 39. APo 1.7 75b8-9. 40. Strictly speaking the ‘why’ is explicated by ‘why it is concordant’. P. says that the explanation for the fact that it is concordant will be that it has a certain ratio of numbers and that this ratio is concordant. Further, knowledge that this ratio is concordant will depend on a more general understanding (also the

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province of arithmetic) of what ratios are concordant, which is a consequence of the definition of concordant he has given above, 117,22-5. 41. The commentary indicates that Philoponus’ text read to de dihoti heteras, tês anô, although the lemma as well as modern texts of APo read to de dihoti tês anô. 42. Namely, the superior science. P’s claim here is consistent with his belief that concordant is primarily an attribute of ratios of numbers (117,22), but I find this interpretation hard to accept. For example, consider an example taken from Euclid’s Optics, which P. uses below (178,20-179,12): a demonstration in optics that proves that ‘things that are seen under a larger angle appear larger’. The attribute proved, ‘appear larger’, has to do with how things appear – and this is not the concern of geometry but of optics. On the other hand, the proof proceeds through middle terms that are purely geometrical, for example, that one angle is larger than another. I suppose these are the terms that Aristotle means when he speaks of attributes that belong per se to the superior science. 43. Ammonius calls this Aristotle’s definition of first philosophy (Commentary on Porphyry’s Introduction, 9,26), but it is not found in any of Aristotle’s surviving works. However, this is how Simplicius characterizes first philosophy (Phys. 47,30-1) and how Michael Psellus (Opuscula Logica, 49, lines 116, 136 and 209), and John of Damascus (Dialectica 3,18 and 66,11 and Fragmenta Philosophica, 8,9, where he identifies it as one of six standard definitions of philosophy) define philosophy . Cf. also Themistius, On Practical Wisdom 300c5 (where both descriptions are applied to practical wisdom). 44. Omitting the full stop. 45. This use of ‘principle’ (also found at 119,13.14.19) to refer to a science is remarkably casual. 46. For the latter instance of subalternate sciences, see APo 76a23-4. The former instance does not occur in APo, but at 79a13-16 Aristotle says that medicine is not subordinate to geometry but gives a case in which the doctor knows the ‘that’ and the geometer knows the ‘why’. 47. See n. 45. 48. See n. 43. 49. If P. is not simply referring to the entire Aristotelian work, he may be thinking specifically of Metaph. 4.1-3, although the expression ‘first philosophy’ is not found there. (In Metaph. it occurs at 993a15, 1026a24, 1061b19.30). 50. P. neglects the case of things we know immediately through perception. 51. P. makes this point several times, notably at 26,9-15, 49,13-15 and 97,25-31. 52. P. wrongly supposes that Euclid, Elements, book 1, proposition 2 is deduced exclusively from the principles (definitions, postulates and common notions) stated at the outset of the book. (He gets this correct at 125,2-3 and 148,19-23.) This is not the case. His claim is true for proposition 1, but beginning with proposition 2 the conclusions of previously proved theorems are used in the proofs: the proofs are not proved exclusively from immediate principles. P. is wrong to suppose that they depend on premises that are not proper, in the sense that their subject or attribute is not in the subject genus of geometry. 53. APo 1.10, 76b10. 54. More precisely, the definitions of point, line, and so forth are principles of geometry as are the hypotheses that points and lines exist. P. would have done better to distinguish between the ‘things’ (the members of the subject

Notes to pages 24-27

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genus of a science) and the definitions and assumptions of existence of such things. The latter are principles, not the former. 55. P. reads ho ti where the manuscripts and modern editions have ti. See 121,21-2. With ti the translation would be ‘what’. 56. APo 1.2, 72a14-24. 57. P. refers to APo 1.10, 76b11-12 (cf. 125,7-8), but there Aristotle says that there are three things ‘with which every science is concerned’, not that demonstrations employ three kinds of things. 58. P. refers to his discussion at 7,18-10,4. 59. The sense in which premises ‘are’, namely, they are the case, is different from that in which the givens ‘are’, namely they exist. P.’s interpretation is probably based on his understanding of the following passage (76a33-4), where the principles, not the givens are said to ‘be’. Contrariwise, Ross (1949) p.538 paraphrases ‘principles’ in the latter passage as ‘primary terms’ (i.e., primitive subjects). 60. Aristotle’s examples at 76a34-6 show that he has in mind not premises and conclusions but the subjects and attributes that the science treats. 61. See n. 55. 62. See n. 59. 63. P. adds ti before trigônon. 64. 9,9-27. 65. 9,9-10 and 9,18-19. 66. Having talked of the unit as a given for which we do not prove that it is, P. now says that what we do not prove is that this is a unit. 67. See n. 65. 68. 75b2-3. See P.’s comment at 98,23-99,4. 69. Aristotle says that they hold ‘in virtue of a proportion’ (76a38-39) not that they hold homonymously, which would contradict Aristotle’s statement (75b23) that it is the same axiom that applies in different sciences. 70. This is not exactly what Aristotle says at 76a42-b1. 71. 10,27-11,3. 72. Aristotle never says that there are axioms that are proper to a single science. This misinterpretation is important to P.’s discussion of scientific principles. See McKirahan (2009). 73. Pace P., difference in subject does not entail homonymy of an attribute. (Humans and birds are both two-footed, and synonymously so.) 74. Euclid defines a line as breadthless length (Elements, book 1, definition 2). The alternative definitions presented by P. are given by Proclus Commentary on the First Book of Euclid’s Elements, 97.7-8 (Friedlein) and go back in essence to Aristotle (de An. 409a4, Metaph. 1016b25-27, 1020a11). 75. The first of these definitions is Euclid’s, the second is found in Plato (Parmenides 137e) and Aristotle (Topics 148b27). 76. At 76a37 Aristotle says ‘some things’ not ‘some axioms’. 77. P. introduces this terminology for subjects and attributes at 7,20-23. 78. P. disregards the distinction between primitive subjects (which are assumed to exist) and derivative subjects (whose existence is proved). 79. This sentence explicates the first sentence of the lemma. 80. Compare Euclid, Elements, book 10, definition 3 (‘ let the assigned straight line be called rational, and those straight lines which are incommensurable with it irrational’). P. does not make it as clear as he might that

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‘irrational’ applies to pairs of magnitudes; a line can only be irrational in relation to another line. 81. That is, that it does not have a ratio to the side. 82. P. is probably referring to the proof referred to by Aristotle (APr 41a26-7) and which was inserted into the text of Euclid (as Elements, book 10, proposition 117 and is printed in Heiberg’s text as book 10, appendix 27). See Heath (1925) vol. 3, p.2 and Knorr (1975) p.227-32. 83. P. reports that some manuscripts known to him had analogon (‘proportional’) instead of alogon (‘irrational’). 84. That is, both arithmetic and geometry make use of proportions: numbers as well as lines and areas can be proportional to one another. 85. P. is referring to Euclid, Elements, book 1, propositions 1-3. See 120,7-14. 86. Astrologia was the usual term for astronomy through the fourth century BC; astronomia is first used in the third century (except for a single occurrence in Aristophanes). 87. Aristotle’s view is that a science posits the definitions (‘what it is’ or ‘what it signifies’) and the existence (‘that it is’) of primitive subjects, while it posits the definition of each non-primitive subject and demonstrates ‘that it is’. P. appears confused on this point. 88. 121,3-19. 89. P. again neglects the derivative subjects, whose existence is proved to follow from the existence of primitive subjects (see n. 78). Also, P. earlier made this claim for ‘givens’ but did not mention the subject genus (or simply the genus) in this connection. 90. 121,7-15. 91. 121,16-18. 92. At 75a37 Aristotle talks of ‘things that they use in demonstrative sciences’ and at 76b11-12 he says that ‘every science is concerned with three things’, but he has not previously identified a trio ‘of which every demonstration consists’. 93. Aristotle has not said in the present chapter that ‘what it signifies’ is assumed for the axioms, but support for this claim can be found at 71a12-14. 94. Aristotle’s important and frequently invoked distinction between what is better known by perception and what is better known in itself (e.g., 71b3372a5) should have made P. more wary. In fact, Aristotle does define what hot and cold signify (GC 325b25-9). Moreover, Aristotle’s present point is about omitting existence claims, not definitions. 95. P. has arithmós esti where Aristotle has arithmòs ésti. 96. This sentence is completed in the next lemma. 97. In this sentence and the next P. inexplicably shifts to indirect discourse. O. Goldin suggests that P. is here reporting the opinion of Ammonius. Alternatively, something may have fallen out of the text. 98. Euclid, Elements, book 7, definitions 1-2 (with the only difference being that in the definition P. has kath’ ho where Euclid has kath’ hên; the translations of both versions are identical, ‘that in virtue of which’). 99. P. has an where Aristotle has ean. 100. See n. 92. 101. P. previously has talked of the genus as consisting of terms that he calls givens. Now it is the genus itself that is called a given. 102. At 125,15 the affections belong per se to the subjects; now they are said to belong per se to the genus.

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103. 76b11-22. 104. This summary of 76b12-15 is derived from P.’s interpretation, but in fact is far from what Aristotle says. 105. Aristotle distinguishes axioms, definitions and hypotheses at 72a14-24; postulates are not mentioned prior to the present passage. P. discusses the differences among axioms, definitions, hypotheses and postulates at 34,6-36,17. 106. This division and the corresponding terminology are not used by Aristotle. 107. This is not Aristotle’s reason for calling this kind of principles ‘common’. For Aristotle they are common to more than one science. Aristotle does not employ this kind of psychologizing approach in distinguishing the kinds of scientific principles. 108. P. interprets einai at 72a20 as predicative, not existential, but in his commentary ad loc. he does not seem to think of hypotheses as being simple predications (37,7-13). 109. In these lines P. identifies three kinds of hypotheses: those ‘called by the common term hypotheses’, those ‘called hypotheses in a special sense’ and those ‘called hypotheses in the strict sense’. Aristotle does not use this terminology, and mentions only two uses of the word ‘hypothesis’, only one of which applies to a class of scientific principles. 110. The genus P. refers to consists of the hypotheses ‘called hypotheses by the common term’, which are immediate premises, namely, scientific principles. What he here refers to as hypotheses in a special sense are not principles (see 76b27 and 131,23-25), and so cannot constitute a species of the genus. 111. P.’s account of the distinction between hypotheses and postulates is flawed because he fails to notice that hypotheses in a special sense are not principles, and so the corresponding kind of postulates are not principles either (76b27-34). 112. This application of the term ‘hypothesis’ is not mentioned by Aristotle. 113. P. holds that some definitions are hypotheses (those that are obviously true) and others (those whose truth is hard to see) are postulates, despite the fact that definitions are unprovable principles while postulates and the relevant kind of hypotheses are provable (76b27). 114. P. paraphrases Euclid’s definition of triangle and circle (Elements, book 1, definitions 19 and 15). 115. Euclid’s definition of circle makes no reference to the use of a compass. 116. Since Aristotle holds that definitions are indemonstrable principles and Euclid does not prove any of his definitions, it is hard to imagine the source of this extraordinary claim. P. may be using the word ‘demonstration’ in a casual way that covers informal explanations. 117. It is hard to understand how these claims could be considered definitions. 118. P. is saying that a definition hypothesizes that the definiens is the same as the definiendum. 119. Sc. of its terms. 120. In the affirmation ‘A belongs to B’ and the negation ‘A does not belong to B’, the terms (A and B) are the same but the relation they have to one another is different. 121. ‘Problem’ (problêma) is a technical term prominent in the Topics. At Top. 101b28-36 problems are said to be identical with (dialectical) premises (protaseis) except for the manner in which they are expressed. ‘Is animal the

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genus of man?’ is a premise, whereas ‘Is animal the genus of man or not?’ is a problem. In the present passage, P. applies the term ‘problem’ to the initial statement in a proof of the conclusion to be proved. This is adapted from mathematical terminology. In his Commentary on the First Book of Euclid’s Elements, Proclus distinguishes between theorems and problems, specifying that a theorem requires us to prove that a given geometrical object has a certain attribute, while a problem requires us to construct a certain geometrical object given certain other geometrical objects (In prim. Eucl. 81,5-22). See also 32,20-23. 122. The problem proposed for demonstration is a proposition identical with the conclusion of the demonstration. The difference between them is that in the latter case we know the proposition scientifically and in the former case not yet. 123. Aristotle does not make this distinction between definitions as found among scientific principles and definitions as they are used in demonstrations, but it is an important distinction nonetheless. See also 130,16-23. This way of conceiving definitions is introduced at 35,2-19. 124. P. quotes Euclid, Elements, book 1, postulates 3 and 1. 125. At 76b32-34 Aristotle applies the term ‘postulate’ to two sorts of cases: 1) something that is opposite to the opinion of the learner (Aristotle does not specify whether such postulates need to be true) and 2) something that can be demonstrated (and therefore is true) and is used without proof. 126. P. is speaking from an Aristotelian viewpoint. Aristotle rejects Democritus’ atomism in Gen. Corr. 1.8. He accepts the existence of homoeomeries, but holds that they are not principles but derived from the four elements (e.g. Meteor. 389b26-28). 127. P. alludes to Euclid’s Fifth Postulate: ‘if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on the side on which the angles are less than two right angles’. 128. Proclus mentions Ptolemy’s book (in Eucl. 365,7-11) and gives some details of his proof (365,14-367,27). He gives his own proof as well (371,24373,2). See discussion in Heath (1925) vol. 1, 204-208. 129. Apparently P. is using this word to describe cases where a hypothesis is apparent to someone. Natural hypotheses were not identified in P.’s taxonomy of hypotheses at 127,26-33. See n. 130. 130. P. does not elsewhere speak of axioms as ‘natural’. However, the context shows what he means. He is here contrasting hypotheses in the generic sense with axioms. Axioms we know ‘from ourselves’ and are ‘evident’ and ‘self-guaranteeing’ (34,10-11, 34,18), but hypotheses are not evident without qualification, but require some attention for their truth to be seen (34,19) and are posited by the teacher (34,21-2). The axioms, then, will be natural notions in the sense that we have them naturally. For the idea cf. P. in APr 49,18-20. 131. For the Neoplatonic doctrine of projection, see Siorvanes (1996) and Sorabji (2004a). 132. This psychologizing approach to the status of principles is alien to Aristotle. See n. 107. 133. P. quotes Euclid, Elements, Book 1, Postulates 1 and 3 (substituting grapsai for graphesthai in Postulate 3). 134. APo 1.2, 72a14-24. 135. Nothing in the Aristotelian text corresponds to ‘yet’. 136. See n. 107.

Notes to pages 31-37

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137. This claim does not hold for Euclid’s construction postulates (Elements, book 1, postulates 1-3), which P. frequently uses as examples of postulates. 138. This is not correct; a definition (or rather definiens) has more than one term. 139. See 128,23-32. 140. At 127,30-3 P. defined hypotheses ‘in the strict sense’ (kuriôs) and at 128,13-15 he distinguished these from hypotheses relative to the learner. Both kinds of hypotheses appear true to the learner, but while the former are immediate principles the latter can be proved. He has not elsewhere spoken of hypotheses ‘without qualification’ (haplôs), which appear to be identical with hypotheses ‘in the strict sense’. Postulates ‘in the strict sense’ are distinguished from postulates ‘relative to the learner’ at 129,5-11. In both cases the postulate does not appear true to the learner but whereas postulates ‘in the strict sense’ are true and require little explanation, postulates ‘relative to the learner’ are either (1) false or (2) true and require more explanation. 141. 129,6-7, 129,11-15. 142. P. has oude  legontai where Ross has ouden  legetai. See n. 143. 143. The first sentence in the lemma can also be rendered ‘for nothing is said to be or not be’, where ‘be’ is used existentially. (This is how it is taken by Ross [1949], p.538, who holds that the hypotheses that are scientific principles are existence-claims [cf. 72a18-20].) P., who holds that all scientific principles that are subject-predicate propositions are hypotheses (127,26-7) and that definitions are not subject-predicate propositions (35,5-7), takes ‘be’ predicatively: definitions do not say that the definiendum has or does not have any predicate. 144. This claim may be surprising, since P. has said that all scientific principles that are subject-predicate propositions are hypotheses (127,26-27), and the proposition ‘man is an animal’ can be used in demonstrations about man. But P. has committed himself to the view that the middle term of a demonstration must be ‘appropriate’, apparently in the sense of being coextensive with the major and minor terms (111,8-15), and clearly man is not the only kind of animal. 145. P. may be referring to Protagoras, whom Aristotle reports to have refuted the geometers, saying that a circle is not tangent to a ruler at a point (Metaph. 998a2-4). 146. In this passage P. consistently speaks of ‘Ideas’ (ideai) rather than ‘Forms’ (eidê) for Plato’s entities, despite the fact that eidos occurs in the lemma, perhaps to reserve the latter word for the Aristotelian notion, which he uses at 133,29. 147. For Plato, no sensible is a perfect instance of a Form. Helen of Troy is both beautiful and not beautiful. 148. This use of ‘primarily’ recalls the sense in which the property of having angles equal to two right angles belongs primarily to triangle, as opposed to isosceles triangle (73b33-74a1). In such cases subject and attribute are coextensive. See n. 151. 149. A typical use of the ‘one in addition to the many’ principle as applied by the Platonists. 150. 83a33. 151. Again P. has in mind the special sense of ‘universal’ defined at 73b2674a3. But the reference to the need for one thing that belongs to many shows that Aristotle has in mind not this kind of universals but the universals that

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are contrasted with particulars, at least one of which is required to form any syllogism. This interpretation of P.’s affects his understanding of the following lemma too. 152. P. refers to the constellation Cancer (in Greek, karkinos, which also is the name of the disease), and to forceps. 153. The sea-animal is the dog-fish, a species of shark found in the Aegean, the star is Sirius, known as the Dog Star, and the philosopher is Diogenes of Sinope, founder of ancient Cynicism (from kuôn, dog). 154. In the Topics Aristotle urges us to be on guard against homonymy, prominently in Top. 1,15. 155. In the Greek text 135,3-23 is a single sentence. I have broken it up to make it more palatable in English. 156. It should be noted that this argument is not strictly speaking a syllogism, since (for example) the predicate of the conclusion (‘equal to B’) is not found in either of the two premises. 157. A remarkable characterization! 158. The Neoplatonists held that the soul is immortal and eternal in its essence but that its external activity, which imparts life to the body, may be discontinuous. See, e.g., Plotinus 6.2.22.22-33 and Philoponus, in De An. 246,23-247,10. 159. The supplement is due to Wallies. 160. See n. 158 with references. 161. 16a29-32. This term is used for terms like ‘not-man’. 162. Accepting Wallies’ conjecture katêgoroumena in place of the MS reading kataphaskomena. 163. P. is casually calling any statement of the form ‘this and not not-this’ an axiom of contradiction. 164. This is an example of the case where the axiom of contradiction applies to the major term. 165. It is hard to see how ‘Callias and not not-Callias’ could figure into a syllogistic argument the same way that ‘animal and not not-animal’ does. According to 77a15-16 the latter is incorporated into the major premise as follows: ‘man is an animal and not a not-animal’, which amounts to ‘man is an animal and man is not a not-animal’ – that is, ‘not’ negates not a term but a proposition. For this to happen, it seems that the ‘not a not-X’ expression must be applied to a term that is a predicate in a premise – as we would expect, given Aristotle’s understanding of the principle of non-contradiction as stating that contradictory predicates cannot apply to a subject. This argument accords with P.’s explanation (just below) of why the axiom cannot be used with either the minor or the middle term. 166. 17a17-18 with 16a29-32. 167. P. does not recognize that this premise is an application of the law of the excluded middle. Also 140,11-12. 168. P. refers to the proof of the incommensurability of the side and diagonal of a square mentioned above,n.82. 169. Themistius, Analyticorum Posteriorum Paraphrasis, ed. M. Wallies, CAG V.1, 1900, 24,3-6. 170. 135,8-27. 171. The text at 139,1-2 is uncertain. I have translated Wallies’ text. 172. See 136,22-137,7 for a discussion of such cases. 173. P. adds pan before mê zôion.

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174. pan (‘every’) is P.’s addition, it is absent in the MSS of Aristotle. 175. This seems to be an impossible interpretation of the Aristotelian text, in which ‘man’ is the predicate, not the subject. 176. Philoponus again fails to recognize that proofs per impossibile exploit the law of the excluded middle and not the principle of non-contradiction. See n. 167. 177. 10,27-11,3; 34,10-13; 36,13-17; 123,15-19. 178. For the terminology and for P.’s application of it to demonstrative sciences, see 7,20-8,5, 8,20-4, 10,7-13. The givens are typically the subject, and the soughts are typically (but not necessarily, cf. 8,21-4) the predicate in the proposition to be proved. In effect, the givens and soughts constitute the subject genus of a science. 179. In the case of these pairs of sciences, the subject genus is the same ‘in a way’ (75b3-16, 76a8-15). 180. Aristotle has discussed this principle only in the present chapter (1.11), where the way he claims it is used in demonstrations can occur in every art and science, even though he does not say so. Aristotle has given the law of the excluded middle as an example of a principle (71a13-14). At 72a16-18 he says that axioms are principles that we need to have in order to learn anything at all, a claim that applies to the principle of non-contradiction, as his discussion in Metaph. book 4 shows. On the other hand, P.’s confusion of the principle of non-contradiction with the law of the excluded middle (see n. 167 and n. 176) makes it possible that P.’s reference is to 71a13-14. 181. E.g., 982a2, 1005b2, 1059a18, 1060a10. 182. P. refers to Metaph. book Gamma (i.e., book 4). 183. In Metaph. book 4, chapter 4 Aristotle argues that this axiom cannot be demonstrated (1006a3-11) and then goes on to support the axiom by ‘demonstrating it by refuting’ (apodeixai elenktikôs), which he explicitly distinguishes from demonstration in the proper sense (1006a15-18). 184. Apprehension (katalêpsis) is assent to an apprehensible impression (katalêptikê phantasia) – an impression on the soul that comes from an existing thing and that guarantees its own truth. 185. 1007a7-10 and context. 186. Keeping the MS reading deon; Wallies prints his own conjecture dein. 187. P.’s outline of Aristotle’s argument is not anything like the kind of demonstrations discussed in APo. 188. 76a38-40. 189. This comment tends to undermine P.’s view that the axioms are homonymous (122,26-123,13). 190. P. refers to his remarkable interpretation of axioms according to which every major premise of a demonstration is an axiom (8,7-8, 10,27-11,3 and 123,15-18). 191. This sentence is completed in the next lemma. 192. See n. 190. 193. The link between dialectic’s use of reputable opinions (endoxa) and its unrestricted scope of inquiry is made in the first sentence of the Topics (100a18-20). 194. See n. 183. 195. See above, 141,3-19 with n. 184. 196. P. is anticipating the following line in the Aristotelian text, which denies that dialectic has any ‘definite’ subjects. This claim is standardly taken

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to mean that dialectic is not restricted to any particular subject matter, but P. disagrees (144,19-145,10). 197. This sentence is completed in the next lemma. 198. Since Aristotle does not make this claim, I take it that it is a parenthetical remark of P.’s. 199. This procedure does not accord with Aristotle’s description of dialectical discussion in Topics book 8, as a conversation in which one person (‘the answerer’, corresponding to P.’s interlocutor) puts forward and attempts to defend a thesis, while the other (‘the questioner’, corresponding to P.’s dialectician), asks a series of questions based on reputable and plausible opinions, with the goal of leading the answerer to give answers contradictory to his thesis. In P. it is the dialectician who attempts to establish a thesis. 200. This is an exaggeration, doubtlessly based on the distinction between what is better known to us and what is better known in itself (71b33-72a5). However, scientific premises are better known to experts (cf. Metaph. 983a1921 and 1029b1-12). 201. P. presumably refers to APr 2.15, especially 63b31-9, although the topic of the chapter has to do with syllogisms generally, not specifically with demonstrations. 202. This is mistaken. Proofs per impossibile proceed by assuming the contradictory of the desired conclusion and showing that it entails something false, not by showing that the desired conclusion follows from the contrary of a premise used in a direct proof of the conclusion. 203. This implausible interpretation of ‘definite’ is an attempt to make sense of the conjunction ‘for’ in 77a32. Shortly below (145,26-8) P. takes ‘definite’ in the sense he here rejects. See also n. 196. 204. 77a32. 205. Keeping the MS reading hôs which Wallies deletes. 206. For the example used in a similar context, see 75b17-20. 207. Aristotle does not make this point, for which see above, 117,15-26. 208. Mousikos (‘musician’). Aristotle once speaks of these people as ‘harmonicians who deal with numbers’ (hoi kata tous arithmous harmonikoi, Top. 107a15-16), but apparently the term did not catch on. 209. P. refers to the subalternate sciences. For these examples, cf. 75b15-17, 76a23-5. 210. P. is speaking vaguely; the doctrine of the four elements is not a single theorem. 211. Ammonius. The solution to the problem is based firmly on what Aristotle says (78b33-79a16), but the context of the Aristotelian passage has nothing to do with the problem P. has raised. 212. 77b1-2. This statement makes it difficult to see how optics can be regarded as anything more than a topic in geometry (like theorems relating to circles or triangles, for example); it neglects the fact that optics has its own principles that are not within the purview of geometry. See the ‘definitions’ at the beginning of Euclid’s Optics. 213. Accepting the suggestion of an anonymous referee . 214. De Int. 20b22-30. 215. For this distinction, see Lausberg (1988) §770. (I am indebted to an anonymous referee for this reference.) 216. 77a29-34.

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217. P. adds ê before deiknutai. 218. P. omits ha after ê. 219. See n. 212. 220. Euclid. See n. 52. 221. Similarly 75b17-19. 222. For the quadratures of Antiphon and Bryson, see 111,17-114,17. 223. Simplicius preserves Hippocrates’ proof (in Phys. 61,5-68,32). For summary and discussion, see Heath (1921) vol. 1, pp.183-200. 224. This recalls Gorgias’ boast to be able to speak more persuasively than experts in any field (Plato, Gorg. 456a7-c7). 225. 77b24-6. The terminology of ‘in respect of negation’ and ‘in respect of disposition’ is not Aristotelian, and P.’s insistence on employing leads to confusion at 153,4-15. 226. Soph. 229b7-8. 227. 77b19-20. 228. If geometers can mistakenly assume false premises, why can they not also make mistakes in reasoning? P. seems to be inflating Aristotle’s claim at 77b27-8. Also, it is bizarre to describe false premises as ‘invalid matter’. 229. Hamartia. P. refers to Top. 148a4-5; the word does not occur in APo. 230. This sentence is continued in the next lemma. 231. This terminology for kinds of mistakes is not found in Aristotle. 232. That is, as an example. Cf. 152,11. 233. This sentence is continued in the next lemma. 234. Aristotle presents three types of deductions due to ignorance: (1) deductions with false geometrical premises; (2) deductions based on illegitimate reasoning; (3) deductions with premises drawn from a different science. P. says that type (1) and (2) mistakes are due to ignorance in respect of disposition and type (3) mistakes are due to ignorance in respect of negation. However, type (3) mistakes do not involve having no opinion on a geometrical matter, but having the false opinion that something irrelevant to geometry is relevant. See n. 228. 235. Aristotle is pointing out that there is a sense in which what P. calls ignorance in respect of disposition can be considered scientific. It is hard to see how P.’s ignorance in respect of negation is relevant here. See n. 225. 236. Pseudaria (to be distinguished from paralogismos, which I translate ‘invalid inference’). Euclid wrote a work with this title. Since Proclus (in Prim. Eucl. 70,1-18) says that Euclid’s Pseudaria train beginners at geometry how to find invalid inferences (paralogismoi) it is not clear that P.’s account of ‘fallacies’ is correct. 237. 77b18-20. 238. 151,24-26, with n. 228. 239. P. has ditton aei where Aristotle has aei to ditton. 240. Namely, the cause of the kind of falsehood in question. 241. This example is found also at 2,13-14. 242. P. seems to assimilate dialectic to sophistical reasoning. (See, e.g., Soph. El. 165b30-166a21). 243. But Aristotle is talking about the absence of illegitimate inferences in mathematics, not in sciences generally. 244. This example is based on the practice of referring to groups of epics as ‘cycles’ – for example, the Trojan Cycle, which included the Iliad and the Odyssey and also other poems that tell the background of the Trojan War and the war before the Iliad and events after the fall of Troy. Proclus (although it

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is not certain that it is the same Proclus as the fifth-century Neoplatonist) wrote an account of this ‘cycle’ which is partly preserved in Photius’ Bibliotheca. 245. De Int. 17b14-16. 246. This sentence is concluded in the next lemma. 247. Who this person is is unclear. The remainder of this comment is a digression that is long even for P. 248. This definition does not apply to epic poems. 249. A closely related version of this epigram (lacking the third verse) is found in Plato (Phaedrus 264d), who remarks that it makes no difference which line is placed first or last, in Diogenes Laertius (1.89-90), who gives it an additional verse, and in other authors. See De Vries (1969) p.212-13. 250. A possible reference to Heraclitus B103. 251. Pseudo-Plutarch Life of Homer, chapter 11. 252. Enkuklia mathêmata, synonymous with enkuklios paideia. 253. The reference is unclear. 254. The subject of comedy. 255. P. refers to Aristotle’s contrast between poetry and history on the grounds that the former (including comedy) treats universals – in the sense of what kinds of things a certain kind of person would do, with an individual name attached to the character who represents that kind of person – while the latter treats particulars (Poet. 1451b5-14). 256. Pisander of Laranda (third century AD). In his work Marriages of Gods and Heroines, he gave a unified account of mythological and human history beginning with the marriage of Zeus and Hera. Apparently his work was so popular that the earlier epics (excepting Iliad and Odyssey) ceased to be read. See RE Bd. 37, 1937, 145-146. 257. For the specialized sense of ‘objection’, see APr 2.26, especially 69a37b1, where Aristotle says that objections may be either universal or particular. However, P. probably correctly identifies the objections under discussion as ‘scientific’ objections, a special type of objection which may be used as premise in a demonstration (77b38-39) that proves not simply that the conclusion of the objectionable proof is not true, but that its opposite is true (158,9-10). 258. ‘These’ corresponds to ‘it’ in the first clause of the lemma. 259. P. is taking the referent of ‘its’ in the first sentence of the lemma to be ‘objection’. 260. Here only syllogisms in Barbara, Celarent, Cesare and Camestres, are under consideration. 261. Aristotle says, rather, that the same propositions are premises and objections. See n. 262. 262. But the claim is that the objection becomes a premise. 263. This discussion does not consider the syllogism: ‘no epic is a circle, every circle is a geometrical figure, therefore no epic is a geometrical figure’, which would avoid the objection in n. 262. Likewise for the following example. 264. Ammonius. 265. See n. 257. 266. Ammonius? 267. P. has ta hepomena amphoterois where Aristotle has amphoterois ta hepomena. This sentence is concluded in the next lemma. 268. See 154,11-20 for this interpretation. Also n. 242. 269. For ‘matter’ and ‘form’ used in this non-Aristotelian fashion, see 152,13153,2.

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270. False. If dog is the genus of fox terrier and beagle, nothing about the relation between fox terriers and beagles follows from the premises ‘dog is predicated of fox terrier’ and ‘dog is predicated of beagle’. P. has another try at 159,8-17. 271. P. correctly defines multiple ratio and correctly states that 200 to 800 is the same multiple ratio as 2 is to 8; his definition of superparticular ratio is non-standard and includes both what are normally called superparticular ratios (where the ratio is n+1 to n, example: 3:2) and superpartient ratios (where the ratio is n+k to n, where k>1, example: 5:3). The property of multiple ratios that he identifies does not hold for what he calls superparticular ratios. (However, see n. 272) For these definitions and discussion, see Nicomachus Introductio Arithmetica 1.18-20. 272. Is 300:200 a superparticular ratio? Not according to the definition given above, according to which 300:299 and 201:200 are superparticular. But 300:200 is equal to 3:2, which is superparticular, and Nicomachus includes this kind of proportion as a species of the superparticular (Introductio Arithmetica 1,19,4). 273. Ammonius. 274. Continuous proportions (here called proportions with multiple ratios) are not multiple proportions as P. defined them at 160,6. In fact, while continuous proportions were probably the ‘fastest growing’ sequence known in antiquity, they do not increase as fast as the sequence of factorials: 1! (= 1), 2! (= 2x1), 3! (= 3x2x1)  n! (= n x (n-1)!). 275. Nicomachus Introductio Arithmetica 1,19,9-15. It should be noted that this sentence is not relevant to the point P. is making in this note. 276. According to Ross (1949), 548, ‘no sophist of the name [Caeneus] is known and P. is no doubt merely guessing’. Ross believes that the reference is to the Lapith Caeneus in Antiphanes’ comedy of the same name. 277. This sentence is concluded in the next lemma. 278. P. has houtôs where Aristotle has houtô. 279. P. has poluplasiôn where Aristotle has pollaplasios. 280. P. has takhistê (nominative) where Aristotle has takhistêi (dative), agreeing with ‘quickest’. 281. This sentence is concluded in the next lemma. 282. That is, we need to indicate that ‘proportion’ does not go grammatically with ‘quickest’. 283. P. does not notice that by changing ‘quickly’ to ‘quickest’ Aristotle has solved the problem. In fact if the original second-figure syllogism had ‘quickest’ instead of ‘quickly’, the syllogism would be valid according to P.’s own criterion (159,7-17). 284. 159,8-10. 285. P. has pseudôn where Aristotle has pseudous. 286. P.’s inaccurate account of the present paragraph is founded in P.’s belief that a chief concern of Aristotle’s in APo 1.11-12 is to distinguish dialectical reasoning from demonstrative. Aristotle’s claim (78a10-11) is about mathematics, not demonstrations in general. 287. P.’s account of analysis is rather vague, not specifying the particular method it employs for discovering premises for a desired conclusion. Pappus’ account of it is more precise, including the sentence ‘Analysis, then, takes that which is sought and passes from it through its successive consequences to

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something which is admitted as the result of synthesis’ (634,11-13 Hultsch, tr. Heath (1921), vol. 2, p.400). 288. This corresponds to Pappus’ definition of synthesis (634,18-23). 289. This corresponds to part of Pappus’ account of analysis, in particular to his reference to ‘retracing our steps’ until ‘we come upon something  belonging to the class of first principles’ (634,13-18), and this is the aspect of analysis that Aristotle has in mind. 290. This state is knowledge(e), i.e., scientific knowledge (epistêmê). 291. APr 2.2-4. 292. Aristotle’s point is that in working back from a desired conclusion to premises that entail it, there is no guarantee that every premise (or set of premises) that we find is true. How P. grafts his distinction between definite premises (characteristic of demonstrations) and indeterminate ones (characteristic of dialectic) is not apparent. For the distinction, see 77a31-32 with 145,4-10. 293. This point is not directly related to the matter at issue, which has to do with false premises. 294. Aristotle says nothing corresponding to ‘proximately’ – an important difference, since the discussion in Aristotle speaks of deductions, not demonstrations, and ‘proximately’ shows that P. thinks that the convertibility of conclusions with premises makes it easy to discover immediate premises – that is, principles. But the fact that two propositions are convertible does not do anything to indicate which, if either, of the two is a principle. 295. Another possibility (adopted by Barnes (1994)) is to take ‘they’ as meaning mathematicians; likewise ‘the  in dialectical conversations’ can mean ‘those who engage in’. 296. This claim has nothing to do with Aristotle’s point. 297. This sentence is concluded in the next lemma. 298. See n. 286. 299. In Aristotle’s example (78a15), C is not the conclusion, but the minor term. The conclusion is ‘A holds of C’. 300. P. here takes his practice of relying on Euclid’s Elements to illustrate APo too far. The second theorem does employ the conclusion of the first, but it does not simply add an additional term at the end. Similarly for the third as regards the second. The fourth theorem does not make use of the third at all. 301. There is no reason why reputable premises cannot be immediate. 302. P.’s belief that deductions with non-immediate premises are dialectical is left unexplained. Whereas previously he seemed to assimilate dialectical with sophistical arguments (n.242), he now seems to suppose that any argument that is not demonstrative is dialectical. A possible line of defense would be to say that the deduction in P.’s example is typical of the work that needs to be done in order to determine the immediate premises that constitute scientific principles, and that this is one of the jobs of dialectic. But P. does not show himself aware of this (positive, from the point of view of science) role of dialectic. 303. The shift from ‘moves itself’ to ‘moves’ is remarkable. 304. ‘Always’ suggests that P. thinks that we can never reach immediate principles by this procedure. Evidently he did not have 84b35 in mind when he wrote this comment, and equally when he reaches that passage in his commentary (269,3-13), he takes no notice of the contradiction between that passage (and his interpretation of it) and what he says here. 305. P. has touto where Aristotle has tout’.

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306. P. is right to have qualms here. The possibility of a science’s having an infinite number of immediate premises, and consequently an infinite number of subjects and/or attributes would raise the possibility of infinitely long proofs, which Aristotle goes to some length to exclude (APo 1,19-22). On the other hand, Aristotle is correct to say that there is no limit to the number of middle terms that can be inserted in chains of deductions (as opposed to demonstrations). 307. That is, he should have said ‘D’ instead of ‘C’. However, I think that Aristotle is not guilty of a slip here. 308. This sentence is continued in the next lemma. 309. This sentence is concluded in the next lemma. 310. Namely, not a particular odd number. 311. P. omits ho before artios. 312. This sentence is concluded in the next lemma. 313. This sentence is concluded in the next lemma. 314. 78b34-79a4. 315. This is the subject of 78a23-26. 316. That is, they are visible for a smaller number of nights. 317. Aratus Phaenomena 1,177-178. 318. 92,9-93,1. 319. This is the subject of 78a26-b11. 320. Proofs from signs are discussed at 21,8-15, 31,8-32,7, 49,5-14, 97,22-31. The examples P. adduces in the present passages are also used in these earlier passages. 321. This kind of case is treated at 78b11-13. 322. At 78b34-79a4. 323. P. omits ei before di’ amesôn. 324. 78a30-33. 325. 78a33-34. 326. The ‘in every case’ relation is defined at 73a28-34. The ‘in [something] as in a whole’ relation is not defined, but is put to use at 79a36-b20. A is predicated ‘in every case’ of B if A is predicated of all B. In such a case B belongs in A ‘as in a whole’. 327. ‘ê’ normally means ‘or’. 328. For this very reason Aristotle rejects the theory of visual rays (De Sensu 438a25-26). 329. Except that according to 171,22-24 they do twinkle for people with weaker visual rays. 330. 168,30-169,6. 331. If it deduces its shape from its phases. 332. The bizarre idea that the relation is accidental is a consequence of the views that the predicate of the conclusion of a demonstration belongs per se to the subject, that the subject and predicate terms must reciprocate (to which P. subscribes – 69,24-70,6 and 134,7-10), and that any predicate of a subject that does not belong to it per se belongs to it accidentally (a consequence of 74b5-12 which P. endorses at 82,26). 333. These are alternative explanations. The former suits the view suggested by the phrase ‘a particular kind of pallor’ (173,5) that one kind of pallor is due to having given birth, another kind to fright, and so on: one kind of pallor for each cause. In that case, it would make sense to speak of the different kinds of pallor being called pallor homonymously. Alternatively, we admit that a

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single effect can be due to more than one cause, and in such cases abandon the doctrine that subject and predicate must reciprocate. 334. Cat. 7b13-14. 335. Namely, the art (tekhnê) of house building. For arts as states (hexeis), see E.N. 1140a6-10. 336. 78a23-26. 337. 78b23-24. 338. 78b28-29. 339. Aristotle, De Resp. 2-3. 340. That is, on the Earth’s equator. 341. This is the reason why at the equator day and night have equal length all year. 342. P. does not recognize that the Anacharsis example (78b29-31) is presented as an argument that fails to qualify as a deduction of the ‘why’ precisely because the middle term is remote from the proximate cause. 343. This sentence refers to the first sentence of the lemma. 344. P. is thinking of syllogisms with universal conclusions. 345. It is hard to reduce this to a second-figure syllogism. 346. Keeping the MS text di’ heautou. (Wallies in his apparatus criticus prefers di’ autou or di’ ekeinou.) The construction is the same as at 176,24-26. 347. I take this as an early use of thelein + infinitive to represent the future. 348. P. holds that the cause of A’s not being B is that A fails to belong to a genus of B, and that the fact that B fails to belong to a genus of A is not the cause of A’s not being B (but rather the cause of B’s not being A). 349. P. has de esti where Aristotle has d’ ésti. 350. Ammonius. 351. This comment is not to be found in Themistius’ Paraphrase of the Posterior Analytics. 352. P. is being careless in identifying these as proximate causes. 353. P. has tôi where Aristotle has to. 354. This lemma is continued in the next lemma. 355. The optical theorem in question is stated and proved by Euclid (Optics, proposition 4), but P. gives a less straightforward proof than Euclid’s, since the proof that angle CBD is larger than CAD is more difficult than the corresponding move in Euclid. Perhaps P.’s mention in the next sentence of puzzles connected with this theorem is an acknowledgment of the difficulty. 356. This sentence is concluded in the next lemma. 357. The general nature of P.’s comment and the absence of examples suggests that P. did not know of any treatises on mechanics that fit Aristotle’s account. At 76a24 Aristotle makes mechanics subordinate to geometry, a claim borne out by the pseudo-Aristotelian Mechanica, which is principally devoted to explaining the properties of the lever and other simple machines by means of properties of circles and straight lines. 358. The reference to sailors comes from Aristotle’s reference to ‘nautical astronomy’ at 79a1. The phenomena in question were known to land-lubbers like Hesiod as well (e.g., Op. 383-387, 414-419, 564-569). At Op. 650-651 Hesiod says that he made only one short voyage, from Aulis to Chalcis, a distance of about a mile. 359. This sentence is concluded in the next lemma. 360. This sentence is concluded in the next lemma. 361. The Pythagoreans based their understanding of harmony on their

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discovery that the intervals of the octave, the fifth and the fourth (which are the fundamental intervals of Greek music) correspond to the ratios 2:1, 3:2 and 4:3. 362. That is, he wants to define them as a class. 363. These properties are taken from the list at Aristotle Meteorologica 372a26-33. 364. These results are due to Zenodorus (who lived perhaps between c. 200 BC and 90 AD). See Heath (1921) vol. 2, 207. 365. In seven days the moon completes approximately one quarter of its cycle – going from new moon to first quarter, from first quarter to full, from full to last quarter, or from last quarter to new moon. 366. 79a25-29. 367. This non-standard use of ‘scientific knowledge(e)’ occurs at 79a24, but Aristotle repudiates it at 92b38 by saying that there is no demonstration of definitions – a point that P. recognizes (183,30). See n. 370. 368. P. would have done well to stop his account at the end of the previous sentence. The present sentence does not accurately represent any remarks of Aristotle’s on how to reach definitions. P.’s account here has some connections with the (unparalleled and problematic) account at 96a24-38, but the remarks on how to go about finding the differentiae are confused. 369. The related passage at 96a24-38 says nothing about how the attributes are discovered. The deductions P. has in mind cannot be demonstrations, but they may be non-demonstrative arguments such as ‘all things that do not twinkle are near, the planets do not twinkle, therefore the planets are near’ (cf. 78a26-b4), in which the conclusion is immediate (and therefore part of the definition of planet) while the minor premise is non-immediate. 370. This subject is not covered in Top. book 8. Wallies (app. crit.) refers to Top. book 7, ch. 3, 153a7ff., which does discuss definitions, but without containing the proof to which P. refers. Perhaps P. is thinking of APo book 2.3-7. 371. Again, this seems to be based on 96a24-38. 372. P. omits the requirement that these are essential attributes. 373. 79a29-31. 374. In this sentence P. has overreached himself. He supposes that something has been proved in the second figure and that it is necessary to demonstrate each of the premises. Since one of the premises is affirmative and universal, it can only be demonstrated in a first-figure syllogism. But the other, which is negative and universal, can be demonstrated through another secondfigure syllogism. The fact that if its premises need to be demonstrated one of them must be demonstrated through a first-figure syllogism is irrelevant, for P. has given no reason to suppose that its premises need to be demonstrated too. 375. For the second figure, see APr 27a15-18; for the third, see 28a1-7. 376. APr 40b17-19. 377. P. adds tou skhêmatos after toutou. 378. This comment applies to the first sentence of the lemma. 379. Aristotelian definitions do not usually contain the word ‘every’. 380. That is, any syllogism with a universal negative conclusion (syllogisms in Celarent, Cesare and Camestres) has one universal affirmative premise. 381. For an example, see 79a41-b1 and 187,28-188,22. 382. ‘Atomically’ is synonymous with ‘immediate’. P. does well to stick with the terminology already established by Aristotle, beginning with 71b21.

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383. 79a30-1, cf. 84b34-5. 384. P. again (see n. 200) neglects the antithesis between what is better known ‘to us’ and what is better known ‘by nature’, which implies that immediate truths are not better known ‘to us’ and are perhaps very difficult for most people to know. The example P. gives in this sentence does not appear well chosen to support this claim. 385. In his comments on this chapter P. supposes that the only immediate negative premises are those involving category terms, such as ‘no substance is a quality’ (187,15-16, 187,20-5, 188,5-6 and 189,3-5). (It is noteworthy that at 191,29-32 he admits another kind of case. See n. 403. Ross points out that Aristotle ‘omits to consider the case in which both are in the same genus’, in particular, ‘the case in which A and B are members of the same infima species’. Ross goes on to say that ‘Aristotle would have, however, to admit that alternative differentiae, no less than summa genera or categories, exclude each other immediately’. (Ross [1949], 557). (P. in fact does refer to alternative differentiae at 199,2-4.) 386. P. makes a slip here; in the first case we have a first figure deduction and in the second, a second figure deduction. 387. An etymological comment. Atomos means ‘unsplittable’, being derived from the root of temnô, the word that means ‘split’. 388. It is disappointing that P. does not see that this same comment applies to ‘no man is irrational’. 389 P.’s comment suits the present lemma, but it does not reflect the generality of Aristotle’s claim at 79b5-11, which does not require the ‘whole’ to be a category, but only a genus to which one of the given terms belongs and the other does not. 390. This sentence is concluded in the next lemma. 391. This unexpectedly narrow interpretation of ‘as in a whole’ governs P.’s discussion of APo 1.15-16. 392. P. here treats terms as immediate or non-immediate, whereas those descriptors apply not to terms but to propositions. He calls ‘substance’ immediate and ‘continuous’ not immediate, apparently following his interpretation that immediate negations occur only in cases where one category term is denied of another and so calling category terms ‘immediate’ and other terms ‘not immediate’. 393. 79b5-12. 394. 79a30. 395. 185,17-20. 396. The assumption that ‘columns’ begin with category terms is not explicit in Aristotle. 397. Camestres. 398. Celarent, Cesare. 399. See APr 41b6. 400. P. refers loosely to APo 1 chs. 2-6, 10, 13, and 15. 401. See n. 226. 402. How this ignorance arises with immediate propositions is the subject of APo 1, ch. 16. Ch. 17 treats non-immediate propositions. P.’s present discussion applies to both cases. 403. P.’s example allows for immediate negations other than those involving category terms (such as ‘substance belongs to every quality’). To judge by his examples at 189,12-13 and 196,13-15, P. regards body as a species falling

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immediately under substance, so that ‘substance belongs to every body’ is immediate. Here he allows that the contrary negation is immediate. He would have done well to incorporate this kind of case into his treatment of APo 1 ch. 15. 404. P. chooses claims which Aristotle refutes. For atoms, see n. 126. He argues that there are more than two principles at Phys. 189a21-b1. 405. That is, it has either an affirmative conclusion or a negative conclusion. 406. If its false conclusion is affirmative and immediate. 407. That is, both premises are universally false. 408. APr 54a4-6, 54b17-23. 409. P. returns to his view that in immediate negations the subject and predicate terms are both category terms. See 191,29-31 and n. 403. 410. The example recalls P.’s view that immediate negations involve category terms. See 186,9-12 with n. 385. 411. When the false conclusion is affirmative and immediate. 412. When the conclusion is negative and both premises are non-immediate, one being affirmative and the other negative. 413. This sentence is completed in the next lemma. 414. 79b12-15 and 189,25-7. 415. This translation represents the ‘philosophical’ imperfect ên. 416. Because it is given that A not-belongs atomically to B (79b29-30). 417. P. describes the following syllogism: substance belongs to all bodies, body belongs to all qualities, therefore substance belongs to all qualities. 418. P. omits ‘primarily’. 419. Because both B and C are category-terms and the members of each category are distinct from the members of the other categories. See 186,10-21 with n. 385. 420. A is affirmed (denied) immediately of B when A belongs (does not belong) immediately to all B. 421. 191,18-192,4. 422. 192,24-7; however Aristotle does not offer the explanation that P. provides. 423. 193,5-9. 424. That is, he has treated the case where ‘A belongs to no B’ is immediate and ‘A belongs to all B’ is the (affirmative) conclusion of a deduction, and has shown how at least one of the premises of the deduction must be false (79b2980a5, cf. 194,20-198,5). 425. That is, he treats cases where ‘A belongs to all B’ is immediate and ‘A belongs to no B’ is the (negative) conclusion of a deduction. P. gives examples where the premises of the erroneous conclusion are immediate (199,4-7). 426. In cases where there are more than two coordinate species, any of them may be used to make this point. 427. P. has in mind the deduction ‘substance belongs to nothing that is bodiless, bodiless belongs to all bodies, therefore substance belongs to no body’. 428. 80a3-4, cf. 196,21-197,21. 429. At 209,19-21 P. will give angels as an example of things that are rational and immortal. 430. That is, it will be true that A (the major term) belongs to none of it and false that A belongs to all of it. 431. That is, the particular terms employed in the example make the general point clear. Aristotle uses this expression in this sense at APr 34b4 and 37b1-2.

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432. This sentence is completed in the next lemma. 433. Given that A belongs to all B and also to all C, a deduction that uses C as a middle term to prove that A belongs to B requires the minor premise to be ‘C belongs to all B’. But further, since the conclusion of the deduction is false and the major premise is true, it follows that the minor premise is false. 434. This is the opposite of the case previously considered (80a15-20), where the major was taken to be true and the minor false. 435. If A is the major term, C the middle and B the minor, the true minor premise is ‘C belongs to every B’ and the false major premise is ‘A belongs to no C’. 436. P. takes thateron hupo thateron (80a23) to mean not ‘one under the other’ but ‘each under the other’. This misunderstanding is the basis of his interpretation of the lemma. 437. These two claims are the given immediate connection and the true minor premise. 438. Aristotle does not restrict his discussion to such cases, which in any case are out of harmony with the strict class-inclusion interpretation of the syllogistic with which P. has been working through his treatment of chs. 15-16. 439. Specifically, cases where ‘A belongs to all B’ is immediate and ‘A belongs to no B’ is the (negative) conclusion of a deduction. 440. 80a9-26. 441. We would expect P. to specify that A belongs to every B atomically (79b25, b29-30, b38, 80a3, and 198,15-19 with n. 424 and n. 425). 442. The lemma says ‘both premises’. P.’s mistake makes his exposition hard to follow. 443. A is affirmed (denied) universally of B when A belongs (does not belong) to all B; A is affirmed (denied) partially of B when A belongs (does not belong) to some B. 444. That is, if it is denied universally of the major term it will have to be denied universally of the minor term too, and if it is denied universally of the minor term it will have to be denied partially of the major term. 445. P. considers only two of the four cases he has mentioned – the cases where the major premise is universally true and the minor premise is universally false. 446. But in the case considered, it is truly predicated universally. This error vitiates the argument in the present paragraph. See also n. 449. 447. One premise is affirmative and one is negative. 448. For the terminology, see 150,29-152,3. In this earlier passage the optimistic view that experts never make mistakes in reasoning was ascribed only to geometers. 449. It has been established that ‘C belongs to no A’ and ‘C belongs to all B’ cannot both be true. (But it is possible for either one of them to be true.) If we take the contraries, we have ‘C belongs to all A’ and ‘C belongs to no B’, which likewise cannot both be true (and it is likewise possible for either one of them to be true). P. mistakenly infers that they must both be false, continuing the error he made at 202,34-203,2. 450. Again, only one of them need be false. See n. 449. 451. P. has huparkhei where Aristotle has huparkhoi. 452. Since P. holds angels to be immortal rational animals (209,19-21), mortal belongs to some animal, but not to all, and to some rational thing, but not to all.

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453. CA, that is, ‘C belongs to all A’. 454. CB, that is, ‘C belongs to no B’. 455. P. comments only on the first of the two cases described in the lemma (80b2-3). 456. After mê atomôs huparkhousin, P., in agreement with MS d of the Posterior Analytics (Laurentianus 72, 5) has ê mê huparkhousin; Ross, following the other principal MSS, omits these words. 457. In fact Aristotle did not state this as his purpose. See 79b27-29. 458. Nor has Aristotle made this assertion. See 79b25 and b29-30. He mentions non-atomic (i.e., non-immediate) premises at 80a5. 459. 79b29-80a5. 460. 80b22-81a4. 461. No first figure syllogism has a negative minor premise. 462. That is, it has a true conclusion. 463. For this term, see 79b7 with 189,11-13. In fact Aristotle does not say that these things are in the same column, although his mention of cases where a middle term is taken from different columns (80b27) implies that he thinks of examples like the one P. gives here as containing terms from the same column. 464. At 189,12-20 P. presents ‘columns’ as if they consist of the genus, species, sub-species, etc. of the subject in question. Here he recognizes columns containing the attributes that can be employed in premises of dialectical arguments concerning the subject. Thus there can be a plurality of columns that include a single term, as Aristotle recognizes at 80b27-8. The notion of columns is an appropriate device for considering genus/species relations, but is inappropriate for considering other per se attributes. 465. This sentence is completed in the next lemma. 466. This is a comment on the first sentence of the lemma. P.’s point is that the error in the deduction with the false conclusion that A belongs to no B cannot be that the minor premise (‘C belongs to all B’) is false. 467. P. follows Aristotle’s unusual usage (in the lemma) of ‘convert’ (antistrephein), which he correctly explains in the first sentence of this comment. 468. This is what Aristotle means by ‘the deduction becomes contrary’: the conclusion of the erroneous deduction is the contrary of the conclusion of the original deduction. 469. See n. 464. 470. Namely, that the major premise must be false and the minor true. 471. P. ignores Aristotle’s more careful ‘almost’. 472. That is, the middle is not ‘appropriate’ as defined at 80b20-21. 473. This is the false conclusion of the deduction of error. 474. The false conclusion of the deduction is ‘rational belongs to no angel’. 475. P. corrects Aristotle’s claim at 80b34. 476. The false conclusion of the deduction is ‘animal belongs to no man’. 477. Sc. in cases where the middle term is not appropriate. 478. Without the guarantee afforded by the middle term’s being appropriate, this combination (that A belongs to all B and to some C) is possible, which makes the argument invalid. 479. The deduction of error is ‘A belongs to no C, C belongs to all B, therefore A belongs to no B’. 480. This claim presupposes that A, B, and C are in the same ‘column’. 481. P. has melloi where Aristotle has mellei.

138

Notes to pages 106-112

482. P. labels the middle term differently from Aristotle: gamma instead of delta. Also at 210,17-19. 483. Namely, they become ‘A belongs to no C’ and ‘C belongs to all B’. 484. P. gives the needed interpretation of Aristotle’s statement that the middle is not under the major term (which usually means that it is not true that the major holds of all of the middle). 485. See n. 480. 486. 80b18-81a4. 487. 80a27-28. 488. This situation was not shown to obtain in the treatment of first figure deductions. 489. P. has hupallêla where Aristotle has hup’ allêla. 490. This sentence is completed in the following lemma. 491. See 196,8-11. 492. This sentence is completed in the following lemma. 493. P. (followed by Ross) here omits a sentence found in the Aristotle manuscripts, which says, ‘Now it is obvious that if the middle term is not under A, both may be false, and either singly’. 494. This is a comment on the first sentence of the lemma. 495. This sentence is completed in the next lemma. 496. Number is one of the ‘common sensibles’ mentioned at de An. 425a16. P.’s account of these is somewhat different from Aristotle’s. 497. Either something has dropped out of the text or the parenthetical provision (‘supposing that) is irrelevant. It points toward the related argument for the point at issue: if there were objects not apprehensible by the senses, there can be no science of them. 498. All these assertions contain universals, but since the last of them is an unprovable principle of geometry, it is not clear what P. has in mind when he says that we learn them through universals. Since he admits shortly below (215,7) that we learn axioms by induction, not by demonstration (which depends on universal principles), I suppose his example is poorly chosen. 499. This is Euclid Elements, book 1 postulate 1. 500. P. assumes that the way a teacher uses informal explanations to convey primitive concepts to students is the same as the way a person who is demonstrating (that is, an expert) grasps universals. 501. This example is taken from Plato’s discussion of recollection at Phaedo 73d. 502. For the Neoplatonic concept of projection, see references in n. 131. 503. Tim. 47a-c. 504. This may be a reference to Damascius in Phaedonem version 2, section 14 (L.G. Westerink, The Greek Commentaries on Plato’s Phaedo, vol. 2: Damascius, Amsterdam/Oxford/New York: North-Holland, 1977, p.294). 505. The failure to refer to definitions may be due to P.’s view that definitions are not scientific premises. See above, 128,26-32. 506. P. holds that the major premise of every demonstration is an axiom (8,7-8). 507. P. (agreeing substantially with MS n of APo) has estai di’ epagôgês gnôrima, ean tis boulêtai gnôrima poiein where Ross, following the other MSS, has estai di’ epagôgês gnôrima poiein. 508. The end of this sentence constitutes the following lemma. 509. The examples are Euclid Elements, book 1, common notions 1 and 3. P.’s

Notes to page 112

139

remark here that a beginner at geometry may not very well understand what these axioms claim is inconsistent with his standard view that axioms appear to everyone and that each person has them from himself and puts them forward even if the teacher does not state them (129,27-28, 130,15). 510. Punctuating with a comma instead of a semicolon. 511. If three-dimensional sensible objects are the basis for the induction that gives us knowledge of surfaces, P. does not make it clear how induction ‘confirms’ that surfaces are two-dimensional, nor does he show that he is aware of the problem that sensible objects are not perfectly flat. 512. Reading katô for natô. 513. A person who lacks one of the senses thereby lacks the kind of sensation that grasps the objects of that sense.

Bibliography Barnes, J. (1984), The Complete Works of Aristotle: The Revised Oxford Translation, Princeton. Barnes, J. (1994), Aristotle, Posterior Analytics, Clarendon Aristotle Series, 2nd edn, Oxford. Bonitz, H. (1870), Index Aristotelicus, Berlin. De Vries, G.J. (1969), A Commentary on the Phaedrus of Plato, Amsterdam. Detel, W. (1993), Aristoteles. Analytica Posteriora, Berlin. D’Ooge, M.L. (1926), Nicomachus of Gerasa. Introduction to Arithmetic, London. Fornaro, S. (2007), ‘Peisander’, in H. Cancik and H. Schneider (eds), Brill’s New Pauly, vol. 10, cols 683-4. Friedlein, G. (1873), Proclus, In primum Euclidis Elementorum librum commentarii, Leipzig. Giardina, G.R. (1999), Giovanni Filopono Matematico, Commentario alla Introduzione Aritmetica di Nicomaco di Gerasa, Catania. Goldin, O. (tr.) (2009), Philoponus(?), On Aristotle Posterior Analytics 2, London. Gratiolus, A. and Theodosius, P. (tr.) (1995), Johannis Philoponi Commentaria in Libros Posteriorum Aristotelis, Neudruck der Ausgabe Venedig 1542 mit einer Einleitung von Koenraad Verrycken und Charles Lohr, Stuttgart-Bad Cannstatt. Grosseteste, R. (1981), Commentarius in posteriorum analyticorum libros, (ed. with intro. by P. Rossi), Florence. Gudeman, A. and Kroll, W. (1916), ‘Ioannes Philoponus’, in A. Pauly, G. Wissowa and W. Kroll (eds), Realencyclopädie der classischen Altertumswissenschaft vol. 9, part 2, cols 1764-95. Heath, T.L. (1921), History of Greek Mathematics, 2 vols, Oxford. Heath, T.L. (1925), The Thirteen Books of Euclid’s Elements, 3 vols, 2nd edn, Oxford. Heath, T.L. (1949), Mathematics in Aristotle, Oxford. Heiberg, I.L. (1916), Euclidis Sectio Canonis, in Euclidis Opera Omnia, vol. 8, Leipzig. Heiberg, I.L. (1969-77), Euclidis Elementa, rev. ed. E.S. Stamatis, 5 vols in 6 parts, Leipzig. Hoche, R. (1866), Nicomachus of Gerasa, Introductio Arithmetica, Leipzig. John Philoponus (1504), Ioannis Grammatici in Posteriora resolutoria Aristotelis commentaria, Venice, 1504 (the editio princeps). Keydell, R. (1937), ‘Peisandros’, in A. Pauly, G. Wissowa and W. Kroll (eds), Realencyclopädie der classischen Altertumswissenschaft vol. 19 part 1, cols 145-6. Knorr, W.R. (1975), The Evolution of the Euclidean Elements, Dordrecht/Boston.

142

Bibliography

Knorr, W.R. (1986), The Ancient Tradition of Geometric Problems, New York. Lamberz, E. (1987), ‘Proklos und die Form des philosophischen Kommentars’, in J. Pépin and H.D. Saffrey (eds), Proclus, lecteur et interprète des anciens, Paris, 1-20. Lausberg, H. (1988), Handbook of Literary Rhetoric. A Foundation for Literary Study, Leiden/Boston/Köln. MacCoull, L.S.B. (1995), ‘Another Look at the Career of John Philoponus’, Journal of Early Christian Studies 3, 269-79. McKirahan, R. (1978), ‘Aristotle’s Subordinate Sciences’, British Journal for the History of Science 11, 199-220. McKirahan, R. (1992), Principles and Proofs. Aristotle’s Theory of Demonstrative Science, Princeton. McKirahan, R. (2000), ‘Philosophy, Science and Mathematics in the fourth century BC’ (in Greek), Deukalion 18, 5-25. McKirahan, R. (tr.) (2008), Philoponus: On Aristotle Posterior Analytics 1.1-8, London. McKirahan, R. (2009), ‘Philoponus’ Account of Scientific Principles in his Commentary on Aristotle’s Posterior Analytics’, Documenti e Studi sulla tradizione filosofica medievale (Pisa: Scuola Normale Superiore) 20, 211-63. Moraux, P. (1979), Le Commentaire d’Alexandre d’Aphrodise aux ‘Seconds Analytiques’ d’Aristote, Peripatoi 13, Berlin. Morrison, D. (1998), ‘Philoponus and Simplicius on Tekmeriodic Proof’, in E. Kessler (ed.), Method and Order in Renaissance Philosophy of Nature, Aldershot/Burlington VT, 1-22. Mueller, I. (1987), ‘Mathematics and Philosophy in Proclus’ Commentary on Book 1 of Euclid’s Elements’, in J. Pépin and H.D. Saffrey (eds), Proclus, lecteur et interprète des anciens. Actes du colloque international du CNRS, Paris, 2-4 octobre 1985, Paris, 305-18. Pellegrin, P. (2005), Aristote. Seconds Analytiques, Paris. Ross, W.D. (1949), Aristotle’s Prior and Posterior Analytics, Oxford. Siorvanes, L. (1996), Proclus. Neo-Platonic Philosophy and Science, New Haven/London. Sorabji, R. (1987a), ‘John Philoponus’, in Sorabji (1987b), 1-40. Sorabji, R. (1987b), Philoponus and the Rejection of Aristotelian Science, London/Ithaca NY. 2nd edn, with two new chapters by Sorabji, Bulletin of the Institute of Classical Studies, 2008. Sorabji, R. (ed.) (1990), Aristotle Transformed: The Ancient Commentators and Their Influence, London/Ithaca NY. Sorabji, R. (2004a), ‘Aristotle’s Perceptual Functions Permeated by Platonist Reason’, in G. Van Riel and C. Macé (eds), Platonic Ideas and Concept Formation in Ancient and Medieval Thought, Leuven, 100-17. Sorabji, R. (2004b), The Philosophy of the Commentators 200-600 AD, vol. 3: Logic and Metaphysics, London/Ithaca NY. Verrycken, K. (1990), ‘The Development of Philoponus’ Thought and its Chronology’, in Sorabji (1990), 233-74. Wallies, M. (1900), Themistii Analyticorum Posteriorum Paraphrasis, CAG 5.1, Berlin. Wallies, M. (1909), Ioannis Philoponi in Aristotelis Analytica Posteriora Commentaria, CAG 13.3, Berlin. Westerink. L.G. (1964), ‘Deux commentaires sur Nicomaque: Asclépius et Jean Philopon’, Revue des études grecques 77, 526-35. Westerink, L.G. (1977), The Greek Commentaries on Plato’s Phaedo, vol. 2: Damascius. Amsterdam/Oxford/New York.

Bibliography

143

Wians, W. (2006), ‘The Philosopher’s Knowledge of Non-Contradiction’, Ancient Philosophy 26, 333-53. Wildberg, C. (1998), ‘Philoponus’, in E. Craig (ed.), Routledge Encyclopedia of Philosophy, London, vol. 7, 371-8.

English-Greek Glossary able to whinny: khremetistikos absent, be: ekleipein, leipein abstract: ex aphaireseôs abstracted: exêirêmenos absurdity: to atopon accident: sumbebêkos accidental: sumbebêkos, kata sumbebêkos accidentally: kata sumbebêkos accomplished: teleios accord: sumphônein account: logos acquaintance: gnôrimos acquire: lambanein activity: energeia actually: energeiai (dat.) acute: oxus acute-angled: oxugônios ad infinitum: ep’ apeiron add: epitithenai, parelkein, proskeisthai, prostithenai, proslambanein addition: prosthesis adjacent: ephexês adjoin: episunaptein admit: sunkhôrein admittedly: homologoumenôs affected, be: paskhein affection: pathos affirm: kataphanai, kataphaskein affirmation: kataphasis affirmative: kataphatikos, katêgorikos affirmative conclusion: kataphasis agree: sunkatatithenai air: aêr alter: alloioun analysis: analusis analyze: analuein ancient: arkhaios angle, acute: hê oxeia angle, adjacent: hê ephexês

angle, exterior: hê ektos angle, horn: hê keratoeidês angle, interior: hê entos angle, rectilinear: hê euthugrammos angle, right: hê orthê angle: gônia animal: zôion answer: apokrinesthai antecedent: hêgoumenon anticipate: prolambanein apex: koruphê apparent: saphês appear true: dokein appear: dokein, emphanizein, phainesthai appearance: dokêsis apply to: huparkhein apply: harmozein, prokheirizesthai apprehension, the view that a. is impossible: akatalêpsia apprehension: antilêpsis, katalêpsis apprehensive: antilêptikos appropriate: oikeios aptitude: hexis arc: periphereia area: embadon argument: kataskeuê, logos arise: ginesthai Aristotle, in: Aristotelikos arithmetic (adj.): arithmêtikos arithmetic (n.): arithmêtikê arithmetic, of: arithmêtikos arithmetical: arithmêtikos art: tekhnê ascend: anatrekhein ash: tephra ask: aitein, epaneresthai, erôtan ask for: erôtan ask questions: erôtan asking questions: erôtêsis assume: lambanein, paralambanein assume in advance: prolambanein

146

English-Greek Glossary

astrologer: astrologos astrology: astrologia, astrologikê astronomer: astronomos astronomy: astronomia atom: atomon atomically: atomôs attempt: peiran attention: epistasia, epistasis attribute: huparkhon, sumbainon, sumbebêkos attribute that belongs: huparkhon attribute, be an a. of: sumbainein Auriga: Êniokhos aware, be a. of: suneidenai axiom: axiôma base: basis based on, be: ginesthai be: einai, ginesthai, huparkhein be a subject: hupokeisthai be inflected: klan become: ginesthai beg the question: ex arkhês aiteisthai begin: arkhesthai beginning: arkhê belief: hupolêpsis believe: doxazein belong: anêkein, huparkhein, prosêkein, sumbainein birth: genetê board: abakion bodiless: asômatos body: sôma book: biblion boundary, be a b.: peratoun breadth: platos breathe: anapnein breathing: anapnein bring forward: propherein bring objections: enistasthai bring together: sunagein can: dunatos, ekhein, endekhesthai cannot: adunatos, ouk endekhesthai cast: poiein category: katêgoria ‘catoptrician’: katoptrikos catoptrics: katoptrikê cause (n.): aitia, aition, aitios cause (v.): poiein celestial equator: orthê sphaira centre: kentron certain kind, of a: pôs ekhousa

change: ameibein characteristic feature: idiotês chief: kurios circle: kuklos, periphereia circular: peripherês circumference: periphereia circumscribe: perigraphein claim against: antilegein clear: dêlos, saphês clear, be: dêlon, dêlon hoti, phainetai clearly: dêlos, dêlon, dêlon hoti, dêlonoti coextensive: sunkatatetagmenos coextensive, be: exisazein coincide: ephaptesthai, epharmozein, epizeugnunai, haptesthai cold: psukhros collect and arrange: sunagein colour: khrôma coloured, be: kekhrôsthai column: sustoikhia combination: sumplokê, suzugia combine: episuntithenai, sumplekein, sunathroizein come to be: ginesthai come to be knowing(g): gnôrizein comedies: kômika comedy: kômôidia comma, place a: hupostizein commensurable: summetros common: koinos commonality: koinotês compass: diabêtês compel: anankazein complete (adj.): holikos complete (v.): sumplêroun, teleioun complete form, in its: entelês complex: poluskhidês compose: poiein, sunkeisthai composed of, be: hupokeisthai comprehend: sunoran comprise: lambanein conceive: noein concept: logos conception: noêsis concerned with, be: ekhein peri, kataginesthai conclude: sunagein conclusion: sumperasma concordant: sumphônos concur: suntrekhein condition, in this c.: houtôs ekhôn confirm: pistousthai

English-Greek Glossary confuse: kakôs khrêsthai connect: sunaptein connection: sunêmmenon consequence, be a c.: hepesthai, akolouthein consequence, be an inevitable c.: hepesthai consequent: hepomenon consider: episkeptein, skopein, theôrein consistent: akolouthos constitute: sunistasthai construct: kataskeuazein, poiein, sunistasthai contain: ekhein, emperiekhein, paremplekein, periekhein containing greater area: polukhôrêtoteros containing the greatest area: polukhôrêtotatos contentious: eristikos continuous: sunekhês continuously: kata sunekheian contradict: antiphaskein contradiction: antiphasis contradiction, member of: morion contradictory: antiphatikos contradictory pair: antikeimena contrary: enantios contrast: antidiairein, paratithenai contribute: sumballein, suntelein conversation: sunousia conversation, engage in: dialegesthai converse: antistrophos conversely: anapalin conversion: antistrophê convert: antistrephein conviction: pistis, to piston conviction, there is c.: piston ekhei convinced of, be: pisteuein coordinate: antidiêrêmenon coordinate claim: apodosis coordinate members of a division: antidiêrêmena coordinate, be: antidiêirêsthai corollary: porisma cosmos: kosmos crab: karkinos critical: krisimos cross over: epallattein cut: temnein day: hêmera

deaf: kôphos deal with: dialambanein, strephesthai deceive: apatan deception: apatê decide: dokein declarative: apophantikos decline: phthisis decrease: meioun deduce: sullogizesthai deduction: sullogismos deduction, in a way suitable for: sullogistikôs deductions, in its: sullogizomenos deductive: sullogistikos defectively: phaulôs define: horizesthai defined: hôrismenos definiendum: horiston definite: hôrismenos definition: horismos, horos Demiurge: dêmiourgos demonstrable: apodeiktos demonstrate: apodeiknunai demonstrate previously: proapodeiknunai demonstrated: apodeiktos demonstrating: apodeixis demonstration: apodeixis demonstration, of: apodeiktos demonstrative: apodeiktikos demonstrator: apodeiktikos denial: anairesis deny: apophanai, apophaskein depend: artasthai depend on: ekhein descend: katagein description, general: hupographê details: ta kata meros determine: aphorizein diagonal: diametros dialectic: dialektikê dialectic, engage in: dialegesthai dialectical: dialektikos dialectical conversation: dialexis dialectician: dialektikos diameter: diametros diametrically: kata diametron differ: diapherein difference: diaphora difference, there is a d.: diapherei different: diaphoros different kind, of a: anomogenês

147

148

English-Greek Glossary

different, be: diapherein differentia: diaphora difficult: duskherês difficulty, have: duskherainein dimension: diastasis, diastatos dimensional: diastatos direct: apoteinein, ep’ eutheias direction: meros directly: autothen discourse: logos, sunousia discover: epheuriskein, exeuriskein, heuriskein discovery: heuresis discuss: dialegesthai discussion: logos disjunctive: diazeuktikos display: deiknunai disposed, be: diakeisthai disposition: diathesis dissimilar: anomoios dissimilar form, of: anomoioskhêmôn dissimilarity: anomoiotês distance: apostasis, diastasis, diastêma distinction: diakrisis distinguish: diairein, diakrinein distinguishing, way of: diakrisis divide: diairein division: diairesis division, be a coordinate member of: antidiêirêsthai do: poiein doctor: iatros doctrine: didaskalia done, be: ginesthai draw: agein, graphein, katagraphein draw a conclusion: sunagein due to: ginomenos kata, ginomenos para due to, be: ginesthai hupo ear: ous earth: gê eclipse: ekleipsis eclipsed, be: ekleipein, eklimpanein education: mathêma education, general: enkuklios effect: aitiatos elaboration: epexergasia element: stoikheion eliminate: anairein employ: khrêsthai, lambanein, paralambanein

enclose: periekhein end: akron, akros, peras, telos end (of a compass): skelos endpoint: peras ensouled: empsukhos entire: holoklêros, holos entirely: holôs, di’ holou, pantêi, pantôs enumeration: katarithmêsis epic: epos epigram: epigramma equal: isos equilateral: isopleuros equivalent: isos erroneous: êpatêmenos error: apatê, planê error, that produces e.: apatôn essence: ousia essential: kat’ ousian establish: kataskeuazein eternal: aïdios even: artios evenly: ex isou evident: enargês, phainomenos, phaneros, prophanês evident, because it is e.: ek tês enargeias exact: êkribômenos examine: exetazein example: paradeigma exceed: huperekhein excessive, be: perieinai excessively: ek periousias exist: huphistanai expert(e): epistêmôn expert in philosophy: philosophos explain: exêgeisthai explanation: exêgêsis, paramuthia expose: ektithenai expound precisely: diarthroun expound: didaskein express: propherein extend: ekballein exterior: ektos external: ektos, prophorikos extravagance: huperbolê extravagantly: kath’ huperbolên, pros huperbolên extreme: akros eye: omma, ophthalmos failure to pay attention: anepistasia fall on: prospiptein

English-Greek Glossary fallacious argument: paralogismos fallacy: pseudarion false: pseudês, pseudos false, the: to pseudos falsehood: to pseudos familiar: gnôrimos far from, be: aphistanai figure: skhêma fill completely: apoplêroun fill in: katapuknoun find: heuriskein find puzzling: aporein finite: peperasmenos fire: pur first: prôtos, proteros first place, in the: prôtôs fix: apereidein fixed: aplanês flow: rhusis flute-girl: aulêtris follow: hepesthai, sumbainein, sunagein follow from: akolouthein following: akolouthos foot long: podiaios for it alone: idiôs foreknowledge: pronoia forget: en lêthêi ginesthai form (n.): eidos, skhêma form (v.): ginesthai, poiein form demonstrations: apodeiknunai form, in its complete: entelês formally: to para to skhêma formed, be: ginesthai former: proteros formula: logos found, be: ginesthai fulgurite: keraunitês full moon: panselênos further down: proienai gaze upward: anablepein general: enkuklios, genikos, katêgorikos, katholikos, katholou, koinos general description: hupographê general education: enkuklios generally: holôs, katholou generate: gennan generated: genêtos generation: gennêsis genus: genos, sungeneia

149

genus, belonging to the same g.: homogenês, sungenês geometer: geômetrês geometrical: geômetrikos geometry: geômetria geometry, ignorant of: ageômetrêtos geometry, of: geômetrikos geometry, that comes from g.: geômetrikos get: ekhein, lambanein give: apodidonai, didonai, tithenai give birth: tiktein given: dedomenon, dotheis glittering: stilpnotês go forth: proienai go on: epagein go on to say: epagein go through precisely: diexerkhesthai go to study with: prosphoitan goal: skopos grant: didonai, sunkhôrein granted: homologoumenos grasp: ekhein, lambanein grow: auxein growth: auxêsis happen: ginesthai, sumbainein ‘harmonician’: harmonikos harmonics: harmonikê harmonics, facts in: harmonica harmony: harmonia have: ekhein have to: dein, pantôs healthy: hugiainein hear: akouein heard: akoustos hearing: akoê, akouein hearing, of: akoustikê heaven: ouranos high: akros hint at: ainittein hold: ginesthai, huparkhein homoeomery: homoiomereia homonymous: homônumos homonymy: homônumia horn angle: hê keratoeidês hot: thermos human: anthrôpos hypothesis: hupothesis hypothesize: hupotithenai ignorance: agnoia ignorant: amathês

150

English-Greek Glossary

ignorant of geometry: ageômetrêtos ignorant, be: agnoein illlustration: hupodeigma illuminate: phôtizein illumination, source of: phôstêr image: phantasia imagination: phantasia immediate: amesos immediately: amesôs, euthus, prosekhôs immortal: athanatos immortality: athanasia imply: sunagein imply, that can i.: sunaktikos impossible: adunatos, ou dunaton impossible, be: ouk endekhesthai impression: tupos in a particular field, in a: kata meros in all cases: pantôs in detail: kata meros in front of, be: epiprosthein in general: katholou in order to establish: pros kataskeuên in what follows: ephexês include: anagein, anapherein, periekhein incommensurable: asummetros incorporeal: asômatos increase (n.): auxêsis increase (v.): auxein indefinite: adioristos, aoristos indemonstrable: anapodeiktos independently: autarkôs indeterminate: adioristos, aoristos indicate: dêloun, endeiknunai individually: en merei induction: epagôgê induction, perform: epagesthai infer: epagein, sumperainein, sunagein infer a conclusion: sunagein infer fallaciously: paralogizesthai infinite: apeiros inflected, be: neuein initial: ex arkhês inner: entos inquiry: pusma inscribe: engraphein, graphein insert: epentithenai, parempiptein, parentithenai insertion: epenthesis inside: entos intelligence: nous

interior: entos interior angle: hê entos interlocutor: prosdialegomenos internal: endiathetos interpreter: exêgetês intersect: sumpiptein introduce: eisagein, suneisagein invalid: asullogistos invalid, be: mê errôsthai invent: porizein investigate: theôrein, zêtein investigation: methodos irrational: alogos irrefutable: alutos isoperimetric: isoperimetros isosceles: isoskelês join essentially with: sunousioun judge: krinein judgment: krisis just (adv.): prosekhôs kind: eidos kind of thing: phusis know(e): epistasthai know(g): ginôskein know(o): eidenai know previously(pg): proginôskein knowledge(g): gnôsis knowledge-producing(g): gnôstikos known(g): gnôrimos lack: ouk ekhein, ekleipein lack of logical sequence: to anakolouthon lamplight: lukhnaion phôs larger: meizôn laughing, capable of: gelastikos lead around: periagein lead astray: apatan learn: manthanein learner: manthanôn learning: to manthanein length: mêkos less: elattôn lie: keisthai, piptein life: zôê light: phôs like: homoios likewise: homoiôs limit: telos line: grammê line, straight: hê eutheia

English-Greek Glossary line of poetry: stikhos list: anagraphein, apographein longer: meizôn look at: prosblepein lung: pneumôn lunule: mêniskos made, be: ginesthai magnitude: megethos major: meizôn major premise: hê meizôn major term: ho meizôn, to meizon make: lambanein, poiein make a difference: diapherein make a distinction: diastellein make a division: diairein make an inference: sumperainein make clear: saphênizein make coextensive: exisazein make coincide: epharmozein make deductions: sullogizesthai make fit: epharmozein make manifest: emphainein make mistakes in hearing: parakouein make mistakes in seeing: paroran make use of: khrêsthai, proskhrasthai makes no difference: adiaphoros man: anthrôpos manifest: emphainein manner: tropos mathematical: mathêmatikos mathematically: kata to mathêma mathematics: mathêmata, mathêmatikê matter: hulê, pragma mean: dêloun, sêmainein meaning: sêmainomenon, sêmasia measure (n.): metron measure (v.): metrein ‘mechanic’: mêkhanikos mechanics: mêkhanikê medical: iatrikos medicine: iatrikê member of a contradiction: morion mention: mimnêskein, mnêmên poieisthai middle: mesos middle term: ho mesos, to meson mind: nous minor: elattôn minor premise: hê elattôn

151

minor term: ho elattôn mirror: katoptron miscellaneous: summiktos mistake: hamartêma, hamartia mistaken, be: hamartanein moon: selênê move: kinein move on: metabainein movement: kinêsis multiple: pollaplasiôn, poluplasiôn music: mousikê musical: mousikos musical interval: harmonia musical theory: mousikê, mousika ‘musician’: mousikos must: anankê, dein, pantôs name: onoma natural: phusikos, kata phusin, sumpephukôs natural philosopher: phusikos natural philosophy: phusikê, theôria phusikê natural philosophy, facts in: phusika natural philosophy, in: phusikos natural philosophy, of: phusikos natural science: phusikê naturally fitting: prosphuês nature: phusis nature, be of a n.: pephukenai nautical: nautikos near, be: parakeisthai necessarily: anankêi (dat.), ex anankês, pantôs necessary: anankaios, anankê necessary, be: dein necessity: to anankaion, anankê need (n.): anankê need (v.): dein, deisthai negation : apophasis negative: apophatikos, arnêtikos next: ephexês non-atomically: mê atomôs non-commensurable: ou summetros non-eternal: ouk aïdios non-expert: anepistêmon non-geometrical: mê geômetrikos non-immediate: mê amesos, emmesos non-immediately: mê amesôs non-proximate: mê prosekhês non-rational: alogos nonsense: teretisma non-twinkling: mê stilbein

152

English-Greek Glossary

non-universally: mê katholou north: boreios northern: boreios not know: agnoein not of the same kind: anomoiogenês not the cause: anaitios not-animal: mê zôion, ou zôion not-belong: mê huparkhein note: episêmeiousthai notice: horan, sunoran not-immortal: ouk athanatos notion: ennoia not-man: mê anthrôpos, ouk anthrôpos not-stone: ou lithos nous: nous number: arithmos object: enistasthai, episkêptein objection: enstasis obliterate: aphanizein oblong: epimêkês observation: paratêrêsis observe: theôrein obtain: lambanein obvious: prodêlos occultation: epiprosthêsis occur: ginesthai occur: sumbainein odd: perittos ophthalmia: ophthalmia opinion: doxa, ennoia opinion, have an: doxazein opposite: anapalin, antikeimenos opposition: antithesis optical theorem: optikon ‘optician’: optikos optics: optikê orbit: kuklos order (n.): eutaxia, taxis order (v.): tattein outer: ektos outside: ektos overlap: epallattein overlook: paroran pale: ôkhros pallor: ôkhriasis paradigm: paradeigma parallel: parallêlos part: meros, morion partially: epi merous particle: morion

particular: merikos, en merei, epi merous, to kata meros pass on: meterkhesthai, metienai passage: khôrion, lexis, rhêton per impossible: di’ adunatou per se: kath’ hauta, kath’ hauto perception: aisthêsis perceptual: aisthêtikos perform an induction: epagesthai perimeter: perimetros perishable: phthartos pertain to: prosêkein pervade: diêkein phase: phôtisma, phôtismos phenomenon: phainomenon philosopher: philosophos philosophy: philosophia phrase: lexis place (n.): khôra place (v.): paratithenai, tithenai place a comma: hupostizein plain: prouptos plainly: enargôs plane: epipedos planet: planômenos, planêtês Platonic Form: eidos Platonic Idea: idea plausible: pithanos plurality: plêthos poem: poiêma poet: poiêtês point: sêmeion pole: polos polygon: polugônion portion: morion pose a puzzle: aporein posit: hupotithenai, tithenai posited, be: hupokeisthai positing: thesis position: thesis possess: ekhein possible: dunatos postulate (n.): aitêma postulate (v.): aitein potentially: dunamei power: dunamis preceding: prosekhês precise: akribês predicate (v.): katêgorein predicated of, be: hepesthai premise: lêmma, protasis present: paradidonai present case: prokeimenon

English-Greek Glossary present discussion: prokeimenon presentation: paradosis pretend: hupoduesthai previously: proteron primary: prôtos principal: kurios principle: arkhê prior: prôtos privative: sterêtikos problem: problêma procedure: methodos proceed: perainesthai, proerhkesthai, proienai proceed backwards in thought: palindromein produce: apotelein, ekballein, poiein, prosekballein produce as evidence: propherein project: proballein proof: deixis proper: idikos, idios, oikeios proportion: analogia, logos proportional: analogos propose: protithenai proposed: prokeimenos proposed, be: prokeisthai proposition: apophansis, protasis prove: deiknunai prove to be: ginesthai provide: paratithenai proximate: prosekhês proximately: prosekhôs pulse: sphugmikos purpose proposed, the: prokeimenon purpose, his present p. is: prokeitai put: tithenai put order into something: tattein puzzle: aporia puzzle, pose a: aporein puzzled about, be: aporein Pythagorean: Puthagoreios qua: hêi quadrature: tetragônismos quadrilateral: tetrapleuros qualification: poiotês quality: poion quantity: poson quartile: tetragônos question: erôtêma, erôtêsis, zêtêsis question, beg the: ex arkhês aiteisthai question, in: prokeimenos questioner: erôtôn

153

radius: diastêma rainbow: iris ranking: tetagmenos ratio: logos ratio of four to three: epitritos ratio of three to two: hêmiolios rational: logikos ray: aktis, opsis reach: anagein, perainesthai reach a conclusion: sumperainein really: alêthês reason: aitia, aitios, logos reason fallaciously: paralogizesthai reasonable, it is: eikotôs reasonably: eikotôs reasoning: logos reasoning fallaciously: paralogizesthai rebut: apokrouein receiving, capable of: dektikos receptive: dektikos reciprocate: antistrephein recognize(g): gnôrizein record: apomnêmoneuma rectangular: orthogônios rectilinear: euthugrammos rectilinear angle: hê euthugrammos rectilinear figure: to euthugrammon reduce: anagein reduction: anagôgê redundant, be: parelkein refer: anagein reflect: anaklan reflection: anaklasis refute: elenkhein region: oikêsis relation: logos, skhesis relevant, be: khôran ekhein remain: leipein remainder: kataleipomenon remote, be: aphistanai removed: apostêsas render: hupekhein repeat: analambanein reply (n.): apantêsis reply (v.): apokrinesthai report: apangellein reputable: endoxos reputable opinion: endoxon require: deisthai restrict: horizesthai result (n): apodosis, apotelesma result (v.): ginesthai

154

English-Greek Glossary

resume: analambanein reverse: anapalin rhetoric: rhêtorikê rhythm: rhuthmos riddling way, in a: ainigmatôdês right: orthos right angles, at: pros orthas rise: epitellein rise before: proanatellein rise together: sunanatellein room for, be: khôran ekhein room for, there is r. for: khôran ekhein round: kukloterês rouse up: anakinein rule: kanôn sailor: nautês same kind, of the: homogenês same way, in the: homoiôs science(e): epistêmê scientific(e): epistêmonikos scientific knowledge(e): epistêmê scientific knowledge(e), have: epistasthai secure: asphalês see: horan seem: dokein self-guaranteeing: autopistos semicircle: hêmikuklion sensation: aisthêsis sense: aisthêsis, dianoia sensible: aisthêtos separate (v.): khôrizein separate, be: apokrinesthai separately: idiai (dat.) sequence, in: ephexês series: seira set: dunein, katabainein set out: ekkeisthai, ektithenai set out in advance: proektithenai shadow: skia shape: skhêma share: epikoinonein share in: metekhein shorter: elattôn show: deiknunai side: meros, pleura sides, having more s.: polugôniôteros sign: sêmeion sign, from a s.: tekmêriôdês signify: sêmainein similar: homoios

similarly: homoiôs simple: haplous simply: haplôs slantwise: eis to plagion smaller: elattôn solar: hêliakos solid: stereos sophist: sophistês sophistical: sophistikos sought: zêtoumenon soul: psukhê soulless: apsukhos sound (adj.): hugiês source of illumination: phôstêr southerly: notios southern: notios spark: spinthêr special area, in some: merikos special sense, in a: idios species: eidos specific: merikos specification: prosdiorismos spherical: sphairoeidês sphericity: to sphairoeides square (n.): tetragônon square (v.): tetragônizein squaring: tetragônismos stamp: diatupoun stand for: lambanein star: astêr, asterôios, astron start, from the s.: euthus starting point: arkhê state: hexis statement: logos stereometry: stereometria stone: lithos straight: euthus straight line: hê eutheia straight line, on a: ep’ eutheias straight sides, has: euthugrammos strict sense, in the: kuriôs strictly: kuriôs strike: prosballein strive for: antipoiein stronger: errômenesteros student: manthanôn study: mathêma subalternate: hupallêloi subject: hupobeblêmenos, hupokeimenon, mathêma subject: pragma subject matter: hupobeblêmena

English-Greek Glossary subject of scientific knowledge(e): epistêton subordinate: hupallêloi substance: ousia substrate: hupokeimenon subtract: aphairein succeed: einai akolouthian succession, in: ephexês successive: ephexês sufficient: autarkês suggestion: huponoia summarize: anakephalaioun summary: suntaxis sun: êelios, hêlios superfluous: perittos superior: huperkeimenos superparticular: epimorios supply: proslambanein suppose: hupolambanein, huponoein, nomizein, oiesthai surface: epiphaneia surprising: thaumastos synonymous: sunônumos synthesis: sunthesis take: lambanein take into account: epilogizein take place: ginesthai take together: sunairein take with: suntattein taste: geusis, khumos teach: didaskein teacher: didaskalos teaching: didaskalia term: horos, onoma terrestrial: khersaios, pezos that distinguishes: diakritikos that judges: kritikos ‘that’, the: to hoti the claim in question: prokeimenon theorem: theôrêma theory: doxa thesis: thesis thing: pragma thing seen: horaton think: dokein, nomizein, oiesthai think about: episkeptein think of: epinoein thought: dianoia, epinoia three halves: hêmiolios through induction: epagomenos tired, be: kamnein tool: organon

155

topic in music: mousikon totality: holotês touch, of: haptikos transfer: metatithenai transform: metalambanein, periagein treat: khrêsthai treatise: pragmateia triangle: trigônon true: alêthês true, be: alêtheuein true, be simultaneously t.: sunalêtheuein truth: alêtheia, to alêthes tune (v.): harmozein tuning (n.): harmonia turn around: periagein turn out to be: ginesthai twinkle: apospinthêrizein, stilbein uncertain, be: aoristainein uncertainty: aoristia understand: akouein, katalambanein, katanoein, lambanein, perilambanein understand: sunepinoein understand: sunienai uneducated: phaulos ungenerated: agenêton uniquely, that holds u.: idios unit: monas universal: katholou universally: katholou unmedical: aniatrikos unqualifiedly: haplôs unrelated: allotrios unrhythmical: arruthmos use: khrêsthai, lambanein useful: khrêsimos useful, be: khrêsimeuein useless: akhrêstos valid: errômenos, hugiês Venus: Aphroditê verbal expression: phônê verse: stikhos visual ray: opsis void: kenon walking erect: orthoperipatêtikos way: tropos white: leukos whiteness: leukotês whole: holos

156

English-Greek Glossary

whole, as a: kath’ holon, to holon wholly: holos, holos di’ holou ‘why’, the: to dioti widely, that holds more w.: koinoteros winged: ptênos wisdom: sophia wise: sophos without being demonstrated: anapodeiktos without demonstration: anapodeiktos

without making the proper distinctions: adiarthrôtos without parts: amerês without qualification: haplôs without significance: asêmos without specifying the distinctions: aprosdioristos word: onoma, phônê work: ergasia write: graphein write in manuscripts: graphein writing: sungramma

Greek-English Index References are the page and line numbers of the CAG text, given in the margins of the translation. * indicates the occurrence of a word from Aristotle’s text that Philoponus quotes in his commentary. ainigmatôdês, in a riddling way, 121,3 abakion, board, 133,2; 156,9 ainittein, hint at, 182,3 adiaphoros, makes no difference, aisthêsis, perception, 125,27; 190,13; 204,3 171,11*; 180,7; 181,7; 192,3; adiarthrôtos, without making the sensation, 214,16; 215,10.16.17; proper distinctions, 122,26 216,13*.14.20.23(bis); sense, adioristos, indefinite, 137,5 213,17*.20.22.24; adunatos, cannot, 184,22; 216,13*; 214,1.2.17.19.22.24; impossible, 112,6.20; 114,6; 138,4; 216,15.16.24.25 139,7; 143,24; 144,3.8; 173,8; aisthêtikos, perceptual, 180,27.28 197,19(bis); 200,16.29; 201,7.9; aisthêtos, sensible, 213,21.22.23; 202,11(bis); 203,2.21.30; 204,9.22; 214,17.19.29.34; 215,3; 210,5; 211,23; 212,13.23; 216,1.14(bis).16.26 214,6.12; 215,9.10; 216,11*; di’ aitein, ask, 130,5; 214,12; postulate, adunatou, per impossibile, 129,20; 149,12; to ex arkhês 137,28.31.32.33; 140,12.13; aiteisthai, beg the question, 144,1.7 aêr, air, 181,15 112,27 agein, draw, 113,5.8.35; 129,8; aitêma, postulate, 127,17.31; 130,7; 214,14 129,6.11.15.17.24; agenêton, ungenerated, 135,12(bis) 130,3.4.9.11.12.13.16.24*.25.27.29; ageômetrêtos, ignorant of 131,21.22.26; 132,22; 133,8.9; geometry, 150,7.17*.18; 214,14; 215,7 151,10.14.17; 152,8.11.12.18(bis); aitia, cause, 162,10; 166,18; 153,6.12*.17.19.20; 154,5.13 167,1.11.15.22.24.25.31; agnoein, be ignorant, 168,1.4.6.7.10.15.21; 169,18; 191,16(bis).23(bis); not know, 170,26; 172,7; 173,1; 159,21; 180,17 174,10(bis).16*.18.19*.21.25.32.34; agnoia, ignorance, 175,10*.28; 177,2.15.22.26; 150,18.21.26.27.28(bis); 178,3.7(bis).8.22; 179,22; 151,1.13.20; 182,14.18; 183,2; 184,30(bis); 152,6.7.12.15.16.19*.20; reason, 121,12; 176,17; 208,3 153,2.2*.4.13; 154,3*.14; aitiatos, effect, 119,19.21; 191,12.13.14.18.19.22.24; 193,32* 168,21.22.24.29(bis).30; aïdios, eternal, 132,11.12; 169,5.6.10(bis).11.12.19(bis).22(bis); 135,11.15(bis).20.22; 145,20; ouk 170,15(bis).17.27; aïdios, non-eternal, 135,23 172,9.14.17.19(bis).20.21.25.26;

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Greek-English Index

173,2.3.7.8(bis).10.13(bis).14.15.18. 19.20.26.27; 175,8.11.18 aition, cause, 119,20; 154,18; 168,10.22.24.28.29.30; 169,5.6.9.10.11(bis).14.19.22(bis); 171,18; 172,9.14.17.19.21.25.26; 173,2.3.4.7.9(bis).10(bis).11.12.13. 14.15.16(bis).19.20.26.27; 174,36*(bis); 175,2(bis).8.11.13.14.15(bis).16(bis). 18; 176,1.24; 177,26.27; 179,16; 183,10.12; 190,12 aitios, cause, 168,5.12.13; 169,1; 173,18; 174,27.30; 176,22.25.29; 177,5.8.11.23; 178,5.11; 182,21; 214,22.23; reason, 164,17; that lead to, 163,3 akatalêpsia, the view that apprehension is impossible, 141,10.15 akhrêstos, useless, 185,25 akoê, hearing, 180,28*; 181,4*; 214,34 akolouthein, be a consequence of, 169,13; follow from, 138,4 akolouthia, einai akolouthian, succeed, 156,13 akolouthos, consistent, 164,7; following,194,2 akouein, hear, 129,31(bis); 132,7; 155,15.30; 180,19; 215,17; hearing, 132,9*; understand, 118,26; 122,11; 152,11.17 akoustikê, of hearing, 213,25 akoustos, heard, 213,19 akribês, precise, 122,26; 132,23; 158,13 akriboun, êkribômenos, exact, 150,24 akron, end, 124,2; extreme, 111,15 akros, end, 113,5.35 akros, extreme, 117,6.13; 158,23.29; 185,16.18; 190,12.22; 192,24.29; 194,28; 204,15; 205,4; 209,23; 211,19; 213,13; high, 162,27 aktis, ray, 178,28; 179,2 alêtheia, truth, 126,5; 127,33; 128,5; 129,33; 130,15; 158,21; 196,29; 209,7 alêthês, true, 111,7.10.16; 112,9; 113,2.4; 120.4.6.8; 123,8.22; 127,27.32; 128,2.14; 129,5.10.13.24(bis); 131,5.12.28;

135,7; 138,30(bis); 139,9.16*.22; 140,21; 143,26; 144,5.14; 151,4.25; 152,21; 154,1(bis).2.7; 162,17.18.29(bis).32; 163,5.15(bis).17; 167,7(bis); 174,29; 175,12; 190,5; 191,19.25.26.29; 192,25; 193,6.7.9.10.13.14.16.20(bis).22.24; 196,6.10.17.22.23.27; 197,25*; 198,3.4.22; 199,26.28.32; 200,3.7.11(bis).15.16(bis).23(bis).29. 30(bis).31; 201,4(bis).7.8.9*.10(bis).11.12.15.28; 202,10.33.34; 203,1.15.18.19.20.21.22.24.25.26.28. 29; 204,1.8.23.24.26(bis).27.32; 205,4.5.11.13.14.16.21.27.31.32; 206,1.7.10.17.21; 207,12.13.14.19.25; 208,1.2.15.16.22; 209,3.22.23.26.27.28; 210,1.3.4.15.16.17.24; 211,7.8(bis).10.12.14.18; 212,1.2.5.6.8.10.13.14.15(bis).16.17; 213,6.8.10; to alêthes, truth, 152,16; 192,12; 209,2; 214,34; alêthôs, really, 141,14 alêtheuein, be true, 123,9; 135,21.27; 136,26; 139,1.2 alloioun, alter, 147,13 allotrios, unrelated, 192,24.30; 195,10; 196,2; 199,27; 200,1.18.20; 205,3.28.29(bis).30.33; 209,18.23.27; 210,16; 211,19.21; 213,5.7.13 alogos, irrational, 124,16.17.19; 142,14; 187,30(bis); 205,33; 206,1; 209,14.16; non-rational, 202,24 alutos, irrefutable, 169,7 amathês, ignorant, 145,3 ameibein, change, 135,14 amerês, without parts, 120,21 amesos, immediate, 111,7.11.16; 120,4.9; 127,21; 134,21.22; 164,24.30; 166,22; 167,17.18.19.30; 168,17.18.20; 170,8.14.18.19; 173,25; 174,12; 184,10.12.23; 185,9.10.31*; 186,1.5.13.19.21.25.31; 187,4.6.12.18.24(bis).25; 188,16.19.29; 190,15.18; 191,7(bis).9.24.28(bis).29.31.32;

Greek-English Index 192,20.24; 193,26; 194,7.21; 198,9.16.19; 201,31; 207,5.7; 210,26.28; amesôs, immediately, 186,3.4.7.12.24.25; 187,2; 189,4.27; 191,25.26; 195,14.15; 196,8.12.15.16.17; 197,1.4.6.9.10.11.17.18.28; 198,1.3.30.31; 199,1.7.9.16; 201,20; mê amesos, non-immediate, 211,3; mê amesôs, non-immediately, 197,29 anablepein, gaze upward, 214,31 anagein, include, 187,30; 189,18.19; 192,29; 195,16; reach, 184,11; reduce, 162,5; refer, 118,23 anagôgê, reduction, 184,27 anagraphein, list, 157,11.17 anairein, eliminate, 133,19; 134,20(bis).21(bis); 141,19; 142,23; 144,3.5; 193,9 anairesis, denial, 135,17; 136,24; 137,4.5 anaitios, not the cause, 172,20* anakephalaioun, summarize, 127,20 anakinein, rouse up, 214,24.29 anaklan, reflect, 182,5 anaklasis, reflection, 181,21; 182,5.7 anakolouthos, to anakolouthon, lack of logical sequence, 177,27 analambanein, repeat, 141,4; 168,23; 174,2; resume, 194,2 analogia, proportion, 123,21*.24; 159,4*.5.6.15.16.17; 160,12.19; 161,4*.15*.19.20 analogos, proportional, 124,20*.21 analuein, analyze, 162,25 analusis, analysis, 162,15.16.19.22.30; 163,11.12.15; 164,8 anankaios, necessary, 121,2(bis).16; 125,24; 138,12; 143,18; 146,27; 162,7; I needed to, 188,13; to anankaion, necessity, 162,9; 184,26 anankazein, compel, 135,17 anankê, must, 117,5*.17; 130,24*.25*.32; 132,19; 134,6*.6; 144,5; 172,21; 173,18; 193,17; necessary, 136,21; 169,9; 173,10; 175,5; 185,14; 189,26; 195,13; 200,28; 201,19*; 203,6; 211,7;

159

213,24; 216,24; necessity, 130,27; 201,21; need, 130,33; 133,25*.26; 134,5.8*; 146,28; is needed,184,20; anankêi (dat.), necessarily, 130,26; ex anankês, necessarily, 130,28.32; 131,7(bis); 163,19.20(bis); 205,17; 210,17; 213,4 anapalin, conversely, 211,27; opposite, 204,7*; reverse, 195,2; vice versa, 113,30 anapherein, include, 188,2(bis) anapnein, breathe, 174,23.26.27.28(bis).31.32; 175,2(bis).5.6; 176,8(bis).13.14(bis).15; 177,29 anapnein, breathing, 174,30.36*.37*; 175,3.10 anapodeiktos, indemonstrable, 120,23; without being demonstrated, 128,22; without demonstration, 129,15 anatrekhein, ascend, 120,13 anêkein, belong, 146,5; 191,3 anepistasia, failure to pay attention, 180,18.24 anepistêmon, non-expert, 150,8 aniatrikos, unmedical, 152,8 anomogenês, of a different kind, 149,13 anomoiogenês, not of the same kind, 113,3.13 anomoios, dissimilar, 114,16 anomoioskhêmôn, of dissimilar form, 203,6 anomoiotês, dissimilarity, 114,13 anthrôpos, human, 216,4; man, 127,22; 132,8.21; 133,12.13; 136,3.23(bis).28.30(bis); 137,1.12.19; 139,16*.17(bis).18.19.23(bis).26; 142,15; 154,26.27(bis).28; 155,25; 159,1.2.11(bis).12.13; 163,3; 164,25.26.28(bis).30.31; 176,19.20.21.22.23; 177,4.5.8.9.10.11(bis).13; 183,26(bis).29; 184,5.7; 185,4(bis).5; 186,30.31; 187,2; 193,8; 201,23.24(bis); 203,26.27; 205,8.9; 207,21.22.23.25; 208,5.9; 209,10.11.24.25; 212,3(bis).27.28; 213,1(bis); mê anthrôpos,

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not-man, 139,23; ouk anthrôpos, not-man, 136,4; 137,2.20 antidiairein, contrast, 140,20; 187,29; antidiêirêsthai, be coordinate, 199,6; be a coordinate member of a division, 209,9; antidiêrêmena, coordinate members of a division, 199,2; antidiêrêmenon, coordinate. 199,3 antikeimenos, opposite, 138,5; 152,19*; 154,4*; 158,10; 191,19; 192,12.15; 193,10; 194,21; 198,13.19; 201,15; 203,22.23.24.26.28; 208,16; 210,24; antikeimena, contradictory pair, 129,21 antilegein, claim against, 113,1 antilêpsis, apprehension, 178,26 antilêptikos, apprehensive, 216,17 antiphasis, contradiction, 127,15; 135,6.9.25.26; 136,2(bis).7.12.19.22; 137,1.3(bis).7.8.11.16.19.23.28; 138,1.3.9.17.20.24.29.31; 139,8; 140,4.5.8.12; 141,1.7.9.12.20.27.29 (ter); 142,22; 143,13.16.22.30; 144,11.12.14.26.30; 145,2(bis).18; 147,18.19.27*.27; 208,14(bis) antiphaskein, contradict, 137,6 antiphatikos, contradictory, 192,15; 193,10; 198,13; 203,24.25 antipoiein, strive for, 157,15 antistrephein, convert, 159,9.12; 161,13; 162,5; 163,16.25; 176,11.14; 183,28; 185,23; 188,12; 208,21; 211,13; reciprocate, 168,24.28; 169,9.20.23; 170,14; 172,5.17.18(bis); 173,14; conversion, 163,30*; not tr., 168,24 antistrophê, conversion, 162,8; 174,37; 175,1.5.6.12.18; 177,3; 185,12 antistrophos, converse, 188,26 antithesis, opposition, 174,37; 175,6 aoristainein, be uncertain, 155,7 aoristia, uncertainty, 155,15; indeterminate, 163,8; is indeterminate, 141,11 aoristos, indefinite, 136,4; 171,2;

indeterminate, 162,32; 163,2.12; 165,20.22.26; 166,4.7.11; apangellein, report, 128,32 apantan, apantôn,in response, 130,35 apantêsis, reply, 132,32 apatan, deceive, 192,9; lead astray, 192,4; apatôn, that produces error, 207,22; êpatêmenôs, erroneous, 192,2; êpatêmenôs, erroneously, 191,22; 194,23 apatê, error, 154,26; 155,3; 91,4.8.10.11.32; 192,5.7.10.13.14.26.27; 193,26.27; 194,1*.9.10.12*.13.16.17*; 198,8.9.12.15.16.31; 203,8; 207,4.8.11.16.26; 208,19.27.30*; 209,2.6; 210,23; 211,3.6.10.17 apeiros, infinite, 165,23; 166,4; naively, 149,21; ep’ apeiron, ad infinitum, 113,15.16.17.19.27; 114,8.9; 165,10* apereidein, fix, 155,7 aphairein, subtract, 126,17; 127,14; 130,35; 215,22 aphairesis, ex aphaireseôs, abstract, 215,15*; 216,3 aphanizein, obliterate, 182,25 aphistanai, be far from, 167,7; be remote, 174,18.20.21.34; 177,21*; 178,2*; be, 168,8.9; apostêsas, removed, 178,7 aphorizein, determine, 137,26 Aphroditê, Venus, 171,17 aplanês, fixed (star), 171,15 apodeiknunai, demonstrate, 111,11; 116,8.21; 117,13; 118,10.11.24.27.28.29.30(bis); 119,9.11.13.14; 120,8.12.19; 127,3; 131,3; 132,28; 141,7.9; 142,6.20; 144,19*; 145,5*.6; 146,10.16.28; 147,6.8; 158,9; 164,19; 167,17.18; 183,17.31; 184,11.14; 195,12; 214,11; form demonstrations, 119,1; apodeiknuôn, in its demonstrations, 144,17 apodeiktikos, demonstrative, 120,3.5.8.16; 133,17; 143,23; 144,21; 146,28; 157,25; 158,12*; 162,14.33; 165,4; 207,25; 208,3.11.25.30; 211,7; demonstrator, 131,3

Greek-English Index apodeiktos, demonstrable, 215,8; demonstrated, 118,11; of demonstration, 111,8 apodeixis, demonstrating, 144,20; demonstration, 111,7.14; 116,2.4; 117,15.17.18.26; 118,7.33; 119,11.24.25; 120,11.23; 121,2.5.6.11.18; 122,1; 123,15; 125,8.13.19; 127,2.9; 128,14.18.22; 129,10; 131,1.10(bis).16.23; 132,26; 133,1.3.20.22.24.26*.26.28; 134,5.6.8.21.22.23*.30; 138,7.16*.17.18; 141,19; 147,16; 162,15; 164,18; 169,8(bis); 175,20*.22*; 177,32; 183,18.30(bis).33; 184,10.30; 210,27; 214,11; 215,9(bis).11 apodidonai, give, 138,6; 146,30; 167,22; 178,8.11.22; 179,22; 180,2.4; 216,12 apodosis, coordinate claim, 178,17; result, 216,10 apographein, list, 179,11 apokrinesthai, answer, 141,11.12.13.16(bis); 145,16; 146,1.4.22.25; 147,20; 149,1.2.3(bis).22; 150,14; 156,8; 178,10; be separate, 157,8; reply, 143,16; apokrinomenos, in reply, 148,2 apokrouein, rebut, 134,1 apomnêmoneuma, record, 178,13 apophanai, deny, 138,15*; 185,18; 186,3; 187,5; 189,5.16; 190,22; 191,25; 196,16; 200,11.15; 202,25.31; 203,18.21; 205,31 apophansis, proposition, 194,7; 198,12 apophantikos, declarative, 147,30; 148,3(bis) apophasis, negation, 128,25; 130,34; 131,5; 135,17.19.21; 136,6.25.27(bis); 137,5; 139,2.10; 143,24; 144,4; 148,4.5; 150,21.22.26; 151,11.13; 152,7; 153,4.5.13; 154,15; 175,9.10*; 186,9.19.21; 187,4.6.12.25; 188,21; 190,16.18; 191,7.9.12*.13.14.27.31; 192,25; 194,21; 198,12 apophaskein, deny, 131,31.32; 136,6; 138,30; 176,25.26.30.31.32;

161

186,4.10.12.17(bis); 187,9.12.18.29; 188,1.3; 189,26.27; 190,12.18; 191,26; 195,1.14; 197,1.10; 200,29.31; 201,27; 202,7.8.9.21.23(bis).24.30.32.34; 203,1.15; 209,27.32; 210,16 apophatikos, negative, 130,18.20; 139,6.7; 145,26; 176,6.7.11.18.29; 177,1.2.3; 183,12.16; 184,18.20.21.23; 185,11.12; 186,2; 188,9.15.17; 190,13.25; 192,10.12; 193,26.27; 194,8; 196,11.13.27(bis); 198,14.18.20; 199,10.11.32; 200,3.7.23; 201,3; 202,1; 204,5.6.8.9.31; 205,13.22.27; 206,2.17; 207,9.19.27; 208,19.20.21.22.23; 209,26; 210,11.24; 211,3.5.9.11; 212,20.24.25.29 apoplêroun, completely fill, 121,11 aporein, be puzzled about, 174,34; find puzzling, 179,10; pose a puzzle, 146,14 aporia, puzzle, 146,26; 147,2.16 apospinthêrizein, twinkle, 171,14 apostasis, distance, 167,25 apoteinein, direct, 131,1; 132,26 apotelein, produce, 178,27; 179,1 apotelesma, result, 173,1 aprosdioristos, without specifying the distinctions, 134,30 apsukhos, soulless, 177,10(bis).12(bis); 189,13; 202,22; 205,2.4.5.31.32; 206,20; 212,5(bis) Aristotelikos, in Aristotle, 158,10 arithmêtikê, arithmetic, 117,18*; 118,10.13.32; 119,1.2.3.27; 122,14; 124,9*.22; 140,16; 146,6; 213,20 arithmêtikos, arithmetic (adj.) 117,22; 123,3.11; 126,8; 140,23; 145,15.28; arithmetical, 145,15; of arithmetic, 146,10 arithmos, number, 121,5; 123,22; 126,4.6.8(bis); 140,17; 141,28.30; 160,8.11.15.23; 165,18.19.20.21.22.25.26; 166,4(bis); 213,21 arkhaios, ancient, 157,8; 178,13 arkhê, beginning, 121,12; 129,27; 130,10; 131,22; 137,33; 138,1;

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156,15; principle, 111,14.19; 115,6.10.14; 116,9*.20.21; 117,15.16.18; 118,4.6.9.10.22.24.27.29.30.31.31*. 32.33; 119,2.3.9.10.11.12(bis).13*.13.14 (bis).17.18.19.20; 120,6.13.19.20.24.27*; 121,25*; 122,1; 127,18; 129,12; 141,6; 142,5; 146,12.14*.15.17(bis).21*.25.29; 147,6.8.11.16; 148,9.11.13.19.20.21.23; 149,8.9.16; 153,24; 154,7; 162,26; 167,18; 179,16; 192,6; starting point, 125,7; 157,29; ex arkhês; at the beginning, 194,10; initial, 183,16; ex arkhês aiteisthai, beg the question, 112,27; tên arkhên, to begin with, 174,4 arkhesthai, begin, 156,23; 162,21; 171,6; 177,24 arnêtikos, negative, 136,6 arruthmos, unrhythmical, 151,7.8.10.14.19; 153,17.18.20; 154,6 artasthai, depend, 147,11; 148,10 artios, even, 129,21; 138,4; 165,19.21.22.25.26(bis); 166,11.13 asêmos, without significance, 141,17 asômatos, bodiless, 199,4; incorporeal, 214,33 asphalês, secure, 150,24 astêr, star, 129,21; 171,15; 173,28 asterôios, star, 134,29 astrologia, astrology, 125,3; 180,3.4.28; 181,4 astrologikê, astrology, 182,28 astrologos, astrologer, 183,2 astron, star, 134,27; 167,3.4.6.8; 168,7; 179,19; 180,21 astronomia, astronomy, 125,4; 179,21; 213,19 astronomos, astronomer, 145,28; 180,20 asullogistos, invalid, 151,4.21.25.26; 153,21; 159,8; 162,9; 190,25; 211,13 asummetros, incommensurable, 124,17; 138,2; 144,2.27 athanasia, immortality, 143,14.17 athanatos, immortal, 127,34; 132,10.12; 135,9.10.11.19.20.22;

144,10(bis); 145,20; 167,17; ouk athanatos, not-immortal, 135,22 atomon, atom, 129,12; 192,5 atomôs, atomically, 186,24*.25*; 196,24*; mê atomôs, non-atomically, 197,26 atopos, to atopon, absurdity, 144,13 aulêtris, flute-girl, 176,9.10; 178,9.10.11 autarkês, sufficient, 139,21.25; 185,9; autarkôs, independently, 117,13 autopistos, self-guaranteeing, 127,22; 129,29; 167,27; 168,13; 184,23 autothen, directly, 148,15; 185,12.15 auxein, grow, 159,4.5(bis).6(bis).15(bis).16(bis).18. 20(bis); 160,10.12.19; 161,4; 185,30*; increase, 113,15.16.17.19(bis).23.27.30(bis); 114,10(bis); 164,9.22.27.29; 165,3 auxêsis, growth, 147,13; 161,14.21; increase, 160,11; 164,9.16; 165,15.27; are increased, 164,22 axiôma, axiom, 112,9; 121,11.14; 123,2.4.7.10.11.12.13.15; 124,6; 125,1.2.12.21; 126,15; 127,5.12.16.24; 129,23.26.27.30; 130,2.3.8.11.14.25.26.32.33; 131,6; 132,22; 135,3.6.25; 136,1.7.12.19; 137,1.11.12.16.19.22.25.27; 138,1.8.17.19.28.31; 139,8; 140,4.11.28.30; 141,1.5.7.9.20.25; 142,5.8.20.22; 164,14.19.20; 214,14; 215,7.11 basis, base, 178,27; 179,2.4 biblion, book, 129,16; 141,8; 183,31 boreios, north, 167,4.5.7.11; 168,8.9; 178,1; northern, 167,3.5.6; 168,8; 180,20 deiknunai, display, 158,26; prove, 111,6.18; 112,32; 113,4.12.25.34; 114,11; 115,4.7.10.12.13; 116,9.10.15; 117,5.10.18.21; 118,5(bis).6.9; 120,5.9.10(bis).13.22; 121,1*.2; 122,9.10.15.17.20; 124,20.29; 125,2.3.5; 134,6.14; 135,26; 137,15.27; 138,3.8.16*.23.28;

Greek-English Index 139,22; 142,10; 143,17.22; 144,3.10.14.27; 145,3.13.17; 146,9; 148,10.13.19.20(bis).21.23; 158,10; 164,12.14.19.21; 167,19.30; 168,31; 169,19; 172,3.9; 174,23; 176,6; 177,29; 178,6(bis).8; 179,7.16; 181,17; 182,12; 183,6.7.21.22; 184,14.15.18(bis).20.21.22(bis).23. 24; 185,11.12.13.16.25; 186,1(bis); 188,15.18; 189,11; 196,23.25; 198,18.24; 199,9; 200,8.22.24; 202,3; 203,2.14.16.21; 204,1.9.13; 207,21.23; 208,5; 209,10; 210,29; 213,17; 214,5; 215,4; show, 164,6; 166,15; 182,11; 188,22; 198,24; 214,28 dein, have to, 203,7; it is necessary, 121,15.26; 122,5; 135,25; 138,16*.18; 147,4; 174,28; 176,24; 184,14; 186,4; 204,6; 205,27; 208,21; 210,10; 212,12; must, 111,13.16; 119,11; 120,5; 121,21; 133,17.28; 136,15; 138,20; 141,12; 146,5.14.18.21; 147,2.20; 157,22; 158,8; 160,14; 168,23; 174,36*; 176,29(bis); 183,31; 190,5; 199,22; 202,6.8; need, 111,8; 152,11; 157,13; 158,9; 160,22; 184,11; should have, 165,29 deisthai, need, 111,11; 129,31.32; 131,23; 134,16.18; 138,24; 140,15(bis); 168,14; 170,19; 184,9(bis); 185,11.20.29; 214,24; require, 127,32; 128,13.14.18.22; 129,6.10.13; 130,9; 147,21; 167,30; 215,16 deixis, proof, 137,33; 140,11; 144,8; 155,30; 168,23; 171,1; eis deixin, to prove, 129,32 dektikos, capable of receiving, 133,13.14; receptive, 159,11.13(bis); 164,26(bis) dêlos, clear, 126,4.7.14*; 127,12; 128,3; 138,10; 174,13; 175,21; 190,5; 195,14; 200,17; 210,9; clearly, 124,16; 125,10; 128,23; 135,10.27; 136,2; 137,2; 153,17; dêlon, clearly, 195,3; 204,7*; it is clear, 113,34.37; dêlon hoti, clearly, 142,16; 147,10; 155,27; 156,7; 158,9; 165,10; 169,16; 173,15; 193,18; 199,6.8;

163

200,18.20; 201,1; 202,8; 203,18; 204,18.28.30; 205,9.26; 207,17; 208,23; 210,20; 212,29; it is clear, 175,31; 176,6.28; 189,3.14; 193,15; 201,24; 202,33; 204,15; 213,22; 214,17; dêlonoti, clearly, 143,18; 161,9; 164,1; 174,24; 175,10; 210,8; 212,22; not tr., 136,8; 149,8 dêloun, indicate, 130,21; 132,8.15; 153,22; 174,18; 186,24; 193,28; mean, 181,19.27 dêmiourgos, Demiurge, 215,2 diabêtês, compass, 128,11 diairein, distinguish, 152,15.16; 153,4; divide, 121,6.10; 135,7; 140,21; 151,1; 194,5; 200,14; make a division, 191,18; division could occur, 192,7 diairesis, division, 125,7; 127,21; 129,25; 191,11; 194,10.18 diakeisthai, be disposed, 150,27; 191,21 diakrinein, distinguish, 127,16; 129,27; 130,10; 131,21.23.30; 143,7; 162,14; 180,6 diakrisis, distinction, 127,19; 130,10; 186,9; way of distinguishing, 133,8 diakritikos, that distinguishes, 214,4 dialambanein, deal with, 124,16.21; 198,9 dialegesthai; discuss, 117,22; 119,16; 141,14; 142,10; 143,2; 146,18; 149,7.9.10.14; 150,7; 182,5.7; engage in conversation, 163,4; engage in dialectic, 208,7 dialektikê, dialectic, 115,5; 134,29; 141,5; 142,10.26; 143,7.8.11; 145,1.9; 147,23; 155,18 dialektikos, dialectical, 147,18.22.25; 154,16.18; 155,17; 156,4; 158,12*.17(bis); 162,15.16.33; 163,1; 164,2.22.24; 165,4.5; 207,28; 208,7.11.26.27; 209,1; 211,7; dialectician, 140,24; 142,11; 144,9.11; 155,2 dialexis, dialectical conversation, 154,23; 158,25.27.28; 164,1*.21 diametros, diagonal, 124,18.19.25; 144,2.27; diameter, 113,5.7.20.26.32.35; kata

164

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diametron, diametrically, 167,23.24; 168,2 dianoia, sense, 196,10; thought, 127,20; 215,15 diapherein, be different, 128,24; 129,25; 131,21; 147,20; 158,27; 162,33; 164,1*.6; 165,4; 166,15; 170,5.13; 178,16; 179,14; 182,10; differ, 118,28; 130,2.3.12.17.25; 132,22; 143,10; 166,17.21; 171,8; 209,3; make a difference, 139,11*; 141,27; 161,20; 165,24; 197,25*.28; 198,4; diapherei, there is a difference, 118,7* diaphora, difference, 118,28; 137,29; 147,23; 168,18; 174,2.11; 182,12; 191,6; differentia, 183,24.27 diaphoros, different, 119,24; 123,2.4; 179,4 diarthroun, expound precisely, 122,26; 150,20; 166,1; 193,5 diastasis, dimension, 214,7.9; distance, 168,8.9 diastatos, dimension, 124,1; dimensional, 120,22; 131,13 diastellein, make a distinction, 122,21 diastêma, distance, 128,11; 182,25; radius, 113,22.33; 129,7; 130,7 diathesis, disposition, 131,9; 150,21.23.27.29; 151,20; 152,7.20; 153,5.6; 154,15; 191,12*.18.19.21.24; 193,32*; 194,5*.13; 198,9 diatupoun, stamp, 155,14 diazeuktikos, disjunctive, 171,11 didaskalia, doctrine, 191,8; teaching, 141,17; 191,3 didaskalos, teacher, 112,1.30; 129,28.31; 130,5 didaskein, expound, 191,9; 193,5; 207,6; teach, 112,33; 121,1; 141,16; 154,25 didonai, give, 139,16*; 146,20; 147,2; 215,2; grant, 143,12; 145,6.7; 149,12; dedosthai, be given, 121,9; 139,19; dedomenon, given, 121,12.14.27.29; 122,4.12; 124,6.7; 125,10.11; 127,3; 140,31; dotheis, given, 111,29 diêkein, pervade, 140,13.25

diexerkhesthai, go through precisely, 132,23 dikha, in two, 113,21; not tr., 194,16 dioti, because, 115,6; 117,20; 118,18.30; 129,26; 130,8; 133,9; 154,9; 157,26; 164,24; 167,24; 172,9.24; 173,3; 174,34; 180,24; 184,19.20; 186,25; 187,3; 188,13; 198,20; 200,8; 205,22; 207,27; 208,10; to dioti, the ‘why’, 118,8.12*.15*.17(bis); 146,29.31; 166,16.18.20.22; 167,3.12*.16.20(bis).23.31; 168,7.17.18.19.21(bis); 169,5.17.21.27; 170,9.13.17.23; 172,6.9.13.24.27; 173,23.24.27; 174,2.12.32; 175,20*.22*.23.28.31; 178,16.20; 180,17.27; 181,17; 182,2*.10; 183,8; 191,6 dokein, appear, 127,34; 130,26.28; 147,24; 150,9; appear true, 127,28.29.30.31; 128,13.16.21; 129,2.3.5.9.21; 130,4.5.13.14.15.16.29(bis).32.33; 131,7.23.25.26; 143,21; decide, 143,26; seem, 118,25; 119,1; 130,25*; 135,8.15.18.20; 143,15; 158,23; 171,14.15; 174,37; 177,24; 182,18; 215,15; think, 115,4; 149,11.22; 157,6.7; 158,8 dokêsis, appearance, 192,16 doxa, opinion, 129,9.18; theory, 133,30 doxazein, believe, 129,19; 192,5; 194,9; have an opinion, 192,2 dunamis, power, 214,33; dunamei, potentially, 147,19; 173,15.19.20 dunatos, can, 185,21; 188,11; 193,21; possible, 111,18; 120,19; 123,1; 136,2; 137,11.15.20; 144,21; 159,10; 160,11; 162,3; 165,12; 169,16; 176,10.18; 183,20*; 184,2; 185,18; 190,15; 199,18; 209,16.21; 212,1.5; 216,18; ou dunaton, impossible, 169,23 dunein, set, 167,3.5.6.8.11.14.15.16; 168,9.10.11(bis); 177,31; 180,21.22.23.24 duskherainein, have difficulty, 182,23 duskherês, difficult, 162,32

Greek-English Index êelios, sun, 156,19 eidenai, know(o) 120,14.16; 129,19; 146,6.10.15.16.18.19(bis).27.28.30; 147,3; 150,8.24; 151,14; 154,24(bis); 155,17; 159,18; 179,20.21; 180,8.14.18.19.20(bis); 181,25; 182,1*; 183,1; 186,5; 191,15.16.21.22.23(bis); 198,23; 202,11; 214,11 eidos, form, 133,24; 151,7; 181,3.5*.6*.14; Form, 133,25*.29; kind, 128,4; 151,1; 173,11; species, 177,13; 187,29(bis); 199,4 eisagein, introduce, 133,18.21.24.30; 134,1; 141,10; 193,6 ekballein, extend, 129,14; produce, 113,32; 116,10; 124,25 ekhein, be, 122,12; 126,5; 127,21; 128,1.3.4; 129,32; 130,27.28; 135,8; 160,21(bis); 162,23; 166,19; 181,17; 195,8; 201,27; 202,32; 204,7*; 205,20; 207,17; 210,8*.11*; 215,20; can, 111,29; contain, 121,8; 153,11; 160,3.4.5; 167,31; 168,6.7; 174,10; depend on, 153,14; get, 129,28; grasp, 131,16; have, 116,7*.21; 117,5.20; 118,31*; 122,15; 124,18.22; 126,19; 127,24; 128,16.18.21; 129,34; 130,1.9.15.22; 134,18.19; 143,1.2; 149,1; 150,22.23.26.27; 151,8.9 (ter); 155,3; 156,12; 159,19; 160,13; 163,2; 167,13.14; 170,9; 173,17; 174,13.31.32; 175,25; 177,26.30.32; 179,26; 181,10.11.15; 182,20; 183,16; 184,26; 186,6.10.15.17.19; 187,11; 191,16.21; 192,16.21(bis); 208,2; 213,25; 214,8.20.21; 216,1.17; possess, 153,18.19*.24*; 154,6(bis).7; 156,2; 214,2, ekhein peri, be concerned with, 144,18; 145,9; 147,12; 155,2; 157,6; 179,15; ekhein pros, be related to, 181,12.13.26; 194,28; 204,14; ekhôn, with, 171,22; 178,27; 179,4; 180,15; 188,19; epi tôn houtôs ekhontôn, in such cases, 172,27; houtôs ekhein, be related in this way, 178,19; houtôs ekhôn, in this condition, 147,6; khôran ekhei, there is

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room for, 135,18; be relevant, 147,15; mê ekhôn, without, 214,16; ouk ekhein, lack, 216,13*.15.23; piston ekhei, there is conviction, 140,6; pôs ekhousa, of a certain kind, 132,22 ekkeisthai, set out, 160,15 ekleipein, be absent, 213,17*.18.23.25; 214,3.18(bis).19; 216,25(bis); be eclipsed, 167,27(bis).29(bis).31; 168,3; lack, 214,1.2 ekleipsis, eclipse, 167,24; 168,1.16 eklimpanein, be eclipsed, 167,22 ektithenai, expose, 191,5; set out, 127,21; 174,1; 175,7; 193,28 ektos, exterior, 113,15.20.23; 114,9; external, 131,17; outer, 113,36; 114,3.7.8.9; outside, 111,24; 113,6.35; 120,22; 149,16; hê ektos, exterior angle, 113,8.9.17(bis).27.28.29.30.31; 116,11 elattôn, less, 129,14; 147,1; 192,1; minor, 136,20; 137,3.10.14; 161,7.18; 171,3.5; 177,1; 190,6(bis); 193,1.15.17.20; 197,21; 200,28; 207,13; shorter, 175,26.28; 179,9; smaller, 111,23.26.27; 112,10.11.12.14.15.23(bis).24; 113,3.10.12.18.20.24.26.29; 114,12.15; 115,3; 149,13; 167,13.14; 177,31; 178,21; 182,20(bis); 191,20; hê elattôn, minor premise, 137,33; 192,34; 193,14; 196,21.24; 199,22.32; 200,16.23; 201,4.8.12.15; 204,23.24.26; 205,2.12.13.14(bis).21.22.26.27; 206,2.7.10.18.21; 207,17.26; 208,2.20; 209,3.13.17.24.26.28; 210,2.12; 211,8.13.14.24.28; 212,3.4.8.10.14.17; 213,6.7.9; ho elattôn, minor term, 136,25; 137,17; 139,5; 140,8; 154,19.22; 155,22; 176,23.25.26.27.28.30.31.32; 193,16.18.22; 195,1.3.9; 198,29; 199,1.3.5.15.23; 200,17.19.21; 201,17.18.28; 202,28(bis).29; 205,1.2.6.8.24.28.29.33;

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209,9.12.18.28; 211,25; 212,4.7.10; 213,7.8 elenkhein, refute, 149,22(bis).23; 150,1.2 (ter); 216,18 embadon, area, 182,21 emmesos, non-immediate, 164,18; 167,2.20.22.26; 173,25; 174,11.13.22; 184,10; 191,24.31.33; 193,27; 198,10; 207,6.7; 210,23 emperiekhein, contain, 147,19 emphainein, make manifest, 171,7; manifest, 171,15 emphanizein, appear, 168,26 empsukhos, ensouled, 186,8; 189,12; 205,25(bis); 208,5; 211,29 enantios, contrary, 127,35; 128,33; 129,9.17; 131,25; 136,21.26; 143,23.29; 144,8.15.27.28; 145,6; 152,21; 154,1(bis).7; 192,16.26; 193,6.8(bis); 194,9; 198,8.12(bis); 203,18.23; 207,26; 208,1.14; 210,8*.10* enargeia, ek tês enargeias, because it is evident, 125,23; 126,7; 127,12.14; 183,31 enargês, evident, 125,26; 127,1; 128,8; 131,2; 138,10; enargôs, plainly, 200,25 endeiknunai, indicate, 118,15 endekhesthai, can, 114,12; 122,4; 131,10; 147,29; 169,13; 187,3; 190,21; 193,13.19; 195,7.8.12; 196,11.17(bis).22; 202,1; 205,11.14; 210,19; 211,18; 213,6; endekhetai, it is possible, 112,29.31; 118,26; 119,9.10.25*; 136,29; 138,22; 144,12; 159,7.9; 164,18; 173,15; 177,1; 184,15; 186,13; 188,14; 189,15; 192,2.3.32; 195,18; 196,2.7; 197,17; 200,7.14.28; 201,11; 202,2.14.24; 203,14.17.21; 205,22; 209,7.29; 210,26.27; 211,23; 212,16; 216,19; ouk endekhesthai, cannot, 141,2; 148,1; 187,5; 193,16; 195,15; 199,25; 199,28; 201,3; be impossible, 119,24; 121,1*; 136,20; 136,25; 138,15*; 144,6; 186,25 endiathetos, internal, 131,4.16 endoxos, reputable, 164,41;

endoxon, reputable opinion, 142,10.13; 143,17; 163,1 energeia, activity, 135,11; energeiai (dat.), actually, 173,16.20(bis) engraphein, inscribe, 111,21.23.25.27.29; 112,3.15.16.18.22; 113,26; 114,15(bis) Êniokhos, Auriga, 167,9; 180,22.23 enistasthai, bring objections, 157,22; object, 157,24.28; 158,1 enkhôrein, enkhôrei, it is possible, 188,6(bis); 188,3*; 189,10*.22; 196,6 enkuklios, general, 157,2.10; general education, 157,8 ennoia, notion, 127,24; 134,1; 153,15; opinion, 150,22.23; think, 214,27.28.32 enstasis, objection, 157,22.26.27.29; 158,1.3.5.6.8.11* entelês, in its complete form, 127,21 entos, inner, 113,37; 114,2.7; inside, 111,24; 113,34; 128,7; 179,8; interior, 113,26; hê entos, interior angle, 113,9.10.16 (ter).18.28.29(bis).30.31; 114,10(bis); 116,11(bis); 117,7 eoikenai, eikotôs, it is reasonable, 168,4; reasonably, 174,34 epagein, go on, 115,13; 118,15; 170,2; 178,2; 194,1.24; go on to say, 216,19; infer, 153,22; epagesthai, perform an induction, 216,1; perform inductions, 216,13*.23(bis); epagomenon, what follows, 153,14; epagomenos, through induction, 215,23 epagôgê, induction, 157,23; 171,10*; 214,12.15(bis).16.17; 215,7.10.16*.19.28; 216,3.7.11*.13.15.17.20(bis).22(bis) epallattein, cross over, 119,24; overlap, 189,13.14.20(bis) epaneresthai, ask, 156,9 epenthesis, insertion, 186,26 epentithenai, insert, 164,16.23.27; 165,2.5.16; 185,10.31 epexergasia, elaboration, 128,14; 170,4; 196,29; 198,23 ephaptesthai, coincide, 113,37

Greek-English Index epharmozein, coincide, 112,19.20; 113,38; 114,7; 123,17; 126,15; 149,13; 156,11; make coincide, 112,5; make fit, 112,4; 165,11 epheuriskein, discover, 165,12 ephexês, adjacent, 116,18; 128,30; in succession, 160,14.16.19; in what follows, 150,19; 152,6; 165,29; 166,19; 174,15; 188,7.24; 194,24; 157,14; 160,6; next; 115,13; 122,20; 134,1; 153,7; 197,26; successive, 160,21; hê ephexês, adjacent angle, 116,11.13.14.15.16.17; 117,7.8.9; 126,19; in sequence, 164,16; kai ephexês, and so on, 156,14; ta ephexês, the following ones, 120,8; what follows, 153,14 epigramma, epigram, 156,12.26; 157,1 epikoinonein, share, 140,31.30*; 141,1.23.24 epilogizein, take into account, 181,6 epimêkês, oblong, 182,14 epimorios, superparticular, 159,22; 160,3.4.5.7.11.20 epinoein, think of, 202,15 epinoia, thought, 216,5 epipedos, plane, 155,13; 179,14 epiphaneia, surface, 128,19; 155,1; 215,27.28 epiprosthein, be in front of, 124,2 epiprosthêsis, occultation, 167,26 episêmeiousthai, note, 188,7.13; 212,11 episkeptein, consider, 181,12; think about, 128,2 episkêptein, object, 112,1 epistasia, attention, 128,13(bis) epistasis, attention, 127,32; 128,2 epistasthai, have scientific knowledge(e), 178,16; 215,10; 216,24; know(e), 119,16* epistêmê, science(e), 117,16.25; 118,5.8.14*.18.24.25(bis).27.32; 119,10.12.13.14.15(bis).18(bis).24; 120,19.24; 123,2.5.9.16*; 124,7.8.14; 125,1.11.20.22.24.23; 127,1.3.10.11; 132,27; 135,3; 138,7.9; 140,15.29.31.32*; 141,2.20.23.26.30; 142,5.6.9.25; 143,6.19.25.29; 144,26; 145,1.14.16.22*.23*.27;

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146,5.12.14.16.22.27; 147,3.16.23; 148,19.20; 150,8.13.17; 151,12; 152,8.9*; 154,14(bis).16.17; 155,3.4.11; 157,6; 158,16.18.25.27; 163,8.12; 164,1.2.6.8.9.20.25.26; 165,8.11.16.18.20.21; 168,19; 170,2*.3.22; 173,24; 174,1; 178,17.18*.18.19; 180,17; 181,2.11.18(bis); 182,11(bis); 183,8.13; 184,8.27.31.32; 185,25.26; 191,4(bis).5.10.11; 213,19.21; 214,1.6.10; 216,25; scientific knowledge(e), 111,6; 116,2; 119,20; 133,12.14; 159,10.12.13; 183,19*.23; 191,14.17; 192,13.14.17; 198,8.13; 203,8; 213,9.10(bis).18.25; 214,3.4.18.20.21.22; 216,19.21.25 epistêmôn, expert(e), 126,15; 127,13.28; 140,23; 142,24; 143,28; 145,14.29; 146,2.6.9; 147,3; 150,9.13.14; 155,31; 191,3.17 epistêmonikos, scientific(e), 118,6; 143,27; 145,21.22*.24.25(bis); 157,27; 163,10; 183,6; 198,11.14 epistêton, subject of scientific knowledge(e), 116,9 episunaptein, adjoin, 164,10.17; 165,8 episuntithenai, combine, 183,24.27 epitellein, rise, 179,19 epitithenai, add, 188,7 epitritos, the ratio of four to three, 117,20.21.23; 160,7.23 epizeugnunai, coincide, 114,1 epos, epic, 155,6.16; 156,10(bis).11.12.21.22.23.24; 158,2(bis).3.4.5.6(bis).7 ergasia, work, 169,16 eristikos, contentious, 131,2 erôtan, ask, 127,34; 143,15; 145,13.16.26.29; 146,2.20.22.30; 147,7.9.10.11; 148,14.25; 149,2.20; 150,6.12; 155,6.16; 156,7; 178,9; ask a question, 145,19; ask for, 145,19; erôtan, ask questions, 143,11.19; 144,17.20*; 145,5(bis).5*; erôtôn, questioner, 149,7 erôtêma, question, 145,13.16.18*.22*.24.25; 147,17*.18.24.28.30; 148,4; 149,1;

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150,12.14.16*.29; 152,8.19; 153,5.19; 154,5.13 erôtêsis, asking questions, 143,25; 144,22; 146,2; question, 128,1; 146,4; 147,21 errômenos, valid, 203,9; errômenesteros, stronger, 171,21 eutaxia, order, 214,32 euthugrammos, has straight sides, 182,20; rectilinear, 111,22.26; 113,10(bis).11.18; hê euthugrammos, rectilinear angle, 113,28; 114,11; to euthugrammon, rectilinear figure, 111,23.25.28.29; 112,3.17.19.22.24; 114,14.17 euthus, from the start, 199,2; immediately, 128,3; 155,7.15.31; 196,13.15; 197,30; 214,28; right at the beginning, 138,1; 194,10; straight, 122,2*.3.11.18.19*; 150,25; ep’ eutheias, direct, 137,28.29.31; 138,18; 144,1.7; on a straight line, 124,24; hê eutheia, straight line, 112,5.20; 113,6.13.21.32.34; 114,2; 116,18(bis); 124,1.24; 128,5.8.11.29.30; 129,8; 130,6; 132,30(bis); 133,1; 140,11; 142,15.16; 146,3; 149,13.21.22; 178,30; 179,8; 181,15; 183,14(bis).16; 214,13 exêgeisthai, explain, 158,7; 160,14; 178,1 exêgêsis, explanation, 112,1.36; 158,13 exêgetês, interpreter, 158,13 exêgêtikos, explains, 177,20 exêirêmenos, abstracted, 133,21 exetazein, examine, 181,17; 193,29 exeuriskein, discover, 163,7 exisazein, be coextensive, 136,22; 159,9; 162,4.6.8.10; 193,18.19; 201,21.23.26; make coextensive, 183,25 gê, earth, 128,20(bis); 154,29.30; 167,13.14(bis).26; 175,25(bis).26.27; 177,32 gelastikos, capable of laughing, 159,11.12; 201,23.25.26 genetê, birth, 214,19.20

genêtos, generated, 132,11; 135,12.13 genikos, general, 118,4.19; 186,11.20; 187,16; 190,18(bis).23; 193,14.18.19; 195,19; 200,19.20 gennan, generate, 112,29; 128,12; 161,4.9* gennêsis, generation, 161,14.21 genos, genus, 120,27*; 125,10.22; 127,2.30; 129,5; 141,22; 142,26; 143,9; 144,18.20.21.24; 145,8*; 146,1; 177,12.13(bis); 183,24.26.28; 186,12.20; 187,16; 189,12.14; 193,15.17; 195,19; 200,9.14.15; not tr., 215,3 geômetrês, geometer, 113,4.25; 119,8; 123,3.5.8; 125,25; 126,18; 129,15; 132,28.29; 133,1; 138,2; 140,18; 142,12; 145,15.27.29; 146,3.13.13*; 147,7; 148,14.21.25; 149,4.7.8.9.10.12.14.20.22*; 150,1(bis).7.12.15; 151,12.23; 155,6.9.10; 156,7; 162,16; 179,7; 181,26; 182,14; 183,13 geômetria, geometry, 115,4.6; 116,20; 118,9; 119,5.26; 120,7.20.22; 122,15; 123,17; 124,3.9*.16; 140,16; 144,26; 146,7.11; 147,5; 148,11; 149,23; 150,7; 151,11.15.22.25; 153,11.20; 154,8.9; 162,27; 179,14; 181,10.13(bis).16; 213,19; 215,17 geômetrikos, geometrical, 112,35; 115,6; 145,15; 146,23; 147,7; 148,9.10.12.14; 149,1.20; 150,15.16*.28; 151,6.15.16.17; 152,10*.11*.12; 153,11*; of geometry; 116,21; 120,13; 147,8; 148,9.13; 149,8.9.16; that come from geometry, 148,15; mê geômetrikos, non-geometrical, 148,11 geusis, taste, 214,3 ginesthai, arise, 192,27; 216,22; be, 131,10.14; 154,23; 157,12; 169,25(bis).26; 175,14; 176,5.25; 177,2; 181,24; 190,25; 215,22; be based on, 111,14; 117,15.17; 119,11; 121,21; 152,2.20; 195,4; be done, 179,19; be formed, 145,24*; 190,11; be found, 155,5; be made, 131,9; become, 113,17.19.28; 114,10.11; 122,3;

Greek-English Index 128,3; 130,22.23; 142,1; 147,28; 157,28.29; 158,1.2.6.12*; 171,19; 176,15; 196,9; 204,8; 210,11; 211,11; 215,17; come to be, 112,33; 127,35; 172,7; 178,29; form, 113,7; happen, 154,23; 160,10; 176,29; hold, 184,2; occur, 120,11; 151,23; 152,16; 154,16; 158,16.25.28; 165,15.16.27.21; 173,6.9.24; 175,23; 176,17; 178,26; 180,3; 181,25; 183,18; 190,4; 191,9.11.24.32; 192,34; 193,26; 194,1*.5.12*.13.18.19.22.24; 198,9.16.18.19; 202,1; 207,5(bis).6; 210,23.25; prove to be, 187,24; 190,8; result, 116,12; 208,23; take place, 118,33; 164,12.16; 174,16*; 184,10; 188,10.19.22.24; 192,11; 211,7; turn out to be, 161,15; 189,6; 197,27; 198,4; 205,5; 210,10.15; not tr., 173,9; 214,27; eis to genesthai, to yield, 139,21; en lêthêi ginesthai, forget, 214,26; ginesthai hupo, be due to, 173,11; ginomenos kata, due to, 158,20; ginomenos para, due to, 158,26 ginôskein, know(g), 115,13; 116,18; 119,17*.18.21; 120,3*; 186,4; 213,23; 216,4.8.15(bis).17.18 gnôrimos, acquaintance, 214,29 gnôrimos, familiar, 183,32; known(g), 168,29; 170,15(bis).20.22; 172,20*; 215,17 gnôrizein, come to be knowing(g), 215,15*; recognize(g), 215,19.22 gnôsis, knowledge(g), 119,19; 180,3; 191,19; 214,25; 215,4; 216,14.22 gnôstikos, knowledge-producing(g), 213,24 gônia, angle, 113,6.9.10.11.14.15.18.20.23; 114,8.9; 116,10.12.14.18.19; 117,7; 118,8; 126,18.19; 128,30; 129,34; 130,1; 134,17.19; 142,13; 178,22.24; 179,5.6.9; 182,24.25; 191,15; 192,1 grammê, line, 120,21(bis); 123,28; 124,1.10*.23; 128,6.9.19; 129,8; 130,7; 133,4; 146,3; 149,2.21; 155,9.12; 214,13

169

graphein, draw, 111,24.25.28; 112,3.17.19; 113,20.22.33; 114,9; 129,7; 130,7; 155,7; inscribe, 155,31; write, 157,11(bis); write in manuscripts, 124,20 hamartanein, be mistaken, 151,6; 158,19.21 hamartêma, mistake, 151,6 hamartia, mistake, 149,16; 151,21.23.26; 152,1.3.16.22; 154,8.9 haplous, simple, 118,25; 164,8; 176,15; 191,14.16; 192,5; 194,21; just one kind, 194,17*; haplôs, simply, 122,1; 132,7.11; 136,9; 137,32.33; 149,7; 167,21.27; 175,4.7; 181,23; 194,16; 196,11; unqualifiedly, 173,3; without qualification, 131,22.(bis).24; 132,20; 141,30; 165,23; 166,4; 182,2*.3 haptesthai, coincide, 113,38 haptikos, of touch, 214,2 harmonia, harmony, 119,2.3; 215,1(bis); (musical) interval, 117,20(bis); tuning, 180,6.9.19 harmonikê, harmonics, 119,27; 180,28; 213,19; 214,1.21 harmonikos, ‘harmonician’, 117,22; harmonika, facts in harmonics, 117,18* harmozein, apply, 140,29; 147,25; tune, 180,8 hêgeisthai, hêgoumenon, antecedent, 174,30; 175,1.3.12.17 hêi, qua, 134,14; 146,13*; 150,1.2; in that, 216,7* hêliakos, solar, 175,26 hêlios, sun, 128,20; 129,2; 167,23; 168,3; 175,25 hêmera, day, 175,26.27 hêmikuklion, semicircle, 113,4.22.33.34.36.37; 114,2.3; 175,25; 181,24 hêmiolios, the ratio of three to two, 160,7.23; three halves, 160,9 hepesthai, be a consequence, 163,21; 168,11.13; 169,22; be an inevitable consequence, 133,20; be predicated of, 158,29.31(bis); 159,1.2; 161,14; follow, 144,3; 172,23.26; 173,1; 174,30;

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175,9(bis); hepomenon, consequent, 174,30; 175,3 heuresis, discovery, 162,16.32; 163,20; to discover, 163,27 heuriskein, discover, 118,24; 149,11; 162,19.21.26; 165,8.9(bis); 166,20; 183,23; find, 113,12; 151,24; 157,17; 160,21.23; 174,3; 185,23; 191,10; 203,14 hexis, aptitude, 159,19; state, 162,27; 173,17 holikos, complete, 121,6 holoklêros, entire, 129,16 holos, entire, 135,14; 136,12; 139,20; whole, 152,11; 160,3; 171,7; 187,28*.31; 188,4*.5.21; 189,4.10*; 190,4*; wholly, 207,20.24.25; 209,13.15; holôs, at all, 112,26; 130,29; 131,10; 150,24; 151,8; 153,18; 154,6; 191,13; entirely, 136,19; 151,17; 185,25; generally, 127,15; not tr., 141,15; di’ holou, entirely, 202,10; 203,16; 204,7.13; holos di’ holou, wholly, 202,6; kath’ holon, as a whole, 135,14; to holon, as a whole, 169,8 holotês, totality, 135,12 homogenês, belonging to the same genus, 159,3; of the same kind, 113,2 homoiomereia, homoeomery, 129,13 homoios, like, 132,6; similar, 166,9; not tr., 177,7; homoiôs, in the same way, 155,28; 215,27; likewise, 112,12; 119,4; 122,4; 123,11; 135,11.23; 139,5; 140,22; 141,7; 146,8.11; 154,13; 160,2.5; 165,20.26; 167,20; 169,2.15; 171,13; 180,20; 184,18; 191,31; 197,9.14; 199,4; 202,20.21.23.30; 203,19.26; 204,8; 215,23; similarly, 118,5; 123,24; 125,25; 126,18; 134,28; 152,9; 165,21; 177,31; 191,29; 214,2; 215,21; kai ta homoia, and the like, 181,24; kai ta toutois homoia, and the like, 129,8 homologoumenos, granted, 120,20; 162,21.24.26; homologoumenôs, admittedly, 123,25 homônumia, homonymy, 123,25;

154,26; 155,4.11; 158,20.23; 173,11 homônumos, homonymous, 123,5.12; 127,30; 129,4; 134,26*.27.29; 154,20.24; 155,19; 158,17.24; 179,26* horan, notice, 113,11; 125,7; 138,8; 156,22; 162,6*.9*; 167,29; 175,23; 187,23; see, 129,23; 155,9.27*.31; 171,12.13(bis); 178,21.22.23.27; 179,2.5(bis); 180,2.14.22; 196,28; 214,32 horaton, thing seen, 178,26.30(bis); 179,3; 213,19 horismos, definition, 117,10; 122,6; 127,18.26; 128,4.23.26.27(bis).32.33; 129,1.2; 130,16.19.21(bis).23; 132,6.14.16; 133,11; 156,2.11; 163,29*.31*; 164,3; 180,2; 183,18.19.20.22.23.25.29.30.32 (ter).33; 184,1(bis).3.6.7; 185,3(bis) horiston, definiendum, 130,22; 132,15; 183,25 horizesthai, define, 125,24.26; 126,6.8.14.18; 155,3.12.17.18; 156,5; 183,25; 185,4; restrict, 144,25.27.30; 145,1; hôrismenos, defined, 155,10.30; definite, 143,2; 144,22*.24.25; 145,4*.9.21.27; 155,1.2; 162,30; 163,11(bis); 164,4; 165,20.21.25.26; 166,9.13 horos, definition, 116,17.20; 128,15.19; 131,30.31; 132,2; 133,8; 164,14; 180,4; term, 111,11.12.15; 117,6; 121,8; 125,11; 134,16; 135,24; 136,1.3.7(bis).14.20.22; 137,3.4.6.7.8.9.10.15.17.23; 138,29.31; 139,5.10; 140,3; 153,11; 154,18.20; 155,12.27.28(bis); 158,20.24; 159,9; 160,10; 161,13; 162,4.11; 163,28; 164,11.12.16.22.27.30; 165,3.5.16; 168,4.6.14; 170,10.19.25; 171,2.5.6; 172,4.18; 174,7.8.17.24.33.34; 175,29; 176,5.22.24; 177,6(bis).22*.23; 178,3*.4.5; 184,11; 185,14.31; 186,4.10.14.15.20.26; 187,3.5.9.24; 188,9.21; 189,15.16(bis); 190,3.6.7.11.21;

Greek-English Index 192,23.28; 193,1(bis).15.16.17.21; 194,22.28; 195,9; 197,29; 198,29.33; 199,2.14.20.23.26; 200,1.4.8.10.15.21; 201,16.24.26; 202,7.15; 203,14(bis).18; 204,31; 205,3; 206,24; 207,21; 208,3.5.8; 209,10.12; 211,24.28; 212,6.27; 213,4 hoti, to hoti, the ‘that’, 118,7.11*.14*.16(bis).17; 146,29.30; 166,15.17.20; 167,2.12.19.26; 168,2.3.12.15.19.20.22; 169,1.6.17.20.26; 170,8.13.16.18.22.25; 171,1; 172,14.24; 173,2.23.25.26; 174,2.12(bis).25.33; 175,20*.22*; 178,16.20; 179,21; 180,8.27; 181,6; 182,1*.10; 191,7 hugiainein, healthy, 182,22.26(bis) hugiês, sound, 182,18; valid, 152,1; 159,7.10.14.16; 162,11 hulê, matter, 151,1.6.21.23.25.26; 152,2.17; 154,3; 181,3; 203,9 hupallêloi, one under the other, 212,25; subalternate, 117,25; 118,18.32; 119,25; 140,32; 148,15; 178,19; 182,11.12; subordinate, 186,21 huparkhein, apply to, 116,1; be, 171,14; 212,9; belong, 111,13; 116,7; 117,3*.11; 119,19; 127,3.4.5; 132,10.15(bis); 133,12; 134,7.9.14.15.16.19.22.23; 142,9.12; 143,19; 154,19; 165,24.25.27; 167,2; 175,9(bis).10*.11*; 176,30; 185,13; 186,6; 187,4.17.18.22.30; 188,13.23.28; 189,25; 190,7.8.14.16.22; 191,27.29.30.31; 192,26.27; 193,23.25; 194,3*(bis).6*(bis).14*.23; 195,2.18; 196,7.8.10.24*.28; 197,2 (ter).3.4.5.7(bis).9.10.11.12.14.16.17. 18.19.27; 198,3.31.32; 199,7.8.11.12.13.16.19(bis).23.24.26. 27; 200,2.3.18.19.21; 201,1(bis).2.3.10.11.20.21; 202,4.14*.15.16(bis).17(bis).18.19. 20(bis).21; 203,20(bis); 204,2.16.18.26.27.28.29 (ter); 205,7.16.17.18.20.23(bis).26.32; 206,11(bis).13.14.19*.22(bis);

171

207,8.9.10(bis).15.20.22.23.24; 209,10.14.18.19; 210,9.10.17.18.19.20(bis); 212,7.16.21*.21.24.26(bis); 213,1; hold, 117,8.9; 130,27; mê huparkhein, not-belong, 197,4; huparkhon, attribute, 115,12; 120,15; 124,29; 163,2.9; 164,3; 183,24; 184,2.6; 208,4; 213,22; attribute that belongs, 164,18 hupekhein, render, 146,12.13.20* huperbolê, extravagance, 178,4; kath’ huperbolên, extravagantly, 174,19*; 177,19*.21*; 178,2*; pros huperbolên, extravagantly, 177,30 huperekhein, be above, 154,29 (ter).30; 155,1; exceed, 160,15 huperkeisthai, huperkeimenos, superior, 146,6.8.15.16.21; 147,3 huphistanai, exist, 215,26 hupobeblêmenos, subject, 146,1; 216,16.26; hupobeblêmena, subject matter; 123,12; 140,16; 155,10.18 hupodeigma, illustration, 122,19; 188,17 hupoduesthai, pretend, 191,14.17; 192,13; 198,13; 203,8 hupographê, general description, 183,33(bis); 184,3 hupokeisthai, be a subject, 137,17; 144,19; be composed of, 113,13; be posited, 193,17; 197,4.9.14; 198,4; 201,2.11; 204,2.25; 205,2.19; 206,13; 209,31; 210,4.20; 211,20; 212,16.21; hupokeimenon, subject, 111,13; 117,4.12; 128,24; 134,7; 138,22; 140,17; 183,24; 186,6; 216,6; substrate, 181,14.15; subject, 111,8; 116,9; 117,15; 121,8.9; 123,23.25; 125,10.15.22; 127,5; 130,22; 133,18; 137,5.6.8.15; 141,22; 143,1; 145,9; 154,21; 155,25; 159,10; 187,5.9.15; 188,23.26 hupolambanein, suppose, 180,15 hupolêpsis, belief, 191,32; 194,17 huponoein, suppose, 133,31; 174,6 huponoia, suggestion, 133,20 hupostizein, place a comma, 161,15

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hupothesis, hypothesis, 127,17.27(bis).30.31.33; 128,3.15.17.23(bis).26.32.33; 129,1.2.3.4.20.23.26.29; 130,2.3(bis).8.11.12.13.16.23.24*. 25.27.28; 131,21.22.24(bis).25.30(bis); 132,1.6.8.9(bis).12.13.14(bis).16. 20(bis).21.22.29; 133,8.9 hupotithenai, hypothesize, 125,22; 132,10; 133,19.23; 144,3; posit, 201,15; 206,7; 212,22 iatrikê, medicine, 119,5; 146,17; 157,5; 182,28 iatrikos, medical, 150,16; in medicine, 151,13 iatros, doctor, 146,18.23.25(bis).31; 147,5; 150,2(bis).12.15; 182,13.24; 183,1; 214,3 idea, (Platonic) Idea, 133,18.20.21.24.27.29.30.31; 134,2.5 idikos, proper, 141,24; 142,1; idikôteron, more properly, 129,4 idios, proper, 115,4; 123,16*.17; 124,2.6.7.8; 140,17; 141,26; 142,6; 147,16; in a special sense, 127,30; that holds uniquely, 184,2.5.6(bis); idiai (dat.), separately, 121,14; idiôs, for it alone, 152,1 idiotês, characteristic feature, 127,16 iris, rainbow, 181,12*.17.20.23; 182,4(bis) isoperimetros, isoperimetric, 148,26; 182,15.16.19 isopleuros, equilateral, 162,18.20.24 isos, equal, 111,27.28.30(bis); 112,10.13.17.23.24.28.29.31.32; 113,3.12.17.19.28; 114,3.4(bis).5.6.10.13.15.17; 115,3; 116,7*.10.12.13.14.15(bis).16.17. 19(bis); 117,5.7.8.9; 118,9; 123,2.3.6(bis).18(bis).19; 126,16.17 (ter).19; 127,14(bis); 128,8.9.29.30; 130,35 (ter); 134,17.19; 135,5(bis).6; 149,1.14; 175,25.26; 183,14; 191,16; 214,10(bis); 215,18(bis).21(bis). 22(ter).27; 216,2(bis); equivalent,

139,16; 152,12; ex isou, evenly, 124,1 isôs, perhaps, 141,3; 160,12; 171,6; 177,28; 194,9; 214,26 isoskelês, isosceles, 134,17.18 kamnein, kekmêkenai, be tired, 169,15(bis).16 kanôn, rule, 186,3 karkinos, crab, 134,26 kata, kath’ hauta, per se, 118,18*.19; kath’ hauto, per se, 115,12; 116,7; 117,3*.8.9.11; 120,15; 124,9.10.13.29; 125,15; 126,14; 127,2.4; 134,8(bis); 142,9.12; 143,18; 163,10.18; 164,3.17; 167,1; 213,22 katabainein, set, 167,10 katagein, descend, 189,12 kataginesthai, be concerned with, 142,26; 143,9; 144,20.21; 180,1; 213,18.22 katagraphein, draw, 112,32; 156,8(bis) katalambanein, understand, 141,18 kataleipein, kataleipomenon, remainder, 126,17; 130,35; 215,22 katalêpsis, apprehension, 141,18; 142,23 katanoein, understand, 128,10 kataphanai, affirm, 191,25; 202,25.28.29; 203,17; 205,30 kataphasis, affirmation, 128,25; 130,34; 131,5; 136,26.27; 139,1; 143,24; 144,4; 148,3.5; 175,9.11*; 183,20; 186,5; 191,7.28.29; 194,21; 195,20; 210,24; affirmative conclusion, 135,17 kataphaskein, affirm, 131,32; 136,6; 138,30; 186,11; 191,26; 193,16; 196,10; 197,6.10.20(bis).28.30; 200,11.13.30; 201,28; 202,7.9(bis).26.27.29.31; 202,33; 203,15; 213,5 kataphatikos, affirmative, 130,18.20; 131,11.13.14; 139,6.7; 145,26; 158,30; 161,5.11.15; 162,3; 171,2; 176,6.26; 183,10.11.21; 184,15.16.17.20.21.24; 186,2; 188,16.19; 190,6;

Greek-English Index 192,9.10.12.17.19.20; 193,25.27; 196,11.13.27(bis); 198,11.15.16(bis).19; 199,1.10.11.22; 200,23; 201,4.6.8; 204,8.30; 205,16; 207,8.17.18.27; 208,21.22; 210,11; 211,4.11(bis).12.17; 212,20.24(bis).28 katapuknoun, fill in, 185,30*; 188,13 katarithmêsis, enumeration, 171,5 kataskeuazein, construct, 112,25.26; 113,1; establish, 127,33; 143,13.14(bis); 144,11; 145,2.6.7.8; 146,22; 161,4; 162,25; 166,16(bis).17.20.22; 167,7.21; 168,3.17.30; 169,18; 216,11 kataskeuê, argument, 129,32; pros kataskeuên, in order to establish, 145,4 katêgorein, predicate, 117,3.11; 119,4(bis); 121,8.9; 127,26; 130,17.19.23; 131,31; 133,11; 134,13; 136,5; 137,6.9.17.18; 138,22; 139,10; 154,21.22; 155,24; 159,9; 166,1; 176,22.27.28; 185,14.15.17; 186,6.7.12.15.16.30; 187,2.4.11.16; 188,9.21.26; 189,14; 193,22; 195,9.13(bis).16; 196,2.3.11.12.15.17.26*; 197,17.18.23*; 198,1; 199,1.10(bis); 200,9.30; 202,34; 203,1.20; 204,15; 209,28; 212,20.28 katêgoria, category, 200,14 katêgorikos, affirmative, 151,5; general, 185,3; 194,8 katholikos, general, 141,24.25; 157,24; 187,10; 188,2.10.23; 189,25.26; 190,5; 196,25; 199,25; 200,5; 211,28; 212,7 katholou, general, 180,18; generally, 196,25; 199,9; in general, 173,8; universal, 133,10; 134,10*; 157,25.26*.27; 180,16; 183,22(bis); 184,17; 185,4; 192,12.13.15.17.20; 198,14.20; 203,23.28; 215,1.8; universally, 130,18.19; 133,17; 134,13.17; 140,13; 141,21; 180,19; 185,5; 192,22(bis).23.33; 193,3.4.5.6.9.18.22; 202,2.26.27.28.29.31.33(bis).

173

34(bis); 203,1(bis).3.22.27.29; 204,9.21.22.24(bis).25; 205,3.4.5(bis).14.15.21.24.26.33; 206,18.20; 207,14; 208,2.15; 209,8.12.17.21.24; 210,26; 211,12.18.22.23.28; 212,12.17; mê katholou, non-universally, 195,13.14; to katholou, universal, 133,23.28; 134,7(bis).15.20; 183,12; 192,14.16; 214,7.11(bis).15; 215,8.9.11; 216,11*.20.21.22.24 katoptrikê, catoptrics, 181,19.20(bis) katoptrikos, ‘catoptrician’, 181,22.25.27; 182,3.3.6 katoptron, mirror, 182,5.6 keisthai, lie, 124,2; 156,17; the text has, 121,22 kekhrôsthai, be coloured, 111,9(bis).12(bis).13 kenon, void, 129,19(bis) kentron, centre, 113,22.32; 114,1.2.3; 128,9.10.20.28; 129,7; 130,7; 148,26 keratoeidês, hê keratoeidês, horn angle, 113,14 keraunitês, fulgurite, 211,25(bis).26.27 khersaios, terrestrial, 134,28 khôra, place, 121,11; khôran ekhein, be room for, 135,18; ouk ekhei khôran, is not relevant, 147,15 khôrion, passage, 158,8.13; 160,14 khôrizein, separate, 154,8; 216,6 khremetistikos, able to whinny, 209,14.15(bis) khrêsimeuein, be useful, 129,20; 136,18(bis); 137,23; 145,28 khrêsimos, useful, 139,27; 141,22 khrêsthai, employ, 119,25.27; 123,2.3.4.8; 134,30; 135,25.26; 136,1.14; 137,11.16.22.23.24.27; 138,1.9.11.12.13.17.20(bis).29.31; 139,8; 140,15.24; 141,3.5.10.24.21; 146,7; 148,12; 179,16; 181,5.14.22; 184,31; 191,11; 192,6; 196,25; 203,8; 207,22; 208,9; 209,6; 214,16; make use of, 115,6; 120,10; 135,3; 143,6; treat, 181,2; use, 123,11.12.15; 136,20.25.31;

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137,12.18.20; 140,4.11.13.22; 141,8.17.20.21; 142,5.8; 182,27; 202,21; 207,26; 208,27.28; 209,1.7; kakôs khrêsthai, confuse, 174,37; 175,6 khrôma, colour, 214,20 khumos, taste, 214,4 kinein, move, 135,13 (ter).14; 161,18.19(bis).20; 163,4; 164,30.31(bis); 165,1 (ter).2; 175,24; 178,29; 208,8; 214,28; 216,4 kinêsis, movement, 161,21 klan, being inflected, 124,22.23 koinos, common, 111,15.18; 115,5; 117,23; 118,23*.31*.32.33; 119,14.17; 122,5; 123,16.18.21*; 124,6.18.21; 125,1; 127,22.23.24.27; 130,15; 140,24.28; 141,23*.26.27.30; 143,6; 151,26; 152,17; 153,1; 179,26; 180,2.3; 189,14; 200,9.13; 213,21; general, 115,11.14; 116,3.8; 121,13; 140,30.32; 142,21; koinoteros, that holds more widely, 184,1.4.5 koinotês, commonality, 123,23 kômikos, kômika, comedies, 157,8 kômôidia, comedy, 157,9 kôphos, deaf, 214,20 koruphê, apex, 178,28; 179,1.4 kosmos, cosmos, 132,11.12; 135,15(bis); 145,20 krinein, judge, 150,8; 180,8 krisimos, critical, 183,1(bis) krisis, judgment, 214,34 kritikos, that judges, 214,18 kuklos, circle, 111,21.22.24.28.30; 112,2.4.16.19.22.24.26.27.28.29.31. 32.33.34; 113,6.20.22.25.26(bis).31.33.36; 114,1.7.8.9.14.17; 115,4; 128,6.7.9.12.27; 129,7; 130,7; 132,30(bis); 149,10; 155,5.6.7.8 (ter).8.12.15.16.30; 156,7.8.10.11(bis).12.22; 157,1.2.11.17.21(bis).24.25; 158,2(bis).4(bis).7(bis); 182,15.17; 216,6; orbit, 175,24.26; not tr., 111,22.23 kukloterês, round, 182,20 kurios, chief, 184,32*; principal, 167,15; 174,10; kuriôs, in the

strict sense, 119,20; 120,8; 123,21; 127,33; 128,3.15.17; 129,6; 130,2(bis); 134,16.20(bis).22.26; 136,27; 137,3.7; 157,10; 158,21; 168,6; 208,14; strictly, 172,7; 173,3; 178,3 lakhanon, cabbage, 154,29.30 lambanein, acquire, 213,20; 214,15; 215,3; 216,19; assume, 114,13; 120,22; 121,15.17.20*; 122,10.16.20.21.22.23; 124,8.14.17.20; 125,5.11.16; 127,11; 129,15.22; 131,23; 132,29(bis); 133,1.4; 135,4; 137,3; 138,2.16*.29; 141,28; 144,18; 151,3.4.23; 153,12; 163,29*.31*; 164,2; 171,11*; 181,23; 199,7.12.16.17.18; 207,20; comprise, 184,1.4.5; employ, 154,19.20; get, 144,22; 203,9; grasp, 181,16; 183,30; 214,11; make, 194,16; obtain, 143,20.25.29; stand for, 187,20; take, 111,7.10.17; 120,4.5.6; 121,4.29; 123,25; 125,7; 127,28.29; 128,21.22.26; 129,3; 130,4; 133,2; 137,9.15.18; 139,6.7.12.19.21.24.25.27; 140,19; 143,12.17.21; 147,28; 150,20; 154,21; 158,29; 160,1.10; 162,23; 163,18; 165,18; 166,4; 168,2; 173,5.12.18; 174,26; 175,11.18; 176,22; 177,28.31; 182,19.28; 184,10.11; 185,13; 187,9.15; 188,26; 189,16; 190,21.24; 192,23.28; 193,1.15.23; 194,22.28; 195,1.19; 196,1.2; 197,25*; 198,29.33; 199,3.4.5(bis).13.20.23(bis).24.25. 27.32; 200,1.7.8.10.12.13.14.16.17.21; 201,16.18.27; 202,6.19.20.25.32; 203,3.16; 204,4*.6.10.13.14.16.19.22.23.27. 30; 205,20.23.27.28; 206,10.19.24.26; 208,4.7.26(bis); 209,8(bis).10.12.14.16.17.23.28; 210,2.10.12.18.27.28; 211,21.22.23(bis).24.28; 212,6.21; understand, 147,26; use, 171,12; not tr., 158,18; 209,1

Greek-English Index leipein, be absent, 214,9; remain, 118,17 lêmma, premise, 138,11.12.14 leukos, white, 140,19 leukotês, whiteness, 193,1.2.3 lexis, passage, 158,11; 177,24; 193,28; 198,23; 202,11; phrase, 132,13(bis) lithos, stone, 111,9.10.11; 136,4; 159,1.2; 176,20(bis).21.22.24; 177,5(bis).7.9.10(bis).11(bis).12.14; 193,8; 200,2 (ter); 203,24; 209,24.25(bis); 211,20.22.25.26(bis).27.29; 212,3.5.9.11(bis).27.29; 213,1.2; ou lithos, not-stone, 136,4 logikos, rational, 133,12.13; 157,6; 184,4.7; 185,5; 187,1.3.30.31; 189,12; 193,11(bis); 198,2; 199,19.21(bis); 200,2(bis).3.18.19.20.22.23; 204,17.18.19.20; 205,1.5.8.9.25.26.32; 206,1.21; 208,5; 209,19(bis) logos, account, 113,2; 123,8.9; 132,13; 133,27; 141,4; 146,12.13*.20.21*; 173,30; 177,24; 178,4; 179,12; 180,2.9.18; 194,2.13; argument, 141,10; 142,23.24; 146,16.19; 149,8; 159,7; 164,9; 175,8; 181,21; 184,7; 196,25.29; 200,4.5; concept, 214,31; discourse, 131,1*.4.8.9.15.17; 132,26*.32; discussion, 155,8.28; 163,13; 164,7; 191,9; formula, 180,16; 184,2.4; 215,1; proportion, 123,22*; ratio, 117,20.21.22; 118,13; 159,20.21.22; 160,13.15.20; 180,6.20; reason, 113,24; 202,21; 216,18.21; reasoning, 129,25; 152,2; 192,6; relation, 128,21; 181,10; statement, 147,21.29; 148,1.2.3(bis); 150,19; not tr., 181,11 manthanein, learn, 157,13; 214,6; 215,7; to manthanein, learning, 154,24; manthanôn, learner, 127,28.29; 128,15.16.23; 129,3.11; 130,4.11.13; 131,21.24.25.26; 154,23.24; student, , 214,12

175

mathêma, education, 157,2; study; 182,4(bis); subject; 157,4; kata to mathêma, mathematically, 182,2*; mathêmata, mathematics, 155,27.28.30; 163,30* mathêmatikê, mathematics, 159,19 mathêmatikos, mathematical, 163,25.27; 180,4.9.27; 181,1.3.6 megethos, magnitude, 113,13; 114,13.16; 122,11*.12.14.16.22.24; 123,7.7*.10.22.28; 124,16.17.21; 125,26; 140,16.20.22; 141,28.30; 154,30; 178,24.27; 179,2; 215,20 meioun, decrease, 113,15.16.27.30.31; 114,10 meizôn, larger, 111,22.27(bis); 112,10.11.12.14.15.22.23.24; 113,2.11.12.29.31; 114,11.12.14; 115,3; 128,20; 142,11.13.18; 144,28; 149,13; 154,29.30; 178,21.22.23(bis); 179,5.6.9; 181,24; 182,19; 214,8; longer, 175,25.27; major, 121,11; 137,31; 138,31; 139,10.26; 140,7; 161,13; 195,9; 196,6.18; 197,27.29; 198,4.29; 199,2.14.28; 200,10; 206,17; 213,5; hê meizôn, major premise, 162,5.9; 193,20; 196,22.23; 199,21.32; 200,7.16.22; 201,3.7.11.16; 204,22.24.25.31; 205,4.5.7.9.12.13.15.16.21; 206,7.22; 207,13.18.27; 208,2.21; 209,2.13.16.23.25.29; 210,1.11; 211,9(bis).12.14.23.27; 212,2.4.8.10.13.17; ho meizôn, major term, 138,30; 154,20.22; 171,3.6; 172,4.5.6.19; 176,23.26.27.30.31; 193,14.15.21.22; 195,1.2; 196,3; 197,30; 198,5; 199,1.3.4.24; 200,15.18; 201,17.18.20(bis).27(bis).28.29; 205,1(bis).7.24.28.29.30; 209,9.18.27; 210,16; 211,29; 212,2.7.10; 213,4.5; to meizon, major term, 134,13.14.15.16; 136,16; 137,12.17.24; 155,22; 209,27 mêkhanikê, mechanics, 119,5.26; 179,15

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mêkhanikos, ‘mechanic’, 146,7.11; 147,5 mêkos, length, 215,28 mêniskos, lunule, 149,15 merikos, particular, 183,12; 184,19; 185,26; 211,5; 214,17; 216,14(bis).21; specific, 209,12; 211,24; 212,9; specifically, 141,21; in some special area, 142,24 meros, direction, 165,15; part, 114,7(bis); 128,26; 135,12.14; 149,16; 157,28; 175,26; 187,20; 202,28; 206,11; side, 183,15; en merei, individually, 130,18.19; particular, 133,10; epi merous, partially, 192,22; 202,27.29.31; 204,15.20; particular, 133,9; kata meros, in a particular field, 127,13; 140,23; 145,14.29; in detail, 198,23; ta kata meros, details, 196,29; to kata meros, particular, 180,17.24; 214,17; 215,23 mesos, middle, 111,11.12.15; 117,5*.6; 124,2; 128,20; 134,16; 136,20; 137,3.11.16; 139,5; 140,3; 154,18; 155,29; 158,20.24; 163,28; 164,16.29; 168,4.6.14; 170,10.19.25; 171,2; 172,4.18; 174,4*.7.8.14.16*.17.24.33.34; 175,29; 176,5.22.24; 177,5.22*.23.25*; 178,3*.4; 184,11; 185,14.31; 186,26; 190,3.7.11.21; 192,23.28.34; 193,16.21; 194,22.28; 198,33; 199,20.23; 200,1.21; 201,16.23; 202,6.7; 204,14.17; 205,3; 208,3.5.8.16*; 209,10.12; 211,24.28; 212,6; 213,4; ho mesos, middle term, 117,12; 139,12; 155,2; 158,17.23.29; 163,31; 164,23; 167,30; 168,5; 172,5.6.7; 174,9.10.20; 176,27.28.30.31.32; 178,7; 185,10; 187,13.18; 189,16; 190,12.24; 193,18; 194,23; 195,3.13.16; 196,1.3; 197,4.8.29.30; 198,5; 199,15; 200,18.19; 201,18.19.20.27.28; 202,8.32; 205,1.2.6.8.25.27; 206,10; 207,12.15.17.22.23.26; 208,1.9.26.27; 209,1.3.6.8.14.17.23.26.29;

210,16; 211,6.17.19; 212,2.4.9; 213,7.13; to meson, middle term, 134,13.14.15(bis).18(bis).20.21.22(b is); 136,31; 137,15; 138,30; 139,9; 155,22.23; 175,19*; 190,16; 195,9; 199,4.5.7; 205,31.33 metabainein, move on, 162,14; 164,6 metalambanein, transform, 207,19.28; 208,19; 211,10 metatithenai, transfer, 117,26 metekhein, share in, 177,12.14 meterkhesthai, pass on, 207,6 methê, drunkenness, 175,30.31; 176,1 methodos, investigation, 154,17; procedure, 158,17.18; 160,22.23; 162,18 methuskesthai, get drunk, 178,12(bis) metienai, pass on, 202,1; 204,14 metrein, measure, 117,24(bis) metron, measure, 117,23; 124,18 mimnêskein, mention, 122,19; 127,15.19; 173,2; 179,12; 191,18; 194,11 mnêmê, mnêmên poieisthai, mention, 191,13 monas, unit, 121,29; 122,12.14.15.19*.22.24; 124,9*; 126,8.9(bis) morion, member of a contradiction, 135,26; 138,3; 143,13.16.21.30; 144,5.6.11.14.26.30; 145,2.18; 147,19.27; part, 113,36.37; 124,24; 141,13; 147,13; 160,4; 182,22.26; particle, 136,6; portion, 205,17 mousikê, music, 118,10.12.33; 119,1; 180,7; 181,4; 213,9.10; musical theory, 180,6 mousikos, musical, 150,29; 153,18; ‘musician’, 117,19; 145,27; 146,6.9; 180,18; mousika, musical theory, 150,2; 151,12; mousikon, a topic in music, 149,3 nautês, sailor, 179,19.21 nautikos, nautical, 180,4; 181,1.4 neuein, be inflected, 124,22(bis) noein, conceive, 133,2; 155,15 noêsis, conception, 133,4 nomizein, suppose, 118,29; think,

Greek-English Index 144,12; 160,12; 171,23; 188,1; 194,12 notios, southern, 167,3.4.6.13.15; 168,7.9; 177,31; 180,21(bis); notiôteros, more southerly, 180,24 nous, intelligence, 133,12.13; 159,13; 164,25.26; mind, 180,10; nous, 135,19.22 oiesthai, suppose, 112,28; 120,4*; 132,28; 150,26; 153,10*; 158,30; 191,21; think, 132,29; 133,18; 180,22 oikeios, appropriate, 111,8.12.14.15.16.18; 115,10.14; 116,8; 117,15; 118,6.29.30; 145,14.16; 146,10.22.28; 150,6.15; 154,14; 183,6.17; 184,8.27.32; 205,6; 208,15*; 209,29; proper, 119,9.10.11; 120,6.9; 146,13.29; 147,6; 150,13; 183,9 oikêsis, region, 175,24.27 oikothen, from within oneself, 127,23; 128,16.18; 129,27.30; 130,15 ôkhriasis, pallor, 173,4.5(bis).7 ôkhros, pale, 169,13(bis) omma, eye, 178,25(bis).28.29(bis); 179,3 onoma, name, 123,23; term, 127,27.30; word, 134,26; 141,17(bis); 154,28(bis); 156,4 ophthalmia, ophthalmia, 147,11 ophthalmos, eye, 215,2 opsis, ray, 178,25; visual ray, 171,19.21.22.24; 181,15.22; 182,5.7 optikê, optics, 119,27; 181,10.11.13(bis).16.18 optikos, ‘optician’, 146,8.11.23; 147,4; 178,20; 179,5.7; 181,14.21.22.25*.26(bis); 182,2*.3; optikon, optical theorem, 147,9; 148,12*; 179,10 organon, tool, 134,28 orthogônios, rectangular, 129,34 orthoperipatêtikos, walking erect, 201,24.25(bis); walks erect, 216,5 orthos, right, 129,34; orthê sphaira, celestial equator, 175,24.27; hê orthê, right angle, 113,19; 114,11;

177

116,7*.10.11.14.15(bis).16(bis). 17(bis).18.19; 117,5.9.10(bis); 118,9; 126,19; 128,30; 129,14; 134,17.19; 191,15; 192,1; pros orthas, at right angles, 113,5.8.35 ouranos, heaven, 135,13; 180,3; 214,31 ous, ear, 180,10; 215,2 ousia, essence, 135,10; substance, 131,12; 154,27; 159,1; 181,2.5.6*; 186,7.30.31; 187,2.21.22(bis).23.24; 188,1.6.18.20.27.28.29.30; 189,4.12.13.18; 191,27.28.29.30; 192,25.26.29.31(bis); 193,2.23.25; 196,14.15(bis); 198,1; 199,3; 200,12(bis); 202,19; 205,1; 207,21(bis).23.24; 212,11; kat’ ousian, essential, 184,2.5; 208,4 oxugônios, acute-angled, 130,1 oxus, acute, 113,9.10.11.18; 130,1; hê oxeia, acute angle, 113,24.28; 114,11 palindromein, proceed backwards in thought, 214,33 panselênos, full moon, 167,28.29; 168,16 pantêi, entirely, 151,11; 179,25.26 pantôs, always, 125,21; 129,32; 130,33; 138,7; 169,13; 173,1.3.12.18; entirely, 113,6.35; 121,11; 173,9; 203,6; 213,24; 214,1; in all cases, 128,33; 129,23.24; 130,4; must, 168,25(bis).27(bis); 169,4.10.11(bis).12.25; 170,19; 172,21.26(bis); 173,20; 192,11.14.17; 197,21; 202,22.24.26; 204,23.27; necessarily, 117,12; 120,12; 129,23; 131,6.16; 132,1; 136,9; 141,12.16.29; 156,13; 168,17.18; 169,22.25; ‘have to’, 195,20; 196,1; 200,13.17.21; 201,8.12; 202,17.19.21.27.30; 207,12.13; 215,21 paradeigma, example, 121,29; 122,12; 131,27; 152,10; 153,7; 165,17; 169,21; 170,18.23; 172,12; 173,27.29.30; 174,3.5.22; 182,27; 188,5; 213,8; paradigm, 214,30

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paradidonai, present, 123,1; 127,16.18; 130,10; 160,23; 186,3; 191,4; 207,4.7 paradosis, presentation, 192,5 parakeisthai, be near, 182,22 parakouein, make mistakes in hearing, 192,3 paralambanein, assume, 122,5.9.13.14.17; employ, 121,2.5; 128,28.31; 130,21; 134,7; 135,28; 136,17; 138,7; 181,15; 183,20.21; 195,20; 202,2 parallêlos, parallel, 150,24.26; 151,2.16.17.18; 153,10* paralogismos, fallacious argument, 152,1.21*; 153,2; 154,4*.16.22; 155,28; 157,20; 158,16.20.25.27.28; 192,4.8.11.17.20.31; 193,26; 194,24; 195,2.3; 197,20; 203,11 paralogizesthai, infer fallaciously, 192,15; reason fallaciously, 155,11.20; reasoning fallaciously, 151,2 paramuthia, explanation, 129,6.14; 130,9 paratêrêsis, observation, 179,18.20.21 paratithenai, contrast, 112,8; place, 171,12; provide, 131,27 parelkein, add, 121,22; be redundant, 139,24 parempiptein, insert, 186,14; 187,3 paremplekein, contain, 157,9 parentithenai, insert, 186,16 paroran, make mistakes in seeing, 192,3; overlook, 127,1; 138,10 paskhein, be affected, 171,19.22 pathos, affection, 118,18*; 124,13(bis); 126,17; 127,4.5; 173,11; not tr., 126,14 peiran, attempt, 112,29; 142,10.21; 143,17; 202,4 perainesthai, proceed, 168,20; 173,25; 211,18; reach, 192,18; peperasmenos, finite, 165,23; 166,5 peras, end, 142,15; 179,8; endpoint, 183,16 peratoun, be a boundary, 214,9(bis) periagein, lead around, 145,3; transform, 162,4; turn around, 128,12

perieinai, be excessive, 177,28; 178,6 periekhein, contain, 168,14; 179,10; 211,26; 212,4; 213,8; enclose, 128,5.6.9.33; 155,9; include, 117,16; 157,3 perigraphein, circumscribe, 111,23.24.25.26.29; 112,3.15.16.19.22; 113,31; 114,14.16 perilambanein, understand, 165,10 perimetros, perimeter, 182,20 periousia, ek periousias, excessively, 177,28.31 periphereia, arc, 167,13.14; 177,32; circle, 146,3; 148,25; 149,2.13.21; circumference, 112,4.5.20; 113,7.8.13; 128,29 peripherês, circular, 182,13*.21 perittos, odd, 129,22; 138,4; 165,19.20.21.24.25(bis); 166,9; superfluous, 179,11 pezos, terrestrial, 198,31(bis); 199,5.6(bis).15.17.18.25(bis).26 phainesthai, appear, 112,36; 171,17; 178,21.23; 182,12; phainetai, it is clear, 151,22; 177,27; phainomena, phenomena, 179,18.18*; phainomenos, evident, 184,26 phaneros, evident, 119,10; 128,4; image, 171,20 phantasia, imagination, 155,7; 156,9 phaulos, uneducated, 150,9; phaulôs, defectively, 153,19*.24*.24; 154,6.7 philosophia, philosophy, 118,23; 119,16; 141,5; 142,20.26; 143,3.7.8.10; 144,17; 215,3 philosophos, philosopher, 111,31; 134,29; 140,24; expert in philosophy; 142,25; ho philosophos, the Philosopher (Ammonius), 146,26; 158,7; 160,14; 177,19; (Aristotle), 126,3 phônê, verbal expression, 132,7(bis); word, 155,19 phôs, light, 171,20; lukhnaion phôs, lamplight, 171,13.24.25 phôstêr, source of illumination, 167,25 phôtisma, phase, 169,2

Greek-English Index phôtismos, phase, 168,26.31(bis); 172,13 phôtizein, illuminate, 129,2; 168,26.28 Phrux, Phrygian, 157,1 phthartos, perishable, 116,1 phthisis, decline, 147,13 phuein, pephukenai, be of a nature, 151,8.9 phusikê, natural philosophy, 147,12; natural science, 119,4; 147,5 phusikos, natural, 126,6; 181,15; 192,6; natural philosopher, 125,25; 126,6; 146,18(bis); 181,25*; 182,1*; in natural philosophy, 146,24; of natural philosophy, 147,11; phusika, facts in natural philosophy, 146,23; hai Phusikai, the Physics, 112,6; hê Meta ta phusika, the Metaphysics, 142,23; ta Meta ta phusika, the Metaphysics, 119,16; 141,6.9 phusis, kind of thing, 120,15; nature, 120,14; 128,31; 131,4; 143,2.20.28; 177,6; 182,23.27; 207,17; 210,8; kata phusin, natural, 129,26 piptein, lie, 113,6.34.36 pisteuein, be convinced of, 216,1 pistis, conviction, 128,16.18; 214,14; eis pistin, to be convinced, 170,20; to be convincing, 168,14 pistos, to piston, conviction, 127,24; 130,8; 140,6 pistousthai, confirm, 132,27; 173,30; 181,27; 188,17; 196,7; 200,5; 215,28 pithanos, plausible, 143,18 plagios, eis to plagion, slantwise, 165,28* planan, planômenos, planet, 171,16(bis) planê, error, 155,5; 173,10 planêtês, planet, 170,24(bis); 172,7.8 platos, breadth, 216,1 plêthos, plurality, 126,9 pleura, side, 114,6; 116,10.13; 124,19.26; 129,33; 138,2; 142,11.18; 144,2(bis).27.28; 178,28; 179,2.4.7.9; 191,20; 214,8 pneumôn, lung, 174,31, 177,30

179

podiaios, a foot long, 133,2*(bis).3 poiein, cast, 167,28.29; 168,16; cause, 136,6; compose, 156,18; 157,14; construct, 116,3; 133,1.3; 135,18; 149,8; 181,21; 203,11; do, 112,33; 113,27; 144,9; 184,13; form, 186,14.16.18; make, 111,7.30; 116,19; 128,30; 141,26; 150,19; 156,15; 159,7.10.14.17; 171,6; 172,6; 175,1.12; 176,11; 177,3; 182,24; 186,27; 190,25; 194,10; 197,21; 207,28; 208,21; 211,13.27; produce, 159,3*; 170,16.23; mnêmên poieisthai, mention, 191,13; not tr., 179,12 poiêma, poem, 157,12.17 poiêtês, poet, 157,7.11.16 poion, quality, 191,27.28; 192,30.32; 193,2; 196,16 poiotês, qualification, 192,25.26.31.32; 193,3.24.25 pollaplasiôn, multiple, 159,4*.5.6.15.16.17.19.21.22; 160,6.12.19; 161,3*.9 polos, pole, 167,5(bis).8.12; 168,8.9; 178,1 polugônion, polygon, 182,17 polugônios, polugôniôteros, having more sides, 182,16 polukhôrêtotatos, containing the greatest area, 148,26; 182,15.17; polukhôrêtoteros, containing greater area, 182,16 poluplasiôn, multiple, 161,14* poluskhidês, complex, 192,8 porisma, corollary, 119,8 porizein, invent, 215,3 poson, quantity, 187,21(bis).23; 188,5.18.20.28 (ter); 189,4; 192,30.31(bis) pragma, matter, 150,22.23.27; subject, 163,2.9; 181,5; thing, 119,21; 120,14; 128,32; 129,12; 131,4; 135,8; 141,18; 142,9; 143,19.20.28; 154,28; 155,2.4; 164,3; 166,18; 167,1; 168,2; 170,26; 173,16; 174,25.33; 180,1; 215,4; not tr., 141,11 pragmateia, treatise, 157,14 proanatellein, rise before, 180,23 proapodeiknunai, demonstrate previously, 125,1; 164,20; 165,12

180

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proballesthai, project, 129,28; 214,30 problêma, problem, 117,3.11; 128,25; 188,9 prodêlos, obvious, 138,10 proektithenai, set out in advance, 122,6 proerhkesthai, proceed, 120,3; 192,27 proginôskein, know previously(pg), 121,26 proienai, 174,8; 193,28; go forth, 171,18; proceed, 169,18; 214,11; further down, 120,14 prokeisthai, be proposed, 183,6; prokeitai, his present purpose is, 137,26; 198,8; prokeimena, present discussion, 137,26; 138,5; prokeimenos, in question, 113,14; 117,17; 132,1; 135,18; 139,11; 157,21; 179,10; 183,28; proposed, 139,22.25; to prokeimenon, the claim in question, 111,19; 115,7; 118,26; 127,20; 142,5; 143,27; 145,7; 176,7; 182,28; the present case, 159,14; 176,17.20; the present discussion, 179,12; 214,5; the purpose proposed, 166,15; 213,17 prokheirizesthai, apply, 123,7.9.30 prolambanein, anticipate, 134,1; assume in advance, 121,14.21; 122,2; 125,20.21.24; 126,23; 127,9.13; 138,14 pronoia, foreknowledge, 144,9 prophanês, evident, 129,32; 142,17.25; 165,3 propherein, bring forward, 136,15; express, 148,1; produce as evidence, 157,23 prophora, en têi prophorai, said out loud, 131,2 prophorikos, external, 131,8.9.15 prosballein, strike, 171,21.25; 178,26.30; 179,3 prosblepein, look at, 171,25 prosdialegomenos, interlocutor, 127,31.34; 128,13.21; 129,2.4.5.9.17.18; 131,8.11; 143,11.15.16.26; 144,13; 145,3 prosdiorismos, specification, 155,24 prosekballein, produce, 116,13

prosêkein, belong to, 151,15; pertain to, 149,23; should, 191,3 prosekhês, preceding, 197,23; proximate, 117,16; 167,22.25.26; 168,1.5; 174,9.10.18.25.28; 176,1; 177,22.25; 178,11; prosekhôs, immediately, 146,5.15.22; 147,3; just, 117,4; 135,3; 147,22; proximately, 125,2; 163,28; 167,2; 198,4; mê prosekhês, non-proximate, 177,23 proskeisthai, add, 134,26 proskhrasthai, make use of, 181,7 proslambanein, added, 164,23; supply, 136,11 prosphoitan, go to study with, 141,14 prosphuês, naturally fitting, 112,36 prospiptein, fall on, 128,8.29 prosthesis, addition, 164,12 prostithenai, add, 111,6; 118,15.28; 135,21; 141,4; 164,11; 188,3; 190,23; 195,8 protasis, premise, 111,7.10.17; 120,4.6.9; 121,7.10(bis).11.15.16.20.25.27; 127,21; 128,25.26.27.31; 130,17.21.22; 132,1*.2.19(bis).20.22; 133,10.17; 134,21.22; 135,4.28; 136,2.5.9.10.11.14.15; 137,7.8.31; 139,19.26; 143,12(bis).20.25.27.29; 144,18.22; 145,19(bis).21.23.25.26.29; 147,17*.27*.28.29; 151,3.4.22.24; 152,21; 153,20; 154,1.2; 157,25*.29; 158,1.6.12*.22; 159,8; 161,7.13.18; 162,7.10.17.19.21.25.28.29.31; 163,3.5.7.9.16.17.18.20.26(bis).27; 164,18.24.30; 167,1.2.28.30; 168,14.20; 170,8.14.18; 171,3.25; 174,11.13.22; 175,4.7; 176,11; 177,1.3; 184,10.11.12.14.20.24; 185,9.11.31; 186,1.2.13.26(bis).31; 187,18; 188,14.15.17.20.29; 190,6.13.25; 192,21.28.33; 193,3.4.6.20; 194,7; 195,7; 196,6.18.21; 197,21.27; 198,4.10.12.15.16.18.19.22.25.33; 199,28; 200,22.28; 201,31; 202,2.5; 203,3.7.22; 205,13;

Greek-English Index 206,17; 207,5.7.13; 210,29; 211,19; 213,5.14; 215,8; proposition, 131,11.13.14; 191,24.32; 208,15 protithenai, propose, 143,13; 162,17.20; 191,9; 202,3; 207,4 prôtos, first, 118,23; 119,15; 120,7; 125,3; 138,29; 140,24; 141,5; 142,20.25.26; 143,2.7.8.10; 144,17; 148,21.22(bis); 156,14; 157,13; 161,15; 162,5; 164,14.15.17; 170,2*.3; 171,2; 173,30; 174,11; 176,6.10.17.18.32; 177,2; 178,16; 183,7(bis).10.17(bis).19.23.26; 184,8.13.15.17.22.26.27; 185,9.21.29.30; 186,19; 188,11.14.23; 190,4.8.17; 192,11.18; 198,17.20.24.32; 201,31; 204,32; 205,19; 206,13.19; 207,9.10(bis).20.23.25; 211,4; primary, 119,17.20; 120,13(bis); 121,19*; 133,17.19; 134,8; 170,9; prior, 120,12; 143,23; 193,5; Protera Analutika, Prior Analytics, 184,24; proteron, first, 152,15; 170,22; 171,5; 173,28; 187,15; 193,28; 207,7; previously, 117.13; 122,18; 123,1.15; proteros, first, 186,18; 193,28; 194,12; 200,25; former, 169,12; prôtôs, in the first place, 194,3*; primarily, 118,19; 133,22; 134,13.15.16.19; 163,10; 164,18; 186,6.30; 187,4.12; 194,6*.14*; 195,14; 197,23* prouptos, plain, 145,9 pseudarion, fallacy, 151,22; 153,20; 154,9 pseudês, false, 112,9; 129,9.11; 132,29; 138,3; 143,27; 151,3.4.20.23.24; 152,21; 153,6.12.20; 154,2.3; 162,31; 163,5.8; 191,19.26.28.30.32; 192,21(bis).22.23.27.33(bis); 193,3.4.6.7.10.11.12.22.24; 194,17; 195,3.7; 196,21.24; 197,21; 198,17.21.22.25.33; 199,9.14.17.18.22(bis).24.25.27.28. 29; 200,1.17.21.29*; 201,8.12.15.27; 202,2.6.8.9.10(bis).33.34; 203,1.3.16.17.18.19.20.22.23.28.

181

29(bis); 204,3.4*.5.7.10.13.14.16.19.21.22. 23.24.25.26.28.31; 205,3.5.7.9.11.14.15.21.22.24.26. 30.32; 206,1.7.18.20.21.24.26; 207,13.14.20.24.28; 208,2.15.16.23; 209,3.8.12.13(bis).15.16.17(bis).20. 21(bis).22.24.26.27.29; 210,2.3.12.15.18.27.28; 211,9.12.13.14.18.22.23.24.26.27. 28.29; 212,1.3.4.6.8.11.12(bis).14.15.17; 213,5.6.8.10; 213,14 pseudos, false, 144,14; 175,23; 193,9; 197,2; 203,24.26; to pseudos, falsehood, 154,17; 192,12.28; 193,13; 202,1; 203,9; 207,12; 208,22; the false, 135,8; 140,22 psukhê, soul, 127,34.35; 131,9.17*; 132,10.11.27*.32; 135,9.20; 143,14.16; 144,10; 145,20; 155,14.15; 156,12; 167,17; 214,23.30; 215,4; not tr., 135,9 psukhros, cold, 125,25; 126,4.5.6 ptênos, winged, 199,5.6.15.16.17; 206,25(bis) pur, fire, 159,4*.5.6; 161,3*.18.19.21; 168,25 (ter); 169,3(bis).4(bis); 170,19 pusma, inquiry, 147,21(bis) Puthagoreios, Pythagorean, 180,6 rhêton, passage, 196,10 rhêtorikê, rhetoric, 157,5 rhônnunai, errômenos, valid, 203,9; errômenesteros, stronger, 171,21; mê errôsthai, be invalid, 153,1 rhusis, flow, 123,28 rhuthmos, rhythm, 151,8.9 saphênizein, make clear, 121,4 saphês, apparent, 129,29; clear, 125,23; 126,5; 127,14; 138,8; 145,8.17.29; 150,19; 170,18; 184,16; 186,24; 200,4 seira, series, 189,12 selênê, moon, 129,1; 156,19; 167,23.27.28.29; 168,3.16.26.30; 169,2; 172,13; 173,28; 183,2 sêmainein, mean, 186,24; signify,

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121,13.17.19*.21.25; 122,1.9; 124,12.13.17; 125,5.16.20.25; 126,15; 127,10.14; 132,14(bis); 140,19; 144,23.25; 174,5.7(bis).18; 177,24; sêmainomenon, meaning, 154,21; 155,19; 156,1 sêmantikos, signify, 141,18 sêmasia, meaning, 127,13; 141,27; 155,12 sêmeion, point, 113,21(bis).38; 114,2; 120,20.21; 123,28; 124,1.10*.25; 128,7.18; 129,7.8; 130,6(bis); 151,8.14; 153,18; 178,25.30; 183,15; 214,13(bis); sign, 169,7 skelos, end (of a compass), 128,10.11 skhêma, figure, 111,22.26; 112,4; 128,5.6; 148,26; 151,5; 155,6(bis).9.14; 156,7.10; 157,22(bis); 158,3.4.5.6(bis).7.30; 159,14.17; 161,5.11.15; 162,3.6; 171,1; 174,5.6.15.16*; 175,21.23; 176,2.7.11.16.17.18.32; 177,2.14; 182,15.18; 183,7.8.9.11.17.19.23; 184,8.9.12.13.15.17.22.25.26.27.31. 33; 185,12.18.20.24.29.30; 186,18; 188,8.10.14.15.19.22.24; 190,4.8.11.13.17; 192,11.18; 198,17.20.24.32; 200,24; 201,2.31; 202,2.4.6; 203,2.6.7; 204,4.6; 205,19; 206,13.19.23; 207,11; 208,20; 210,10.23.25; 211,5; 212,27; form, 136,15; 152,1.17.22; 153,1; 154,8.10; 158,19.21.26.28; 159,10; 162,4; 203,9; 207,18; 211,13; shape, 172,13; 182,24; 183,3; 214,20; to para to skhêma, formally, 203,11 skhesis, relation, 128,24; 171,8 skholê, ou skholê hêmin, we don’t have time, 141,14 skia, shadow, 167,28(bis); 168,16 skopein, consider, 133,30; 192,13 skopos, goal, 137,26; 138,5 Skuthês, Scythian, 175,30(bis); 176,9(bis); 178,8.9.10 sôma, body, 111,10(bis).12 131,13; 146,31; 171,21.23; 181,16; 186,7; 187,1.2; 189,3.5.12.13.18; 191,30(bis); 192,6.23(bis).24; 196,15; 198,1; 199,3.4; 212,9(bis).11; 214,7.33 sophia, wisdom, 141,6

sophistês, sophist, 159,3; 161,3 sophistikos, sophistical, 203,10 sophos, wise, 154,23.24 sphaira, hê orthê sphaira, the celestial equator, 175,24.27 sphairoeidês, spherical, 168,27(bis).30.31; to sphairoeides, sphericity, 169,2 sphugmikos, pulse, 214,3 spinthêr, spark, 214,25 spoudazein, aim, 145,3 stereometria, stereometry, 179,14.15 stereos, solid, 179,15 sterêtikos, privative, 204,4 stikhos, line of poetry, 151,10.18; 153,19; verse, 156,13.15.23; 157,12 stilbein, twinkle, 170,24.27; 171,10.12.13.14.15.17.18.23.26; 172,3.4.6.8(bis).10; mê stilbein, non-twinkling, 170,27; 171,3; mê stilbon, non-twinkling, 170,24 stilpnotês, glittering, 171,20.25 stoikheion, element, 135,11; 146,17.19.24.31 strephesthai, deal with, 157,4 sullogismos, deduction, 116,12; 120,3.11.16; 131,11.14; 135,24; 136,1.23; 137,22.27; 138,11.12.20; 139,11; 145,21*.23*; 147,26.28; 151,2.7; 152,19*.20*.22; 154,3.4.4*; 157,26*.28.29; 158,3.9.19.22; 159,7; 162,15.33; 163,25; 164,9.21.22.27.29; 165,3.4.15; 166,16; 167,2.6.12.16.21.24.31; 168,1.7.10.15.17.20; 169,1.5.17.18.20.26; 170,9(bis).16.23.25; 171,1(bis); 172,7.14.22.24; 173,23; 174,12.14.15*.25.32; 175,23.28.31; 176,11.18.26; 177,2.15; 178,5.20; 182,10; 184,25.30; 186,14.16; 188,7.10.19.22.23; 189,6; 190,3.22; 191,6.33; 192,2.3.4.7.8.10.24.27; 193,32*; 194,1.4*.5.6.8.9.10.11.11*.16.17.19. 22; 198,10.11; 203,8(bis); 207,4.8.11.16.26.28; 208,27.30; 209,6; 210,24; 211,6.9.10.17; not tr., 192,1

Greek-English Index sullogistikos, deductive, 145,17*; 147,17*.22.25; 148,4; 151,3; 152,2; 158,23; 162,4; 203,7; 207,18; 210,10; sullogistikôs, in a way suitable for deduction, 145,19 sullogizesthai, deduce, 120,8; 135,19; 159,3; 168,12.21; 169,2.6(bis); 170,16; 172,13.27; 173,26; 174,31; 183,9.26.27; 184,3.31; 194,22; 204,5; make deductions, 142,8; sullogizomenos, doing the deduction, 129,22; 138,11; in its deductions, 138,9 sumbainein, be an attribute of, 118,20; belong to, 124,9; 125,15; follow, 208,28; happen; 127,11; 158,26; 204,2; 210,29; happen accidentally, 150,1; occur, 194,4*; 198,23; 201,16; 210,29; sumbainon, attribute, 142,1; 181,21.23; 182,7; sumbebêkos, accident, 164,25; 214,18; accidental, 163,30*; attribute, 118,19; 124,10.13; 126,14; kata sumbebêkos, accidental, 115,12; 163,1; 164,2; accidentally, 115,13; 116,2.3; 149,23*; 163,9; 173,4 sumballein, contribute, 191,8 summetros, commensurable, 138,2.3; 140,18.20; 144,2; ou summetros, non-commensurable, 140,18(bis) summiktos, miscellaneous, 179,11 sumperainein, infer, 132,31; 135,16; make an inference, 136,12; reach a conclusion, 129,18; 167,1 sumperasma, conclusion, 111,16; 121,7(bis).16.21.27; 128,25; 132,19; 135,22.24.25.28; 136,9.11.12.18.19; 137,21.22.24.25.30.32; 138,13.16*.18*.19.21.25.28; 139,11.21.25.27; 140,3.5.6.7; 148,22; 151,4.24.25; 157,28.29; 158,2.4.9; 162,8.17.18.21.28.29.30.31; 163,2.5.6.10.16.17.21.26(bis).28; 164,10(bis); 168,5; 176,5.12.14.19; 177,4.6; 178,5.6; 183,13; 184,19;

183

185,23; 188,12; 201,9*; 204,5; 208,11.16; 211,11; 212,12.13 sumphônein, accord, 158,11 sumphônos, concordant, 117,19.21(bis).22.23; 118,11.12.13; 180,8.9.19(bis).20 sumphunai, sumpephukôs, natural, 127,22.23.25; 129,27 sumpiptein, intersect, 124,25; 129,15; 150,25.26; 151,2.16.17.18; 153,10*; 155,13 sumplekein, combine, 151,3.5; 159,8; 162,11 sumplêroun, complete, 136,8 sumplokê, combination, 151,22; 162,9.10; 190,26 sunagein, bring together, 182,23.27; collect and arrange, 157,15; conclude, 153,21; 207,16; draw a conclusion, 121,7.27; 136,21; 137,24; 143,24.30; 144,6.8; 151,4.21.24; 159,17; 198,14; 211,5; follow, 212,15; imply, 163,17.28; infer, 119,8; 135,27; 136,26; 137,13.25; 138,5.13.19.21.24; 143,22.27.23; 154,9; 156,10; 162,8.17.19.28.29.31; 163,3.5.10; 172,14.25; 176,13.16.19; 178,6; 183,11(bis).13; 184,16; 185,22.26; 188,11; 190,15; 192,19(bis); 193,13.24; 198,21.33; 199,17.19; 201,2; 202,5; 205,19; 207,12(bis).14; 208,1(bis).4.7.9.11.16.22; 209,2; 212,22; infer a conclusion, 163,15; to sunagomenon, what follows, 129,23 sunairein, take together, 152,11 sunaktikos, that can imply, 162,30; 163,6 sunalêtheuein, be simultaneously true, 127,15; 135,9; 136,21.27; 137,2.10; 138,23; 139,8; 140,5.7.8(bis); 141,2; 144,12 sunanatellein, rise together, 180,21 sunaptein, connect, 158,23; sunêmmenon, connection, 163,29 sunathroizein, combine, 183,28 suneidenai, be aware of, 132,28 suneisagein, introduce, 133,27 sunekdokhê, kata sunekdokhên,

184

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if you take the particular as standing for the whole, 157,10 sunekheia, kata sunekheian, continuously, 160,10.11 sunekhês, continuous, 187,20.22(bis).23; 188,5.18.20.21.27.28.29.30; 189,3.5.17.19(bis) sunepinoein, understand, 136,8 sungeneia, genus, 117,6*.12* sungenês, belonging to the same genus, 158,31 sungramma, writing, 157,16 sunienai, understand, 132,3.7; 154,25; 215,18 sunistasthai, constitute, 127,2; construct, 111,30; 162,20; 179,8(bis); 183,14 sunkatatattein, sunkatatetagmenos, coextensive, 133,24 sunkatatithenai, agree, 128,1 sunkeisthai, compose, 126,9 sunkhôrein, admit, 131,7; 141,12; 144,14; grant, 112,31.36; 129,12.18.19.22.24; 130,5.6.14.34; 131,11.14.15; 143,26; 149,12; 155,16; 156,10; 214,13 sunônumos, synonymous, 179,25* sunoran, comprehend, 162,27; notice, 159,8; 186,5 sunousia, conversation, 154,18; 155,17; 156,4; discourse, 147,24 sunousioun, join essentially with, 214,30 suntattein, take with, 136,7; 155,24; 163,29* suntaxis, summary, 216,10.12 suntelein, contribute, 139,25; 141,25 sunthesis, synthesis, 162,22.23; 164,8 suntrekhein, concur, 112,2 sustoikhia, column, 189,11.17.20; 208,6.10.25*; 209,4 suzugia, combination, 189,13 tattein, order, 197,30; put order into something, 214,32; tetagmenos, ranking, 119,26 taxis, order, 157,13.15; 172,5; not tr., 137,8 tekhnê, art, 118,24.25; 119,15(bis); 141,2; 148,11; 153,5*

tekmêriôdês, from a sign, 169,8; 170,27; from signs, 168,22 teleios, accomplished, 147,4 teleioun, complete, 184,26 telos, end, 137,32; 156,15; limit, 182,17 temnein, cut, 113,20.26; 114,7; 182,23 tephra, ash, 168,25(bis); 169,3(bis).4.5; 170,18 teretisma, nonsense, 134,1 tetragônismos, quadrature, 111,18.19; 112,2.7.26; 115,4.10; squaring, 149,11 tetragônizein, square, 111,21; 112,1.8.21.27.33.34; 149,15 tetragônon, square, 111,30(bis); 112,28.29.31.32; 124,18 tetragônos, quartile, 183,2 tetrapleuros, quadrilateral, 129,34 thaumastos, surprising, 144,10; 147,6 theôrein, consider, 136,3.7; 178,18*; find, 136,10(bis); investigate, 216,11*; observe, 127,33; 215,10 theôrêma, theorem, 113,25; 120,7; 122,15; 125,2; 128,28; 144,29; 146,7.8.24.29; 147,4.15; 148,9.10.21.22; 150,3; 151,12; 164,14.19; 165.8.11; 179,10.11; 181,12.17.20; 183,9.17 theôria, theôria phusikê, natural philosophy, 147,12 thermos, hot, 125,25; 126,4.5.7 thesis, positing, 137,4; position, 177,26; thesis, 127,25(bis) tiktein, give birth, 169,12.13.23.24; 172,22(bis).23.25.26; 173,4.5.6.8; 175,13.14 tithenai, give, 122,12.20; 136,23; 153,7; 165,17; 173,27.29; 174,3.6.14.15.22; place, 142,14.15.17; 164,13; 168,4; 171,13; 174,4*.7.8.9.17; 175,19*.20; 176,5; 177,25*; posit, 133,20; 138,1; 188,24; 190,11; put, 172,5; 188,18 trigônon, triangle, 114,5; 116,7.9.12.14.21; 117,4.7.8; 122,2*.3.18(bis).19*.23(bis); 128,5; 129,33; 134,18; 142,11.12; 144,28; 162,18.20.24; 178,27;

Greek-English Index 179,1.3.7.9; 191,15.20; 192,1; 214,8; 216,6 tropos, manner, 144,8; 153,13; 154,8; 174,1.12; way, 113,25; 129,29; 136,17; 152,15.18; 159,4; 165,17; 166,17.22; 170,4.13.22; 173,2.24.27.30; 178,17*; 191,9.10; 193,29; 200,24; 201,7; 207,5; 214,6; not tr., 123,5; 153,13; ton tropon touton; as follows, 111,21; 112,21; 113,31; touton ton tropon, this is how, 168,18 tupos, impression, 155,15; 214,29 zêtein, investigate, 112,27; 116,21; 124,8.10.14; 125,5; 135,23; 136,13; 149,10; 150,16; 162,24; 176,16; 183,26; is sought, 121,9; to zêtoumenon, the sought, 121,13.17; 122,4.15; 127,4; 140,31; 162,22.23; thing sought, 122,2; thing that is sought, 122,14.21; what is sought, 137,21; zêtoumenon, sought, 121,18 zêtêsis, question, 196,9 zôê, life, 167,18

185

zôion, animal, 123,24; 131,12; 132,21; 133,12.13.29; 134,27; 136,13.23.24; 137,1.13.14; 139,17.18.19.20.21.23.26; 140,19; 142,14; 154,26(bis); 155,25; 163,3; 164,26.27.28.29.31(bis); 165,1.2(bis); 174,24(bis).26.27.28(bis).29.33. 36*(bis); 175,2.6; 176,8(bis).12(bis).13.19.20.21(bis). 23; 177,4.5.7.8(bis).9.30; 183,27; 184,4.6; 185,5; 187,1.31; 189,12.17.18.20; 193,11.12; 198,2.30.31; 199,5.7.14.15.16.17.19.20.21; 200,2(bis).3; 202,18.21.22.24; 203,26.27; 204,17.18.19.20; 205,1.6.24.25.32; 206,1.20(bis).25(bis).26; 207,21.22.24(bis); 208,5.10; 209,9.11.14(bis).24(bis); 211,20.21.25.29; 212,3.5.8.27.28.29; 213,9(bis); mê zôion, not-animal, 175,2.3.5; ou zôion, not-animal, 136,13.24(bis); 137,13.14; 139,20.22.26

Index of Passages References are to the Notes to the Translation. AMMONIUS

in Isag. 9,26: n.43

ARATUS

Phaenomena 1,177-178: n.317

ARISTOTLE

Analytica Posteriora 1.2: n.400; 1.3: n.400; 1.4: n.400; 1.5: n.400; 1.6: n.400; 1.10: n.400; 1.11: n.180; n.286; 1.12: n.286; 1.13: n.400; 1.15: n.391; n.400; n.403; n.438; 1.16: n.391; n.402; n.438; 1.17: n.402; 1.19: n.306; 1.20: n.306; 1.21: n.306; 1.22: n.306; 2.3: n.370; 2.4: n.370; 2.5: n.370; 2.6: n.370; 2.7: n.370; 71a12-14: n.93; 71a13-14: n.180; 71b21: n.382; 71b33-72a5: n.94; n.200; 72a14-24: n.56; n.105; n.134; 72a16-18: n.180; 72a18-20: n.143; 72a20: n.108; 73a28-34: n.326; 73b26-74a3: n.151; 73b26-74a32: n.3; 73b33-74a1: n.148; 74b5-12: n.332; 75a37: n.92; 75b2-3: n.68; n.69; 75b3-16: n.179; 75b8-12: n.36; 75b8-9: n.39; 75b15-17: n.209; 75b17-19: n.221; 75b17-20: n.206; 75b42: n.28; 76a1: n.3; n.28; 76a3: n.28; 6a8-15: n.179; 76a23-24: n.46; 76a23-25: n.209; 76a24: n.357; 76a33-34: n.59; 76a34-36: n.60; 76a37: n.76; 76a38-39: n.69; 76a38-40: n.188; 76a42-b1: n.70; 76b10: n.53; 76b11-12: n.57; n.92; 76b11-22: n.103; 76b12-15: n.104; 76b27: n.110; n.113; 76b27-34: n.111; 76b32-34: n.125; 77a15-16: n.165; 77a29-34: n.216; 77a31-32: n.292; 77a32: n.203; n.204;

77b1-2: n.212; 77b18-20: n.237; 77b19-20: n.227; 77b24-26: n.225; 77b27-28: n.228; 77b38-39: n.257; 78a10-11: n.286; 78a15: n.299; 78a23-26: n.315; n.336; 78a26-b11: n.319; 78a26-b4: n.369; 78a30-33: n.324; 78a33-34: n.325; 78b11-13: n.321; 78b23-24: n.337; 78b28-29: n.338; 78b29-31: n.342; 78b33-79a16: n.211; 78b34-79a4: n.314; n.322; 79a1: n.358; 79a1-2: n.37; 79a10-13: n.37; 79a13-16: n.46; 79a24: n.367; 79a25-29: n.366; 79a29-31: n.373; 79a30: n.394; 79a30-31: n.383; 79a36-b20: n.326; 79a41-b1: n.381; 79b5-11: n.389; 79b5-12: n.393; 79b7: n.463; 79b12-15: n.414; 79b25: n.441; n.458; 79b27-29: n.457; 79b29-30: n.416; n.441; n.458; 79b29-80a5: n.424; n.459; 79b38: n.441; 80a3: n.441; 80a3-4: n.428; 80a5: n.458; 80a9-26: n.440; 80a15-20: n.434; 80a23: n.436; 80a27-28: n.487; 80b2-3: n.455; 80b18-81a4: n.486; 80b20-21: n.472; 80b22-81a4: n.460; 80b27: n.463; 80b27-28: n.464; 80b34: n.475; 83a33: n.150; 84b34-35: n.383; 84b35: n.304; 92b38: n.367; 96a24-38: n.368; n.369; n.371 Analytica Priora 2.2: n.291; 2.3: n.291; 2.4: n.291; 2.15: n.201; 2.26: n.257; 27a15-18: n.375; 28a1-7: n.375; 34b4: n.431; 37b1-2: n.431; 40b17-19: n.376;

188

Index of Passages

41a26-27: n.82; 41b6: n.399; 54a4-6: n.408; 54b17-23: n.408; 63b31-9: n.201; 69a37-b1: n.257 Categoriae 7b13-14: n.334 De Anima 409a4: n.74; 425a16: n.496; De Generatione et Corruptione 1.8: n.126; 325b25-29: n.94; De Interpretatione 16a29-32: n.161; n.166; 17a17-18: n.166; 17b14-16: n.245; 20b22-30: n.214 De Respiratione 2: n.339; 3: n.339 De Sensu 438a25-26: n.328 De Sophisticis Elenchis 165b30-166a21: n.242; 172a3-8: n.7; Ethica Nicomachea 1140a6-10: n.335 Metaphysica 4: n.180; 4: n.182; 4.1: n.49; 4.2: n.49; 4.3: n.49; 4.4: n.183; 982a2: n.181; 983a19-21: n.200; 993a15: n.49; 998a2-4: n.145; 1005b2: n.181; 1006a3-11: n.183; 1006a15-18: n.183; 1007a7-10: n.185; 1016b25-27: n.74; 1020a11: n.74; 1026a24: n.49; 1029b1-12: n.200; 1059a18: n.181; 1060a10: n.181; 1061b19: n.49; 1061b30: n.49 Meteorologica 372a26-33: n.363; 389b26-28: n.126 Physica 185a16-17: n.6; 189a21-b1: n.404 Poetica 1451b5-14: n.255 Topica 1.15: n.154; 8: n.199; n.370; 100a18-20: n.193; 101b28-36: n.121; 107a15-16: n.208; 148a4-5: n.229; 148b27: n.76; 153a7ff.: n.370 DAMASCIUS

in Phaed. vers. 2.14: n.504

DIOGENES LAERTIUS

Vitae Philosophorum 1.89-90: n.249

EUCLID

Elements 1, C.N. 1: n.509; 1, C.N. 3: n.509; 1, Def. 2: n.74; 1, Def. 10: n.32; 1, Def. 15: n.114; n.115; 1, Def. 19: n.114; 1, Post. 1: n.124; n.133; n.137;

n.499; 1, Post. 2: n.137; 1, Post. 3: n.124; n.133; n.137; 1, Post. 5: n.127; 1, Prop. 1: n.52; n.85; n.300; 1, Prop. 2: n.52; n.85; n.300; 1, Prop. 3: n.85; n.300; 1, Prop. 4: n.300; 1, Prop. 13: n.32; 1, Prop. 32: n.29; 3, Prop. 16: n.17; n.21; 7, Def. 1: n.98; 7, Def. 2: 98; 10, Def. 3: 80; 10, Prop. 117: n.82 Optics Prop. 4: n.355 HERACLITUS

fr. 103: n.250

HESIOD

Opera et Dies 383-387: n.358; 414-419: n.358; 564-569: n.358; 650-651: n.358

JOHN OF DAMASCUS

Dialectica 3,18: n.43; 66,11: n.43 Fragmenta Philosophica 8,9: n.43

MICHAEL PSELLUS

Opuscula Logica 49, lines 116, 136 and 209: n.43

NICOMACHUS

Introductio Arithmetica 1,18-20: n.271; 1,19,4: n.272; 1,19,9-15: n.275

PAPPUS

Collectio 634,11-13: n.287; 634,13-18: n.289; 634,18-23: n.288

PHILOPONUS

in An. Post. 2,13-14: n.241; 7,18-10,4: n.58; 7,20-23: n.77; 7,20-8,5: n.178; 8,7-8: n.190; n.506; 8,20-24: n.178; 8,21-24: n.178; 9,9-10: n.65; 9,9-27: n.64; 9,18-19: n.65; 10,7-13: n.178; 10,27-11,3: n.71; n.177; n.190; 21,8-15: n.320; 26,9-15: n.51; 31,8-32,7: n.320; 34,6-36,17: n.105; 34,10-11: n.130; 34,10-13: n.177; 34,18: n.130; 34,19: n.130; 34,21-22: n.130; 35,2-19: n.123; 35,5-7: n.143; 36,13-17: n.177; 37,7-13: n.108; 49,5-14: n.320; 49,13-15: n.51; 69,24-70,6: n.332; 82,26: n.332; 92,9-93,1: n.318; 97,22-31: n.320; 97,25-31: n.51; 98,23-99,4: n.68; 111,8-15:

Index of Passages n.144; 111,17-114,17: n.222; 112,25-36: n.26; 113,13-14: n.12; 117,15-26: n.207; 117,22: n.42; 117,22-25: n.40; 119,13: n.45; 119,14: n.45; 119,19: n.45; 120,7-14: n.85; 121,3-19: n.88; 121,7-15: n.90; 121,16-18: n.91; 121,21-22: n.55; 122,26-123,13: n.189; 123,15-18: n.190; 123,15-19: n.177; 125,2-3: n.52; 125,7-8: n.57; 125,15: n.102; 127,26-27: n.143; n.144; 127,26-33: n.129; 127,30-33: n.140; 128,13-15: n.140; 128,23-32: n.139; 128,26-32: n.505; 129,5-11: n.140; 129,6-7: n.141; 129,11-15: n.141; 129,27-28: n.509; 130,15: n.509; 130,16-23: n.123; 131,23-25: n.110; 133,29: n.146; 134,7-10: n.332; 135,3-23: n.155; 135,8-27: n.170; 136,22-137,7: n.172; 139,1-2: n.171; 140,11-12: n.167; 141,3-19: n.195; 144,19-145,10: n.196; 145,4-10: n.292; 145,26-28: n.203; 148,19-23: n.52; 150,29-152,3: n.448; 151,24-26: n.238; 152,11: n.232; 152,13-153,2: n.269; 153,4-15: n.225; 154,11-20: n.268; 159,7-17: n.283; 159,8-10: n.284; 159,8-17: n.270; 160,6: n.274; 168,30-169,6: n.330; 171,22-24: n.329; 173,5: n.333; 176,24-26: n.346; 178,20-179,12: n.42; 183,30: n.367; 185,17-20: n.395; 186,9-12: n.410; 186,10-21: n.419; 187,15-16: n.385; 187,20-25: n.385; 187,28-188,22: n.381; 188,5-6: n.385; 189,3-5: n.385;

189

189,11-13: n.463; 189,12-13: n.403; 189,12-20: n.464; 189,25-27: n.414; 191,18-192,4: n.421; 191,29-31: n.409; 191,29-32: n.395; 192,24-27: n.422; 193,5-9: n.423; 194,20-198,5: n.424; 196,8-11: n.491; 196,13-15: n.403; 196,21-197,21: n.428; 198,15-19: n.441; 199,2-4: n.385; 199,4-7: n.425; 202,34-203,2: n.449; 209,19-21: n.429; n.452; 210,17-19: n.482; 215,7: n.498; 269,3-13: n.304 in An. Pr. 49,18-20: n.130 in DA 246,23-247,10: n.158 in Phys. 31,9-32,3: n.6 PLATO

Gorg. 456a7-c7: n.224 Parm. 137e: n.75 Phdo. 73d: n.501 Phdr. 264d: n.249 Soph. 229b7-8: n.226 Tim. 47a-c n.503

PLOTINUS

Enn. 6.2.22.22-33: n.158

PROCLUS

in Prim.Eucl. 32,20-23: n.121; 70,1-18: n.236; 81,5-22: n.121; 97.7-8: 74; 365,7-11: n.128; 365,14-367,27: n.128; 371,24-373,2: n.128

PS.- ARISTOTLE

Problemata 921b1-13: n.38;

PS.-PLUTARCH

Life of Homer 11: n.251

SIMPLICIUS

in Phys. 47,30-31: n.43; 61,5-68,32: n.223

THEMISTIUS

An. Post. Paraphr. 24,3-6: n.169 On Practical Wisdom 300c5: n.43

Index of Names References are to the page and line numbers of the CAG text, which appear in the margins of the translation. Alexander, 111,20.31; 112,1; 122,11; 126,3; 139,9; 159,18; 160,8.13; 174,4.8; 181,11; 196,9 Ammonius (‘our teacher’), 112,30; (‘the Philosopher’), 146,26; 158,7; 160,14; 177,19 Anacharsis, 178,8 Anaxagoras, 129,13 Antiphon, 112,2.4.7.18; 149,11 Aratus, 167,8 Aristotle, 111,19; 112,6.34; 137,14; 141,8; 142,22; 147,7; 151,26; 174,14.20.35; 177,20; 181,27; 182,1; 191,5; 200,24; (‘the Philosopher’) 126,3 Bryson, 111,17.19.20; 112,2.7.8.21.25.35; 113,1; 114,13; 115,4; 149,11

Democritus, 129,11 Euclid (‘the Geometer’), 113,25; 148,21 Herodotus, 156,26 Hippocrates, 149,15 Homer, 156,26(bis) Caeneus, 161,3 Callias, 136,23.24.28(bis).29(bis).30; 137,13.19; 139,21.24 Midas, 156,17.21; 157,1 Nicomachus, 160,24 Pisander, 157,14 Plato, 136,31; 150,28; 215,1 Proclus, 111,31; 112,9.20.25.30.36; 129,16; 160,13; 181,19.27 Ptolemy, 129,16 Socrates, 136,31 Themistius, 138,5; 177,27

Subject Index References are to the page and line numbers of the CAG text, which appear in the margins of the translation. abstraction, 133,21-2; 215,15-24; 216,2-8 accidents, 115,10-116,4; 162,33-163,10; 163,30-164,3; 164,24-5 affirmation, 128,23-5; 143,22-4; 147,28-148,5; 183,20-1 affirmation, immediate, 186,1-8; 191,25-31 Alexander, 111,20-112,2; 122,10-20; 126,3-4; 139,9-12; 159,17-160,13; 174,3-10; 181,11-19; 196,8-18 Ammonius (‘the Philosopher’), 146,26-147,1; 157,20-158,13; 160,15; 177,19-23 Anacharsis, 175,29-176,1; 178,8-10 analysis; 162,14-32; 163,8-164,8 angels, 209,16-21 Antiphon, 112,2-8; 149,9-13 apprehension, 141,9-19; 142,22-5 (see also sensation, apprehensive) arguments, fallacious, see arguments, invalid arguments, invalid, 150,29-151,5; 151,20-152,3; 152,15-153,2; 153,17-22; 154,3-155,20; 157,20-158,7; 158,16-159,17; 192,4.7-193,26; 194,24-198,5; 203,8-10 arguments, sophistical, 203,10-11 arithmetic (= number theory), 117,18-26; 118,10-13; 119,1-4.27; 122,14-15; 123,2-13; 124,8-10.20-2; 126,7-10; 140,16-17; 145,15.26-8; 146,5-10; 213,20-1 astrology, 125,3-4 astronomy, 125,3-4; 145,26-8;

179,18-180,4; 180,20-181,7; 182,27-183,3; 213,19 attributes, accidental, 115,10; 162,33-163,12; 163,30-164,4 attributes, essential, 183,33-184,7; 208,3-6 (see also attributes, per se) attributes, per se, 115,11; 116,7-9; 117,3-13; 118,18-20; 120,14-16; 124,6-14.29-125,3; 125,15-16; 126,14; 127,1-6; 134,6-9; 142,9-18; 143,14-19; 163,9-11.15-21; 164,1-3; 164,17-21; 166,22-167,2; 213,21-2 axioms (see also principles, principle of non-contradiction), 121,9-15; 123,1-26; 124,29-125,3; 125,12-13.21-2; 126,14-15; 127,5-6.12-18.21-4; 129,23-4.26-131,9; 135,3-6; 140,28-141,3; 141,20-142,7; 142,20-1; 164,13-16.19-21; 214,14; 215,7-8.11 Bryson, 111,17-115,7; 149,9-14 catoptrics, 181,10-182,7 cause (see also cause and effect, reciprocation of), 119,18-21; 166,18; 167,1-2; 168,1-169,21; 170,8-10; 172,17-173,29; 174,37-175,18; 176,17-177,15; 179,15-16; 184,30-2; 204,19-23 cause and effect, reciprocation of, 168,23-169,27; 170,13-20; 172,4-14 cause, proximate, 166,22-168,18; 174,3-34; 175,28-9; 177,19-178,13 comedy, 157,7-9

194

Subject Index

common notions, 127,21-4 (see also axioms) conclusions, 121,5-27; 128,23-5; 135,16-136,14; 138,15-25; 143,19-24.29-144,15; 148,19-23; 151,2-5.20-6; 157,26-158,10; 162,6-11.16-33; 163,8-10.15-21.25-8; 164,9-17; 168,5; 177,4-11; 178,4-13; 183,9-17; 184,13-22; 198,11-15.18-22; 207,11-14 contradiction (see also premise of a contradiction, principle of non-contradiction), 137,2-10; 143,10-14.29-145,3; 147,17-28; 208,14-16 contradictories, 198,8-15; 203,23-4 contraries, 143,22-4.29-144,15; 193,5-9; 203,22-30; 210,8-13 conversion (= changing a negative proposition to a positive proposition or vice versa), 174,37-175,18 conversion of terms in conclusions, 176,10-15.31-177,4; 185,22-6; 188,12-13 conversion of terms in premises, 159,8-17; 161,13-162,11; 172,4-10; 176,10-15; 208,19-23; 211,11-14 conversions of premises with conclusions, 163,25-164,4 deductions, demonstrative, see demonstrations deductions, dialectical, see dialectic deductions, mathematical deductions per impossibile, 137,26-138,5; 140,11-25; 143,29-144,8 definitions (see also principles), 116,17-22; 117,9-10; 122,4-6; 123,28-124,3; 125,24-7; 127,15-18.24-7; 128,4-129,3; 130,16-23; 131,30-132,16; 133,8-14; 155,6-14; 163,28-164,4; 183,18-184,7; 185,3-5 demonstrations, 111,6-17; 116,1-4; 117,3-18.25-118,13; 119,8-12.24-6; 120,3-16;

121,1-18; 125,7-8.12-13; 127,1-6; 131,1-17; 133,20-4; 134,5-24; 134,29-30; 138,6-11; 162,27-163,10; 164,17-21; 165,6-166,2; 183,6-185,25; 207,25-208,11; 208,25-209,4; 211,6-9 dialectic, 115,3-7; 134,29-30; 140,23-5; 141,3-5; 142,8-18; 143,6-145,10; 154,15-155,20; 156,4-5; 158,16-28; 162,32-163,13; 164,21-165,6; 207,25-208,11; 208,25-209,4; 211,6-9 differentiae, 183,22-9 discourse, internal and external, 131,1-17; 132,26-133,5 discussions, dialectical, see dialectic effects (see also cause and effect, reciprocation of), 119,19-21 epic poetry; 155,5-17; 156,9-157,17 errors, scientific (see also ignorance due to disposition), 151,5-152,3; 155,1-17; 191,3-213,14 Euclid (‘the Geometer’), 113,4-11; 148,19-23 expert (epistêmôn), 126,14-15; 127,12-15; 142,24-5; 143,25-9; 145,13-16.28-146,14; 150,6-16; 191,3-5 fallacies, see arguments, invalid form, 181,1-7.13-16; 203,8-10 Forms (Platonic), (see Ideas) formula, universal, 180,14-24; 214,33-215,1 general education, 157,2-11 genus (subject of a science), 117,3-13; 125,10-12.27; 127,2-3; 142,25; 143,9-10; 144,17-26; 145,9-11.28-146,1 genus (taxonomic), 183,22-9; 189,10-20; 195,18-196,3; 200,9-15 geometry, 116,7-22; 119,5.26; 120,20-3; 123,2-13.17-19.8-124,3; 124,8-10.16-26; 125,25-7; 126,18-19; 132,29-133,5;

Subject Index 140,16-22; 145,15-16.26-9; 146,2-4.11-14.21-3; 147,2-8; 148,9-150,18; 151,10-26; 152,6-12; 153,4-22; 155,3-17; 156,7-12; 162,16-28; 178,24-179,12; 179,11-12; 181,10-18; 182,13-18; 213,19 given, 121,7-18.29; 122,3-6.12-24; 124,6-14; 125,10-12; 127,2-3; 140,31-2 harmonics, 117,18-25; 118,10-13; 119,1-4.27; 145,26-8; 146,5-10; 149,3-4; 150,2-3; 151,11-13; 180,6-10.18-20.28-181,7; 213,19.25-214,2; 214,20-1; 214,34-215,1 Hippocrates of Chios, 149,15-17 homonymy, 123,4-13.21-6; 134,26-30; 154,20-155,20; 158,19-26; 173,1-13; 179,25-180,4 hypotheses (see also principles), 127,15-18.24-130,29; 131,21-132,23; 133,8-11 Ideas (Platonic), 133,16-134,10 ignorance due to disposition, 150,20-152,9; 153,4-6.24-154,10; 191,18-24; 193,32-194,6; 194,9-14; 198,8-15 ignorance due to negation, 150,20-6; 151,5-15; 153,4-5.12-14; 191,10-18 imagination, 155,6-9; 156,7-9 impressions, 155,12-16; 214,25-31 induction, 157,20-4; 171,10-15; 214,11-17; 215,9-216,26 inflection. 124,22-4 intervals (musical), 117,19-25; 118,11-13 knowledge, scientific, 111,6-8; 116,1-4; 119,20-1; 183,18-25; 192,13-17; 198,8-15; 203,7-11; 213,17-18.24-214,5; 214,17-25; 215,9-11; 216,16-26 magnitudes, irrational, 124,16-20 mathematics (see also arithmetic,

195

geometry, harmonics, optics), 155,27-156,2; 163,25-8; 181,1-7 matter, 150,29-152,3; 181,1-7; 203,8-10 mechanics, 119,5.26; 146,5-11; 147,2-5; 179,15-16 medicine, 119,5; 146,17-147,5; 147,10-14; 150,2-3.14-16; 151,11-13; 182,13-183,3; 214,2-3 middle, appropriate, 208,15-16; 209,28-31 mistakes, see error music, see harmonics music in respect to hearing, 180,6-10.26-181,7 musical theory, see harmonics negation, 147,28-148,5 negation, immediate, 186,1-21.30-187,25; 189,25-7; 191,5-10.24-31; 194,21-3 Nicomachus, 160,22-4 number (see also arithmetic), 126,3-10 objections, scientific, 157,20-158,13 observation, 179,18-21 optics, 119,27; 146,5-11.21-3; 147,2-9; 148,9-16; 178,20-179,12; 181,10-182,2 opinions, reputable, 142,10-11; 143,14-19; 162,33-163,4; 164,21-5 particulars, 180,14-24; 214,16-17; 216,13-16.21 perception, 125,24-7; 171,10-15; 180,6-10; 181,1-7; 192,1-3 (see also sensation) perishables, no demonstration of, 116,1-4 ‘phenomena’, 179,18-22 philosophy, first, 118,22-5; 119,13-16; 140,23-5; 141,3-19; 142,20-143,14; 144,18-26 philosophy, natural, 119,4; 125,24-5; 126,6-7; 146,17-26; 147,2-5.10-12; 181,22-182,2 Plato, 150,27-8; 215,1-5 (see also Ideas)

196

Subject Index

postulates (see also principles), 127,15-18.27-31; 129,5-25; 130,2-29; 131,21-7; 133,8-11; 214,12-14; 215,7-8 premises, demonstrative, 111,6-17; 120,6-14; 121,5-26; 128,24-32; 130,16-23; 131,30-132,3; 132,19-22; 133,17-18; 134,20-3; 135,3-6; 138,8-11; 143,19-29; 144,17-22; 145,17-146,1; 157,25-6; 158,11-12; 162,14-32; 163,8-11.15-28; 164,17-21; 166,22-167,18; 167,26-168,18; 170,8-20; 174,7-13; 184,9-24; 185,9-11.29-186,27 premises, dialectical (see also dialectic), 158,11-12; 163,31-164,3; 164,21-165,6; 208,7-11.25-8 premises, immediate, 120,4-11; 127,21-7; 166,22-168,23; 170,8-20; 173,23-5; 184,9-24; 185,9-11.29-186,3; 186,9-21.30-187,6; 194,6-9; 196,29-198,5; 198,15-207,6; 210,23-9 premises of a contradiction, 147,26-30 premises, scientific, see principles, demonstrative principle of non-contradiction 127,15; 135,6-142,2 principles, appropriate, 111,6-19; 115,10-11; 116,7-22; 117,15-18; 118,4-8.27-33 principles, common (see also axioms), 118,31-119,5; 119,16-21; 123,1-26; 124,29-125,2; 127,21-4; 130,14-15; 141,22-142,2; 143,6-7 principles, demonstrative (see also axioms, definitions, hypotheses, postulates, premises, demonstrative, premises, immediate, principles, common), 116,19-22; 119,2-4; 120,19-122,24; 148,19-23 principles, proper, 119,8-12; 120,3-10; 123,15-18.28-124,26; 141,22-142,6; 146,11-147,16 problems, 117,3-4.11-13; 128,23-5

Proclus, 111,31-112,36; 129,13-17; 160,13-24; 181,19-182,2 proportions, 123,21-6; 124,20-2; 159,17-160,24; 161,3-21 Ptolemy, 129,13-17 Pythagoreans, 180,6-7 questions, deductive, see questions, scientific questions, dialectical, 143,10-19; 145,3-10; 147,18-148,5 questions, scientific, 145,13-146,14; 147,17-148,16; 148,25-149,4; 150,11-154,10 rainbow, 181,10-182,7 ratios, see proportions ratios, concordant, 117,19-26; 118,10-13; 180,18-20 refutation, 149,20-150,3 science, see knowledge, scientific sciences, mathematical (see also arithmetic, geometry, harmonics, mechanics, stereometry), 180,27-181,7 sciences, perceptual, 180,27-181,7 sciences, subalternate, 117,15-119,5; 119,24-7; 140,31-2; 146,1-147,16; 148,9-16; 178,16-180,10; 182,10-11 sciences, subordinate, see sciences, subalternate sciences, superior, see sciences, subalternate sensation, 213,22-5; 214,15-17; 215,9-11.15-17; 216,10-16.20-6 sensation, apprehensive, 216,16-26 senses, 213,17-25; 214,15-215,5; 216,13-26 sensibles, 213,17-25; 214,15-20.25-215,5; 215,27-216,1; 216,13-16 signs, proofs from, 168,22-169,9; 170,22-171,1 sought, 121,7-18; 122,2-4.13-16.20-4; 125,15-16; 127,4; 140,31-2; 162,20-7 soul, 214,23-215,5 stereometry, 179,14-16 subject, see genus

Subject Index substance, 181,4-7 substrate, 181,14-16 synthesis, 162,20-2; 164,7-9 terms, 125,10-12 terms, extreme, 117,12-13 terms, middle, 117,3-13; 134,13-24; 137,14-21; 154,17-155,1; 155,22-5; 158,19-159,6; 163,25-164,4; 170,8-171,3; 174,3-21; 176,5-6.16-177,15; 184,10-12; 185,29-186,1; 192,22-193,4; 193,14-25; 194,21-196,3; 204,14-17; 207,11-14.25-208,6; 208,14-16.25-210,5; 210,15-20; 211,6-9.17-212,17; 213,4-14 ‘that it is’, 121,14-18.25-7; 122,1-6.9-24; 124,12-14; 125,4-5.20-1; 127,9-11 the ‘that’, 118,7-17; 146,26-147,1; 166,15-169,27; 170,8-171,1; 172,4-174,13; 174,23-34; 175,19-176,2; 178,16-179,12; 179,18-22; 180,6-10.26-183,3; 191,5-7

197

the ‘why’, 118,7-17; 146,26-147,1; 166,15-169,27; 170,8-171,1; 172,4-174,13; 174,31-4; 175,19-176,2; 178,16-179,12; 179,18-22; 180,6-10; 180,14-183,3; 183,7-12 Themistius, 138,6-11; 177,27-8 theorems, 124,29-125,3; 146,5-9.21-147,16; 148,19-23; 164,9-21; 165,8-12; 179,10-12; 183,6-9 theses, 127,24-7 thought. 215,15 tuning, concordant, 180,6-10.18-20 twinkling (explanation of), 171,12-26 universals, 133,22-134,23; 214,6-19; 215,7-11; 216,10-12.18-26 verging, 124,22-6 ‘what it signifies’, 121,9-122,2; 122,9; 124,12-20; 125,4-5.15-16.19-20.24-7; 126,15-16; 127,9-15; 132,14-16