Authors/Aristotle/posteriora/L/L1

Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11

Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 Chapter 20 Chapter 21 Chapter 22

Chapter 23 Chapter 24 Chapter 25 Chapter 26 Chapter 27 Chapter 28 Chapter 29 Chapter 30 Chapter 31 Chapter 32 Chapter 33 Chapter 34

Greek	Latin	English
c1	LIBER PRIMUS. CAPUT PRIMUM. De praecognitis.	Posterior Analytics Translated by G. R. G. Mure Chapter 1
71a1 Πᾶσα διδασκαλία καὶ πᾶσα μάθησις διανοητικὴ ἐκ προϋπαρχούσης γίνεται γνώσεως. φανερὸν δὲ τοῦτο θεωροῦσιν ἐπὶ πασῶν·	Omnis doctrina et omnis disciplina intellectiva ex praeexistente fit cognitione. Manifestum autem hoc speculantibus in omnibus,	ALL instruction given or received by way of argument proceeds from pre-existent knowledge. This becomes evident upon a survey of all the species of such instruction.
↵ αἵ τε γὰρ μαθηματικαὶ τῶν ἐπιστημῶν διὰ τούτου ↵ τοῦ τρόπου παραγίνονται καὶ τῶν ἄλλων ἑκάστη τεχνῶν.	mathematicae enim scientiae per hunc modum fiunt, et aliarum unaquaeque artium.	The mathematical sciences and all other speculative disciplines are acquired in this way,
71a5 ὁμοίως δὲ καὶ περὶ τοὺς λόγους οἵ τε διὰ συλλογισμῶν καὶ οἱ δι᾽ ἐπαγωγῆς· ἀμφότεροι γὰρ διὰ προγινωσκομένων ποιοῦνται τὴν διδασκαλίαν, οἱ μὲν λαμβάνοντες ὡς παρὰ ξυνιέν↵ των, οἱ δὲ δεικνύντες τὸ καθόλου διὰ τοῦ δῆλον εἶναι τὸ καθ᾽ ἕκαστον.	Similiter autem et circa orationes, quae per syllogismos et quae per inductionem: utraeque enim per prius nota faciunt doctrinam, hae quidem accipientes tanquam a notis, illae vero monstrantes universale, per id (quod manifestum est) singulare.	and so are the two forms of dialectical reasoning, syllogistic and inductive; for each of these latter make use of old knowledge to impart new, the syllogism assuming an audience that accepts its premisses, induction exhibiting the universal as implicit in the clearly known particular.
ὡς δ᾽ αὔτως καὶ οἱ ῥητορικοὶ συμπείθουσιν· ἢ γὰρ↵ διὰ παραδειγμάτων, ὅ ἐστιν ἐπαγωγή, ἢ δι᾽ ἐνθυμημάτων, ↵ ὅπερ ἐστὶ συλλογισμός.	Similiter autem et rhetoricae persuadent, aut enim per exempla, quod est inductio, aut per enthymemata, quod quidem est syllogismus.	Again, the persuasion exerted by rhetorical arguments is in principle the same, since they use either example, a kind of induction, or enthymeme, a form of syllogism.
διχῶς δ᾽ ἀναγκαῖον προγινώσκειν· τὰ μὲν γάρ, ὅτι ἔστι, προϋπολαμβάνειν ἀναγκαῖον, τὰ δέ, τί τὸ λεγόμενόν ἐστι, ξυνιέναι δεῖ, τὰ δ᾽ ἄμφω, οἷον ὅτι μὲν ἅπαν ἢ φῆσαι ἢ ἀποφῆσαι ἀληθές, ὅτι ἔστι, τὸ δὲ τρίγωνον, ὅτι τοδὶ σημαίνει, τὴν δὲ μονάδα ἄμφω, καὶ τί ση↵ μαίνει καὶ ὅτι ἔστιν· οὐ γὰρ ὁμοίως τούτων ἕκαστον δῆλον ἡμῖν.	Dupliciter autem est necessarium praecognoscere, alia numque quia sunt praeopinari necesse est, alia vero quid est quod dicitur intelligere oportet, quaedam autem utraque. Ut quoniam quidem omne aut affirmare, aut negare verum est, quia est, triangulum autem, quoniam hoc significat, sed unitatem utraque, et quid significat quidem, et quia est, non enim similiter horum unumquodque manifestum est nobis.	The pre-existent knowledge required is of two kinds. In some cases admission of the fact must be assumed, in others comprehension of the meaning of the term used, and sometimes both assumptions are essential. Thus, we assume that every predicate can be either truly affirmed or truly denied of any subject, and that ‘triangle’ means so and so; as regards ‘unit’ we have to make the double assumption of the meaning of the word and the existence of the thing. The reason is that these several objects are not equally obvious to us.
Ἔστι δὲ γνωρίζειν τὰ μὲν πρότερον γνωρίσαντα, τῶν δὲ καὶ ἅμα λαμβάνοντα τὴν γνῶσιν, οἷον ὅσα τυγχάνει ὄντα ὑπὸ τὸ καθόλου οὗ ἔχει τὴν γνῶσιν.	Est autem cognoscere alia quidem prius cognoscentem, quorumdam autem simul accipere notitiam, ut quaecunque contingunt esse sub universalibus quorum habent cognitionem;	Recognition of a truth may in some cases contain as factors both previous knowledge and also knowledge acquired simultaneously with that recognition-knowledge, this latter, of the particulars actually falling under the universal and therein already virtually known.
ὅτι μὲν γὰρ πᾶν τρί↵ γωνον ἔχει δυσὶν ὀρθαῖς ἴσας, προήιδει· ὅτι δὲ τόδε τὸ ἐν τῶι ἡμικυκλίωι τρίγωνόν ἐστιν, ἅμα ἐπαγόμενος ἐγνώρισεν. (ἐνίων γὰρ τοῦτον τὸν τρόπον ἡ μάθησίς ἐστι, καὶ οὐ διὰ τοῦ μέσου τὸ ἔσχατον γνωρίζεται, ὅσα ἤδη τῶν καθ᾽ ἕκαστα τυγχάνει ὄντα καὶ μὴ καθ᾽ ὑποκειμένου τινός.)	quod enim omnis triangulus habeat tres angulos duobus rectis aequales, praescivit, quod vero hic qui est in semicirculo triangulus sit, simul inducens cognovit. Quorumdam enim hoc modo disciplina est, et non per medium ultimum cognoscitur, ut quaecunque iam singularium contingunt esse, et non de subiecto aliquo.	For example, the student knew beforehand that the angles of every triangle are equal to two right angles; but it was only at the actual moment at which he was being led on to recognize this as true in the instance before him that he came to know ‘this figure inscribed in the semicircle’ to be a triangle. For some things (viz. the singulars finally reached which are not predicable of anything else as subject) are only learnt in this way, i.e. there is here no recognition through a middle of a minor term as subject to a major.
71a24 πρὶν δ᾽ ἐπαχθῆναι ↵ ἢ λαβεῖν συλλογισμὸν τρόπον μέν τινα ἴσως φατέον ἐπίστασθαι, τρόπον δ᾽ ἄλλον οὔ. ὁ γὰρ μὴ ἤιδει εἰ ἔστιν ἁπλῶς, τοῦτο πῶς ἤιδει ὅτι δύο ὀρθὰς ἔχει ἁπλῶς; ἀλλὰ δῆλον ὡς ὡδὶ μὲν ἐπίσταται, ὅτι καθόλου ἐπίσταται, ἁπλῶς δ᾽ οὐκ ἐπίσταται.	Antequam autem sit inducere, aut accipere syllogismum, quodam quidem modo fortasse dicendum est scire, modo autem alio, non quod enim nescivit si est simpliciter, hoc quodam modo scivit quod duos habet rectos simpliciter, sed manifestum est quod sic quidem scit quoniam universaliter scivit, simpliciter autem non scit;	Before he was led on to recognition or before he actually drew a conclusion, we should perhaps say that in a manner he knew, in a manner not. If he did not in an unqualified sense of the term know the existence of this triangle, how could he know without qualification that its angles were equal to two right angles? No: clearly he knows not without qualification but only in the sense that he knows universally.
71a29 εἰ δὲ μή, τὸ ἐν τῶι Μένωνι ἀπόρημα συμβήσεται·	si vero non, Menonis ambiguitas continget,	If this distinction is not drawn, we are faced with the dilemma in the Meno:
71a30 ἢ γὰρ οὐδὲν μαθήσεται ἢ ἃ οἶδεν. οὐ γὰρ δή, ὥς γέ τινες ἐγχειροῦσι λύειν, λεκτέον. ἆρ᾽ οἶδας ἅπασαν δυάδα ὅτι ἀρτία ἢ οὔ; φήσαντος δὲ προήνεγκάν τινα δυάδα ἣν οὐκ ὤιετ᾽ εἶναι, ὥστ᾽ οὐδ᾽ ἀρτίαν. λύουσι γὰρ οὐ φάσκοντες εἰδέναι πᾶσαν δυάδα ἀρτίαν οὖσαν, ἀλλ᾽ ἣν ἴσασιν ὅτι δυάς.	aut enim nihil discet, aut quae novit. Non enim, sicut quidam conantur solvere, dicendum est, nunquid scivisti omnem dualitatem quoniam par est, aut non? dicente autem, attulerat quamdam dualitatem, quam non opinatus est esse, quare neque parem. Solvunt enim dicentes non cognoscere omnem dualitatem parem esse, sed quam sciunt quod dualitas fit.	either a man will learn nothing or what he already knows; for we cannot accept the solution which some people offer. A man is asked, ‘Do you, or do you not, know that every pair is even?’ He says he does know it. The questioner then produces a particular pair, of the existence, and so a fortiori of the evenness, of which he was unaware. The solution which some people offer is to assert that they do not know that every pair is even, but only that everything which they know to be a pair is even:
71b1 καίτοι ἴσασι μὲν οὗπερ τὴν ἀπόδειξιν ἔχουσι καὶ οὗ ἔλαβον, ἔλαβον δ᾽ οὐχὶ παντὸς οὗ ἂν εἰδῶσιν ὅτι τρίγωνον ἢ ὅτι ἀριθμός, ἀλλ᾽ ἁπλῶς κατὰ παντὸς ἀριθμοῦ καὶ τριγώνου· οὐδεμία γὰρ πρότασις λαμβάνεται τοιαύτη, ὅτι ὃν σὺ οἶδας ἀριθ↵ μὸν ἢ ὁ σὺ οἶδας εὐθύγραμμον, ἀλλὰ κατὰ παντός.	Attamen sciunt quidem cuius vere demonstrationem habent, et cuius acceperunt: acceperunt autem non de omni cuius utique sciunt, quod triangulus aut quod numerus sit, sed simpliciter de omni numero et triangulo, neque enim una propositio accipitur huius quod quem tu nosti numerum, aut quod tu nosti rectilineum, sed de omni.	yet what they know to be even is that of which they have demonstrated evenness, i.e. what they made the subject of their premiss, viz. not merely every triangle or number which they know to be such, but any and every number or triangle without reservation. For no premiss is ever couched in the form ‘every number which you know to be such’, or ‘every rectilinear figure which you know to be such’: the predicate is always construed as applicable to any and every instance of the thing.
71b5 ἀλλ᾽ οὐδέν (οἶμαι) κωλύει, ὁ μανθάνει, ἔστιν ὡς ἐπίστασθαι, ἔστι δ᾽ ὡς ἀγνοεῖν· ἄτοπον γὰρ οὐκ εἰ οἶδέ πως ὁ μανθάνει, ἀλλ᾽ εἰ ὡδί, οἷον ἧι μανθάνει καὶ ὥς.	Sed nihil (ut opinor) prohibet quod discit quis, sit ut scire, sit etiam ut ignorare, inconveniens enim non est, si scit quodam modo quod discit, sed si hoc modo, ut in quantum discit, et scit.	On the other hand, I imagine there is nothing to prevent a man in one sense knowing what he is learning, in another not knowing it. The strange thing would be, not if in some sense he knew what he was learning, but if he were to know it in that precise sense and manner in which he was learning it.
c2	CAPUT II. De modis sciendi, et demonstratione.	Chapter 2
71b8 Ἐπίστασθαι δὲ οἰόμεθ᾽ ἕκαστον ἁπλῶς, ἀλλὰ μὴ τὸν ↵ σοφιστικὸν τρόπον τὸν κατὰ συμβεβηκός, ὅταν τήν τ᾽ αἰτίαν οἰώμεθα γινώσκειν δι᾽ ἣν τὸ πρᾶγμά ἐστιν, ὅτι ἐκείνου αἰτία ἐστί, καὶ μὴ ἐνδέχεσθαι τοῦτ᾽ ἄλλως ἔχειν.	Scire autem opinamur unumquodque simpliciter, sed non sophistico modo, qui est secundum accidens, cum causam arbitramur cognoscere, propter quam res est, quoniam illius causa est, et non est contingere hoc aliter se habere.	We suppose ourselves to possess unqualified scientific knowledge of a thing, as opposed to knowing it in the accidental way in which the sophist knows, when we think that we know the cause on which the fact depends, as the cause of that fact and of no other, and, further, that the fact could not be other than it is.
71b12 δῆλον τοίνυν ὅτι τοιοῦτόν τι τὸ ἐπίστασθαί ἐστι· καὶ γὰρ οἱ μὴ ἐπιστάμενοι καὶ οἱ ἐπιστάμενοι οἱ μὲν οἴονται αὐτοὶ οὕτως ἔχειν,	Manifestum igitur est quoniam huiusmodi aliquid scire sit, et namque non scientes, et scientes quidem opinantur ipsi sic se habere,	Now that scientific knowing is something of this sort is evident-witness both those who falsely claim it and those who actually possess it, since the former merely imagine themselves to be,
71b14 οἱ δ᾽ ἐπιστά↵ μενοι καὶ ἔχουσιν, ὥστε οὗ ἁπλῶς ἔστιν ἐπιστήμη, τοῦτ᾽ ἀδύνατον ἄλλως ἔχειν. 71b16 Εἰ μὲν οὖν καὶ ἕτερος ἔστι τοῦ ἐπίστασθαι τρόπος, ὕστερον ἐροῦμεν,	scientes autem etiam, habent scientiam. Quare cuius simpliciter est scientia, hoc est impossibile aliter se habere. Si quidem igitur et alius est sciendi modus, posterius dicemus.	while the latter are also actually, in the condition described. Consequently the proper object of unqualified scientific knowledge is something which cannot be other than it is. There may be another manner of knowing as well-that will be discussed later.
71b17 φαμὲν δὲ καὶ δι᾽ ἀποδείξεως εἰδέναι.	Dicimus autem scire, et per demonstrationem intelligere.	What I now assert is that at all events we do know by demonstration.
71b18 ἀπόδειξιν δὲ λέγω συλλογισμὸν ἐπιστημονικόν· ἐπιστημονικὸν δὲ λέγω καθ᾽ ὃν τῶι ἔχειν αὐτὸν ἐπιστάμεθα.	Demonstrationem autem dico syllogismum epistemonicon, id est facientem scire, sed epistemonicon dico, secundum quem (in habendo ipsum) scimus.	By demonstration I mean a syllogism productive of scientific knowledge, a syllogism, that is, the grasp of which is eo ipso such knowledge.
71b20 εἰ τοίνυν ἐστὶ τὸ ἐπί↵ στασθαι οἷον ἔθεμεν, ἀνάγκη καὶ τὴν ἀποδεικτικὴν ἐπιστήμην ἐξ ἀληθῶν τ᾽ εἶναι καὶ πρώτων καὶ ἀμέσων καὶ γνωριμωτέρων καὶ προτέρων καὶ αἰτίων τοῦ συμπεράσματος·	Si igitur est scire ut posuimus, necesse est et demonstrativam scientiam ex veris esse, et primis, et immediatis, et notioribus, et prioribus, causisque conclusionis.	Assuming then that my thesis as to the nature of scientific knowing is correct, the premisses of demonstrated knowledge must be true, primary, immediate, better known than and prior to the conclusion, which is further related to them as effect to cause.
71b22 οὕτω γὰρ ἔσονται καὶ αἱ ἀρχαὶ οἰκεῖαι τοῦ δεικνυμένου.	Sic enim principia erunt propria eius quod demonstratur,	Unless these conditions are satisfied, the basic truths will not be ‘appropriate’ to the conclusion.
71b23 συλλογισμὸς μὲν γὰρ ἔσται καὶ ἄνευ τούτων, ἀπόδειξις δ᾽ οὐκ ἔσται· οὐ γὰρ ↵ ποιήσει ἐπιστήμην.	nam syllogismus quidem erit, et sine his, demonstratio autem non erit, non enim faciet scientiam.	Syllogism there may indeed be without these conditions, but such syllogism, not being productive of scientific knowledge, will not be demonstration.
71b24 ἀληθῆ μὲν οὖν δεῖ εἶναι, ὅτι οὐκ ἔστι τὸ μὴ ὂν ἐπίστασθαι, οἷον ὅτι ἡ διάμετρος σύμμετρος.	Vera quidem igitur oportet esse, quoniam quod non est, non est scire, ut quod diameter sit symeter.	The premisses must be true: for that which is non-existent cannot be known-we cannot know, e.g. that the diagonal of a square is commensurate with its side.
71b27 ἐκ πρώτων δ᾽ ἀναποδείκτων, ὅτι οὐκ ἐπιστήσεται μὴ ἔχων ἀπόδειξιν αὐτῶν· τὸ γὰρ ἐπίστασθαι ὧν ἀπόδειξις ἔστι μὴ κατὰ συμβεβηκός, τὸ ἔχειν ἀπόδειξίν ἐστιν.	Ex primis autem et indemonstrabilibus est, quia non sciet, non habens demonstrationem ipsorum, scire enim quorum demonstratio est non secundum accidens, est habere demonstrationem.	The premisses must be primary and indemonstrable; otherwise they will require demonstration in order to be known, since to have knowledge, if it be not accidental knowledge, of things which are demonstrable, means precisely to have a demonstration of them.
71b29 αἴτιά τε καὶ γνωριμώτερα ↵ δεῖ εἶναι καὶ πρότερα, αἴτια μὲν ὅτι τότε ἐπιστάμεθα ὅταν τὴν αἰτίαν εἰδῶμεν, καὶ πρότερα, εἴπερ αἴτια, καὶ προγινωσκόμενα οὐ μόνον τὸν ἕτερον τρόπον τῶι ξυνιέναι, ἀλλὰ καὶ τῶι εἰδέναι ὅτι ἔστιν.	Causas quoque, et notiora oportet esse, et priora. Causas quidem, quoniam tunc scimus, cum causam cognoscimus, et priora, siquidem causae sunt, et notiora, non solum altero modo intelligendo, sed in sciendo quoniam sunt.	The premisses must be the causes of the conclusion, better known than it, and prior to it; its causes, since we possess scientific knowledge of a thing only when we know its cause; prior, in order to be causes; antecedently known, this antecedent knowledge being not our mere understanding of the meaning, but knowledge of the fact as well.
πρότερα δ᾽ ἐστὶ καὶ γνωριμώτερα διχῶς· οὐ γὰρ ταὐτὸν πρότερον τῆι φύσει καὶ πρὸς ἡμᾶς πρότερον, οὐδὲ γνωριμώτερον καὶ ἡμῖν γνωριμώτερον. λέγω δὲ πρὸς ἡμᾶς μὲν πρότερα καὶ γνωριμώτερα τὰ ἐγγύτερον τῆς αἰσθήσεως, ἁπλῶς δὲ πρότερα καὶ γνωριμώτερα τὰ πορρώτερον.	Priora autem et notiora dupliciter sunt, non enim idem est natura prius, et ad nos prius, neque notius natura, et nobis notius. Dico autem ad nos priora, et notiora, propinquiora sensui. Simpliciter autem priora, et notiora, quae longius sunt.	Now ‘prior’ and ‘better known’ are ambiguous terms, for there is a difference between what is prior and better known in the order of being and what is prior and better known to man. I mean that objects nearer to sense are prior and better known to man; objects without qualification prior and better known are those further from sense.
ἔστι δὲ πορρωτάτω μὲν τὰ καθόλου μάλιστα, ἐγγυτάτω ↵ δὲ τὰ καθ᾽ ἕκαστα· καὶ ἀντίκειται ταῦτ᾽ ἀλλήλοις. ἐκ πρώτων δ᾽ ἐστὶ τὸ ἐξ ἀρχῶν οἰκείων· ταὐτὸ γὰρ λέγω πρῶτον καὶ ἀρχήν.	Sunt autem longinquissima quidem, universalia maxime. Proxima autem, singularia, et opponuntur haec ad se invicem. Ex primis autem est quod ex propriis principiis est, idem enim dico primum et principium.	Now the most universal causes are furthest from sense and particular causes are nearest to sense, and they are thus exactly opposed to one another. In saying that the premisses of demonstrated knowledge must be primary, I mean that they must be the ‘appropriate’ basic truths, for I identify primary premiss and basic truth.
72a8 ἀρχὴ δ᾽ ἐστὶν ἀποδείξεως πρότασις ἄμεσος, ἄμεσος δὲ ἧς μὴ ἔστιν ἄλλη προτέρα.	Est autem principium demonstrationis propositio immediata. Immediata autem est qua non est alia prior.	A ‘basic truth’ in a demonstration is an immediate proposition. An immediate proposition is one which has no other proposition prior to it.
72a9 πρότασις δ᾽ ἐστὶν ἀποφάνσεως τὸ ἕτερον μόριον, ἓν καθ᾽ ἑνός,	Propositio autem est enuntiationis altera pars, unum de uno.	A proposition is either part of an enunciation, i.e. it predicates a single attribute of a single subject.
72a10 διαλεκτικὴ μὲν ἡ ὁμοίως λαμβάνουσα ὁποτερονοῦν, ἀποδεικτικὴ δὲ ἡ ὡρισμένως θάτερον, ὅτι ἀληθές.	Dialectica quidem est similiter accipiens quamlibet. Demonstrativa autem determinatae alterum quoniam verum est.	If a proposition is dialectical, it assumes either part indifferently; if it is demonstrative, it lays down one part to the definite exclusion of the other because that part is true.
72a11 ἀπόφανσις δὲ ἀντιφάσεως ὁποτερονοῦν μόριον, ἀντίφασις δὲ ἀντίθεσις ἧς οὐκ ἔστι μεταξὺ καθ᾽ αὑτήν, μόριον δ᾽ ἀντιφάσεως τὸ μὲν τὶ κατὰ τινὸς κατάφασις, τὸ δὲ τὶ ἀπὸ τινὸς ἀπόφασις.	Enuntiatio autem contradictionis quaelibet pars. Contradictio autem est oppositio cuius non est medium secundum se. Pars autem contradictionis, quae quidem aliquid de aliquo est, affirmatio est. Quae vero est aliquid ab aliquo, negatio est.	The term ‘enunciation’ denotes either part of a contradiction indifferently. A contradiction is an opposition which of its own nature excludes a middle. The part of a contradiction which conjoins a predicate with a subject is an affirmation; the part disjoining them is a negation.
72a15 Ἀμέσου δ᾽ ἀρ↵ χῆς συλλογιστικῆς θέσιν μὲν λέγω ἣν μὴ ἔστι δεῖξαι, μηδ᾽ ἀνάγκη ἔχειν τὸν μαθησόμενόν τι· ἣν δ᾽ ἀνάγκη ἔχειν τὸν ὁτιοῦν μαθησόμενον, ἀξίωμα· ἔστι γὰρ ἔνια τοιαῦτα· τοῦτο γὰρ μάλιστ᾽ ἐπὶ τοῖς τοιούτοις εἰώθαμεν ὄνομα λέγειν.	Immediati autem principii syllogistici, positionem quidem dico, quam non est monstrare, nec necesse est habere docendum aliquid. Quam vero necesse est habere quemlibet docendum, dignitatem. Sunt enim quaedam huiusmodi, hoc enim maxime in huiuscemodi consuevimus nomen dicere.	I call an immediate basic truth of syllogism a ‘thesis’ when, though it is not susceptible of proof by the teacher, yet ignorance of it does not constitute a total bar to progress on the part of the pupil: one which the pupil must know if he is to learn anything whatever is an axiom. I call it an axiom because there are such truths and we give them the name of axioms par excellence.
72a19 θέσεως δ᾽ ἡ μὲν ὁποτερονοῦν τῶν μορίων τῆς ἀντιφάσεως λαμβά↵ νουσα, οἷον λέγω τὸ εἶναί τι ἢ τὸ μὴ εἶναί τι, ὑπόθεσις, ἡ δ᾽ ἄνευ τούτου ὁρισμός. ὁ γὰρ ὁρισμὸς θέσις μέν ἐστι· τίθεται γὰρ ὁ ἀριθμητικὸς μονάδα τὸ ἀδιαίρετον εἶναι κατὰ τὸ ποσόν· ὑπόθεσις δ᾽ οὐκ ἔστι· τὸ γὰρ τί ἐστι μονὰς καὶ τὸ εἶναι μονάδα οὐ ταὐτόν.	Positionis autem, quae quidem est quamlibet partium enuntiationis accipiens, ut dico aliquid esse, aut non esse, suppositio est. Quae vero sine hoc, definitio est, definitio enim positio quaedam est. Ponit enim arithmeticus unitatem, indivisibile esse secundum quantitatem, suppositio autem non est, id enim quod quid est unitas, et esse unitatem, non idem est.	If a thesis assumes one part or the other of an enunciation, i.e. asserts either the existence or the non-existence of a subject, it is a hypothesis; if it does not so assert, it is a definition. Definition is a ‘thesis’ or a ‘laying something down’, since the arithmetician lays it down that to be a unit is to be quantitatively indivisible; but it is not a hypothesis, for to define what a unit is is not the same as to affirm its existence.
72a25 Ἐπεὶ δὲ δεῖ πιστεύειν τε καὶ εἰδέναι τὸ πρᾶγμα τῶι τοιοῦτον ἔχειν συλλογισμὸν ὃν καλοῦμεν ἀπόδειξιν, ἔστι δ᾽ οὗτος τῶι ταδὶ εἶναι ἐξ ὧν ὁ συλλογισμός, ἀνάγκη μὴ μόνον προγινώσκειν τὰ πρῶτα, ἢ πάντα ἢ ἔνια, ἀλλὰ καὶ μᾶλλον·	Quoniam autem oportet credere, et scire rem, in huiusmodi habendo syllogismum quem vocamus demonstrationem. Est autem hic eo quod ea sunt, ex quibus est syllogismus, necesse est non solum praecognoscere prima, aut omnia, aut quamdam, sed et magis.	Now since the required ground of our knowledge-i.e. of our conviction-of a fact is the possession of such a syllogism as we call demonstration, and the ground of the syllogism is the facts constituting its premisses, we must not only know the primary premisses – some if not all of them – beforehand, but know them better than the conclusion:
72a28 αἰεὶ γὰρ δι᾽ ὁ ὑπάρχει ἕκαστον, ἐκείνωι μᾶλλον ὑπάρ↵χει, οἷον δι᾽ ὁ φιλοῦμεν, ἐκεῖνο φίλον μᾶλλον. ὥστ᾽ εἴπερ ἴσμεν διὰ τὰ πρῶτα καὶ πιστεύομεν, κἀκεῖνα ἴσμεν τε καὶ πιστεύομεν μᾶλλον, ὅτι δι᾽ ἐκεῖνα καὶ τὰ ὕστερα.	Semper enim propter quod unumquodque est, illud est, ut propter quod amamus, illud magis amicum est, quare siquidem scimus propter prima et credimus, et illa scimus et credimus magis, quoniam propter illa et posteriora.	for the cause of an attribute’s inherence in a subject always itself inheres in the subject more firmly than that attribute; e.g. the cause of our loving anything is dearer to us than the object of our love. So since the primary premisses are the cause of our knowledge-i.e. of our conviction-it follows that we know them better-that is, are more convinced of them-than their consequences, precisely because of our knowledge of the latter is the effect of our knowledge of the premisses.
οὐχ οἷόν τε δὲ πιστεύειν μᾶλλον ὧν οἶδεν ἃ μὴ τυγχάνει μήτε εἰδὼς μήτε βέλτιον διακείμενος ἢ εἰ ἐτύγχανεν εἰδώς.	Non potest autem credere magis quae scit, quae non contingit neque sciens, neque melius dispositus quam si contigerit sciens,	Now a man cannot believe in anything more than in the things he knows, unless he has either actual knowledge of it or something better than actual knowledge.
72a33 συμβήσεται δὲ τοῦτο, εἰ μή τις προγνώσεται τῶν δι᾽ ἀπόδειξιν πιστευόντων· μᾶλλον γὰρ ἀνάγκη πιστεύειν ταῖς ἀρχαῖς ἢ πάσαις ἢ τισὶ τοῦ συμπεράσματος.	accidet autem hoc nisi aliquis praecognoverit propter demonstrationem credentium, magis enim necesse est credere principiis, aut omnibus, aut quibusdam, quam conclusioni.	But we are faced with this paradox if a student whose belief rests on demonstration has not prior knowledge; 72a36 a man must believe in some, if not in all, of the basic truths more than in the conclusion.
τὸν δὲ μέλλοντα ἕξειν τὴν ἐπιστήμην τὴν δι᾽ ἀποδείξεως οὐ μόνον δεῖ τὰς ἀρχὰς μᾶλλον γνωρίζειν καὶ μᾶλλον αὐταῖς πιστεύειν ἢ τῶι δεικνυμένωι,	Debentem autem habere scientiam per demonstrationem, non solum oportet principia magis cognoscere et magis ipsis credere quam ei quod demonstratur,	Moreover, if a man sets out to acquire the scientific knowledge that comes through demonstration, 72a38 he must not only have a better knowledge of the basic truths and a firmer conviction of them than of the connexion which is being demonstrated:
ἀλλὰ μηδ᾽ ἄλλο αὐτῶι πιστότερον εἶναι μηδὲ γνωριμώτερον τῶν ἀντικειμένων ταῖς ἀρχαῖς ἐξ ὧν ἔσται συλλογισμὸς ὁ τῆς ἐναντίας ἀπάτης, εἴπερ δεῖ τὸν ἐπιστάμενον ἁπλῶς ἀμετάπειστον εἶναι.	sed neque aliud ipsi credibilius esse, neque notius oppositis principiis, ex quibus erit syllogismus contrariae deceptionis, si quidem oportet simpliciter scientem, immutabilem esse.	more than this, nothing must be more certain or better known to him than these basic truths in their character as contradicting the fundamental premisses which lead to the opposed and erroneous conclusion. For indeed the conviction of pure science must be unshakable.
c3	CAPUT III. Quod non omnium sit demonstrativa scientia.	Chapter 3
72b5 Ἐνίοις μὲν οὖν διὰ τὸ δεῖν τὰ πρῶτα ἐπίστασθαι οὐ δοκεῖ ἐπιστήμη εἶναι, τοῖς δ᾽ εἶναι μέν, πάντων μέντοι ἀπόδειξις εἶναι· ὧν οὐδέτερον οὔτ᾽ ἀληθὲς οὔτ᾽ ἀναγκαῖον.	Quibusdam autem igitur propter id quod oportet prima scire, non videtur scientia esse. Quibusdam autem esse quidem, omnium tamen demonstrationes esse, quorum neutrum neque verum neque necessarium.	Some hold that, owing to the necessity of knowing the primary premisses, there is no scientific knowledge. Others think there is, but that all truths are demonstrable. Neither doctrine is either true or a necessary deduction from the premisses.
72b8 οἱ μὲν γὰρ ὑποθέμενοι μὴ εἶναι ὅλως ἐπίστασθαι, οὗτοι εἰς ἄπειρον ἀξιοῦσιν ἀνάγεσθαι ὡς οὐκ ἂν ἐπισταμένους τὰ ὕστερα διὰ τὰ ↵ πρότερα, ὧν μὴ ἔστι πρῶτα, ὀρθῶς λέγοντες· ἀδύνατον γὰρ τὰ ἄπειρα διελθεῖν. εἴ τε ἵσταται καὶ εἰσὶν ἀρχαί, ταύτας ἀγνώστους εἶναι ἀποδείξεώς γε μὴ οὔσης αὐτῶν, ὅπερ φασὶν εἶναι τὸ ἐπίστασθαι μόνον· εἰ δὲ μὴ ἔστι τὰ πρῶτα εἰδέναι, οὐδὲ τὰ ἐκ τούτων εἶναι ἐπίστασθαι ἁπλῶς οὐδὲ κυρίως, ἀλλ᾽ ↵ ἐξ ὑποθέσεως, εἰ ἐκεῖνα ἔστιν.	Ponentes autem non esse omnino scire, hi ad infinitum volunt reduci, tanquam non sit utique scientes posteriora propter priora, quorum non sint prima, recte dicentes. Impossibile enim est infinita pertransire, et si stent et sint principia haec, ignota esse, cum demonstratio non sit ipsorum, quod quidem dicunt esse scire solum. Si vero non est prima scire, neque quae ex eis sunt, est scire, neque simpliciter neque proprie, sed ex conditione, si illa sint.	The first school, assuming that there is no way of knowing other than by demonstration, maintain that an infinite regress is involved, on the ground that if behind the prior stands no primary, we could not know the posterior through the prior (wherein they are right, for one cannot traverse an infinite series): if on the other hand-they say-the series terminates and there are primary premisses, yet these are unknowable because incapable of demonstration, which according to them is the only form of knowledge. And since thus one cannot know the primary premisses, knowledge of the conclusions which follow from them is not pure scientific knowledge nor properly knowing at all, but rests on the mere supposition that the premisses are true.
72b15 οἱ δὲ περὶ μὲν τοῦ ἐπίστασθαι ὁμολογοῦσι· δι᾽ ἀποδείξεως γὰρ εἶναι μόνον· ἀλλὰ πάντων εἶναι ἀπόδειξιν οὐδὲν κωλύειν· ἐνδέχεσθαι γὰρ κύκλωι γίνεσθαι τὴν ἀπόδειξιν καὶ ἐξ ἀλλήλων.	Hi autem de eo quod quidem est scire, sic confitentur, per demonstrationem enim esse solum, sed omnium esse demonstrationem nihil prohibet, contingit enim circulo fieri demonstrationem, et ex iis quae sunt ad invicem.	The other party agree with them as regards knowing, holding that it is only possible by demonstration, but they see no difficulty in holding that all truths are demonstrated, on the ground that demonstration may be circular and reciprocal.
72b18 Ἡμεῖς δέ φαμεν οὔτε πᾶσαν ἐπιστήμην ἀποδεικτικὴν εἶναι, ἀλλὰ τὴν τῶν ἀμέσων ↵ ἀναπόδεικτον (καὶ τοῦθ᾽ ὅτι ἀναγκαῖον, φανερόν· εἰ γὰρ ἀνάγκη μὲν ἐπίστασθαι τὰ πρότερα καὶ ἐξ ὧν ἡ ἀπόδειξις, ἵσταται δέ ποτε τὰ ἄμεσα, ταῦτ᾽ ἀναπόδεικτα ἀνάγκη εἶναι) – ταῦτά τ᾽ οὖν οὕτω λέγομεν, καὶ οὐ μόνον ἐπιστήμην ἀλλὰ καὶ ἀρχὴν ἐπιστήμης εἶναί τινά φαμεν, ἧι τοὺς ὅρους γνω↵ρίζομεν.	Nos autem dicemus neque omnem scientiam demonstrativam esse, sed immediatorum esse indemonstrabilem, et quod hoc necessarium sit, manifestum est. Si enim necesse est quidem scire priora, ex quibus est demonstratio, stant autem aliquando immediata, haec quidem indemonstrabilia necesse est esse, et hoc igitur sic dicimus, et non solum scientiam, sed et principium scientiae esse quoddam dicimus, in quantum terminos cognoscimus.	Our own doctrine is that not all knowledge is demonstrative: on the contrary, knowledge of the immediate premisses is independent of demonstration. (The necessity of this is obvious; for since we must know the prior premisses from which the demonstration is drawn, and since the regress must end in immediate truths, those truths must be indemonstrable.) Such, then, is our doctrine, and in addition we maintain that besides scientific knowledge there is its originative source which enables us to recognize the definitions.
72b25 κύκλωι τε ὅτι ἀδύνατον ἀποδείκνυσθαι ἁπλῶς, δῆ- λον, εἴπερ ἐκ προτέρων δεῖ τὴν ἀπόδειξιν εἶναι καὶ γνωριμωτέρων· ἀδύνατον γάρ ἐστι τὰ αὐτὰ τῶν αὐτῶν ἅμα πρότερα καὶ ὕστερα εἶναι, εἰ μὴ τὸν ἕτερον τρόπον, οἷον τὰ μὲν πρὸς ἡμᾶς τὰ δ᾽ ἁπλῶς, ὅνπερ τρόπον ἡ ἐπαγωγὴ ποιεῖ γνώρι↵ μον. εἰ δ᾽ οὕτως, οὐκ ἂν εἴη τὸ ἁπλῶς εἰδέναι καλῶς ὡρισμένον, ἀλλὰ διττόν· ἢ οὐχ ἁπλῶς ἡ ἑτέρα ἀπόδειξις, γινομένη γ᾽ ἐκ τῶν ἡμῖν γνωριμωτέρων.	Circulo autem quod impossibile sit demonstrare simpliciter manifestum est, si quidem ex prioribus oportet demonstrationem esse, et notioribus, impossibile enim est eadem sibi invicem simul priora, et posteriora esse, nisi altero modo, ut haec quidem ad nos, illa vero simpliciter, quo certe modo inductio facit notum. Si autem sic est, non utique erit simpliciter scire bene definitum, sed dupliciter. An non simpliciter altera demonstratio fit ex nobis notioribus?	Now demonstration must be based on premisses prior to and better known than the conclusion; and the same things cannot simultaneously be both prior and posterior to one another: so circular demonstration is clearly not possible in the unqualified sense of ‘demonstration’, but only possible if ‘demonstration’ be extended to include that other method of argument which rests on a distinction between truths prior to us and truths without qualification prior, i.e. the method by which induction produces knowledge. But if we accept this extension of its meaning, our definition of unqualified knowledge will prove faulty; for there seem to be two kinds of it. Perhaps, however, the second form of demonstration, that which proceeds from truths better known to us, is not demonstration in the unqualified sense of the term.
72b33 συμβαίνει δὲ τοῖς λέγουσι κύκλωι τὴν ἀπόδειξιν εἶναι οὐ μόνον τὸ νῦν εἰρημένον, ἀλλ᾽ οὐδὲν ἄλλο λέγειν ἢ ὅτι τοῦτ᾽ ἔστιν εἰ τοῦτ᾽ ἔστιν· οὕτω δὲ πάντα ↵ ῥάιδιον δεῖξαι. δῆλον δ᾽ ὅτι τοῦτο συμβαίνει τριῶν ὅρων τεθέντων. τὸ μὲν γὰρ διὰ πολλῶν ἢ δι᾽ ὀλίγων ἀνακάμπτειν φάναι οὐδὲν διαφέρει, δι᾽ ὀλίγων δ᾽ ἢ δυοῖν.	Accidit vero dicentibus circulo demonstrationem esse, non solum quod nunc dictum est, sed nihil aliud dicere quam quoniam hoc est, si hoc est, sic autem facile est demonstrare omnia. Manifestum autem est quod hoc accidit tribus terminis positis, per multos enim, aut per paucos reflectere dicere nihil differt, per paucos autem, aut per duos.	The advocates of circular demonstration are not only faced with the difficulty we have just stated: in addition their theory reduces to the mere statement that if a thing exists, then it does exist-an easy way of proving anything. That this is so can be clearly shown by taking three terms, for to constitute the circle it makes no difference whether many terms or few or even only two are taken.
72b38 ὅταν γὰρ τοῦ Α ὄντος ἐξ ἀνάγκης ἦι τὸ Β, τούτου δὲ τὸ Γ, τοῦ Α ὄντος ἔσται τὸ Γ. εἰ δὴ τοῦ Α ὄντος ἀνάγκη τὸ Β εἶναι, τούτου δ᾽ 73a1 ὄντος τὸ Α	Cum enim sit a, sit ex necessitate b, hoc autem cum sit, et c, cum igitur a sit, erit et c; si igitur cum sit a, necesse est b esse, hoc autem cum sit, a est.	Thus by direct proof, if A is, B must be; if B is, C must be; therefore if A is, C must be.
(τοῦτο γὰρ ἦν τὸ κύκλωι), κείσθω τὸ Α ἐφ᾽ οὗ τὸ Γ. τὸ οὖν τοῦ Β ὄντος τὸ Α εἶναι λέγειν ἐστὶ τὸ Γ εἶναι λέγειν, τοῦτο δ᾽ ὅτι τοῦ Α ὄντος τὸ Γ ἔστι· τὸ δὲ Γ τῶι Α τὸ αὐτό.	Hoc enim erit circulo: ponatur autem a in quo c, b igitur cum sit, a esse dicere, est et ipsum c dicere esse. Hoc autem dicere est, quoniam cum sit a est c, c autem ipsi a idem est,	Since then-by the circular proof-if A is, B must be, and if B is, A must be, A may be substituted for C above. Then ‘if B is, A must be’=’if B is, C must be’, which above gave the conclusion ‘if A is, C must be’: but C and A have been identified.
73a6 ὥστε συμβαίνει λέγειν τοὺς κύκλωι φάσκοντας εἶναι ↵ τὴν ἀπόδειξιν οὐδὲν ἕτερον πλὴν ὅτι τοῦ Α ὄντος τὸ Α ἔστιν. οὕτω δὲ πάντα δεῖξαι ῥάιδιον.	quare accidit dicere, circulo dicentes esse demonstrationem nihil aliud nisi cum sit a est a, sic autem omnia demonstrare leve est.	Consequently the upholders of circular demonstration are in the position of saying that if A is, A must be-a simple way of proving anything.
Οὐ μὴν ἀλλ᾽ οὐδὲ τοῦτο δυνατόν, πλὴν ἐπὶ τούτων ὅσα ἀλλήλοις ἕπεται, ὥσπερ τὰ ἴδια. ἑνὸς μὲν οὖν κειμένου δέδεικται ὅτι οὐδέποτ᾽ ἀνάγκη τι εἶναι ἕτερον (λέγω δ᾽ ἑνός, ὅτι οὔτε ὅρου ἑνὸς οὔτε θέσεως μιᾶς τεθεί↵ σησ), ἐκ δύο δὲ θέσεων πρώτων καὶ ἐλαχίστων ἐνδέχεται, εἴπερ καὶ συλλογίσασθαι.	At vero neque hoc possibile, nisi in iis quae alternatim se consequuntur, sicut sunt propria. Uno quidem igitur posito ostensum est quod nequaquam necesse aliquid esse alterum. Dico autem uno, quoniam nec termino uno, nec positione una posita, ex duabus autem positionibus primis et minimis, siquidem contingit et syllogizare.	Moreover, even such circular demonstration is impossible except in the case of attributes that imply one another, viz. ‘peculiar’ properties. Now, it has been shown that the positing of one thing-be it one term or one premiss-never involves a necessary consequent: two premisses constitute the first and smallest foundation for drawing a conclusion at all and therefore a fortiori for the demonstrative syllogism of science.
ἐὰν μὲν οὖν τό τε Α τῶι Β καὶ τῶι Γ ἕπηται, καὶ ταῦτ᾽ ἀλλήλοις καὶ τῶι Α, οὕτω μὲν ἐνδέχεται ἐξ ἀλλήλων δεικνύναι πάντα τὰ αἰτηθέντα ἐν τῶι πρώτωι σχήματι, ὡς δέδεικται ἐν τοῖς περὶ συλλογισμοῦ.	Si igitur et a ipsi b, et c sequatur, et haec ad invicem, et ipsi a, si quidem igitur contingit ex alternis monstrare omnia quaesita in prima figura, sicut ostensum est in iis qui de syllogismo sunt.	If, then, A is implied in B and C, and B and C are reciprocally implied in one another and in A, it is possible, as has been shown in my writings on the syllogism, to prove all the assumptions on which the original conclusion rested, by circular demonstration in the first figure.
↵ δέδεικται δὲ καὶ ὅτι ἐν τοῖς ἄλλοις σχήμασιν ἢ οὐ γίνεται συλλογισμὸς ἢ οὐ περὶ τῶν ληφθέντων. τὰ δὲ μὴ ἀντικατηγορούμενα οὐδαμῶς ἔστι δεῖξαι κύκλωι, ὥστ᾽ ἐπειδὴ ὀλίγα τοιαῦτα ἐν ταῖς ἀποδείξεσι, φανερὸν ὅτι κενόν τε καὶ ἀδύνα- τον τὸ λέγειν ἐξ ἀλλήλων εἶναι τὴν ἀπόδειξιν καὶ διὰ τοῦτο ↵ πάντων ἐνδέχεσθαι εἶναι ἀπόδειξιν.	Ostensum est autem quod in aliis figuris aut non fit syllogismus, aut non de acceptis quae autem non mutuo praedicantur, nequaquam est demonstrare circulo, quare quoniam pauca quidem huiusmodi in demonstrationibus sunt, manifestum est quod vanum quidem et impossibile sit dicere ex iis quae sunt ad invicem esse demonstrationem, et propter hoc contingere omnium esse demonstrationem.	But it has also been shown that in the other figures either no conclusion is possible, or at least none which proves both the original premisses. Propositions the terms of which are not convertible cannot be circularly demonstrated at all, and since convertible terms occur rarely in actual demonstrations, it is clearly frivolous and impossible to say that demonstration is reciprocal and that therefore everything can be demonstrated.
c4	CAPUT IV. Quid de omni, quid per se, et per universale.	Chapter 4
73a21 Ἐπεὶ δ᾽ ἀδύνατον ἄλλως ἔχειν οὗ ἔστιν ἐπιστήμη ἁπλῶς, ἀναγκαῖον ἂν εἴη τὸ ἐπιστητὸν τὸ κατὰ τὴν ἀποδεικτικὴν ἐπιστήμην· ἀποδεικτικὴ δ᾽ ἐστὶν ἣν ἔχομεν τῶι ἔχειν ἀπόδειξιν. ἐξ ἀναγκαίων ἄρα συλλογισμός ἐστιν ἡ ἀπόδειξις. ληπτέον ↵ ἄρα ἐκ τίνων καὶ ποίων αἱ ἀποδείξεις εἰσίν.	Quoniam autem impossibile est aliter se habere id cuius est scientia simpliciter, necessarium utique erit id esse scibile, quod est secundum demonstrativam scientiam. Demonstrativa autem est, quam habemus in habendo demonstrationem: ex necessariis itaque syllogismis est demonstratio. Accipiendum igitur est ex quibus et qualibus demonstrationes sunt,	Since the object of pure scientific knowledge cannot be other than it is, the truth obtained by demonstrative knowledge will be necessary. And since demonstrative knowledge is only present when we have a demonstration, it follows that demonstration is an inference from necessary premisses. So we must consider what are the premisses of demonstration-i.e. what is their character:
73a25 πρῶτον δὲ διορίσωμεν τί λέγομεν τὸ κατὰ παντὸς καὶ τί τὸ καθ᾽ αὑτὸ καὶ τί τὸ καθόλου.	primum quidem determinabimus quid dicimus de omni, et quid per se, et quid universale.	and as a preliminary, let us define what we mean by an attribute ‘true in every instance of its subject’, an ‘essential’ attribute, and a ‘commensurate and universal’ attribute.
73a28 Κατὰ παντὸς μὲν οὖν τοῦτο λέγω ὁ ἂν ἦι μὴ ἐπὶ τινὸς μὲν τινὸς δὲ μή, μηδὲ ποτὲ μὲν ποτὲ δὲ μή, οἷον εἰ κατὰ ↵ παντὸς ἀνθρώπου ζῶιον, εἰ ἀληθὲς τόνδ᾽ εἰπεῖν ἄνθρωπον, ἀληθὲς καὶ ζῶιον, καὶ εἰ νῦν θάτερον, καὶ θάτερον, καὶ εἰ ἐν πάσηι γραμμῆι στιγμή, ὡσαύτως.	De omni quidem hoc dico, quod utique est non in quodam quidem, in quodam autem non, neque quod aliquando quidem, aliquando vero non, ut si de omni homine animal, si verum est quidem dicere hominem, verum est et dicere animal, et si nunc alterum, et alterum, et si in omni linea punctum, similiter est,	I call ‘true in every instance’ what is truly predicable of all instances-not of one to the exclusion of others-and at all times, not at this or that time only; e.g. if animal is truly predicable of every instance of man, then if it be true to say ‘this is a man’, ‘this is an animal’ is also true, and if the one be true now the other is true now. A corresponding account holds if point is in every instance predicable as contained in line.
73a32 σημεῖον δέ· καὶ γὰρ τὰς ἐνστάσεις οὕτω φέρομεν ὡς κατὰ παντὸς ἐρωτώμενοι, ἢ εἰ ἐπί τινι μή, ἢ εἴ ποτε μή.	signum autem est, namque instantias sic ferimus, ut de omni etiam interrogati, aut si in quodam non, aut si aliquando non.	There is evidence for this in the fact that the objection we raise against a proposition put to us as true in every instance is either an instance in which, or an occasion on which, it is not true.
73a34 Καθ᾽ αὑτὰ δ᾽ ὅσα ὑπάρχει τε ↵ ἐν τῶι τί ἐστιν, οἷον τριγώνωι γραμμὴ καὶ γραμμῆι στιγμή (ἡ γὰρ οὐσία αὐτῶν ἐκ τούτων ἐστί, καὶ ἐν τῶι λόγωι τῶι λέγοντι τί ἐστιν ἐνυπάρχει),	Per se autem sunt quaecunque sunt in eo quod quid est, ut triangulo inest linea, et punctum lineae, substantia enim ipsorum ex his est, et in ratione dicenti quid est, insunt.	Essential attributes are (1) such as belong to their subject as elements in its essential nature (e.g. line thus belongs to triangle, point to line; for the very being or ‘substance’ of triangle and line is composed of these elements, which are contained in the formulae defining triangle and line):
73a38 καὶ ὅσοις τῶν ὑπαρχόντων αὐτοῖς αὐτὰ ἐν τῶι λόγωι ἐνυπάρχουσι τῶι τί ἐστι δηλοῦντι, οἷον τὸ εὐθὺ ὑπάρχει γραμμῆι καὶ τὸ περιφερές, καὶ τὸ περιττὸν καὶ ἄρτιον ἀριθμῶι, καὶ τὸ πρῶτον καὶ σύνθετον, καὶ ἰσόπλευρον καὶ ἑτερόμηκες· καὶ πᾶσι τούτοις ἐνυπάρχουσιν ἐν τῶι λόγωι τῶι τί ἐστι λέγοντι ἔνθα μὲν γραμμὴ ἔνθα δ᾽ ἀριθμός. ὁμοίως δὲ καὶ ἐπὶ τῶν ἄλλων τὰ τοιαῦθ᾽ ἑκάστοις καθ᾽ αὑτὰ λέγω, ὅσα δὲ μηδετέρως ὑπάρχει, συμβεβηκότα, ↵ οἷον τὸ μουσικὸν ἢ λευκὸν τῶι ζώιωι.	Et quibuscunque eorum quae insunt ipsis, ipsa in ratione insunt quid est monstranti, ut rectum inest lineae, et circulare, et par, et impar numero, et primum, et compositum, et isopleurum, et altera parte longius, et quae omnibus his insunt, in ratione quid est dicente, illinc quidem linea, hinc vero numerus, similiter et in aliis huiusmodi, unicuique per se esse dico. Quaecunque vero neutraliter insunt, accidentia sunt, ut musicum, aut album animal.	(2) such that, while they belong to certain subjects, the subjects to which they belong are contained in the attribute’s own defining formula. Thus straight and curved belong to line, odd and even, prime and compound, square and oblong, to number; and also the formula defining any one of these attributes contains its subject-e.g. line or number as the case may be. Extending this classification to all other attributes, I distinguish those that answer the above description as belonging essentially to their respective subjects; whereas attributes related in neither of these two ways to their subjects I call accidents or ‘coincidents’; e.g. musical or white is a ‘coincident’ of animal.
73b5 ἔτι ὁ μὴ καθ᾽ ὑποκειμένου λέγεται ἄλλου τινός, οἷον τὸ βαδίζον ἕτερόν τι ὂν βαδίζον ἐστὶ καὶ τὸ λευκὸν ‹λευκόν›, ἡ δ᾽ οὐσία, καὶ ὅσα τόδε τι σημαίνει, οὐχ ἕτερόν τι ὄντα ἐστὶν ὅπερ ἐστίν. τὰ μὲν δὴ μὴ καθ᾽ ὑποκειμένου καθ᾽ αὑτὰ λέγω, τὰ δὲ καθ᾽ ὑποκειμένου συμ↵ βεβηκότα.	Amplius, quod non de subiecto alio quodam dicitur, ut ambulans, aut album, cum et alterum quiddam sit ambulans et album. Substantia autem et quaecunque hoc aliquid significant, non alterum aliquid sunt quam quod quidem sunt; quae quidem igitur non de subiecto alio sunt, per se dico, quae vero de subiecto, accidentia.	Further (a) that is essential which is not predicated of a subject other than itself: e.g. ‘the walking [thing]’ walks and is white in virtue of being something else besides; whereas substance, in the sense of whatever signifies a ‘this somewhat’, is not what it is in virtue of being something else besides. Things, then, not predicated of a subject I call essential; things predicated of a subject I call accidental or ‘coincidental’.
73b10 ἔτι δ᾽ ἄλλον τρόπον τὸ μὲν δι᾽ αὑτὸ ὑπάρχον ἑκάστωι καθ᾽ αὑτό, τὸ δὲ μὴ δι᾽ αὑτὸ συμβεβηκός, οἷον εἰ βαδίζοντος ἤστραψε, συμβεβηκός· οὐ γὰρ διὰ τὸ βαδίζειν ἤστραψεν, ἀλλὰ συνέβη, φαμέν, τοῦτο.	Item alio modo quod quidem propter ipsum inest unicuique, per se dico, quod vero non propter ipsum, accidens est, ut si ambulans coruscavit, accidens est, non enim propter id quod ambulavit, coruscavit, sed accidens dicimus hoc.	In another sense again (b) a thing consequentially connected with anything is essential; one not so connected is ‘coincidental’. An example of the latter is ‘While he was walking it lightened’: the lightning was not due to his walking; it was, we should say, a coincidence.
εἰ δὲ δι᾽ αὑτό, καθ᾽ αὑτό, οἷον εἴ τι σφαττόμενον ἀπέθανε, καὶ κατὰ τὴν ↵ σφαγήν, ὅτι διὰ τὸ σφάττεσθαι, ἀλλ᾽ οὐ συνέβη σφαττόμενον ἀποθανεῖν.	Si vero propter ipsum, per se, ut si aliquid interfectum interiit, secundum interfectionem, quoniam propter id quod interfectum est interiit, sed non quod accidat interfectum interire.	If, on the other hand, there is a consequential connexion, the predication is essential; e.g. if a beast dies when its throat is being cut, then its death is also essentially connected with the cutting, because the cutting was the cause of death, not death a ‘coincident’ of the cutting.
73b16 τὰ ἄρα λεγόμενα ἐπὶ τῶν ἁπλῶς ἐπιστητῶν καθ᾽ αὑτὰ οὕτως ὡς ἐνυπάρχειν τοῖς κατηγορουμένοις ἢ ἐνυπάρχεσθαι δι᾽ αὑτά τέ ἐστι καὶ ἐξ ἀνάγκης. οὐ γὰρ ἐνδέχεται μὴ ὑπάρχειν ἢ ἁπλῶς ἢ τὰ ἀντικείμενα, οἷον ↵ γραμμῆι τὸ εὐθὺ ἢ τὸ καμπύλον καὶ ἀριθμῶι τὸ περιττὸν ἢ τὸ ἄρτιον. ἔστι γὰρ τὸ ἐναντίον ἢ στέρησις ἢ ἀντίφασις ἐν τῶι αὐτῶι γένει, οἷον ἄρτιον τὸ μὴ περιττὸν ἐν ἀριθμοῖς ἧι ἕπεται. ὥστ᾽ εἰ ἀνάγκη φάναι ἢ ἀποφάναι, ἀνάγκη καὶ τὰ καθ᾽ αὑτὰ ὑπάρχειν.	Quae ergo dicuntur in simpliciter scibilibus per se sic sunt, sicut esse praedicatis, aut inesse, propter ipsaque sunt et ex necessitate, non enim contingunt non inesse, aut simpliciter, aut opposita, ut lineae aut rectum, aut obliquum, et numero aut par aut impar, est enim contrariorum, aut privatio, aut contradictio in eodem genere, ut par aut impar in numeris secundum quod consequitur; quare si necesse est affirmare, aut negare, necesse est et quae sunt per se, inesse.	So far then as concerns the sphere of connexions scientifically known in the unqualified sense of that term, all attributes which (within that sphere) are essential either in the sense that their subjects are contained in them, or in the sense that they are contained in their subjects, are necessary as well as consequentially connected with their subjects. For it is impossible for them not to inhere in their subjects either simply or in the qualified sense that one or other of a pair of opposites must inhere in the subject; e.g. in line must be either straightness or curvature, in number either oddness or evenness. For within a single identical genus the contrary of a given attribute is either its privative or its contradictory; e.g. within number what is not odd is even, inasmuch as within this sphere even is a necessary consequent of not-odd. So, since any given predicate must be either affirmed or denied of any subject, essential attributes must inhere in their subjects of necessity.
73b25 Τὸ μὲν οὖν κατὰ παντὸς καὶ καθ᾽ αὑτὸ διωρίσθω τὸν τρόπον τοῦτον·	De omni quidem, et per se, sed determinatum sit hoc modo.	Thus, then, we have established the distinction between the attribute which is ‘true in every instance’ and the ‘essential’ attribute.
73b27 καθόλου δὲ λέγω ὁ ἂν κατὰ παντός τε ὑπάρχηι καὶ καθ᾽ αὑτὸ καὶ ἧι αὐτό. φανερὸν ἄρα ὅτι ὅσα καθόλου, ἐξ ἀνάγκης ὑπάρχει τοῖς πράγμασιν.	Universale autem dico, quod cum de omni sit, et per se est, et secundum quod ipsum est. Manifestum igitur est quod quaecunque sunt universalia ex necessitate insunt rebus.	I term ‘commensurately universal’ an attribute which belongs to every instance of its subject, and to every instance essentially and as such; 73b28 from which it clearly follows that all commensurate universals inhere necessarily in their subjects.
73b29 τὸ καθ᾽ αὑτὸ δὲ καὶ ἧι αὐτὸ ταὐτόν, οἷον καθ᾽ αὑτὴν τῆι γραμμῆι ↵ ὑπάρχει στιγμὴ καὶ τὸ εὐθύ (καὶ γὰρ ἧι γραμμή), καὶ τῶι τριγώνωι ἧι τρίγωνον δύο ὀρθαί (καὶ γὰρ καθ᾽ αὑτὸ τὸ τρίγωνον δύο ὀρθαῖς ἴσον).	Per se autem et secundum quod ipsum est, ut per se lineae inest punctum, et rectitudo, et namque [inest enim] secundum quod linea est, et triangulo secundum quod triangulus est, insunt duo recti, etenim per se triangulus duobus rectis aequalis est.	The essential attribute, and the attribute that belongs to its subject as such, are identical. E.g. point and straight belong to line essentially, for they belong to line as such; and triangle as such has two right angles, for it is essentially equal to two right angles.
73b33 τὸ καθόλου δὲ ὑπάρχει τότε, ὅταν ἐπὶ τοῦ τυχόντος καὶ πρώτου δεικνύηται.	Universale autem est tunc, cum in quolibet, et primo monstretur,	An attribute belongs commensurately and universally to a subject when it can be shown to belong to any random instance of that subject and when the subject is the first thing to which it can be shown to belong.
73b34 οἷον τὸ δύο ὀρθὰς ἔχειν οὔτε τῶι σχήματί ἐστι καθόλου (καίτοι ἔστι δεῖξαι ↵ κατὰ σχήματος ὅτι δύο ὀρθὰς ἔχει, ἀλλ᾽ οὐ τοῦ τυχόντος σχήματος, οὐδὲ χρῆται τῶι τυχόντι σχήματι δεικνύς· τὸ γὰρ τετράγωνον σχῆμα μέν, οὐκ ἔχει δὲ δύο ὀρθαῖς ἴσασ) – τὸ δ᾽ ἰσοσκελὲς ἔχει μὲν τὸ τυχὸν δύο ὀρθαῖς ἴσας, ἀλλ᾽ οὐ πρῶτον, ἀλλὰ τὸ τρίγωνον πρότερον. ὁ τοίνυν τὸ τυχὸν πρῶτον δείκνυται δύο ὀρθὰς ἔχον ἢ ὁτιοῦν ἄλλο, τούτωι πρώτωι ὑπάρχει καθόλου,	ut duos rectos habere, neque cuilibet figurae inest universaliter, et tamen est monstrare de figura quod duos rectos habet, sed non de figura qualibet, nec utitur qualibet figura monstrans, quadrangulus enim figura quidem est, non habet autem duobus rectis aequales, sed isosceles habet quidem quodlibet duobus rectis aequales, sed non primum, sed triangulus prius. Quodcunque igitur primum monstratur duos habere rectos, aut quodcunque aliud, huic primo inest universale,	Thus, e.g. (1) the equality of its angles to two right angles is not a commensurately universal attribute of figure. For though it is possible to show that a figure has its angles equal to two right angles, this attribute cannot be demonstrated of any figure selected at haphazard, nor in demonstrating does one take a figure at random-a square is a figure but its angles are not equal to two right angles. On the other hand, any isosceles triangle has its angles equal to two right angles, yet isosceles triangle is not the primary subject of this attribute but triangle is prior. So whatever can be shown to have its angles equal to two right angles, or to possess any other attribute, in any random instance of itself and primarily-that is the first subject to which the predicate in question belongs commensurately and universally,
74a1 καὶ ἡ ἀπόδειξις καθ᾽ αὑτὸ τούτου καθόλου ἐστί, τῶν δ᾽ ἄλλων τρόπον τινὰ οὐ καθ᾽ αὑτό, οὐδὲ τοῦ ἰσοσκελοῦς οὐκ ἔστι καθόλου ἀλλ᾽ ἐπὶ πλέον.	et demonstratio per se huius universalis est, aliorum autem quodammodo, et non per se, neque isoscelis est universaliter, sed in plus.	and the demonstration, in the essential sense, of any predicate is the proof of it as belonging to this first subject commensurately and universally: while the proof of it as belonging to the other subjects to which it attaches is demonstration only in a secondary and unessential sense. Nor again (2) is equality to two right angles a commensurately universal attribute of isosceles; it is of wider application.
c5	CAPUT V. Quo pacto contingat allucinatio circa universale primum.	Chapter 5
74a4 Δεῖ δὲ μὴ λανθάνειν ὅτι πολλάκις συμβαίνει διαμαρ↵ τάνειν καὶ μὴ ὑπάρχειν τὸ δεικνύμενον πρῶτον καθόλου, ἧι δοκεῖ δείκνυσθαι καθόλου πρῶτον.	Oportet autem non latere quoniam multoties contingit peccare, et non esse quod demonstratur primum universale, secundum quod videtur demonstrari universale primum;	We must not fail to observe that we often fall into error because our conclusion is not in fact primary and commensurately universal in the sense in which we think we prove it so.
74a6 ἀπατώμεθα δὲ ταύτην τὴν ἀπάτην, ὅταν ἢ μηδὲν ἦι λαβεῖν ἀνώτερον παρὰ τὸ καθ᾽ ἕκαστον [ἢ τὰ καθ᾽ ἕκαστα], ἢ ἦι μέν, ἀλλ᾽ ἀνώνυμον ἦι ἐπὶ διαφόροις εἴδει πράγμασιν, ἢ τυγχάνηι ὂν ὡς ἐν μέρει ὅλον ↵ ἐφ᾽ ὧι δείκνυται· τοῖς γὰρ ἐν μέρει ὑπάρξει μὲν ἡ ἀπόδειξις, καὶ ἔσται κατὰ παντός, ἀλλ᾽ ὅμως οὐκ ἔσται τούτου πρώτου καθόλου ἡ ἀπόδειξις. λέγω δὲ τούτου πρώτου, ἧι τοῦτο, ἀπόδειξιν, ὅταν ἦι πρώτου καθόλου.	oberramus autem hac deceptione, cum aut aliud nihil sit accipere a superiori extra singulare, vel singularia, aut si sit quidem, sed innominatum sit in differentibus specie rebus, aut contingat esse sicut in parte, totum in quo monstratur: iis enim quae sunt in parte, inest quidem demonstratio, et erit de omni, sed tamen non huius erit primi universalis demonstratio, dico autem huius primi secundum quod huius demonstrationem, cum sit primi universalis.	We make this mistake (1) when the subject is an individual or individuals above which there is no universal to be found: (2) when the subjects belong to different species and there is a higher universal, but it has no name: (3) when the subject which the demonstrator takes as a whole is really only a part of a larger whole; for then the demonstration will be true of the individual instances within the part and will hold in every instance of it, yet the demonstration will not be true of this subject primarily and commensurately and universally. When a demonstration is true of a subject primarily and commensurately and universally, that is to be taken to mean that it is true of a given subject primarily and as such.
74a13 εἰ οὖν τις δείξειεν ὅτι αἱ ὀρθαὶ οὐ συμπίπτουσι, δόξειεν ἂν τούτου εἶναι ἡ ἀπόδειξις διὰ τὸ ↵ ἐπὶ πασῶν εἶναι τῶν ὀρθῶν. οὐκ ἔστι δέ, εἴπερ μὴ ὅτι ὡδὶ ἴσαι γίνεται τοῦτο, ἀλλ᾽ ἧι ὁπωσοῦν ἴσαι.	Si ergo aliquis monstrabit quidem quod recte non intercidant, videbitur utique huius esse demonstratio, propter id quod in omnibus est rectis. Non autem est, nisi quidem (quoniam sic aequales sint) fiat hoc, sed secundum quod quomodolibet aequales.	Case (3) may be thus exemplified. If a proof were given that perpendiculars to the same line are parallel, it might be supposed that lines thus perpendicular were the proper subject of the demonstration because being parallel is true of every instance of them. But it is not so, for the parallelism depends not on these angles being equal to one another because each is a right angle, but simply on their being equal to one another.
74a17 καὶ εἰ τρίγωνον μὴ ἦν ἄλλο ἢ ἰσοσκελές, ἧι ἰσοσκελὲς ἂν ἐδόκει ὑπάρχειν.	Et si triangulus non esset alius quam isosceles, secundum quod isosceles videretur utique inesse.	An example of (1) would be as follows: if isosceles were the only triangle, it would be thought to have its angles equal to two right angles qua isosceles.
74a18 καὶ τὸ ἀνάλογον ὅτι καὶ ἐναλλάξ, ἧι ἀριθμοὶ καὶ ἧι γραμμαὶ καὶ ἧι στερεὰ καὶ ἧι χρόνοι, ὥσπερ ἐδείκνυτό ποτε χωρίς, ἐνδε↵ χόμενόν γε κατὰ πάντων μιᾶι ἀποδείξει δειχθῆναι·	Et proportionale quod communicabiliter est, secundum quod numeri sunt, et secundum quod lineae, et secundum quod solida, et secundum quod tempora sunt, quemadmodum demonstratum est aliquando seorsum, contingens autem de omnibus una demonstratione monstrari,	An instance of (2) would be the law that proportionals alternate. Alternation used to be demonstrated separately of numbers, lines, solids, and durations, though it could have been proved of them all by a single demonstration.
ἀλλὰ διὰ τὸ μὴ εἶναι ὠνομασμένον τι ταῦτα πάντα ἓν, ἀριθμοί μήκη χρόνοι στερεά, καὶ εἴδει διαφέρειν ἀλλήλων, χωρὶς ἐλαμβάνετο. νῦν δὲ καθόλου δείκνυται·	sed propter id quod non est nominatum aliquid secundum quod haec omnia unum sunt, numeri, longitudines, tempora, solida, et specie differentia, seorsum ab invicem accepta sunt, nunc autem universale monstratur.	Because there was no single name to denote that in which numbers, lengths, durations, and solids are identical, and because they differed specifically from one another, this property was proved of each of them separately. To-day, however, the proof is commensurately universal,
οὐ γὰρ ἧι γραμμαὶ ἢ ἧι ἀριθμοὶ ὑπῆρχεν, ἀλλ᾽ ἧι τοδί, ὁ καθόλου ὑποτίθενται ↵ ὑπάρχειν.	Non enim secundum quod lineae sunt, aut secundum quod numeri, inerat, sed secundum quod hoc est, quod universale supponunt esse.	for they do not possess this attribute qua lines or qua numbers, but qua manifesting this generic character which they are postulated as possessing universally.
74a25 διὰ τοῦτο οὐδ᾽ ἄν τις δείξηι καθ᾽ ἕκαστον τὸ τρίγωνον ἀποδείξει ἢ μιᾶι ἢ ἑτέραι ὅτι δύο ὀρθὰς ἔχει ἕκαστον, τὸ ἰσόπλευρον χωρὶς καὶ τὸ σκαληνὲς καὶ τὸ ἰσοσκελές, οὔπω οἶδε τὸ τρίγωνον ὅτι δύο ὀρθαῖς, εἰ μὴ τὸν σοφιστικὸν τρόπον, οὐδὲ καθ᾽ ὅλου τριγώνου, οὐδ᾽ εἰ μηδὲν ἔστι παρὰ ταῦτα ↵ τρίγωνον ἕτερον. οὐ γὰρ ἧι τρίγωνον οἶδεν, οὐδὲ πᾶν τρίγωνον, ἀλλ᾽ ἢ κατ᾽ ἀριθμόν· κατ᾽ εἶδος δ᾽ οὐ πᾶν, καὶ εἰ μηδὲν ἔστιν ὁ οὐκ οἶδεν.	Propter hoc nec si aliquis monstret unumquemque triangulum demonstratione aut una, aut altera, quod duos rectos habet unusquisque, isopleuron seorsum, et scalenon, et isosceles, nondum cognovit triangulum quod duos rectos habet, nisi sophistico modo, neque universaliter triangulum, ne quidem si nullus est praeter haec triangulus alter, non enim secundum quod triangulus est, cognovit, neque omnem triangulum, sed secundum numerum, secundum speciem autem non omnem, et si nullus est quem non novit.	Hence, even if one prove of each kind of triangle that its angles are equal to two right angles, whether by means of the same or different proofs; still, as long as one treats separately equilateral, scalene, and isosceles, one does not yet know, except sophistically, that triangle has its angles equal to two right angles, nor does one yet know that triangle has this property commensurately and universally, even if there is no other species of triangle but these. For one does not know that triangle as such has this property, nor even that ‘all’ triangles have it-unless ‘all’ means ‘each taken singly’: if ‘all’ means ‘as a whole class’, then, though there be none in which one does not recognize this property, one does not know it of ‘all triangles’.
74a33 Πότ᾽ οὖν οὐκ οἶδε καθόλου, καὶ πότ᾽ οἶδεν ἁπλῶς; δῆλον δὴ ὅτι εἰ ταὐτὸν ἦν τριγώνωι εἶναι καὶ ἰσοπλεύρωι ἢ ἑκάστωι ἢ πᾶσιν. εἰ δὲ μὴ ταὐτὸν ἀλλ᾽ ἕτερον, ↵ ὑπάρχει δ᾽ ἧι τρίγωνον, οὐκ οἶδεν.	Quando ergo non novit universaliter, et quando novit simpliciter, manifestum est, quoniam si idem erit triangulo esse, et isopleuro, aut unicuique, aut omnibus, si vero non idem, sed alterum, est autem secundum quod est triangulus, non novit.	When, then, does our knowledge fail of commensurate universality, and when it is unqualified knowledge? If triangle be identical in essence with equilateral, i.e. with each or all equilaterals, then clearly we have unqualified knowledge: if on the other hand it be not, and the attribute belongs to equilateral qua triangle; then our knowledge fails of commensurate universality.
74a35 πότερον δ᾽ ἧι τρίγωνον ἢ ἧι ἰσοσκελὲς ὑπάρχει; καὶ πότε κατὰ τοῦθ᾽ ὑπάρχει πρῶτον; καὶ καθόλου τίνος ἡ ἀπόδειξις; δῆλον ὅτι ὅταν ἀφαιρουμένων ὑπάρχηι πρώτωι.	Utrum autem secundum quod est triangulus, aut secundum quod est isosceles ipsit, et quando de hoc est primum et universale, cuius est demonstratio, manifestum est, quando remotis insit primum,	‘But’, it will be asked, ‘does this attribute belong to the subject of which it has been demonstrated qua triangle or qua isosceles? What is the point at which the subject. to which it belongs is primary? (i.e. to what subject can it be demonstrated as belonging commensurately and universally?)’ Clearly this point is the first term in which it is found to inhere as the elimination of inferior differentiae proceeds.
οἷον τῶι ἰσοσκελεῖ χαλκῶι τριγώνωι ὑπάρξουσι δύο ὀρθαί, ἀλλὰ καὶ τοῦ χαλκοῦν εἶναι ἀφαιρεθέντος καὶ τοῦ ἰσοσκελές. ἀλλ᾽ οὐ τοῦ σχήματος ἢ πέρατος. ἀλλ᾽ οὐ πρώτων. τίνος οὖν πρώτου; εἰ δὴ τριγώνου, κατὰ τοῦτο ὑπάρχει καὶ τοῖς ἄλλοις, καὶ τούτου καθόλου ἐστὶν ἡ ἀπόδειξις.	ut isosceli aeneo triangulo insunt duo recti, sed aeneum esse remoto et isoscele, sed non figura aut termino, sed non primis. Quo igitur primo? si itaque triangulo est, et secundum hoc inest, et aliis, et huius universalis est demonstratio.	Thus the angles of a brazen isosceles triangle are equal to two right angles: but eliminate brazen and isosceles and the attribute remains. ‘But’-you may say-’eliminate figure or limit, and the attribute vanishes.’ True, but figure and limit are not the first differentiae whose elimination destroys the attribute. ‘Then what is the first?’ If it is triangle, it will be in virtue of triangle that the attribute belongs to all the other subjects of which it is predicable, and triangle is the subject to which it can be demonstrated as belonging commensurately and universally.
c6	CAPUT VI. Demonstrationem ex necessariis et propositionibus per se esse.	Chapter 6
6 74b5 Εἰ οὖν ἐστιν ἡ ἀποδεικτικὴ ἐπιστήμη ἐξ ἀναγκαίων ἀρχῶν (ὁ γὰρ ἐπίσταται, οὐ δυνατὸν ἄλλως ἔχειν),	Si igitur est demonstrativa scientia, et ex necessariis principiis, quod enim scitur non potest se aliter habere.	Demonstrative knowledge must rest on necessary basic truths; for the object of scientific knowledge cannot be other than it is.
74b6 τὰ δὲ καθ᾽ αὑτὰ ὑπάρχοντα ἀναγκαῖα τοῖς πράγμασιν (τὰ μὲν γὰρ ἐν τῶι τί ἐστιν ὑπάρχει· τοῖς δ᾽ αὐτὰ ἐν τῶι τί ἐστιν ὑπάρχει κατηγορουμένοις αὐτῶν, ὧν θάτερον τῶν ἀντικειμένων ἀνάγκη ↵ ὑπάρχειν),	Quae autem per se sunt necessario insunt rebus, haec enim insunt in eo quod quid est, quibusdam autem haec insunt in eo quod quid est, praedicantibus de ipsis, quorum alterum oppositorum necesse est inesse.	Now attributes attaching essentially to their subjects attach necessarily to them: for essential attributes are either elements in the essential nature of their subjects, or contain their subjects as elements in their own essential nature. (The pairs of opposites which the latter class includes are necessary because one member or the other necessarily inheres.)
φανερὸν ὅτι ἐκ τοιούτων τινῶν ἂν εἴη ὁ ἀποδεικτικὸς συλλογισμός· ἅπαν γὰρ ἢ οὕτως ὑπάρχει ἢ κατὰ συμβεβηκός, τὰ δὲ συμβεβηκότα οὐκ ἀναγκαῖα.	Manifestum est igitur quod ex huiusmodi quibusdam utique fit demonstrativus syllogismus, omne enim aut sic inest, aut secundum accidens, accidentia autem necessaria non sunt.	It follows from this that premisses of the demonstrative syllogism must be connexions essential in the sense explained: for all attributes must inhere essentially or else be accidental, and accidental attributes are not necessary to their subjects.
74b13 Η δὴ οὕτω λεκτέον, ἢ ἀρχὴν θεμένοις ὅτι ἡ ἀπόδειξις ἀναγκαίων ἐστί, καὶ εἰ ἀποδέδεικται, οὐχ οἷόν τ᾽ ἄλλως ↵ ἔχειν· ἐξ ἀναγκαίων ἄρα δεῖ εἶναι τὸν συλλογισμόν. ἐξ ἀληθῶν μὲν γὰρ ἔστι καὶ μὴ ἀποδεικνύντα συλλογίσασθαι, ἐξ ἀναγκαίων δ᾽ οὐκ ἔστιν ἀλλ᾽ ἢ ἀποδεικνύντα· τοῦτο γὰρ ἤδη ἀποδείξεώς ἐστιν.	Aut igitur sic dicendum, aut principium ponentibus quod demonstratio necessaria sit, et si demonstretur non aliter habere posse, ex necessariis igitur oportet esse syllogismum, ex veris quidem est, et non demonstrantem syllogizare, ex necessariis autem non est, sed aut demonstrantem, hoc enim proprium iam demonstrationis est.	We must either state the case thus, or else premise that the conclusion of demonstration is necessary and that a demonstrated conclusion cannot be other than it is, and then infer that the conclusion must be developed from necessary premisses. For though you may reason from true premisses without demonstrating, yet if your premisses are necessary you will assuredly demonstrate-in such necessity you have at once a distinctive character of demonstration.
74b18 σημεῖον δ᾽ ὅτι ἡ ἀπόδειξις ἐξ ἀναγκαίων, ὅτι καὶ τὰς ἐνστάσεις οὕτω φέρομεν πρὸς τοὺς οἰομένους ἀπο↵ δεικνύναι, ὅτι οὐκ ἀνάγκη, ἂν οἰώμεθα ἢ ὅλως ἐνδέχεσθαι ἄλλως ἢ ἕνεκά γε τοῦ λόγου.	Signum autem est quod demonstratio ex necessariis sit, quoniam et instantias sic ferimus ad opinantes demonstrare. Quoniam non sit necesse si opinamur, aut omnino contingere aliter, aut orationis causa.	That demonstration proceeds from necessary premisses is also indicated by the fact that the objection we raise against a professed demonstration is that a premiss of it is not a necessary truth-whether we think it altogether devoid of necessity, or at any rate so far as our opponent’s previous argument goes.
74b22 δῆλον δ᾽ ἐκ τούτων καὶ ὅτι εὐήθεις οἱ λαμβάνειν οἰόμενοι καλῶς τὰς ἀρχάς, ἐὰν ἔνδοξος ἦι ἡ πρότασις καὶ ἀληθής, οἷον οἱ σοφισταὶ ὅτι τὸ ἐπίστασθαι τὸ ἐπιστήμην ἔχειν. οὐ γὰρ τὸ ἔνδοξον ἡμῖν ἀρχή ἐστιν, ↵ ἀλλὰ τὸ πρῶτον τοῦ γένους περὶ ὁ δείκνυται· καὶ τἀληθὲς οὐ πᾶν οἰκεῖον.	Manifestum autem ex iis est, et quoniam stulti qui opinati sunt accipere bene principia, si probabilis sit propositio, et vera, ut sophistae quoniam scire, scientiam est habere, non enim quod probabile est aut non, principium est, sed primum in genere circa quod demonstratur, et verum non omne, proprium.	This shows how naive it is to suppose one’s basic truths rightly chosen if one starts with a proposition which is (1) popularly accepted and (2) true, such as the sophists’ assumption that to know is the same as to possess knowledge. For (1) popular acceptance or rejection is no criterion of a basic truth, which can only be the primary law of the genus constituting the subject matter of the demonstration; and (2) not all truth is ‘appropriate’.
74b27 ὅτι δ᾽ ἐξ ἀναγκαίων εἶναι δεῖ τὸν συλλογισμόν, φανερὸν καὶ ἐκ τῶνδε. εἰ γὰρ ὁ μὴ ἔχων λόγον τοῦ διὰ τί οὔσης ἀποδείξεως οὐκ ἐπιστήμων, εἴη δ᾽ ἂν ὥστε τὸ Α κατὰ τοῦ Γ ἐξ ἀνάγκης ὑπάρχειν, τὸ δὲ Β τὸ μέσον, δι᾽ ↵ οὗ ἀπεδείχθη, μὴ ἐξ ἀνάγκης, οὐκ οἶδε διότι. οὐ γάρ ἐστι τοῦτο διὰ τὸ μέσον· τὸ μὲν γὰρ ἐνδέχεται μὴ εἶναι, τὸ δὲ συμπέρασμα ἀναγκαῖον.	Quod autem ex necessariis oportet esse syllogismum, manifestum ex his est, si enim non est habens rationem propter quid existente demonstratione, non est sciens, sit autem utique ut quod a, de c ex necessitate esse, b autem medium per quod demonstratum est non ex necessitate, non scivit propter quod, non enim est hoc, propter medium, hoc quidem contingit non inesse, conclusio autem necessaria.	A further proof that the conclusion must be the development of necessary premisses is as follows. Where demonstration is possible, one who can give no account which includes the cause has no scientific knowledge. If, then, we suppose a syllogism in which, though A necessarily inheres in C, yet B, the middle term of the demonstration, is not necessarily connected with A and C, then the man who argues thus has no reasoned knowledge of the conclusion, since this conclusion does not owe its necessity to the middle term; for though the conclusion is necessary, the mediating link is a contingent fact.
74b32 ἔτι εἴ τις μὴ οἶδε νῦν ἔχων τὸν λόγον καὶ σωιζόμενος, σωιζομένου τοῦ πράγματος, μὴ ἐπιλελησμένος, οὐδὲ πρότερον ἤιδει. φθαρείη δ᾽ ἂν τὸ μέσον, εἰ μὴ ↵ ἀναγκαῖον, ὥστε ἕξει μὲν τὸν λόγον σωιζόμενος σωιζομένου τοῦ πράγματος, οὐκ οἶδε δέ. οὐδ᾽ ἄρα πρότερον ἤιδει. εἰ δὲ μὴ ἔφθαρται, ἐνδέχεται δὲ φθαρῆναι, τὸ συμβαῖνον ἂν εἴη δυνατὸν καὶ ἐνδεχόμενον. ἀλλ᾽ ἔστιν ἀδύνατον οὕτως ἔχοντα εἰδέναι.	Amplius si aliquis nescit, nunc habens rationem, et salvatus est, et salva re, nec oblitus est neque prius scivit, corrumpetur autem utique medium nisi sit necessarium. Quare habebit quidem rationem salvus, salva re, nescit autem, nec ergo prius scivit, si vero non corruptum est, contingit autem corrumpi quod accidit, utique erit possibile, et contingens, sed est impossibile, sic se habentem scire.	Or again, if a man is without knowledge now, though he still retains the steps of the argument, though there is no change in himself or in the fact and no lapse of memory on his part; then neither had he knowledge previously. But the mediating link, not being necessary, may have perished in the interval; and if so, though there be no change in him nor in the fact, and though he will still retain the steps of the argument, yet he has not knowledge, and therefore had not knowledge before. Even if the link has not actually perished but is liable to perish, this situation is possible and might occur. But such a condition cannot be knowledge.
75a1 Ὅταν μὲν οὖν τὸ συμπέρασμα ἐξ ἀνάγκης ἦι, οὐδὲν κωλύει τὸ μέσον μὴ ἀναγκαῖον εἶναι δι᾽ οὗ ἐδείχθη (ἔστι γὰρ τὸ ἀναγκαῖον καὶ μὴ ἐξ ἀναγκαίων συλλογίσασθαι, ὥσπερ καὶ ἀληθὲς μὴ ἐξ ἀληθῶν)· ὅταν δὲ τὸ μέσον ἐξ ἀνάγκης, ↵ καὶ τὸ συμπέρασμα ἐξ ἀνάγκης, ὥσπερ καὶ ἐξ ἀληθῶν ἀληθὲς ἀεί (ἔστω γὰρ τὸ Α κατὰ τοῦ Β ἐξ ἀνάγκης, καὶ τοῦτο κατὰ τοῦ Γ· ἀναγκαῖον τοίνυν καὶ τὸ Α τῶι Γ ὑπάρχειν)· ὅταν δὲ μὴ ἀναγκαῖον ἦι τὸ συμπέρασμα, οὐδὲ τὸ μέσον ἀναγκαῖον οἷόν τ᾽ εἶναι (ἔστω γὰρ τὸ Α τῶι Γ μὴ ἐξ ἀνάγ↵ κης ὑπάρχειν, τῶι δὲ Β, καὶ τοῦτο τῶι Γ ἐξ ἀνάγκης· καὶ τὸ Α ἄρα τῶι Γ ἐξ ἀνάγκης ὑπάρξει· ἀλλ᾽ οὐχ ὑπέκειτο).	Cum igitur conclusio quidem ex necessitate est, nihil prohibet medium non esse necessarium, per quod monstrata est, est enim necessarium et ex non necessariis syllogizare, sit et verum ex non veris. Cum autem medium ex necessitate est, et conclusio est ex necessitate, sicut ex veris verum est semper. Sit enim a de b ex necessitate, et hoc de c, necesse est ergo et a c inesse, sed cum non ex necessitate sit conclusio, neque medium necessarium esse possibile est, sit enim a in c non ex necessitate inesse, in b autem a, et hoc in c ex necessitate erit, sed non esse supponebatur.	When the conclusion is necessary, the middle through which it was proved may yet quite easily be non-necessary. You can in fact infer the necessary even from a non-necessary premiss, just as you can infer the true from the not true. On the other hand, when the middle is necessary the conclusion must be necessary; just as true premisses always give a true conclusion. Thus, if A is necessarily predicated of B and B of C, then A is necessarily predicated of C. But when the conclusion is non-necessary the middle cannot be necessary either. Thus: let A be predicated non-necessarily of C but necessarily of B, and let B be a necessary predicate of C; then A too will be a necessary predicate of C, which by hypothesis it is not.
75a13 Ἐπεὶ τοίνυν εἰ ἐπίσταται ἀποδεικτικῶς, δεῖ ἐξ ἀνάγκης ὑπάρχειν, δῆλον ὅτι καὶ διὰ μέσου ἀναγκαίου δεῖ ἔχειν τὴν ἀπόδειξιν· ἢ οὐκ ἐπιστήσεται οὔτε διότι οὔτε ὅτι ἀνάγκη ἐκεῖνο εἶ↵ ναι, ἀλλ᾽ ἢ οἰήσεται οὐκ εἰδώς, ἐὰν ὑπολάβηι ὡς ἀναγκαῖον τὸ μὴ ἀναγκαῖον, ἢ οὐδ᾽ οἰήσεται, ὁμοίως ἐάν τε τὸ ὅτι εἰδῆι διὰ μέσων ἐάν τε τὸ διότι καὶ δι᾽ ἀμέσων.	Quoniam igitur si scit demonstrative, oportet ex necessitate inesse, manifestum quoniam et per medium necessarium oportet habere demonstrationem, aut non sciet, neque enim propter quid, neque quia, quare necesse est illud esse, sed aut opinabitur, nesciens, si opinabitur non necessarium tanquam necessarium, aut neque opinabitur similiter, sive quoniam sciat per media, sive propter quid, et per immediata.	To sum up, then: demonstrative knowledge must be knowledge of a necessary nexus, and therefore must clearly be obtained through a necessary middle term; otherwise its possessor will know neither the cause nor the fact that his conclusion is a necessary connexion. Either he will mistake the non-necessary for the necessary and believe the necessity of the conclusion without knowing it, or else he will not even believe it-in which case he will be equally ignorant, whether he actually infers the mere fact through middle terms or the reasoned fact and from immediate premisses.
75a18 Τῶν δὲ συμβεβηκότων μὴ καθ᾽ αὑτά, ὃν τρόπον διωρίσθη τὰ καθ᾽ αὑτά, οὐκ ἔστιν ἐπιστήμη ἀποδεικτική. οὐ γὰρ ↵ ἔστιν ἐξ ἀνάγκης δεῖξαι τὸ συμπέρασμα· τὸ συμβεβηκὸς γὰρ ἐνδέχεται μὴ ὑπάρχειν· περὶ τοῦ τοιούτου γὰρ λέγω συμβεβηκότος.	Accidentium autem non per se quo modo definitum est, per se quidem non est scientia demonstrativa, non enim est ex necessitate monstrare conclusionem, accidens enim contingit non esse, de tali autem dico accidente.	Of accidents that are not essential according to our definition of essential there is no demonstrative knowledge; for since an accident, in the sense in which I here speak of it, may also not inhere, it is impossible to prove its inherence as a necessary conclusion.
75a21 καίτοι ἀπορήσειεν ἄν τις ἴσως τίνος ἕνεκα ταῦτα δεῖ ἐρωτᾶν περὶ τούτων, εἰ μὴ ἀνάγκη τὸ συμπέρασμα εἶναι· οὐδὲν γὰρ διαφέρει εἴ τις ἐρόμενος τὰ τυχόντα εἶτα εἴπειεν τὸ ↵ συμπέρασμα.	Et tamen ambiget fortasse aliquis, cuius causa haec oportet interrogare de his, si non necesse est conclusionem esse, nihil enim differt si aliquis interrogatus contingentia, postea dicat conclusionem,	A difficulty, however, might be raised as to why in dialectic, if the conclusion is not a necessary connexion, such and such determinate premisses should be proposed in order to deal with such and such determinate problems. Would not the result be the same if one asked any questions whatever and then merely stated one’s conclusion?
75a24 δεῖ δ᾽ ἐρωτᾶν οὐχ ὡς ἀναγκαῖον εἶναι διὰ τὰ ἠρωτημένα, ἀλλ᾽ ὅτι λέγειν ἀνάγκη τῶι ἐκεῖνα λέγοντι, καὶ ἀληθῶς λέγειν, ἐὰν ἀληθῶς ἦι ὑπάρχοντα.	oportet autem interrogare non tanquam necessarium esse propter interrogata, sed quod dicere necesse est illa dicenti, et vere dicere si verae sunt quae insunt.	The solution is that determinate questions have to be put, not because the replies to them affirm facts which necessitate facts affirmed by the conclusion, but because these answers are propositions which if the answerer affirm, he must affirm the conclusion and affirm it with truth if they are true.
	CAPUT VII. Demonstrationes ex iis quae per se sunt et ex perpetuis esse.
75a28 Ἐπεὶ δ᾽ ἐξ ἀνάγκης ὑπάρχει περὶ ἕκαστον γένος ὅσα καθ᾽ αὑτὰ ὑπάρχει καὶ ἧι ἕκαστον, φανερὸν ὅτι περὶ τῶν ↵ καθ᾽ αὑτὰ ὑπαρχόντων αἱ ἐπιστημονικαὶ ἀποδείξεις καὶ ἐκ τῶν τοιούτων εἰσίν.	Quoniam autem ex necessitate sunt circa unum quodque genus quaecunque per se sunt, et secundum quod unum quodque est, manifestum est quoniam de iis quae sunt per se, scientificae demonstrationes, et ex talibus sunt.	Since it is just those attributes within every genus which are essential and possessed by their respective subjects as such that are necessary it is clear that both the conclusions and the premisses of demonstrations which produce scientific knowledge are essential.
τὰ μὲν γὰρ συμβεβηκότα οὐκ ἀναγκαῖα, ὥστ᾽ οὐκ ἀνάγκη τὸ συμπέρασμα εἰδέναι διότι ὑπάρχει, οὐδ᾽ εἰ ἀεὶ εἴη, μὴ καθ᾽ αὑτὸ δέ, οἷον οἱ διὰ σημείων συλλογισμοί. τὸ γὰρ καθ᾽ αὑτὸ οὐ καθ᾽ αὑτὸ ἐπιστήσεται, οὐδὲ διότι ↵ (τὸ δὲ διότι ἐπίστασθαί ἐστι τὸ διὰ τοῦ αἰτίου ἐπίστασθαι). δι᾽ αὑτὸ ἄρα δεῖ καὶ τὸ μέσον τῶι τρίτωι καὶ τὸ πρῶτον τῶι μέσωι ὑπάρχειν.	Accidentia enim non necessaria sunt. Quare non necessarium conclusionem scire propter quid sit, neque si semper sint, non per se autem, ut sunt per signa syllogismi, hoc enim per se, non per se sciet, neque propter quod. Propter quid autem scire est per causam scire. Propter hoc ipsum ergo oportet et tertio medium, et primum medio inesse.	For accidents are not necessary: and, further, since accidents are not necessary one does not necessarily have reasoned knowledge of a conclusion drawn from them (this is so even if the accidental premisses are invariable but not essential, as in proofs through signs; for though the conclusion be actually essential, one will not know it as essential nor know its reason); but to have reasoned knowledge of a conclusion is to know it through its cause. We may conclude that the middle must be consequentially connected with the minor, and the major with the middle.
c7		Chapter 7
75a38 Οὐκ ἄρα ἔστιν ἐξ ἄλλου γένους μεταβάντα δεῖξαι, οἷον τὸ γεωμετρικὸν ἀριθμητικῆι.	Non ergo est ex alio genere descendentem demonstrare, ut geometricum in arithmeticam.	It follows that we cannot in demonstrating pass from one genus to another. We cannot, for instance, prove geometrical truths by arithmetic.
75a39 τρία γάρ ἐστι τὰ ἐν ταῖς ἀπο↵ δείξεσιν, ἓν μὲν τὸ ἀποδεικνύμενον, τὸ συμπέρασμα (τοῦτο δ᾽ ἐστὶ τὸ ὑπάρχον γένει τινὶ καθ᾽ αὑτό), ἓν δὲ τὰ ἐξιώματα (ἀξιώματα δ᾽ ἐστὶν ἐξ ὧν)· τρίτον τὸ γένος τὸ ὑποκείμενον, οὗ τὰ πάθη καὶ τὰ καθ᾽ αὑτὰ συμβεβηκότα δηλοῖ ἡ ἀπόδειξις.	Tria enim sunt in demonstrationibus: unum quidem quae demonstratur conclusio, hoc autem est quod inest alicui generi per se; unum autem dignitates, dignitates autem sunt ex quibus est demonstratio; tertium autem genus subiectum, cuius passiones, et per se accidentia ostendit demonstratio.	For there are three elements in demonstration: (1) what is proved, the conclusion-an attribute inhering essentially in a genus; (2) the axioms, i.e. axioms which are premisses of demonstration; (3) the subject-genus whose attributes, i.e. essential properties, are revealed by the demonstration.
75b2 ἐξ ὧν μὲν οὖν ἡ ἀπόδειξις, ἐνδέχεται τὰ αὐτὰ εἶναι· ὧν δὲ τὸ γένος ἕτερον, ὥσπερ ἀριθμητικῆς καὶ γεωμετρίας, οὐκ ἔστι τὴν ἀριθμητικὴν ἀπόδειξιν ἐφαρμόσαι ἐπὶ ↵ τὰ τοῖς μεγέθεσι συμβεβηκότα, εἰ μὴ τὰ μεγέθη ἀριθμοί εἰσι· τοῦτο δ᾽ ὡς ἐνδέχεται ἐπί τινων, ὕστερον λεχθήσεται. ἡ δ᾽ ἀριθμητικὴ ἀπόδειξις ἀεὶ ἔχει τὸ γένος περὶ ὁ ἡ ἀπόδειξις,	Ex quibus igitur demonstratio fit, contingit eadem esse. Quorum autem genus alterum est, sicut arithmeticae et geometriae, non est arithmeticam demonstrationem convenire in magnitudinibus accidentia, nisi magnitudines numeri sint (hoc autem quo modo contingit in quibusdam posterius dicetur), sed arithmetica demonstratio semper habet genus circa quod fit demonstratio, et aliae similiter.	The axioms which are premisses of demonstration may be identical in two or more sciences: but in the case of two different genera such as arithmetic and geometry you cannot apply arithmetical demonstration to the properties of magnitudes unless the magnitudes in question are numbers. How in certain cases transference is possible I will explain later. Arithmetical demonstration and the other sciences likewise possess, each of them, their own genera;
75b8 καὶ αἱ ἄλλαι ὁμοίως. ὥστ᾽ ἢ ἁπλῶς ἀνάγκη τὸ αὐτὸ εἶναι γένος ἢ πῆι, εἰ μέλλει ἡ ἀπόδειξις μεταβαίνειν.	Quare aut simpliciter necesse est idem esse genus, aut aliquo modo, si debet demonstratio descendere,	so that if the demonstration is to pass from one sphere to another, the genus must be either absolutely or to some extent the same.
75b10 ↵ ἄλλως δ᾽ ὅτι ἀδύνατον, δῆλον· ἐκ γὰρ τοῦ αὐτοῦ γένους ἀνάγκη τὰ ἄκρα καὶ τὰ μέσα εἶναι. εἰ γὰρ μὴ καθ᾽ αὑτά, συμβεβηκότα ἔσται.	aliter autem quoniam impossibile, manifestum est, ex eodem enim genere necesse est ultima, et media esse, si namque non sunt per se, accidentia erunt.	If this is not so, transference is clearly impossible, because the extreme and the middle terms must be drawn from the same genus: otherwise, as predicated, they will not be essential and will thus be accidents.
75b13 διὰ τοῦτο τῆι γεωμετρίαι οὐκ ἔστι δεῖξαι ὅτι τῶν ἐναντίων μία ἐπιστήμη, ἀλλ᾽ οὐδ᾽ ὅτι οἱ δύο κύβοι κύβος· οὐδ᾽ ἄλληι ἐπιστήμηι τὸ ἑτέρας, ἀλλ᾽ ἢ ὅσα οὕτως ↵ ἔχει πρὸς ἄλληλα ὥστ᾽ εἶναι θάτερον ὑπὸ θάτερον, οἷον τὰ ὀπτικὰ πρὸς γεωμετρίαν καὶ τὰ ἁρμονικὰ πρὸς ἀριθμητικήν.	Propter hoc geometriae non est demonstrare quod contrariorum una sit scientia, sed neque quod duo cubi sit unus cubus, neque alterius scientiae, quod alterius est, sed aut quaecunque sic se habent ad invicem, ut quod alterum sit sub altero, ut perspectiva ad geometriam, et consonantia ad aritmethicam,	That is why it cannot be proved by geometry that opposites fall under one science, nor even that the product of two cubes is a cube. Nor can the theorem of any one science be demonstrated by means of another science, unless these theorems are related as subordinate to superior (e.g. as optical theorems to geometry or harmonic theorems to arithmetic).
75b17 οὐδ᾽ εἴ τι ὑπάρχει ταῖς γραμμαῖς μὴ ἧι γραμμαὶ καὶ ἧι ἐκ τῶν ἀρχῶν τῶν ἰδίων, οἷον εἰ καλλίστη τῶν γραμμῶν ἡ εὐθεῖα ἢ εἰ ἐναντίως ἔχει τῆι περιφερεῖ· οὐ γὰρ ἧι τὸ ↵ ἴδιον γένος αὐτῶν, ὑπάρχει, ἀλλ᾽ ἧι κοινόν τι.	neque si aliquid inest lineis non secundum quod lineae sunt, et non in quantum ex propriis principiis, ut si pulcherrima linearum recta est, aut si contrario modo se habeat circularis, non enim secundum quod proprium ipsarum genus est, sed in quantum commune quoddam.	Geometry again cannot prove of lines any property which they do not possess qua lines, i.e. in virtue of the fundamental truths of their peculiar genus: it cannot show, for example, that the straight line is the most beautiful of lines or the contrary of the circle; for these qualities do not belong to lines in virtue of their peculiar genus, but through some property which it shares with other genera.
c8		Chapter 8
75b21 Φανερὸν δὲ καὶ ἐὰν ὦσιν αἱ προτάσεις καθόλου ἐξ ὧν ὁ συλλογισμός, ὅτι ἀνάγκη καὶ τὸ συμπέρασμα ἀΐδιον εἶναι τῆς τοιαύτης ἀποδείξεως καὶ τῆς ἁπλῶς εἰπεῖν ἀποδείξεως.	Manifestum autem est, et si sint propositiones universales ex quibus est syllogismus, quod necesse est et conclusionem perpetuam esse huiusmodi demonstrationis, et simpliciter (ut est dicere) demonstrationis. Non est ergo demonstratio corruptibilium, neque scientia simpliciter,	t is also clear that if the premisses from which the syllogism proceeds are commensurately universal, the conclusion of such i.e. in the unqualified sense-must also be eternal. Therefore no attribute can be demonstrated nor known by strictly scientific knowledge to inhere in perishable things.
75b24 οὐκ ἔστιν ἄρα ἀπόδειξις τῶν φθαρτῶν οὐδ᾽ ἐπιστήμη ↵ ἁπλῶς, ἀλλ᾽ οὕτως ὥσπερ κατὰ συμβεβηκός, ὅτι οὐ καθ᾽ ὅλου αὐτοῦ ἐστιν ἀλλὰ ποτὲ καὶ πώς. ὅταν δ᾽ ἦι, ἀνάγκη τὴν ἑτέραν μὴ καθόλου εἶναι πρότασιν καὶ φθαρτήν – φθαρτὴν μὲν ὅτι ἔσται καὶ τὸ συμπέρασμα οὔσης, μὴ καθόλου δὲ ὅτι τῶι μὲν ἔσται τῶι δ᾽ οὐκ ἔσται ἐφ᾽ ὧν – ὥστ᾽ οὐκ ἔστι συλ↵ λογίσασθαι καθόλου, ἀλλ᾽ ὅτι νῦν.	sed sic est sicut secundum accidens, et non universalis ipsius est, sed aliquando et sic; cum autem ita sit, necesse est alteram non universalem esse propositionem et corruptibilem: corruptibilem quidem, quoniam et conclusio est; non universalem autem, quod hoc quidem erit; hoc autem non erit ex quibus est, quare non est syllogizare universaliter, sed quoniam nunc est.	The proof can only be accidental, because the attribute’s connexion with its perishable subject is not commensurately universal but temporary and special. If such a demonstration is made, one premiss must be perishable and not commensurately universal (perishable because only if it is perishable will the conclusion be perishable; not commensurately universal, because the predicate will be predicable of some instances of the subject and not of others); so that the conclusion can only be that a fact is true at the moment-not commensurately and universally.
75b32 ὁμοίως δ᾽ ἔχει καὶ περὶ ὁρισμούς, ἐπείπερ ἐστὶν ὁ ὁρισμὸς ἢ ἀρχὴ ἀποδείξεως ἢ ἀπόδειξις θέσει διαφέρουσα ἢ συμπέρασμά τι ἀποδείξεως. αἱ δὲ τῶν πολλάκις γινομένων ἀποδείξεις καὶ ἐπιστῆμαι, οἷον σελήνης ἐκλείψεως, δῆλον ὅτι ἧι μὲν τοιοῦδ᾽ εἰσίν, ἀεὶ εἰσίν, ↵ ἧι δ᾽ οὐκ ἀεί, κατὰ μέρος εἰσίν. ὥσπερ δ᾽ ἡ ἔκλειψις, ὡσαύτως τοῖς ἄλλοις.	Similiter se habet de definitione, quoniam quidem est definitio aut principium demonstrationis, aut demonstratio positione differens, aut conclusio quaedam demonstrationis. Eorum autem quae saepe fiunt, sunt demonstrationes et scientiae (ut lunae defectus), manifestum est quoniam secundum quod huiusmodi sunt, semper sunt, in quantum autem non semper secundum partem sunt. Sicut autem defectus est, similiter et in aliis.	The same is true of definitions, since a definition is either a primary premiss or a conclusion of a demonstration, or else only differs from a demonstration in the order of its terms. Demonstration and science of merely frequent occurrences-e.g. of eclipse as happening to the moon-are, as such, clearly eternal: whereas so far as they are not eternal they are not fully commensurate. Other subjects too have properties attaching to them in the same way as eclipse attaches to the moon.
c9		Chapter 9
75b37 Ἐπεὶ δὲ φανερὸν ὅτι ἕκαστον ἀποδεῖξαι οὐκ ἔστιν ἀλλ᾽ ἢ ἐκ τῶν ἑκάστου ἀρχῶν, ἂν τὸ δεικνύμενον ὑπάρχηι ἧι ἐκεῖνο, οὐκ ἔστι τὸ ἐπίστασθαι τοῦτο, ἂν ἐξ ἀληθῶν καὶ ἀναποδείκτων ↵ δειχθῆι καὶ ἀμέσων.	Quoniam autem manifestum est quod demonstrare unumquodque non est, sed aut ex unoquoque principiorum, si id quod demonstratur sit secundum quod est illud, non autem est scire hoc quidem, si ex veris et indemonstrabilibus monstretur, et immediatis.	It is clear that if the conclusion is to show an attribute inhering as such, nothing can be demonstrated except from its ‘appropriate’ basic truths. Consequently a proof even from true, indemonstrable, and immediate premisses does not constitute knowledge.
75b40 ἔστι γὰρ οὕτω δεῖξαι, ὥσπερ Βρύσων τὸν τετραγωνισμόν. κατὰ κοινόν τε γὰρ δεικνύουσιν οἱ τοιοῦτοι λόγοι, ὁ καὶ ἑτέρωι ὑπάρξει· διὸ καὶ ἐπ᾽ ἄλλων ἐφαρμόττουσιν οἱ λόγοι οὐ συγγενῶν. οὐκοῦν οὐχ ἧι ἐκεῖνο ἐπίσταται, ἀλλὰ κατὰ συμβεβηκός· οὐ γὰρ ἂν ἐφήρμοττεν ἡ ἀπόδειξις καὶ ἐπ᾽ ἄλλο γένος.	Est enim sic monstrare sicut Bryson tetragonismon; secundum commune enim monstrant rationes huiusmodi quod et alteri inest, unde et in aliis conveniunt hae rationes non congeneis. Non itaque secundum illud scit, sed secundum accidens, non enim convenit demonstratio et in aliud genus.	Such proofs are like Bryson’s method of squaring the circle; for they operate by taking as their middle a common character-a character, therefore, which the subject may share with another-and consequently they apply equally to subjects different in kind. They therefore afford knowledge of an attribute only as inhering accidentally, not as belonging to its subject as such: otherwise they would not have been applicable to another genus.
Ἕκαστον δ᾽ ἐπιστάμεθα μὴ κατὰ συμβεβηκός, ὅταν ↵ κατ᾽ ἐκεῖνο γινώσκωμεν καθ᾽ ὁ ὑπάρχει, ἐκ τῶν ἀρχῶν τῶν ἐκείνου ἧι ἐκεῖνο, οἷον τὸ δυσὶν ὀρθαῖς ἴσας ἔχειν, ὧι ὑπάρχει καθ᾽ αὑτὸ τὸ εἰρημένον, ἐκ τῶν ἀρχῶν τῶν τούτου. ὥστ᾽ εἰ καθ᾽ αὑτὸ κἀκεῖνο ὑπάρχει ὧι ὑπάρχει, ἀνάγκη τὸ μέσον ἐν τῆι αὐτῆι συγγενείαι εἶναι.	Unumquodque autem scimus non secundum accidens, cum secundum illud cognoscimus, secundum quod est ex principiis illius in quantum illud est, ut duobus rectis aequales habere, cui inest per se quod dictum est, ex principiis illius. Quare si per se et illud inest cui inest, necesse est medium in eadem proximitate esse;	Our knowledge of any attribute’s connexion with a subject is accidental unless we know that connexion through the middle term in virtue of which it inheres, and as an inference from basic premisses essential and ‘appropriate’ to the subject-unless we know, e.g. the property of possessing angles equal to two right angles as belonging to that subject in which it inheres essentially, and as inferred from basic premisses essential and ‘appropriate’ to that subject: so that if that middle term also belongs essentially to the minor, the middle must belong to the same kind as the major and minor terms.
76a8 εἰ δὲ μή, ἀλλ᾽ ὡς ↵ τὰ ἁρμονικὰ δι᾽ ἀριθμητικῆς. τὰ δὲ τοιαῦτα δείκνυται μὲν ὡσαύτως, διαφέρει δέ· τὸ μὲν γὰρ ὅτι ἑτέρας ἐπιστήμης (τὸ γὰρ ὑποκείμενον γένος ἕτερον), τὸ δὲ διότι τῆς ἄνω, ἧς καθ᾽ αὑτὰ τὰ πάθη ἐστίν.	si vero non sit, sed sicut harmonica per arithmeticam, huiusmodi autem demonstrantur quidem similiter, sed differunt. Nam ipsum quidem quia alterius quidem scientiae est, subiectum enim genus alterum est, sed propter quid, est superioris, cuius per se passiones sunt.	The only exceptions to this rule are such cases as theorems in harmonics which are demonstrable by arithmetic. Such theorems are proved by the same middle terms as arithmetical properties, but with a qualification-the fact falls under a separate science (for the subject genus is separate), but the reasoned fact concerns the superior science, to which the attributes essentially belong.
ὥστε καὶ ἐκ τούτων φανερὸν ὅτι οὐκ ἔστιν ἀποδεῖξαι ἕκαστον ἁπλῶς ἀλλ᾽ ἢ ἐκ τῶν ἑκά↵ στου ἀρχῶν. ἀλλὰ τούτων αἱ ἀρχαὶ ἔχουσι τὸ κοινόν.	Quare ex his manifestum est quod non sit demonstrare unumquodque simpliciter, aliter quam ex propriis uniuscuiusque principiis, sed horum principia habent commune.	Thus, even these apparent exceptions show that no attribute is strictly demonstrable except from its ‘appropriate’ basic truths, which, however, in the case of these sciences have the requisite identity of character.
76a17 Εἰ δὲ φανερὸν τοῦτο, φανερὸν καὶ ὅτι οὐκ ἔστι τὰς ἑκάστου ἰδίας ἀρχὰς ἀποδεῖξαι· ἔσονται γὰρ ἐκεῖναι ἁπάντων ἀρχαί, καὶ ἐπιστήμη ἡ ἐκείνων κυρία πάντων.	Si autem manifestum hoc, manifestum et quoniam non est uniuscuiusque propria principia demonstrare. Erunt enim illa omnium principia, et scientia eorum domina omnium,	It is no less evident that the peculiar basic truths of each inhering attribute are indemonstrable; for basic truths from which they might be deduced would be basic truths of all that is, and the science to which they belonged would possess universal sovereignty.
76a19 καὶ γὰρ ἐπίσταται μᾶλλον ὁ ἐκ τῶν ἀνώτερον αἰτίων εἰδώς· ἐκ τῶν ↵ προτέρων γὰρ οἶδεν, ὅταν ἐκ μὴ αἰτιατῶν εἰδῆι αἰτίων. ὥστ᾽ εἰ μᾶλλον οἶδε καὶ μάλιστα, κἂν ἐπιστήμη ἐκείνη εἴη καὶ μᾶλλον καὶ μάλιστα.	et namque scit magis ex superioribus causis sciens, ex prioribus enim scit, cum non ex causatis sciat causis; quare si magis scit, et maxime, et scientia illa erit et magis, et maxime.	This is so because he knows better whose knowledge is deduced from higher causes, for his knowledge is from prior premisses when it derives from causes themselves uncaused: hence, if he knows better than others or best of all, his knowledge would be science in a higher or the highest degree.
76a23 ἡ δ᾽ ἀπόδειξις οὐκ ἐφαρμόττει ἐπ᾽ ἄλλο γένος, ἀλλ᾽ ἢ ὡς εἴρηται αἱ γεωμετρικαὶ ἐπὶ τὰς μηχανικὰς ἢ ὀπτικὰς καὶ αἱ ἀριθμητικαὶ ἐπὶ τὰς ἁρ↵ μονικάς. Χαλεπὸν δ᾽ ἐστὶ τὸ γνῶναι εἰ οἶδεν ἢ μή.	Sed demonstratio non convenit in aliud genus aliter quam, ut dictum est, geometricae in machinativas, aut perspectivas, et arithmeticae in harmonicas.	But, as things are, demonstration is not transferable to another genus, with such exceptions as we have mentioned of the application of geometrical demonstrations to theorems in mechanics or optics, or of arithmetical demonstrations to those of harmonics.
76a26 Χαλεπὸν γὰρ τὸ γνῶναι εἰ ἐκ τῶν ἑκάστου ἀρχῶν ἴσμεν ἢ μή· ὅπερ ἐστὶ τὸ εἰδέναι. οἰόμεθα δ᾽, ἂν ἔχωμεν ἐξ ἀληθινῶν τινῶν συλλογισμὸν καὶ πρώτων, ἐπίστασθαι. τὸ δ᾽ οὐκ ἔστιν, ἀλλὰ ↵ συγγενῆ δεῖ εἶναι τοῖς πρώτοις.	Difficile autem est nosse si ex uniuscuiusque principiis scimus, aut non, quod quidem est scire, opinamur autem hoc, si habeamus ex veris aliquibus syllogismum et primis scire; sed hoc non est, sed congenea oportet esse primis.	It is hard to be sure whether one knows or not; for it is hard to be sure whether one’s knowledge is based on the basic truths appropriate to each attribute-the differentia of true knowledge. We think we have scientific knowledge if we have reasoned from true and primary premisses. But that is not so: the conclusion must be homogeneous with the basic facts of the science.
c10	CAPUT VIII. De principiis tum vagis communibusque, tum propriis ac addictis.	Chapter 10
76a31 Λέγω δ᾽ ἀρχὰς ἐν ἑκάστωι γένει ταύτας ἃς ὅτι ἔστι μὴ ἐνδέχεται δεῖξαι.	Dico autem principia in unoquoque genere, illa quae quoniam sint non contingit demonstrare.	call the basic truths of every genus those clements in it the existence of which cannot be proved.
76a32 τί μὲν οὖν σημαίνει καὶ τὰ πρῶτα καὶ τὰ ἐκ τούτων, λαμβάνεται, ὅτι δ᾽ ἔστι, τὰς μὲν ἀρχὰς ἀνάγκη λαμβάνειν, τὰ δ᾽ ἄλλα δεικνύναι· οἷον τί μονὰς ↵ ἢ τί τὸ εὐθὺ καὶ τρίγωνον, εἶναι δὲ τὴν μονάδα λαβεῖν καὶ μέγεθος, τὰ δ᾽ ἕτερα δεικνύναι.	Quid quidem igitur significent et prima et quae sunt ex primis, accipiendum; quod [F. quoad] autem principia, quidem [ an. quaedam?] necesse est accipere, alia vero demonstrare; ut quid unitas, aut quid rectum, et quid triangulus, esse tem unitatem accipere, et magnitudinem, altera vero monstrare.	As regards both these primary truths and the attributes dependent on them the meaning of the name is assumed. The fact of their existence as regards the primary truths must be assumed; but it has to be proved of the remainder, the attributes. Thus we assume the meaning alike of unity, straight, and triangular; but while as regards unity and magnitude we assume also the fact of their existence, in the case of the remainder proof is required.
76a37 Ἔστι δ᾽ ὧν χρῶνται ἐν ταῖς ἀποδεικτικαῖς ἐπιστήμαις τὰ μὲν ἴδια ἑκάστης ἐπιστήμης τὰ δὲ κοινά, κοινὰ δὲ κατ᾽ ἀναλογίαν, ἐπεὶ χρήσιμόν γε ὅσον ἐν τῶι ὑπὸ τὴν ἐπιστήμην ↵ γένει·	Sunt autem quibus utuntur in demonstrativis scientiis, alia quidem propria uniuscuiusque scientiae, alia vero communia. Communia vero secundum analogiam, quoniam utile est quantumcunque est in eo (quod est sub scientia) genere.	Of the basic truths used in the demonstrative sciences some are peculiar to each science, and some are common, but common only in the sense of analogous, being of use only in so far as they fall within the genus constituting the province of the science in question.
76a40 ἴδια μὲν οἷον γραμμὴν εἶναι τοιανδὶ καὶ τὸ εὐθύ, κοινὰ δὲ οἷον τὸ ἴσα ἀπὸ ἴσων ἂν ἀφέληι, ὅτι ἴσα τὰ λοιπά.	Propria principia quidem, ut lineam esse eiusmodi, et rectum. Communia autem, ut, aequalia ab aequalibus si auferas, quod aequalia reliqua sunt.	Peculiar truths are, e.g. the definitions of line and straight; common truths are such as ‘take equals from equals and equals remain’.
76a42 ἱκανὸν δ᾽ ἕκαστον τούτων ὅσον ἐν τῶι γένει· ταὐτὸ γὰρ ποιήσει, κἂν μὴ κατὰ πάντων λάβηι ἀλλ᾽ ἐπὶ μεγεθῶν μόνον, τῶι δ᾽ ἀριθμητικῶι ἐπ᾽ ἀριθμῶν.	Sufficientes autem unumquodque horum, quantumcunque in genere est. Idem enim faciet, etsi non de omnibus accipiat, sed in magnitudinibus solum, arithmeticae autem in numeris.	Only so much of these common truths is required as falls within the genus in question: for a truth of this kind will have the same force even if not used generally but applied by the geometer only to magnitudes, or by the arithmetician only to numbers.
76b2 Ἔστι δ᾽ ἴδια μὲν καὶ ἃ λαμβάνεται εἶναι, περὶ ἃ ἡ ἐπιστήμη θεωρεῖ τὰ ὑπάρχοντα καθ᾽ αὑτά, οἷον μονάδας ἡ ↵ ἀριθμητική, ἡ δὲ γεωμετρία σημεῖα καὶ γραμμάς. ταῦτα γὰρ λαμβάνουσι τὸ εἶναι καὶ τοδὶ εἶναι. τὰ δὲ τούτων πάθη καθ᾽ αὑτά, τί μὲν σημαίνει ἕκαστον, λαμβάνουσιν, οἷον ἡ μὲν ἀριθμητικὴ τί περιττὸν ἢ ἄρτιον ἢ τετράγωνον ἢ κύβος, ἡ δὲ γεωμετρία τί τὸ ἄλογον ἢ τὸ κεκλάσθαι ἢ νεύειν, ὅτι ↵ δ᾽ ἔστι, δεικνύουσι διά τε τῶν κοινῶν καὶ ἐκ τῶν ἀποδεδειγμένων. καὶ ἡ ἀστρολογία ὡσαύτως.	Sunt autem propria quidem, et quae accipiuntur esse, circa quae scientia speculatur, quae sunt per se, ut arithmetica unitates, geometria autem signa et lineas, haec enim recipiunt esse, et hoc esse, horum autem passiones per se. Quid quidem unaquaeque significet, accipiunt, ut arithmetica quidem, quid par, aut impar, aut quadratus, aut cubus, geometria vero quid irrationale, aut inflecti, aut concurrere, quod autem sint, demonstrant per communia, et ex iisque demonstrantur, et astrologia similiter.	Also peculiar to a science are the subjects the existence as well as the meaning of which it assumes, and the essential attributes of which it investigates, e.g. in arithmetic units, in geometry points and lines. Both the existence and the meaning of the subjects are assumed by these sciences; but of their essential attributes only the meaning is assumed. For example arithmetic assumes the meaning of odd and even, square and cube, geometry that of incommensurable, or of deflection or verging of lines, whereas the existence of these attributes is demonstrated by means of the axioms and from previous conclusions as premisses. Astronomy too proceeds in the same way.
πᾶσα γὰρ ἀποδεικτικὴ ἐπιστήμη περὶ τρία ἐστίν, ὅσα τε εἶναι τίθεται (ταῦτα δ᾽ ἐστὶ τὸ γένος, οὗ τῶν καθ᾽ αὑτὰ παθημάτων ἐστὶ θεωρητική), καὶ τὰ κοινὰ λεγόμενα ἀξιώματα, ἐξ ὧν πρώτων ἀποδεί↵ κνυσι, καὶ τρίτον τὰ πάθη, ὧν τί σημαίνει ἕκαστον λαμβάνει.	Omnis enim demonstrativa scientia circa tria est, quaecunque esse ponuntur. Haec autem sunt genus, cuius per se passionum speculativa est, et quae communes dicuntur dignitates, ex quibus primis demonstrant, et tertium passiones, quarum quid significet unaquaeque accipit.	For indeed every demonstrative science has three elements: (1) that which it posits, the subject genus whose essential attributes it examines; (2) the so-called axioms, which are primary premisses of its demonstration; (3) the attributes, the meaning of which it assumes.
76b16 ἐνίας μέντοι ἐπιστήμας οὐδὲν κωλύει ἔνια τούτων παρορᾶν, οἷον τὸ γένος μὴ ὑποτίθεσθαι εἶναι, ἂν ἦι φανερὸν ὅτι ἔστιν (οὐ γὰρ ὁμοίως δῆλον ὅτι ἀριθμὸς ἔστι καὶ ὅτι ψυχρὸν καὶ θερμόν), καὶ τὰ πάθη μὴ λαμβάνειν τί σημαίνει, ἂν ἦι δῆ↵ λα· ὥσπερ οὐδὲ τὰ κοινὰ οὐ λαμβάνει τί σημαίνει τὸ ἴσα ἀπὸ ἴσων ἀφελεῖν, ὅτι γνώριμον. ἀλλ᾽ οὐδὲν ἧττον τῆι γε φύσει τρία ταῦτά ἐστι, περὶ ὅ τε δείκνυσι καὶ ἃ δείκνυσι καὶ ἐξ ὧν.	Quasdam tamen scientias nihil prohibet quaedam eorum despicere: ut genus non supponere esse, si sit manifestum quoniam est, non enim similiter manifestum est quoniam numerus sit, et quod calidum, et frigidum, et passiones non est recipere quid significent, si sint manifestae, sicut neque communia non recipit quid significent quod est aequalia ab aequalibus demere, quoniam notum est. Sed nihil minus natura tria haec sint, circa quod demonstrat, et quae demonstrat, et ex quibus.	Yet some sciences may very well pass over some of these elements; e.g. we might not expressly posit the existence of the genus if its existence were obvious (for instance, the existence of hot and cold is more evident than that of number); or we might omit to assume expressly the meaning of the attributes if it were well understood. In the way the meaning of axioms, such as ‘Take equals from equals and equals remain’, is well known and so not expressly assumed. Nevertheless in the nature of the case the essential elements of demonstration are three: the subject, the attributes, and the basic premisses.
76b23 Οὐκ ἔστι δ᾽ ὑπόθεσις οὐδ᾽ αἴτημα, ὁ ἀνάγκη εἶναι δι᾽ αὑτὸ καὶ δοκεῖν ἀνάγκη. οὐ γὰρ πρὸς τὸν ἔξω λόγον ἡ ἀπό↵ δειξις, ἀλλὰ πρὸς τὸν ἐν τῆι ψυχῆι, ἐπεὶ οὐδὲ συλλογισμός. ἀεὶ γὰρ ἔστιν ἐνστῆναι πρὸς τὸν ἔξω λόγον, ἀλλὰ πρὸς τὸν ἔσω λόγον οὐκ ἀεί.	Non est suppositio, neque petitio, quod necesse est propter seipsum esse, et videri necesse est: non est enim ad exterius orationem demonstratio, sed ad eam quae est in anima, quoniam neque syllogismus. Semper enim est instare ad exterius orationem, sed ad interius orationem non semper.	That which expresses necessary self-grounded fact, and which we must necessarily believe, is distinct both from the hypotheses of a science and from illegitimate postulate-I say ‘must believe’, because all syllogism, and therefore a fortiori demonstration, is addressed not to the spoken word, but to the discourse within the soul, and though we can always raise objections to the spoken word, to the inward discourse we cannot always object.
76b27 ὅσα μὲν οὖν δεικτὰ ὄντα λαμβάνει αὐτὸς μὴ δείξας, ταῦτ᾽, ἐὰν μὲν δοκοῦντα λαμβάνηι τῶι μανθάνοντι, ὑποτίθεται, καὶ ἔστιν οὐχ ἁπλῶς ὑπόθεσις ἀλλὰ ↵ πρὸς ἐκεῖνον μόνον, ἂν δὲ ἢ μηδεμιᾶς ἐνούσης δόξης ἢ καὶ ἐναντίας ἐνούσης λαμβάνηι τὸ αὐτό, αἰτεῖται.	Quaecunque ergo quidem demonstrabilia esse accipit ipse non demonstrans, haec si quidem quae videntur accipiat, discenti suppositio, et non est simpliciter suppositio, sed ad illum solum, si vero neque unius opinionis, aut contraria est, accipiat, idem petit.	That which is capable of proof but assumed by the teacher without proof is, if the pupil believes and accepts it, hypothesis, though only in a limited sense hypothesis-that is, relatively to the pupil; if the pupil has no opinion or a contrary opinion on the matter, the same assumption is an illegitimate postulate.
καὶ τούτωι διαφέρει ὑπόθεσις καὶ αἴτημα· ἔστι γὰρ αἴτημα τὸ ὑπεναντίον τοῦ μανθάνοντος τῆι δόξηι, ἢ ὁ ἄν τις ἀποδεικτὸν ὂν λαμβάνηι καὶ χρῆται μὴ δείξας.	Et in hoc differt suppositio, et petitio, est enim petitio in contrarium discentis opinioni, aut quodcunque aliquis demonstrabile cum sit, accipiat, et utatur non demonstrans.	Therein lies the distinction between hypothesis and illegitimate postulate: the latter is the contrary of the pupil’s opinion, demonstrable, but assumed and used without demonstration.
76b35 ↵ Οἱ μὲν οὖν ὅροι οὐκ εἰσὶν ὑποθέσεις (οὐδὲν γὰρ εἶναι ἢ μὴ λέγεται), ἀλλ᾽ ἐν ταῖς προτάσεσιν αἱ ὑποθέσεις, τοὺς δ᾽ ὅρους μόνον ξυνίεσθαι δεῖ· τοῦτο δ᾽ οὐχ ὑπόθεσις (εἰ μὴ καὶ τὸ ἀκούειν ὑπόθεσίν τις εἶναι φήσει), ἀλλ᾽ ὅσων ὄντων τῶι ἐκεῖνα εἶναι γίνεται τὸ συμπέρασμα.	Termini igitur non sunt suppositiones, nihil enim esse aut non esse dicunt, sed in propositionibus sunt suppositiones. Terminos solum intelligere oportet, hoc autem non est suppositio, nisi et audire aliquis suppositionem esse dicat, sed quibuscunque existentibus in eo quod illa sunt, fit conclusio.	The definition-viz. those which are not expressed as statements that anything is or is not-are not hypotheses: but it is in the premisses of a science that its hypotheses are contained. Definitions require only to be understood, and this is not hypothesis-unless it be contended that the pupil’s hearing is also an hypothesis required by the teacher. Hypotheses, on the contrary, postulate facts on the being of which depends the being of the fact inferred.
76b39 οὐδ᾽ ὁ γεωμέτρης ψευδῆ ↵ ὑποτίθεται, ὥσπερ τινὲς ἔφασαν, λέγοντες ὡς οὐ δεῖ τῶι ψεύδει χρῆσθαι, τὸν δὲ γεωμέτρην ψεύδεσθαι λέγοντα ποδιαίαν τὴν οὐ ποδιαίαν ἢ εὐθεῖαν τὴν γεγραμμένην οὐκ εὐθεῖαν οὖσαν. ὁ δὲ γεωμέτρης οὐδὲν συμπεραίνεται τῶι τήνδε εἶναι γραμμὴν ἣν αὐτὸς ἔφθεγκται, ἀλλὰ τὰ διὰ τούτων δηλούμενα.	Neque autem geometra falsa supponit, sicut quidam affirmant dicentes quod oportet non falso ut geometram, mentiri autem dicentem lineam esse unius pedis, quae non est unius pedis aut rectam lineam, non rectam existentem, sed geometra nihil secundum hanc lineam concludit, quam ipse posuit, sed quae per haec ostenduntur.	Nor are the geometer’s hypotheses false, as some have held, urging that one must not employ falsehood and that the geometer is uttering falsehood in stating that the line which he draws is a foot long or straight, when it is actually neither. The truth is that the geometer does not draw any conclusion from the being of the particular line of which he speaks, but from what his diagrams symbolize.
77a3 ἔτι τὸ αἴτημα καὶ ὑπόθεσις πᾶσα ἢ ὡς ὅλον ἢ ὡς ἐν μέρει, οἱ δ᾽ ὅροι οὐδέτερον τούτων.	Amplius. Petitio et suppositio omnis, aut sicut totum est, aut in parte, termini autem neutrum horum.	A further distinction is that all hypotheses and illegitimate postulates are either universal or particular, whereas a definition is neither.
c11		Chapter 11
77a5 Εἴδη μὲν οὖν εἶναι ἢ ἕν τι παρὰ τὰ πολλὰ οὐκ ἀνάγκη, εἰ ἀπόδειξις ἔσται, εἶναι μέντοι ἓν κατὰ πολλῶν ἀληθὲς εἰπεῖν ἀνάγκη· οὐ γὰρ ἔσται τὸ καθόλου, ἂν μὴ τοῦτο ἦι· ἐὰν δὲ τὸ καθόλου μὴ ἦι, τὸ μέσον οὐκ ἔσται, ὥστ᾽ οὐδ᾽ ἀπόδειξις. δεῖ ἄρα τι ἓν καὶ τὸ αὐτὸ ἐπὶ πλειόνων εἶναι μὴ ὁμώνυμον.	Species quidem igitur esse, aut unum aliquid extra multa non necesse est esse, si demonstratio erit, esse tamen unum de multis verum dicere, necesse est. Non enim erit universale nisi hoc sit. Si vero universale non sit, medium non erit, quare neque demonstratio, oportet itaque aliquid unum, et idem de pluribus esse non aequivocum.	So demonstration does not necessarily imply the being of Forms nor a One beside a Many, but it does necessarily imply the possibility of truly predicating one of many; since without this possibility we cannot save the universal, and if the universal goes, the middle term goes witb. it, and so demonstration becomes impossible. We conclude, then, that there must be a single identical term unequivocally predicable of a number of individuals.
77a10 ↵ τὸ δὲ μὴ ἐνδέχεσθαι ἅμα φάναι καὶ ἀποφάναι οὐδεμία λαμβάνει ἀπόδειξις, ἀλλ᾽ ἢ ἐὰν δέηι δεῖξαι καὶ τὸ συμπέρασμα οὕτως. δείκνυται δὲ λαβοῦσι τὸ πρῶτον κατὰ τοῦ μέσου, ὅτι ἀληθές, ἀποφάναι δ᾽ οὐκ ἀληθές. τὸ δὲ μέσον οὐδὲν διαφέρει εἶναι καὶ μὴ εἶναι λαβεῖν, ὡς δ᾽ αὔτως καὶ ↵ τὸ τρίτον. εἰ γὰρ ἐδόθη, καθ᾽ οὗ ἄνθρωπον ἀληθὲς εἰπεῖν, εἰ καὶ μὴ ἄνθρωπον ἀληθές, ἀλλ᾽ εἰ μόνον ἄνθρωπον ζῶιον εἶναι, μὴ ζῶιον δὲ μή, ἔσται [γὰρ] ἀληθὲς εἰπεῖν Καλλίαν, εἰ καὶ μὴ Καλλίαν, ὅμως ζῶιον, μὴ ζῶιον δ᾽ οὔ. αἴτιον δ᾽ ὅτι τὸ πρῶτον οὐ μόνον κατὰ τοῦ μέσου λέγεται ἀλλὰ καὶ κατ᾽ ↵ ἄλλου διὰ τὸ εἶναι ἐπὶ πλειόνων, ὥστ᾽ οὐδ᾽ εἰ τὸ μέσον καὶ αὐτό ἐστι καὶ μὴ αὐτό, πρὸς τὸ συμπέρασμα οὐδὲν διαφέρει.	Non contingere autem idem simul affirmare, et negare, neque una recipit demonstratio, sed aut si indigeat monstrare conclusionem sic, ostenditur autem accipientibus primum de medio quod verum, negare autem non verum, medium autem nihil differt esse, et non est accipere, similiter autem et tertium, si enim assignetur de aliquo hominem, verum est dicere, et si non hominem, verum, sed si solum hominem animal esse, omne, non animal autem non. Erit enim verum dicere Calliam, sive non Calliam esse animal, non animal autem, non. Causa autem est quod primum non solum de medio dicitur, sed de alio, propter id quod de pluribus, quare neque si medium et idem est, et non idem, ad conclusionem nihil differt.	The law that it is impossible to affirm and deny simultaneously the same predicate of the same subject is not expressly posited by any demonstration except when the conclusion also has to be expressed in that form; in which case the proof lays down as its major premiss that the major is truly affirmed of the middle but falsely denied. It makes no difference, however, if we add to the middle, or again to the minor term, the corresponding negative. For grant a minor term of which it is true to predicate man-even if it be also true to predicate not-man of it — still grant simply that man is animal and not not-animal, and the conclusion follows: for it will still be true to say that Callias — even if it be also true to say that not-Callias — is animal and not not-animal. The reason is that the major term is predicable not only of the middle, but of something other than the middle as well, being of wider application; so that the conclusion is not affected even if the middle is extended to cover the original middle term and also what is not the original middle term.
77a22 τὸ δ᾽ ἅπαν φάναι ἢ ἀποφάναι ἡ εἰς τὸ ἀδύνατον ἀπόδειξις λαμβάνει, καὶ ταῦτα οὐδ᾽ ἀεὶ καθόλου, ἀλλ᾽ ὅσον ἱκανόν, ἱκανὸν δ᾽ ἐπὶ τοῦ γένους. λέγω δ᾽ ἐπὶ τοῦ γένους οἷον περὶ ↵ ὁ γένος τὰς ἀποδείξεις φέρει, ὥσπερ εἴρηται καὶ πρότερον.	Commune autem affirmare, aut negare, quae est ad impossibile demonstratio, accipit, et hoc neque semper, universaliter, sed quantum sufficiens est, sufficiens autem est in genere; dico autem in genere, ut circa quod genus demonstrationes fert, sicut dictum est prius.	The law that every predicate can be either truly affirmed or truly denied of every subject is posited by such demonstration as uses reductio ad impossibile, and then not always universally, but so far as it is requisite; within the limits, that is, of the genus-the genus, I mean (as I have already explained), to which the man of science applies his demonstrations.
77a26 Ἐπικοινωνοῦσι δὲ πᾶσαι αἱ ἐπιστῆμαι ἀλλήλαις κατὰ τὰ κοινά (κοινὰ δὲ λέγω οἷς χρῶνται ὡς ἐκ τούτων ἀποδεικνύντες, ἀλλ᾽ οὐ περὶ ὧν δεικνύουσιν οὐδ᾽ ὁ δεικνύουσιν),	Communicant autem omnes scientiae secundum communia. Communia autem dico, quibus utuntur tanquam ex eis demonstrantes, sed non ex quibus demonstrant, neque quod demonstrant.	In virtue of the common elements of demonstration-I mean the common axioms which are used as premisses of demonstration, not the subjects nor the attributes demonstrated as belonging to them-all the sciences have communion with one another,
77a29 καὶ ἡ διαλεκτικὴ πάσαις, καὶ εἴ τις καθόλου πειρῶιτο δει↵ κνύναι τὰ κοινά, οἷον ὅτι ἅπαν φάναι ἢ ἀποφάναι, ἢ ὅτι ἴσα ἀπὸ ἴσων, ἢ τῶν τοιούτων ἄττα. ἡ δὲ διαλεκτικὴ οὐκ ἔστιν οὕτως ὡρισμένων τινῶν, οὐδὲ γένους τινὸς ἑνός. οὐ γὰρ ἂν ἠρώτα· ἀποδεικνύντα γὰρ οὐκ ἔστιν ἐρωτᾶν διὰ τὸ τῶν ἀντικειμένων ὄντων μὴ δείκνυσθαι τὸ αὐτό. δέδεικται δὲ τοῦτο ἐν τοῖς ↵ περὶ συλλογισμοῦ.	Et dialectica quidem de omnibus, et si aliqua universaliter tentet monstrare communia, ut quod est affirmare omne aut negare, aut quod est aequalia ab aequalibus demere, aut talium quaelibet, sed dialectica non est definitorum sic quorumdam, neque generis alicuius unius, non enim interrogaret, demonstrantem autem non est interrogare, propter id quod oppositorum esse non monstrat idem. Ostensum autem est hoc in iis quae de syllogismo.	and in communion with them all is dialectic and any science which might attempt a universal proof of axioms such as the law of excluded middle, the law that the subtraction of equals from equals leaves equal remainders, or other axioms of the same kind. Dialectic has no definite sphere of this kind, not being confined to a single genus. Otherwise its method would not be interrogative; for the interrogative method is barred to the demonstrator, who cannot use the opposite facts to prove the same nexus. This was shown in my work on the syllogism.
c12	CAPUT IX. Cuiusque disciplinae proprietas, accommodatasque esse interrogationes.	Chapter 12
77a36 Εἰ δὲ τὸ αὐτό ἐστιν ἐρώτημα συλλογιστικὸν καὶ πρότασις ἀντιφάσεως, προτάσεις δὲ καθ᾽ ἑκάστην ἐπιστήμην ἐξ ὧν ὁ συλλογισμὸς ὁ καθ᾽ ἑκάστην, εἴη ἄν τι ἐρώτημα ἐπιστημονικόν, ἐξ ὧν ὁ καθ᾽ ἑκάστην οἰκεῖος γίνεται συλλο↵ γισμός. δῆλον ἄρα ὅτι οὐ πᾶν ἐρώτημα γεωμετρικὸν ἂν εἴη οὐδ᾽ ἰατρικόν, ὁμοίως δὲ καὶ ἐπὶ τῶν ἄλλων· ἀλλ᾽ ἐξ	Si autem idem est interrogatio syllogistica, et propositio contradictionis, propositiones autem sunt secundum unamquamque scientiam, ex quibus est syllogismus, secundum unamquamque erunt utique aliquae interrogationes scientiales, ex quibus qui est secundum unamquamque proprius fit syllogismus. Manifestum itaque quod non omnis interrogatio geometrica erit, neque medicinalis, similiter autem et in aliis,	If a syllogistic question is equivalent to a proposition embodying one of the two sides of a contradiction, and if each science has its peculiar propositions from which its peculiar conclusion is developed, then there is such a thing as a distinctively scientific question, and it is the interrogative form of the premisses from which the ‘appropriate’ conclusion of each science is developed. Hence it is clear that not every question will be relevant to geometry, nor to medicine, nor to any other science:
77a42 [77b]ὧν δείκνυταί τι περὶ ὧν ἡ γεωμετρία ἐστίν, ἢ ἃ ἐκ τῶν αὐτῶν δείκνυται τῆι γεωμετρίαι, ὥσπερ τὰ ὀπτικά. ὁμοίως δὲ καὶ ἐπὶ τῶν ἄλλων.	sed ex quibus aut monstratur aliquid de quibus geometrica est, aut quod ex eisdem monstratur geometriae, ut visibilia, similiter autem et in aliis.	only those questions will be geometrical which form premisses for the proof of the theorems of geometry or of any other science, such as optics, which uses the same basic truths as geometry. Of the other sciences the like is true.
77b2 καὶ περὶ μὲν τούτων καὶ λόγον ὑφε↵ κτέον ἐκ τῶν γεωμετρικῶν ἀρχῶν καὶ συμπερασμάτων, περὶ δὲ τῶν ἀρχῶν λόγον οὐχ ὑφεκτέον τῶι γεωμέτρηι ἧι γεωμέτρης· ὁμοίως δὲ καὶ ἐπὶ τῶν ἄλλων ἐπιστημῶν.	Et de iis quidem et rationem ponendam esse ex geometricis principiis et conclusionibus, sed principiorum rationem non ponendam esse geometrae secundum quod est geometria, similiter autem et in aliis scientiis.	Of these questions the geometer is bound to give his account, using the basic truths of geometry in conjunction with his previous conclusions; of the basic truths the geometer, as such, is not bound to give any account. The like is true of the other sciences.
77b6 οὔτε πᾶν ἄρα ἕκαστον ἐπιστήμονα ἐρώτημα ἐρωτητέον, οὔθ᾽ ἅπαν τὸ ἐρωτώμενον ἀποκριτέον περὶ ἑκάστου, ἀλλὰ τὰ κατὰ τὴν ἐπιστήμην διορισθέντα.	Neque omne est utique unumquemque scientem interrogandum, neque secundum omne interrogatum esse respondendum de unoquoque, sed quae sunt secundum scientiam determinata.	There is a limit, then, to the questions which we may put to each man of science; nor is each man of science bound to answer all inquiries on each several subject, but only such as fall within the defined field of his own science.
77b8 εἰ δὲ διαλέξεται γεωμέτρηι ἧι γεω↵ μέτρης οὕτως, φανερὸν ὅτι καὶ καλῶς, ἐὰν ἐκ τούτων τι δεικνύηι· εἰ δὲ μή, οὐ καλῶς. δῆλον δ᾽ ὅτι οὐδ᾽ ἐλέγχει γεωμέτρην ἀλλ᾽ ἢ κατὰ συμβεβηκός· ὥστ᾽ οὐκ ἂν εἴη ἐν ἀγεωμετρήτοις περὶ γεωμετρίας διαλεκτέον· λήσει γὰρ ὁ φαύλως διαλεγόμενος. ὁμοίως δὲ καὶ ἐπὶ τῶν ἄλλων ἔχει ↵ ἐπιστημῶν.	Si autem disputet cum geometra, secundum quod est geometra, sic manifestum est quod et bene, si ex iis aliquid demonstret, si vere non, non bene, manifestum autem est quod non arguit geometram, sed aut secundum accidens. Quare non utique erit in non geometricis de geometria disputandum, latebit enim prave disputans, similiter autem et in aliis se habet scientiis.	If, then, in controversy with a geometer qua geometer the disputant confines himself to geometry and proves anything from geometrical premisses, he is clearly to be applauded; if he goes outside these he will be at fault, and obviously cannot even refute the geometer except accidentally. One should therefore not discuss geometry among those who are not geometers, for in such a company an unsound argument will pass unnoticed. This is correspondingly true in the other sciences.
77b16 Ἐπεὶ δ᾽ ἔστι γεωμετρικὰ ἐρωτήματα, ἆρ᾽ ἔστι καὶ ἀγεωμέτρητα;	Quoniam autem sunt geometricae interrogationes, nonne sunt et non geometricae?	Since there are ‘geometrical’ questions, does it follow that there are also distinctively ‘ungeometrical’ questions?
77b17 καὶ παρ᾽ ἑκάστην ἐπιστήμην τὰ κατὰ τὴν ἄγνοιαν τὴν ποίαν γεωμετρικά ἐστιν;	Et quae sunt secundum unamquamque scientiam, et quae secundum ignorantiam qualis geometrica est?	Further, in each special science-geometry for instance-what kind of error is it that may vitiate questions, and yet not exclude them from that science?
77b19 καὶ πότερον ὁ κατὰ τὴν ἄγνοιαν συλλογισμὸς ὁ ἐκ τῶν ἀντικει↵ μένων συλλογισμός, ἢ ὁ παραλογισμός, κατὰ γεωμετρίαν δέ, ἢ ‹ὁ› ἐξ ἄλλης τέχνης,	Et utrum qui secundum ignorantiam syllogismus, qui est ex oppositis syllogismus, an paralogismus? Secundum geometriam autem, aut ex alia arte	Again, is the erroneous conclusion one constructed from premisses opposite to the true premisses, or is it formal fallacy though drawn from geometrical premisses?
bk77b21 οἷον τὸ μουσικόν ἐστιν ἐρώτημα ἀγεωμέτρητον περὶ γεωμετρίας, 77b23 τὸ δὲ τὰς παραλλήλους συμπίπτειν οἴεσθαι γεωμετρικόν πως καὶ ἀγεωμέτρητον ἄλλον τρόπον;	ut musica, est interrogatio non geometrica, de geometrica autem, ut ad invicem parallelas concurrere opinari, geometrica quodammodo est, et non geometrica alio modo,	r, perhaps, the erroneous conclusion is due to the drawing of premisses from another science; e.g. in a geometrical controversy a musical question is distinctively ungeometrical, whereas the notion that parallels meet is in one sense geometrical, being ungeometrical in a different fashion:
διττὸν γὰρ τοῦτο, ὥσπερ τὸ ἄρρυθμον, καὶ τὸ ↵ μὲν ἕτερον ἀγεωμέτρητον τῶι μὴ ἔχειν [ὥσπερ τὸ ἄρρυθμον], τὸ δ᾽ ἕτερον τῶι φαύλως ἔχειν· καὶ ἡ ἄγνοια αὕτη καὶ ἡ ἐκ τῶν τοιούτων ἀρχῶν ἐναντία.	dupliciter enim hoc est sicut arithmon, alterum quidem non geometricum est in non habendo, alterum vero in prave habendo, et ignorantia haec quae est ex eius principiis, contraria est.	the reason being that ‘ungeometrical’, like ‘unrhythmical’, is equivocal, meaning in the one case not geometry at all, in the other bad geometry? It is this error, i.e. error based on premisses of this kind-’of’ the science but false-that is the contrary of science.
77b27 ἐν δὲ τοῖς μαθήμασιν οὐκ ἔστιν ὁμοίως ὁ παραλογισμός, ὅτι τὸ μέσον ἐστὶν ἀεὶ τὸ διττόν· κατά τε γὰρ τούτου παντός, καὶ τοῦτο πάλιν κατ᾽ ἄλλου ↵ λέγεται παντός (τὸ δὲ κατηγορούμενον οὐ λέγεται πᾶν), ταῦτα δ᾽ ἔστιν οἷον ὁρᾶν τῆι νοήσει, ἐν δὲ τοῖς λόγοις λανθάνει. ἆρα πᾶς κύκλος σχῆμα; ἂν δὲ γράψηι, δῆλον. τί δέ; τὰ ἔπη κύκλος; φανερὸν ὅτι οὐκ ἔστιν.	In doctrinis autem non est similiter paralogismus, quia medium semper est dupliciter, etenim de hoc omni, et hoc iterum de alio dicitur omni, quod autem praedicatur, non dicitur omne. Haec autem sunt ut est videre in intellectu; sed in orationibus latet, utrum omnis circulus figura sit, si scribatur autem manifestum est. Quid autem, sintne carmina circulus? manifestum quoniam non sunt.	In mathematics the formal fallacy is not so common, because it is the middle term in which the ambiguity lies, since the major is predicated of the whole of the middle and the middle of the whole of the minor (the predicate of course never has the prefix ‘all’); and in mathematics one can, so to speak, see these middle terms with an intellectual vision, while in dialectic the ambiguity may escape detection. E.g. ‘Is every circle a figure?’ A diagram shows that this is so, but the minor premiss ‘Are epics circles?’ is shown by the diagram to be false.
77b34 Οὐ δεῖ δ᾽ ἔνστασιν εἰς αὐτὸ φέρειν, ἂν ἦι ἡ πρότασις ↵ ἐπακτική. ὥσπερ γὰρ οὐδὲ πρότασίς ἐστιν ἣ μὴ ἔστιν ἐπὶ πλειόνων (οὐ γὰρ ἔσται ἐπὶ πάντων, ἐκ τῶν καθόλου δ᾽ ὁ συλλογισμόσ), δῆλον ὅτι οὐδ᾽ ἔνστασις. αἱ αὐταὶ γὰρ προτάσεις καὶ ἐνστάσεις· ἣν γὰρ φέρει ἔνστασιν, αὕτη γένοιτ᾽ ἂν πρότασις ἢ ἀποδεικτικὴ ἢ διαλεκτική.	Non oportet autem instantiam in ipsum ferre si sit propositio inductiva, sicut enim neque propositio est, quae non est in pluribus, non enim erit in omnibus, ex universalibus autem syllogismus manifestum est quod neque instantia est, eaedem enim sunt propositiones et instantiae, quam enim fert instantiam, haec fiet utique propositio demonstrativa, aut dialectica.	If a proof has an inductive minor premiss, one should not bring an ‘objection’ against it. For since every premiss must be applicable to a number of cases (otherwise it will not be true in every instance, which, since the syllogism proceeds from universals, it must be), then assuredly the same is true of an ‘objection’; since premisses and ‘objections’ are so far the same that anything which can be validly advanced as an ‘objection’ must be such that it could take the form of a premiss, either demonstrative or dialectical.
77b40 ↵ Συμβαίνει δ᾽ ἐνίους ἀσυλλογίστως λέγειν διὰ τὸ λαμβάνειν ἀμφοτέροις τὰ ἑπόμενα, οἷον καὶ ὁ Καινεὺς ποιεῖ, [78a]ὅτι τὸ πῦρ ἐν τῆι πολλαπλασίαι ἀναλογίαι· καὶ γὰρ τὸ πῦρ ταχὺ γεννᾶται, ὥς φησι, καὶ αὕτη ἡ ἀναλογία. οὕτω δ᾽ οὐκ ἔστι συλλογισμός· ἀλλ᾽ εἰ τῆι ταχίστηι ἀναλογίαι ἕπεται ἡ πολλαπλάσιος καὶ τῶι πυρὶ ἡ ταχίστη ἐν τῆι κινήσει ↵ ἀναλογία.	Contingit autem quosdam non syllogistice dicere propter id quod accipiunt utrisque consequentia, ut et Caeneus facit quod ignis in multiplicata analogia fit, et namque ignis cito generatur, sicut dicit, et haec est analogia. Sic autem non est syllogismus, sed si velocissima analogia sequitur multiplicata, et ignem velocissima in mutatione analogia.	On the other hand, arguments formally illogical do sometimes occur through taking as middles mere attributes of the major and minor terms. An instance of this is Caeneus’ proof that fire increases in geometrical proportion: ‘Fire’, he argues, ‘increases rapidly, and so does geometrical proportion’. There is no syllogism so, but there is a syllogism if the most rapidly increasing proportion is geometrical and the most rapidly increasing proportion is attributable to fire in its motion.
78a5 ἐνίοτε μὲν οὖν οὐκ ἐνδέχεται συλλογίσασθαι ἐκ τῶν εἰλημμένων, ὁτὲ δ᾽ ἐνδέχεται, ἀλλ᾽ οὐχ ὁρᾶται.	Aliquando quidem igitur syllogizare non contingit ex acceptis, aliquando vero contingit, sed non videtur.	Sometimes, no doubt, it is impossible to reason from premisses predicating mere attributes: but sometimes it is possible, though the possibility is overlooked.
78a8 Εἰ δ᾽ ἦν ἀδύνατον ἐκ ψεύδους ἀληθὲς δεῖξαι, ῥάιδιον ἂν ἦν τὸ ἀναλύειν· ἀντέστρεφε γὰρ ἂν ἐξ ἀνάγκης. ἔστω γὰρ τὸ Α ὄν· τούτου δ᾽ ὄντος ταδὶ ἔστιν, ἃ οἶδα ὅτι ἔστιν, οἷον τὸ Β. ἐκ ↵ τούτων ἄρα δείξω ὅτι ἔστιν ἐκεῖνο.	Si autem esset impossibile ex falsis verum monstrare, facile utique esset resolvere, converterentur enim ex necessitate. Sit enim a ens, hoc autem cum sit, ea utique sunt quae novi quoniam sunt ut b, ex his igitur monstrabo quoniam illud est.	If false premisses could never give true conclusions ‘resolution’ would be easy, for premisses and conclusion would in that case inevitably reciprocate. I might then argue thus: let A be an existing fact; let the existence of A imply such and such facts actually known to me to exist, which we may call B. I can now, since they reciprocate, infer A from B.
78a10 ἀντιστρέφει δὲ μᾶλλον τὰ ἐν τοῖς μαθήμασιν, ὅτι οὐδὲν συμβεβηκὸς λαμβάνουσιν (ἀλλὰ καὶ τούτωι διαφέρουσι τῶν ἐν τοῖς διαλόγοισ) ἀλλ᾽ ὁρισμούς.	Convertuntur autem magis quae sunt in mathematicis, quoniam nullum recipiunt accidens (et in hoc differunt ab iis quae sunt in dialogis), sed definitiones.	Reciprocation of premisses and conclusion is more frequent in mathematics, because mathematics takes definitions, but never an accident, for its premisses-a second characteristic distinguishing mathematical reasoning from dialectical disputations.
78a13 Αὔξεται δ᾽ οὐ διὰ τῶν μέσων, ἀλλὰ τῶι προσλαμ↵ βάνειν, οἷον τὸ Α τοῦ Β, τοῦτο δὲ τοῦ Γ, πάλιν τοῦτο τοῦ Δ, καὶ τοῦτ᾽ εἰς ἄπειρον· καὶ εἰς τὸ πλάγιον, οἷον τὸ Α καὶ κατὰ τοῦ Γ καὶ κατὰ τοῦ Ε, οἷον ἔστιν ἀριθμὸς ποσὸς ἢ καὶ ἄπειρος τοῦτο ἐφ᾽ ὧι Α, ὁ περιττὸς ἀριθμὸς ποσὸς ἐφ᾽ οὗ Β, ἀριθμὸς περιττὸς ἐφ᾽ οὗ Γ· ἔστιν ἄρα τὸ Α κατὰ ↵ τοῦ Γ. καὶ ἔστιν ὁ ἄρτιος ποσὸς ἀριθμὸς ἐφ᾽ οὗ Δ, ὁ ἄρτιος ἀριθμὸς ἐφ᾽ οὗ Ε· ἔστιν ἄρα τὸ Α κατὰ τοῦ Ε.	Augentur autem non per media, sed in assumendo, ut a de b, hoc autem de c. Item hoc de d, et hoc in infinitum, et in latus, ut a de c, et de e, ut est numerus quantus, vel infinitus. Hoc autem in quo sit a numerus impar quantus in quo b, numerus impar in quo c, est et itaque a de c, et est par quantus numerus in quo sit d, par numerus in quo est e, est ergo a de e.	A science expands not by the interposition of fresh middle terms, but by the apposition of fresh extreme terms. E.g. A is predicated of B, B of C, C of D, and so indefinitely. Or the expansion may be lateral: e.g. one major A, may be proved of two minors, C and E. Thus let A represent number-a number or number taken indeterminately; B determinate odd number; C any particular odd number. We can then predicate A of C. Next let D represent determinate even number, and E even number. Then A is predicable of E.
c13	CAPUT X. De demonstratione propter quid, et quod est.	Chapter 13
78a22 Τὸ δ᾽ ὅτι διαφέρει καὶ τὸ διότι ἐπίστασθαι, πρῶτον μὲν ἐν τῆι αὐτῆι ἐπιστήμηι, καὶ ἐν ταύτηι διχῶς, ἕνα μὲν τρόπον ἐὰν μὴ δι᾽ ἀμέσων γίνηται ὁ συλλογισμός (οὐ γὰρ ↵ λαμβάνεται τὸ πρῶτον αἴτιον, ἡ δὲ τοῦ διότι ἐπιστήμη κατὰ τὸ πρῶτον αἴτιον), ἄλλον δὲ εἰ δι᾽ ἀμέσων μέν, ἀλλὰ μὴ διὰ τοῦ αἰτίου ἀλλὰ τῶν ἀντιστρεφόντων διὰ τοῦ γνωριμωτέρου. κωλύει γὰρ οὐδὲν τῶν ἀντικατηγορουμένων γνωριμώτερον εἶναι ἐνίοτε τὸ μὴ αἴτιον, ὥστ᾽ ἔσται διὰ τούτου ἡ ↵ ἀπόδειξις,	Sed quia differt, et propter quid scire, primum in eadem scientia, et in hac dupliciter. Uno quidem modo si non per immediata fiat syllogismus, non enim accipitur prima causa, sed quae propter quid scientia, est secundum primam causam. Alio modo si per immediata quidem, sed non per causam, sed per convertentia, et per notius, prohibet enim nihil aeque praedicantium, notius aliquando esse non causa, quare per hanc erit demonstratio.	Knowledge of the fact differs from knowledge of the reasoned fact. To begin with, they differ within the same science and in two ways: (1) when the premisses of the syllogism are not immediate (for then the proximate cause is not contained in them-a necessary condition of knowledge of the reasoned fact): (2) when the premisses are immediate, but instead of the cause the better known of the two reciprocals is taken as the middle; for of two reciprocally predicable terms the one which is not the cause may quite easily be the better known and so become the middle term of the demonstration.
78a30 οἷον ὅτι ἐγγὺς οἱ πλάνητες διὰ τοῦ μὴ στίλβειν. ἔστω ἐφ᾽ ὧι Γ πλάνητες, ἐφ᾽ ὧι Β τὸ μὴ στίλβειν, ἐφ᾽ ὧι Α τὸ ἐγγὺς εἶναι. ἀληθὲς δὴ τὸ Β κατὰ τοῦ Γ εἰπεῖν· οἱ γὰρ πλάνητες οὐ στίλβουσιν. ἀλλὰ καὶ τὸ Α κατὰ τοῦ Β· τὸ γὰρ μὴ στίλβον ἐγγύς ἐστι· τοῦτο δ᾽ εἰλήφθω δι᾽ ἐπαγω↵ γῆς ἢ δι᾽ αἰσθήσεως. ἀνάγκη οὖν τὸ Α τῶι Γ ὑπάρχειν, ὥστ᾽ ἀποδέδεικται ὅτι οἱ πλάνητες ἐγγύς εἰσιν. οὗτος οὖν ὁ συλλογισμὸς οὐ τοῦ διότι ἀλλὰ τοῦ ὅτι ἐστίν· οὐ γὰρ διὰ τὸ μὴ στίλβειν ἐγγύς εἰσιν, ἀλλὰ διὰ τὸ ἐγγὺς εἶναι οὐ στίλβουσιν.	Ut quod prope sint planetae, per illud quod non scintillare, sit in quo c planetae, in quo b non scintillare, in quo a prope esse, verum igitur est de c b dicere, planetae enim non scintillant, sed et a de b, non scintillans enim prope est. Hoc autem accipitur per inductionem, aut per sensum, necesse ergo a ipsi c inesse, quare demonstratum est quod planetae prope sunt. Hic ergo syllogismus non est propter quid, sed quia, non enim ex eo quod non scintillant prope sunt, sed propter illud quod prope sunt, non scintillant.	Thus (2) (a) you might prove as follows that the planets are near because they do not twinkle: let C be the planets, B not twinkling, A proximity. Then B is predicable of C; for the planets do not twinkle. But A is also predicable of B, since that which does not twinkle is near — we must take this truth as having been reached by induction or sense-perception. Therefore A is a necessary predicate of C; so that we have demonstrated that the planets are near. This syllogism, then, proves not the reasoned fact but only the fact; since they are not near because they do not twinkle, but, because they are near, do not twinkle.
78a39 ἐγχωρεῖ δὲ καὶ διὰ θατέρου θάτερον δειχθῆναι, καὶ ἔσται τοῦ διότι ἡ ἀπόδειξις, οἷον ἔστω τὸ Γ πλάνητες, ἐφ᾽ ὧι Β [78b]τὸ ἐγγὺς εἶναι, τὸ Α τὸ μὴ στίλβειν· ὑπάρχει δὴ καὶ τὸ Β τῶι Γ καὶ τὸ Α τῶι Β, ὥστε καὶ τῶι Γ τὸ Α [τὸ μὴ στίλβειν]. καὶ ἔστι τοῦ διότι ὁ συλλογισμός· εἴληπται γὰρ τὸ πρῶτον αἴτιον.	Possibile est autem et per alterum, alterum monstrare, et erit propter quid demonstratio, ut sit c planetae, in quo b prope esse, a non scintillare, est igitur b in c, quod est non scintillare, quare et in c, et erit propter quid syllogismus, accepta enim est prima causa.	The major and middle of the proof, however, may be reversed, and then the demonstration will be of the reasoned fact. Thus: let C be the planets, B proximity, A not twinkling. Then B is an attribute of C, and A-not twinkling-of B. Consequently A is predicable of C, and the syllogism proves the reasoned fact, since its middle term is the proximate cause.
78b3 πάλιν ὡς τὴν σελήνην δεικνύουσιν ὅτι σφαι↵ ροειδής, διὰ τῶν αὐξήσεων – εἰ γὰρ τὸ αὐξανόμενον οὕτω σφαιροειδές, αὐξάνει δ᾽ ἡ σελήνη, φανερὸν ὅτι σφαιροειδής – οὕτω μὲν οὖν τοῦ ὅτι γέγονεν ὁ συλλογισμός, ἀνάπαλιν δὲ τεθέντος τοῦ μέσου τοῦ διότι· οὐ γὰρ διὰ τὰς αὐξήσεις σφαιροειδής ἐστιν, ἀλλὰ διὰ τὸ σφαιροειδὴς εἶναι λαμβά↵ νει τὰς αὐξήσεις τοιαύτας. σελήνη ἐφ᾽ ὧι Γ, σφαιροειδὴς ἐφ᾽ ὧι Β, αὔξησις ἐφ᾽ ὧι Α.	Iterum sic lunam demonstrant quod quidem circularis sit per incrementa, si enim quod augetur sic circulare quidem sit, augetur autem sic luna, manifestum quoniam circularis, sic igitur ipsius, quia factus est syllogismus, e converso autem posito medio ipsius propter quod syllogismus fit. Non enim propter augmenta ipsius circularis est, sed quia circularis est, accipit augmenta huiusmodi. Luna sit in quo c, in quo b sit augmentum, sit in quo a circulare.	Another example is the inference that the moon is spherical from its manner of waxing. Thus: since that which so waxes is spherical, and since the moon so waxes, clearly the moon is spherical. Put in this form, the syllogism turns out to be proof of the fact, but if the middle and major be reversed it is proof of the reasoned fact; since the moon is not spherical because it waxes in a certain manner, but waxes in such a manner because it is spherical. (Let C be the moon, B spherical, and A waxing.)
78b10 ἐφ᾽ ὧν δὲ τὰ μέσα μὴ ἀντιστρέφει καὶ ἔστι γνωριμώτερον τὸ ἀναίτιον, τὸ ὅτι μὲν δείκνυται, τὸ διότι δ᾽ οὔ. Ἔτι ἐφ᾽ ὧν τὸ μέσον ἔξω τίθεται.	In quibus autem media non convertuntur, et est notius quod non est causa, quia monstratur, sed et propter quid, non amplius in quibus medium extra ponitur,	Again (b), in cases where the cause and the effect are not reciprocal and the effect is the better known, the fact is demonstrated but not the reasoned fact. 78b13 This also occurs (1) when the middle falls outside the major and minor,
καὶ γὰρ ἐν τούτοις τοῦ ὅτι καὶ οὐ τοῦ διότι ἡ ἀπόδειξις· οὐ ↵ γὰρ λέγεται τὸ αἴτιον.	etenim in his non propter quid, sed ipsius quia demonstratio, non enim dicitur causa.	for here too the strict cause is not given, and so the demonstration is of the fact, not of the reasoned fact.
78b14 οἷον διὰ τί οὐκ ἀναπνεῖ ὁ τοῖχος; ὅτι οὐ ζῶιον. εἰ γὰρ τοῦτο τοῦ μὴ ἀναπνεῖν αἴτιον, ἔδει τὸ ζῶιον εἶναι αἴτιον τοῦ ἀναπνεῖν, οἷον εἰ ἡ ἀπόφασις αἰτία τοῦ μὴ ὑπάρχειν, ἡ κατάφασις τοῦ ὑπάρχειν, ὥσπερ εἰ τὸ ἀσύμμετρα εἶναι τὰ θερμὰ καὶ τὰ ψυχρὰ τοῦ μὴ ὑγιαίνειν, τὸ ↵ σύμμετρα εἶναι τοῦ ὑγιαίνειν, – ὁμοίως δὲ καὶ εἰ ἡ κατάφασις τοῦ ὑπάρχειν, ἡ ἀπόφασις τοῦ μὴ ὑπάρχειν. ἐπὶ δὲ τῶν οὕτως ἀποδεδομένων οὐ συμβαίνει τὸ λεχθέν· οὐ γὰρ ἅπαν ἀναπνεῖ ζῶιον.	Ut propter quid non respirat paries, quia non est animal, si enim non respirandi causa est hoc, oportet esse animal causa respirandi. Ut si negatio causa est ipsius non esse, affirmatio causa est ipsius esse, sicut si sine mensura esse calida et frigida, causa est non sanandi, et mensura huius causa erit sanandi. Similiter autem et si affirmatio est causa ipsius esse, et negatio ipsius non esse. In his autem sic demonstratis non contingit quod dictum est, non enim omne animal respirat.	For example, the question ‘Why does not a wall breathe?’ might be answered, ‘Because it is not an animal’; but that answer would not give the strict cause, because if not being an animal causes the absence of respiration, then being an animal should be the cause of respiration, according to the rule that if the negation of causes the non-inherence of y, the affirmation of x causes the inherence of y; e.g. if the disproportion of the hot and cold elements is the cause of ill health, their proportion is the cause of health; and conversely, if the assertion of x causes the inherence of y, the negation of x must cause y’s non-inherence. But in the case given this consequence does not result; for not every animal breathes.
78b24 ὁ δὲ συλλογισμὸς γίνεται τῆς τοιαύτης αἰτίας ἐν τῶι μέσωι σχήματι. οἷον ἔστω τὸ Α ζῶιον, ἐφ᾽ ↵ ὧι Β τὸ ἀναπνεῖν, ἐφ᾽ ὧι Γ τοῖχος. τῶι μὲν οὖν Β παντὶ ὑπάρχει τὸ Α (πᾶν γὰρ τὸ ἀναπνέον ζῶιον), τῶι δὲ Γ οὐθενί, ὥστε οὐδὲ τὸ Β τῶι Γ οὐθενί· οὐκ ἄρα ἀναπνεῖ ὁ τοῖχος.	Syllogismus autem huius causae est in media figura, ut sit a animal, in quo b respirare, in quo c paries, in quo b quidem igitur omni est a, omne enim respirans est animal, in c autem nullo, quare neque b in c nullo est, non igitur respirat, paries.	A syllogism with this kind of cause takes place in the second figure. Thus: let A be animal, B respiration, C wall. Then A is predicable of all B (for all that breathes is animal), but of no C; and consequently B is predicable of no C; that is, the wall does not breathe.
78b27 ἐοίκασι δ᾽ αἱ τοιαῦται τῶν αἰτιῶν τοῖς καθ᾽ ὑπερβολὴν εἰρημένοις· τοῦτο δ᾽ ἔστι τὸ πλέον ἀποστήσαντα τὸ μέ↵ σον εἰπεῖν, οἷον τὸ τοῦ Ἀναχάρσιος, ὅτι ἐν Σκύθαις οὐκ εἰσὶν αὐλητρίδες, οὐδὲ γὰρ ἄμπελοι.	Comparantur autem huiusmodi causae secundum excellentiam dictis, hoc autem est plurimum distans medium dicere, sicut enim illud est quod Anarcharsidis, quod in Scythis non sunt sibilatores, neque enim vites.	Such causes are like far-fetched explanations, which precisely consist in making the cause too remote, as in Anacharsis’ account of why the Scythians have no flute-players; namely because they have no vines.
78b32 Κατὰ μὲν δὴ τὴν αὐτὴν ἐπιστήμην καὶ κατὰ τὴν τῶν μέσων θέσιν αὗται διαφοραί εἰσι τοῦ ὅτι πρὸς τὸν τοῦ διότι συλλογισμόν·	Secundum igitur eamdem scientiam, et secundum mediorum positionem, hae differentiae sunt ipsius, quia ad eum qui propter quid est syllogismum.	Thus, then, do the syllogism of the fact and the syllogism of the reasoned fact differ within one science and according to the position of the middle terms.
78b34 ἄλλον δὲ τρόπον διαφέρει τὸ διότι τοῦ ὅτι ↵ τῶι δι᾽ ἄλλης ἐπιστήμης ἑκάτερον θεωρεῖν.	Alio autem modo differt propter quid ab ipso quia, quod est per aliam scientiam utrumque speculari.	But there is another way too in which the fact and the reasoned fact differ, and that is when they are investigated respectively by different sciences.
78b35 τοιαῦτα δ᾽ ἐστὶν ὅσα οὕτως ἔχει πρὸς ἄλληλα ὥστ᾽ εἶναι θάτερον ὑπὸ θάτερον, οἷον τὰ ὀπτικὰ πρὸς γεωμετρίαν καὶ τὰ μηχανικὰ πρὸς στερεομετρίαν καὶ τὰ ἁρμονικὰ πρὸς ἀριθμητικὴν καὶ τὰ φαινόμενα πρὸς ἀστρολογικήν.	Huiusmodi autem sunt, quaecunque sic se habent ad invicem, quod alterum sub altero est, ut perspectiva ad geometriam, et machinativa ad stereometriam, et harmonica ad arithmeticam, et apparentia ad astrologiam.	This occurs in the case of problems related to one another as subordinate and superior, as when optical problems are subordinated to geometry, mechanical problems to stereometry, harmonic problems to arithmetic, the data of observation to astronomy.
79a1 σχεδὸν δὲ συνώνυμοί εἰσιν ἔνιαι τούτων τῶν ἐπιστημῶν, οἷον ἀστρολογία ἥ τε μα[79a]θηματικὴ καὶ ἡ ναυτική, καὶ ἁρμονικὴ ἥ τε μαθηματικὴ καὶ ἡ κατὰ τὴν ἀκοήν.	Fere quidem univocae sunt harum quaedam scientiarum, ut astrologia mathematica, et quae navalis est, et harmonica mathematica, quae est et secundum auditum.	(Some of these sciences bear almost the same name; e.g. mathematical and nautical astronomy, mathematical and acoustical harmonics.)
79a3 ἐνταῦθα γὰρ τὸ μὲν ὅτι τῶν αἰσθητικῶν εἰδέναι, τὸ δὲ διότι τῶν μαθηματικῶν· οὗτοι γὰρ ἔχουσι τῶν αἰτίων τὰς ἀποδείξεις, καὶ πολλάκις οὐκ ἴσασι τὸ ὅτι, κα↵ θάπερ οἱ τὸ καθόλου θεωροῦντες πολλάκις ἔνια τῶν καθ᾽ ἕκαστον οὐκ ἴσασι δι᾽ ἀνεπισκεψίαν.	Hic enim ipsum, quia sensibilium est scire, sed propter quid, mathematicorum. Hi enim habent causarum demonstrationes, et frequenter nesciunt ipsum, quia sicut illi universale considerantes, saepe quaedam singularium nesciunt, propter id quod non intendunt.	Here it is the business of the empirical observers to know the fact, of the mathematicians to know the reasoned fact; for the latter are in possession of the demonstrations giving the causes, and are often ignorant of the fact: just as we have often a clear insight into a universal, but through lack of observation are ignorant of some of its particular instances.
ἔστι δὲ ταῦτα ὅσα ἕτερόν τι ὄντα τὴν οὐσίαν κέχρηται τοῖς εἴδεσιν. τὰ γὰρ μαθήματα περὶ εἴδη ἐστίν· οὐ γὰρ καθ᾽ ὑποκειμένου τινός· εἰ γὰρ καὶ καθ᾽ ὑποκειμένου τινὸς τὰ γεωμετρικά ἐστιν, ἀλλ᾽ οὐχ ἧι γε καθ᾽ ὑποκειμέ↵ νου.	Sunt autem haec quaecunque alterum quiddam sunt secundum substantiam, et utuntur speciebus. Mathematicae enim circa species sunt, non enim de subiecto aliquo, si enim de aliquo subiecto geometrica sunt, sed non sunt secundum quod geometrica sunt.	These connexions have a perceptible existence though they are manifestations of forms. For the mathematical sciences concern forms: they do not demonstrate properties of a substratum, since, even though the geometrical subjects are predicable as properties of a perceptible substratum, it is not as thus predicable that the mathematician demonstrates properties of them.
79a10 ἔχει δὲ καὶ πρὸς τὴν ὀπτικήν, ὡς αὕτη πρὸς τὴν γεωμετρίαν, ἄλλη πρὸς ταύτην, οἷον τὸ περὶ τῆς ἴριδος· τὸ μὲν γὰρ ὅτι φυσικοῦ εἰδέναι, τὸ δὲ διότι ὀπτικοῦ, ἢ ἁπλῶς ἢ τοῦ κατὰ τὸ μάθημα.	Habet autem se ad perspectivam sicut haec ad geometriam, et alia ad istam, ut id quod est de iride ipsum quidem quia est scire physici, sed propter quid perspectivi, aut simpliciter, aut secundum disciplinam.	As optics is related to geometry, so another science is related to optics, namely the theory of the rainbow. Here knowledge of the fact is within the province of the natural philosopher, knowledge of the reasoned fact within that of the optician, either qua optician or qua mathematical optician.
79a13 πολλαὶ δὲ καὶ τῶν μὴ ὑπ᾽ ἀλλήλας ἐπιστημῶν ἔχουσιν οὕτως, οἷον ἰατρικὴ πρὸς γεωμετρίαν· ὅτι ↵ μὲν γὰρ τὰ ἕλκη τὰ περιφερῆ βραδύτερον ὑγιάζεται, τοῦ ἰατροῦ εἰδέναι, διότι δὲ τοῦ γεωμέτρου.	Multae autem, et non sub se invicem scientiarum, habent sic, ut medicina ad geometriam, quod enim vulnera circularia tardius sanantur, medici est scire quia, propter quid autem geometrae.	Many sciences not standing in this mutual relation enter into it at points; e.g. medicine and geometry: it is the physician’s business to know that circular wounds heal more slowly, the geometer’s to know the reason why.
c14	CAPUT XI. Primam figuram maximae scientiae esse accommodatam.	Chapter 14
79a17 Τῶν δὲ σχημάτων ἐπιστημονικὸν μάλιστα τὸ πρῶτόν ἐστιν. αἵ τε γὰρ μαθηματικαὶ τῶν ἐπιστημῶν διὰ τούτου φέρουσι τὰς ἀποδείξεις, οἷον ἀριθμητικὴ καὶ γεωμετρία καὶ ↵ ὀπτική, καὶ σχεδὸν ὡς εἰπεῖν ὅσαι τοῦ διότι ποιοῦνται τὴν σκέψιν· ἢ γὰρ ὅλως ἢ ὡς ἐπὶ τὸ πολὺ καὶ ἐν τοῖς πλείστοις διὰ τούτου τοῦ σχήματος ὁ τοῦ διότι συλλογισμός.	Figurarum autem faciens scire maxime, prima est. Mathematicae enim scientiarum per hanc demonstrationes ferunt, ut arithmetica, et geometria, et perspectiva, et fere est dicere quaecunque propter quid speculari faciunt considerationem, aut enim omnino, aut sicut frequentius, et in plurimis, per hanc figuram (qui est propter quid) fit syllogismus,	Of all the figures the most scientific is the first. Thus, it is the vehicle of the demonstrations of all the mathematical sciences, such as arithmetic, geometry, and optics, and practically all of all sciences that investigate causes: for the syllogism of the reasoned fact is either exclusively or generally speaking and in most cases in this figure
ὥστε κἂν διὰ τοῦτ᾽ εἴη μάλιστα ἐπιστημονικόν· κυριώτατον γὰρ τοῦ εἰδέναι τὸ διότι θεωρεῖν. εἶτα τὴν τοῦ τί ἐστιν ἐπιστήμην ↵ διὰ μόνου τούτου θηρεῦσαι δυνατόν.	quare et propter hoc erit maxime faciens scire, propriissimum enim scire, propter quid speculari. Postea ipsius quod quid est scientiam, per hanc solam venari possibile est.
ἐν μὲν γὰρ τῶι μέσωι σχήματι οὐ γίνεται κατηγορικὸς συλλογισμός, ἡ δὲ τοῦ τί ἐστιν ἐπιστήμη καταφάσεως· ἐν δὲ τῶι ἐσχάτωι γίνεται μὲν ἀλλ᾽ οὐ καθόλου, τὸ δὲ τί ἐστι τῶν καθόλου ἐστίν· οὐ γὰρ πῆι ἐστι ζῶιον δίπουν ὁ ἄνθρωπος.	In media enim figura non fit praedicativus syllogismus, sed ipsius quod quid est scientia, affirmationis est, in ultima autem fit quidem, sed non universaliter, sed quod quid est, universalium est, non enim quodammodo est animal bipes homo.	In the second figure no affirmative conclusion is possible, and knowledge of a thing’s essence must be affirmative; while in the third figure the conclusion can be affirmative, but cannot be universal, and essence must have a universal character: e.g. man is not two-footed animal in any qualified sense, but universally.
79a30 ἔτι τοῦτο μὲν ἐκείνων ↵ οὐδὲν προσδεῖται, ἐκεῖνα δὲ διὰ τούτου καταπυκνοῦται καὶ αὔξεται, ἕως ἂν εἰς τὰ ἄμεσα ἔλθηι. φανερὸν οὖν ὅτι κυριώτατον τοῦ ἐπίστασθαι τὸ πρῶτον σχῆμα.	Amplius haec quidem illis nihil indiget, illae autem per hanc densantur, et augmentantur quousque utique ad immediata veniant, manifestum igitur est quod maxime propria scientiae est prima figura.	Finally, the first figure has no need of the others, while it is by means of the first that the other two figures are developed, and have their intervals closepacked until immediate premisses are reached. Clearly, therefore, the first figure is the primary condition of knowledge.
c15		Chapter 15
79a33 Ὥσπερ δὲ ὑπάρχειν τὸ Α τῶι Β ἐνεδέχετο ἀτόμως, οὕτω καὶ μὴ ὑπάρχειν ἐγχωρεῖ. λέγω δὲ τὸ ἀτόμως ὑπάρχειν ἢ ↵ μὴ ὑπάρχειν τὸ μὴ εἶναι αὐτῶν μέσον· οὕτω γὰρ οὐκέτι ἔσται κατ᾽ ἄλλο τὸ ὑπάρχειν ἢ μὴ ὑπάρχειν.	Sicut autem a esse in b contingit indivisibiliter, sic et non esse possibile est; dico autem indivisibiliter esse, vel non esse eo quod non est aliquid eorum medium, sic enim non erit secundum aliud esse.	Just as an attribute A may (as we saw) be atomically connected with a subject B, so its disconnexion may be atomic. I call ‘atomic’ connexions or disconnexions which involve no intermediate term; since in that case the connexion or disconnexion will not be mediated by something other than the terms themselves.
79a36 ὅταν μὲν οὖν ἢ τὸ Α ἢ τὸ Β ἐν ὅλωι τινὶ ἦι, ἢ καὶ ἄμφω, οὐκ ἐνδέχεται τὸ Α τῶι Β πρώτως μὴ ὑπάρχειν. ἔστω γὰρ τὸ Α ἐν ὅλωι τῶι Γ. οὐκοῦν εἰ τὸ Β μὴ ἔστιν ἐν ὅλωι τῶι Γ (ἐγχωρεῖ γὰρ τὸ μὲν ↵ Α εἶναι ἔν τινι ὅλωι, τὸ δὲ Β μὴ εἶναι ἐν τούτωι), συλλογισμὸς ἔσται τοῦ μὴ ὑπάρχειν τὸ Α τῶι Β· εἰ γὰρ τῶι μὲν [79b]Α παντὶ τὸ Γ, τῶι δὲ Β μηδενί, οὐδενὶ τῶι Β τὸ Α. ὁμοίως δὲ καὶ εἰ τὸ Β ἐν ὅλωι τινί ἐστιν, οἷον ἐν τῶι Δ· τὸ μὲν γὰρ Δ παντὶ τῶι Β ὑπάρχει, τὸ δὲ Α οὐδενὶ τῶι Δ, ὥστε τὸ Α οὐδενὶ τῶι Β ὑπάρξει διὰ συλλογισμοῦ.	Cum igitur aut a aut b in toto aliquo sit, aut etiam ambo, non contingit a in b primo non esse. Sit enim a in toto c, igitur si b non est in toto c (potest enim a quidem esse in quodam toto, sed b non esse in hoc), syllogismus erit quod non sit a in b; si enim in a quidem omni est c, in b autem nullo est, in nullo b est a. Similiter autem et si b in aliquo toto est, ut in b, d enim in omni b est, in a autem nullo, d, quare a in nullo b erit per syllogismum. Eodem autem modo monstrabitur etsi utraque in toto aliquo sit.	It follows that if either A or B, or both A and B, have a genus, their disconnexion cannot be primary. Thus: let C be the genus of A. Then, if C is not the genus of B-for A may well have a genus which is not the genus of B-there will be a syllogism proving A’s disconnexion from B thus: all A is C, no B is C, therefore no B is A. Or if it is B which has a genus D, we have all B is D, no D is A, therefore no B is A, by syllogism; and the proof will be similar if both A and B have a genus.
79b5 τὸν αὐτὸν ↵ δὲ τρόπον δειχθήσεται καὶ εἰ ἄμφω ἐν ὅλωι τινί ἐστιν. ὅτι δ᾽ ἐνδέχεται τὸ Β μὴ εἶναι ἐν ὧι ὅλωι ἐστὶ τὸ Α, ἢ πάλιν τὸ Α ἐν ὧι τὸ Β, φανερὸν ἐκ τῶν συστοιχιῶν, ὅσαι μὴ ἐπαλλάττουσιν ἀλλήλαις. εἰ γὰρ μηδὲν τῶν ἐν τῆι Α Γ Δ συστοιχίαι κατὰ μηδενὸς κατηγορεῖται τῶν ἐν τῆι Β Ε Ζ, τὸ ↵ δ᾽ Α ἐν ὅλωι ἐστὶ τῶι Θ συστοίχωι ὄντι, φανερὸν ὅτι τὸ Β οὐκ ἔσται ἐν τῶι Θ· ἐπαλλάξουσι γὰρ αἱ συστοιχίαι.	Quod autem contingit b non esse in quo toto est a, aut iterum a in quo est b, manifestum est ex coordinationibus, quaecunque non commutantur ad invicem, si enim nihil eorum quae sunt in a c d coordinatione, de nullo praedicatur eorum quae sunt in b e f, a autem in toto g, sic coordinatione existente, manifestum est quod b non erit in g, commutarentur enim coordinationes, similiter autem est et si b in toto aliquo est.	That the genus of A need not be the genus of B and vice versa, is shown by the existence of mutually exclusive coordinate series of predication. If no term in the series ACD...is predicable of any term in the series BEF...,and if G-a term in the former series-is the genus of A, clearly G will not be the genus of B; since, if it were, the series would not be mutually exclusive. So also if B has a genus, it will not be the genus of A.
79b12 ὁμοίως δὲ καὶ εἰ τὸ Β ἐν ὅλωι τινί ἐστιν. ἐὰν δὲ μηδέτερον ἦι ἐν ὅλωι μηδενί, μὴ ὑπάρχηι δὲ τὸ Α τῶι Β, ἀνάγκη ἀτόμως μὴ ὑπάρχειν. εἰ γὰρ ἔσται τι μέσον, ἀνάγκη θάτερον αὐ↵ τῶν ἐν ὅλωι τινὶ εἶναι. ἢ γὰρ ἐν τῶι πρώτωι σχήματι ἢ ἐν τῶι μέσωι ἔσται ὁ συλλογισμός. εἰ μὲν οὖν ἐν τῶι πρώτωι, τὸ Β ἔσται ἐν ὅλωι τινί (καταφατικὴν γὰρ δεῖ τὴν πρὸς τοῦτο γενέσθαι πρότασιν), εἰ δ᾽ ἐν τῶι μέσωι, ὁπότερον ἔτυχεν (πρὸς ἀμφοτέροις γὰρ ληφθέντος τοῦ στερητικοῦ γίνεται συλ↵ λογισμός· ἀμφοτέρων δ᾽ ἀποφατικῶν οὐσῶν οὐκ ἔσται).	Si vero neutrum sit in toto aliquo nullo, non sit autem a in b, necesse est indivisibiliter non esse; si enim erit aliquod medium, necesse est alterum horum in quodam toto esse, aut enim in prima figura, aut in media erit syllogismus, si quidem igitur in prima figura b, erit in toto aliquo, affirmativam enim ad hoc oportet fieri propositionem. Si vero in media, utrum contingit, ad utrumque enim posito privativo, erit syllogismus, cum autem utraque privativa sit, non erit,	If, on the other hand, neither A nor B has a genus and A does not inhere in B, this disconnexion must be atomic. If there be a middle term, one or other of them is bound to have a genus, for the syllogism will be either in the first or the second figure. If it is in the first, B will have a genus-for the premiss containing it must be affirmative: if in the second, either A or B indifferently, since syllogism is possible if either is contained in a negative premiss, but not if both premisses are negative.
79b21 Φανερὸν οὖν ὅτι ἐνδέχεταί τε ἄλλο ἄλλωι μὴ ὑπάρχειν ἀτόμως, καὶ πότ᾽ ἐνδέχεται καὶ πῶς, εἰρήκαμεν.	manifestum igitur est quod contingit et aliud in alio non esse indivisibiliter, et quando contingit, et quo modo diximus.	Hence it is clear that one thing may be atomically disconnected from another, and we have stated when and how this is possible.
c16	CAPUT XII. De ignorantia et syllogismo imperitiae eorum quae primo immediateque insunt.	Chapter 16
79b23 Ἄγνοια δ᾽ ἡ μὴ κατ᾽ ἀπόφασιν ἀλλὰ κατὰ διάθεσιν λεγομένη ἔστι μὲν ἡ διὰ συλλογισμοῦ γινομένη ἀπάτη, ↵ αὕτη δ᾽ ἐν μὲν τοῖς πρώτως ὑπάρχουσιν ἢ μὴ ὑπάρχουσι συμβαίνει διχῶς· ἢ γὰρ ὅταν ἁπλῶς ὑπολάβηι ὑπάρχειν ἢ μὴ ὑπάρχειν, ἢ ὅταν διὰ συλλογισμοῦ λάβηι τὴν ὑπόληψιν.	Ignorantia autem non secundum negationem, sed secundum dispositionem dicta, est quidem per syllogismum facta deceptio. Haec autem in iis, quae insunt primo, aut non insunt, contingit dupliciter, aut enim est cum simpliciter accipiat esse, vel non esse, aut cum per syllogismum accipiat opinionem,	Ignorance-defined not as the negation of knowledge but as a positive state of mind-is error produced by inference. (1) Let us first consider propositions asserting a predicate’s immediate connexion with or disconnexion from a subject. Here, it is true, positive error may befall one in alternative ways; for it may arise where one directly believes a connexion or disconnexion as well as where one’s belief is acquired by inference.
79b28 τῆς μὲν οὖν ἁπλῆς ὑπολήψεως ἁπλῆ ἡ ἀπάτη, τῆς δὲ διὰ συλλογισμοῦ πλείους. μὴ ὑπαρχέτω γὰρ τὸ Α μη↵ δενὶ τῶι Β ἀτόμως· οὐκοῦν ἐὰν συλλογίζηται ὑπάρχειν τὸ Α τῶι Β, μέσον λαβὼν τὸ Γ, ἠπατημένος ἔσται διὰ συλλογισμοῦ.	simplicis quidem ergo opinionis simplex deceptio, sed quae est per syllogismum plures sunt. Non sit enim a in nullo b indivisibiliter, ergo si syllogizet a esse in b medium accipiens c, deceptus erit per syllogismum,	The error, however, that consists in a direct belief is without complication; but the error resulting from inference-which here concerns us-takes many forms. Thus, let A be atomically disconnected from all B: then the conclusion inferred through a middle term C, that all B is A, will be a case of error produced by syllogism.
79b31 ἐνδέχεται μὲν οὖν ἀμφοτέρας τὰς προτάσεις εἶναι ψευδεῖς, ἐνδέχεται δὲ τὴν ἑτέραν μόνον. εἰ γὰρ μήτε τὸ Α μηδενὶ τῶν Γ ὑπάρχει μήτε τὸ Γ μηδενὶ τῶν Β, εἴ↵ ληπται δ᾽ ἑκατέρα ἀνάπαλιν, ἄμφω ψευδεῖς ἔσονται.	contingit igitur utrasque propositiones esse falsas, contingit autem alteram solum esse falsam, si enim neque a in nullo c erit, neque c in nullo b erit, accepta autem utraque e contrario, utraeque enim erunt falsae.	Now, two cases are possible. Either (a) both premisses, or (b) one premiss only, may be false. (a) If neither A is an attribute of any C nor C of any B, whereas the contrary was posited in both cases, both premisses will be false.
79b34 ἐγχωρεῖ δ᾽ οὕτως ἔχειν τὸ Γ πρὸς τὸ Α καὶ Β ὥστε μήτε ὑπὸ τὸ Α εἶναι μήτε καθόλου τῶι Β. τὸ μὲν γὰρ Β ἀδύνατον εἶναι ἐν ὅλωι τινί (πρώτως γὰρ ἐλέγετο αὐτῶι τὸ Α μὴ ὑπάρχειν), τὸ δὲ Α οὐκ ἀνάγκη πᾶσι τοῖς οὖσιν εἶναι καθόλου, ↵ ὥστ᾽ ἀμφότεραι ψευδεῖς.	Potest autem sic se habere c ad a, et ad b, quod neque c sub a sit, neque in b universaliter: b quidem impossibile est esse in aliquo toto, primum enim dictum est in ipso a b non esse, a autem non necesse est omnibus inesse, quae sunt c universaliter, quare utraeque falsae sunt.	(C may quite well be so related to A and B that C is neither subordinate to A nor a universal attribute of B: for B, since A was said to be primarily disconnected from B, cannot have a genus, and A need not necessarily be a universal attribute of all things. Consequently both premisses may be false.)
79b40 ἀλλὰ καὶ τὴν ἑτέραν ἐνδέχεται ἀληθῆ λαμβάνειν, οὐ μέντοι ὁποτέραν ἔτυχεν, ἀλλὰ τὴν [80a]Α Γ· ἡ γὰρ Γ Β πρότασις ἀεὶ ψευδὴς ἔσται διὰ τὸ ἐν μηδενὶ εἶναι τὸ Β, τὴν δὲ Α Γ ἐγχωρεῖ, οἷον εἰ τὸ Α καὶ τῶι Γ καὶ τῶι Β ὑπάρχει ἀτόμως (ὅταν γὰρ πρώτως κατηγορῆται ταὐτὸ πλειόνων, οὐδέτερον ἐν οὐδετέρωι ἔσται). διαφέ↵ ρει δ᾽ οὐδέν, οὐδ᾽ εἰ μὴ ἀτόμως ὑπάρχει.	Sed alteram contingit veram accipere, non tamen quamlibet contingit, sed quae est a c, nam c b propositio semper falsa erit, propter id quod c in nullo b est, sed quae est a c potest, ut si a et in c, et in b indivisibiliter est, cum enim primum praedicetur idem de pluribus, neutrum in neutro erit. Differt autem nihil, neque si non indivisibiliter insit, ipsius quidem esse,	On the other hand, (b) one of the premisses may be true, though not either indifferently but only the major A-C since, B having no genus, the premiss C-B will always be false, while A-C may be true. This is the case if, for example, A is related atomically to both C and B; because when the same term is related atomically to more terms than one, neither of those terms will belong to the other. It is, of course, equally the case if A-C is not atomic.
80a6 Ἡ μὲν οὖν τοῦ ὑπάρχειν ἀπάτη διὰ τούτων τε καὶ οὕτω γίνεται μόνως (οὐ γὰρ ἦν ἐν ἄλλωι σχήματι τοῦ ὑπάρχειν συλλογισμόσ), ἡ δὲ τοῦ μὴ ὑπάρχειν ἔν τε τῶι πρώτωι καὶ ἐν τῶι μέσωι σχήματι.	igitur deceptio per ista, et sic fit solum, non enim erat in alia figura ipsius esse syllogismus,	Error of attribution, then, occurs through these causes and in this form only-for we found that no syllogism of universal attribution was possible in any figure but the first.
80a8 πρῶτον οὖν εἴπωμεν ποσα↵ χῶς ἐν τῶι πρώτωι γίνεται, καὶ πῶς ἐχουσῶν τῶν προτάσεων.	qui vero ipsius non esse, in prima figura, et in media est, primum igitur dicamus quot modis in prima fit, et quomodo se habentibus propositionibus.	On the other hand, an error of non-attribution may occur either in the first or in the second figure. Let us therefore first explain the various forms it takes in the first figure and the character of the premisses in each case.
80a11 ἐνδέχεται μὲν οὖν ἀμφοτέρων ψευδῶν οὐσῶν, οἷον εἰ τὸ Α καὶ τῶι Γ καὶ τῶι Β ὑπάρχει ἀτόμως· ἐὰν γὰρ ληφθῆι τὸ μὲν Α τῶι Γ μηδενί, τὸ δὲ Γ παντὶ τῶι Β, ψευδεῖς αἱ προτάσεις.	Contingit igitur utrisque falsis, ut si a et in b, et in c indivisibiliter sit, si enim accipiatur a in c nullo esse, c autem in omni b, falsae sunt propositiones.	(c) It may occur when both premisses are false; e.g. supposing A atomically connected with both C and B, if it be then assumed that no C is and all B is C, both premisses are false.
80a14 ἐνδέχεται δὲ καὶ τῆς ἑτέρας ψευδοῦς οὔσης, ↵ καὶ ταύτης ὁποτέρας ἔτυχεν. ἐγχωρεῖ γὰρ τὴν μὲν Α Γ ἀληθῆ εἶναι, τὴν δὲ Γ Β ψευδῆ, τὴν μὲν Α Γ ἀληθῆ ὅτι οὐ πᾶσι τοῖς οὖσιν ὑπάρχει τὸ Α, τὴν δὲ Γ Β ψευδῆ ὅτι ἀδύνατον ὑπάρχειν τῶι Β τὸ Γ, ὧι μηδενὶ ὑπάρχει τὸ Α· οὐ γὰρ ἔτι ἀληθὴς ἔσται ἡ Α Γ πρότασις· ἅμα δέ, εἰ καὶ ↵ εἰσὶν ἀμφότεραι ἀληθεῖς, καὶ τὸ συμπέρασμα ἔσται ἀληθές.	Contingit autem et altera falsa, et hac quacunque contingente, potest enim quae est a c vera esse, quae vero b c falsa, sed quae est a c vera, quoniam non in omnibus quae sunt, inest a, sed quae est c b falsa esse, quoniam impossibile est esse in b c, in quo nullo est a, non enim etiam vera erit, quae est a c propositio, simul autem et si sint utraeque verae, et conclusio vera erit, sed etiam quae est c b.	(d) It is also possible when one is false. This may be either premiss indifferently. A-C may be true, C-B false-A-C true because A is not an attribute of all things, C-B false because C, which never has the attribute A, cannot be an attribute of B; for if C-B were true, the premiss A-C would no longer be true, and besides if both premisses were true, the conclusion would be true.
80a20 ἀλλὰ καὶ τὴν Γ Β ἐνδέχεται ἀληθῆ εἶναι τῆς ἑτέρας οὔσης ψευδοῦς, οἷον εἰ τὸ Β καὶ ἐν τῶι Γ καὶ ἐν τῶι Α ἐστίν· ἀνάγκη γὰρ θάτερον ὑπὸ θάτερον εἶναι, ὥστ᾽ ἂν λάβηι τὸ Α μηδενὶ τῶι Γ ὑπάρχειν, ψευδὴς ἔσται ἡ πρότασις. φα↵ νερὸν οὖν ὅτι καὶ τῆς ἑτέρας ψευδοῦς οὔσης καὶ ἀμφοῖν ἔσται ψευδὴς ὁ συλλογισμός.	Contingit veram esse cum altera sit falsa, ut si b in c sit, et in a, necesse est enim alterum sub altero esse, quare si accipiatur a in nullo c esse, falsa erit propositio, palam igitur quoniam et cum altera sit falsa, et in utrisque, falsus syllogismus erit.	Or again, C-B may be true and A-C false; e.g. if both C and A contain B as genera, one of them must be subordinate to the other, so that if the premiss takes the form No C is A, it will be false. This makes it clear that whether either or both premisses are false, the conclusion will equally be false.
80a27 Ἐν δὲ τῶι μέσωι σχήματι ὅλας μὲν εἶναι τὰς προτάσεις ἀμφοτέρας ψευδεῖς οὐκ ἐνδέχεται· ὅταν γὰρ τὸ Α παντὶ τῶι Β ὑπάρχηι, οὐδὲν ἔσται λαβεῖν ὁ τῶι μὲν ἑτέρωι παντὶ θατέρωι ↵ δ᾽ οὐδενὶ ὑπάρξει· δεῖ δ᾽ οὕτω λαμβάνειν τὰς προτάσεις ὥστε τῶι μὲν ὑπάρχειν τῶι δὲ μὴ ὑπάρχειν, εἴπερ ἔσται συλλογισμός. εἰ οὖν οὕτω λαμβανόμεναι ψευδεῖς, δῆλον ὡς ἐναντίως ἀνάπαλιν ἕξουσι· τοῦτο δ᾽ ἀδύνατον.	Sed in media quidem figura totas propositiones esse utrasque falsas non contingit, cum enim in omni b a sit, nihil erit accipere quod in altero erit omni, in altero vero nullo erit, oportet autem sic in media figura accipere propositiones, et quod in hoc quidem sit, et quod in hoc non sit, si quidem erit syllogismus, si igitur accipiantur sic falsae, palam quod contrariae e converso se habebunt, hoc autem impossibile est.	In the second figure the premisses cannot both be wholly false; for if all B is A, no middle term can be with truth universally affirmed of one extreme and universally denied of the other: but premisses in which the middle is affirmed of one extreme and denied of the other are the necessary condition if one is to get a valid inference at all. Therefore if, taken in this way, they are wholly false, their contraries conversely should be wholly true. But this is impossible.
ἐπί τι δ᾽ ἑκατέραν οὐδὲν κωλύει ψευδῆ εἶναι, οἷον εἰ τὸ Γ καὶ τῶι Α καὶ ↵ τῶι Β τινὶ ὑπάρχοι· ἂν γὰρ τῶι μὲν Α παντὶ ληφθῆι ὑπάρχον, τῶι δὲ Β μηδενί, ψευδεῖς μὲν ἀμφότεραι αἱ προτάσεις, οὐ μέντοι ὅλαι ἀλλ᾽ ἐπί τι. 80a38 καὶ ἀνάπαλιν δὲ τεθέντος τοῦ στερητικοῦ ὡσαύτως.	In aliquo autem utramque propositionem nihil prohibet falsam esse, ut si c in a, et in b aliquo est; si enim a in omni c accipiatur esse, b autem in nullo, falsae quidem utraeque propositiones, non tamen totae, sed in aliquo, e converso autem posito privativo similiter.	On the other hand, there is nothing to prevent both premisses being partially false; e.g. if actually some A is C and some B is C, then if it is premised that all A is C and no B is C, both premisses are false, yet partially, not wholly, false. The same is true if the major is made negative instead of the minor.
τὴν δ᾽ ἑτέραν εἶναι ψευδῆ καὶ ὁποτερανοῦν ἐνδέχεται. ὁ γὰρ ὑπάρχει τῶι Α παντί, καὶ τῶι Β ὑπάρχει· ἐὰν οὖν ληφθῆι τῶι μὲν Α ὅλωι ὑπάρχειν [80b]τὸ Γ, τῶι δὲ Β ὅλωι μὴ ὑπάρχειν, ἡ μὲν Γ Α ἀληθὴς ἔσται, ἡ δὲ Γ Β ψευδής.	Alteram autem falsam, et quamlibet contingit, quod enim est in a omni, et in b est, si igitur accipiatur in a quidem toto esse c, in b autem toto non esse, quae quidem est c a erit vera, sed quae est c b falsa.	Or one premiss may be wholly false, and it may be either of them. Thus, supposing that actually an attribute of all A must also be an attribute of all B, then if C is yet taken to be a universal attribute of all but universally non-attributable to B, C-A will be true but C-B false.
πάλιν ὁ τῶι Β μηδενὶ ὑπάρχει, οὐδὲ τῶι Α παντὶ ὑπάρξει· εἰ γὰρ τῶι Α, καὶ τῶι Β· ἀλλ᾽ οὐχ ὑπῆρχεν. ἐὰν οὖν ληφθῆι τὸ Γ τῶι μὲν Α ὅλωι ὑπάρχειν, τῶι δὲ ↵ Β μηδενί, ἡ μὲν Γ Β πρότασις ἀληθής, ἡ δ᾽ ἑτέρα ψευδής.	Iterum quod in b nullo est, neque in a omni erit, si enim in a est, et in b, sed non inerat, si igitur accipiatur c in toto quidem a esse, in b autem nullo quidem, quae est b c propositio vera est, altera vero falsa.	Again, actually that which is an attribute of no B will not be an attribute of all A either; for if it be an attribute of all A, it will also be an attribute of all B, which is contrary to supposition; but if C be nevertheless assumed to be a universal attribute of A, but an attribute of no B, then the premiss C-B is true but the major is false.
80b6 ὁμοίως δὲ καὶ μετατεθέντος τοῦ στερητικοῦ. ὁ γὰρ μηδενὶ ὑπάρχει τῶι Α, οὐδὲ τῶι Β οὐδενὶ ὑπάρξει· ἐὰν οὖν ληφθῆι τὸ Γ τῶι μὲν Α ὅλωι μὴ ὑπάρχειν, τῶι δὲ Β ὅλωι ὑπάρχειν, ἡ μὲν Γ Α πρότασις ἀληθὴς ἔσται, ἡ ἑτέρα δὲ ↵ ψευδής. καὶ πάλιν, ὁ παντὶ τῶι Β ὑπάρχει, μηδενὶ λαβεῖν τῶι Α ὑπάρχον ψεῦδος. ἀνάγκη γάρ, εἰ τῶι Β παντί, καὶ τῶι Α τινὶ ὑπάρχειν· ἐὰν οὖν ληφθῆι τῶι μὲν Β παντὶ ὑπάρχειν τὸ Γ, τῶι δὲ Α μηδενί, ἡ μὲν Γ Β ἀληθὴς ἔσται, ἡ δὲ Γ Α ψευδής.	Similiter autem fit transposito privativo, quod enim in nullo est a, neque in b nullo erit, si igitur accipiatur c in toto quidem a non esse, in b autem esse toto, quae quidem est a c propositio vera est, altera autem falsa. Et iterum quod in omni b est, in nullo accipere a esse, falsum est, necesse enim est si in omni b, et in quodam a esse, si igitur accipiatur in omni quidem b esse c, in a autem nullo, quae quidem est c b, vera erit, quae autem est a c, falsa.	The case is similar if the major is made the negative premiss. For in fact what is an attribute of no A will not be an attribute of any B either; and if it be yet assumed that C is universally non-attributable to A, but a universal attribute of B, the premiss C-A is true but the minor wholly false. Again, in fact it is false to assume that that which is an attribute of all B is an attribute of no A, for if it be an attribute of all B, it must be an attribute of some A. If then C is nevertheless assumed to be an attribute of all B but of no A, C-B will be true but C-A false.
80b14 φανερὸν οὖν ὅτι καὶ ἀμφοτέρων οὐσῶν ↵ ψευδῶν καὶ τῆς ἑτέρας μόνον ἔσται συλλογισμὸς ἀπατητικὸς ἐν τοῖς ἀτόμοις.	Manifestum igitur quod utrisque falsis, et altera solum, erit syllogismus deceptivus in indivisionibus.	It is thus clear that in the case of atomic propositions erroneous inference will be possible not only when both premisses are false but also when only one is false.
c17	CAPUT XIII. De syllogismo imperitiae eorum quae insunt mediate.	Chapter 17
80b17 Ἐν δὲ τοῖς μὴ ἀτόμως ὑπάρχουσιν [ἢ μὴ ὑπάρχουσιν], ὅταν μὲν διὰ τοῦ οἰκείου μέσου γίνηται τοῦ ψεύδους ὁ συλλογισμός, οὐχ οἷόν τε ἀμφοτέρας ψευδεῖς εἶναι τὰς ↵ προτάσεις, ἀλλὰ μόνον τὴν πρὸς τῶι μείζονι ἄκρωι. (λέγω δ᾽ οἰκεῖον μέσον δι᾽ οὗ γίνεται τῆς ἀντιφάσεως ὁ συλλογισμός.)	In iis autem quae non indivisibiliter insunt, aut non insunt, cum quidem per proprium medium falsitatis fiat syllogismus, non possibile est falsas utrasque esse propositiones, sed solum quae ante maius extremum. Dico autem proprium medium, per quod fit contradictionis syllogismus:	In the case of attributes not atomically connected with or disconnected from their subjects, (a) (i) as long as the false conclusion is inferred through the ‘appropriate’ middle, only the major and not both premisses can be false. By ‘appropriate middle’ I mean the middle term through which the contradictory-i.e. the true-conclusion is inferrible.
ὑπαρχέτω γὰρ τὸ Α τῶι Β διὰ μέσου τοῦ Γ. ἐπεὶ οὖν ἀνάγκη τὴν Γ Β καταφατικὴν λαμβάνεσθαι συλλογισμοῦ γινομένου, δῆλον ὅτι ἀεὶ αὕτη ἔσται ἀληθής· οὐ ↵ γὰρ ἀντιστρέφεται. 80b27 ἡ δὲ Α Γ ψευδής· ταύτης γὰρ ἀντιστρεφομένης ἐναντίος γίνεται ὁ συλλογισμός.	sit enim a in b per medium c, quoniam igitur necesse est quae est c b affirmativam accipi syllogismo facto, manifestum quod semper haec est vera, non enim convertitur, sed quae est a c falsa, hac enim conversa, e contrario fiet syllogismus.	Thus, let A be attributable to B through a middle term C: then, since to produce a conclusion the premiss C-B must be taken affirmatively, it is clear that this premiss must always be true, for its quality is not changed. But the major A-C is false, for it is by a change in the quality of A-C that the conclusion becomes its contradictory-i.e. true.
ὁμοίως δὲ καὶ εἰ ἐξ ἄλλης συστοιχίας ληφθείη τὸ μέσον, οἷον τὸ Δ εἰ καὶ ἐν τῶι Α ὅλωι ἐστι καὶ κατὰ τοῦ Β κατηγορεῖται παντός· ἀνάγκη γὰρ τὴν μὲν Δ Β πρότασιν μένειν, τὴν δ᾽ ↵ ἑτέραν ἀντιστρέφεσθαι, ὥσθ᾽ ἡ μὲν ἀεὶ ἀληθής, ἡ δ᾽ ἀεὶ ψευδής. καὶ σχεδὸν ἥ γε τοιαύτη ἀπάτη ἡ αὐτή ἐστι τῆι διὰ τοῦ οἰκείου μέσου.	Similiter autem est, et si ex alia coordinatione accipiatur medium, ut d, si et in toto a, et de b praedicatur omni, necesse est enim (quae quidem est d b ) propositionem manere, alteram autem converti. Quare haec quidem semper vera, illa vero semper falsa, et fere huiusmodi deceptio, eadem est ei quae fit per proprium medium.	Similarly (ii) if the middle is taken from another series of predication; e.g. suppose D to be not only contained within A as a part within its whole but also predicable of all B. Then the premiss D-B must remain unchanged, but the quality of A-D must be changed; so that D-B is always true, A-D always false. Such error is practically identical with that which is inferred through the ‘appropriate’ middle.
80b32 ἐὰν δὲ μὴ διὰ τοῦ οἰκείου μέσου γίνηται ὁ συλλογισμός, ὅταν μὲν ὑπὸ τὸ Α ἦι τὸ μέσον, τῶι δὲ Β μηδενὶ ὑπάρχηι, ἀνάγκη ψευδεῖς εἶναι ἀμφοτέρας. ↵ ληπτέαι γὰρ ἐναντίως ἢ ὡς ἔχουσιν αἱ προτάσεις, εἰ μέλλει συλλογισμὸς ἔσεσθαι· οὕτω δὲ λαμβανομένων ἀμφότεραι γίνονται ψευδεῖς. οἷον εἰ τὸ μὲν Α ὅλωι τῶι Δ ὑπάρχει, τὸ δὲ Δ μηδενὶ τῶν Β· ἀντιστραφέντων γὰρ τούτων συλλογισμός τ᾽ ἔσται καὶ αἱ προτάσεις ἀμφότεραι ψευδεῖς.	Si vero non per proprium medium fiat syllogismus, cum quidem sub a sit medium, in b autem nullo est, necesse est utrasque falsas esse, sumendae enim e contrario sunt quam ut se habeant propositiones, si debeat syllogismus esse: sic autem acceptis, utraeque fiunt falsae, ut si a quidem in toto d est, d autem in nullo b, conversis enim iis, syllogismus erit, et propositiones utraeque falsae.	On the other hand, (b) if the conclusion is not inferred through the ‘appropriate’ middle-(i) when the middle is subordinate to A but is predicable of no B, both premisses must be false, because if there is to be a conclusion both must be posited as asserting the contrary of what is actually the fact, and so posited both become false: e.g. suppose that actually all D is A but no B is D; then if these premisses are changed in quality, a conclusion will follow and both of the new premisses will be false.
ὅταν δὲ μὴ ἦι ὑπὸ τὸ Α τὸ μέσον, οἷον τὸ Δ, ἡ [81a]μὲν Α Δ ἀληθὴς ἔσται, ἡ δὲ Δ Β ψευδής. ἡ μὲν γὰρ Α Δ ἀληθής, ὅτι οὐκ ἦν ἐν τῶι Α τὸ Δ, ἡ δὲ Δ Β ψευδής, ὅτι εἰ ἦν ἀληθής, κἂν τὸ συμπέρασμα ἦν ἀληθές· ἀλλ᾽ ἦν ψεῦδος.	Cum vero non sit sub a medium ut d, quae quidem est a d vera erit, quae vero est d b falsa, quae enim est a d vera est, quia non erit in a, d, quae vero est d b falsa, quia si esset vera, et conclusio esset vera, sed falsa erat.	When, however, (ii) the middle D is not subordinate to A, A-D will be true, D-B false-A-D true because A was not subordinate to D, D-B false because if it had been true, the conclusion too would have been true; but it is ex hypothesi false.
81a5 ↵ Διὰ δὲ τοῦ μέσου σχήματος γινομένης τῆς ἀπάτης, ἀμφοτέρας μὲν οὐκ ἐνδέχεται ψευδεῖς εἶναι τὰς προτάσεις ὅλας (ὅταν γὰρ ἦι τὸ Β ὑπὸ τὸ Α, οὐδὲν ἐνδέχεται τῶι μὲν παντὶ τῶι δὲ μηδενὶ ὑπάρχειν, καθάπερ ἐλέχθη καὶ πρότερον), τὴν ἑτέραν δ᾽ ἐγχωρεῖ, καὶ ὁποτέραν ἔτυχεν.	Sed per mediam figuram facta deceptione, utrasque quidem non contingit falsas esse propositiones totas, cum enim sit b quidem sub a, nihil contingit in hoc quidem omni, in illo autem nullo esse, sicut dictum est et prius. Altera vero potest esse falsa, et quaecunque contingit.	When the erroneous inference is in the second figure, both premisses cannot be entirely false; since if B is subordinate to A, there can be no middle predicable of all of one extreme and of none of the other, as was stated before. One premiss, however, may be false, and it may be either of them.
εἰ γὰρ ↵ τὸ Γ καὶ τῶι Α καὶ τῶι Β ὑπάρχει, ἐὰν ληφθῆι τῶι μὲν Α ὑπάρχειν τῶι δὲ Β μὴ ὑπάρχειν, ἡ μὲν Γ Α ἀληθὴς ἔσται, ἡ δ᾽ ἑτέρα ψευδής. πάλιν δ᾽ εἰ τῶι μὲν Β ληφθείη τὸ Γ ὑπάρχον, τῶι δὲ Α μηδενί, ἡ μὲν Γ Β ἀληθὴς ἔσται, ἡ δ᾽ ἑτέρα ψευδής. ↵ Ἐὰν μὲν οὖν στερητικὸς ἦι τῆς ἀπάτης ὁ συλλογισμός, εἴρηται πότε καὶ διὰ τίνων ἔσται ἡ ἀπάτη·	Si enim c et in b, et in a est, si accipiatur in a quidem esse, in b vero non esse, quae quidem a c vera erit, altera autem falsa. Item si in b accipiatur c esse, in a autem nullo est, quae quidem est c b vera erit, altera autem falsa, siquidem igitur privativus sit deceptionis syllogismus, dictum est quando et per quae erit deceptio.	Thus, if C is actually an attribute of both A and B, but is assumed to be an attribute of A only and not of B, C-A will be true, C-B false: or again if C be assumed to be attributable to B but to no A, C-B will be true, C-A false. We have stated when and through what kinds of premisses error will result in cases where the erroneous conclusion is negative.
ἐὰν δὲ καταφατικός, ὅταν μὲν διὰ τοῦ οἰκείου μέσου, ἀδύνατον ἀμφοτέρας εἶναι ψευδεῖς· ἀνάγκη γὰρ τὴν Γ Β μένειν, εἴπερ ἔσται συλλογισμός, καθάπερ ἐλέχθη καὶ πρότερον. 81a20 ὥστε ἡ Α Γ ↵ ἀεὶ ἔσται ψευδής· αὕτη γάρ ἐστιν ἡ ἀντιστρεφομένη.	Si vero sit affirmativus, tunc cum per proprium medium, impossibile est utrasque esse falsas, necesse est enim quae est c b manere, siquidem erit syllogismus sicut dictum est et prius, quare c a semper erit falsa, haec enim est conversa.	If the conclusion is affirmative, (a) (i) it may be inferred through the ‘appropriate’ middle term. In this case both premisses cannot be false since, as we said before, C-B must remain unchanged if there is to be a conclusion, and consequently A-C, the quality of which is changed, will always be false.
ὁμοίως δὲ καὶ εἰ ἐξ ἄλλης συστοιχίας λαμβάνοιτο τὸ μέσον, ὥσπερ ἐλέχθη καὶ ἐπὶ τῆς στερητικῆς ἀπάτης· ἀνάγκη γὰρ τὴν μὲν Δ Β μένειν, τὴν δ᾽ Α Δ ἀντιστρέφεσθαι, καὶ ἡ ἀπάτη ἡ αὐτὴ τῆι πρότερον.	Similiter autem est et si ex alia ordinatione accipiatur medium, sicut dictum est et in privativa deceptione, necesse enim est quae quidem est d b manere, quae vero est a d converti, et haec deceptio eadem est priori.	This is equally true if (ii) the middle is taken from another series of predication, as was stated to be the case also with regard to negative error; for D-B must remain unchanged, while the quality of A-D must be converted, and the type of error is the same as before.
81a25 ὅταν δὲ μὴ διὰ τοῦ οἰκείου, ἐὰν ↵ μὲν ἦι τὸ Δ ὑπὸ τὸ Α, αὕτη μὲν ἔσται ἀληθής, ἡ ἑτέρα δὲ ψευδής· ἐγχωρεῖ γὰρ τὸ Α πλείοσιν ὑπάρχειν ἃ οὐκ ἔστιν ὑπ᾽ ἄλληλα.	Cum vero sic non per proprium, si quidem sit d sub a, haec quidem erit vera, altera vero falsa, potest enim a in pluribus esse, quae non sunt sub se invicem.	(b) The middle may be inappropriate. Then (i) if D is subordinate to A, A-D will be true, but D-B false; since A may quite well be predicable of several terms no one of which can be subordinated to another.
ἐὰν δὲ μὴ ἦι τὸ Δ ὑπὸ τὸ Α, αὕτη μὲν ἀεὶ δῆλον ὅτι ἔσται ψευδής (καταφατικὴ γὰρ λαμβάνεται), τὴν δὲ Δ Β ἐνδέχεται καὶ ἀληθῆ εἶναι καὶ ψευδῆ· οὐδὲν ↵ γὰρ κωλύει τὸ μὲν Α τῶι Δ μηδενὶ ὑπάρχειν, τὸ δὲ Δ τῶι Β παντί, οἷον ζῶιον ἐπιστήμηι, ἐπιστήμη δὲ μουσικῆι. οὐδ᾽ αὖ μήτε τὸ Α μηδενὶ τῶν Δ μήτε τὸ Δ μηδενὶ τῶν Β. [φανερὸν οὖν ὅτι μὴ ὄντος τοῦ μέσου ὑπὸ τὸ Α καὶ ἀμφοτέρας ἐγχωρεῖ ψευδεῖς εἶναι καὶ ὁποτέραν ἔτυχεν.]	Si vero non sit d sub a, haec quidem semper manifestum est quoniam est falsa, affirmativa enim accipiatur. Quae vero est d b, contingit et veram esse, et falsam, nihil enim prohibet a quidem in nullo d esse, d autem in omni b, ut animal in scientia, scientia autem ut in musica, neque utique a in nullo d, neque d in nullo b esse. Manifestum igitur est quoniam cum non sit medium proprium, et utrae possunt simul esse falsae, et quaecunque contingit.	If, however, (ii) D is not subordinate to A, obviously A-D, since it is affirmed, will always be false, while D-B may be either true or false; for A may very well be an attribute of no D, whereas all B is D, e.g. no science is animal, all music is science. Equally well A may be an attribute of no D, and D of no B. It emerges, then, that if the middle term is not subordinate to the major, not only both premisses but either singly may be false.
81a35 ↵ Ποσαχῶς μὲν οὖν καὶ διὰ τίνων ἐγχωρεῖ γίνεσθαι τὰς κατὰ συλλογισμὸν ἀπάτας ἔν τε τοῖς ἀμέσοις καὶ ἐν τοῖς δι᾽ ἀποδείξεως, φανερόν.	Quot quidem igitur modis, et per quae possunt fieri secundum syllogismum deceptiones, et in aliis quae sunt sine medio, et in iis quae sunt per demonstrationem manifestum est.	Thus we have made it clear how many varieties of erroneous inference are liable to happen and through what kinds of premisses they occur, in the case both of immediate and of demonstrable truths.
c18	CAPUT XIV. Si sensus a natura deficit, et scientiam propriorum sensibilium illius deficere.	Chapter 18
81a38 Φανερὸν δὲ καὶ ὅτι, εἴ τις αἴσθησις ἐκλέλοιπεν, ἀνάγκη καὶ ἐπιστήμην τινὰ ἐκλελοιπέναι,	Manifestum est autem, et si aliquis sensus defecerit, necesse est aliquam scientiam deficere,	It is also clear that the loss of any one of the senses entails the loss of a corresponding portion of knowledge, and that,
81a39 ἣν ἀδύνατον λαβεῖν, εἴπερ μανθάνομεν ἢ ἐπαγωγῆι ἢ ἀποδείξει, ἔστι δ᾽ ἡ μὲν ἀπόδει[81b]ξις ἐκ τῶν καθόλου, ἡ δ᾽ ἐπαγωγὴ ἐκ τῶν κατὰ μέρος,	quam impossibile est accipere, si quidem discimus, aut per inductionem, aut per demonstrationem. Est autem demonstratio ex universalibus, inductio autem ex iis quae sunt particularia.	since we learn either by induction or by demonstration, this knowledge cannot be acquired. Thus demonstration develops from universals, induction from particulars;
81b2 ἀδύνατον δὲ τὰ καθόλου θεωρῆσαι μὴ δι᾽ ἐπαγωγῆς (ἐπεὶ καὶ τὰ ἐξ ἀφαιρέσεως λεγόμενα ἔσται δι᾽ ἐπαγωγῆς γνώριμα ποιεῖν, ὅτι ὑπάρχει ἑκάστωι γένει ἔνια, καὶ εἰ μὴ χωριστά ἐστιν, ἧι τοιονδὶ ἕκαστον),	Impossibile autem est universalia speculari, nisi per inductionem, quoniam et quae ex abstractione dicuntur, est per inductionem nota facere, si quis vult nota facere, quia sunt in unoquoque genere quaedam, et si non separabilia sint secundum quod huiusmodi unumquodque est.	but since it is possible to familiarize the pupil with even the so-called mathematical abstractions only through induction-i.e. only because each subject genus possesses, in virtue of a determinate mathematical character, certain properties which can be treated as separate even though they do not exist in isolation-it is consequently impossible to come to grasp universals except through induction.
81b5 ἐπαχθῆναι δὲ μὴ ἔχοντας αἴσθησιν ἀδύνατον. τῶν γὰρ καθ᾽ ἕκαστον ἡ αἴσθησις· οὐ γὰρ ἐνδέχεται λαβεῖν αὐτῶν τὴν ἐπιστήμην· οὔτε γὰρ ἐκ τῶν καθόλου ἄνευ ἐπαγωγῆς, οὔτε δι᾽ ἐπαγωγῆς ἄνευ τῆς αἰσθήσεως.	Inducere autem non habentes sensum, impossibile est, singularium enim sensus est, non enim contingit accipere ipsorum scientiam, neque enim est ex universalibus sine inductione, neque per inductionem sine sensu.	But induction is impossible for those who have not sense-perception. For it is sense-perception alone which is adequate for grasping the particulars: they cannot be objects of scientific knowledge, because neither can universals give us knowledge of them without induction, nor can we get it through induction without sense-perception.
c19	CAPUT XV. Ex quot et qualibus constat syllogismus, et an sursum deorsumve fiat in infinitum abitio.	Chapter 19
81b10 Ἔστι δὲ πᾶς συλλογισμὸς διὰ τριῶν ὅρων, καὶ ὁ μὲν δεικνύναι δυνάμενος ὅτι ὑπάρχει τὸ Α τῶι Γ διὰ τὸ ὑπάρχειν τῶι Β καὶ τοῦτο τῶι Γ, ὁ δὲ στερητικός, τὴν μὲν ἑτέραν πρότασιν ἔχων ὅτι ὑπάρχει τι ἄλλο ἄλλωι, τὴν δ᾽ ἑτέραν ὅτι οὐχ ὑπάρχει.	Est autem omnis syllogismus per tres terminos, et quidem monstrare possibilis est quoniam a est in c, propter id quod est in b, et hoc in c. Sed privativus est quidem alteram propositionem habens quoniam est aliquid aliud in alio, alteram autem quoniam non est.	Every syllogism is effected by means of three terms. One kind of syllogism serves to prove that A inheres in C by showing that A inheres in B and B in C; the other is negative and one of its premisses asserts one term of another, while the other denies one term of another.
81b14 φανερὸν οὖν ὅτι αἱ μὲν ἀρχαὶ καὶ αἱ λε↵ γόμεναι ὑποθέσεις αὗταί εἰσι· λαβόντα γὰρ ταῦτα οὕτως ἀνάγκη δεικνύναι, οἷον ὅτι τὸ Α τῶι Γ ὑπάρχει διὰ τοῦ Β, πάλιν δ᾽ ὅτι τὸ Α τῶι Β δι᾽ ἄλλου μέσου, καὶ ὅτι τὸ Β τῶι Γ ὡσαύτως.	Manifestum igitur est quod principia, et suppositiones dicta haec sunt, accipientem enim haec sic, necesse est monstrare ut quod a sit in c per b, iterum autem quod a sit in b per aliud medium, et quod b sit in c similiter.	It is clear, then, that these are the fundamentals and so-called hypotheses of syllogism. Assume them as they have been stated, and proof is bound to follow-proof that A inheres in C through B, and again that A inheres in B through some other middle term, and similarly that B inheres in C.
81b18 κατὰ μὲν οὖν δόξαν συλλογιζομένοις καὶ μόνον διαλεκτικῶς δῆλον ὅτι τοῦτο μόνον σκεπτέον, εἰ ἐξ ὧν ↵ ἐνδέχεται ἐνδοξοτάτων γίνεται ὁ συλλογισμός, ὥστ᾽ εἰ καὶ μὴ ἔστι τι τῆι ἀληθείαι τῶν Α Β μέσον, δοκεῖ δὲ εἶναι, ὁ διὰ τούτου συλλογιζόμενος συλλελόγισται διαλεκτικῶς· πρὸς δ᾽ ἀλήθειαν ἐκ τῶν ὑπαρχόντων δεῖ σκοπεῖν.	Secundum igitur opinionem syllogizantibus, et solum dialectice, manifestum est quod hoc solum intendendum, si ex quibus contingit maxime probabilibus fiat syllogismus. Quare et si est aliquid in veritate eorum quae sunt a b medium, videtur autem non esse per hoc syllogizans, syllogizat dialectice. Ad veritatem autem, ex iis quae sunt oportet intendere, habet autem se sic.	If our reasoning aims at gaining credence and so is merely dialectical, it is obvious that we have only to see that our inference is based on premisses as credible as possible: so that if a middle term between A and B is credible though not real, one can reason through it and complete a dialectical syllogism. If, however, one is aiming at truth, one must be guided by the real connexions of subjects and attributes.
81b24 ἔχει δ᾽ οὕτως· ἐπειδὴ ἔστιν ὁ αὐτὸ μὲν κατ᾽ ἄλλου κατηγορεῖται μὴ κατὰ ↵ συμβεβηκός – λέγω δὲ τὸ κατὰ συμβεβηκός, οἷον τὸ λευκόν ποτ᾽ ἐκεῖνό φαμεν εἶναι ἄνθρωπον, οὐχ ὁμοίως λέγοντες καὶ τὸν ἄνθρωπον λευκόν· ὁ μὲν γὰρ οὐχ ἕτερόν τι ὢν λευκός ἐστι, τὸ δὲ λευκόν, ὅτι συμβέβηκε τῶι ἀνθρώπωι εἶναι λευκῶι – ἔστιν οὖν ἔνια τοιαῦτα ὥστε καθ᾽ αὑτὰ κατηγορεῖσθαι.	Quoniam autem est quod ipsum quidem de alia praedicatur, non secundum accidens. Dico autem secundum accidens, ut album aliquando dicimus illud esse hominem, non similiter dicentes et hominem album, cum enim non sit alterum, aliquod album est, album autem est homo, quoniam accidit homini esse album, sunt igitur quaedam huiusmodi, quaecunque secundum se praedicantur.	Thus: since there are attributes which are predicated of a subject essentially or naturally and not coincidentally-not, that is, in the sense in which we say ‘That white (thing) is a man’, which is not the same mode of predication as when we say ‘The man is white’: the man is white not because he is something else but because he is man, but the white is man because ‘being white’ coincides with ‘humanity’ within one substratum-therefore there are terms such as are naturally subjects of predicates.
81b30 ↵ Ἔστω δὴ τὸ Γ τοιοῦτον ὁ αὐτὸ μὲν μηκέτι ὑπάρχει ἄλλωι, τούτωι δὲ τὸ Β πρώτωι, καὶ οὐκ ἔστιν ἄλλο μεταξύ. καὶ πάλιν τὸ Ε τῶι Ζ ὡσαύτως, καὶ τοῦτο τῶι Β. ἆρ᾽ οὖν τοῦτο ἀνάγκη στῆναι, ἢ ἐνδέχεται εἰς ἄπειρον ἰέναι;	Si igitur c huiusmodi quod ipsum quidem non in alio sit, in hoc autem b sit primo, et non per aliud medium, iterum e de d sit, et similiter et hoc in b, nunquid igitur hoc necesse est stare, an contingit in infinitum ire?	Suppose, then, C such a term not itself attributable to anything else as to a subject, but the proximate subject of the attribute B— i.e. so that B-C is immediate; suppose further E related immediately to F, and F to B. The first question is, must this series terminate, or can it proceed to infinity?
81b34 καὶ πάλιν εἰ τοῦ μὲν Α μηδὲν κατηγορεῖται καθ᾽ αὑτό, τὸ δὲ Α τῶι Θ ↵ ὑπάρχει πρώτωι, μεταξὺ δὲ μηδενὶ προτέρωι, καὶ τὸ Θ τῶι Η, καὶ τοῦτο τῶι Β, ἆρα καὶ τοῦτο ἵστασθαι ἀνάγκη, ἢ καὶ τοῦτ᾽ ἐνδέχεται εἰς ἄπειρον ἰέναι; διαφέρει δὲ τοῦτο τοῦ πρότερον τοσοῦτον, ὅτι τὸ μέν ἐστιν, ἆρα ἐνδέχεται ἀρξαμένωι ἀπὸ τοιούτου ὁ μηδενὶ ὑπάρχει ἑτέρωι ἀλλ᾽ ἄλλο ἐκείνωι, ἐπὶ τὸ ἄνω εἰς ἄπειρον ἰέναι, θάτερον δὲ ἀρξάμενον ἀπὸ τοιούτου	Et iterum si de a quidem nihil praedicatur per se, a autem in f est primo, medium autem in nullo priori, et f in a, et hoc in b, nunquid et hoc stare necesse est? an et hic contingit in infinitum abire? Differt autem hoc a priori intantum, quoniam hoc quidem est nunquid, contingit incepturum ad huiusmodi quod in nullo est altero, sed aliud in illo, in sursum in infinitum abire, alterum autem incoepturum ab huiusmodi quod ipsum quidem de alio, de illo autem nihil praedicatur, in deorsum intendentem, si contingit in infinitum ire.	The second question is as follows: Suppose nothing is essentially predicated of A, but A is predicated primarily of H and of no intermediate prior term, and suppose H similarly related to G and G to B; then must this series also terminate, or can it too proceed to infinity? There is this much difference between the questions: the first is, is it possible to start from that which is not itself attributable to anything else but is the subject of attributes, and ascend to infinity? The second is the problem whether one can start from that which is a predicate but not itself a subject of predicates, and descend to infinity?
82a2 [82a]ὁ αὐτὸ μὲν ἄλλου, ἐκείνου δὲ μηδὲν κατηγορεῖται, ἐπὶ τὸ κάτω σκοπεῖν εἰ ἐνδέχεται εἰς ἄπειρον ἰέναι. Ἔτι τὰ μεταξὺ ἆρ᾽ ἐνδέχεται ἄπειρα εἶναι ὡρισμένων τῶν ἄκρων; λέγω δ᾽ οἷον εἰ τὸ Α τῶι Γ ὑπάρχει, μέσον δ᾽ αὐτῶν τὸ Β, τοῦ ↵ δὲ Β καὶ τοῦ Α ἕτερα, τούτων δ᾽ ἄλλα, ἆρα καὶ ταῦτα εἰς ἄπειρον ἐνδέχεται ἰέναι, ἢ ἀδύνατον;	Amplius media nunquid contingit infinita esse determinatis terminis? Dico autem ut si a in c sit, medium ipsorum sit b, ab ipso autem b, et ab a altera, sed horum alia, nunquid et haec in infinitum contingit abire? an impossibile est?	A third question is, if the extreme terms are fixed, can there be an infinity of middles? I mean this: suppose for example that A inheres in C and B is intermediate between them, but between B and A there are other middles, and between these again fresh middles; can these proceed to infinity or can they not?
82a7 ἔστι δὲ τοῦτο σκοπεῖν ταὐτὸ καὶ εἰ αἱ ἀποδείξεις εἰς ἄπειρον ἔρχονται, καὶ εἰ ἔστιν ἀπόδειξις ἅπαντος, ἢ πρὸς ἄλληλα περαίνεται.	est autem hoc quidem intendere idem et si demonstrationes in infinitum veniunt, et si est demonstratio omnis rei, an ad invicem concludantur.	This is the equivalent of inquiring, do demonstrations proceed to infinity, i.e. is everything demonstrable? Or do ultimate subject and primary attribute limit one another?
82a9 Ὁμοίως δὲ λέγω καὶ ἐπὶ τῶν στερητικῶν συλλογισμῶν ↵ καὶ προτάσεων, οἷον εἰ τὸ Α μὴ ὑπάρχει τῶι Β μηδενί, ἤτοι πρώτωι, ἢ ἔσται τι μεταξὺ ὧι προτέρωι οὐχ ὑπάρχει (οἷον εἰ τῶι Η, ὁ τῶι Β ὑπάρχει παντί), καὶ πάλιν τούτου ἔτι ἄλλωι προτέρωι, οἷον εἰ τῶι Θ, ὁ τῶι Η παντὶ ὑπάρχει. καὶ γὰρ ἐπὶ τούτων ἢ ἄπειρα οἷς ὑπάρχει προτέροις, ἢ ἵσταται.	Similiter autem dico et in privativis syllogismis et propositionibus, ut si a non inest b nulli aut primo, aut est aliquid medium, cui priori non inest a, ut si sit g quod omni b inest, et iterum hoc non etiam alii priori, ut si h est quod sit in omni b, et namque in his aut infinita sunt, in quibus non est in prioribus, aut statur.	I hold that the same questions arise with regard to negative conclusions and premisses: viz. if A is attributable to no B, then either this predication will be primary, or there will be an intermediate term prior to B to which a is not attributable-G, let us say, which is attributable to all B-and there may still be another term H prior to G, which is attributable to all G. The same questions arise, I say, because in these cases too either the series of prior terms to which a is not attributable is infinite or it terminates.
82a15 ↵ Ἐπὶ δὲ τῶν ἀντιστρεφόντων οὐχ ὁμοίως ἔχει. οὐ γὰρ ἔστιν ἐν τοῖς ἀντικατηγορουμένοις οὗ πρώτου κατηγορεῖται ἢ τελευταίου πάντα γὰρ πρὸς πάντα ταύτηι γε ὁμοίως ἔχει, εἴτ᾽ ἐστὶν ἄπειρα τὰ κατ᾽ αὐτοῦ κατηγορούμενα, εἴτ᾽ ἀμφότερά ἐστι τὰ ἀπορηθέντα ἄπειρα· πλὴν εἰ μὴ ὁμοίως ἐνδέχεται ἀντι↵ στρέφειν, ἀλλὰ τὸ μὲν ὡς συμβεβηκός, τὸ δ᾽ ὡς κατηγορίαν.	Sed inconvertibilibus non similiter se habet, non enim est in aeque praedicabilibus de quo primo praedicatur, aut ultimo, omnia enim ad omnia sic similiter se habent, sive sunt infinita de ipso praedicata, sive utraque sunt dubitata infinita, nisi similiter converti contingat, sed hoc quidem sicut accidens, illud vero sicut praedicatum.	One cannot ask the same questions in the case of reciprocating terms, since when subject and predicate are convertible there is neither primary nor ultimate subject, seeing that all the reciprocals qua subjects stand in the same relation to one another, whether we say that the subject has an infinity of attributes or that both subjects and attributes-and we raised the question in both cases-are infinite in number. These questions then cannot be asked-unless, indeed, the terms can reciprocate by two different modes, by accidental predication in one relation and natural predication in the other.
c20	CAPUT XVI. Determinatis extremis, summo imoque, media non esse infinita.	Chapter 20
82a21 Ὅτι μὲν οὖν τὰ μεταξὺ οὐκ ἐνδέχεται ἄπειρα εἶναι, εἰ ἐπὶ τὸ κάτω καὶ τὸ ἄνω ἵστανται αἱ κατηγορίαι, δῆλον. λέγω δ᾽ ἄνω μὲν τὴν ἐπὶ τὸ καθόλου μᾶλλον, κάτω δὲ τὴν ἐπὶ τὸ κατὰ μέρος.	Quoniam igitur media non contingit infinita esse, si in sursum et deorsum stent praedicata, manifestum est. Dico autem sursum quidem, quod universale magis est, deorsum autem, quod particulare est.	Now, it is clear that if the predications terminate in both the upward and the downward direction (by ‘upward’ I mean the ascent to the more universal, by ‘downward’ the descent to the more particular), the middle terms cannot be infinite in number.
82a24 εἰ γὰρ τοῦ Α κατηγορουμένου κατὰ ↵ τοῦ Ζ ἄπειρα τὰ μεταξύ, ἐφ᾽ ὧν Β, δῆλον ὅτι ἐνδέχοιτ᾽ ἂν ὥστε καὶ ἀπὸ τοῦ Α ἐπὶ τὸ κάτω ἕτερον ἑτέρου κατηγορεῖσθαι εἰς ἄπειρον (πρὶν γὰρ ἐπὶ τὸ Ζ ἐλθεῖν, ἄπειρα τὰ μεταξύ) καὶ ἀπὸ τοῦ Ζ ἐπὶ τὸ ἄνω ἄπειρα, πρὶν ἐπὶ τὸ Α ἐλθεῖν. ὥστ᾽ εἰ ταῦτα ἀδύνατα, καὶ τοῦ Α καὶ Ζ ἀδύνατον ↵ ἄπειρα εἶναι μεταξύ.	Si enim a praedicante de c, infinita sunt media, in quibus est b, manifestum est quod continget utrique, ut ab a in deorsum alterum de altero contingit praedicari in infinitum, antequam enim in c veniat, infinita sunt media, et ab c in sursum infinita, antequam in a veniat. Quare si haec impossibilia sunt, et ipsius a et c impossibile est infinita esse media.	For suppose that A is predicated of F, and that the intermediates-call them BB’B”...-are infinite, then clearly you might descend from and find one term predicated of another ad infinitum, since you have an infinity of terms between you and F; and equally, if you ascend from F, there are infinite terms between you and A. It follows that if these processes are impossible there cannot be an infinity of intermediates between A and F.
82a30 οὐδὲ γὰρ εἴ τις λέγοι ὅτι τὰ μέν ἐστι τῶν Α Β Ζ ἐχόμενα ἀλλήλων ὥστε μὴ εἶναι μεταξύ, τὰ δ᾽ οὐκ ἔστι λαβεῖν, οὐδὲν διαφέρει. ὁ γὰρ ἂν λάβω τῶν Β, ἔσται πρὸς τὸ Α ἢ πρὸς τὸ Ζ ἢ ἄπειρα τὰ μεταξὺ ἢ οὔ. ἀφ᾽ οὗ δὴ πρῶτον ἄπειρα, εἴτ᾽ εὐθὺς εἴτε μὴ εὐθύς, οὐδὲν διαφέ↵ ρει· τὰ γὰρ μετὰ ταῦτα ἄπειρά ἐστιν.	Neque enim si aliquis dicat quod haec quidem quae a b c contingentia sunt ad invicem, ut et non sint media, illa vero non esse accipere, nihil differt. Quodcunque enim accipio eorum quae sunt b, erunt ab a, aut c infinita media. An non a quo iam prima sint infinita sive statim, sive non statim, nihil differt? quae enim sunt post haec, infinita sunt.	Nor is it of any effect to urge that some terms of the series AB...F are contiguous so as to exclude intermediates, while others cannot be taken into the argument at all: whichever terms of the series B...I take, the number of intermediates in the direction either of A or of F must be finite or infinite: where the infinite series starts, whether from the first term or from a later one, is of no moment, for the succeeding terms in any case are infinite in number.
c21	CAPUT XVII. Propositionis negativae mediatae media, quibus revocetur ad immediatam, non esse infinita.	Chapter 21
82a37 Φανερὸν δὲ καὶ ἐπὶ τῆς στερητικῆς ἀποδείξεως ὅτι στήσεται, εἴπερ ἐπὶ τῆς κατηγορικῆς ἵσταται ἐπ᾽ ἀμφότερα. ἔστω γὰρ μὴ ἐνδεχόμενον μήτε ἐπὶ τὸ ἄνω ἀπὸ τοῦ ὑστάτου εἰς ἄπειρον ἰέναι (λέγω δ᾽ ὕστατον ὁ αὐτὸ μὲν ἄλλωι [82b]μηδενὶ ὑπάρχει, ἐκείνωι δὲ ἄλλο, οἷον τὸ Ζ) μήτε ἀπὸ τοῦ πρώτου ἐπὶ τὸ ὕστατον (λέγω δὲ πρῶτον ὁ αὐτὸ μὲν κατ᾽ ἄλλου, κατ᾽ ἐκείνου δὲ μηδὲν ἄλλο). εἰ δὴ ταῦτ᾽ ἔστι, καὶ ἐπὶ τῆς ἀποφάσεως στήσεται.	Manifestum autem et in privata demonstratione quoniam statur, si quidem in praedicativa statur in utrisque. Sit enim non contingens neque in sursum ab ultimo in infinitum ire (dico autem in quo statur, quod ipsum quidem in alio nullo est, sed in illo aliud ut c ) neque a primo in ultimum. Dico autem primum quod ipsum quidem de alio, sed de illo nullum aliud dicitur: si igitur haec erunt, manifestum quod et in negatione stabitur.	Further, if in affirmative demonstration the series terminates in both directions, clearly it will terminate too in negative demonstration. Let us assume that we cannot proceed to infinity either by ascending from the ultimate term (by ‘ultimate term’ I mean a term such as was, not itself attributable to a subject but itself the subject of attributes), or by descending towards an ultimate from the primary term (by ‘primary term’ I mean a term predicable of a subject but not itself a subject). If this assumption is justified, the series will also terminate in the case of negation.
82b4 τριχῶς γὰρ δείκνυται μὴ ↵ ὑπάρχον. ἢ γὰρ ὧι μὲν τὸ Γ, τὸ Β ὑπάρχει παντί, ὧι δὲ τὸ Β, οὐδενὶ τὸ Α. τοῦ μὲν τοίνυν Β Γ, καὶ ἀεὶ τοῦ ἑτέρου διαστήματος, ἀνάγκη βαδίζειν εἰς ἄμεσα· κατηγορικὸν γὰρ τοῦτο τὸ διάστημα.	Tripliciter autem monstratur non esse, aut enim in quo quidem est c, b inest omni, sed in quo est b nulli, inest a, ipsum igitur b c, et semper alterum spatium necesse est ire in immediata, praedicativum enim est hoc spatium.	For a negative conclusion can be proved in all three figures. In the first figure it is proved thus: no B is A, all C is B. In packing the interval B-C we must reach immediate propositions — as is always the case with the minor premiss — since B-C is affirmative.
τὸ δ᾽ ἕτερον δῆλον ὅτι εἰ ἄλλωι οὐχ ὑπάρχει προτέρωι, οἷον τῶι Δ, τοῦτο δεήσει τῶι Β παντὶ ὑπάρ↵ χειν. καὶ εἰ πάλιν ἄλλωι τοῦ Δ προτέρωι οὐχ ὑπάρχει, ἐκεῖνο δεήσει τῶι Δ παντὶ ὑπάρχειν. ὥστ᾽ ἐπεὶ ἡ ἐπὶ τὸ ἄνω ἵσταται ὁδός, καὶ ἡ ἐπὶ τὸ Α στήσεται, καὶ ἔσται τι πρῶτον ὧι οὐχ ὑπάρχει.	Sed alterum manifestum est, quod si in alio non est priori, ut in d, hoc indigebit in omni b esse, et si iterum a in alio priore quam d non fuerit, illud indigebit in omni d esse: quare quoniam in deorsum stat via, et quae in sursum stabit, et erit aliquid in quo primo non erit a.	As regards the other premiss it is plain that if the major term is denied of a term D prior to B, D will have to be predicable of all B, and if the major is denied of yet another term prior to D, this term must be predicable of all D. Consequently, since the ascending series is finite, the descent will also terminate and there will be a subject of which A is primarily non-predicable.
82b13 Πάλιν εἰ τὸ μὲν Β παντὶ τῶι Α, τῶι δὲ Γ μηδενί, τὸ Α τῶν Γ οὐδενὶ ὑπάρχει. πάλιν τοῦτο εἰ δεῖ δεῖ↵ ξαι, δῆλον ὅτι ἢ διὰ τοῦ ἄνω τρόπου δειχθήσεται ἢ διὰ τούτου ἢ τοῦ τρίτου. ὁ μὲν οὖν πρῶτος εἴρηται, ὁ δὲ δεύτερος δειχθήσεται. οὕτω δ᾽ ἂν δεικνύοι, οἷον τὸ Δ τῶι μὲν Β παντὶ ὑπάρχει, τῶι δὲ Γ οὐδενί, εἰ ἀνάγκη ὑπάρχειν τι τῶι Β. καὶ πάλιν εἰ τοῦτο τῶι Γ μὴ ὑπάρξει, ἄλλο τῶι Δ ↵ ὑπάρχει, ὁ τῶι Γ οὐχ ὑπάρχει. οὐκοῦν ἐπεὶ τὸ ὑπάρχειν ἀεὶ τῶι ἀνωτέρω ἵσταται, στήσεται καὶ τὸ μὴ ὑπάρχειν.	Item si b quidem in omni a, in c autem nullo, a in c nullo erit. Iterum hoc si oportet monstrare, manifestum est, quod aut per superiorem modum sursum monstrabitur, aut per hunc, aut tertium. Primus quidem igitur dictus est, secundus autem demonstrabitur. Sic autem utique monstrabitur, ut quod d in b omni est, in c autem nullo, si necesse est aliquid esse in b, et iterum si hoc in c non erit, aliud vero in d est quod in c non est, igitur quoniam esse semper superiori stat, stabit et non esse.	In the second figure the syllogism is, all A is B, no C is B,..no C is A. If proof of this is required, plainly it may be shown either in the first figure as above, in the second as here, or in the third. The first figure has been discussed, and we will proceed to display the second, proof by which will be as follows: all B is D, no C is D..., since it is required that B should be a subject of which a predicate is affirmed. Next, since D is to be proved not to belong to C, then D has a further predicate which is denied of C. Therefore, since the succession of predicates affirmed of an ever higher universal terminates, the succession of predicates denied terminates too.
82b23 Ὁ δὲ τρίτος τρόπος ἦν· εἰ τὸ μὲν Α τῶι Β παντὶ ὑπάρχει, τὸ δὲ Γ μὴ ὑπάρχει, οὐ παντὶ ὑπάρχει τὸ Γ ὧι τὸ Α. πάλιν δὲ τοῦτο ἢ διὰ τῶν ἄνω εἰρημένων ἢ ὁμοίως δειχθήσεται. ↵ ἐκείνως μὲν δὴ ἵσταται, εἰ δ᾽ οὕτω, πάλιν λήψεται τὸ Β τῶι Ε ὑπάρχειν, ὧι τὸ Γ μὴ παντὶ ὑπάρχει. καὶ τοῦτο πάλιν ὁμοίως. ἐπεὶ δ᾽ ὑπόκειται ἵστασθαι καὶ ἐπὶ τὸ κάτω, δῆλον ὅτι στήσεται καὶ τὸ Γ οὐχ ὑπάρχον.	Tertius autem modus est, si a b omni insit, c vero in nullo b sit, non in omni c, in quo est a. Iterum autem hoc per superius dicta aut similiter demonstrabitur. Illis igitur modis statur. Si vero sic est, iterum accipietur b in e esse, in quo c non in omni e, et hoc iterum similiter. Quoniam autem suppositum est stare et in deorsum, manifestum est quod stabit et quod in c non est.	The third figure shows it as follows: all B is A, some B is not C. Therefore some A is not C. This premiss, i.e. C-B, will be proved either in the same figure or in one of the two figures discussed above. In the first and second figures the series terminates. If we use the third figure, we shall take as premisses, all E is B, some E is not C, and this premiss again will be proved by a similar prosyllogism. But since it is assumed that the series of descending subjects also terminates, plainly the series of more universal non-predicables will terminate also.
82b28 Φανερὸν δ᾽ ὅτι καὶ ἐὰν μὴ μιᾶι ὁδῶι δεικνύηται ἀλλὰ πά↵ σαις, ὁτὲ μὲν ἐκ τοῦ πρώτου σχήματος, ὁτὲ δὲ ἐκ τοῦ δευτέρου ἢ τρίτου, ὅτι καὶ οὕτω στήσεται· πεπερασμέναι γάρ εἰσιν αἱ ὁδοί, τὰ δὲ πεπερασμένα πεπερασμενάκις ἀνάγκη πεπεράνθαι πάντα.	Manifestum autem est quoniam etsi non una via monstretur, sed omnibus, aliquando quidem ex prima figura, aliquando vero ex secunda, aut ex tertia, quoniam et sic stabitur, finitae enim sunt viae. Finita autem finite sumpta pluries, necesse est finiri omnia.	Even supposing that the proof is not confined to one method, but employs them all and is now in the first figure, now in the second or third-even so the regress will terminate, for the methods are finite in number, and if finite things are combined in a finite number of ways, the result must be finite.
82b34 Ὅτι μὲν οὖν ἐπὶ τῆς στερήσεως, εἴπερ καὶ ἐπὶ τοῦ ὑπάρ↵ χειν, ἵσταται, δῆλον.	Quod quidem igitur in privatione si quidem et inesse statim, manifestum est.	Thus it is plain that the regress of middles terminates in the case of negative demonstration, if it does so also in the case of affirmative demonstration. That in fact the regress terminates in both these cases may be made clear by the following dialectical considerations.
c22	CAPUT XVIII. Propositionis affirmativae mediatae, media non esse infinita.	Chapter 22
ὅτι δ᾽ ἐπ᾽ ἐκείνων, λογικῶς μὲν θεωροῦσιν ὧδε φανερόν. 83a1 Ἐπὶ μὲν οὖν τῶν ἐν τῶι τί ἐστι κατηγορουμένων δῆλον· εἰ γὰρ ἔστιν ὁρίσασθαι ἢ εἰ γνωστὸν τὸ τί ἦν εἶναι, τὰ δ᾽ ἄπειρα μὴ ἔστι διελθεῖν, ἀνάγκη πεπεράνθαι τὰ ἐν τῶι τί [83a]ἐστι κατηγορούμενα.	Sed quid in illis qui quidem logice speculantur, sic manifestum fit. In iis quidem igitur quae quidem in eo quod quid est praedicantur, manifestum est; si est enim definire, aut si notum est quod quid erat esse, infinita autem non est transire, necesse est finiri in eo quod quid est praedicata.	In the case of predicates constituting the essential nature of a thing, it clearly terminates, seeing that if definition is possible, or in other words, if essential form is knowable, and an infinite series cannot be traversed, predicates constituting a thing’s essential nature must be finite in number.
καθόλου δὲ ὧδε λέγομεν. ἔστι γὰρ εἰπεῖν ἀληθῶς τὸ λευκὸν βαδίζειν καὶ τὸ μέγα ἐκεῖνο ξύλον εἶναι, καὶ πάλιν τὸ ξύλον μέγα εἶναι καὶ τὸν ἄνθρωπον βαδίζειν. ἕτερον δή ἐστι τὸ οὕτως εἰπεῖν καὶ τὸ ἐκείνως. ὅταν ↵ μὲν γὰρ τὸ λευκὸν εἶναι φῶ ξύλον, τότε λέγω ὅτι ὧι συμβέβηκε λευκῶι εἶναι ξύλον ἐστίν, ἀλλ᾽ οὐχ ὡς τὸ ὑποκείμενον τῶι ξύλωι τὸ λευκόν ἐστι· καὶ γὰρ οὔτε λευκὸν ὂν οὔθ᾽ ὅπερ λευκόν τι ἐγένετο ξύλον, ὥστ᾽ οὐκ ἔστιν ἀλλ᾽ ἢ κατὰ συμβεβηκός.	Universaliter autem sic dicimus, est enim vere dicere album ambulare, et magnum illud lignum esse, et iterum lignum magnum esse, et hominem ambulare, sed alterum est sic dicere, aut illo modo, cum enim album quidem esse dico quod cui accidit album esse, lignum est, sed non quod subiectum signo, album sit, et namque neque quod album est, neque quod quidem album aliquod est, factum lignum est, quare non est aliter quam secundum accidens.	But as regards predicates generally we have the following prefatory remarks to make. (1) We can affirm without falsehood ‘the white (thing) is walking’, and that big (thing) is a log’; or again, ‘the log is big’, and ‘the man walks’. But the affirmation differs in the two cases. When I affirm ‘the white is a log’, I mean that something which happens to be white is a log-not that white is the substratum in which log inheres, for it was not qua white or qua a species of white that the white (thing) came to be a log, and the white (thing) is consequently not a log except incidentally.
ὅταν δὲ τὸ ξύλον λευκὸν εἶναι φῶ, οὐχ ὅτι ἕτερόν ↵ τί ἐστι λευκόν, ἐκείνωι δὲ συμβέβηκε ξύλωι εἶναι, οἷον ὅταν τὸ μουσικὸν λευκὸν εἶναι φῶ (τότε γὰρ ὅτι ὁ ἄνθρωπος λευκός ἐστιν, ὧι συμβέβηκεν εἶναι μουσικῶι, λέγω), ἀλλὰ τὸ ξύλον ἐστὶ τὸ ὑποκείμενον, ὅπερ καὶ ἐγένετο, οὐχ ἕτερόν τι ὂν ἢ ὅπερ ξύλον ἢ ξύλον τί.	Cum autem lignum album esse dico, non quod aliquod alterum sit album, illi autem accidit lignum esse, ut cum musicum album esse dico, tunc enim quoniam homo est albus, cui accidit esse musicum, dico, sed lignum est subiectum, quod quidem et factum est, non cum alterum aliquid sit quam quod quidem lignum est, aut lignum aliquod.	On the other hand, when I affirm ‘the log is white’, I do not mean that something else, which happens also to be a log, is white (as I should if I said 'the musician is white,' which would mean 'the man who happens also to be a musician is white'); on the contrary, log is here the substratum-the substratum which actually came to be white, and did so qua wood or qua a species of wood and qua nothing else.
εἰ δὴ δεῖ νομοθετῆσαι, ἔστω ↵ τὸ οὕτω λέγειν κατηγορεῖν, τὸ δ᾽ ἐκείνως ἤτοι μηδαμῶς κατηγορεῖν, ἢ κατηγορεῖν μὲν μὴ ἁπλῶς, κατὰ συμβεβηκὸς δὲ κατηγορεῖν. ἔστι δ᾽ ὡς μὲν τὸ λευκὸν τὸ κατηγορούμενον, ὡς δὲ τὸ ξύλον τὸ οὗ κατηγορεῖται. ὑποκείσθω δὴ τὸ κατηγορούμενον κατηγορεῖσθαι ἀεί, οὗ κατηγορεῖται, ↵ ἁπλῶς, ἀλλὰ μὴ κατὰ συμβεβηκός· οὕτω γὰρ αἱ ἀποδείξεις ἀποδεικνύουσιν.	Si igitur oportet nomina ponere, sit sic dicere praedicari, sed illo modo, aut nullo modo praedicari, aut praedicari quidem non simpliciter, sed secundum accidens praedicatur, est autem tanquam album quidem quod praedicatur, sed sicut lignum est de quo praedicatur. Supponatur ergo praedicatum praedicari semper de quo praedicatur, simpliciter, sed non secundum accidens. Sic enim demonstrationes demonstrant.	If we must lay down a rule, let us entitle the latter kind of statement predication, and the former not predication at all, or not strict but accidental predication. 'White' and 'log' will thus serve as types respectively of predicate and subject. We shall assume, then, that the predicate is invariably predicated strictly and not accidentally of the subject, for on such predication demonstrations depend for their force.
83a21 ὥστε ἢ ἐν τῶι τί ἐστιν ἢ ὅτι ποιὸν ἢ ποσὸν ἢ πρός τι ἢ ποιοῦν τι ἢ πάσχον ἢ ποὺ ἢ ποτέ, ὅταν ἓν καθ᾽ ἑνὸς κατηγορηθῆι.	Quare in eo quod quid est, aut quoniam quale, aut quantum, aut ad aliquid, aut faciens, aut patiens, aut ubi, aut quando, cum unum de uno praedicabitur.	It follows from this that when a single attribute is predicated of a single subject, the predicate must affirm of the subject either some element constituting its essential nature, or that it is in some way qualified, quantified, essentially related, active, passive, placed, or dated.
83a24 Ἔτι τὰ μὲν οὐσίαν σημαίνοντα ὅπερ ἐκεῖνο ἢ ὅπερ ↵ ἐκεῖνό τι σημαίνει καθ᾽ οὗ κατηγορεῖται· ὅσα δὲ μὴ οὐσίαν σημαίνει, ἀλλὰ κατ᾽ ἄλλου ὑποκειμένου λέγεται ὁ μὴ ἔστι μήτε ὅπερ ἐκεῖνο μήτε ὅπερ ἐκεῖνό τι, συμβεβηκότα,	Amplius, substantiam quidem significantia, quod quidem illud est, aut aliquod illud quidem significant de quo praedicantur. Quaecunque vero non substantiam significant, sed de aliquo subiecto dicuntur, quod non est, neque quod illud est, neque quod quidem illud aliquid est, accidentia sunt,	(2) Predicates which signify substance signify that the subject is identical with the predicate or with a species of the predicate. Predicates not signifying substance which are predicated of a subject not identical with themselves or with a species of themselves are accidental or coincidental;
83a27 οἷον κατὰ τοῦ ἀνθρώπου τὸ λευκόν. οὐ γάρ ἐστιν ὁ ἄνθρωπος οὔτε ὅπερ λευκὸν οὔτε ὅπερ λευκόν τι, ἀλλὰ ζῶιον ↵ ἴσως· ὅπερ γὰρ ζῶιόν ἐστιν ὁ ἄνθρωπος. ὅσα δὲ μὴ οὐσίαν σημαίνει, δεῖ κατά τινος ὑποκειμένου κατηγορεῖσθαι, καὶ μὴ εἶναί τι λευκὸν ὁ οὐχ ἕτερόν τι ὂν λευκόν ἐστιν.	ut de homine album. Neque enim est homo, neque quod quidem album est, neque quod quidem album est aliquid, sed animal forsan, quod quidem enim animal est, homo est. Quaecunque vero non substantiam significant, oportet de aliquo subiecto praedicari, et non esse quid album, quod non cum alterum aliquod sit, album est.	e.g. white is a coincident of man, seeing that man is not identical with white or a species of white, but rather with animal, since man is identical with a species of animal. These predicates which do not signify substance must be predicates of some other subject, and nothing can be white which is not also other than white.
83a33 τὰ γὰρ εἴδη χαιρέτω· τερετίσματά τε γάρ ἐστι, καὶ εἰ ἔστιν, οὐδὲν πρὸς τὸν λόγον ἐστίν· αἱ γὰρ ἀποδείξεις περὶ τῶν τοι↵ ούτων εἰσίν.	Species enim valeant, et genera, monstra enim sunt, et si sint, nihil ad rationem sunt, demonstrationes enim de huiusmodi sunt.	The Forms we can dispense with, for they are mere sound without sense; and even if there are such things, they are not relevant to our discussion, since demonstrations are concerned with predicates such as we have defined.
83a36 Ἔτι εἰ μὴ ἔστι τόδε τοῦδε ποιότης κἀκεῖνο τούτου, μηδὲ ποιότητος ποιότης,	Amplius, si non est hoc huius qualitas, et illud huius, neque qualitatis qualitas,	(3) If A is a quality of B, B cannot be a quality of A-a quality of a quality.
83a37 ἀδύνατον ἀντικατηγορεῖσθαι ἀλλήλων οὕτως, ἀλλ᾽ ἀληθὲς μὲν ἐνδέχεται εἰπεῖν,	impossibile est aeque praedicari ad invicem sic, sed verum quidem contingit dicere sic,	Therefore A and B cannot be predicated reciprocally of one another in strict predication: they can be affirmed without falsehood of one another,
ἀντικατηγορῆσαι δ᾽ ἀληθῶς οὐκ ἐνδέχεται. ἢ γάρ τοι ὡς οὐσία κατηγορηθή[83b]σεται, οἷον ἢ γένος ὂν ἢ διαφορὰ τοῦ κατηγορουμένου.	aeque autem vere praedicari non contingit. An enim sicut substantia praedicabitur, aut ut genus, aut ut differentia praedicari?	but not genuinely predicated of each other. 83a39 For one alternative is that they should be substantially predicated of one another, i.e. B would become the genus or differentia of A-the predicate now become subject.
83b1 ταῦτα δὲ δέδεικται ὅτι οὐκ ἔσται ἄπειρα, οὔτ᾽ ἐπὶ τὸ κάτω οὔτ᾽ ἐπὶ τὸ ἄνω (οἷον ἄνθρωπος δίπουν, τοῦτο ζῶιον, τοῦτο δ᾽ ἕτερον· οὐδὲ τὸ ζῶιον κατ᾽ ἀνθρώπου, τοῦτο δὲ κατὰ Καλλίου, τοῦτο ↵ δὲ κατ᾽ ἄλλου ἐν τῶι τί ἐστιν), τὴν μὲν γὰρ οὐσίαν ἅπασαν ἔστιν ὁρίσασθαι τὴν τοιαύτην, τὰ δ᾽ ἄπειρα οὐκ ἔστι διεξελθεῖν νοοῦντα. ὥστ᾽ οὔτ᾽ ἐπὶ τὸ ἄνω οὔτ᾽ ἐπὶ τὸ κάτω ἄπειρα· ἐκείνην γὰρ οὐκ ἔστιν ὁρίσασθαι ἧς τὰ ἄπειρα κατηγορεῖται.	haec autem ostensa sunt, quoniam non erunt infinita, neque in sursum, neque in deorsum, ut homo bipes, hoc animal, hoc autem alterum est, neque animal de homine, hoc autem de Callia, hoc autem de alio in eo quod quid est, substantiam enim omnem est definire huiusmodi, infinita autem non est transire intelligentem, quare neque in sursum, neque in deorsum infinita sunt, illam enim non est definire, de qua infinita praedicantur.	But it has been shown that in these substantial predications neither the ascending predicates nor the descending subjects form an infinite series; e.g. neither the series, man is biped, biped is animal, &c., nor the series predicating animal of man, man of Callias, Callias of a further. subject as an element of its essential nature, is infinite. For all such substance is definable, and an infinite series cannot be traversed in thought: consequently neither the ascent nor the descent is infinite, since a substance whose predicates were infinite would not be definable.
83b8 ὡς μὲν δὴ γένη ἀλλήλων οὐκ ἀντικατηγορηθήσεται· ἔσται ↵ γὰρ αὐτὸ ὅπερ αὐτό τι.	Sic igitur genera ad invicem, non aequaliter praedicantur, erit enim ipsum, quod quidem ipsum aliquid est,	Hence they will not be predicated each as the genus of the other; for this would equate a genus with one of its own species.
83b10 οὐδὲ μὴν τοῦ ποιοῦ ἢ τῶν ἄλλων οὐδέν, ἂν μὴ κατὰ συμβεβηκὸς κατηγορηθῆι· πάντα γὰρ ταῦτα συμβέβηκε καὶ κατὰ τῶν οὐσιῶν κατηγορεῖται.	neque tamen de qualitate, aut aliorum nullo, nisi secundum accidens praedicabitur, omnia enim haec accidunt, et de substantiis praedicantur,	Nor (the other alternative) can a quale be reciprocally predicated of a quale, nor any term belonging to an adjectival category of another such term, except by accidental predication; for all such predicates are coincidents and are predicated of substances.
83b13 ἀλλὰ δὴ ὅτι οὐδ᾽ εἰς τὸ ἄνω ἄπειρα ἔσται· ἑκάστου γὰρ κατηγορεῖται ὁ ἂν σημαίνηι ἢ ποιόν τι ἢ ποσόν τι ἤ τι τῶν τοιούτων ↵ ἢ τὰ ἐν τῆι οὐσίαι· ταῦτα δὲ πεπέρανται, καὶ τὰ γένη τῶν κατηγοριῶν πεπέρανται· ἢ γὰρ ποιὸν ἢ ποσὸν ἢ πρός τι ἢ ποιοῦν ἢ πάσχον ἢ ποὺ ἢ ποτέ.	sed quoniam neque in sursum, infinita sunt. De unoquoque enim praedicatur quod significat aut quantum aliquid, aut quale aliquid, aut huiusmodi quae sunt in substantia, haec autem finita sunt, et genera praedicamentorum finita, aut enim quale, aut quantum, aut ad aliquid, aut facere, aut pati, aut ubi, aut quando.	On the other hand - in proof of the impossibility of an infinite ascending series-every predication displays the subject as somehow qualified or quantified or as characterized under one of the other adjectival categories, or else is an element in its substantial nature: these latter are limited in number, and the number of the widest kinds under which predications fall is also limited, for every predication must exhibit its subject as somehow qualified, quantified, essentially related, acting or suffering, or in some place or at some time.
Ὑπόκειται δὴ ἓν καθ᾽ ἑνὸς κατηγορεῖσθαι, αὐτὰ δὲ αὑτῶν, ὅσα μὴ τί ἐστι, μὴ κατηγορεῖσθαι. συμβεβηκότα γάρ ἐστι πάντα, ἀλλὰ τὰ μὲν ↵ καθ᾽ αὑτά, τὰ δὲ καθ᾽ ἕτερον τρόπον· ταῦτα δὲ πάντα καθ᾽ ὑποκειμένου τινὸς κατηγορεῖσθαί φαμεν, τὸ δὲ συμβεβηκὸς οὐκ εἶναι ὑποκείμενόν τι· οὐδὲν γὰρ τῶν τοιούτων τίθεμεν εἶναι ὁ οὐχ ἕτερόν τι ὂν λέγεται ὁ λέγεται, ἀλλ᾽ αὐτὸ ἄλλου καὶ τοῦτο καθ᾽ ἑτέρου.	Suppositum autem est unum de uno praedicari, ipsa autem de ipsis quaecunque non ad aliquid sunt, praedicari, non dicimus, accidentia enim sunt omnia, sed haec quidem secundum seipsa, alia vero secundum alterum modum. Haec autem omnia de subiecto quodam praedicari dicimus, accidens autem non esse subiectum aliquod, nihil enim talium ponimus esse, non quod alterum aliquod esse dicitur, sed ipsum de aliis, et alia quidem de alio,	I assume first that predication implies a single subject and a single attribute, and secondly that predicates which are not substantial are not predicated of one another. We assume this because such predicates are all coincidents, and though some are essential coincidents, others of a different type, yet we maintain that all of them alike are predicated of some substratum and that a coincident is never a substratum-since we do not class as a coincident anything which does not owe its designation to its being something other than itself, but always hold that any coincident is predicated of some substratum other than itself, and that another group of coincidents may have a different substratum.
83b24 οὔτ᾽ εἰς τὸ ἄνω ↵ ἄρα ἓν καθ᾽ ἑνὸς οὔτ᾽ εἰς τὸ κάτω ὑπάρχειν λεχθήσεται. καθ᾽ ὧν μὲν γὰρ λέγεται τὰ συμβεβηκότα, ὅσα ἐν τῆι οὐσίαι ἑκάστου, ταῦτα δὲ οὐκ ἄπειρα· ἄνω δὲ ταῦτά τε καὶ τὰ συμβεβηκότα, ἀμφότερα οὐκ ἄπειρα. ἀνάγκη ἄρα εἶναί τι οὗ πρῶτόν τι κατηγορεῖται καὶ τούτου ἄλλο, καὶ τοῦτο ↵ ἵστασθαι καὶ εἶναί τι ὁ οὐκέτι οὔτε κατ᾽ ἄλλου προτέρου οὔτε κατ᾽ ἐκείνου ἄλλο πρότερον κατηγορεῖται.	neque in sursum, ergo unum de uno, neque in deorsum esse dicetur, de quibus enim dicuntur accidentia, quaecunque in substantia uniuscuiusque sunt, haec autem non sunt infinita, sed sursum ipsa quoque et accidentia utraque non infinita sunt. Necesse est ergo esse aliquid, de quo primum praedicetur, et de hoc aliud, et hoc stare, et esse aliquid quod non amplius, neque de alio priori, neque de illo aliud prius praedicetur.	Subject to these assumptions then, neither the ascending nor the descending series of predication in which a single attribute is predicated of a single subject is infinite. For the subjects of which coincidents are predicated are as many as the constitutive elements of each individual substance, and these we have seen are not infinite in number, while in the ascending series are contained those constitutive elements with their coincidents-both of which are finite. We conclude that there is a given subject (D) of which some attribute (C) is primarily predicable; that there must be an attribute (B) primarily predicable of the first attribute, and that the series must end with a term (A) not predicable of any term prior to the last subject of which it was predicated (B), and of which no term prior to it is predicable.
83b33 Εἷς μὲν οὖν τρόπος λέγεται ἀποδείξεως οὗτος, ἔτι δ᾽ ἄλλος, εἰ ὧν πρότερα ἄττα κατηγορεῖται, ἔστι τούτων ἀπόδειξις, ὧν δ᾽ ἔστιν ἀπόδειξις, οὔτε βέλτιον ἔχειν ἐγχωρεῖ ↵ πρὸς αὐτὰ τοῦ εἰδέναι, οὔτ᾽ εἰδέναι ἄνευ ἀποδείξεως, εἰ δὲ τόδε διὰ τῶνδε γνώριμον,	Unus quidem igitur modus demonstrationis, dicitur hic. Adhuc autem alius, si de quibus priora quaedam praedicantur, est horum demonstratio, quorum autem est demonstratio, neque potius habere possibile est ad ipsa, quam scire, neque scire est sine demonstratione, si autem hoc, per haec sit notum.	The argument we have given is one of the so-called proofs; an alternative proof follows. Predicates so related to their subjects that there are other predicates prior to them predicable of those subjects are demonstrable; but of demonstrable propositions one cannot have something better than knowledge, nor can one know them without demonstration.
τάδε δὲ μὴ ἴσμεν μηδὲ βέλτιον ἔχομεν πρὸς αὐτὰ τοῦ εἰδέναι, οὐδὲ τὸ διὰ τούτων γνώριμον ἐπιστησόμεθα.	Haec autem nescimus, neque melius habemus ad ipsa quam scire, neque per haec notum sciemus.	Secondly, if a consequent is only known through an antecedent (viz. premisses prior to it) and we neither know this antecedent nor have something better than knowledge of it, then we shall not have scientific knowledge of the consequent.
εἰ οὖν ἔστι τι εἰδέναι δι᾽ ἀποδείξεως ἁπλῶς καὶ μὴ ἐκ τινῶν μηδ᾽ ἐξ ὑποθέσεως, ἀνάγκη ἵστασθαι τὰς [84a]κατηγορίας τὰς μεταξύ. εἰ γὰρ μὴ ἵστανται, ἀλλ᾽ ἔστιν ἀεὶ τοῦ ληφθέντος ἐπάνω, ἁπάντων ἔσται ἀπόδειξις· ὥστ᾽ εἰ τὰ ἄπειρα μὴ ἐγχωρεῖ διελθεῖν, ὧν ἔστιν ἀπόδειξις, ταῦτ᾽ οὐκ εἰσόμεθα δι᾽ ἀποδείξεως.	Si igitur est aliquid scire per demonstrationem simpliciter, et non ex aliquibus, neque ex suppositione, necessarium est stare praedicationes mediorum, si enim non steterint, sed est semper acceptio in superius, omnium erit demonstratio, quare si infinita non possibile est pertransire, quorum est demonstratio, haec non sciemus per demonstrationem.	Therefore, if it is possible through demonstration to know anything without qualification and not merely as dependent on the acceptance of certain premisses-i.e. hypothetically-the series of intermediate predications must terminate. If it does not terminate, and beyond any predicate taken as higher than another there remains another still higher, then every predicate is demonstrable. Consequently, since these demonstrable predicates are infinite in number and therefore cannot be traversed, we shall not know them by demonstration.
εἰ οὖν μηδὲ βέλτιον ἔχομεν πρὸς ↵ αὐτὰ τοῦ εἰδέναι, οὐκ ἔσται οὐδὲν ἐπίστασθαι δι᾽ ἀποδείξεως ἁπλῶς, ἀλλ᾽ ἐξ ὑποθέσεως. Λογικῶς μὲν οὖν ἐκ τούτων ἄν τις πιστεύσειε περὶ τοῦ λεχθέντος,	Si igitur neque melius habemus ad ipsa quam scire, non erit scire per demonstrationem simpliciter, sed ex suppositione. Logice quidem igitur ex his utique aliquis credat, de eo quod dictum est.	If, therefore, we have not something better than knowledge of them, we cannot through demonstration have unqualified but only hypothetical science of anything. As dialectical proofs of our contention these may carry conviction,
84a8 ἀναλυτικῶς δὲ διὰ τῶνδε φανερὸν συντομώτερον, ὅτι οὔτ᾽ ἐπὶ τὸ ἄνω οὔτ᾽ ἐπὶ τὸ κάτω ἄπειρα τὰ κατ↵ ηγορούμενα ἐνδέχεται εἶναι ἐν ταῖς ἀποδεικτικαῖς ἐπιστήμαις, περὶ ὧν ἡ σκέψις ἐστίν.	Analytice autem manifestum est per haec velocius, neque in sursum, neque in deorsum infinita praedicata contingit esse in demonstrativi scientiis, de quibus intentio est;	but an analytic process will show more briefly that neither the ascent nor the descent of predication can be infinite in the demonstrative sciences which are the object of our investigation.
84a10 ἡ μὲν γὰρ ἀπόδειξίς ἐστι τῶν ὅσα ὑπάρχει καθ᾽ αὑτὰ τοῖς πράγμασιν.	demonstratio enim est ex his quaecunque ipsa quidem insunt secundum seipsa rebus,	Demonstration proves the inherence of essential attributes in things.
84a11 καθ᾽ αὑτὰ δὲ διττῶς· ὅσα τε γὰρ [ἐν] ἐκείνοις ἐνυπάρχει ἐν τῶι τί ἐστι, καὶ οἷς αὐτὰ ἐν τῶι τί ἐστιν ὑπάρχουσιν αὐτοῖς· οἷον τῶι ἀριθμῶι τὸ περιτ↵ τόν, ὁ ὑπάρχει μὲν ἀριθμῶι, ἐνυπάρχει δ᾽ αὐτὸς ὁ ἀριθμὸς ἐν τῶι λόγωι αὐτοῦ, καὶ πάλιν πλῆθος ἢ τὸ διαιρετὸν ἐν τῶι λόγωι τῶι τοῦ ἀριθμοῦ ἐνυπάρχει.	secundum seipsa vero dupliciter; quaecunque enim in illis insunt in eo quod quid est, et in quibus ipsa in eo quod quid est insunt ipsis, ut in numero impar, quod inest quidem numero, est autem ipse numerus in ratione ipsius, et iterum multitudo, aut divisibile, in ratione numeri,	Now attributes may be essential for two reasons: either because they are elements in the essential nature of their subjects, or because their subjects are elements in their essential nature. An example of the latter is odd as an attribute of number-though it is number’s attribute, yet number itself is an element in the definition of odd; of the former, multiplicity or the indivisible, which are elements in the definition of number.
84a18 τούτων δ᾽ οὐδέτερα ἐνδέχεται ἄπειρα εἶναι, οὔθ᾽ ὡς τὸ περιττὸν τοῦ ἀριθμοῦ (πάλιν γὰρ ἂν τῶι περιττῶι ἄλλο εἴη ὧι ἐνυπῆρχεν ὑπάρ↵ χοντι· τοῦτο δ᾽ εἰ ἔστι, πρῶτον ὁ ἀριθμὸς ἐνυπάρξει ὑπάρχουσιν αὐτῶι· εἰ οὖν μὴ ἐνδέχεται ἄπειρα τοιαῦτα ὑπάρχειν ἐν τῶι ἑνί, οὐδ᾽ ἐπὶ τὸ ἄνω ἔσται ἄπειρα·	horum autem neutra contingit infinita esse, neque ut impar numeri. Iterum enim si impar aliud insit, cui inerat existenti hoc, si est primum numerus, erit iis quae insunt ipsi. Si igitur non contingit infinita huiusmodi esse in uno, neque in sursum erunt infinita,	In neither kind of attribution can the terms be infinite. They are not infinite where each is related to the term below it as odd is to number, for this would mean the inherence in odd of another attribute of odd in whose nature odd was an essential element: but then number will be an ultimate subject of the whole infinite chain of attributes, and be an element in the definition of each of them. Hence, since an infinity of attributes such as contain their subject in their definition cannot inhere in a single thing, the ascending series is equally finite.
84a23 ἀλλὰ μὴν ἀνάγκη γε πάντα ὑπάρχειν τῶι πρώτωι, οἷον τῶι ἀριθμῶι, κἀκείνοις τὸν ἀριθμόν, ὥστ᾽ ἀντιστρέφοντα ἔσται, ἀλλ᾽ οὐχ ↵ ὑπερτείνοντα)·	at vero necesse est omnia inesse primo, ut numero, et in illis numerum, quare convertibilia erunt, sed non excedentia.	Note, moreover, that all such attributes must so inhere in the ultimate subject-e.g. its attributes in number and number in them-as to be commensurate with the subject and not of wider extent.
84a25 οὐδὲ μὴν ὅσα ἐν τῶι τί ἐστιν ἐνυπάρχει, οὐδὲ ταῦτα ἄπειρα· οὐδὲ γὰρ ἂν εἴη ὁρίσασθαι. ὥστ᾽ εἰ τὰ μὲν κατηγορούμενα καθ᾽ αὑτὰ πάντα λέγεται, ταῦτα δὲ μὴ ἄπειρα, ἵσταιτο ἂν τὰ ἐπὶ τὸ ἄνω, ὥστε καὶ ἐπὶ τὸ κάτω.	Neque tamen quaecunque sunt in eo quod quid est, neque haec infinita sunt, neque esset definire: quare si praedicata per se quidem omnia dicuntur, haec autem non infinita sunt, stabunt utique in sursum, quare et in deorsum,	Attributes which are essential elements in the nature of their subjects are equally finite: otherwise definition would be impossible. Hence, if all the attributes predicated are essential and these cannot be infinite, the ascending series will terminate, and consequently the descending series too.
84a28 Εἰ δ᾽ οὕτω, καὶ τὰ ἐν τῶι μεταξὺ δύο ὅρων ἀεὶ πε↵ περασμένα.	si autem sic est, et quae sunt in medio duorum terminorum semper sunt finita.	If this is so, it follows that the intermediates between any two terms are also always limited in number.
84a29 εἰ δὲ τοῦτο, δῆλον ἤδη καὶ τῶν ἀποδείξεων ὅτι ἀνάγκη ἀρχάς τε εἶναι, καὶ μὴ πάντων εἶναι ἀπόδειξιν, ὅπερ ἔφαμέν τινας λέγειν κατ᾽ ἀρχάς.	Si vero hoc est, manifestum iam est, et demonstratio non quod necesse est principia esse, et non omnium esse demonstrationem, quod quidem diximus quosdam dicere iuxta principium,	An immediately obvious consequence of this is that demonstrations necessarily involve basic truths, and that the contention of some-referred to at the outset-that all truths are demonstrable is mistaken.
εἰ γὰρ εἰσὶν ἀρχαί, οὔτε πάντ᾽ ἀποδεικτὰ οὔτ᾽ εἰς ἄπειρον οἷόν τε βαδίζειν· τὸ γὰρ εἶναι τούτων ὁποτερονοῦν οὐδὲν ἄλλο ἐστὶν ἢ τὸ εἶναι μη- ↵ δὲν διάστημα ἄμεσον καὶ ἀδιαίρετον, ἀλλὰ πάντα διαιρετά. τῶι γὰρ ἐντὸς ἐμβάλλεσθαι ὅρον, ἀλλ᾽ οὐ τῶι προσλαμβάνεσθαι ἀποδείκνυται τὸ ἀποδεικνύμενον, ὥστ᾽ εἰ τοῦτ᾽ εἰς ἄπειρον ἐνδέχεται ἰέναι, ἐνδέχοιτ᾽ ἂν δύο ὅρων ἄπειρα μεταξὺ εἶναι μέσα. ἀλλὰ τοῦτ᾽ ἀδύνατον, εἰ ἵστανται αἱ κατ[84b]ηγορίαι ἐπὶ τὸ ἄνω καὶ τὸ κάτω. ὅτι δὲ ἵστανται, δέδεικται λογικῶς μὲν πρότερον, ἀναλυτικῶς δὲ νῦν.	si enim principia sunt. Non omnia sunt demonstrabilia, neque in infinitum possibile ire, esse enim horum quodlibet, nihil est aliud quam esse nullum spatium sine medio, et indivisibile, sed omnia divisibilia. Intus enim immittendo terminum, sed non assumendo demonstratur quod demonstratur, quare si hoc in infinitum contingit ire, contingit utique duorum terminorum infinita esse interius media, sed hoc impossibile est, si praedicationes steterint in superius, et in inferius: quod autem stent, monstratum est logice prius, analytice vero nunc.	For if there are basic truths, (a) not all truths are demonstrable, and (b) an infinite regress is impossible; since if either (a) or (b) were not a fact, it would mean that no interval was immediate and indivisible, but that all intervals were divisible. This is true because a conclusion is demonstrated by the interposition, not the apposition, of a fresh term. If such interposition could continue to infinity there might be an infinite number of terms between any two terms; but this is impossible if both the ascending and descending series of predication terminate; and of this fact, which before was shown dialectically, analytic proof has now been given.
c23		Chapter 23
84b3 Δεδειγμένων δὲ τούτων φανερὸν ὅτι, ἐάν τι τὸ αὐτὸ δυσὶν ὑπάρχηι, οἷον τὸ Α τῶι τε Γ καὶ τῶι Δ, μὴ κατ↵ ηγορουμένου θατέρου κατὰ θατέρου, ἢ μηδαμῶς ἢ μὴ κατὰ παντός, ὅτι οὐκ ἀεὶ κατὰ κοινόν τι ὑπάρξει.	CAPUT XIX. Elementa monstrandarum mediatarum non infinita esse. Monstratis autem his, manifestum est si aliquid idem insit duobus, ut a et in c, et in d, non praedicante altero de altero, aut nullo modo, aut non de omni, quod non semper secundum commune aliquid inerit,	It is an evident corollary of these conclusions that if the same attribute A inheres in two terms C and D predicable either not at all, or not of all instances, of one another, it does not always belong to them in virtue of a common middle term.
84b6 οἷον τῶι ἰσοσκελεῖ καὶ τῶι σκαληνεῖ τὸ δυσὶν ὀρθαῖς ἴσας ἔχειν κατὰ κοινόν τι ὑπάρχει (ἧι γὰρ σχῆμά τι, ὑπάρχει, καὶ οὐχ ἧι ἕτερον), τοῦτο δ᾽ οὐκ ἀεὶ οὕτως ἔχει.	ut isosceli, et scaleno, aequales duobus rectis habere, secundum commune aliquid inest, secundum enim quod figura quaedam sunt, et non secundum alterum. Hoc autem non semper sic se habet.	Isosceles and scalene possess the attribute of having their angles equal to two right angles in virtue of a common middle; for they possess it in so far as they are both a certain kind of figure, and not in so far as they differ from one another. But this is not always the case:
84b9 ἔστω γὰρ τὸ Β καθ᾽ ↵ ὁ τὸ Α τῶι Γ Δ ὑπάρχει. δῆλον τοίνυν ὅτι καὶ τὸ Β τῶι Γ καὶ Δ κατ᾽ ἄλλο κοινόν, κἀκεῖνο καθ᾽ ἕτερον, ὥστε δύο ὅρων μεταξὺ ἄπειροι ἂν ἐμπίπτοιεν ὅροι. ἀλλ᾽ ἀδύνατον. κατὰ μὲν τοίνυν κοινόν τι ὑπάρχειν οὐκ ἀνάγκη ἀεὶ τὸ αὐτὸ πλείοσιν, εἴπερ ἔσται ἄμεσα διαστήματα.	Sit enim b secundum quod a, in c et d, erit manifestum igitur quod b in c d secundum aliud commune est, et illud secundum alterum, quare duorum terminorum medii infiniti utique inciderunt termini, sed hoc est impossibile, secundum igitur aliquid commune inesse, non necesse est semper idem pluribus, quoniam quidem erunt immediata spatia.	for, were it so, if we take B as the common middle in virtue of which A inheres in C and D, clearly B would inhere in C and D through a second common middle, and this in turn would inhere in C and D through a third, so that between two terms an infinity of intermediates would fall-an impossibility. Thus it need not always be in virtue of a common middle term that a single attribute inheres in several subjects, since there must be immediate intervals.
84b15 ἐν μέν↵ τοι τῶι αὐτῶι γένει καὶ ἐκ τῶν αὐτῶν ἀτόμων ἀνάγκη τοὺς ὅρους εἶναι, εἴπερ τῶν καθ᾽ αὑτὸ ὑπαρχόντων ἔσται τὸ κοινόν· οὐ γὰρ ἦν ἐξ ἄλλου γένους εἰς ἄλλο διαβῆναι τὰ δεικνύμενα.	In eodem tamen genere, et ex eisdem atomis necesse est terminos esse, siquidem iis quae per sunt, erit commune, non enim erat ex alio genere in aliud genus descendere quae demonstrantur.	Yet if the attribute to be proved common to two subjects is to be one of their essential attributes, the middle terms involved must be within one subject genus and be derived from the same group of immediate premisses; for we have seen that processes of proof cannot pass from one genus to another.
84b19 Φανερὸν δὲ καὶ ὅτι, ὅταν τὸ Α τῶι Β ὑπάρχηι, εἰ ↵ μὲν ἔστι τι μέσον, ἔστι δεῖξαι ὅτι τὸ Α τῶι Β ὑπάρχει, καὶ στοιχεῖα τούτου ἔστι ταὐτὰ καὶ τοσαῦθ᾽ ὅσα μέσα ἐστίν· αἱ γὰρ ἄμεσοι προτάσεις στοιχεῖα, ἢ πᾶσαι ἢ αἱ καθόλου. εἰ δὲ μὴ ἔστιν, οὐκέτι ἔστιν ἀπόδειξις, ἀλλ᾽ ἡ ἐπὶ τὰς ἀρχὰς ὁδὸς αὕτη ἐστίν.	Manifestum est autem quoniam et cum a in b sit, siquidem est aliquid medium, est demonstrare quod a in b sit, et elementa huius sunt haec et tot quot media sunt, immediatae enim propositiones sunt elementa, aut omnes, aut universales; si vero non est medium, non amplius erit demonstratio, sed in principia via est haec.	It is also clear that when A inheres in B, this can be demonstrated if there is a middle term. Further, the ‘elements’ of such a conclusion are the premisses containing the middle in question, and they are identical in number with the middle terms, seeing that the immediate propositions-or at least such immediate propositions as are universal-are the ‘elements’. If, on the other hand, there is no middle term, demonstration ceases to be possible: we are on the way to the basic truths.
84b24 ὁμοίως δὲ καὶ εἰ τὸ Α τῶι Β μὴ ὑπάρχει, ↵ εἰ μὲν ἔστιν ἢ μέσον ἢ πρότερον ὧι οὐχ ὑπάρχει, ἔστιν ἀπόδειξις, εἰ δὲ μή, οὐκ ἔστιν, ἀλλ᾽ ἀρχή, καὶ στοιχεῖα τοσαῦτ᾽ ἔστιν ὅσοι ὅροι· αἱ γὰρ τούτων προτάσεις ἀρχαὶ τῆς ἀπο- δείξεώς εἰσιν. καὶ ὥσπερ ἔνιαι ἀρχαί εἰσιν ἀναπόδεικτοι, ὅτι ἐστὶ τόδε τοδὶ καὶ ὑπάρχει τόδε τωιδί, οὕτω καὶ ὅτι οὐκ ἔστι ↵ τόδε τοδὶ οὐδ᾽ ὑπάρχει τόδε τωιδί, ὥσθ᾽ αἱ μὲν εἶναί τι, αἱ δὲ μὴ εἶναί τι ἔσονται ἀρχαί.	Similiter autem erit et si a in b non sit, siquidem enim aut medium est, aut prius cui non inest, erit demonstratio. Si vero non sit medium, non est demonstratio, sed principia, et elementa sunt tot quot sunt termini, horum enim propositiones principia demonstrationis sunt, et sicut quaedam principia sunt indemonstrabilia quod sit hoc illud, et quod sit hoc in illo, sic quod non erit hoc illud, neque quod sit hoc in illo. Quare haec quidem esse aliquid, alia non esse aliquid, erunt principia.	Similarly if A does not inhere in B, this can be demonstrated if there is a middle term or a term prior to B in which A does not inhere: otherwise there is no demonstration and a basic truth is reached. There are, moreover, as many ‘elements’ of the demonstrated conclusion as there are middle terms, since it is propositions containing these middle terms that are the basic premisses on which the demonstration rests; and as there are some indemonstrable basic truths asserting that ‘this is that’ or that ‘this inheres in that’, so there are others denying that ‘this is that’ or that ‘this inheres in that’-in fact some basic truths will affirm and some will deny being.
84b32 Ὅταν δὲ δέηι δεῖξαι, ληπτέον ὁ τοῦ Β πρῶτον κατηγορεῖται. ἔστω τὸ Γ, καὶ τούτου ὁμοίως τὸ Δ. καὶ οὕτως ἀεὶ βαδίζοντι οὐδέποτ᾽ ἐξωτέρω πρότασις οὐδ᾽ ὑπάρχον λαμβάνεται τοῦ Α ἐν τῶι δεικνύναι, ἀλλ᾽ ἀεὶ ↵ τὸ μέσον πυκνοῦται, ἕως ἀδιαίρετα γένηται καὶ ἕν. ἔστι δ᾽ ἓν ὅταν ἄμεσον γένηται, καὶ μία πρότασις ἁπλῶς ἡ ἄμεσος.	Cum ergo indigeat monstrare aliquid, accipiendum quod de b primum praedicetur. Sit c, et de hoc similiter a, et sic semper eunti nunquam extra, erit propositio, neque si esse ipsius a accipiatur ut demonstretur, sed semper medium densetur quousque indivisibilia fiant, et unum, est autem unum, cum immediatum fiat, et una propositio simplex est immediata,	When we are to prove a conclusion, we must take a primary essential predicate-suppose it C-of the subject B, and then suppose A similarly predicable of C. If we proceed in this manner, no proposition or attribute which falls beyond A is admitted in the proof: the interval is constantly condensed until subject and predicate become indivisible, i.e. one. We have our unit when the premiss becomes immediate, since the immediate premiss alone is a single premiss in the unqualified sense of ‘single’.
84b38 καὶ ὥσπερ ἐν τοῖς ἄλλοις ἡ ἀρχὴ ἁπλοῦν, τοῦτο δ᾽ οὐ ταὐτὸ πανταχοῦ, ἀλλ᾽ ἐν βάρει μὲν μνᾶ, ἐν δὲ μέλει δίεσις, ἄλλο δ᾽ ἐν ἄλλωι, οὕτως ἐν συλλογισμῶι τὸ ἓν [85a]πρότασις ἄμεσος, ἐν δ᾽ ἀποδείξει καὶ ἐπιστήμηι ὁ νοῦς.	et quemadmodum in aliis est principium simplex. Hoc autem non idem ubique est, sed in gravi quidem uncia, in melodia autem diesis, aliud autem in alio, sic est in syllogismo unum, propositio immediata, in demonstratione autem, et scientia, intellectus.	And as in other spheres the basic element is simple but not identical in all-in a system of weight it is the mina, in music the quarter-tone, and so on — so in syllogism the unit is an immediate premiss, and in the knowledge that demonstration gives it is an intuition.
85a2 ἐν μὲν οὖν τοῖς δεικτικοῖς συλλογισμοῖς τοῦ ὑπάρχοντος οὐδὲν ἔξω πίπτει,	In ostensivis igitur syllogismis eius quod est, nihil cadit extra.	In syllogisms, then, which prove the inherence of an attribute, nothing falls outside the major term.
85a3 ἐν δὲ τοῖς στερητικοῖς, ἔνθα μὲν ὁ δεῖ ὑπάρχειν, οὐδὲν τούτου ἔξω πίπτει, οἷον εἰ τὸ Α τῶι Β διὰ τοῦ Γ μή ↵ (εἰ γὰρ τῶι μὲν Β παντὶ τὸ Γ, τῶι δὲ Γ μηδενὶ τὸ Α)· πάλιν ἂν δέηι ὅτι τῶι Γ τὸ Α οὐδενὶ ὑπάρχει, μέσον ληπτέον τοῦ Α καὶ Γ, καὶ οὕτως ἀεὶ πορεύσεται.	Sed in privativis, ubi quidem quod oportet esse, nihil cadit extra hoc, ut si a in b per c non inest, si enim in b quidem omni c, est autem a in nullo c. Iterum si indigeat quod in c, a nullo sit, medium accipiendum est ipsius a et c, et sic semper procedet.	In the case of negative syllogisms on the other hand, (1) in the first figure nothing falls outside the major term whose inherence is in question; e.g. to prove through a middle C that A does not inhere in B the premisses required are, all B is C, no C is A. Then if it has to be proved that no C is A, a middle must be found between and C; and this procedure will never vary.
85a7 ἐὰν δὲ δέηι δεῖξαι ὅτι τὸ Δ τῶι Ε οὐχ ὑπάρχει τῶι τὸ Γ τῶι μὲν Δ παντὶ ὑπάρχειν, τῶι δὲ Ε μηδενί [ἢ μὴ παντί], τοῦ Ε οὐδέποτ᾽ ἔξω ↵ πεσεῖται· τοῦτο δ᾽ ἐστὶν ὧι δεῖ ὑπάρχειν.	Si vero indigeat monstrare quod d in e non sit, eo quod est c in d quidem omni, in e autem nullo, aut non in omni, e nunquam extra cadit, hoc autem est, cui non oportet inesse.	(2) If we have to show that E is not D by means of the premisses, all D is C; no E, or not all E, is C; then the middle will never fall beyond E, and E is the subject of which D is to be denied in the conclusion.
85a10 ἐπὶ δὲ τοῦ τρίτου τρόπου, οὔτε ἀφ᾽ οὗ δεῖ οὔτε ὁ δεῖ στερῆσαι οὐδέποτ᾽ ἔξω βαδιεῖται.	In tertio autem modo, neque a quo oportet, neque quod oportet privari, nequaquam extra ibit.	(3) In the third figure the middle will never fall beyond the limits of the subject and the attribute denied of it.
c24	CAPUT XX. Quod demonstratio universalis praestantior sit particulari.	Chapter 24
85a12 Οὔσης δ᾽ ἀποδείξεως τῆς μὲν καθόλου τῆς δὲ κατὰ μέρος, καὶ τῆς μὲν κατηγορικῆς τῆς δὲ στερητικῆς, ἀμφι↵ σβητεῖται ποτέρα βελτίων· ὡς δ᾽ αὔτως καὶ περὶ τῆς ἀποδεικνύναι λεγομένης καὶ τῆς εἰς τὸ ἀδύνατον ἀγούσης ἀποδείξεως.	Cum autem sit omnis demonstratio, alia quidem universalis, alia vero particularis, et haec quidem cathegorica, illa vero privativa, dubitabitur qualis potior sit, similiter autem et de ea, quae demonstrare dicitur, et deducenti ad impossibile demonstratione.	Since demonstrations may be either commensurately universal or particular, and either affirmative or negative; the question arises, which form is the better? And the same question may be put in regard to so-called ‘direct’ demonstration and reductio ad impossibile.
85a17 πρῶτον μὲν οὖν ἐπισκεψώμεθα περὶ τῆς καθόλου καὶ τῆς κατὰ μέρος· δηλώσαντες δὲ τοῦτο, καὶ περὶ τῆς δεικνύναι λεγομένης καὶ τῆς εἰς τὸ ἀδύνατον εἴπωμεν.	Primum quidem igitur intendamus de universali et particulari. Ostendentes autem hoc, et de ea quae demonstrare dicitur, et quae est ad impossibile, dicemus.	Let us first examine the commensurately universal and the particular forms, and when we have cleared up this problem proceed to discuss ‘direct’ demonstration and reductio ad impossibile.
85a20 ↵ Δόξειε μὲν οὖν τάχ᾽ ἄν τισιν ὡδὶ σκοποῦσιν ἡ κατὰ μέρος εἶναι βελτίων. εἰ γὰρ καθ᾽ ἣν μᾶλλον ἐπιστάμεθα ἀπόδειξιν βελτίων ἀπόδειξις (αὕτη γὰρ ἀρετὴ ἀποδείξεωσ), μᾶλλον δ᾽ ἐπιστάμεθα ἕκαστον ὅταν αὐτὸ εἰδῶμεν καθ᾽ αὑτὸ ἢ ὅταν κατ᾽ ἄλλο (οἷον τὸν μουσικὸν Κορίσκον ὅταν ↵ ὅτι ὁ Κορίσκος μουσικὸς ἢ ὅταν ὅτι ἅνθρωπος μουσικός· ὁμοίως δὲ καὶ ἐπὶ τῶν ἄλλων), ἡ δὲ καθόλου ὅτι ἄλλο, οὐχ ὅτι αὐτὸ τετύχηκεν ἐπιδείκνυσιν (οἷον ὅτι τὸ ἰσοσκελὲς οὐχ ὅτι ἰσοσκελὲς ἀλλ᾽ ὅτι τρίγωνον), ἡ δὲ κατὰ μέρος ὅτι αὐτό· – εἰ δὴ βελτίων μὲν ἡ καθ᾽ αὑτό, τοιαύτη δ᾽ ἡ κατὰ μέρος τῆς ↵ καθόλου μᾶλλον, καὶ βελτίων ἂν ἡ κατὰ μέρος ἀπόδειξις εἴη.	Videbitur igitur fortassis utique quibusdam sic intendentibus, quod particularis est potior, si enim secundum quam magis scimus demonstrationem, potior demonstratio est, haec enim virtus demonstrationis. Magis autem scimus unumquodque cum ipsum cognoscimus secundum ipsum, quam secundum aliud, ut musicum Coriscum, quando cognoscemus quod Coriscus musicus est, quam quod homo musicus sit. Similiter autem et in aliis. Sed universalis quoniam aliud, non quoniam ipsum fortasse demonstrat, ut quoniam isosceles habet tres angulos aequales duobus rectis, non quoniam isosceles, sed quoniam triangulus, sed particularis quoniam ipsum est; si igitur potior est quae est secundum ipsum, huiusmodi autem est particularis, et universali magis, et potior utique secundum partem demonstratio erit.	The following considerations might lead some minds to prefer particular demonstration. (1) The superior demonstration is the demonstration which gives us greater knowledge (for this is the ideal of demonstration), and we have greater knowledge of a particular individual when we know it in itself than when we know it through something else; e.g. we know Coriscus the musician better when we know that Coriscus is musical than when we know only that man is musical, and a like argument holds in all other cases. But commensurately universal demonstration, instead of proving that the subject itself actually is x, proves only that something else is x — e.g. in attempting to prove that isosceles is x, it proves not that isosceles but only that triangle is x — whereas particular demonstration proves that the subject itself is x. The demonstration, then, that a subject, as such, possesses an attribute is superior. If this is so, and if the particular rather than the commensurately universal forms demonstrates, particular demonstration is superior.
85a31 ἔτι εἰ τὸ μὲν καθόλου μὴ ἔστι τι παρὰ τὰ καθ᾽ ἕκαστα, ἡ δ᾽ ἀπόδειξις δόξαν ἐμποιεῖ εἶναί τι τοῦτο καθ᾽ ὁ ἀποδείκνυσι, καί τινα φύσιν ὑπάρχειν ἐν τοῖς οὖσι ταύτην, οἷον τριγώνου παρὰ τὰ τινὰ καὶ σχήματος παρὰ τὰ τινὰ καὶ ↵ ἀριθμοῦ παρὰ τοὺς τινὰς ἀριθμούς, βελτίων δ᾽ ἡ περὶ ὄντος ἢ μὴ ὄντος καὶ δι᾽ ἣν μὴ ἀπατηθήσεται ἢ δι᾽ ἥν, ἔστι δ᾽ ἡ μὲν καθόλου τοιαύτη (προϊόντες γὰρ δεικνύουσιν ὥσπερ περὶ τοῦ ἀνὰ λόγον, οἷον ὅτι ὁ ἂν ἦι τι τοιοῦτον ἔσται ἀνὰ λόγον ὁ οὔτε γραμμὴ οὔτ᾽ ἀριθμὸς οὔτε στερεὸν οὔτ᾽ ἐπί[85b]πεδον, ἀλλὰ παρὰ ταῦτά τι)· – εἰ οὖν καθόλου μὲν μᾶλλον αὕτη, περὶ ὄντος δ᾽ ἧττον τῆς κατὰ μέρος καὶ ἐμποιεῖ δόξαν ψευδῆ, χείρων ἂν εἴη ἡ καθόλου τῆς κατὰ μέρος.	Amplius, si universale quidem non est aliquid praeter singularia, demonstratio autem opinionem conficit esse aliquid hoc de quo demonstrat, et quamdam naturam esse hanc in iis quae sunt (ut trianguli praeter quosdam, et figurae praeter quasdam, et numeri praeter quosdam numeros), potior autem est quae est de esse quam de non esse, et propter quam non errabitur, quam propter quam errabitur; est autem universalis huiusmodi (procedentes enim demonstrant universale quemadmodum de eo quod est proportionale ut quod sit tale, erit proportionale quod neque linea, neque numerus, neque solidum, neque planum est, sed praeter haec aliquid), si igitur universalis magis haec est, et de eo quidem quod est minus, universalis quam particularis, et fecit opinionem falsam, indignior utique erit universalis particulari.	(2) The universal has not a separate being over against groups of singulars. Demonstration nevertheless creates the opinion that its function is conditioned by something like this-some separate entity belonging to the real world; that, for instance, of triangle or of figure or number, over against particular triangles, figures, and numbers. But demonstration which touches the real and will not mislead is superior to that which moves among unrealities and is delusory. Now commensurately universal demonstration is of the latter kind: if we engage in it we find ourselves reasoning after a fashion well illustrated by the argument that the proportionate is what answers to the definition of some entity which is neither line, number, solid, nor plane, but a proportionate apart from all these. Since, then, such a proof is characteristically commensurate and universal, and less touches reality than does particular demonstration, and creates a false opinion, it will follow that commensurate and universal is inferior to particular demonstration.
85b4 Η πρῶτον μὲν οὐδὲν μᾶλλον ἐπὶ τοῦ καθόλου ἢ τοῦ κατὰ ↵ μέρος ἅτερος λόγος ἐστίν; εἰ γὰρ τὸ δυσὶν ὀρθαῖς ὑπάρχει μὴ ἧι ἰσοσκελὲς ἀλλ᾽ ἧι τρίγωνον, ὁ εἰδὼς ὅτι ἰσοσκελὲς ἧττον οἶδεν ἧι αὐτὸ ἢ ὁ εἰδὼς ὅτι τρίγωνον. ὅλως τε, εἰ μὲν μὴ ὄντος ἧι τρίγωνον εἶτα δείκνυσιν, οὐκ ἂν εἴη ἀπόδειξις, εἰ δὲ ὄντος, ὁ εἰδὼς ἕκαστον ἧι ἕκαστον ὑπάρχει μᾶλλον οἶδεν.	Et primum quidem nihil magis in universali quam in particulari altera ratio est, si enim quod duobus rectis inest, non est secundum quod est isosceles, sed secundum quod triangulus est, cognoscens quoniam isosceles habet tres inquantum ipsum est, minus cognovit quam cognoscensa, quoniam triangulus est, et omnino si non quidem secundum quod sit triangulus, et postea monstrat, non erit utique demonstratio; si vero sit cognoscens unumquodque secundum quod unumquodque est, magis cognovit,	We may retort thus. (1) The first argument applies no more to commensurate and universal than to particular demonstration. If equality to two right angles is attributable to its subject not qua isosceles but qua triangle, he who knows that isosceles possesses that attribute knows the subject as qua itself possessing the attribute, to a less degree than he who knows that triangle has that attribute. To sum up the whole matter: if a subject is proved to possess qua triangle an attribute which it does not in fact possess qua triangle, that is not demonstration: but if it does possess it qua triangle the rule applies that the greater knowledge is his who knows the subject as possessing its attribute qua that in virtue of which it actually does possess it.
εἰ δὴ ↵ τὸ τρίγωνον ἐπὶ πλέον ἐστί, καὶ ὁ αὐτὸς λόγος, καὶ μὴ καθ᾽ ὁμωνυμίαν τὸ τρίγωνον, καὶ ὑπάρχει παντὶ τριγώνωι τὸ δύο, οὐκ ἂν τὸ τρίγωνον ἧι ἰσοσκελές, ἀλλὰ τὸ ἰσοσκελὲς ἧι τρίγωνον, ἔχοι τοιαύτας τὰς γωνίας. ὥστε ὁ καθόλου εἰδὼς μᾶλλον οἶδεν ἧι ὑπάρχει ἢ ὁ κατὰ μέρος. βελτίων ἄρα ἡ καθό- ↵ λου τῆς κατὰ μέρος.	si igitur triangulus in plus est, et eadem ratio, et non secundum aequivocationem triangulus est, et inest omni triangulo quod est duobus rectis aequales habens, non utique est triangulus inquantum isosceles, sed isosceles secundum quod triangulus huiusmodi habet tres angulos. Quare universaliter sciens magis cognovit secundum quod est, quam particulariter, potior est ergo universalis quam particularis.	Since, then, triangle is the wider term, and there is one identical definition of triangle-i.e. the term is not equivocal-and since equality to two right angles belongs to all triangles, it is isosceles qua triangle and not triangle qua isosceles which has its angles so related. It follows that he who knows a connexion universally has greater knowledge of it as it in fact is than he who knows the particular; and the inference is that commensurate and universal is superior to particular demonstration.
85b15 ἔτι εἰ μὲν εἴη τις λόγος εἷς καὶ μὴ ὁμωνυμία τὸ καθόλου, εἴη τ᾽ ἂν οὐδὲν ἧττον ἐνίων τῶν κατὰ μέρος, ἀλλὰ καὶ μᾶλλον, ὅσωι τὰ ἄφθαρτα ἐν ἐκείνοις ἐστί, τὰ δὲ κατὰ μέρος φθαρτὰ μᾶλλον,	Amplius, siquidem sit quaedam ratio una, et non aequivocatio, universale erit utique nil minus secundum partem quibusdam, sed magis est quanto incorruptibiliora sunt illis, quae vero secundum partem sunt corruptibilia magis.	(2) If there is a single identical definition i.e. if the commensurate universal is unequivocal-then the universal will possess being not less but more than some of the particulars, inasmuch as it is universals which comprise the imperishable, particulars that tend to perish.
85b18 ἔτι τε οὐδεμία ἀνάγκη ὑπολαμβάνειν τι εἶναι τοῦτο παρὰ ταῦτα, ὅτι ἓν δη↵ λοῖ, οὐδὲν μᾶλλον ἢ ἐπὶ τῶν ἄλλων ὅσα μὴ τὶ σημαίνει ἀλλ᾽ ἢ ποιὸν ἢ πρός τι ἢ ποιεῖν. εἰ δὲ ἄρα, οὐχ ἡ ἀπόδειξις αἰτία ἀλλ᾽ ὁ ἀκούων.	Amplius, nec una necessitas est opinari aliquid esse hoc praeter haec, quoniam ostendunt unum nihil magis quam in aliis quaecunque non aliquid significant, sed aut quale, aut ad aliquid, aut augere; si ergo, non demonstratio causa est, sed audiens.	(3) Because the universal has a single meaning, we are not therefore compelled to suppose that in these examples it has being as a substance apart from its particulars-any more than we need make a similar supposition in the other cases of unequivocal universal predication, viz. where the predicate signifies not substance but quality, essential relatedness, or action. If such a supposition is entertained, the blame rests not with the demonstration but with the hearer.
85b22 Ἔτι εἰ ἡ ἀπόδειξις μέν ἐστι συλλογισμὸς δεικτικὸς αἰτίας καὶ τοῦ διὰ τί, τὸ καθόλου δ᾽ αἰτιώτερον (ὧι γὰρ καθ᾽ ↵ αὑτὸ ὑπάρχει τι, τοῦτο αὐτὸ αὑτῶι αἴτιον· τὸ δὲ καθόλου πρῶτον· αἴτιον ἄρα τὸ καθόλου)· ὥστε καὶ ἡ ἀπόδειξις βελτίων· μᾶλλον γὰρ τοῦ αἰτίου καὶ τοῦ διὰ τί ἐστιν.	Amplius, si demonstratio est syllogismus demonstrativus qui fit causae et propter quod, universale magis causa; cui enim per se inest aliquid, hoc idem ipsi causa est: universale autem primum, causa ergo universale est, quare et demonstratio dignior est, magis enim causae est universale, et eius quod propter quid est.	(4) Demonstration is syllogism that proves the cause, i.e. the reasoned fact, and it is rather the commensurate universal than the particular which is causative (as may be shown thus: that which possesses an attribute through its own essential nature is itself the cause of the inherence, and the commensurate universal is primary; hence the commensurate universal is the cause). Consequently commensurately universal demonstration is superior as more especially proving the cause, that is the reasoned fact.
85b28 Ἔτι μέχρι τούτου ζητοῦμεν τὸ διὰ τί, καὶ τότε οἰόμεθα εἰδέναι, ὅταν μὴ ἦι ὅτι τι ἄλλο τοῦτο ἢ γινόμενον ἢ ὄν· τέλος γὰρ καὶ ↵ πέρας τὸ ἔσχατον ἤδη οὕτως ἐστίν. οἷον τίνος ἕνεκα ἦλθεν; ὅπως λάβηι τἀργύριον, τοῦτο δ᾽ ὅπως ἀποδῶι ὁ ὤφειλε, τοῦτο δ᾽ ὅπως μὴ ἀδικήσηι· καὶ οὕτως ἰόντες, ὅταν μηκέτι δι᾽ ἄλλο μηδ᾽ ἄλλου ἕνεκα, διὰ τοῦτο ὡς τέλος φαμὲν ἐλθεῖν καὶ εἶναι καὶ γίνεσθαι, καὶ τότε εἰδέναι μάλιστα διὰ τί ↵ ἦλθεν.	Amplius, usque ad hoc quaerimus propter quid, etenim tunc opinamur scire, cum non sit aliquid aliud quam hoc aut quod fiat, aut quod sit, finis enim et terminus ultimus iam sic est, ut cuius causa venit? ut accipiat argentum. Hoc autem est quatenus reddat cui debuit. Hoc autem ut non iniuste agat, et sic procedentes cum non sit amplius propter quid, nec alterius causa, propter hoc sicut propter finem dicimus venire, et esse, et fieri, et tunc est scire maxime propter quid venit;	(5) Our search for the reason ceases, and we think that we know, when the coming to be or existence of the fact before us is not due to the coming to be or existence of some other fact, for the last step of a search thus conducted is eo ipso the end and limit of the problem. Thus: ‘Why did he come?’ ‘To get the money-wherewith to pay a debt-that he might thereby do what was right.’ When in this regress we can no longer find an efficient or final cause, we regard the last step of it as the end of the coming-or being or coming to be-and we regard ourselves as then only having full knowledge of the reason why he came.
εἰ δὴ ὁμοίως ἔχει ἐπὶ πασῶν τῶν αἰτιῶν καὶ τῶν διὰ τί, ἐπὶ δὲ τῶν ὅσα αἴτια οὕτως ὡς οὗ ἕνεκα οὕτως ἴσμεν μάλιστα, καὶ ἐπὶ τῶν ἄλλων ἄρα τότε μάλιστα ἴσμεν, ὅταν μηκέτι ὑπάρχηι τοῦτο ὅτι ἄλλο.	si igitur se habet similiter, et in omnibus causis, et quae sunt propter quid, in iis autem quaecunque sic sunt causae, sicut quae est cuius causa sic scimus maxime, et in aliis igitur tunc maxime scimus, cum non amplius sit hoc quoniam aliud est:	If, then, all causes and reasons are alike in this respect, and if this is the means to full knowledge in the case of final causes such as we have exemplified, it follows that in the case of the other causes also full knowledge is attained when an attribute no longer inheres because of something else.
ὅταν μὲν οὖν γινώσκωμεν ὅτι τέτταρσιν αἱ ἔξω ἴσαι ὅτι ἰσοσκελές, ἔτι λείπεται διὰ [86a]τί τὸ ἰσοσκελές – ὅτι τρίγωνον, καὶ τοῦτο, ὅτι σχῆμα εὐθύγραμμον. εἰ δὲ τοῦτο μηκέτι διότι ἄλλο, τότε μάλιστα ἴσμεν. καὶ καθόλου δὲ τότε· ἡ καθόλου ἄρα βελτίων.	cum igitur cognoscimus quidem quod quatuor qui extra sunt aequales sunt, quoniam isosceles, adhuc deest propter quid isosceles? quia triangulus, et hoc, quia est figura rectis lineis contenta. Si autem hoc non amplius propter quod aliud, tunc maxime scimus, universale est autem tunc, universalis igitur potior.	Thus, when we learn that exterior angles are equal to four right angles because they are the exterior angles of an isosceles, there still remains the question 'Why has isosceles this attribute?' and its answer 'Because it is a triangle, and a triangle has it because a triangle is a rectilinear figure.' If rectilinear figure possesses the property for no further reason, at this point we have full knowledge-but at this point our knowledge has become commensurately universal, and so we conclude that commensurately universal demonstration is superior.
Ἔτι ὅσωι ἂν μᾶλλον κατὰ μέρος ἦι, εἰς τὰ ἄπειρα ἐμπίπτει, ἡ ↵ δὲ καθόλου εἰς τὸ ἁπλοῦν καὶ τὸ πέρας.	Amplius quantocunque utique magis secundum partes est, in infinita cadit, universale autem in simplex, et in finem,	86a3 (6) The more demonstration becomes particular the more it sinks into an indeterminate manifold, while universal demonstration tends to the simple and determinate.
ἔστι δ᾽, ἧι μὲν ἄπειρα, οὐκ ἐπιστητά, ἧι δὲ πεπέρανται, ἐπιστητά. ἧι ἄρα καθόλου, μᾶλλον ἐπιστητὰ ἢ ἧι κατὰ μέρος. ἀποδεικτὰ ἄρα μᾶλλον τὰ καθόλου. τῶν δ᾽ ἀποδεικτῶν μᾶλλον μᾶλλον ἀπόδειξις· ἅμα γὰρ μᾶλλον τὰ πρός τι. βελτίων ἄρα ἡ ↵ καθόλου, ἐπείπερ καὶ μᾶλλον ἀπόδειξις.	sunt autem secundum quod infinita non scibilia, sed secundum quod finita scibilia sunt, secundum utique quod universalia, magis scibilia sunt, quam quae sunt secundum partem demonstrabilia, ergo magis universalia. De magis demonstrabilibus autem magis est demonstratio, simul enim magis ad aliquid sunt, dignior igitur universalis est, quoniam quidem et magis demonstratio est.	But objects so far as they are an indeterminate manifold are unintelligible, so far as they are determinate, intelligible: they are therefore intelligible rather in so far as they are universal than in so far as they are particular. From this it follows that universals are more demonstrable: but since relative and correlative increase concomitantly, of the more demonstrable there will be fuller demonstration. Hence the commensurate and universal form, being more truly demonstration, is the superior.
86a11 Ἔτι εἰ αἱρετωτέρα καθ᾽ ἣν τοῦτο καὶ ἄλλο ἢ καθ᾽ ἣν τοῦτο μόνον οἶδεν· ὁ δὲ τὴν καθόλου ἔχων οἶδε καὶ τὸ κατὰ μέρος, οὗτος δὲ τὴν καθόλου οὐκ οἶδεν· ὥστε κἂν οὕτως αἱρετωτέρα εἴη. Ἔτι δὲ ὧδε.	Amplius, si magis eligenda est secundum quam hoc et aliud, quam secundum quam hoc solum cognovit, universale autem habens, cognovit particulare, hoc autem, universale non scivit, quare et sic utique magis eligenda erit universalis. Amplius autem et sic.	(7) Demonstration which teaches two things is preferable to demonstration which teaches only one. He who possesses commensurately universal demonstration knows the particular as well, but he who possesses particular demonstration does not know the universal. So that this is an additional reason for preferring commensurately universal demonstration. And there is yet this further argument:
86a14 τὸ γὰρ καθόλου μᾶλλον δεικνύναι ἐστὶ τὸ διὰ μέσου δει↵ κνύναι ἐγγυτέρω ὄντος τῆς ἀρχῆς. ἐγγυτάτω δὲ τὸ ἄμεσον· τοῦτο δ᾽ ἀρχή. εἰ οὖν ἡ ἐξ ἀρχῆς τῆς μὴ ἐξ ἀρχῆς, ἡ μᾶλλον ἐξ ἀρχῆς τῆς ἧττον ἀκριβεστέρα ἀπόδειξις. ἔστι δὲ τοιαύτη ἡ καθόλου μᾶλλον· κρείττων ‹ἄρ᾽› ἂν εἴη ἡ καθόλου. οἷον εἰ ἔδει ἀποδεῖξαι τὸ Α κατὰ τοῦ Δ· μέσα τὰ ↵ ἐφ᾽ ὧν Β Γ· ἀνωτέρω δὴ τὸ Β, ὥστε ἡ διὰ τούτου καθόλου μᾶλλον. Ἀλλὰ τῶν μὲν εἰρημένων ἔνια λογικά ἐστι·	Universale enim magis scire est eo quod est per medium demonstrare, cum propius sit principio, proxime autem immediatum est, hoc autem est principium. Si igitur quae ex principio est ea quae non ex principio, quae magis ex principio ea quae minus est, certior est demonstratio, est autem huiusmodi universalis magis, dignior utique erit universalis, ut si oportet monstrare a de d, media autem sint in quibus est b c, magis autem sursum sit b quam c, quare si per b magis est universalis, sed eorum quae dicta sunt quaedam logica sunt.	(8) Proof becomes more and more proof of the commensurate universal as its middle term approaches nearer to the basic truth, and nothing is so near as the immediate premiss which is itself the basic truth. If, then, proof from the basic truth is more accurate than proof not so derived, demonstration which depends more closely on it is more accurate than demonstration which is less closely dependent. But commensurately universal demonstration is characterized by this closer dependence, and is therefore superior. Thus, if A had to be proved to inhere in D, and the middles were B and C, B being the higher term would render the demonstration which it mediated the more universal. Some of these arguments, however, are dialectical.
86a21 μάλιστα δὲ δῆλον ὅτι ἡ καθόλου κυριωτέρα, ὅτι τῶν προτάσεων τὴν μὲν προτέραν ἔχοντες ἴσμεν πως καὶ τὴν ὑστέραν καὶ ἔχομεν ↵ δυνάμει, οἷον εἴ τις οἶδεν ὅτι πᾶν τρίγωνον δυσὶν ὀρθαῖς, οἶδέ πως καὶ τὸ ἰσοσκελὲς ὅτι δύο ὀρθαῖς, δυνάμει, καὶ εἰ μὴ οἶδε τὸ ἰσοσκελὲς ὅτι τρίγωνον· ὁ δὲ ταύτην ἔχων τὴν πρότασιν τὸ καθόλου οὐδαμῶς οἶδεν, οὔτε δυνάμει οὔτ᾽ ἐνεργείαι.	Maxime autem manifestum est quod universalis magis praecipua sit, quoniam propositionum quidem hanc priorem habentes, scimus quodammodo, et posteriorem et habemus potentia, ut si aliquis cognoverit quod omnis triangulus habeat tres duobus rectis aequales, scivit quodam modo et quod isosceles duobus rectis potentia sit, et si non cognovit isoscelem quod triangulus sit, hanc autem habens propositionem, nullo modo universale cognovit, neque potentia, neque actu.	The clearest indication of the precedence of commensurately universal demonstration is as follows: if of two propositions, a prior and a posterior, we have a grasp of the prior, we have a kind of knowledge-a potential grasp-of the posterior as well. For example, if one knows that the angles of all triangles are equal to two right angles, one knows in a sense-potentially-that the isosceles’ angles also are equal to two right angles, even if one does not know that the isosceles is a triangle; but to grasp this posterior proposition is by no means to know the commensurate universal either potentially or actually.
86a28 καὶ ἡ μὲν καθόλου νοητή, ἡ δὲ κατὰ μέρος εἰς ↵ αἴσθησιν τελευτᾶι.	Et universalis quidem intelligibilis est, sed particularis in sensu perficitur.	Moreover, commensurately universal demonstration is through and through intelligible; particular demonstration issues in sense-perception.
c25		Chapter 25
Ὅτι μὲν οὖν ἡ καθόλου βελτίων τῆς κατὰ μέρος, τοσαῦθ᾽ ἡμῖν εἰρήσθω·	Quod igitur universalis dignior sit particulari, tot nobis dicta sint.	The preceding arguments constitute our defence of the superiority of commensurately universal to particular demonstration.
	CAPUT XXI. Quod demonstratio affirmativa praestantior evadat negativa.
86a32 ὅτι δ᾽ ἡ δεικτικὴ τῆς στερητικῆς, ἐντεῦθεν δῆλον. ἔστω γὰρ αὕτη ἡ ἀπόδειξις βελτίων τῶν ἄλλων τῶν αὐτῶν ὑπαρχόντων, ἡ ἐξ ἐλαττόνων αἰτημάτων ἢ ὑπο↵ θέσεων ἢ προτάσεων. εἰ γὰρ γνώριμοι ὁμοίως, τὸ θᾶττον γνῶναι διὰ τούτων ὑπάρξει· τοῦτο δ᾽ αἱρετώτερον.	Quod autem monstrativa sit dignior privativa, hinc manifestum est, sit enim haec demonstratio dignior (aliis eisdem existentibus) aut ex minoribus quaestionibus, aut suppositionibus, aut propositionibus. Si enim notae sunt similiter velocius cognoscere per haec erit, hoc autem appetibilius est.	That affirmative demonstration excels negative may be shown as follows. (1) We may assume the superiority ceteris paribus of the demonstration which derives from fewer postulates or hypotheses-in short from fewer premisses; for, given that all these are equally well known, where they are fewer knowledge will be more speedily acquired, and that is a desideratum.
λόγος δὲ τῆς προτάσεως, ὅτι βελτίων ἡ ἐξ ἐλαττόνων, καθόλου ὅδε· εἰ γὰρ ὁμοίως εἴη τὸ γνώριμα εἶναι τὰ μέσα, τὰ δὲ πρότερα γνωριμώτερα, ἔστω ἡ μὲν διὰ μέσων ἀπόδειξις τῶν [86b]Β Γ Δ ὅτι τὸ Α τῶι Ε ὑπάρχει, ἡ δὲ διὰ τῶν Ζ Η ὅτι τὸ Α τῶι Ε. ὁμοίως δὴ ἔχει τὸ ὅτι τὸ Α τῶι Δ ὑπάρχει καὶ τὸ Α τῶι Ε. τὸ δ᾽ ὅτι τὸ Α τῶι Δ πρότερον καὶ γνωριμώτερον ἢ ὅτι τὸ Α τῶι Ε· διὰ γὰρ τούτου ἐκεῖνο ἀπο↵ δείκνυται, πιστότερον δὲ τὸ δι᾽ οὗ.	Ratio autem propositionis quod melior sit ex minoribus, universaliter est sic, si enim contingit similiter cognita esse mediae, priora autem notiora sunt. Si autem per media demonstratio eorum, quae sunt b c d quod a in e sit, altera autem demonstratio quod a in d sit per b c similiter igitur se habet hoc quod a in d sit, et quod a in e sit, sed quod a in d sit prius est, et cognoscibilius quam quod a in e, per hoc enim illud demonstratur: credibilius autem est per quod.	The argument implied in our contention that demonstration from fewer assumptions is superior may be set out in universal form as follows. Assuming that in both cases alike the middle terms are known, and that middles which are prior are better known than such as are posterior, we may suppose two demonstrations of the inherence of A in E, the one proving it through the middles B, C and D, the other through F and G. Then A-D is known to the same degree as A-E (in the second proof), but A-D is better known than and prior to A-E (in the first proof); since A-E is proved through A-D, and the ground is more certain than the conclusion.
καὶ ἡ διὰ τῶν ἐλαττόνων ἄρα ἀπόδειξις βελτίων τῶν ἄλλων τῶν αὐτῶν ὑπαρχόντων. ἀμφότεραι μὲν οὖν διά τε ὅρων τριῶν καὶ προτάσεων δύο δείκνυνται, ἀλλ᾽ ἡ μὲν εἶναί τι λαμβάνει, ἡ δὲ καὶ εἶναι καὶ μὴ εἶναί τι· διὰ πλειόνων ἄρα, ὥστε χείρων.	Et quae per pauciora demonstratio, potior aliis eisdem existentibus. Utraeque quidem per terminos tres et propositiones duas monstrant, sed haec quidem esse aliquid accipit, illa vero et esse, et non esse aliquid. Per plura itaque, quare dignior est.	Hence demonstration by fewer premisses is ceteris paribus superior. Now both affirmative and negative demonstration operate through three terms and two premisses, but whereas the former assumes only that something is, the latter assumes both that something is and that something else is not, and thus operating through more kinds of premiss is inferior.
86b10 ↵ Ἔτι ἐπειδὴ δέδεικται ὅτι ἀδύνατον ἀμφοτέρων οὐσῶν στερητικῶν τῶν προτάσεων γενέσθαι συλλογισμόν,	Amplius, quoniam ostensum est impossibile per utramque privativarum propositionum fieri syllogismum,	(2) It has been proved that no conclusion follows if both premisses are negative,
ἀλλὰ τὴν μὲν δεῖ τοιαύτην εἶναι, τὴν δ᾽ ὅτι ὑπάρχει, ἔτι πρὸς τούτωι δεῖ τόδε λαβεῖν. τὰς μὲν γὰρ κατηγορικὰς αὐξανομένης τῆς ἀποδείξεως ἀναγκαῖον γίνεσθαι πλείους, τὰς δὲ στερητικὰς ↵ ἀδύνατον πλείους εἶναι μιᾶς ἐν ἅπαντι συλλογισμῶι.	sed oportere quidem huiusmodi esse unam, aliam vero quoniam est. Amplius, praeter hoc oportet et hoc accipere, praedicativas enim augmentata demonstratione necesse est fieri plures, privativas autem impossibile plures una in omni syllogismo esse.	but that one must be negative, the other affirmative. 86b12 So we are compelled to lay down the following additional rule: as the demonstration expands, the affirmative premisses must increase in number, but there cannot be more than one negative premiss in each complete proof.
ἔστω γὰρ μηδενὶ ὑπάρχον τὸ Α ἐφ᾽ ὅσων τὸ Β, τῶι δὲ Γ ὑπάρχον παντὶ τὸ Β. ἂν δὴ δέηι πάλιν αὔξειν ἀμφοτέρας τὰς προτάσεις, μέσον ἐμβλητέον. τοῦ μὲν Α Β ἔστω τὸ Δ, τοῦ δὲ Β Γ τὸ Ε. τὸ μὲν δὴ Ε φανερὸν ὅτι κατηγορικόν, τὸ δὲ Δ ↵ τοῦ μὲν Β κατηγορικόν, πρὸς δὲ τὸ Α στερητικὸν κεῖται. τὸ μὲν γὰρ Δ παντὸς τοῦ Β, τὸ δὲ Α οὐδενὶ δεῖ τῶν Δ ὑπάρχειν. γίνεται οὖν μία στερητικὴ πρότασις ἡ τὸ Α Δ. ὁ δ᾽ αὐτὸς τρόπος καὶ ἐπὶ τῶν ἑτέρων συλλογισμῶν.	Sit enim in nullo esse a in quibus est b, in c autem omni sit b, si igitur opus est rursus augere utrasque propositiones, medium iniiciendum est, huius quidem a b sit d, sed b c sit e, e igitur manifestum est praedicativum esse, sed d de b quidem praedicativum, a autem de d tanquam privativum ponitur: d enim de omni b, sed a oportet in nullo d esse, fit ergo una privativa propositio.	Thus, suppose no B is A, and all C is B. Then if both the premisses are to be again expanded, a middle must be interposed. Let us interpose D between A and B, and E between B and C. Then clearly E is affirmatively related to B and C, while D is affirmatively related to B but negatively to A; for all B is D, but there must be no D which is A. Thus there proves to be a single negative premiss, A-D.
ἀεὶ γὰρ τὸ μέσον τῶν κατηγορικῶν ὅρων κατηγορικὸν ἐπ᾽ ἀμφότερα· ↵ τοῦ δὲ στερητικοῦ ἐπὶ θάτερα στερητικὸν ἀναγκαῖον εἶναι, ὥστε αὕτη μία τοιαύτη γίνεται πρότασις, αἱ δ᾽ ἄλλαι κατηγορικαί. εἰ δὴ γνωριμώτερον δι᾽ οὗ δείκνυται καὶ πιστότερον, δείκνυται δ᾽ ἡ μὲν στερητικὴ διὰ τῆς κατηγορικῆς, αὕτη δὲ δι᾽ ἐκείνης οὐ δείκνυται, προτέρα καὶ γνωριμωτέρα οὖσα ↵ καὶ πιστοτέρα βελτίων ἂν εἴη.	Idem autem modus est et in aliis syllogismis. Semper enim medium praedicativorum terminorum, praedicativum in utraque est, sed privativi in altera privativum necesse est esse, quare haec una huiusmodi fit propositio, aliae vero praedicativae. Si igitur notius est per quod demonstratur, et credibilius, demonstratur autem privativa quidem per praedicativam, haec autem per illam non demonstratur, prior ergo, et notior, et credibilior cum sit, melior itaque erit.	In the further prosyllogisms too it is the same, because in the terms of an affirmative syllogism the middle is always related affirmatively to both extremes; in a negative syllogism it must be negatively related only to one of them, and so this negation comes to be a single negative premiss, the other premisses being affirmative. If, then, that through which a truth is proved is a better known and more certain truth, and if the negative proposition is proved through the affirmative and not vice versa, affirmative demonstration, being prior and better known and more certain, will be superior.
86b30 ἔτι εἰ ἀρχὴ συλλογισμοῦ ἡ καθόλου πρότασις ἄμεσος, ἔστι δ᾽ ἐν μὲν τῆι δεικτικῆι καταφατικὴ ἐν δὲ τῆι στερητικῆι ἀποφατικὴ ἡ καθόλου πρότασις, ἡ δὲ καταφατικὴ τῆς ἀποφατικῆς προτέρα καὶ γνωριμωτέρα (διὰ γὰρ τὴν κατάφασιν ἡ ἀπόφασις γνώ- ↵ ριμος, καὶ προτέρα ἡ κατάφασις, ὥσπερ καὶ τὸ εἶναι τοῦ μὴ εἶναι)· ὥστε βελτίων ἡ ἀρχὴ τῆς δεικτικῆς ἢ τῆς στερητικῆς· ἡ δὲ βελτίοσιν ἀρχαῖς χρωμένη βελτίων.	Amplius, si principium syllogismi propositio universalis sit immediata, est autem ut in monstrativa praedicativa, in privativa autem, negativa propositio universalis, affirmativa autem negativa prior, et notior (per affirmativam enim negativa nota) et prior affirmativa est, sicut esse prius est non esse. Quare potius est principium monstrativae quam privativae, dignioribus autem principiis utitur dignior.	(3) The basic truth of demonstrative syllogism is the universal immediate premiss, and the universal premiss asserts in affirmative demonstration and in negative denies: and the affirmative proposition is prior to and better known than the negative (since affirmation explains denial and is prior to denial, just as being is prior to not-being). It follows that the basic premiss of affirmative demonstration is superior to that of negative demonstration, and the demonstration which uses superior basic premisses is superior.
86b39 ἔτι ἀρχοειδεστέρα· ἄνευ γὰρ τῆς δεικνυούσης οὐκ ἔστιν ἡ στερητική.	Adhuc principalior est, sine enim monstrativa non est privativa.	(4) Affirmative demonstration is more of the nature of a basic form of proof, because it is a sine qua non of negative demonstration.
c26	CAPUT XXII. Demonstrationem ostensivam potiorem esse ea quae ducit ad incommodum.	Chapter 26
87a1 Ἐπεὶ δ᾽ ἡ κατηγορικὴ τῆς στερητικῆς βελτίων, δῆλον ὅτι καὶ τῆς εἰς τὸ ἀδύνατον ἀγούσης.	Quod quidem praedicativa, privativa dignior sit, manifestum est. Et ad impossibile ducente,	Since affirmative demonstration is superior to negative, it is clearly superior also to reductio ad impossibile.
87a2 δεῖ δ᾽ εἰδέναι τίς ἡ διαφορὰ αὐτῶν. ἔστω δὴ τὸ Α μηδενὶ ὑπάρχον τῶι Β, τῶι δὲ Γ τὸ Β παντί· ἀνάγκη δὴ τῶι Γ μηδενὶ ὑπάρχειν τὸ Α. ↵ οὕτω μὲν οὖν ληφθέντων δεικτικὴ ἡ στερητικὴ ἂν εἴη ἀπόδειξις ὅτι τὸ Α τῶι Γ οὐχ ὑπάρχει.	oportet autem scire quae differentia sit ipsarum. Si igitur a in nullo b, in c autem omni b, necesse est in nullo c esse a; sic igitur acceptis, ostensiva privativa erit demonstratio, quoniam a in c non erit.	We must first make certain what is the difference between negative demonstration and reductio ad impossibile. Let us suppose that no B is A, and that all C is B: the conclusion necessarily follows that no C is A. If these premisses are assumed, therefore, the negative demonstration that no C is A is direct.
87a7 ἡ δ᾽ εἰς τὸ ἀδύνατον ὧδ᾽ ἔχει. εἰ δέοι δεῖξαι ὅτι τὸ Α τῶι Β οὐχ ὑπάρχει, ληπτέον ὑπάρχειν, καὶ τὸ Β τῶι Γ, ὥστε συμβαίνει τὸ Α τῶι Γ ὑπάρχειν. τοῦτο δ᾽ ἔστω γνώριμον καὶ ὁμολογούμενον ὅτι ↵ ἀδύνατον. οὐκ ἄρα οἷόν τε τὸ Α τῶι Β ὑπάρχειν. εἰ οὖν τὸ Β τῶι Γ ὁμολογεῖται ὑπάρχειν, τὸ Α τῶι Β ἀδύνατον ὑπάρχειν.	Quae vero est ad impossibile sic se habet, si opus est demonstrare quod a in b non sit, accipiendum est a esse in b, et b in c, quare accidit a in c esse. Hoc autem si notum, et concessum quod ipsum est impossibile esse, non igitur possibile est a in b esse. Si ergo b in c concessum est inesse, a in b impossibile est esse,	Reductio ad impossibile, on the other hand, proceeds as follows. Supposing we are to prove that does not inhere in B, we have to assume that it does inhere, and further that B inheres in C, with the resulting inference that A inheres in C. This we have to suppose a known and admitted impossibility; and we then infer that A cannot inhere in B. Thus if the inherence of B in C is not questioned, A’s inherence in B is impossible.
87a13 οἱ μὲν οὖν ὅροι ὁμοίως τάττονται, διαφέρει δὲ τὸ ὁποτέρα ἂν ἦι γνωριμωτέρα ἡ πρότασις ἡ στερητική, πότερον ὅτι τὸ Α τῶι Β οὐχ ὑπάρχει ἢ ὅτι τὸ Α τῶι Γ. ὅταν μὲν ↵ οὖν ἦι τὸ συμπέρασμα γνωριμώτερον ὅτι οὐκ ἔστιν, ἡ εἰς τὸ ἀδύνατον γίνεται ἀπόδειξις, ὅταν δ᾽ ἡ ἐν τῶι συλλογισμῶι, ἡ ἀποδεικτική.	termini igitur similiter ordinantur. Differt autem quo, qualis sit notior privativa propositio, utrum igitur quia a b non inest, an quia a c, cum igitur est conclusio notior quoniam non est, quod est impossibile, fit demonstratio, cum autem in syllogismo sit, demonstrativa est.	The order of the terms is the same in both proofs: they differ according to which of the negative propositions is the better known, the one denying A of B or the one denying A of C. When the falsity of the conclusion is the better known, we use reductio ad impossible; when the major premiss of the syllogism is the more obvious, we use direct demonstration.
87a17 φύσει δὲ προτέρα ἡ ὅτι τὸ Α τῶι Β ἢ ὅτι τὸ Α τῶι Γ. πρότερα γάρ ἐστι τοῦ συμπεράσματος ἐξ ὧν τὸ συμπέρασμα· ἔστι δὲ τὸ μὲν Α τῶι Γ μὴ ὑπάρχειν συμ↵ πέρασμα, τὸ δὲ Α τῶι Β ἐξ οὗ τὸ συμπέρασμα.	Natura autem prior est, quae est quod a in b non sit, quam a in c non sit, priora enim conclusione sunt, ex quibus est conclusio, est autem quae est a in c non esse, conclusio, a autem in b ex quibus est conclusio.	All the same the proposition denying A of B is, in the order of being, prior to that denying A of C; for premisses are prior to the conclusion which follows from them, and ‘no C is A’ is the conclusion, ‘no B is A’ one of its premisses.
87a19 οὐ γὰρ εἰ συμβαίνει ἀναιρεῖσθαί τι, τοῦτο συμπέρασμά ἐστιν, ἐκεῖνα δὲ ἐξ ὧν, ἀλλὰ τὸ μὲν ἐξ οὗ συλλογισμός ἐστιν ὁ ἂν οὕτως ἔχηι ὥστε ἢ ὅλον πρὸς μέρος ἢ μέρος πρὸς ὅλον ἔχειν, αἱ δὲ τὸ Α Γ καὶ Β Γ προτάσεις οὐκ ἔχουσιν οὕτω πρὸς ↵ ἀλλήλας.	Non enim si contingit removeri aliquid, hoc conclusio est, illa autem ex quibus sunt, sed hoc quidem ex quo syllogismus est, utique sic se habet, ut aut sicut totum ad partem, aut ut pars ad totum se habet, sed quae sunt a c, et a b propositiones, non sic se habent ad invicem.	For the destructive result of reductio ad impossibile is not a proper conclusion, nor are its antecedents proper premisses. On the contrary: the constituents of syllogism are premisses related to one another as whole to part or part to whole, whereas the premisses A-C and A-B are not thus related to one another.
εἰ οὖν ἡ ἐκ γνωριμωτέρων καὶ προτέρων κρείττων, εἰσὶ δ᾽ ἀμφότεραι ἐκ τοῦ μὴ εἶναί τι πισταί, ἀλλ᾽ ἡ μὲν ἐκ προτέρου ἡ δ᾽ ἐξ ὑστέρου, βελτίων ἁπλῶς ἂν εἴη τῆς εἰς τὸ ἀδύνατον ἡ στερητικὴ ἀπόδειξις, ὥστε καὶ ἡ ταύτης βελτίων ἡ κατηγορικὴ δῆλον ὅτι καὶ τῆς εἰς τὸ ἀδύνατόν ↵ ἐστι βελτίων.	Si igitur ex dignioribus et credibilioribus dignior est, sunt autem utraeque ex non esse aliquid, credibiles, sed haec quidem ex priori, illa vero ex posteriori, potior utique simpliciter erit ea quae est ad impossibile, privativa demonstratio, quare et hac dignior praedicativa. Manifestum est ergo quod et ea quae est ad impossibile, potior est.	Now the superior demonstration is that which proceeds from better known and prior premisses, and while both these forms depend for credence on the not-being of something, yet the source of the one is prior to that of the other. Therefore negative demonstration will have an unqualified superiority to reductio ad impossibile, and affirmative demonstration, being superior to negative, will consequently be superior also to reductio ad impossibile.
c27	CAPUT XXIII. Quae scientia certior, quae una, quae altera, et eiusdem plures esse posse demonstrationes.	Chapter 27
87a31 Ἀκριβεστέρα δ᾽ ἐπιστήμη ἐπιστήμης καὶ προτέρα ἥ τε τοῦ ὅτι καὶ διότι ἡ αὐτή, ἀλλὰ μὴ χωρὶς τοῦ ὅτι τῆς τοῦ διότι, καὶ ἡ μὴ καθ᾽ ὑποκειμένου τῆς καθ᾽ ὑποκειμένου, οἷον ἀριθμητικὴ ἁρμονικῆς, καὶ ἡ ἐξ ἐλαττόνων τῆς ἐκ προσ↵ θέσεως, οἷον γεωμετρίας ἀριθμητική. λέγω δ᾽ ἐκ προσθέσεως, οἷον μονὰς οὐσία ἄθετος, στιγμὴ δὲ οὐσία θετός· ταύτην ἐκ προσθέσεως.	Certior autem est scientia, scientia et prior, quae ipsius quia et propter quid eadem est quam non, extra eam quae est propter quid. Et quae non est de subiecto, ea quae est de subiecto, ut arithmetica, harmonica. Et quae est ex minoribus, ea quae ex appositione, ut arithmetica, geometria: dico autem ex additione, ut unitas substantia est sine positione, punctum autem substantia posita, hoc autem ex appositione.	The science which is knowledge at once of the fact and of the reasoned fact, not of the fact by itself without the reasoned fact, is the more exact and the prior science. A science such as arithmetic, which is not a science of properties qua inhering in a substratum, is more exact than and prior to a science like harmonics, which is a science of properties inhering in a substratum; and similarly a science like arithmetic, which is constituted of fewer basic elements, is more exact than and prior to geometry, which requires additional elements. What I mean by ‘additional elements’ is this: a unit is substance without position, while a point is substance with position; the latter contains an additional element.
c28		Chapter 28
87a38 Μία δ᾽ ἐπιστήμη ἐστὶν ἡ ἑνὸς γένους, ὅσα ἐκ τῶν πρώτων σύγκειται καὶ μέρη ἐστὶν ἢ πάθη τούτων καθ᾽ αὑτά.	Una autem scientia est, quae est unius generis, quaecunque ex primis componitur, et partes sunt aut passiones horum, quae sunt per se.	A single science is one whose domain is a single genus, viz. all the subjects constituted out of the primary entities of the genus-i.e. the parts of this total subject-and their essential properties.
87a41 ἑτέρα δ᾽ ἐπιστήμη ἐστὶν ἑτέρας, ὅσων αἱ ἀρχαὶ μήτ᾽ ἐκ τῶν αὐ[87b]τῶν μήθ᾽ ἅτεραι ἐκ τῶν ἑτέρων.	Altera autem scientia est ab altera, quarumcunque principia neque ex eisdem, neque ex alteris sunt.	One science differs from another when their basic truths have neither a common source nor are derived those of the one science from those the other.
87b1 τούτου δὲ σημεῖον, ὅταν εἰς τὰ ἀναπόδεικτα ἔλθηι· δεῖ γὰρ αὐτὰ ἐν τῶι αὐτῶι γένει εἶναι τοῖς ἀποδεδειγμένοις. σημεῖον δὲ καὶ τούτου, ὅταν τὰ δεικνύμενα δι᾽ αὐτῶν ἐν ταὐτῶι γένει ὦσι καὶ συγγενῆ.	Huius autem est signum, cum in demonstrabilia veniant, oportet enim in eodem genere esse cum iis quae demonstrantur, signum autem est et huius, cum demonstrabilia per ipsa in eodem genere sunt, et proxima.	This is verified when we reach the indemonstrable premisses of a science, for they must be within one genus with its conclusions: and this again is verified if the conclusions proved by means of them fall within one genus-i.e. are homogeneous.
c29		Chapter 29
87b5 Πλείους δ᾽ ἀποδείξεις εἶναι τοῦ αὐτοῦ ἐγχωρεῖ οὐ μόνον ἐκ τῆς αὐτῆς συστοιχίας λαμβάνοντι μὴ τὸ συνεχὲς μέσον, οἷον τῶν Α Β τὸ Γ καὶ Δ καὶ Ζ, ἀλλὰ καὶ ἐξ ἑτέρας. οἷον ἔστω τὸ Α μεταβάλλειν, τὸ δ᾽ ἐφ᾽ ὧι Δ κινεῖσθαι, τὸ δὲ Β ἥδεσθαι, καὶ πάλιν τὸ Η ἠρεμίζεσθαι. ἀληθὲς οὖν καὶ τὸ Δ ↵ τοῦ Β καὶ τὸ Α τοῦ Δ κατηγορεῖν· ὁ γὰρ ἡδόμενος κινεῖται καὶ τὸ κινούμενον μεταβάλλει. πάλιν τὸ Α τοῦ Η καὶ τὸ Η τοῦ Β ἀληθὲς κατηγορεῖν· πᾶς γὰρ ὁ ἡδόμενος ἠρεμίζεται καὶ ὁ ἠρεμιζόμενος μεταβάλλει. ὥστε δι᾽ ἑτέρων μέσων καὶ οὐκ ἐκ τῆς αὐτῆς συστοιχίας ὁ συλλογισμός. οὐ μὴν ὥστε μη↵ δέτερον κατὰ μηδετέρου λέγεσθαι τῶν μέσων· ἀνάγκη γὰρ τῶι αὐτῶι τινι ἄμφω ὑπάρχειν. ἐπισκέψασθαι δὲ καὶ διὰ τῶν ἄλλων σχημάτων ὁσαχῶς ἐνδέχεται τοῦ αὐτοῦ γενέσθαι συλλογισμόν.	Plures autem demonstrationes eiusdem possibile est esse, non solum ex eodem ordine accipiendi non continuum medium (ut eorum quae sunt a b, c et d et e ), sed et ex altero, ut sit a transmutari, in quo autem d moveri, sed laetari sit in quo b, et iterum e quiescere, verum igitur est et d de b, et de d praedicari, laetans enim movetur, et quod movetur transmutatur, iterum a de e, et e de b, verum praedicari, omnis enim laetans quiescit, et quiescens transmutatur, quare per altera media, et non ex eodem ordine syllogismus est, non tamen est neutrum de neutro dici mediorum. Necesse est enim idem e alicui utraque inesse, intendere autem est per alias figuras, quot modis contingit eiusdem fieri syllogismum.	One can have several demonstrations of the same connexion not only by taking from the same series of predication middles which are other than the immediately cohering term e.g. by taking C, D, and F severally to prove A-B— but also by taking a middle from another series. Thus let A be change, D alteration of a property, B feeling pleasure, and G relaxation. We can then without falsehood predicate D of B and A of D, for he who is pleased suffers alteration of a property, and that which alters a property changes. Again, we can predicate A of G without falsehood, and G of B; for to feel pleasure is to relax, and to relax is to change. So the conclusion can be drawn through middles which are different, i.e. not in the same series-yet not so that neither of these middles is predicable of the other, for they must both be attributable to some one subject.A further point worth investigating is how many ways of proving the same conclusion can be obtained by varying the figure.
c30	CAPUT XXIV. Non esse scientiam fortuitorum, neque in sensuum functione.	Chapter 30
87b19 Τοῦ δ᾽ ἀπὸ τύχης οὐκ ἔστιν ἐπιστήμη δι᾽ ἀποδείξεως. ↵ οὔτε γὰρ ὡς ἀναγκαῖον οὔθ᾽ ὡς ἐπὶ τὸ πολὺ τὸ ἀπὸ τύχης ἐστίν, ἀλλὰ τὸ παρὰ ταῦτα γινόμενον· ἡ δ᾽ ἀπόδειξις θατέρου τούτων.	Sed eius quod est a fortuna non est scientia per demonstrationem. Neque enim sicut necessarium, neque sicut frequenter quod est a fortuna, sed extra hoc fit, sed demonstratio alterius horum.	There is no knowledge by demonstration of chance conjunctions; for chance conjunctions exist neither by necessity nor as general connexions but comprise what comes to be as something distinct from these. Now demonstration is concerned only with one or other of these two;
πᾶς γὰρ συλλογισμὸς ἢ δι᾽ ἀναγκαίων ἢ διὰ τῶν ὡς ἐπὶ τὸ πολὺ προτάσεων· καὶ εἰ μὲν αἱ προτάσεις ἀναγκαῖαι, καὶ τὸ συμπέρασμα ἀναγκαῖον, εἰ δ᾽ ὡς ↵ ἐπὶ τὸ πολύ, καὶ τὸ συμπέρασμα τοιοῦτον. ὥστ᾽ εἰ τὸ ἀπὸ τύχης μήθ᾽ ὡς ἐπὶ τὸ πολὺ μήτ᾽ ἀναγκαῖον, οὐκ ἂν εἴη αὐτοῦ ἀπόδειξις.	Omnis enim syllogismus aut per necessarias, aut per eas quae sunt tanquam frequenter propositiones, et siquidem propositiones necessariae sunt, et conclusio erit necessaria, si vero sint sicut frequenter, et conclusio huiusmodi. Quare si id quod est a fortuna neque est sicut frequenter, neque necessarium, neque utique erit ipsius demonstratio.	for all reasoning proceeds from necessary or general premisses, the conclusion being necessary if the premisses are necessary and general if the premisses are general. Consequently, if chance conjunctions are neither general nor necessary, they are not demonstrable.
c31		Chapter 31
87b28 Οὐδὲ δι᾽ αἰσθήσεως ἔστιν ἐπίστασθαι. εἰ γὰρ καὶ ἔστιν ἡ αἴσθησις τοῦ τοιοῦδε καὶ μὴ τοῦδέ τινος, ἀλλ᾽ αἰσθάνεσθαί ↵ γε ἀναγκαῖον τόδε τι καὶ ποὺ καὶ νῦν. τὸ δὲ καθόλου καὶ ἐπὶ πᾶσιν ἀδύνατον αἰσθάνεσθαι· οὐ γὰρ τόδε οὐδὲ νῦν· οὐ γὰρ ἂν ἦν καθόλου· τὸ γὰρ ἀεὶ καὶ πανταχοῦ καθόλου φαμὲν εἶναι. ἐπεὶ οὖν αἱ μὲν ἀποδείξεις καθόλου, ταῦτα δ᾽ οὐκ ἔστιν αἰσθάνεσθαι, φανερὸν ὅτι οὐδ᾽ ἐπίστασθαι δι᾽ αἰσθή↵ σεως ἔστιν,	Neque per sensum est scire, si enim est sensus talis huius, et non huius alicuius, sed sentire hoc aliquid est necesse et ubi et nunc. Universale autem quod est in omnibus, impossibile est sentire, neque enim hoc aliquid est, neque nunc, neque ubi, neque enim utique esset universale, quod enim semper est, et ubique, universale dicimus esse; quoniam igitur demonstrationes universales sunt, haec autem non est sentire, manifestum est quod neque scire per sensum est.	Scientific knowledge is not possible through the act of perception. Even if perception as a faculty is of ‘the such’ and not merely of a ‘this somewhat’, yet one must at any rate actually perceive a ‘this somewhat’, and at a definite present place and time: but that which is commensurately universal and true in all cases one cannot perceive, since it is not ‘this’ and it is not ‘now’; if it were, it would not be commensurately universal-the term we apply to what is always and everywhere. Seeing, therefore, that demonstrations are commensurately universal and universals imperceptible, we clearly cannot obtain scientific knowledge by the act of perception:
87b34 ἀλλὰ δῆλον ὅτι καὶ εἰ ἦν αἰσθάνεσθαι τὸ τρίγωνον ὅτι δυσὶν ὀρθαῖς ἴσας ἔχει τὰς γωνίας, ἐζητοῦμεν ἂν ἀπόδειξιν καὶ οὐχ ὥσπερ φασί τινες ἠπιστάμεθα· αἰσθάνεσθαι μὲν γὰρ ἀνάγκη καθ᾽ ἕκαστον, ἡ δ᾽ ἐπιστήμη τὸ τὸ καθόλου γνωρίζειν ἐστίν.	Sed manifestum quoniam si esset sentire triangulum, quod duobus rectis haberet aequales angulos, quaereremus utique demonstrationem, et non, sicut quidam fatentur, sciremus. Sentire enim necesse est singulariter, scientia autem est in cognoscendo universale.	nay, it is obvious that even if it were possible to perceive that a triangle has its angles equal to two right angles, we should still be looking for a demonstration-we should not (as some say) possess knowledge of it; for perception must be of a particular, whereas scientific knowledge involves the recognition of the commensurate universal.
διὸ καὶ εἰ ἐπὶ τῆς σελήνης ὄντες ἑωρῶμεν ἀντιφράττουσαν τὴν γῆν, οὐκ ἂν ἤιδειμεν τὴν αἰτίαν [88a]τῆς ἐκλείψεως. ἠισθανόμεθα γὰρ ἂν ὅτι νῦν ἐκλείπει, καὶ οὐ διότι ὅλως· οὐ γὰρ ἦν τοῦ καθόλου αἴσθησις.	Unde et si super lunam essemus, et videremus obiectam terram, non utique sciremus causam defectus, sentiremus enim quoniam deficeret, sed non propter quid omnino, non enim universalis, sensus.	So if we were on the moon, and saw the earth shutting out the sun’s light, we should not know the cause of the eclipse: we should perceive the present fact of the eclipse, but not the reasoned fact at all, since the act of perception is not of the commensurate universal.
οὐ μὴν ἀλλ᾽ ἐκ τοῦ θεωρεῖν τοῦτο πολλάκις συμβαῖνον τὸ καθόλου ἂν θηρεύσαντες ἀπόδειξιν εἴχομεν· ἐκ γὰρ τῶν καθ᾽ ἕκαστα πλει↵ όνων τὸ καθόλου δῆλον.	Sed ex considerare hoc multoties accidere universale venantes, demonstrationem habemus, ex singularibus enim pluribus universale manifestum est.	I do not, of course, deny that by watching the frequent recurrence of this event we might, after tracking the commensurate universal, possess a demonstration, for the commensurate universal is elicited from the several groups of singulars.
88a5 τὸ δὲ καθόλου τίμιον, ὅτι δηλοῖ τὸ αἴτιον· ὥστε περὶ τῶν τοιούτων ἡ καθόλου τιμιωτέρα τῶν αἰσθήσεων καὶ τῆς νοήσεως, ὅσων ἕτερον τὸ αἴτιον· περὶ δὲ τῶν πρώτων ἄλλος λόγος. Φανερὸν οὖν ὅτι ἀδύνατον τῶι αἰσθάνεσθαι ἐπίστασθαί τι ↵ τῶν ἀποδεικτῶν, εἰ μή τις τὸ αἰσθάνεσθαι τοῦτο λέγει, τὸ ἐπιστήμην ἔχειν δι᾽ ἀποδείξεως.	Universale autem honorabile, quoniam ostendit causam: quare de huiusmodi universalis honorabilior est sensibus et cognitione quorumcunque altera causa est, sed de primis alia ratio est. Manifestum igitur est quod impossibile sit sentiendo scire aliud demonstratorum, nisi aliquis dicat sentire, scientiam habere per demonstrationem.	The commensurate universal is precious because it makes clear the cause; so that in the case of facts like these which have a cause other than themselves universal knowledge is more precious than sense-perceptions and than intuition. (As regards primary truths there is of course a different account to be given.) Hence it is clear that knowledge of things demonstrable cannot be acquired by perception, unless the term perception is applied to the possession of scientific knowledge through demonstration.
88a11 ἔστι μέντοι ἔνια ἀναγόμενα εἰς αἰσθήσεως ἔκλειψιν ἐν τοῖς προβλήμασιν. ἔνια γὰρ εἰ ἑωρῶμεν οὐκ ἂν ἐζητοῦμεν, οὐχ ὡς εἰδότες τῶι ὁρᾶν, ἀλλ᾽ ὡς ἔχοντες τὸ καθόλου ἐκ τοῦ ὁρᾶν. οἷον εἰ τὴν ὕαλον τετρυπη↵ μένην ἑωρῶμεν καὶ τὸ φῶς διιόν, δῆλον ἂν ἦν καὶ διὰ τί καίει, τῶι ὁρᾶν μὲν χωρὶς ἐφ᾽ ἑκάστης, νοῆσαι δ᾽ ἅμα ὅτι ἐπὶ πασῶν οὕτως.	Sunt tamen quaedam reducta ad sensus defectum in propositis, quaedam enim si videremus, non utique quaereremus. Sed non tanquam scientes in videndo, sed tanquam habentes universale ex eo quod videmus, ut si vitrum foratum videremus, et lumen pertransiens, manifestum utique erit et propter quid illuminat, propter id quod videremus quidem seorsum in unoquoque, intelligere autem simul est, quoniam in omnibus sic est.	Nevertheless certain points do arise with regard to connexions to be proved which are referred for their explanation to a failure in sense-perception: there are cases when an act of vision would terminate our inquiry, not because in seeing we should be knowing, but because we should have elicited the universal from seeing; if, for example, we saw the pores in the glass and the light passing through, the reason of the kindling would be clear to us because we should at the same time see it in each instance and intuit that it must be so in all instances.
c32	CAPUT XXV. Non omnium syllogismorum eadem principia esse posse.	Chapter 32
88a18 Τὰς δ᾽ αὐτὰς ἀρχὰς ἁπάντων εἶναι τῶν συλλογισμῶν ἀδύνατον, πρῶτον μὲν λογικῶς θεωροῦσιν. οἱ μὲν γὰρ ἀλη↵ θεῖς εἰσι τῶν συλλογισμῶν, οἱ δὲ ψευδεῖς.	Eadem autem esse principia omnium syllogismorum impossibile est, primum quidem logice speculantibus, hi enim veri sunt syllogismi, alii autem falsi.	All syllogisms cannot have the same basic truths. This may be shown first of all by the following dialectical considerations. (1) Some syllogisms are true and some false:
88a20 καὶ γὰρ εἰ ἔστιν ἀληθὲς ἐκ ψευδῶν συλλογίσασθαι, ἀλλ᾽ ἅπαξ τοῦτο γίνεται, οἷον εἰ τὸ Α κατὰ τοῦ Γ ἀληθές, τὸ δὲ μέσον τὸ Β ψεῦδος· οὔτε γὰρ τὸ Α τῶι Β ὑπάρχει οὔτε τὸ Β τῶι Γ. ἀλλ᾽ ἐὰν τούτων μέσα λαμβάνηται τῶν προτάσεων, ψευδεῖς ↵ ἔσονται διὰ τὸ πᾶν συμπέρασμα ψεῦδος ἐκ ψευδῶν εἶναι, τὰ δ᾽ ἀληθῆ ἐξ ἀληθῶν, ἕτερα δὲ τὰ ψευδῆ καὶ τἀληθῆ.	Et si enim sit verum ex falsis syllogizare, sed semel hoc fit, ut si a de c verum sit, medium autem b falsum, neque enim a in b, neque b in c; sed si harum media accipiantur propositionum, falsae erunt, ex eo quod omnis conclusio falsa ex falsis est, vera autem ex veris, altera autem sunt vera et falsa.	for though a true inference is possible from false premisses, yet this occurs once only-I mean if A for instance, is truly predicable of C, but B, the middle, is false, both A-B and B-C being false; nevertheless, if middles are taken to prove these premisses, they will be false because every conclusion which is a falsehood has false premisses, while true conclusions have true premisses, and false and true differ in kind.
88a26 εἶτα οὐδὲ τὰ ψευδῆ ἐκ τῶν αὐτῶν ἑαυτοῖς· ἔστι γὰρ ψευδῆ ἀλλήλοις καὶ ἐναντία καὶ ἀδύνατα ἅμα εἶναι, οἷον τὸ τὴν δικαιοσύνην εἶναι ἀδικίαν ἢ δειλίαν, καὶ τὸν ἄνθρωπον ἵππον ↵ ἢ βοῦν, ἢ τὸ ἴσον μεῖζον ἢ ἔλαττον.	Postea neque falsa ex eisdem sunt, est enim falsa ad invicem, et contraria, et impossibilia simul esse, ut iustitiam esse iniustitiam, aut timorem audaciam, aut hominem equum aut bovem, aut aequale, maius et minus.	Then again, (2) falsehoods are not all derived from a single identical set of principles: there are falsehoods which are the contraries of one another and cannot coexist, e.g. ‘justice is injustice’, and ‘justice is cowardice’; ‘man is horse’, and ‘man is ox’; ‘the equal is greater’, and ‘the equal is less.’ From established principles we may argue the case as follows, confining-ourselves therefore to true conclusions.
88a31 Ἐκ δὲ τῶν κειμένων ὧδε· οὐδὲ γὰρ τῶν ἀληθῶν αἱ αὐταὶ ἀρχαὶ πάντων. ἕτεραι γὰρ πολλῶν τῶι γένει αἱ ἀρχαί, καὶ οὐδ᾽ ἐφαρμόττουσαι, οἷον αἱ μονάδες ταῖς στιγμαῖς οὐκ ἐφαρμόττουσιν· αἱ μὲν γὰρ οὐκ ἔχουσι θέσιν, αἱ δὲ ἔχουσιν. ἀνάγκη δέ γε ἢ εἰς ↵ μέσα ἁρμόττειν ἢ ἄνωθεν ἢ κάτωθεν, ἢ τοὺς μὲν εἴσω ἔχειν τοὺς δ᾽ ἔξω τῶν ὅρων.	Ex oppositis autem sic est, neque enim verorum eadem principia omnium sunt, altera enim multorum genere principia sunt, et neque conveniunt, ut unitates punctis non conveniunt, hae enim non habent positionem, illa autem habent. Necesse autem est aut in media convenire, aut in sursum, aut deorsum, aut hos interius habere, illos autem exterius terminorum.	Not even all these are inferred from the same basic truths; many of them in fact have basic truths which differ generically and are not transferable; units, for instance, which are without position, cannot take the place of points, which have position. The transferred terms could only fit in as middle terms or as major or minor terms, or else have some of the other terms between them, others outside them.
88a37 ἀλλ᾽ οὐδὲ τῶν κοινῶν ἀρχῶν οἷόν τ᾽ εἶναί τινας ἐξ ὧν ἅπαντα δειχθήσεται· λέγω δὲ κοινὰς [88b]οἷον τὸ πᾶν φάναι ἢ ἀποφάναι. τὰ γὰρ γένη τῶν ὄντων ἕτερα, καὶ τὰ μὲν τοῖς ποσοῖς τὰ δὲ τοῖς ποιοῖς ὑπάρχει μόνοις, μεθ᾽ ὧν δείκνυται διὰ τῶν κοινῶν.	Sed neque communium principiorum possunt esse aliqua ex quibus omnia demonstrabuntur, dico autem communia, ut omne affirmare aut negare, genera enim eorum quae sunt, altera sunt, et alia quidem in quantitatibus, alia vero in qualitatibus sunt solum, cum quibus demonstrantur per communia.	Nor can any of the common axioms-such, I mean, as the law of excluded middle-serve as premisses for the proof of all conclusions. For the kinds of being are different, and some attributes attach to quanta and some to qualia only; and proof is achieved by means of the common axioms taken in conjunction with these several kinds and their attributes.
88b4 ἔτι αἱ ἀρχαὶ οὐ πολλῶι ἐλάττους τῶν συμπερασμάτων· ἀρχαὶ μὲν γὰρ αἱ ↵ προτάσεις, αἱ δὲ προτάσεις ἢ προσλαμβανομένου ὅρου ἢ ἐμβαλλομένου εἰσίν. ἔτι τὰ συμπεράσματα ἄπειρα, οἱ δ᾽ ὅροι πεπερασμένοι.	Amplius, principia non multo minora sunt conclusionibus, principia enim propositiones sunt, propositiones autem assumpti termini, aut immissi termini sunt. Adhuc conclusiones sunt infinitae, termini autem finiti.	Again, it is not true that the basic truths are much fewer than the conclusions, for the basic truths are the premisses, and the premisses are formed by the apposition of a fresh extreme term or the interposition of a fresh middle. Moreover, the number of conclusions is indefinite, though the number of middle terms is finite;
88b8 ἔτι αἱ ἀρχαὶ αἱ μὲν ἐξ ἀνάγκης, αἱ δ᾽ ἐνδεχόμεναι. Οὕτω μὲν οὖν σκοπουμένοις ἀδύνατον τὰς αὐτὰς εἶναι ↵ πεπερασμένας, ἀπείρων ὄντων τῶν συμπερασμάτων.	Amplius, principia haec quidem ex necessitate, illa contingentia. Sic igitur considerantibus impossibile est eadem principia esse, aut finita, cum infinitae sunt conclusiones.	and lastly some of the basic truths are necessary, others variable. Looking at it in this way we see that, since the number of conclusions is indefinite, the basic truths cannot be identical or limited in number.
88b10 εἰ δ᾽ ἄλλως πως λέγοι τις, οἷον ὅτι αἱδὶ μὲν γεωμετρίας αἱδὶ δὲ λογισμῶν αἱδὶ δὲ ἰατρικῆς, τί ἂν εἴη τὸ λεγόμενον ἄλλο πλὴν ὅτι εἰσὶν ἀρχαὶ τῶν ἐπιστημῶν; τὸ δὲ τὰς αὐτὰς φά- ναι γελοῖον, ὅτι αὐταὶ αὑταῖς αἱ αὐταί· πάντα γὰρ οὕτω ↵ γίγνεται ταὐτά.	Si vero aliter quodammodo dicat quis, quod haec quidem geometriae, illa vero numerorum, illa autem medicinae, quid utique erit aliud quod dicitur, nisi quod sunt principia scientiarum diversa? sed eadem dicere derisio est, quoniam eadem eisdem eadem erunt, omnia namque sic fiunt eadem.	If, on the other hand, identity is used in another sense, and it is said, e.g. ‘these and no other are the fundamental truths of geometry, these the fundamentals of calculation, these again of medicine’; would the statement mean anything except that the sciences have basic truths? To call them identical because they are self-identical is absurd, since everything can be identified with everything in that sense of identity.
88b15 ἀλλὰ μὴν οὐδὲ τὸ ἐξ ἁπάντων δείκνυσθαι ὁτιοῦν, τοῦτ᾽ ἐστὶ τὸ ζητεῖν ἁπάντων εἶναι τὰς αὐτὰς ἀρχάς· λίαν γὰρ εὔηθες.	At vero neque quod est ex omnibus demonstrare quodlibet, est quaerere omnium esse eadem principia, multum enim insipiens est.	Nor again can the contention that all conclusions have the same basic truths mean that from the mass of all possible premisses any conclusion may be drawn. That would be exceedingly naive,
οὔτε γὰρ ἐν τοῖς φανεροῖς μαθήμασι τοῦτο γίνεται, οὔτ᾽ ἐν τῆι ἀναλύσει δυνατόν· αἱ γὰρ ἄμεσοι προτάσεις ἀρχαί, ἕτερον δὲ συμπέρασμα προσληφθείσης γίνε↵ ται προτάσεως ἀμέσου.	Neque enim in manifestis doctrinis hoc fit, neque in resolutione hoc est possibile, immediatae enim propositiones sunt principia, altera autem conclusio fit accepta propositione immediata.	for it is not the case in the clearly evident mathematical sciences, nor is it possible in analysis, since it is the immediate premisses which are the basic truths, and a fresh conclusion is only formed by the addition of a new immediate premiss:
88b20 εἰ δὲ λέγοι τις τὰς πρώτας ἀμέσους προτάσεις, ταύτας εἶναι ἀρχάς, μία ἐν ἑκάστωι γένει ἐστίν.	Si autem dicat aliquis primas immediatas propositiones eadem esse principia, una in unoquoque genere est.	but if it be admitted that it is these primary immediate premisses which are basic truths, each subject-genus will provide one basic truth.
88b22 εἰ δὲ μήτ᾽ ἐξ ἁπασῶν ὡς δέον δείκνυσθαι ὁτιοῦν μήθ᾽ οὕτως ἑτέρας ὥσθ᾽ ἑκάστης ἐπιστήμης εἶναι ἑτέρας, λείπεται εἰ συγγενεῖς αἱ ἀρχαὶ πάντων, ἀλλ᾽ ἐκ τωνδὶ μὲν ταδί, ἐκ δὲ ↵ τωνδὶ ταδί.	Si vero neque ex omnibus ut opus est demonstrari contingit quodlibet, neque sic ex altero tanquam erunt uniuscuiusque scientiae altera, relinquitur quod proxima sint principia omnium, et ex his quidem haec, ex illis autem illa.	If, however, it is not argued that from the mass of all possible premisses any conclusion may be proved, nor yet admitted that basic truths differ so as to be generically different for each science, it remains to consider the possibility that, while the basic truths of all knowledge are within one genus, special premisses are required to prove special conclusions.
88b25 φανερὸν δὲ καὶ τοῦθ᾽ ὅτι οὐκ ἐνδέχεται· δέδεικται γὰρ ὅτι ἄλλαι ἀρχαὶ τῶι γένει εἰσὶν αἱ τῶν διαφόρων τῶι γένει. αἱ γὰρ ἀρχαὶ διτταί, ἐξ ὧν τε καὶ περὶ ὅ· αἱ μὲν οὖν ἐξ ὧν κοιναί, αἱ δὲ περὶ ὁ ἴδιαι, οἷον ἀριθμός, μέγεθος.	Manifestum autem et hoc est, quoniam non contingit, monstratum est enim quod altera principia genere sunt differentium genere. Principia enim duplicia sunt, ex quibus et circa quod, ex quibus quidem igitur, communia sunt, quae autem sunt circa quod, propria sunt, ut numerus, magnitudo.	But that this cannot be the case has been shown by our proof that the basic truths of things generically different themselves differ generically. For fundamental truths are of two kinds, those which are premisses of demonstration and the subject-genus; and though the former are common, the latter – number, for instance, and magnitude – are peculiar.
c33	CAPUT XXVI. Quod scientia et Scibile ab opinatione et opinabili discrepent.	Chapter 33
88b30 Τὸ δ᾽ ἐπιστητὸν καὶ ἐπιστήμη διαφέρει τοῦ δοξαστοῦ καὶ δόξης, ὅτι ἡ μὲν ἐπιστήμη καθόλου καὶ δι᾽ ἀναγκαίων, τὸ δ᾽ ἀναγκαῖον οὐκ ἐνδέχεται ἄλλως ἔχειν.	Scibile autem et scientia differunt ab opinabili et opinione, quoniam scientia universalis et pernecessaria est, necessarium autem non contingit aliter se habere.	Scientific knowledge and its object differ from opinion and the object of opinion b3l in that scientific knowledge is commensurately universal and proceeds by necessary connexions, and that which is necessary cannot be otherwise.
88b33 ἔστι δέ τινα ἀληθῆ μὲν καὶ ὄντα, ἐνδεχόμενα δὲ καὶ ἄλλως ἔχειν. δῆλον οὖν ὅτι περὶ μὲν ταῦτα ἐπιστήμη οὐκ ἔστιν· εἴη γὰρ ἂν ἀδύνατα ↵ ἄλλως ἔχειν τὰ δυνατὰ ἄλλως ἔχειν.	Sunt autem quaedam vera, quae contingit aliter se habere, manifestum est igitur quod circa haec scientia non est, essent enim utique impossibilia aliter se habere.	So though there are things which are true and real and yet can be otherwise, scientific knowledge clearly does not concern them: if it did, things which can be otherwise would be incapable of being otherwise.
ἀλλὰ μὴν οὐδὲ νοῦς (λέγω γὰρ νοῦν ἀρχὴν ἐπιστήμησ) οὐδ᾽ ἐπιστήμη ἀναπόδεικτος· τοῦτο δ᾽ ἐστὶν ὑπόληψις τῆς ἀμέσου προτάσεως. ἀληθὴς δ᾽ [89a]ἐστὶ νοῦς καὶ ἐπιστήμη καὶ δόξα καὶ τὸ διὰ τούτων λεγόμενον·	At vero neque intellectus, dico enim intellectum principium esse scientiae, neque scientia indemonstrabilis, haec autem est acceptio immediatae propositionis, verus enim est intellectus, et scientia, et opinio, et quid per haec dicitur.	Nor are they any concern of rational intuition-by rational intuition I mean an originative source of scientific knowledge-nor of indemonstrable knowledge, which is the grasping of the immediate premiss. Since then rational intuition, science, and opinion, and what is revealed by these terms, are the only things that can be ‘true’,
ὥστε λείπεται δόξαν εἶναι περὶ τὸ ἀληθὲς μὲν ἢ ψεῦδος, ἐνδεχόμενον δὲ καὶ ἄλλως ἔχειν. τοῦτο δ᾽ ἐστὶν ὑπόληψις τῆς ἀμέσου προτάσεως καὶ μὴ ἀναγκαίας.	Quare relinquitur opinionem esse circa verum quidem, aut falsum, contingens autem est et aliter se habere. Hoc autem est acceptio immediatae propositionis, et non necessariae.	it follows that it is opinion that is concerned with that which may be true or false, and can be otherwise: opinion in fact is the grasp of a premiss which is immediate but not necessary.
89a5 καὶ ὁμο↵ λογούμενον δ᾽ οὕτω τοῖς φαινομένοις· ἥ τε γὰρ δόξα ἀβέβαιον, καὶ ἡ φύσις ἡ τοιαύτη.	Certum autem est sic apparentibus, opinio ei incertum est, et natura huiusmodi est.	This view also fits the observed facts, for opinion is unstable, and so is the kind of being we have described as its object.
89a7 πρὸς δὲ τούτοις οὐδεὶς οἴεται δοξάζειν, ὅταν οἴηται ἀδύνατον ἄλλως ἔχειν, ἀλλ᾽ ἐπίστασθαι· ἀλλ᾽ ὅταν εἶναι μὲν οὕτως, οὐ μὴν ἀλλὰ καὶ ἄλλως οὐδὲν κωλύειν, τότε δοξάζειν, ὡς τοῦ μὲν τοιούτου δόξαν οὖσαν, ↵ τοῦ δ᾽ ἀναγκαίου ἐπιστήμην.	Adhuc autem nullus arbitratur opinari, cum opinetur impossibile aliter se habere, sed scire, sed quando esse quidem sic, sed tamen aliter, nihil prohibet et tunc opinari, tanquam huiusmodi quidem opinionem esse, necessariam autem scientiam.	Besides, when a man thinks a truth incapable of being otherwise he always thinks that he knows it, never that he opines it. He thinks that he opines when he thinks that a connexion, though actually so, may quite easily be otherwise; for he believes that such is the proper object of opinion, while the necessary is the object of knowledge.
89a11 Πῶς οὖν ἔστι τὸ αὐτὸ δοξάσαι καὶ ἐπίστασθαι, καὶ διὰ τί οὐκ ἔσται ἡ δόξα ἐπιστήμη, εἴ τις θήσει ἅπαν ὁ οἶδεν ἐνδέχεσθαι δοξάζειν; ἀκολουθήσει γὰρ ὁ μὲν εἰδὼς ὁ δὲ δοξάζων διὰ τῶν μέσων, ἕως εἰς τὰ ἄμεσα ἔλθηι, ὥστ᾽ εἴπερ ↵ ἐκεῖνος οἶδε, καὶ ὁ δοξάζων οἶδεν. ὥσπερ γὰρ καὶ τὸ ὅτι δοξάζειν ἔστι, καὶ τὸ διότι· τοῦτο δὲ τὸ μέσον.	Quomodo est igitur non idem opinari, et scire et quare non erit opinio scientia, si quis posuerit omne quod scit contingere opinari? consequitur enim hic quidem sciens, ille vero opinans per media, quousque ad immediata veniat, quare si ille quidem scivit, et opinans scivit, sicut enim et quia opinari, et propter quid, hoc autem medium est.	In what sense, then, can the same thing be the object of both opinion and knowledge? And if any one chooses to maintain that all that he knows he can also opine, why should not opinion be knowledge? For he that knows and he that opines will follow the same train of thought through the same middle terms until the immediate premisses are reached; because it is possible to opine not only the fact but also the reasoned fact, and the reason is the middle term; 89a16 so that, since the former knows, he that opines also has knowledge.
ἢ εἰ μὲν οὕτως ὑπολήψεται τὰ μὴ ἐνδεχόμενα ἄλλως ἔχειν ὥσπερ (εχει) τοὺς ὁρισμοὺς δι᾽ ὧν αἱ ἀποδείξεις, οὐ δοξάσει ἀλλ᾽ ἐπιστήσεται·	An si quis sic arbitrabitur non contingentia aliter se habere, sicut se habent definitiones per quas sunt demonstrationes, non opinabitur sed sciet?	The truth perhaps is that if a man grasp truths that cannot be other than they are, in the way in which he grasps the definitions through which demonstrations take place, he will have not opinion but knowledge:
εἰ δ᾽ ἀληθῆ μὲν εἶναι, οὐ μέντοι ταῦτά γε αὐτοῖς ↵ ὑπάρχειν κατ᾽ οὐσίαν καὶ κατὰ τὸ εἶδος, δοξάσει καὶ οὐκ ἐπιστήσεται ἀληθῶς, καὶ τὸ ὅτι καὶ τὸ διότι, ἐὰν μὲν διὰ τῶν ἀμέσων δοξάσηι· ἐὰν δὲ μὴ διὰ τῶν ἀμέσων, τὸ ὅτι μόνον δοξάσει; τοῦ δ᾽ αὐτοῦ δόξα καὶ ἐπιστήμη οὐ πάντως ἐστίν, ἀλλ᾽ ὥσπερ καὶ ψευδὴς καὶ ἀληθὴς τοῦ αὐτοῦ τρό↵ πον τινά, οὕτω καὶ ἐπιστήμη καὶ δόξα τοῦ αὐτοῦ.	Si autem vera quidem esse, non tamen haec ipsis inesse secundum substantiam, et secundum speciem opinabitur, et non sciet vere, et quia et propter quid, si quidem per immediata opinabitur, si vero non per immediata opinabitur, solum ipsum quia opinabitur. Eiusdem autem opinio, et scientia non penitus est, sed sicut vera, et falsa eiusdem quodam modo est, sic et scientia et opinio eiusdem,	if on the other hand he apprehends these attributes as inhering in their subjects, but not in virtue of the subjects’ substance and essential nature possesses opinion and not genuine knowledge; and his opinion, if obtained through immediate premisses, will be both of the fact and of the reasoned fact; if not so obtained, of the fact alone. 89a23 The object of opinion and knowledge is not quite identical; it is only in a sense identical, just as the object of true and false opinion is in a sense identical.
καὶ γὰρ δόξαν ἀληθῆ καὶ ψευδῆ ὡς μέν τινες λέγουσι τοῦ αὐτοῦ εἶναι, ἄτοπα συμβαίνει αἱρεῖσθαι ἄλλα τε καὶ μὴ δοξάζειν ὁ δοξάζει ψευδῶς·	et opinionem ei veram, et falsam (sicut dicunt quidam) eiusdem esse, inconvenientia accidunt appetere aliaque, et non opinari quae opinantur falsae.	The sense in which some maintain that true and false opinion can have the same object leads them to embrace many strange doctrines, particularly the doctrine that what a man opines falsely he does not opine at all.
ἐπεὶ δὲ τὸ αὐτὸ πλεοναχῶς λέγεται, ἔστιν ὡς ἐνδέχεται, ἔστι δ᾽ ὡς οὔ. τὸ μὲν γὰρ ↵ σύμμετρον εἶναι τὴν διάμετρον ἀληθῶς δοξάζειν ἄτοπον· ἀλλ᾽ ὅτι ἡ διάμετρος, περὶ ἣν αἱ δόξαι, τὸ αὐτό, οὕτω τοῦ αὐτοῦ, τὸ δὲ τί ἦν εἶναι ἑκατέρωι κατὰ τὸν λόγον οὐ τὸ αὐτό.	Quoniam autem idem multipliciter dicitur, est autem sicut contingit, est autem ut non, commensuratam enim diametrum esse vere, opinari inconveniens est, sed quod diameter (circa quam sunt opiniones) idem sic eiusdem est, sed quod quid erat esse, unicuique secundum rationem non est idem.	There are really many senses of ‘identical’, and in one sense the object of true and false opinion can be the same, in another it cannot. Thus, to have a true opinion that the diagonal is commensurate with the side would be absurd: but because the diagonal with which they are both concerned is the same, the two opinions have objects so far the same: on the other hand, as regards their essential definable nature these objects differ.
ὁμοίως δὲ καὶ ἐπιστήμη καὶ δόξα τοῦ αὐτοῦ. ἡ μὲν γὰρ οὕτως τοῦ ζώιου ὥστε μὴ ἐνδέχεσθαι μὴ εἶναι ζῶιον, ἡ δ᾽ ↵ ὥστ᾽ ἐνδέχεσθαι, οἷον εἰ ἡ μὲν ὅπερ ἀνθρώπου ἐστίν, ἡ δ᾽ ἀνθρώπου μέν, μὴ ὅπερ δ᾽ ἀνθρώπου. τὸ αὐτὸ γὰρ ὅτι ἄνθρωπος, τὸ δ᾽ ὡς οὐ τὸ αὐτό.	Similiter autem et scientia, et opinio eiusdem est, haec enim si animalis est quod non est contingere non esse animal, sed illa quidem quae est contingere, ut si haec quidem quod hominis quidem est, illa vero hominis quidem, non autem quod quidem hominis est, idem enim est, quia homo, hoc autem sicut non idem.	The identity of the objects of knowledge and opinion is similar. Knowledge is the apprehension of, e.g. the attribute ‘animal’ as incapable of being otherwise, opinion the apprehension of 'animal' as capable of being otherwise-e.g. the apprehension that animal is an element in the essential nature of man is knowledge; the apprehension of animal as predicable of man but not as an element in man's essential nature is opinion: man is the subject in both judgements, but the mode of inherence differs.
Φανερὸν δ᾽ ἐκ τούτων ὅτι οὐδὲ δοξάζειν ἅμα τὸ αὐτὸ καὶ ἐπίστασθαι ἐνδέχεται. ἅμα γὰρ ἂν ἔχοι ὑπόληψιν τοῦ [89b]ἄλλως ἔχειν καὶ μὴ ἄλλως τὸ αὐτό· ὅπερ οὐκ ἐνδέχεται. ἐν ἄλλωι μὲν γὰρ ἑκάτερον εἶναι ἐνδέχεται τοῦ αὐτοῦ ὡς εἴρηται, ἐν δὲ τῶι αὐτῶι οὐδ᾽ οὕτως οἷόν τε· ἕξει γὰρ ὑπόληψιν ἅμα, οἷον ὅτι ὁ ἄνθρωπος ὅπερ ζῶιον (τοῦτο γὰρ ἦν τὸ ↵ μὴ ἐνδέχεσθαι εἶναι μὴ ζῶιον) καὶ μὴ ὅπερ ζῶιον· τοῦτο γὰρ ἔστω τὸ ἐνδέχεσθαι.	Manifestum autem ex his est quod neque opinari simul idem, et scire contingit, simul enim haberet utique opinionem aliter habendi, et non aliter idem, quod quidem non contingit, in alio enim unumquodque esse contingit eiusdem, sicut dictum est, sed in eodem nihil sic potest esse, haberet enim opinionem simul. Ut quod homo esset secundum quod est animal, hoc enim fuerit non contingere esse non animal, et non secundum quod animal, hoc enim sicut contingere.	This also shows that one cannot opine and know the same thing simultaneously; for then one would apprehend the same thing as both capable and incapable of being otherwise-an impossibility. Knowledge and opinion of the same thing can co-exist in two different people in the sense we have explained, but not simultaneously in the same person. That would involve a man's simultaneously apprehending, e.g. (1) that man is essentially animal-i.e. cannot be other than animal-and (2) that man is not essentially animal, that is, we may assume, may be other than animal.
89b6 Τὰ δὲ λοιπὰ πῶς δεῖ διανεῖμαι ἐπί τε διανοίας καὶ νοῦ καὶ ἐπιστήμης καὶ τέχνης καὶ φρονήσεως καὶ σοφίας, τὰ μὲν φυσικῆς τὰ δὲ ἠθικῆς θεωρίας μᾶλλόν ἐστιν.	Reliqua autem quomodo oportet distribuere in rationem, et intellectum, et scientiam, et artem, et prudentiam, et sapientiam, haec quidem physicae est, illa vero ethicae speculationis magis.	Further consideration of modes of thinking and their distribution under the heads of discursive thought, intuition, science, art, practical wisdom, and metaphysical thinking, belongs rather partly to natural science, partly to moral philosophy.
c34	CAPUT XXVII. De solertia. Solertia autem est subtilitas quaedam in non perspecto tempore medii.	Chapter 34
89b10 Ἡ δ᾽ ἀγχίνοιά ἐστιν εὐστοχία τις ἐν ἀσκέπτωι χρόνωι τοῦ μέσου, οἷον εἴ τις ἰδὼν ὅτι ἡ σελήνη τὸ λαμπρὸν ἀεὶ ἔχει πρὸς τὸν ἥλιον, ταχὺ ἐνενόησε διὰ τί τοῦτο, ὅτι διὰ τὸ λάμπειν ἀπὸ τοῦ ἡλίου· ἢ διαλεγόμενον πλουσίωι ἔγνω διότι δανείζεται· ἢ διότι φίλοι, ὅτι ἐχθροὶ τοῦ αὐτοῦ. πάντα γὰρ ↵ τὰ αἴτια τὰ μέσα [ὁ] ἰδὼν τὰ ἄκρα ἐγνώρισεν.	Ut si aliquis videns quod luna splendorem semper habet ad solem, statim intellexerit propter quid hoc sit, quia propter id quod illustratur a sole, aut disputantem cum divite, cognovit quoniam commodatum est, aut propter id quod amici sunt, quia inimici eiusdem sunt. Omnes enim causas medias videns, cognovit et ultima.	Quick wit is a faculty of hitting upon the middle term instantaneously. It would be exemplified by a man who saw that the moon has her bright side always turned towards the sun, and quickly grasped the cause of this, namely that she borrows her light from him; or observed somebody in conversation with a man of wealth and divined that he was borrowing money, or that the friendship of these people sprang from a common enmity. In all these instances he has seen the major and minor terms and then grasped the causes, the middle terms.
τὸ λαμπρὸν εἶναι τὸ πρὸς τὸν ἥλιον ἐφ᾽ οὗ Α, τὸ λάμπειν ἀπὸ τοῦ ἡλίου Β, σελήνη τὸ Γ. ὑπάρχει δὴ τῆι μὲν σελήνηι τῶι Γ τὸ Β, τὸ λάμπειν ἀπὸ τοῦ ἡλίου· τῶι δὲ Β τὸ Α, τὸ πρὸς τοῦτ᾽ εἶναι τὸ λαμπρόν, ἀφ᾽ οὗ λάμπει· ὥστε καὶ τῶι Γ τὸ Α ↵ διὰ τοῦ Β.	Splendidum esse ad solem sit, in quo a lucere a sole, b: luna, c, inest igitur lunae quidem ipsi c b, quod quidem est lucere a sole, ipsi autem b a, quod est ad hoc esse splendidum, a quo splendet, quare et ipsi c, inest a per b.	Let A represent ‘bright side turned sunward’, B ‘lighted from the sun’, C the moon. Then B, ‘lighted from the sun’ is predicable of C, the moon, and A, ‘having her bright side towards the source of her light’, is predicable of B. So A is predicable of C through B.