§2.1.10Λέγω δʼ ἀρχὰς ἐν ἑκάστῳ γένει ταύτας ἃς ὅτι ἔστι μὴ ἐνδέχεται δεῖξαι.
I call principles in each genus those things which it is not possible to prove that they exist.
τί μὲν οὖν σημαίνει καὶ τὰ πρῶτα καὶ τὰ ἐκ τούτων, λαμβάνεται, ὅτι δʼ ἔστι, τὰς μὲν ἀρχὰς ἀνάγκη λαμβάνειν, τὰ δʼ ἄλλα δεικνύναι·
Therefore, what both the primary things and those derived from them signify is assumed, but as to their existence, it is necessary to assume that of the principles, and to prove that of the others.
οἷον τί μονὰς ἢ τί τὸ εὐθὺ καὶ τρίγωνον, εἶναι δὲ τὴν μονάδα λαβεῖν καὶ μέγεθος, τὰ δʼ ἕτερα δεικνύναι.
For example, what a unit is, or what straight and triangle are [is assumed], but that the unit and magnitude exist is assumed, whereas the others are proved.
Ἔστι δʼ ὧν χρῶνται ἐν ταῖς ἀποδεικτικαῖς ἐπιστήμαις τὰ μὲν ἴδια ἑκάστης ἐπιστήμης τὰ δὲ κοινά, κοινὰ δὲ κατʼ ἀναλογίαν, ἐπεὶ χρήσιμόν γε ὅσον ἐν τῷ ὑπὸ τὴν ἐπιστήμην γένει·
Of the things they use in the demonstrative sciences, some are proper to each science and others are common, though common by analogy, since they are useful only so far as they apply to the genus under the science.
ἴδια μὲν οἷον γραμμὴν εἶναι τοιονδὶ καὶ τὸ εὐθύ, κοινὰ δὲ οἷον τὸ ἴσα ἀπὸ ἴσων ἂν ἀφείη, ὅτι ἴσα τὰ λοιπά.
Proper, for instance, are that a line is of such a kind, and that straight is so; common, for instance, are that if equals are taken from equals, the remainders are equal.
ἱκανὸν δʼ ἕκαστον τούτων ὅσον ἐν τῷ γένει· ταὐτὸ γὰρ ποιήσει, κἂν μὴ κατὰ πάντων λάβῃ ἀλλʼ ἐπὶ μεγεθῶν μόνον, τῷ δʼ ἀριθμητικῷ ἐπʼ ἀριθμῶν.
Each of these is sufficient in so far as it is in the genus; for it will produce the same result even if one does not assume it of all things but only of magnitudes, or, for the arithmetician, only of numbers.
Ἔστι δʼ ἴδια μὲν καὶ ἃ λαμβάνεται εἶναι, περὶ ἃ ἡ ἐπιστήμη θεωρεῖ τὰ ὑπάρχοντα καθʼ αὑτά, οἷον μονάδας ἡ ἀριθμητική, ἡ δὲ γεωμετρία σημεῖα καὶ γραμμάς.
Proper also are those things whose existence is assumed, about which the science investigates the attributes that belong to them in themselves—for instance, units in arithmetic, and points and lines in geometry.
ταῦτα γὰρ λαμβάνουσι τὸ εἶναι καὶ τοδὶ εἶναι.
For they assume that these exist and are this particular thing.
τὰ δὲ τούτων πάθη καθʼ 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 δʼ ἔστι, δεικνύουσι διά τε τῶν κοινῶν καὶ ἐκ τῶν ἀποδεδειγμένων.
But as to the attributes of these in themselves, they assume what each signifies—for instance, arithmetic, what odd or even or square or cube is, and geometry, what irrational or inflected or verging is—but that they exist, they prove through the common principles and from what has been demonstrated.
καὶ ἡ ἀστρολογία ὡσαύτως.
And similarly for astronomy.
πᾶσα γὰρ ἀποδεικτικὴ ἐπιστήμη περὶ τρία ἐστίν, ὅσα τε εἶναι τίθεται (ταῦτα δʼ ἐστὶ τὸ γένος, οὗ τῶν καθʼ αὑτὰ παθημάτων ἐστὶ θεωρητική), καὶ τὰ κοινὰ λεγόμενα ἀξιώματα, ἐξ ὧν πρώτων ἀποδείκνυσι, καὶ τρίτον τὰ πάθη, ὧν τί σημαίνει ἕκαστον λαμβάνει.
For every demonstrative science is concerned with three things: first, that which it posits to exist (this is the genus, of whose in-itself attributes the science is speculative); second, the so-called common axioms, from which as primary premises it demonstrates; and third, the attributes, of which it assumes what each signifies.
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 δῆλα·
Nevertheless, nothing prevents some sciences from overlooking some of these things: for instance, not hypothesizing that the genus exists, if it is manifest that it does (for it is not equally obvious that number exists and that cold and hot exist); and not assuming what the attributes signify, if they are clear.
ὥσπερ οὐδὲ τὰ κοινὰ οὐ λαμβάνει τί σημαίνει τὸ ἴσα ἀπὸ ἴσων ἀφελεῖν, ὅτι γνώριμον.
Just as in the case of the common principles, it does not assume what it means to subtract equals from equals, because this is already known.
ἀλλʼ οὐδὲν ἧττον τῇ γε φύσει τρία ταῦτά ἐστι, περὶ ὅ τε δείκνυσι καὶ ἃ δείκνυσι καὶ ἐξ ὧν.
But nonetheless, by nature these three things exist: that about which the science demonstrates, that which it demonstrates, and those things from which it demonstrates.
Οὐκ ἔστι δʼ ὑπόθεσις οὐδʼ αἴτημα, ὃ ἀνάγκη εἶναι διʼ αὑτὸ καὶ δοκεῖν ἀνάγκη.
What is necessarily the case in itself and must necessarily be thought to be so is neither a hypothesis nor a petition.
οὐ γὰρ πρὸς τὸν ἔξω λόγον ἡ ἀπόδειξις, ἀλλὰ πρὸς τὸν ἐν τῇ ψυχῇ, ἐπεὶ οὐδὲ συλλογισμός.
For demonstration is not addressed to external discourse, but to discourse in the soul, since this is true of syllogism too.
ἀεὶ γὰρ ἔστιν ἐνστῆναι πρὸς τὸν ἔξω λόγον, ἀλλὰ πρὸς τὸν ἔσω λόγον οὐκ ἀεί.
For it is always possible to object to external discourse, but not always to internal discourse.
ὅσα μὲν οὖν δεικτὰ ὄντα λαμβάνει αὐτὸς μὴ δείξας, ταῦτʼ, ἐὰν μὲν δοκοῦντα λαμβάνῃ τῷ μανθάνοντι, ὑποτίθεται, καὶ ἔστιν οὐχ ἀπλῶς ὑπόθεσις ἀλλὰ πρὸς ἐκεῖνον μόνον, ἂν δὲ ἢ μηδεμιᾶς ἐνούσης δόξης ἢ καὶ ἐναντίας ἐνούσης λαμβάνῃ τὸ αὐτό, αἰτεῖται.
Therefore, whatever things that are demonstrable a person assumes without proving them himself, if he assumes them when they seem true to the learner, he hypothesizes them; and this is not a hypothesis absolutely, but only relative to that learner. But if he assumes the same thing when there is either no opinion or even a contrary opinion in the learner, he petitions it.
καὶ τούτῳ διαφέρει ὑπόθεσις καὶ αἴτημα· ἔστι γὰρ αἴτημα τὸ ὑπεναντίον τοῦ μανθάνοντος τῇ δόξη, ἢ ὃ ἄν τις ἀποδεικτὸν ὂν λαμβάνῃ καὶ χρῆται μὴ δείξας.
And in this respect hypothesis and petition differ; for a petition is what is contrary to the learner’s opinion, or whatever demonstrable thing one assumes and uses without having proved it.
Οἱ μὲν οὖν ὅροι οὐκ εἰσὶν ὑποθέσεις (οὐδὲν γὰρ εἶναι ἢ μὴ λέγεται), ἀλλʼ ἐν ταῖς προτάσεσιν αἱ ὑποθέσεις, τοὺς δʼ ὅρους μόνον ξυνίεσθαι δεῖ·
Therefore, definitions are not hypotheses (for they do not assert that anything exists or does not exist), but hypotheses are among the premises, whereas definitions need only be understood.
τοῦτο δʼ οὐχ ὑπόθεσις (εἰ μὴ καὶ τὸ ἀκούειν ὑπόθεσίν τις εἶναι φήσει), ἀλλʼ ὅσων ὄντων τῷ ἐκεῖνα εἶναι γίνεται τὸ συμπέρασμα. (οὐδʼ ὁ γεωμέτρης φευδῆ ὑποτίθεται, ὥσπερ τινὲς ἔφασαν, λέγοντες ὡς οὐ δεῖ τῷ φεύδει χρῆσθαι, τὸν δὲ γεωμέτρην ψεύδεσθαι λέγοντα ποδιαίαν τὴν οὐ ποδιαίαν ἢ εὐθεῖαν τὴν γεγραμμέντην οὐκ εὐθεῖαν οὖσαν.
And this is not a hypothesis (unless one is to say that even hearing is a hypothesis), but rather [hypotheses are] those things which, by being the case, the conclusion comes to be. (Nor does the geometer hypothesize what is false, as some have claimed, saying that one ought not to use falsehood, and that the geometer speaks falsely when he calls a line a foot long which is not a foot long, or straight a line drawn which is not straight.
ὁ δὲ γεωμέτρης οὐδὲν συμπεραίνεται τῷ τήνδε εἶναι γραμμὴν ἣν αὐτὸς ἔφθεγκται, ἀλλὰ τὰ διὰ τούτων δηλούμενα. ) ἔτι τὸ αἴτημα καὶ ὑπόθεσις πᾶσα ἢ ὡς ὅλον ἢ ὡς ἐν μέρει, οἱ δʼ ὅροι οὐδέτερον τούτων.
But the geometer concludes nothing from the fact that this particular line exists which he himself has spoken of, but rather from the things signified by these.) Furthermore, every petition and hypothesis is either universal or particular, but definitions are neither of these.