§6#2ὁ μὲν ἕτερος αὐτῶν ἀρχέσθω κατὰ τὴν πρώτην ἡμέραν, ὁ δὲ ἕτερος κατὰ τὴν δευτέραν νύκτα.
Let one of them begin on the first day, and the other during the second night.
ἕκτη συζυγία δυοῖν πεμπταίοιν. ὁ μὲν ἕτερος ἀρχέσθω κατὰ τὴν πρώτην ἡμέραν, ὁ δ’ ἄλλος κατὰ τὴν τρίτην νύκτα.
A sixth pairing of two quintans: let one begin on the first day, and the other during the third night.
ἑβδόμη συζυγία δυοῖν πεμπταίοιν. ὁ μὲν ἕτερος ἀρχέσθω κατὰ τὴν πρώτην ἡμέραν, ὁ δ’ ἄλλος κατὰ τὴν τετάρτην νύκτα.
A seventh pairing of two quintans: let one begin on the first day, and the other during the fourth night.
ταύτας οὖν τὰς ἑπτὰ συζυγίας ἐπλέξαμεν ὑποθέμενοι τὸν πρότερον τῶν πεμπταίων ἐν τῇ πρώτῃ τῶν ἡμερῶν ἄρχεσθαι.
These seven pairings, therefore, we have complicated, assuming that the former of the quintans begins on the first of the days.
πάλιν δὲ ἄλλας ζ΄ ποιήσομεν, ὑποθέμενοι τὸν πρότερον αὐτῶν ἄρχεσθαι τῆς πρώτης νυκτός.
And again we shall make another seven, assuming that the former of them begins during the first night.
ἐὰν οὖν οὕτως ἐπιπλέξωμεν αὐτῷ καὶ τὸν ἑκταῖον, εὔδηλον ὡς πλείους αἱ συζυγίαι γενήσονται· καὶ πλείους ἔτι, τὸν ἑβδομαῖον ἐπιπλεκόντων, καὶ πολὺ πλείους ἔτι τὸν ὀγδοαῖον, ἢ καὶ τὸν δεκαταῖον, ἵνα μηδὲν εἴπω περὶ τῶν εἰκοσταίων καὶ τριακοσταίων καὶ τεσσαρακοσταίων, ὅσοι τε πρὸς τούτοις ἐδείχθησαν ἐξ ἀνάγκης ἑπόμενοι ταῖς ὑποθέσεσιν αὐτῶν.
If, then, we complicate the sextan with it in this way, it is obvious that the pairings will become more; and still more, when we complicate the septan, and still much more the octan, or indeed the decan, not to say anything about the twentieth-day, thirtieth-day, and fortieth-day [fevers], and as many besides these as were shown to follow of necessity from their hypotheses.
ἐκεῖνοι μὲν οὖν ἐαθήτωσαν ἐν τῷ παρόντι, γινέσθω δὲ ἡμῖν ἄχρι τοῦ δεκαταίου.
Let those, therefore, be left aside for the present, and let our [discussion] proceed up to the decan.
κᾀμοὶ πρῶτον ἐννόησον, ὡς νῦν ὁπότε διπλοῦς ὁ πεμπταῖος γίγνεται, τὰς συζυγίας εἴπομεν, ἐὰν δὲ τριπλοῦς γενηθείη, δῆλον ὅτι πλῆθος οὐκ ὀλίγον ἄλλο γίνεται συζυγιῶν.
And first consider for me, how just now when the quintan becomes double, we spoke of the pairings, but if it should become triple, it is clear that another not small multitude of pairings arises.
ἐγχωρεῖ δήπου καὶ τετραπλοῦν αὐτὸν γενέσθαι, καθ’ ἑκάστην ἡμέραν ἑνὸς εἰσβάλλοντος παροξυσμοῦ· πενταπλοῦς δ’ ἂν γένοιτο μιχθέντος τῷ τετραπλῷ κατὰ μίαν νύκτα πέμπτου τινὸς ἐπὶ τοῖς εἰρημένοις τέτταρσι πεμπταίοις, καὶ τούτου πάλιν αὐτοῦ τετραχῶς· ἢ γὰρ ἐν τῇ πρώτῃ νυκτὶ τὴν ἀρχὴν δύναται λαβεῖν, ἤ τινι τῶν ἐφεξῆς, ἤτοι τῇ δευτέρᾳ, ἢ τῇ τρίτῃ, ἢ τῇ τετάρτῃ, ἢ τῇ πέμπτῃ.
It is indeed possible for it to become even quadruple, with one paroxysm attacking on each day; and it would become quintuple if, during one night, a certain fifth [quintan] were mixed with the quadruple one, in addition to the four quintans mentioned, and this very thing again in four ways; for it can take its beginning either in the first night, or in one of those following, namely the second, or the third, or the fourth, or the fifth.
ἡ δ’ αὐτὴ ποικίλη τούτοις ἐστὶ καὶ τοὺς ἄλλους τύπους, καὶ μᾶλλον ἔτι κατὰ τοὺς πολυημερωτέρους αὐτῶν.
And the same variety belongs to them also for the other types, and still more so according to those of them of more days.
ἐὰν οὖν ἐπιπλέκῃς ἅπασαν τὴν τοῦ πεμπταίου ποικιλίαν τῇ κατὰ τὸν ἑκταῖον, ἐννοεῖς, οἶμαι, πόσον ἔσται πλῆθος· ἐὰν δὲ μίξῃς αὐταῖς καὶ τὸν ἑβδομαῖον, ἢ ὀγδοαῖον, ἤ τινα τῶν ἐφεξῆς, εἶτα πάλιν ἐκ τεττάρων τύπων οὕτως ἀλλήλοις ἐπιπεπλεγμένων ἑτέρας ποιήσῃς συζυγίας, εἶτ’ ἐκ πέντε, κᾄπειτ’ αὖθις ἐκ πλειόνων, οἶμαί σε λογίζεσθαι πόσος ἀριθμὸς ἔσται συζυγιῶν, ἢ πόσων δεήσει βιβλίων εἰς τὴν διδασκαλίαν αὐτῶν.
If, then, you complicate the whole variety of the quintan with that according to the sextan, you understand, I think, how great the multitude will be; and if you mix with them also the septan, or octan, or any of those following, and then again you make other pairings from four types thus complicated with one another, and then from five, and then again from more, I think you calculate how great the number of pairings will be, or how many books will be needed for the teaching of them.
ὁποία γάρ τις ἡ κατ’ αὐτὰς ἔσται πλοκὴ, πειράσομαί σοι διὰ βραχέων ἐνδείξασθαι, τὴν ἀρχὴν τῷ λόγῳ τήνδε τιθέμενος.
For what kind of weaving there will be according to them, I shall try to show you briefly, laying down this beginning for the discourse.
καθ’ ἕκαστον τῶν τύπων πρότερον δεήσει τὴν ποικιλίαν τῶν συζυγιῶν διελθεῖν, ὡς ἐπὶ τοῦ πεμπταίου σμικρὸν ἔμπροσθεν ἔδειξα· μετὰ ταῦτα ἐπιπλέκειν ἀλλήλαις αὐτὰς, πρῶτον μὲν μίαν συζυγίαν τῶν ἐκ τοῦ πεμπταίου, φέρε εἰπεῖν, μιᾷ τῇ κατὰ τὸν ἑκταῖον, ὡς ἐκ δυοῖν συζυγιῶν γενέσθαι τινὰ τρίτην ἄλλην, εἶθ’ ἑξῆς ἁπάσαις ἁπάσας ἐπιπλέκειν, εἶθ’ οὕτω μίαν δυοῖν, εἶτα τρισὶν, εἶτα πλείοσιν ἄχρι περ ἂν ἁπάσας τοῦ συμπλεκομένου τύπου τὰς συζυγίας ἀναμίξῃς τῇ μιᾷ· μετὰ δὲ τοῦτο δύο συζυγίας ἑνὸς τύπου τοῦ πεμπταίου μιᾷ καθ’ ἑκάστην τῶν ἄλλων μιγνύειν, εἶτα δυοῖν, ἢ τριῶν, ἢ τέτταρσιν, ἄχρι περ ἂν ἁπάσας ἀλλήλαις ἐπιπλέξῃς· εἶθ’ οὕτω πάλιν τέτταρας συζυγίας ταῖς καθ’ ἕκαστον τύπον μιγνύειν, εἶτα πέντε, κᾄπειτα ἓξ, μέχρι περ ἂν ἁπάσας τοῦ πεμπταίου τὰς συζυγίας ἁπάσαις τῶν ἄλλων ἀναμίξῃς, οὐχ ὁμοῦ δηλονότι πάντων, ἀλλ’ ἑκάστου κατὰ μόνας, οἷον τὰς τοῦ πεμπταίου ταῖς κατὰ τὸν ἑβδομαῖον, εἶθ’ οὕτω ταῖς κατὰ τὸν ὀγδοαῖον, ἢ τὸν ἐνναταῖον, ἢ τὸν δεκαταῖον, ἤ τινα τῶν ἐφεξῆς· εἶθ’ ὥσπερ τὸν πεμπταῖον ἑκάστῳ τῶν ἄλλων ἔμιξας, οὕτως ἤδη τὸν δεκαταῖον ἑκάστῳ τῶν ἄλλων μιγνύειν, ἔπειθ’ ἑβδομαῖόν τε καὶ ὀγδοαῖον καὶ τοὺς ἄλλους.
First it will be necessary to go through the variety of pairings according to each of the types, as I showed a little before in the case of the quintan; after this to complicate them with one another, first indeed one pairing of those from the quintan, say, with one of those according to the sextan, so that from two pairings some other third arises, then in succession to complicate all with all, then in this way one with two, then with three, then with more, until you mix all the pairings of the complicated type with the one; and after this to mix two pairings of one type, [namely] of the quintan, with one of each of the others, then with two, or three, or four, until you complicate all with one another; then in this way again to mix four pairings with those according to each type, then five, and then six, until you mix all the pairings of the quintan with all those of the others, obviously not of all together, but of each individually, for example those of the quintan with those according to the septan, then in this way with those according to the octan, or the nonan, or the decan, or any of those following; then just as you mixed the quintan with each of the others, so now to mix the decan with each of the others, then the septan and the octan and the others.
ὅταν δὲ τοῦτο ποιήσῃς, ἀναγκαῖον ἔσται τριῶν τύπον τὰς συζυγίας ἁπάσας ἁπάσαις ἐπιπλέκειν, εἶτα τεττάρων, εἶτα πέντε, κᾄπειθ’ ἓξ, εἶτα ἑπτὰ, καὶ οὕτω καθεξῆς, ἄχρι περ ἂν ἁπάσας ἁπάντων ἐπιπλέξωμεν.
And when you do this, it will be necessary to complicate all the pairings of three types with all [the others], then of four, then of five, and then of six, then of seven, and so on in order, until we complicate all [the pairings] of all [the types].
ὅτι δ’ οὕτω πολὺ πλῆθος ἔσται τῶν συζυγιῶν, ὡς δυσαριθμήτους γενέσθαι μυριάδας, ἤδη νομίζω κατανοεῖν τινα, κᾂν ἄπειρος ἀριθμοῦ ᾖ.
And that there will be so great a multitude of pairings as to become countless myriads, I think anyone already understands, even if they are inexperienced in numbers.
δύναται δὲ καὶ τοῖς ἀριθμητικοῖς κοινωσάμενος τὸν λόγον, ἀκοῦσαι παρ’ αὐτῶν τὸ πλῆθος.
And one can also, by communicating the calculation to arithmeticians, hear from them the total.
ἐγὼ γὰρ ἐὰν ἐπιδεικνύναι νυνὶ αὐτὸ βουληθῶ, βιβλίου μοι δεήσει μείζονος ὧν οἱ ταῦτα ληροῦντες ἔγραψαν, οὐκ ἐννοήσαντες ὅτι μυριοστοῦ μορίου τῶν ἐπιπλοκῶν ἥψαντο, καίτοι γ’ ἀδολεσχήσαντες πολλά.
For if I should wish to demonstrate it now, I would need a book larger than those which these babblers wrote, they not having perceived that they touched on a ten-thousandth part of the complications, although they babbled much.
εἰ μὲν οὖν χρήσιμον εἶναι νομίζουσιν ἐπιπλέκειν ἀλλήλοις τύπους, ἴστωσαν τοῦ χρησίμου τὸ μυριοστὸν μέρος εὑρηκότες· εἰ δ’ ἄχρηστον, αἰδεσθέντες ἤδη ποτὲ καλέσωσιν, ἐπὶ τὰς βίβλους τὸν Ἥφαιστον, ἢ κᾂν αὐτοὶ μὴ καλέσωσιν, ὁ χρόνος αὐτὰς διαφθερεῖ.
If, then, they think that it is useful to complicate types with one another, let them know that they have found a ten-thousandth part of the useful; but if useless, let them, feeling shame at last, call Hephaestus to their books, or even if they themselves do not call [him], time will destroy them.
τίς γὰρ ἐξ αὐτῶν ἄλλος ἔτι γράψει;
For who else will still write from them?
τίς δὲ οὐ καὶ ταύταις ἤδη κατατεμὼν εἰς τὰς ἐφημέρους χρείας ἀναλώσει;
And who will not, cutting even these up already, consume them for daily needs?