§7#2οὕτω δὲ καὶ εἰ ἕνδεκα ὡρῶν ᾖ προβεβλημένη περίοδος, κδ΄ εἶναι τοὺς τύπους ἑνδεκαημέρους ἐρεῖς, ὅπερ ἐστι δωδεκαταίους.
In this way also, if a period of eleven hours is presented, you will say that there are twenty-four eleven-day period types, which is "duodeciman" (twelfth-day fevers).
οὕτω δὲ καὶ εἰ τρισκαίδεκα ὡρῶν ἡ περίοδος ᾖ, τέσσαρας ἐπὶ τοῖς εἴκοσι τεσσαρεσκαιδεκαταίους.
And in this way also, if the period is of thirteen hours, twenty-four "quatuordeciman" (fourteenth-day fevers).
ὡσαύτως δὲ κᾂν μείζων ᾖ προβεβλημένη περίοδος τῶν κδ΄ ὡρῶν, εἰ μὲν μηδὲν ἔχοι μέτρον κοινὸν πρὸς τὸν τῶν κδ΄, ὥσπερ ἐπὶ τοῦ τῶν ε΄ καὶ κ΄ συμβέβηκε, τέτταρας καὶ κ΄ ἐρεῖς εἶναι εἰκοστοεκταίους τύπους. (ὥσπέρ γε καὶ κ΄.
Likewise, even if the presented period is greater than twenty-four hours, if indeed it has no common measure with that of twenty-four—just as it has happened in the case of twenty-five—you will say that there are twenty-four "vicesimosextan" (twenty-sixth-day) types (just as indeed also twenty).
) εἰ δ’ εἴη τι κοινὸν μέτρον ἀμφοῖν, ὥσπερ ἐπὶ τῶν ζ΄ καὶ κ΄ τὸ τρίτον ἐστὶν, ὀκτὼ φήσεις εἶναι δεκαταίους, ἐπειδὴ τῶν μὲν κ΄ καὶ δ΄ τὸ τρίτον ἐστὶν η΄, τῶν δὲκζ΄, θ΄.
But if there is some common measure of both, just as in the case of twenty-seven the third is [the common measure], you will say that there are eight "deciman" (tenth-day fevers), since the third of twenty-four is eight, and of twenty-seven, nine.
δῆλον γὰρ ὅτι τοῦ δεκαταίου τύπου περίοδός ἐστιν ἡμερῶν ἐννέα συμπληρουμένων, αἵτινες ὡρῶν ἀριθμὸν ἐργάζονται ἑκκαίδεκα καὶ διακοσίων, οὗ μέρος ὄγδοόν ἐστιν ὁ προβεβλημένος ἀριθμὸς εἰκοσιεπτὰ ὡρῶν.
For it is clear that the period of the deciman type is of nine completed days, which produce a number of two hundred and sixteen hours, of which the presented number of twenty-seven hours is an eighth part.
ὑποπέπτωκεν οὖν ἡ μέθοδος αὕτη τῇ κατὰ τὸ διάγραμμα κοινῇ προειρημένῃ, τοσοῦτον ἐκείνης διαλλάττουσα, τῷ δύνασθαι καὶ χωρὶς τοῦ διαγράμματος εὑρεῖν τινα πρῶτον ἀριθμὸν ὡρῶν ἡμερησίαις περιόδοις μετρούμενον, οὗ μέρος ὁ προβεβλημένος ἐστίν.
Therefore, this method is subordinate to the common one mentioned before according to the diagram, differing from that one by this much: by the ability, even without the diagram, to find some first number of hours measured by daily periods, of which the presented [number] is a part.
τὸ δὲ θεώρημα τοῦτο δέδεικται καὶ πρὸς Εὐκλείδου κατὰ τὸ τῶν στοιχείων, ἕβδομον δὲ αὐτοῦ ἐστι κεφάλαιον.
And this theorem has been demonstrated also by Euclid in the *Elements*, and it is its seventh book.
τὸ δὲ μεῖζον κοινὸν ἀμφοτέρων τῶν ἀριθμῶν χρὴ λαμβάνοντας σκέψασθαι ποσάκις ἑκάτερον μετρεῖ τὸν ἀριθμὸν τὸ κοινὸν τοῦτο μέτρον, οἷον ἐπὶ τοῦ πεντεκαίδεκα καὶ τοῦ τῶν εἰκοσιτεσσάρων πρῶτον κοινὸν μέτρον ἐστὶν ὁ τῶν τριῶν ἀριθμός· οὗτος δὲ τὸν ε΄ καὶ ι΄ μετρεῖ κατὰ τὸν ε΄, τὸν δὲ εἰκοσιτέσσαρα κατὰ τὸν ὀκτώ.
And one must, taking the greater common [measure] of both numbers, examine how many times this common measure measures each of them; for example, in the case of fifteen and twenty-four, the first common measure is the number three; and this measures fifteen by five, and twenty-four by eight.
ἐὰν οὖν τὸν ε΄ πολλαπλασιάσῃς ἐπὶ τὸν τῶν κδ΄, ἐάν τε τὸν η΄ ἐπὶ τῶν ιε΄, τὸν αὐτὸν ἀριθμὸν εὑρήσεις γινόμενον, ὥσπερ ἑκατὸν καὶ εἴκοσιν ὡρῶν, ἡμερῶν δὲ δηλονότι ε΄.
If then you multiply five by twenty-four, or eight by fifteen, you will find the same number produced, as of one hundred and twenty hours, and clearly of five days.
καὶ διὰ τοῦτο τὸν τύπον ἑκταῖον ἐρεῖς.
And because of this, you will call the type "sextan" (sixth-day fever).
κατὰ δὲ τὸν αὐτὸν λόγον, εἰ τρισκαίδεκα ὡρῶν ὁ προβεβλημένος ἀριθμὸς εἴη, κοινῇ μετρήσουσιν οὗτός τε καὶ ὁ τῶν κδ΄ τὸν ἐξ ἀμφοτέρων ἀλλήλων πολλαπλασίων γινόμενον, ὅσπερ ἐστιν ὁ τῶν τιβ΄. (ἐπειδὴ τῶν πρώτων καλουμένων ἀριθμῶν ἐστιν ὁ τρισκαίδεκα.
According to the same argument, if the presented number is of thirteen hours, both this and that of twenty-four will measure in common the multiple arising from both of each other, which is that of three hundred and twelve. (Since thirteen is one of the numbers called "prime".
προσαγορεύουσιν γὰρ οὕτως ὧν οὐκ ἔστι κοινὸν μέτρον πλὴν μονάδος. ) ὁ μὲν οὖν ἀριθμὸς, ὅσον ἀμφότεροι μετροῦσιν, ἔσται τιβ΄.
For they address in this way those of which there is no common measure except for the unit.) The number, therefore, which both measure will be three hundred and twelve.
τρισκαιδεκάκις γὰρ οὗτος ἔχει τὰς κδ΄ ὥρας.
For this has the twenty-four hours thirteen times.
ὁ δὲ τύπος ἡμερῶν ἔσται τελείων τριῶν καὶ ι΄, τουτέστι τεσσαρεσκαιδεκαταῖος· ἐπεὶ δὲ τετράκις τε καὶ εἰκοσάκις οὗτος μετρεῖ τὸν τῶν τιβ΄, τὸ πλῆθος τῶν τύπων ἔσται δ΄ ἐπὶ τοῖς κ΄.
And the type will be of thirteen complete days, that is, "quatuordeciman" (fourteenth-day fever); and since this [number, thirteen] measures three hundred and twelve twenty-four times, the multitude of the types will be twenty-four.
ἔχετε γοῦν ἤδη καὶ ταύτην ἀπόδειξιν τῆς μεθόδου, καίτοι γ’ οὐ βουληθέντος ἐμοῦ διδάσκειν ὑμᾶς ἄχρηστα πράγματα· κάλλιον γὰρ ἦν αὐτῶν ὅλως καταφρονεῖν μεμνημένους τοῦ Πυθικοῦ παραγγέλματος, ὡς φείδεσθαι προσήκει χρόνου.
At any rate, you now have also this proof of the method, although I did not wish to teach you useless things; for it would have been more beautiful to look down upon them entirely, remembering the Pythian command that it is proper to spare time.