§7.prop.37ἐὰν ἀριθμὸς ὑπό τινος ἀριθμοῦ μετρῆται, ὁ μετρούμενος ὁμώνυμον μέρος ἕξει τῷ μετροῦντι. ἀριθμὸς γὰρ ὁ Α ὑπό τινος ἀριθμοῦ τοῦ Β μετρείσθω· λέγω, ὅτι ὁ Α ὁμώνυμον μέρος ἔχει τῷ Β.
ὁσάκις γὰρ ὁ Β τὸν Α μετρεῖ, τοσαῦται μονάδες ἔστωσαν ἐν τῷ Γ. ἐπεὶ ὁ Β τὸν Α μετρεῖ κατὰ τὰς ἐν τῷ Γ μονάδας, μετρεῖ δὲ καὶ ἡ Δ μονὰς τὸν Γ ἀριθμὸν κατὰ τὰς ἐν αὐτῷ μονάδας, ἰσάκις ἄρα ἡ Δ μονὰς τὸν Γ ἀριθμὸν μετρεῖ καὶ ὁ Β τὸν Α. ἐναλλὰξ ἄρα ἰσάκις ἡ Δ μονὰς τὸν Β ἀριθμὸν μετρεῖ καὶ ὁ Γ τὸν Α· ὃ ἄρα μέρος ἐστὶν ἡ Δ μονὰς τοῦ Β ἀριθμοῦ, τὸ αὐτὸ μέρος ἐστὶ καὶ ὁ Γ τοῦ Α. ἡ δὲ Δ μονὰς τοῦ Β ἀριθμοῦ μέρος ἐστὶν ὁμώνυμον αὐτῷ· καὶ ὁ Γ ἄρα τοῦ Α μέρος ἐστὶν ὁμώνυμον τῷ Β. ὥστε ὁ Α μέρος ἔχει τὸν Γ ὁμώνυμον ὄντα τῷ Β· ὅπερ ἔδει δεῖξαι.
If a number be measured by some number, the measured number will have a part homonymous with the measuring number. For let some number A be measured by some number B; For let some number A be measured by some number B; I say that A has a part homonymous with B. I say that A has a part homonymous with B. For as many times as B measures A, let there be so many units in Γ. Since B measures A according to the units in Γ, and the unit Δ also measures the number Γ according to the units in it, therefore the unit Δ measures the number Γ as many times as B measures A. Therefore, alternately, the unit Δ measures the number B as many times as Γ measures A; therefore, what part the unit Δ is of the number B, the same part is Γ also of A. But the unit Δ is a part of the number B homonymous with it; therefore Γ is also a part of A homonymous with B. Therefore A has the part Γ which is homonymous with B. For as many times as B measures A, let there be so many units in Γ. Since B measures A according to the units in Γ, and the unit Δ also measures the number Γ according to the units in it, therefore the unit Δ measures the number Γ as many times as B measures A. Therefore, alternately, the unit Δ measures the number B as many times as Γ measures A. Therefore, what part the unit Δ is of the number B, the same part is Γ also of A. But the unit Δ is a part of the number B homonymous with it; therefore Γ is also a part of A homonymous with B. Therefore A has the part Γ which is homonymous with B; which was to be proved.
§7.prop.38ἐὰν ἀριθμὸς μέρος ἔχῃ ὁτιοῦν, ὑπὸ ὁμωνύμου ἀριθμοῦ μετρηθήσεται τῷ μέρει. ἀριθμὸς γὰρ ὁ Α μέρος ἐχέτω ὁτιοῦν τὸν Β, καὶ τῷ Β μέρει ὁμώνυμος ἔστω ὁ Γ· λέγω, ὅτι ὁ Γ τὸν Α μετρεῖ. ἐπεὶ γὰρ ὁ Β τοῦ Α μέρος ἐστὶν ὁμώνυμον τῷ Γ, ἔστι δὲ καὶ ἡ Δ μονὰς τοῦ Γ μέρος ὁμώνυμον αὐτῷ, ὃ ἄρα μέρος ἐστὶν ἡ Δ μονὰς τοῦ Γ ἀριθμοῦ, τὸ αὐτὸ μέρος ἐστὶ καὶ ὁ Β τοῦ Α· ἰσάκις ἄρα ἡ Δ μονὰς τὸν Γ ἀριθμὸν μετρεῖ καὶ ὁ Β τὸν Α. ἐναλλὰξ ἄρα ἰσάκις ἡ Δ μονὰς τὸν Β ἀριθμὸν μετρεῖ καὶ ὁ Γ τὸν Α. ὁ Γ ἄρα τὸν Α μετρεῖ· ὅπερ ἔδει δεῖξαι.
If a number have any part whatever, it will be measured by a number homonymous with the part. For let the number A have any part whatever B, and let Γ be a number homonymous with the part B; For let the number A have any part whatever B, and let Γ be a number homonymous with the part B; I say that Γ measures A. For since B is a part of A homonymous with Γ, and the unit Δ is also a part of Γ homonymous with it, therefore what part the unit Δ is of the number Γ, the same part is B also of A; therefore the unit Δ measures the number Γ as many times as B measures A. Therefore, alternately, the unit Δ measures the number B as many times as Γ measures A. Therefore Γ measures A; For since B is a part of A homonymous with Γ, and the unit Δ is also a part of Γ homonymous with it, therefore what part the unit Δ is of the number Γ, the same part is B also of A; therefore the unit Δ measures the number Γ as many times as B measures A. Therefore, alternately, the unit Δ measures the number B as many times as Γ measures A. Therefore Γ measures A; which was to be proved.
§7.prop.39ἀριθμὸν εὑρεῖν, ὃς ἐλάχιστος ὢν ἕξει τὰ δοθέντα μέρη. ἔστω τὰ δοθέντα μέρη τὰ Α, Β, Γ· δεῖ δὴ ἀριθμὸν εὑρεῖν, ὃς ἐλάχιστος ὢν ἕξει τὰ Α, Β, Γ μέρη. ἔστωσαν γὰρ τοῖς Α, Β, Γ μέρεσιν ὁμώνυμοι ἀριθμοὶ οἱ Δ, Ε, Ζ, καὶ εἰλήφθω ὑπὸ τῶν Δ, Ε, Ζ ἐλάχιστος μετρούμενος ἀριθμὸς ὁ Η.
ὁ Η ἄρα ὁμώνυμα μέρη ἔχει τοῖς Δ, Ε, Ζ. τοῖς δὲ Δ, Ε, Ζ ὁμώνυμα μέρη ἐστὶ τὰ Α, Β, Γ· ὁ Η ἄρα ἔχει τὰ Α, Β, Γ μέρη. λέγω δή, ὅτι καὶ ἐλάχιστος ὤν. εἰ γὰρ μή, ἔσται τις τοῦ Η ἐλάσσων ἀριθμός, ὃς ἕξει τὰ Α, Β, Γ μέρη. ἔστω ὁ Θ. ἐπεὶ ὁ Θ ἔχει τὰ Α, Β, Γ μέρη, ὁ Θ ἄρα ὑπὸ ὁμωνύμων ἀριθμῶν μετρηθήσεται τοῖς Α, Β, Γ μέρεσιν. τοῖς δὲ Α, Β, Γ μέρεσιν ὁμώνυμοι ἀριθμοί εἰσιν οἱ Δ, Ε, Ζ· ὁ Θ ἄρα ὑπὸ τῶν Δ, Ε, Ζ μετρεῖται. καί ἐστιν ἐλάσσων τοῦ Η· ὅπερ ἐστὶν ἀδύνατον. οὐκ ἄρα ἔσται τις τοῦ Η ἐλάσσων ἀριθμός, ὃς ἕξει τὰ Α, Β, Γ μέρη· ὅπερ ἔδει δεῖξαι.
To find a number which, being the least, will have the given parts. Let the given parts be A, B, Γ; it is then required to find a number which, being the least, will have the parts A, B, Γ. It is then required to find a number which, being the least, will have the parts A, B, Γ. For let the numbers Δ, E, Z be homonymous with the parts A, B, Γ, and let H be taken as the least number measured by Δ, E, Z. Therefore H has parts homonymous with Δ, E, Z. But the parts homonymous with Δ, E, Z are A, B, Γ; For let the numbers Δ, E, Z be homonymous with the parts A, B, Γ, and let H be taken as the least number measured by Δ, E, Z. Therefore H has parts homonymous with Δ, E, Z. But the parts homonymous with Δ, E, Z are A, B, Γ; therefore H has the parts A, B, Γ. I say then that it is also the least. For if not, there will be some number less than H which will have the parts A, B, Γ. Let it be Θ. Since Θ has the parts A, B, Γ, therefore Θ will be measured by numbers homonymous with the parts A, B, Γ. But the numbers homonymous with the parts A, B, Γ are Δ, E, Z; therefore Θ is measured by Δ, E, Z. And it is less than H; which is impossible. Therefore there will not be any number less than H which will have the parts A, B, Γ; which was to be proved. Therefore H has parts homonymous with Δ, E, Z. But the parts homonymous with Δ, E, Z are A, B, Γ; therefore H has the parts A, B, Γ. I say then that it is also the least. For if not, there will be some number less than H which will have the parts A, B, Γ. Let it be Θ. Since Θ has the parts A, B, Γ, therefore Θ will be measured by numbers homonymous with the parts A, B, Γ. But the numbers homonymous with the parts A, B, Γ are Δ, E, Z; therefore Θ is measured by Δ, E, Z. And it is less than H; which is impossible. Therefore there will not be any number less than H which will have the parts A, B, Γ; which was to be proved.