ELEMENTS BOOK 7. Elementary Number Theory. The propositions contained in Books 7 9 are generally attributed to the school of Pythagoras.

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1 LMNTS OOK 7 lementary Number Theory The propositions contained in ooks 7 9 are generally attributed to the school of Pythagoras. 193

2 ÎÇÖÓ. efinitions αʹ.μονάςἐστιν,καθ ἣνἕκαστοντῶνὄντωνἓνλέγεται. 1. unit is (that) according to which each existing βʹ. Ἀριθμὸς δὲ τὸ ἐκ μονάδων συγκείμενον πλῆθος. (thing) is said (to be) one. γʹ. Μέρος ἐστὶν ἀριθμὸς ἀριθμοῦ ὁ ἐλάσσων τοῦ 2. nd a number (is) a multitude composed of units. μείζονος, ὅταν καταμετρῇ τὸν μείζονα. 3. number is part of a(nother) number, the lesser of δʹ.μέρηδέ,ὅτανμὴκαταμετρῇ. the greater, when it measures the greater. εʹ. Πολλαπλάσιος δὲ ὁ μείζων τοῦ ἐλάσσονος, ὅταν κα- 4. ut (the lesser is) parts (of the greater) when it ταμετρῆταιὑπὸτοῦἐλάσσονος. does not measure it. ϛʹ. ρτιος ἀριθμός ἐστιν ὁ δίχα διαιρούμενος. 5. nd the greater (number is) a multiple of the lesser ζʹ. Περισσὸς δὲ ὁ μὴ διαιρούμενος δίχα ἢ[ὁ] μονάδι when it is measured by the lesser. διαφέρων ἀρτίου ἀριθμοῦ. 6. n even number is one (which can be) divided in ηʹ. Ἀρτιάκις ἄρτιος ἀριθμός ἐστιν ὁ ὑπὸ ἀρτίου ἀριθμοῦ half. μετρούμενος κατὰ ἄρτιον ἀριθμόν. 7. nd an odd number is one (which can)not (be) θʹ. ρτιάκιςδὲπερισσόςἐστινὁὑπὸἀρτίουἀριθμοῦ divided in half, or which differs from an even number by μετρούμενος κατὰ περισσὸν ἀριθμόν. a unit. ιʹ.περισσάκιςδὲπερισσὸςἀριθμόςἐστινὁὑπὸπερισσοῦ 8. n even-times-even number is one (which is) mea- ἀριθμοῦμετρούμενοςκατὰπερισσὸνἀριθμόν. sured by an even number according to an even number. ιαʹ. Πρῶτος ἀριθμός ἐστιν ὁ μονάδι μόνῃ μετρούμενος. 9. nd an even-times-odd number is one (which ιβʹ. Πρῶτοι πρὸς ἀλλήλους ἀριθμοί εἰσιν οἱ μονάδι μόνῃ is) measured by an even number according to an odd μετρούμενοικοινῷμέτρῳ. number. ιγʹ. Σύνθετος ἀριθμός ἐστιν ὁ ἀριθμῷ τινι μετρούμενος. 10. nd an odd-times-odd number is one (which ιδʹ. Σύνθετοι δὲ πρὸς ἀλλήλους ἀριθμοί εἰσιν οἱ ἀριθμῷ is) measured by an odd number according to an odd τινιμετρούμενοικοινῷμέτρῳ. number. $ ιεʹ. Ἀριθμὸς ἀριθμὸν πολλαπλασιάζειν λέγεται, ὅταν, 11. prime number is one (which is) measured by a ὅσαι εἰσὶν ἐν αὐτῷ μονάδες, τοσαυτάκις συντεθῇ ὁ πολ- unit alone. λαπλασιαζόμενος, καὶ γένηταί τις. 12. Numbers prime to one another are those (which ιϛʹ. Οτανδὲδύοἀριθμοὶπολλαπλασιάσαντεςἀλλήλους are) measured by a unit alone as a common measure. ποιῶσί τινα, ὁ γενόμενος ἐπίπεδος καλεῖται, πλευραὶ δὲ 13. composite number is one (which is) measured αὐτοῦ οἱ πολλαπλασιάσαντες ἀλλήλους ἀριθμοί. by some number. ιζʹ. Οταν δὲ τρεῖς ἀριθμοὶ πολλαπλασιάσαντες ἀλλήλους 14. nd numbers composite to one another are those ποιῶσί τινα, ὁ γενόμενος στερεός ἐστιν, πλευραὶ δὲ αὐτοῦ (which are) measured by some number as a common οἱ πολλαπλασιάσαντες ἀλλήλους ἀριθμοί. measure. ιηʹ.τετράγωνοςἀριθμόςἐστινὁἰσάκιςἴσοςἢ[ὁ]ὑπὸ 15. number is said to multiply a(nother) number δύο ἴσων ἀριθμῶν περιεχόμενος. when the (number being) multiplied is added (to itself) ιθʹ.κύβοςδὲὁἰσάκιςἴσοςἰσάκιςἢ[ὁ]ὑπὸτριῶνἴσων as many times as there are units in the former (number), ἀριθμῶνπεριεχόμενος. and (thereby) some (other number) is produced. κʹ. Ἀριθμοὶ ἀνάλογόν εἰσιν, ὅταν ὁ πρῶτος τοῦ δευτέρου 16. nd when two numbers multiplying one another καὶὁτρίτοςτοῦτετάρτουἰσάκιςᾖπολλαπλάσιοςἢτὸαὐτὸ make some (other number) then the (number so) creμέροςἢτὰαὐτὰμέρηὦσιν. ated is called plane, and its sides (are) the numbers which καʹ. Ομοιοι ἐπίπεδοι καὶ στερεοὶ ἀριθμοί εἰσιν οἱ multiply one another. ανάλογον ἔχοντες τὰς πλευράς. 17. nd when three numbers multiplying one another κβʹ. Τέλειος ἀριθμός ἐστιν ὁ τοῖς ἑαυτοῦ μέρεσιν ἴσος make some (other number) then the (number so) created ὤν. is (called) solid, and its sides (are) the numbers which multiply one another. 18. square number is an equal times an equal, or (a plane number) contained by two equal numbers. 19. nd a cube (number) is an equal times an equal times an equal, or (a solid number) contained by three equal numbers. 194

3 In other words, a number is a positive integer greater than unity. In other words, a number a is part of another number b if there exists some number n such that n a = b. 20. Numbers are proportional when the first is the same multiple, or the same part, or the same parts, of the second that the third (is) of the fourth. 21. Similar plane and solid numbers are those having proportional sides. 22. perfect number is that which is equal to its own parts. In other words, a number a is parts of another number b (where a < b) if there exist distinct numbers, m and n, such that n a = m b. In other words. an even-times-even number is the product of two even numbers. In other words, an even-times-odd number is the product of an even and an odd number. $ In other words, an odd-times-odd number is the product of two odd numbers. Literally, first. In other words, a perfect number is equal to the sum of its own factors.. Proposition 1 Δύο ἀριθμῶν ἀνίσων ἐκκειμένων, ἀνθυφαιρουμένου δὲ Two unequal numbers (being) laid down, and the ἀεὶ τοῦ ἐλάσσονος ἀπὸ τοῦ μείζονος, ἐὰν ὁ λειπόμενος lesser being continually subtracted, in turn, from the μηδέποτε καταμετρῇ τὸν πρὸ ἑαυτοῦ, ἕως οὗ λειφθῇ μονάς, greater, if the remainder never measures the (number) οἱ ἐξ ἀρχῆς ἀριθμοὶ πρῶτοι πρὸς ἀλλὴλους ἔσονται. preceding it, until a unit remains, then the original numbers will be prime to one another. Ζ Θ Η H F G Δύο γὰρ [ἀνίσων] ἀριθμῶν τῶν, Δ ἀνθυφαι- For two [unequal] numbers, and, the lesser ρουμένου ἀεὶ τοῦ ἐλάσσονος ἀπὸ τοῦ μείζονος ὁ λειπόμενος being continually subtracted, in turn, from the greater, μηδέποτε καταμετρείτω τὸν πρὸ ἑαυτοῦ, ἕως οὗ λειφθῇ let the remainder never measure the (number) preceding μονάς λέγω, ὅτι οἱ, Δ πρῶτοι πρὸς ἀλλήλους εἰσίν, it, until a unit remains. I say that and are prime τουτέστιν ὅτι τοὺς, Δ μονὰς μόνη μετρεῖ. to one another that is to say, that a unit alone measures ἰ γὰρ μή εἰσιν οἱ, Δ πρῶτοι πρὸς ἀλλήλους, (both) and. μετρήσειτιςαὐτοὺςἀριθμός.μετρείτω,καὶἔστωὁ καὶὁ For if and are not prime to one another then μὲν Δ τὸν Ζ μετρῶν λειπέτω ἑαυτοῦ ἐλάσσονα τὸν Ζ, some number will measure them. Let (some number) ὁδὲζτὸνδημετρῶνλειπέτωἑαυτοῦἐλάσσονατὸνη, measure them, and let it be. nd let measuring ὁδὲητὸνζθμετρῶνλειπέτωμονάδατὴνθ. F leave F less than itself, and let F measuring G πεὶοὖνὁτὸνδμετρεῖ,ὁδὲδτὸνζμετρεῖ, leave G less than itself, and let G measuring FH leave καὶὁἄρατὸνζμετρεῖ μετρεῖδὲκαὶὅλοντὸν a unit, H. καὶλοιπὸνἄρατὸνζμετρήσει.ὁδὲζτὸνδημετρεῖ In fact, since measures, and measures F, καὶὁἄρατὸνδημετρεῖ μετρεῖδὲκαὶὅλοντὸνδ thus also measures F. nd () also measures the καὶ λοιπὸν ἄρα τὸν Η μετρήσει. ὁ δὲ Η τὸν ΖΘ μετρεῖ whole of. Thus, () will also measure the remainder 195

4 καὶὁἄρατὸνζθμετρεῖ μετρεῖδὲκαὶὅλοντὸνζ F. nd F measures G. Thus, also measures G. καὶλοιπὴνἄρατὴνθμονάδαμετρήσειἀριθμὸςὤν ὅπερ nd () also measures the whole of. Thus, () will ἐστὶν ἀδύνατον. οὐκ ἄρα τοὺς, Δ ἀριθμοὺς μετρήσει also measure the remainder G. nd G measures F H. τιςἀριθμός οἱ,δἄραπρῶτοιπρὸςἀλλήλουςεἰσίν Thus, also measures FH. nd () also measures the ὅπερἔδειδεῖξαι. whole of F. Thus, () will also measure the remaining unit H, (despite) being a number. The very thing is impossible. Thus, some number does not measure (both) the numbers and. Thus, and are prime to one another. (Which is) the very thing it was required to show. Here, use is made of the unstated common notion that if a measures b, and b measures c, then a also measures c, where all symbols denote numbers. Here, use is made of the unstated common notion that if a measures b, and a measures part of b, then a also measures the remainder of b, where all symbols denote numbers.. Proposition 2 Δύο ἀριθμῶν δοθέντων μὴ πρώτων πρὸς ἀλλήλους τὸ To find the greatest common measure of two given μέγιστοναὐτῶνκοινὸνμέτρονεὑρεῖν. numbers (which are) not prime to one another. Ζ F Η στωσαν οἱ δοθέντες δύο ἀριθμοὶ μὴ πρῶτοι πρὸς Let and be the two given numbers (which ἀλλήλουςοἱ,δ.δεῖδὴτῶν,δτὸμέγιστονκοινὸν are) not prime to one another. So it is required to find μέτρονεὑρεῖν. the greatest common measure of and. ἰμὲνοὖνὁδτὸνμετρεῖ,μετρεῖδὲκαὶἑαυτόν,ὁ In fact, if measures, is thus a common Δ ἄρα τῶν Δ, κοινὸν μέτρον ἐστίν. καὶ φανερόν, ὅτι measure of and, (since ) also measures itself. καὶμέγιστον οὐδεὶςγὰρμείζωντοῦδτὸνδμετρήσει. nd (it is) manifest that (it is) also the greatest (com- ἰδὲοὐμετρεῖὁδτὸν,τῶν,δἀνθυφαι- mon measure). For nothing greater than can meaρουμένου ἀεὶ τοῦ ἐλάσσονος ἀπὸ τοῦ μείζονος λειφθήσεταί sure. τις ἀριθμός, ὃς μετρήσει τὸν πρὸ ἑαυτοῦ. μονὰς μὲν ut if does not measure then some number γὰροὐλειφθήσεται εἰδὲμή,ἔσονταιοἱ,δπρῶτοι will remain from and, the lesser being continπρὸς ἀλλήλους ὅπερ οὐχ ὑπόκειται. λειφθήσεταί τις ἄρα ually subtracted, in turn, from the greater, which will ἀριθμὸς,ὃςμετρήσειτὸνπρὸἑαυτοῦ. καὶὁμὲνδτὸν measure the (number) preceding it. For a unit will not be μετρῶν λειπέτω ἑαυτοῦ ἐλάσσονα τὸν, ὁ δὲ τὸν left. ut if not, and will be prime to one another ΔΖ μετρῶν λειπέτω ἑαυτοῦ ἐλάσσονα τὸν Ζ, ὁ δὲ Ζ τὸν [Prop. 7.1]. The very opposite thing was assumed. Thus, μετρείτω. ἐπεὶοὖνὁζτὸνμετρεῖ,ὁδὲτὸν some number will remain which will measure the (num- ΔΖμετρεῖ,καὶὁΖἄρατὸνΔΖμετρήσει. μετρεῖδὲκαὶ ber) preceding it. nd let measuring leave ἑαυτόν καὶὅλονἄρατὸνδμετρήσει. ὁδὲδτὸν less than itself, and let measuring F leave F less μετρεῖ καὶὁζἄρατὸνμετρεῖ μετρεῖδὲκαὶτὸν than itself, and let F measure. Therefore, since F καὶὅλονἄρατὸνμετρήσει μετρεῖδὲκαὶτὸνδ ὁζ measures, and measures F, F will thus also ἄρατοὺς,δμετρεῖ. ὁζἄρατῶν,δκοινὸν measure F. nd it also measures itself. Thus, it will G 196

5 μέτρονἐστίν.λέγωδή,ὅτικαὶμέγιστον.εἰγὰρμήἐστινὁ also measure the whole of. nd measures. Ζτῶν,Δμέγιστονκοινὸνμέτρον,μετρήσειτιςτοὺς Thus, F also measures. nd it also measures.,δἀριθμοὺςἀριθμὸςμείζωνὢντοῦζ.μετρείτω, Thus, it will also measure the whole of. nd it also καὶἔστωὁη.καὶἐπεὶὁητὸνδμετρεῖ,ὁδὲδτὸν measures. Thus, F measures (both) and. μετρεῖ, καὶ ὁ Η ἄρα τὸν μετρεῖ μετρεῖ δὲ καὶ ὅλον Thus, F is a common measure of and. So I say τὸν καὶλοιπὸνἄρατὸνμετρήσει. ὁδὲτὸν that (it is) also the greatest (common measure). For if ΔΖμετρεῖ καὶὁηἄρατὸνδζμετρήσει μετρεῖδὲκαὶ F is not the greatest common measure of and ὅλοντὸνδ καὶλοιπὸνἄρατὸνζμετρήσειὁμείζων then some number which is greater than F will meaτὸνἐλάσσονα ὅπερἐστὶνἀδύνατον οὐκἄρατοὺς,δ sure the numbers and. Let it (so) measure ( ἀριθμοὺςἀριθμόςτιςμετρήσειμείζωνὢντοῦζ ὁζἄρα and ), and let it be G. nd since G measures, τῶν, Δ μέγιστόν ἐστι κοινὸν μέτρον[ὅπερ ἔδει δεῖξαι]. and measures, G thus also measures. nd it also measures the whole of. Thus, it will also measure the remainder. nd measures F. Thus, G will also measure F. nd it also measures the whole of. Thus, it will also measure the remainder F, the greater (measuring) the lesser. The very thing is impossible. Thus, some number which is greater than F cannot measure the numbers and. Thus, F is the greatest common measure of and. [(Which is) the very thing it was required to show]. È Ö Ñ. orollary κ δὴ τούτου φανερόν, ὅτι ἐὰν ἀριθμὸς δύο ἀριθμοὺς So it is manifest, from this, that if a number measures μετρῇ, καὶ τὸ μέγιστον αὐτῶν κοινὸν μέτρον μετρήσει ὅπερ two numbers then it will also measure their greatest com- ἔδει δεῖξαι. mon measure. (Which is) the very thing it was required to show.. Proposition 3 Τριῶν ἀριθμῶν δοθέντων μὴ πρώτων πρὸς ἀλλήλους τὸ To find the greatest common measure of three given μέγιστοναὐτῶνκοινὸνμέτρονεὑρεῖν. numbers (which are) not prime to one another. Ζ F στωσαν οἱ δοθέντες τρεῖς ἀριθμοὶ μὴ πρῶτοι πρὸς Let,, and be the three given numbers (which ἀλλήλουςοἱ,, δεῖδὴτῶν,,τὸμέγιστονκοινὸν are) not prime to one another. So it is required to find μέτρονεὑρεῖν. the greatest common measure of,, and. ἰλήφθωγὰρδύοτῶν,τὸμέγιστονκοινὸνμέτρονὁ For let the greatest common measure,, of the two Δ ὁδὴδτὸνἤτοιμετρεῖἢοὐμετρεῖ.μετρείτωπρότερον (numbers) and have been taken [Prop. 7.2]. So μετρεῖδέκαὶτοὺς, ὁδἄρατοὺς,,μετρεῖ ὁ either measures, or does not measure,. First of all, let Δἄρατῶν,,κοινὸνμέτρονἐστίν. λέγωδή,ὅτικαὶ it measure (). nd it also measures and. Thus, 197

6 μέγιστον.εἰγὰρμήἐστινὁδτῶν,,μέγιστονκοινὸν measures,, and. Thus, is a common measure μέτρον,μετρήσειτιςτοὺς,,ἀριθμοὺςἀριθμὸςμείζων of,, and. So I say that (it is) also the greatest ὢν τοῦ Δ. μετρείτω, καὶ ἔστω ὁ. ἐπεὶ οὖν ὁ τοὺς,, (common measure). For if is not the greatest common μετρεῖ,καὶτοὺς,ἄραμετρήσει καὶτὸτῶν,ἄρα measure of,, and then some number greater than μέγιστονκοινὸνμέτρονμετρήσει.τὸδὲτῶν,μέγιστον will measure the numbers,, and. Let it (so) κοινὸνμέτρονἐστὶνὁδ ὁἄρατὸνδμετρεῖὁμείζων measure (,, and ), and let it be. Therefore, since τὸν ἐλάσσονα ὅπερ ἐστὶν ἀδύνατον. οὐκ ἄρα τοὺς,, measures,, and, it will thus also measure and ἀριθμοὺςἀριθμόςτιςμετρήσειμείζωνὢντοῦδ ὁδἄρα. Thus, it will also measure the greatest common meaτῶν,,μέγιστόνἐστικοινὸνμέτρον. sure of and [Prop. 7.2 corr.]. nd is the greatest ΜὴμετρείτωδὴὁΔτὸν λέγωπρῶτον,ὅτιοἱ,δ common measure of and. Thus, measures, the οὔκ εἰσι πρῶτοι πρὸς ἀλλήλους. ἐπεὶ γὰρ οἱ,, οὔκ greater (measuring) the lesser. The very thing is impossiεἰσι πρῶτοι πρὸς ἀλλήλους, μετρήσει τις αὐτοὺς ἀριθμός. ὁ ble. Thus, some number which is greater than cannot δὴ τοὺς,, μετρῶν καὶ τοὺς, μετρήσει, καὶ τὸ measure the numbers,, and. Thus, is the greatτῶν, μέγιστον κοινὸν μέτρον τὸν Δ μετρήσει μετρεῖ est common measure of,, and. δὲκαὶτὸν τοὺςδ,ἄραἀριθμοὺςἀριθμόςτιςμετρήσει So let not measure. I say, first of all, that οἱδ,ἄραοὔκεἰσιπρῶτοιπρὸςἀλλήλους.εἰλήφθωοὖν and are not prime to one another. For since,, αὐτῶν τὸ μέγιστον κοινὸν μέτρον ὁ. καὶ ἐπεὶ ὁ τὸν Δ are not prime to one another, some number will measure μετρεῖ,ὁδὲδτοὺς,μετρεῖ,καὶὁἄρατοὺς, them. So the (number) measuring,, and will also μετρεῖ μετρεῖδὲκαὶτὸν ὁἄρατοὺς,,μετρεῖ. measure and, and it will also measure the greatest ὁἄρατῶν,,κοινόνἐστιμέτρον.λέγωδή,ὅτικαὶ common measure,, of and [Prop. 7.2 corr.]. nd μέγιστον. εἰ γὰρ μή ἐστιν ὁ τῶν,, τὸ μέγιστον it also measures. Thus, some number will measure the κοινὸνμέτρον,μετρήσειτιςτοὺς,,ἀριθμοὺςἀριθμὸς numbers and. Thus, and are not prime to one μείζωνὢντοῦ.μετρείτω,καὶἔστωὁζ.καὶἐπεὶὁζτοὺς another. Therefore, let their greatest common measure,,,μετρεῖ,καὶτοὺς,μετρεῖ καὶτὸτῶν,ἄρα, have been taken [Prop. 7.2]. nd since measures μέγιστονκοινὸνμέτρονμετρήσει.τὸδὲτῶν,μέγιστον, and measures and, thus also measures κοινὸνμέτρονἐστὶνὁδ ὁζἄρατὸνδμετρεῖ μετρεῖδὲ and. nd it also measures. Thus, measures,, καὶτὸν ὁζἄρατοὺςδ,μετρεῖ καὶτὸτῶνδ,ἄρα and. Thus, is a common measure of,, and. So μέγιστον κοινὸν μέτρον μετρήσει. τὸ δὲ τῶν Δ, μέγιστον I say that (it is) also the greatest (common measure). For κοινὸνμέτρονἐστὶνὁ ὁζἄρατὸνμετρεῖὁμείζων if is not the greatest common measure of,, and τὸνἐλάσσονα ὅπερἐστὶνἀδύνατον.οὐκἄρατοὺς,, then some number greater than will measure the num- ἀριθμοὺςἀριθμόςτιςμετρήσειμείζωνὢντοῦ ὁἄρα bers,, and. Let it (so) measure (,, and ), and τῶν,,μέγιστόνἐστικοινὸνμέτρον ὅπερἔδειδεῖξαι. let it be F. nd since F measures,, and, it also measures and. Thus, it will also measure the greatest common measure of and [Prop. 7.2 corr.]. nd is the greatest common measure of and. Thus, F measures. nd it also measures. Thus, F measures and. Thus, it will also measure the greatest common measure of and [Prop. 7.2 corr.]. nd is the greatest common measure of and. Thus, F measures, the greater (measuring) the lesser. The very thing is impossible. Thus, some number which is greater than does not measure the numbers,, and. Thus, is the greatest common measure of,, and. (Which is) the very thing it was required to show.. Proposition 4 πας ἀριθμὸς παντὸς ἀριθμοῦ ὁ ἐλάσσων τοῦ μείζονος ny number is either part or parts of any (other) num- ἤτοι μέρος ἐστὶν ἢ μέρη. ber, the lesser of the greater. στωσανδύοἀριθμοὶοἱ,,καὶἔστωἐλάσσωνὁ Let and be two numbers, and let be the λέγω, ὅτι ὁ τοῦ ἤτοι μέρος ἐστὶν ἢ μέρη. lesser. I say that is either part or parts of. 198

7 Οἱ,γὰρἤτοιπρῶτοιπρὸςἀλλήλουςεἰσὶνἢοὔ. For and are either prime to one another, or not. ἔστωσανπρότερονοἱ,πρῶτοιπρὸςἀλλήλους. διαι- Let and, first of all, be prime to one another. So ρεθέντοςδὴτοῦεἰςτὰςἐναὐτῷμονάδαςἔσταιἑκάστη separating into its constituent units, each of the units μονὰς τῶν ἐν τῷ μέρος τι τοῦ ὥστε μέρη ἐστὶν ὁ in will be some part of. Hence, is parts of. τοῦ. Ζ F Μὴ ἔστωσαν δὴ οἱ, πρῶτοι πρὸς ἀλλήλους ὁ δὴ So let and be not prime to one another. So τὸνἤτοιμετρεῖἢοὐμετρεῖ. εἰμὲνοὖνὁτὸν either measures, or does not measure,. Therefore, if μετρεῖ,μέροςἐστὶνὁτοῦ.εἰδὲοὔ,εἰλήφθωτῶν measures then is part of. nd if not, let the,μέγιστονκοινὸνμέτρονὁδ,καὶδιῃρήσθωὁεἰς greatest common measure,, of and have been τοὺςτῷδἴσουςτοὺς,ζ,ζ.καὶἐπεὶὁδτὸν taken [Prop. 7.2], and let have been divided into, μετρεῖ,μέροςἐστὶνὁδτοῦ ἴσοςδὲὁδἑκάστῳτῶν F, and F, equal to. nd since measures, is, Ζ, Ζ καὶ ἕκαστος ἄρα τῶν, Ζ, Ζ τοῦ μέρος a part of. nd is equal to each of, F, and F. ἐστίν ὥστεμέρηἐστὶνὁτοῦ. Thus,, F, and F are also each part of. Hence, πας ἄρα ἀριθμὸς παντὸς ἀριθμοῦ ὁ ἐλάσσων τοῦ is parts of. μείζονοςἤτοιμέροςἐστὶνἢμέρη ὅπερἔδειδεῖξαι. Thus, any number is either part or parts of any (other) number, the lesser of the greater. (Which is) the very thing it was required to show.. Proposition 5 ὰνἀριθμὸςἀριθμοῦμέροςᾖ, καὶἕτεροςἑτέρουτὸ If a number is part of a number, and another (numαὐτὸμέροςᾖ, καὶσυναμφότεροςσυναμφοτέρουτὸαὐτὸ ber) is the same part of another, then the sum (of the μέροςἔσται,ὅπερὁεἷςτοῦἑνός. leading numbers) will also be the same part of the sum (of the following numbers) that one (number) is of another. Η Θ G H Ζ Ἀριθμὸςγὰρὁ[ἀριθμοῦ]τοῦμέροςἔστω,καὶ For let a number be part of a [number], and F 199

8 ἕτεροςὁδἑτέρουτοῦζτὸαὐτὸμέρος,ὅπερὁτοῦ another (number) (be) the same part of another (num- λέγω,ὅτικαὶσυναμφότεροςὁ,δσυναμφοτέρουτοῦ ber) F that (is) of. I say that the sum, is also, Ζ τὸ αὐτὸ μέρος ἐστίν, ὅπερ ὁ τοῦ. the same part of the sum, F that (is) of. πεὶγάρ,ὃμέροςἐστὶνὁτοῦ,τὸαὐτὸμέροςἐστὶ For since which(ever) part is of, is the same καὶὁδτοῦζ,ὅσοιἄραεἰσὶνἐντῷἀριθμοὶἴσοιτῷ part of F, thus as many numbers as are in equal,τοσοῦτοίεἰσικαὶἐντῷζἀριθμοὶἴσοιτῷδ.διῇρήσθω to, so many numbers are also in F equal to. Let ὁμὲνεἰςτοὺςτῷἴσουςτοὺςη,η,ὁδὲζεἰς have been divided into G and G, equal to, and τοὺςτῷδἴσουςτοὺςθ,θζ ἔσταιδὴἴσοντὸπλῆθος F into H and HF, equal to. So the multitude of τῶνη,ητῷπλήθειτῶνθ,θζ.καὶἐπεὶἴσοςἐστὶν (divisions) G, G will be equal to the multitude of (di- ὁμὲνητῷ,ὁδὲθτῷδ,καὶοἱη,θἄρατοῖς visions) H, HF. nd since G is equal to, and H,Δἴσοι. διὰτὰαὐτὰδὴκαὶοἱη,θζτοῖς,δ.ὅσοι to, thus G, H (is) also equal to,. So, for the ἄρα[εἰσὶν] ἐν τῷ ἀριθμοὶ ἴσοι τῷ, τοσοῦτοί εἰσι καὶ same (reasons), G, HF (is) also (equal) to,. Thus, ἐντοῖς,ζἴσοιτοῖς,δ.ὁσαπλασίωνἄραἐστὶνὁ as many numbers as [are] in equal to, so many are τοῦ,τοσαυταπλασίωνἐστὶκαὶσυναμφότεροςὁ,ζ also in, F equal to,. Thus, as many times as συναμφοτέρουτοῦ,δ.ὃἄραμέροςἐστὶνὁτοῦ,τὸ is (divisible) by, so many times is the sum, F αὐτὸ μέρος ἐστὶ καὶ συναμφότερος ὁ, Δ συναμφοτέρου also (divisible) by the sum,. Thus, which(ever) part τοῦ,ζ ὅπερἔδειδεῖξαι. is of, the sum, is also the same part of the sum, F. (Which is) the very thing it was required to show. In modern notation, this proposition states that if a = (1/n) b and c = (1/n) d then (a+c) = (1/n) (b+d), where all symbols denote numbers.. Proposition 6 ὰν ἀριθμὸς ἀριθμοῦ μέρη ᾖ, καὶ ἕτερος ἑτέρου τὰ If a number is parts of a number, and another (numαὐτὰ μέρη ᾖ, καὶ συναμφότερος συναμφοτέρου τὰ αὐτὰ ber) is the same parts of another, then the sum (of the μέρηἔσται,ὅπερὁεἷςτοῦἑνός. leading numbers) will also be the same parts of the sum (of the following numbers) that one (number) is of another. Η Θ G H Ζ Ἀριθμὸς γὰρ ὁ ἀριθμοῦ τοῦ μέρη ἔστω, καὶ ἕτερος For let a number be parts of a number, and an- ὁδἑτέρουτοῦζτὰαὐτὰμέρη,ἅπερὁτοῦ λέγω, other (number) (be) the same parts of another (num- ὅτικαὶσυναμφότεροςὁ,δσυναμφοτέρουτοῦ,ζ ber) F that (is) of. I say that the sum, is τὰαὐτὰμέρηἐστίν,ἅπερὁτοῦ. also the same parts of the sum, F that (is) of. πεὶγάρ,ἃμέρηἐστὶνὁτοῦ,τὰαὐτὰμέρηκαὶ For since which(ever) parts is of, (is) also ὁδτοῦζ,ὅσαἄραἐστὶνἐντῷμέρητοῦ,τοσαῦτά the same parts of F, thus as many parts of as are in, ἐστικαὶἐντῷδμέρητοῦζ.διῃρήσθωὁμὲνεἰςτὰ so many parts of F are also in. Let have been τοῦμέρητὰη,η,ὁδὲδεἰςτὰτοῦζμέρητὰ divided into the parts of, G and G, and into the ΔΘ,Θ ἔσταιδὴἴσοντὸπλῆθοςτῶνη,ητῷπλήθει parts of F, H and H. So the multitude of (divisions) τῶν ΔΘ,Θ.καὶἐπεί, ὃμέροςἐστὶν ὁη τοῦ,τὸ G, G will be equal to the multitude of (divisions) H, F 200

9 αὑτὸμέροςἐστὶκαὶὁδθτοῦζ,ὃἄραμέροςἐστὶνὁη H. nd since which(ever) part G is of, H is also τοῦ,τὸαὐτὸμέροςἐστὶκαὶσυναμφότεροςὁη,δθ the same part of F, thus which(ever) part G is of, συναμφοτέρουτοῦ,ζ.διὰτὰαὐτὰδὴκαὶὃμέροςἐστὶνὁ the sum G, H is also the same part of the sum, F Η τοῦ, τὸ αὐτὸ μέρος ἐστὶ καὶ συναμφότερος ὁ Η, Θ [Prop. 7.5]. nd so, for the same (reasons), which(ever) συναμφοτέρουτοῦ,ζ.ἃἄραμέρηἐστὶνὁτοῦ,τὰ part G is of, the sum G, H is also the same part αὐτὰμέρηἐστὶκαὶσυναμφότεροςὁ,δσυναμφοτέρου of the sum, F. Thus, which(ever) parts is of, τοῦ,ζ ὅπερἔδειδεῖξαι. the sum, is also the same parts of the sum, F. (Which is) the very thing it was required to show. In modern notation, this proposition states that if a = (m/n) b and c = (m/n) d then (a + c) = (m/n) (b + d), where all symbols denote numbers. Þ. Proposition 7 ὰνἀριθμὸςἀριθμοῦμέροςᾖ,ὅπερἀφαιρεθεὶςἀφαι- If a number is that part of a number that a (part) ρεθέντος,καὶὁλοιπὸςτοῦλοιποῦτὸαὐτὸμέροςἔσται, taken away (is) of a (part) taken away then the remain- ὅπερὁὅλοςτοῦὅλου. der will also be the same part of the remainder that the whole (is) of the whole. Η Ζ G F ἈριθμὸςγὰρὁἀριθμοῦτοῦΔμέροςἔστω,ὅπερ For let a number be that part of a number ἀφαιρεθεὶςὁἀφαιρεθέντοςτοῦζ λέγω,ὅτικαὶλοιπὸς that a (part) taken away (is) of a part taken away ὁ λοιποῦ τοῦ ΖΔ τὸ αὐτὸ μέρος ἐστίν, ὅπερ ὅλος ὁ F. I say that the remainder is also the same part of ὅλουτοῦδ. the remainder F that the whole (is) of the whole ΟγὰρμέροςἐστὶνὁτοῦΖ,τὸαὐτὸμέροςἔστω. καὶὁτοῦη.καὶἐπεί,ὃμέροςἐστὶνὁτοῦζ,τὸ For which(ever) part is of F, let also be the αὐτὸμέροςἐστὶκαὶὁτοῦη,ὃἄραμέροςἐστὶνὁ same part of G. nd since which(ever) part is of τοῦζ,τὸαὐτὸμέροςἐστὶκαὶὁτοῦηζ.ὃδὲμέρος F, is also the same part of G, thus which(ever) ἐστὶνὁτοῦζ,τὸαὐτὸμέροςὑπόκειταικαὶὁτοῦ part is of F, is also the same part of GF Δ ὃἄραμέροςἐστὶκαὶὁτοῦηζ,τὸαὐτὸμέροςἐστὶ [Prop. 7.5]. nd which(ever) part is of F, is καὶτοῦδ ἴσοςἄραἐστὶνὁηζτῷδ.κοινὸςἀφῃρήσθω also assumed (to be) the same part of. Thus, also, ὁζ λοιπὸςἄραὁηλοιπῷτῷζδἐστινἴσος.καὶἐπεί, which(ever) part is of GF, () is also the same ὃμέροςἐστὶνὁτοῦζ,τὸαὐτὸμέρος[ἐστὶ]καὶὁ part of. Thus, GF is equal to. Let F have been τοῦη,ἴσοςδὲὁητῷζδ,ὃἄραμέροςἐστὶνὁτοῦ subtracted from both. Thus, the remainder G is equal Ζ,τὸαὐτὸμέροςἐστὶκαὶὁτοῦΖΔ.ἀλλὰὃμέρος to the remainder F. nd since which(ever) part is ἐστὶνὁτοῦζ,τὸαὐτὸμέροςἐστὶκαὶὁτοῦδ of F, [is] also the same part of G, and G (is) καὶλοιπὸςἄραὁλοιποῦτοῦζδτὸαὐτὸμέροςἐστίν, equal to F, thus which(ever) part is of F, is ὅπερὅλοςὁὅλουτοῦδ ὅπερἔδειδεῖξαι. also the same part of F. ut, which(ever) part is of F, is also the same part of. Thus, the remainder is also the same part of the remainder F that the whole (is) of the whole. (Which is) the very thing it was required to show. In modern notation, this proposition states that if a = (1/n) b and c = (1/n) d then (a c) = (1/n) (b d), where all symbols denote numbers.. Proposition 8 ὰν ἀριθμὸς ἀριθμοῦ μέρη ᾖ, ἅπερ ἀφαιρεθεὶςἀφαι- If a number is those parts of a number that a (part) ρεθέντος, καὶὁλοιπὸςτοῦλοιποῦτὰ αὐτὰμέρη ἔσται, taken away (is) of a (part) taken away then the remain- ἅπερὁὅλοςτοῦὅλου. der will also be the same parts of the remainder that the 201

10 Ζ whole (is) of the whole. F Η Μ Κ ΝΘ G M K N H Λ L ἈριθμὸςγὰρὁἀριθμοῦτοῦΔμέρηἔστω,ἅπερ For let a number be those parts of a number ἀφαιρεθεὶςὁἀφαιρεθέντοςτοῦζ λέγω,ὅτικαὶλοιπὸς that a (part) taken away (is) of a (part) taken away ὁλοιποῦτοῦζδτὰαὐτὰμέρηἐστίν,ἅπερὅλοςὁ F. I say that the remainder is also the same parts ὅλουτοῦδ. of the remainder F that the whole (is) of the whole ΚείσθωγὰρτῷἴσοςὁΗΘ,ἃἄραμέρηἐστὶνὁΗΘ. τοῦ Δ, τὰ αὐτὰ μέρη ἐστὶ καὶ ὁ τοῦ Ζ. διῃρήσθω ὁ For let GH be laid down equal to. Thus, μὲνηθεἰςτὰτοῦδμέρητὰηκ,κθ,ὁδὲεἰςτὰτοῦ which(ever) parts GH is of, is also the same ΖμέρητὰΛ,Λ ἔσταιδὴἴσοντὸπλῆθοςτῶνηκ,κθ parts of F. Let GH have been divided into the parts τῷπλήθειτῶνλ,λ.καὶἐπεί,ὃμέροςἐστὶνὁηκτοῦ of, GK and KH, and into the part of F, L Δ,τὸαὐτὸμέροςἐστὶκαὶὁΛτοῦΖ,μείζωνδὲὁΔ and L. So the multitude of (divisions) GK, KH will be τοῦζ,μείζωνἄρακαὶὁηκτοῦλ.κείσθωτῷλἴσος equal to the multitude of (divisions) L, L. nd since ὁημ.ὃἄραμέροςἐστὶνὁηκτοῦδ,τὸαὐτὸμέροςἐστὶ which(ever) part GK is of, L is also the same part καὶὁημτοῦζ καὶλοιπὸςἄραὁμκλοιποῦτοῦζδ of F, and (is) greater than F, GK (is) thus also τὸαὐτὸμέροςἐστίν,ὅπερὅλοςὁηκὅλουτοῦδ.πάλιν greater than L. Let GM be made equal to L. Thus, ἐπεί,ὃμέροςἐστὶνὁκθτοῦδ,τὸαὐτὸμέροςἐστὶκαὶὁ which(ever) part GK is of, GM is also the same part ΛτοῦΖ,μείζωνδὲὁΔτοῦΖ,μείζωνἄρακαὶὁΘΚ of F. Thus, the remainder MK is also the same part of τοῦλ.κείσθωτῷλἴσοςὁκν.ὃἄραμέροςἐστὶνὁκθ the remainder F that the whole GK (is) of the whole τοῦδ,τὸαὐτὸμέροςἐστὶκαὶὁκντοῦζ καὶλοιπὸς [Prop. 7.5]. gain, since which(ever) part KH is of ἄραὁνθλοιποῦτοῦζδτὸαὐτὸμέροςἐστίν,ὅπερὅλοςὁ, L is also the same part of F, and (is) greater ΚΘὅλουτοῦΔ.ἐδείχθηδὲκαὶλοιπὸςὁΜΚλοιποῦτοῦ than F, HK (is) thus also greater than L. Let KN be ΖΔτὸαὐτὸμέροςὤν,ὅπερὅλοςὁΗΚὅλουτοῦΔ καὶ made equal to L. Thus, which(ever) part KH (is) of συναμφότεροςἄραὁμκ,νθτοῦδζτὰαὐτὰμέρηἐστίν,, KN is also the same part of F. Thus, the remain- ἅπερὅλοςὁθηὅλουτοῦδ.ἴσοςδὲσυναμφότεροςμὲν der NH is also the same part of the remainder F that ὁμκ,νθτῷ,ὁδὲθητῷ καὶλοιπὸςἄραὁ the whole KH (is) of the whole [Prop. 7.5]. nd the λοιποῦτοῦζδτὰαὐτὰμέρηἐστίν,ἅπερὅλοςὁὅλου remainder MK was also shown to be the same part of τοῦδ ὅπερἔδειδεῖξαι. the remainder F that the whole GK (is) of the whole. Thus, the sum MK, NH is the same parts of F that the whole HG (is) of the whole. nd the sum MK, NH (is) equal to, and HG to. Thus, the remainder is also the same parts of the remainder F that the whole (is) of the whole. (Which is) the very thing it was required to show. In modern notation, this proposition states that if a = (m/n) b and c = (m/n) d then (a c) = (m/n) (b d), where all symbols denote numbers.. Proposition 9 ὰνἀριθμὸςἀριθμοῦμέροςᾖ, καὶἕτεροςἑτέρουτὸ If a number is part of a number, and another (numαὐτὸ μέρος ᾖ, καὶ ἐναλλάξ, ὃ μέρος ἐστὶν ἢ μέρη ὁ πρῶτος ber) is the same part of another, also, alternately, τοῦ τρίτου, τὸ αὐτὸ μέρος ἔσται ἢ τὰ αὐτὰμέρη καὶ ὁ which(ever) part, or parts, the first (number) is of the δεύτεροςτοῦτετάρτου. third, the second (number) will also be the same part, or 202

11 the same parts, of the fourth. Η Θ G H Ζ Ἀριθμὸςγὰρὁἀριθμοῦτοῦμέροςἔστω,καὶἕτε- For let a number be part of a number, and anροςὁδἑτέρουτοῦζτὸαὐτὸμέρος,ὅπερὁτοῦ other (number) (be) the same part of another F that λέγω,ὅτικαὶἐναλλάξ,ὃμέροςἐστὶνὁτοῦδἢμέρη,τὸ (is) of. I say that, also, alternately, which(ever) αὐτὸμέροςἐστὶκαὶὁτοῦζἢμέρη. part, or parts, is of, is also the same part, or πεὶγὰρὃμέροςἐστὶνὁτοῦ,τὸαὐτὸμέροςἐστὶ parts, of F. καὶὁδτοῦζ,ὅσοιἄραεἰσὶνἐντῷἀριθμοὶἴσοιτῷ For since which(ever) part is of, is also the,τοσοῦτοίεἰσικαὶἐντῷζἴσοιτῷδ.διῃρήσθωὁμὲν same part of F, thus as many numbers as are in εἰςτοὺςτῷἴσουςτοὺςη,η,ὁδὲζεἰςτοὺςτῷ equal to, so many are also in F equal to. Let Δ ἴσους τοὺς Θ, ΘΖ ἔσται δὴ ἴσον τὸ πλῆθος τῶν Η, have been divided into G and G, equal to, and F ΗτῷπλήθειτῶνΘ,ΘΖ. into H and HF, equal to. So the multitude of (di- ΚαὶἐπεὶἴσοιεἰσὶνοἱΗ,Ηἀριθμοὶἀλλήλοις,εἰσὶ visions) G, G will be equal to the multitude of (diviδὲκαὶοἱθ,θζἀριθμοὶἴσοιἀλλήλοις,καίἐστινἴσοντὸ sions) H, HF. πλῆθοςτῶνη,ητῷπλήθειτῶνθ,θζ,ὃἄραμέρος nd since the numbers G and G are equal to one ἐστὶνὁητοῦθἢμέρη,τὸαὐτὸμέροςἐστὶκαὶὁη another, and the numbers H and HF are also equal to τοῦθζἢτὰαὐτὰμέρη ὥστεκαὶὃμέροςἐστὶνὁητοῦ one another, and the multitude of (divisions) G, G Θἢμέρη,τὸαὐτὸμέροςἐστὶκαὶσυναμφότεροςὁ is equal to the multitude of (divisions) H, H, thus συναμφοτέρουτοῦζἢτὰαὐτὰμέρη. ἴσοςδὲὁμὲνη which(ever) part, or parts, G is of H, G is also τῷ,ὁδὲθτῷδ ὃἄραμέροςἐστὶνὁτοῦδἢμέρη, the same part, or the same parts, of HF. nd hence, τὸαὐτὸμέροςἐστὶκαὶὁτοῦζἢτὰαὐτὰμέρη ὅπερ which(ever) part, or parts, G is of H, the sum ἔδειδεῖξαι. is also the same part, or the same parts, of the sum F [Props. 7.5, 7.6]. nd G (is) equal to, and H to. Thus, which(ever) part, or parts, is of, is also the same part, or the same parts, of F. (Which is) the very thing it was required to show. In modern notation, this proposition states that if a = (1/n) b and c = (1/n) d then if a = (k/l) c then b = (k/l) d, where all symbols denote numbers. F. Proposition 10 ὰνἀριθμὸςἀριθμοῦμέρηᾖ,καὶἕτεροςἑτέρουτὰαὐτὰ If a number is parts of a number, and another (numμέρη ᾖ, καὶ ἐναλλάξ, ἃ μέρη ἐστὶν ὁ πρῶτος τοῦ τρίτου ἢ ber) is the same parts of another, also, alternately, μέρος,τὰαὐτὰμέρηἔσταικαὶὁδεύτεροςτοῦτετάρτουἢ which(ever) parts, or part, the first (number) is of the τὸαὐτὸμέρος. third, the second will also be the same parts, or the same Ἀριθμὸς γὰρ ὁ ἀριθμοῦ τοῦ μέρη ἔστω, καὶ ἕτερος part, of the fourth. ὁδἑτέρουτοῦζτὰαὐτὰμέρη λέγω,ὅτικαὶἐναλλάξ, For let a number be parts of a number, and ἃ μέρη ἐστὶν ὁ τοῦ Δ ἢ μέρος, τὰ αὐτὰ μέρη ἐστὶ καὶ another (number) (be) the same parts of another F. ὁτοῦζἢτὸαὐτὸμέρος. I say that, also, alternately, which(ever) parts, or part, 203

12 is of, is also the same parts, or the same part, of F. Η Θ G H Ζ πεὶγάρ,ἃμέρηἐστὶνὁτοῦ,τὰαὐτὰμέρηἐστὶ For since which(ever) parts is of, is also καὶὁδτοῦζ,ὅσαἄραἐστὶνἐντῷμέρητοῦ, the same parts of F, thus as many parts of as are in τοσαῦτακαὶἐντῷδμέρητοῦζ.διῃρήσθωὁμὲνεἰς, so many parts of F (are) also in. Let have τὰτοῦμέρητὰη,η,ὁδὲδεἰςτὰτοῦζμέρητὰ been divided into the parts of, G and G, and ΔΘ,Θ ἔσταιδὴἴσοντὸπλῆθοςτῶνη,ητῷπλήθει into the parts of F, H and H. So the multitude of τῶν ΔΘ, Θ. καὶ ἐπεί, ὃ μέρος ἐστὶν ὁ Η τοῦ, τὸ αὐτὸ (divisions) G, G will be equal to the multitude of (diμέροςἐστὶκαὶὁδθτοῦζ,καὶἐναλλάξ,ὃμέροςἐστὶνὁ visions) H, H. nd since which(ever) part G is ΗτοῦΔΘἢμέρη,τὸαὐτὸμέροςἐστὶκαὶὁτοῦΖἢ of, H is also the same part of F, also, alternately, τὰαὐτὰμέρη.διὰτὰαὐτὰδὴκαί,ὃμέροςἐστὶνὁητοῦ which(ever) part, or parts, G is of H, is also the Θἢμέρη,τὸαὐτὸμέροςἐστὶκαὶὁτοῦΖἢτὰαὐτὰ same part, or the same parts, of F [Prop. 7.9]. nd so, μέρη ὥστε καί[ὃ μέρος ἐστὶν ὁ Η τοῦ ΔΘ ἢ μέρη, τὸ for the same (reasons), which(ever) part, or parts, G is αὐτὸμέροςἐστὶκαὶὁητοῦθἢτὰαὐτὰμέρη καὶὃ of H, is also the same part, or the same parts, of F ἄραμέροςἐστὶνὁητοῦδθἢμέρη,τὸαὐτὸμέροςἐστὶ [Prop. 7.9]. nd so [which(ever) part, or parts, G is of καὶὁτοῦδἢτὰαὐτὰμέρη ἀλλ ὃμέροςἐστὶνὁη H, G is also the same part, or the same parts, of H. τοῦδθἢμέρη,τὸαὐτὸμέροςἐδείχθηκαὶὁτοῦζἢτὰ nd thus, which(ever) part, or parts, G is of H, is αὐτὰ μέρη, καὶ] ἃ[ἄρα] μέρη ἐστὶν ὁ τοῦ Δ ἢ μέρος, also the same part, or the same parts, of [Props. 7.5, τὰαὐτὰμέρηἐστὶκαὶὁτοῦζἢτὸαὐτὸμέρος ὅπερἔδει 7.6]. ut, which(ever) part, or parts, G is of H, δεῖξαι. was also shown (to be) the same part, or the same parts, of F. nd, thus] which(ever) parts, or part, is of, is also the same parts, or the same part, of F. (Which is) the very thing it was required to show. In modern notation, this proposition states that if a = (m/n) b and c = (m/n) d then if a = (k/l) c then b = (k/l) d, where all symbols denote numbers.. Proposition 11 ανᾖὡςὅλοςπρὸςὅλον,οὕτωςἀφαιρεθεὶςπρὸςἀφαι- If as the whole (of a number) is to the whole (of anρεθέντα,καὶὁλοιπὸςπρὸςτὸνλοιπὸνἔσται,ὡςὅλοςπρὸς other), so a (part) taken away (is) to a (part) taken away, ὅλον. then the remainder will also be to the remainder as the στω ὡς ὅλος ὁ πρὸς ὅλον τὸν Δ, οὕτως ἀφαι- whole (is) to the whole. ρεθεὶςὁπρὸςἀφαιρεθέντατὸνζ λέγω,ὅτικαὶλοιπὸς Let the whole be to the whole as the (part) ὁπρὸςλοιπὸντὸνζδἐστιν,ὡςὅλοςὁπρὸςὅλον taken away (is) to the (part) taken away F. I say τὸνδ. that the remainder is to the remainder F as the whole (is) to the whole. F 204

13 Ζ F πείἐστινὡςὁπρὸςτὸνδ,οὕτωςὁπρὸς (For) since as is to, so (is) to F, thus τὸνζ,ὃἄραμέροςἐστὶνὁτοῦδἢμέρη,τὸαὐτὸ which(ever) part, or parts, is of, is also the μέροςἐστὶκαὶὁτοῦζἢτὰαὐτὰμέρη. καὶλοιπὸς same part, or the same parts, of F [ef. 7.20]. Thus, ἄραὁλοιποῦτοῦζδτὸαὐτὸμέροςἐστὶνἢμέρη,ἅπερ the remainder is also the same part, or parts, of the ὁτοῦδ.ἔστινἄραὡςὁπρὸςτὸνζδ,οὕτωςὁ remainder F that (is) of [Props. 7.7, 7.8]. πρὸςτὸνδ ὅπερἔδειδεῖξαι. Thus, as is to F, so (is) to [ef. 7.20]. (Which is) the very thing it was required to show. In modern notation, this proposition states that if a : b :: c : d then a : b :: a c : b d, where all symbols denote numbers.. Proposition 12 ὰν ὦσιν ὁποσοιοῦν ἀριθμοὶ ἀνάλογον, ἔσται ὡς εἷς If any multitude whatsoever of numbers are proporτῶνἡγουμένωνπρὸςἕνατῶνἑπομένων,οὕτωςἅπαντεςοἱ tional then as one of the leading (numbers is) to one of ἡγούμενοι πρὸς ἅπαντας τοὺς ἑπομένους. the following so (the sum of) all of the leading (numbers) will be to (the sum of) all of the following. στωσαν ὁποσοιοῦν ἀριθμοὶ ἀνάλογον οἱ,,, Δ, Let any multitude whatsoever of numbers,,,, ὡςὁπρὸςτὸν,οὕτωςὁπρὸςτὸνδ λέγω,ὅτιἐστὶν, be proportional, (such that) as (is) to, so (is) ὡςὁπρὸςτὸν,οὕτωςοἱ,πρὸςτοὺς,δ. to. I say that as is to, so, (is) to,. πεὶγάρἐστινὡςὁπρὸςτὸν,οὕτωςὁπρὸς For since as is to, so (is) to, thus which(ever) τὸνδ,ὃἄραμέροςἐστὶνὁτοῦἢμέρη,τὸαὐτὸμέρος part, or parts, is of, is also the same part, or parts, ἐστὶκαὶὁτοῦδἢμέρη. καὶσυναμφότεροςἄραὁ, of [ef. 7.20]. Thus, the sum, is also the same συναμφοτέρουτοῦ,δτὸαὐτὸμέροςἐστὶνἢτὰαὐτὰ part, or the same parts, of the sum, that (is) of μέρη,ἅπερὁτοῦ.ἔστινἄραὡςὁπρὸςτὸν,οὕτως [Props. 7.5, 7.6]. Thus, as is to, so, (is) to, οἱ,πρὸςτοὺς,δ ὅπερἔδειδεῖξαι. [ef. 7.20]. (Which is) the very thing it was required to show. 205

14 In modern notation, this proposition states that if a : b :: c : d then a : b :: a + c : b + d, where all symbols denote numbers.. Proposition 13 ὰν τέσσαρες ἀριθμοὶ ἀνάλογον ὦσιν, καὶ ἐναλλὰξ If four numbers are proportional then they will also ἀνάλογον ἔσονται. be proportional alternately. στωσαντέσσαρεςἀριθμοὶἀνάλογονοἱ,,,δ, Let the four numbers,,, and be proportional, ὡςὁπρὸςτὸν,οὕτωςὁπρὸςτὸνδ λέγω,ὅτικαὶ (such that) as (is) to, so (is) to. I say that they ἐναλλὰξ ἀνάλογον ἔσονται, ὡς ὁ πρὸς τὸν, οὕτως ὁ will also be proportional alternately, (such that) as (is) πρὸςτὸνδ. to, so (is) to. πεὶγάρἐστινὡςὁπρὸςτὸν,οὕτωςὁπρὸςτὸν For since as is to, so (is) to, thus which(ever) Δ, ὃ ἄρα μέρος ἐστὶν ὁ τοῦ ἢ μέρη, τὸ αὐτὸ μέρος ἐστὶ part, or parts, is of, is also the same part, καὶὁτοῦδἢτὰαὐτὰμέρη.ἐναλλὰξἄρα,ὃμέροςἐστὶν or the same parts, of [ef. 7.20]. Thus, alterately, ὁτοῦἢμέρη,τὸαὐτὸμέροςἐστὶκαὶὁτοῦδἢτὰ which(ever) part, or parts, is of, is also the same αὐτὰ μέρη. ἔστιν ἄρα ὡς ὁ πρὸς τὸν, οὕτως ὁ πρὸς part, or the same parts, of [Props. 7.9, 7.10]. Thus, as τὸνδ ὅπερἔδειδεῖξαι. is to, so (is) to [ef. 7.20]. (Which is) the very thing it was required to show. In modern notation, this proposition states that if a : b :: c : d then a : c :: b : d, where all symbols denote numbers.. Proposition 14 ὰν ὦσιν ὁποσοιοῦν ἀριθμοὶ καὶ ἄλλοι αὐτοῖς ἴσοι τὸ If there are any multitude of numbers whatsoever, πλῆθος σύνδυο λαμβανόμενοι καὶ ἐν τῷ αὐτῷ λόγῳ, καὶ δι and (some) other (numbers) of equal multitude to them, ἴσουἐντῷαὐτῷλόγῷἔσονται. (which are) also in the same ratio taken two by two, then they will also be in the same ratio via equality. Ζ στωσαν ὁποσοιοῦν ἀριθμοὶ οἱ,, καὶ ἄλλοι αὐτοῖς Let there be any multitude of numbers whatsoever,, ἴσοιτὸπλῆθοςσύνδυολαμβανόμενοιἐντῷαὐτῷλόγῳοἱ,, and (some) other (numbers),,, F, of equal Δ,,Ζ,ὡςμὲνὁπρὸςτὸν,οὕτωςὁΔπρὸςτὸν, multitude to them, (which are) in the same ratio taken ὡςδὲὁπρὸςτὸν,οὕτωςὁπρὸςτὸνζ λέγω,ὅτι two by two, (such that) as (is) to, so (is) to, καὶδι ἴσουἐστὶνὡςὁπρὸςτὸν,οὕτωςὁδπρὸςτὸν and as (is) to, so (is) to F. I say that also, via Ζ. equality, as is to, so (is) to F. πεὶγάρἐστινὡςὁπρὸςτὸν,οὕτωςὁδπρὸς For since as is to, so (is) to, thus, alternately, τὸν,ἐναλλὰξἄραἐστὶνὡςὁπρὸςτὸνδ,οὕτωςὁ as is to, so (is) to [Prop. 7.13]. gain, since πρὸςτὸν.πάλιν,ἐπείἐστινὡςὁπρὸςτὸν,οὕτωςὁ as is to, so (is) to F, thus, alternately, as is F 206

15 πρὸςτὸνζ,ἐναλλὰξἄραἐστὶνὡςὁπρὸςτὸν,οὕτως to, so (is) to F [Prop. 7.13]. nd as (is) to, ὁπρὸςτὸνζ.ὡςδὲὁπρὸςτὸν,οὕτωςὁπρὸς so (is) to. Thus, also, as (is) to, so (is) to F. τὸνδ καὶὡςἄραὁπρὸςτὸνδ,οὕτωςὁπρὸςτὸν Thus, alternately, as is to, so (is) to F [Prop. 7.13]. Ζ ἐναλλὰξ ἄρα ἐστὶν ὡς ὁ πρὸς τὸν, οὕτως ὁ Δ πρὸς (Which is) the very thing it was required to show. τὸνζ ὅπερἔδειδεῖξαι. In modern notation, this proposition states that if a : b :: d : e and b : c :: e : f then a : c :: d : f, where all symbols denote numbers.. Proposition 15 ὰν μονὰς ἀριθμόν τινα μετρῇ, ἰσακις δὲ ἕτερος ἀριθμὸς If a unit measures some number, and another num- ἄλλον τινὰ ἀριθμὸν μετρῇ, καὶ ἐναλλὰξ ἰσάκις ἡ μονὰς τὸν ber measures some other number as many times, then, τρίτον ἀριθμὸν μετρήσει καὶ ὁ δεύτερος τὸν τέταρτον. also, alternately, the unit will measure the third number as many times as the second (number measures) the fourth. Η Θ Κ Λ Ζ K L F Μονὰςγὰρἡἀριθμόντινατὸνμετρείτω,ἰσάκιςδὲ For let a unit measure some number, and let ἕτεροςἀριθμὸςὁδἄλλοντινὰἀριθμὸντὸνζμετρείτω another number measure some other number F as λέγω, ὅτικαὶἐναλλὰξἰσάκιςἡμονὰςτὸνδἀριθμὸν many times. I say that, also, alternately, the unit also μετρεῖκαὶὁτὸνζ. measures the number as many times as (measures) πεὶγὰρἰσάκιςἡμονὰςτὸνἀριθμὸνμετρεῖκαὶὁ F. ΔτὸνΖ,ὅσαιἄραεἰσὶνἐντῷμονάδες,τοσοῦτοίεἰσι For since the unit measures the number as καὶἐντῷζἀριθμοὶἴσοιτῷδ.διῃρήσθωὁμὲνεἰςτὰς many times as (measures) F, thus as many units as ἐνἑαυτῷμονάδαςτὰςη,ηθ,θ,ὁδὲζεἰςτοὺςτῷδ are in, so many numbers are also in F equal to ἴσουςτοὺςκ,κλ,λζ.ἔσταιδὴἴσοντὸπλῆθοςτῶνη,. Let have been divided into its constituent units, ΗΘ,ΘτῷπλήθειτῶνΚ,ΚΛ,ΛΖ.καὶἐπεὶἴσαιεἰσὶναἱ G, GH, and H, and F into the (divisions) K, KL, Η,ΗΘ,Θμονάδεςἀλλήλαις,εἰσὶδὲκαὶοἱΚ,ΚΛ,ΛΖ and LF, equal to. So the multitude of (units) G, ἀριθμοὶἴσοιἀλλήλοις,καίἐστινἴσοντὸπλῆθοςτῶνη, GH, H will be equal to the multitude of (divisions) ΗΘ,ΘμονάδωντῷπλήθειτῶνΚ,ΚΛ,ΛΖἀριθμῶν, K, KL, LF. nd since the units G, GH, and H ἔσταιἄραὡςἡημονὰςπρὸςτὸνκἀριθμόν,οὕτωςἡ are equal to one another, and the numbers K, KL, and ΗΘμονὰςπρὸςτὸνΚΛἀριθμὸνκαὶἡΘμονὰςπρὸςτὸν LF are also equal to one another, and the multitude of ΛΖἀριθμόν.ἔσταιἄρακαὶὡςεἷςτῶνἡγουμένωνπρὸςἕνα the (units) G, GH, H is equal to the multitude of the τῶνἑπομένων,οὕτωςἅπαντεςοἱἡγούμενοιπρὸςἅπαντας numbers K, KL, LF, thus as the unit G (is) to the τοὺςἑπομένους ἔστινἄραὡςἡημονὰςπρὸςτὸνκ number K, so the unit GH will be to the number KL, ἀριθμόν,οὕτωςὁπρὸςτὸνζ.ἴσηδὲἡημονὰςτῇ and the unit H to the number LF. nd thus, as one of μονάδι,ὁδὲκἀριθμὸςτῷδἀριθμῷ. ἔστινἄραὡςἡ the leading (numbers is) to one of the following, so (the μονὰςπρὸςτὸνδἀριθμόν,οὕτωςὁπρὸςτὸνζ. sum of) all of the leading will be to (the sum of) all of ἰσάκιςἄραἡμονὰςτὸνδἀριθμὸνμετρεῖκαὶὁτὸν the following [Prop. 7.12]. Thus, as the unit G (is) to Ζ ὅπερἔδειδεῖξαι. This proposition is a special case of Prop G the number K, so (is) to F. nd the unit G (is) equal to the unit, and the number K to the number. Thus, as the unit is to the number, so (is) to F. Thus, the unit measures the number as many times as (measures) F [ef. 7.20]. (Which is) the very thing it was required to show. H 207

16 . Proposition 16 ὰν δύο ἀριθμοὶ πολλαπλασιάσαντες ἀλλήλους ποιῶσί If two numbers multiplying one another make some τινας, οἱ γενόμενοι ἐξ αὐτῶν ἴσοι ἀλλήλοις ἔσονται. (numbers) then the (numbers) generated from them will be equal to one another. στωσαν δύο ἀριθμοὶ οἱ,, καὶ ὁ μὲν τὸν πολ- Let and be two numbers. nd let make (by) λαπλασιάσαςτὸνποιείτω,ὁδὲτὸνπολλαπλασιάσας multiplying, and let make (by) multiplying. I τὸν Δ ποιείτω λέγω, ὅτι ἴσος ἐστὶν ὁ τῷ Δ. say that is equal to. πεὶγὰρὁτὸνπολλαπλασιάσαςτὸνπεποίηκεν, For since has made (by) multiplying, thus ὁἄρατὸνμετρεῖκατὰτὰςἐντῷμονάδας.μετρεῖδὲ measures according to the units in [ef. 7.15]. nd καὶἡμονὰςτὸνἀριθμὸνκατὰτὰςἐναὐτῷμονάδας the unit also measures the number according to the ἰσάκιςἄραἡμονὰςτὸνἀριθμὸνμετρεῖκαὶὁτὸν units in it. Thus, the unit measures the number as.ἐναλλὰξἄραἰσάκιςἡμονὰςτὸνἀριθμὸνμετρεῖκαὶ many times as (measures). Thus, alternately, the ὁτὸν.πάλιν,ἐπεὶὁτὸνπολλαπλασιάσαςτὸνδ unit measures the number as many times as (meaπεποίηκεν,ὁἄρατὸνδμετρεῖκατὰτὰςἐντῷμονάδας. sures) [Prop. 7.15]. gain, since has made (by) μετρεῖ δὲ καὶ ἡ μονὰς τὸν κατὰ τὰς ἐν αὐτῷ μονάδας multiplying, thus measures according to the units ἰσάκιςἄραἡμονὰςτὸνἀριθμὸνμετρεῖκαὶὁτὸνδ. in [ef. 7.15]. nd the unit also measures ac- ἰσάκιςδὲἡμονὰςτὸνἀριθμὸνἐμέτρεικαὶὁτὸν cording to the units in it. Thus, the unit measures the ἰσάκιςἄραὁἑκάτεροντῶν,δμετρεῖ. ἴσοςἄραἐστὶν number as many times as (measures). nd the ὁτῷδ ὅπερἔδειδεῖξαι. unit was measuring the number as many times as (measures). Thus, measures each of and an equal number of times. Thus, is equal to. (Which is) the very thing it was required to show. In modern notation, this proposition states that a b = b a, where all symbols denote numbers. Þ. Proposition 17 ὰν ἀριθμὸς δύο ἀριθμοὺς πολλαπλασιάσας ποιῇ τινας, If a number multiplying two numbers makes some οἱ γενόμενοι ἐξ αὐτῶν τὸν αὐτὸν ἕξουσι λόγον τοῖς πολλα- (numbers) then the (numbers) generated from them will πλασιασθεῖσιν. have the same ratio as the multiplied (numbers). Ζ Ἀριθμὸς γὰρ ὁ δύο ἀριθμοὺς τοὺς, πολλα- For let the number make (the numbers) and πλασιάσαςτοὺςδ,ποιείτω λέγω,ὅτιἐστὶνὡςὁπρὸς (by) multiplying the two numbers and (respecτὸν,οὕτωςὁδπρὸςτὸν. tively). I say that as is to, so (is) to. πεὶγὰρὁτὸνπολλαπλασιάσαςτὸνδπεποίηκεν, For since has made (by) multiplying, thus ὁἄρατὸνδμετρεῖκατὰτὰςἐντῷμονάδας. μετρεῖ measures according to the units in [ef. 7.15]. nd F 208

17 δὲκαὶἡζμονὰςτὸνἀριθμὸνκατὰτὰςἐναὐτῷμονάδας the unit F also measures the number according to the ἰσάκιςἄραἡζμονὰςτὸνἀριθμὸνμετρεῖκαὶὁτὸνδ. units in it. Thus, the unit F measures the number as ἔστινἄραὡςἡζμονὰςπρὸςτὸνἀριθμόν,οὕτωςὁ many times as (measures). Thus, as the unit F is πρὸςτὸνδ.διὰτὰαὐτὰδὴκαὶὡςἡζμονὰςπρὸςτὸν to the number, so (is) to [ef. 7.20]. nd so, for ἀριθμόν,οὕτωςὁπρὸςτὸν καὶὡςἄραὁπρὸςτὸν the same (reasons), as the unit F (is) to the number, Δ,οὕτωςὁπρὸςτὸν.ἐναλλὰξἄραἐστὶνὡςὁπρὸς so (is) to. nd thus, as (is) to, so (is) to. τὸν,οὕτωςὁδπρὸςτὸν ὅπερἔδειδεῖξαι. Thus, alternately, as is to, so (is) to [Prop. 7.13]. (Which is) the very thing it was required to show. In modern notation, this proposition states that if d = a b and e = a c then d : e :: b : c, where all symbols denote numbers.. Proposition 18 ὰν δύο ἀριθμοὶ ἀριθμόν τινα πολλαπλασιάσαντες If two numbers multiplying some number make some ποιῶσί τινας, οἱ γενόμενοι ἐξ αὐτῶν τὸν αὐτὸν ἕξουσι λόγον (other numbers) then the (numbers) generated from τοῖς πολλαπλασιάσασιν. them will have the same ratio as the multiplying (numbers). Δύογὰρἀριθμοὶοἱ,ἀριθμόντινατὸνπολλα- For let the two numbers and make (the numbers) πλασιάσαντες τοὺς Δ, ποιείτωσαν λέγω, ὅτι ἐστὶν ὡς ὁ and (respectively, by) multiplying some number. πρὸςτὸν,οὕτωςὁδπρὸςτὸν. I say that as is to, so (is) to. πεὶγὰρὁτὸνπολλαπλασιάσαςτὸνδπεποίηκεν, For since has made (by) multiplying, has καὶὁἄρατὸνπολλαπλασιάσαςτὸνδπεποίηκεν. διὰ thus also made (by) multiplying [Prop. 7.16]. So, for τὰαὐτὰδὴκαὶὁτὸνπολλαπλασιάσαςτὸνπεποίηκεν. the same (reasons), has also made (by) multiplying ἀριθμὸς δὴ ὁ δύο ἀριθμοὺς τοὺς, πολλαπλασιάσας. So the number has made and (by) multiplying τοὺςδ,πεποίηκεν.ἔστινἄραὡςὁπρὸςτὸν,οὕτως the two numbers and (respectively). Thus, as is to ὁ Δ πρὸς τὸν ὅπερ ἔδει δεῖξαι., so (is) to [Prop. 7.17]. (Which is) the very thing it was required to show. In modern notation, this propositions states that if a c = d and b c = e then a : b :: d : e, where all symbols denote numbers.. Proposition 19 ὰν τέσσαρες ἀριθμοὶ ἀνάλογον ὦσιν, ὁ ἐκ πρώτου καὶ If four number are proportional then the number creτετάρτουγενόμενοςἀριθμὸςἴσοςἔσταιτῷἐκδευτέρουκαὶ ated from (multiplying) the first and fourth will be equal τρίτου γενομένῳ ἀριθμῷ καὶ ἐὰν ὁ ἐκ πρώτου καὶ τετάρτου to the number created from (multiplying) the second and γενόμενος ἀριθμὸς ἴσος ᾖ τῷ ἐκ δευτέρου καὶ τρίτου, οἱ third. nd if the number created from (multiplying) the τέσσασρες ἀριθμοὶ ἀνάλογον ἔσονται. first and fourth is equal to the (number created) from στωσαν τέσσαρες ἀριθμοὶ ἀνάλογον οἱ,,, Δ, (multiplying) the second and third then the four num- ὡςὁπρὸςτὸν,οὕτωςὁπρὸςτὸνδ,καὶὁμὲν bers will be proportional. τὸνδπολλαπλασιάσαςτὸνποιείτω,ὁδὲτὸνπολ- Let,,, and be four proportional numbers, λαπλασιάσαςτὸνζποιείτω λέγω,ὅτιἴσοςἐστὶνὁτῷζ. (such that) as (is) to, so (is) to. nd let make (by) multiplying, and let make F (by) multiplying. I say that is equal to F. 209

18 Ζ Η F G ΟγὰρτὸνπολλαπλασιάσαςτὸνΗποιείτω. ἐπεὶ For let make G (by) multiplying. Therefore, since οὖνὁτὸνπολλαπλασιάσαςτὸνηπεποίηκεν,τὸνδὲ has made G (by) multiplying, and has made (by) Δ πολλαπλασιάσας τὸν πεποίηκεν, ἀριθμὸς δὴ ὁ δύο multiplying, the number has made G and by mul- ἀριθμοὺς τοὺς, Δ πολλαπλασιάσας τούς Η, πεποίηκεν. tiplying the two numbers and (respectively). Thus, ἔστινἄραὡςὁπρὸςτὸνδ,οὕτωςὁηπρὸςτὸν.ἀλλ as is to, so G (is) to [Prop. 7.17]. ut, as (is) to ὡςὁπρὸςτὸνδ,οὕτωςὁπρὸςτὸν καὶὡςἄρα, so (is) to. Thus, also, as (is) to, so G (is) to ὁπρὸςτὸν,οὕτωςὁηπρὸςτὸν.πάλιν,ἐπεὶὁ. gain, since has made G (by) multiplying, but, τὸνπολλαπλασιάσαςτὸνηπεποίηκεν,ἀλλὰμὴνκαὶὁ in fact, has also made F (by) multiplying, the two τὸν πολλαπλασιάσας τὸν Ζ πεποίηκεν, δύο δὴ ἀριθμοὶ numbers and have made G and F (respectively, by) οἱ,ἀριθμόντινατὸνπολλαπλασιάσαντεςτοὺςη,ζ multiplying some number. Thus, as is to, so G (is) πεποιήκασιν.ἔστινἄραὡςὁπρὸςτὸν,οὕτωςὁηπρὸς to F [Prop. 7.18]. ut, also, as (is) to, so G (is) to τὸνζ.ἀλλὰμὴνκαὶὡςὁπρὸςτὸν,οὕτωςὁηπρὸς. nd thus, as G (is) to, so G (is) to F. Thus, G has τὸν καὶὡςἄραὁηπρὸςτὸν,οὕτωςὁηπρὸςτὸν the same ratio to each of and F. Thus, is equal to F Ζ.ὁΗἄραπρὸςἑκάτεροντῶν,Ζτὸναὐτὸνἔχειλόγον [Prop. 5.9]. ἴσοςἄραἐστὶνὁτῷζ. So, again, let be equal to F. I say that as is to, στωδὴπάλινἴσοςὁτῷζ λέγω,ὅτιἐστὶνὡςὁ so (is) to. πρὸςτὸν,οὕτωςὁπρὸςτὸνδ. For, with the same construction, since is equal to F, Τῶνγὰραὐτῶνκατασκευασθέντων,ἐπεὶἴσοςἐστὶνὁ thus as G is to, so G (is) to F [Prop. 5.7]. ut, as G τῷζ,ἔστινἄραὡςὁηπρὸςτὸν,οὕτωςὁηπρὸςτὸν (is) to, so (is) to [Prop. 7.17]. nd as G (is) to F, Ζ.ἀλλ ὡςμὲνὁηπρὸςτὸν,οὕτωςὁπρὸςτὸνδ,ὡς so (is) to [Prop. 7.18]. nd, thus, as (is) to, so δὲὁηπρὸςτὸνζ,οὕτωςὁπρὸςτὸν.καὶὡςἄραὁ (is) to. (Which is) the very thing it was required to πρὸςτὸν,οὕτωςὁπρὸςτὸνδ ὅπερἔδειδεῖξαι. show. In modern notation, this proposition reads that if a : b :: c : d then a d = bc, and vice versa, where all symbols denote numbers.. Proposition 20 Οἱ ἐλάχιστοι ἀριθμοὶ τῶν τὸν αὐτὸν λόγον ἐχόντων The least numbers of those (numbers) having the αὐτοῖς μετροῦσι τοὺς τὸν αὐτὸν λόγον ἔχοντας ἰσάκις ὅ same ratio measure those (numbers) having the same raτεμείζωντὸνμείζονακαὶὁἐλάσσωντὸνἐλάσσονα. tio as them an equal number of times, the greater (mea- στωσαν γὰρ ἐλάχιστοι ἀριθμοὶ τῶν τὸν αὐτὸν λόγον suring) the greater, and the lesser the lesser. ἐχόντωντοῖς,οἱδ,ζ λέγω,ὅτιἰσάκιςὁδτὸν For let and F be the least numbers having the μετρεῖκαὶὁζτὸν. same ratio as and (respectively). I say that measures the same number of times as F (measures). 210

19 Η Θ Ζ G H F ΟΔγὰρτοῦοὔκἐστιμέρη.εἰγὰρδυνατόν,ἔστω For is not parts of. For, if possible, let it be καὶὁζἄρατοῦτὰαὐτὰμέρηἐστίν,ἅπερὁδτοῦ (parts of ). Thus, F is also the same parts of that.ὅσαἄραἐστὶνἐντῷδμέρητοῦ,τοσαῦτάἐστικαὶ (is) of [ef. 7.20, Prop. 7.13]. Thus, as many parts ἐντῷζμέρητοῦ.διῃρήσθωὁμὲνδεἰςτὰτοῦ of as are in, so many parts of are also in F. Let μέρητὰη,ηδ,ὁδὲζεἰςτὰτοῦμέρητὰθ,θζ have been divided into the parts of, G and G, ἔσταιδὴἴσοντὸπλῆθοςτῶνη,ηδτῷπλήθειτῶνθ, and F into the parts of, H and HF. So the multi- ΘΖ.καὶἐπεὶἴσοιεἰσὶνοἱΗ,ΗΔἀριθμοὶἀλλήλοις,εἰσὶ tude of (divisions) G, G will be equal to the multitude δὲκαὶοἱθ,θζἀριθμοὶἴσοιἀλλήλοις,καίἐστινἴσοντὸ of (divisions) H, HF. nd since the numbers G and πλῆθοςτῶνη,ηδτῷπλήθειτῶνθ,θζ,ἔστινἄραὡς G are equal to one another, and the numbers H and ὁηπρὸςτὸνθ,οὕτωςὁηδπρὸςτὸνθζ.ἔσταιἄρα HF are also equal to one another, and the multitude of καὶὡςεἷςτῶνἡγουμένωνπρὸςἕνατῶνἑπομένων,οὕτως (divisions) G, G is equal to the multitude of (divi- ἅπαντεςοἱἡγούμενοιπρὸςἅπανταςτοὺςἑπομένους.ἔστιν sions) H, HF, thus as G is to H, so G (is) to HF. ἄραὡςὁηπρὸςτὸνθ,οὕτωςὁδπρὸςτὸνζ οἱ Thus, as one of the leading (numbers is) to one of the Η,ΘἄρατοῖςΔ,Ζἐντῷαὐτῷλόγῳεἰσὶνἐλάσσονες following, so will (the sum of) all of the leading (num- ὄντεςαὐτῶν ὅπερἐστὶνἀδύνατον ὑπόκεινταιγὰροἱδ, bers) be to (the sum of) all of the following [Prop. 7.12]. Ζἐλάχιστοιτῶντὸναὐτὸνλόγονἐχόντωναὐτοῖς. οὐκ Thus, as G is to H, so (is) to F. Thus, G ἄραμέρηἐστὶνὁδτοῦ μέροςἄρα.καὶὁζτοῦτὸ and H are in the same ratio as and F, being less αὐτὸμέροςἐστίν,ὅπερὁδτοῦ ἰσάκιςἄραὁδτὸν than them. The very thing is impossible. For and μετρεῖκαὶὁζτὸν ὅπερἔδειδεῖξαι. F were assumed (to be) the least of those (numbers) having the same ratio as them. Thus, is not parts of. Thus, (it is) a part (of ) [Prop. 7.4]. nd F is the same part of that (is) of [ef. 7.20, Prop 7.13]. Thus, measures the same number of times that F (measures). (Which is) the very thing it was required to show.. Proposition 21 Οἱ πρῶτοι πρὸς ἀλλήλους ἀριθμοὶ ἐλάχιστοί εἰσι τῶν τὸν Numbers prime to one another are the least of those αὐτὸνλόγονἐχόντωναὐτοῖς. (numbers) having the same ratio as them. στωσανπρῶτοιπρὸςἀλλήλουςἀριθμοὶοἱ, λέγω, Let and be numbers prime to one another. I say ὅτι οἱ, ἐλάχιστοί εἰσι τῶν τὸν αὐτὸν λόγον ἐχόντων that and are the least of those (numbers) having the αὐτοῖς. same ratio as them. ἰγὰρμή,ἔσονταίτινεςτῶν,ἐλάσσονεςἀριθμοὶ For if not then there will be some numbers less than ἐντῷαὐτῷλόγῳὄντεςτοῖς,.ἔστωσανοἱ,δ. and which are in the same ratio as and. Let them be and. 211