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Euclid: Elementa

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ιβ῾ 
PROPOSITION 12. 
12 
 
 
 
Ἐὰν ἀπὸ μονάδος ὁποσοιοῦν ἀριθμοὶ ἑξῆς ἀνάλογον ὦσιν, ὑφ᾽ ὅσων ἂν ὁ ἔσχατος πρώτων ἀριθμῶν μετρῆται, ὑπὸ τῶν αὐτῶν καὶ ὁ παρὰ τὴν μονάδα μετρηθήσεται. 
If as many numbers as we please beginning from an unit be in continued proportion, by however many prime numbers the last is measured, the next to the unit will also be measured by the same. 
Si ab unitate quotlibet numeri deinceps proportionales fuerint, sub quotcumque primis numeris extremus mensuratur, sub eisdem et qui iuxta unitatem mensurabitur. 
 
 
 
Ἔστωσαν ἀπὸ μονάδος ὁποσοιδηποτοῦν ἀριθμοὶ ἀνάλογον οἱ Α, Β, Γ, Δ:  λέγω, ὅτι ὑφ᾽ ὅσων ἂν ὁ Δ πρώτων ἀριθμῶν μετρῆται, ὑπὸ τῶν αὐτῶν καὶ ὁ Α μετρηθήσεται. 
Let there be as many numbers as we please, A, B, C, D, beginning from an unit, and in continued proportion;  I say that, by however many prime numbers D is measured, A will also be measured by the same. 
Sint ab unitate quotlibet numeri proportiona1es A, B, G, D.  Dico quoniam sub quotcumque numerus D primis numeris mensuratur, sub eisdem et numerus A mensurabitur. 
   
   
   
Μετρείσθω γὰρ ὁ Δ ὑπό τινος πρώτου ἀριθμοῦ τοῦ Ε:  λέγω, ὅτι ὁ Ε τὸν Α μετρεῖ. 
For let D be measured by any prime number E;  I say that E measures A. 
Mensuretur enim numerus D sub aliquo primo numero E.  Dico quoniam numerus E numerum A metitur. 
   
   
   
μὴ γάρ:  καί ἐστιν ὁ Ε πρῶτος, ἅπας δὲ πρῶτος ἀριθμὸς πρὸς ἅπαντα, ὃν μὴ μετρεῖ, πρῶτός ἐστιν:  οἱ Ε, Α ἄρα πρῶτοι πρὸς ἀλλήλους εἰσίν.  καὶ ἐπεὶ ὁ Ε τὸν Δ μετρεῖ, μετρείτω αὐτὸν κατὰ τὸν Ζ:  ὁ Ε ἄρα τὸν Ζ πολλαπλασιάσας τὸν Δ πεποίηκεν.  πάλιν, ἐπεὶ ὁ Α τὸν Δ μετρεῖ κατὰ τὰς ἐν τῷ Γ μονάδας,  ὁ Α ἄρα τὸν Γ πολλαπλασιάσας τὸν Δ πεποίηκεν.  ἀλλὰ μὴν καὶ ὁ Ε τὸν Ζ πολλαπλασιάσας τὸν Δ πεποίηκεν:  ὁ ἄρα ἐκ τῶν Α, Γ ἴσος ἐστὶ τῷ ἐκ τῶν Ε, Ζ.  ἔστιν ἄρα ὡς ὁ Α πρὸς τὸν Ε, ὁ Ζ πρὸς τὸν Γ.  οἱ δὲ Α, Ε πρῶτοι, οἱ δὲ πρῶτοι καὶ ἐλάχιστοι, οἱ δὲ ἐλάχιστοι μετροῦσι τοὺς τὸν αὐτὸν λόγον ἔχοντας ἰσάκις ὅ τε ἡγούμενος τὸν ἡγούμενον καὶ ὁ ἑπόμενος τὸν ἑπόμενον:  μετρεῖ ἄρα ὁ Ε τὸν Γ. 
For suppose it does not;  now E is prime, and any prime number is prime to any which it does not measure; [VII. 29]  therefore E, A are prime to one another.  And, since E measures D, let it measure it according to F,  therefore E by multiplying F has made D.  Again, since A measures D according to the units in C, [IX. 11 and Por.]  therefore A by multiplying C has made D.  But, further, E has also by multiplying F made D;  therefore the product of A, C is equal to the product of E, F.  Therefore, as A is to E, so is F to C. [VII. 19]  But A, E are prime, primes are also least, [VII. 21] and the least measure those which have the same ratio the same number of times, the antecedent the antecedent and the consequent the consequent; [VII. 20]  therefore E measures C. 
Non enim metiatur numerus E numerum A  et est numerus E primus. Omnis autem primus ad omnem quem non metitur est primus.  Numeri ergo E, A primi ad se invicem sunt.  Et quoniam numerus E numerum D metitur, metiatur autem secundum numerum Z,  numerus ergo E numerum Z multiplicans numerum D fecerit.  Rursum quoniam numerus A numerum D metitur secundum eas que in numero G unitates,  numerus ergo A numerum G multiplicans numerum D fecit.  Necnon et numerus E numerum Z multiplicans numerum D fecit.  Qui ergo ex numeris A, G equalis est ei qui ex numeris E, Z.  Est ergo sicut A ad E ita Z ad G.  Numeri autem A, E primi. Qui autem primi, et minimi. Qui autem minimi, metiuntur eandem proportionem habentes equaliter antecedens antecedentem et consequens consequentem.  Metitur ergo numerus E numerum G. 
                       
                       
                       
μετρείτω αὐτὸν κατὰ τὸν Η:  ὁ Ε ἄρα τὸν Η πολλαπλασιάσας τὸν Γ πεποίηκεν.  ἀλλὰ μὴν διὰ τὸ πρὸ τούτου καὶ ὁ Α τὸν Β πολλαπλασιάσας τὸν Γ πεποίηκεν.  ὁ ἄρα ἐκ τῶν Α, Β ἴσος ἐστὶ τῷ ἐκ τῶν Ε, Η.  ἔστιν ἄρα ὡς ὁ Α πρὸς τὸν Ε, ὁ Η πρὸς τὸν Β.  οἱ δὲ Α, Ε πρῶτοι, οἱ δὲ πρῶτοι καὶ ἐλάχιστοι,  οἱ δὲ ἐλάχιστοι ἀριθμοὶ μετροῦσι τοὺς τὸν αὐτὸν λόγον ἔχοντας αὐτοῖς ἰσάκις ὅ τε ἡγούμενος τὸν ἡγούμενον καὶ ὁ ἑπόμενος τὸν ἑπόμενον:  μετρεῖ ἄρα ὁ Ε τὸν Β. 
Let it measure it according to G;  therefore E by multiplying G has made C.  But, further, by the theorem before this, A has also by multiplying B made C. [IX. 11 and Por.]  Therefore the product of A, B is equal to the product of E, G.  Therefore, as A is to E, so is G to B. [VII. 19]  But A, E are prime, primes are also least, [VII. 21]  and the least numbers measure those which have the same ratio with them the same number of times, the antecedent the antecedent and the consequent the consequent: [VII. 20]  therefore E measures B. 
Metitur ipsum secundum numerum I.  Numerus ergo E numerum I multiplicans numerum G fecit.  Necnon propter id quod ante hoc et numerus A numerum B multiplicans numerum G fecit.  Qui ergo ex numeris A, B equalis est ei qui ex numeris E, I.  Est ergo sicut numerus A ad numerum E ita numerus I ad numerum B.  Numeri autem A, E primi. Qui autem primi, et minimi.  Minimi autem numeri metiuntur eandem proportionem habentes equaliter antecedens antecedentem et consequens consequentem.  Metitur ergo numerus E numerum B. 
               
               
               
μετρείτω αὐτὸν κατὰ τὸν Θ:  ὁ Ε ἄρα τὸν Θ πολλαπλασιάσας τὸν Β πεποίηκεν.  ἀλλὰ μὴν καὶ ὁ Α ἑαυτὸν πολλαπλασιάσας τὸν Β πεποίηκεν:  ὁ ἄρα ἐκ τῶν Ε, Θ ἴσος ἐστὶ τῷ ἀπὸ τοῦ Α.  ἔστιν ἄρα ὡς ὁ Ε πρὸς τὸν Α, ὁ Α πρὸς τὸν Θ.  οἱ δὲ Α, Ε πρῶτοι, οἱ δὲ πρῶτοι καὶ ἐλάχιστοι,  οἱ δὲ ἐλάχιστοι μετροῦσι τοὺς τὸν αὐτὸν λόγον ἔχοντας ἰσάκις ὅ τε ἡγούμενος τὸν ἡγούμενον καὶ ὁ ἑπόμενος τὸν ἑπόμενον:  μετρεῖ ἄρα ὁ Ε τὸν Α ὡς ἡγούμενος ἡγούμενον.  ἀλλὰ μὴν καὶ οὐ μετρεῖ: ὅπερ ἀδύνατον.  οὐκ ἄρα οἱ Ε, Α πρῶτοι πρὸς ἀλλήλους εἰσίν.  σύνθετοι ἄρα.  οἱ δὲ σύνθετοι ὑπὸ [πρώτου] ἀριθμοῦ τινος μετροῦνται.  καὶ ἐπεὶ ὁ Ε πρῶτος ὑπόκειται, ὁ δὲ πρῶτος ὑπὸ ἑτέρου ἀριθμοῦ οὐ μετρεῖται ἢ ὑφ᾽ ἑαυτοῦ,  ὁ Ε ἄρα τοὺς Α, Ε μετρεῖ: ὥστε ὁ Ε τὸν Α μετρεῖ.  μετρεῖ δὲ καὶ τὸν Δ:  ὁ Ε ἄρα τοὺς Α, Δ μετρεῖ.  ὁμοίως δὴ δείξομεν, ὅτι ὑφ᾽ ὅσων ἂν ὁ Δ πρώτων ἀριθμῶν μετρῆται, ὑπὸ τῶν αὐτῶν καὶ ὁ Α μετρηθήσεται:  ὅπερ ἔδει δεῖξαι. 
Let it measure it according to H;  therefore E by multiplying H has made B.  But further A has also by multiplying itself made B; [IX. 8]  therefore the product of E, H is equal to the square on A.  Therefore, as E is to A, so is A to H. [VII. 19]  But A, E are prime, primes are also least, [VII. 21]  and the least measure those which have the same ratio the same number of times, the antecedent the antecedent and the consequent the consequent; [VII. 20]  therefore E measures A, as antecedent antecedent.  But, again, it also does not measure it: which is impossible.  Therefore E, A are not prime to one another.  Therefore they are composite to one another.  But numbers composite to one another are measured by some number. [VII. Def. 14]  And, since E is by hypothesis prime, and the prime is not measured by any number other than itself,  therefore E measures A, E, so that E measures A.  [But it also measures D;  therefore E measures A, D.]  Similarly we can prove that, by however many prime numbers D is measured, A will also be measured by the same.  Q. E. D. 
Metiatur ipsum secundum numerum T.  Numerus ergo E numerum T multiplicans numerum B fecit.  Necnon et numerus A se ipsum multiplicans numerum B fecit.  Qui ergo ex numeris E, T equalis est ei qui a numero A.  Est ergo sicut E ad A ita A ad T.  Numeri autem A, E primi. Qui autem primi, et minimi.  Qui autem minimi metiuntur eandem proportionem habentes equaliter, antecedens antecedentem et consequens consequentem.  Metitur ergo numerus E numerum A sicut antecedens antecedentem.  Sed et non metitur. [Quod impossibile.  Non ergo numeri E, A primi ad se invicem sunt.  Compositi ergo.  Compositi autemnumeri ab aliquo primo numero mensurantur. Numeri ergo A, E ab aliquo primo numero mensurantur. Subiacebat autem et non metiens. Quod impossibile. Non ergo numerus E numerum A non metitur. Metitur ergo.  Et quoniam numerus E primus subiacet qui autem primus sub altero numero non mensuratur quam sub se ipso.  Numerus ergo E numeros A, E metitur. Quare et numerus E numerum A metitur. Subiacebat autem et non metiens.] Quod impossibile. Non ergo numerus E numerum A non metitur. Metitur ergo.  Metitur autem et numerum D.  Numerus ergo E numeros A, D metitur.  Similiter ergo ostendemus quoniam sub quotcumque primis numeris numerus D mensuratur, sub eisdem et numerus A mensurabitur.  Quod oportebat ostendere. 
                                   
                                   
                                   
 
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