Euclid · Elements §7.prop.16-7.prop.18

Commutativity of Multiplication and Ratios of Products

Passage 118 of 316 · Greek

Summary

Proposition 16 proves the commutative law of multiplication (a × b = b × a), Proposition 17 shows that the ratio of products equals the ratio of the multiplied numbers (a × b : a × c = b : c), and Proposition 18 shows that the ratio of products equals the ratio of the multipliers (a × c : b × c = a : b).

§7.prop.16ἐὰν δύο ἀριθμοὶ πολλαπλασιάσαντες ἀλλήλους ποιῶσί τινας, οἱ γενόμενοι ἐξ αὐτῶν ἴσοι ἀλλήλοις ἔσονται.

If two numbers by multiplying one another make certain numbers, the numbers produced from them will be equal to one another.

ἔστωσαν δύο ἀριθμοὶ οἱ Α, Β, καὶ ὁ μὲν Α τὸν Β πολλαπλασιάσας τὸν Γ ποιείτω, ὁ δὲ Β τὸν Α πολλαπλασιάσας τὸν Δ ποιείτω· λέγω, ὅτι ἴσος ἐστὶν ὁ Γ τῷ Δ. ἐπεὶ γὰρ ὁ Α τὸν Β πολλαπλασιάσας τὸν Γ πεποίηκεν, ὁ Β ἄρα τὸν Γ μετρεῖ κατὰ τὰς ἐν τῷ Α μονάδας.

Let there be two numbers A, B, and let A by multiplying B make Γ, and B by multiplying A make Δ; I say that Γ is equal to Δ. For since A by multiplying B has made Γ, therefore B measures Γ according to the units in A.

μετρεῖ δὲ καὶ ἡ Ε μονὰς τὸν Α ἀριθμὸν κατὰ τὰς ἐν αὐτῷ μονάδας· ἰσάκις ἄρα ἡ Ε μονὰς τὸν Α ἀριθμὸν μετρεῖ καὶ ὁ Β τὸν Γ. ἐναλλὰξ ἄρα ἰσάκις ἡ Ε μονὰς τὸν Β ἀριθμὸν μετρεῖ καὶ ὁ Α τὸν Γ. πάλιν, ἐπεὶ ὁ Β τὸν Α πολλαπλασιάσας τὸν Δ πεποίηκεν, ὁ Α ἄρα τὸν Δ μετρεῖ κατὰ τὰς ἐν τῷ Β μονάδας.

And the unit E also measures the number A according to the units in it; therefore the unit E measures the number A the same number of times that B measures Γ. Therefore, alternately, the unit E measures the number B the same number of times that A measures Γ. Again, since B by multiplying A has made Δ, therefore A measures Δ according to the units in B.

μετρεῖ δὲ καὶ ἡ Ε μονὰς τὸν Β κατὰ τὰς ἐν αὐτῷ μονάδας· ἰσάκις ἄρα ἡ Ε μονὰς τὸν Β ἀριθμὸν μετρεῖ καὶ ὁ Α τὸν Δ. ἰσάκις δὲ ἡ Ε μονὰς τὸν Β ἀριθμὸν ἐμέτρει καὶ ὁ Α τὸν Γ· ἰσάκις ἄρα ὁ Α ἑκάτερον τῶν Γ, Δ μετρεῖ.

And the unit E also measures the number B according to the units in it; therefore the unit E measures the number B the same number of times that A measures Δ. But the unit E measured the number B the same number of times that A measured Γ; therefore A measures each of Γ, Δ the same number of times.

ἴσος ἄρα ἐστὶν ὁ Γ τῷ Δ· ὅπερ ἔδει δεῖξαι.

Therefore Γ is equal to Δ; which was to be proved.

§7.prop.17ἐὰν ἀριθμὸς δύο ἀριθμοὺς πολλαπλασιάσας ποιῇ τινας, οἱ γενόμενοι ἐξ αὐτῶν τὸν αὐτὸν ἕξουσι λόγον τοῖς πολλαπλασιασθεῖσιν.

If a number by multiplying two numbers make certain numbers, the numbers produced from them will have the same ratio as the multiplied numbers.

ἀριθμὸς γὰρ ὁ Α δύο ἀριθμοὺς τοὺς Β, Γ πολλαπλασιάσας τοὺς Δ, Ε ποιείτω· λέγω, ὅτι ἐστὶν ὡς ὁ Β πρὸς τὸν Γ, οὕτως ὁ Δ πρὸς τὸν Ε. ἐπεὶ γὰρ ὁ Α τὸν Β πολλαπλασιάσας τὸν Δ πεποίηκεν, ὁ Β ἄρα τὸν Δ μετρεῖ κατὰ τὰς ἐν τῷ Α μονάδας.

For let the number A by multiplying two numbers B, Γ make Δ, E; I say that, as B is to Γ, so is Δ to E. For since A by multiplying B has made Δ, therefore B measures Δ according to the units in A.

μετρεῖ δὲ καὶ ἡ Ζ μονὰς τὸν Α ἀριθμὸν κατὰ τὰς ἐν αὐτῷ μονάδας· ἰσάκις ἄρα ἡ Ζ μονὰς τὸν Α ἀριθμὸν μετρεῖ καὶ ὁ Β τὸν Δ. ἔστιν ἄρα ὡς ἡ Ζ μονὰς πρὸς τὸν Α ἀριθμόν, οὕτως ὁ Β πρὸς τὸν Δ. διὰ τὰ αὐτὰ δὴ καὶ ὡς ἡ Ζ μονὰς πρὸς τὸν Α ἀριθμόν, οὕτως ὁ Γ πρὸς τὸν Ε· καὶ ὡς ἄρα ὁ Β πρὸς τὸν Δ, οὕτως ὁ Γ πρὸς τὸν Ε. ἐναλλὰξ ἄρα ἐστὶν ὡς ὁ Β πρὸς τὸν Γ, οὕτως ὁ Δ πρὸς τὸν Ε· ὅπερ ἔδει δεῖξαι.

And the unit Z also measures the number A according to the units in it; therefore the unit Z measures the number A the same number of times that B measures Δ. Therefore, as the unit Z is to the number A, so is B to Δ. For the same reason also, as the unit Z is to the number A, so is Γ to E; therefore also, as B is to Δ, so is Γ to E. Therefore, alternately, as B is to Γ, so is Δ to E; which was to be proved.

§7.prop.18ἐὰν δύο ἀριθμοὶ ἀριθμόν τινα πολλαπλασιάσαντες ποιῶσί τινας, οἱ γενόμενοι ἐξ αὐτῶν τὸν αὐτὸν ἕξουσι λόγον τοῖς πολλαπλασιάσασιν.

If two numbers by multiplying any number make certain numbers, the numbers produced from them will have the same ratio as the multiplying numbers.

δύο γὰρ ἀριθμοὶ οἱ Α, Β ἀριθμόν τινα τὸν Γ πολλαπλασιάσαντες τοὺς Δ, Ε ποιείτωσαν· λέγω, ὅτι ἐστὶν ὡς ὁ Α πρὸς τὸν Β, οὕτως ὁ Δ πρὸς τὸν Ε. ἐπεὶ γὰρ ὁ Α τὸν Γ πολλαπλασιάσας τὸν Δ πεποίηκεν, καὶ ὁ Γ ἄρα τὸν Α πολλαπλασιάσας τὸν Δ πεποίηκεν.

For let two numbers A, B by multiplying any number Γ make Δ, E; I say that, as A is to B, so is Δ to E. For since A by multiplying Γ has made Δ, therefore Γ also by multiplying A has made Δ.

διὰ τὰ αὐτὰ δὴ καὶ ὁ Γ τὸν Β πολλαπλασιάσας τὸν Ε πεποίηκεν.

For the same reason also, Γ by multiplying B has made E.

ἀριθμὸς δὴ ὁ Γ δύο ἀριθμοὺς τοὺς Α, Β πολλαπλασιάσας τοὺς Δ, Ε πεποίηκεν.

Therefore the number Γ by multiplying two numbers A, B has made Δ, E.

ἔστιν ἄρα ὡς ὁ Α πρὸς τὸν Β, οὕτως ὁ Δ πρὸς τὸν ε· ὅπερ ἔδει δεῖξαι.

Therefore, as A is to B, so is Δ to E; which was to be proved.

Notes

§7.prop.16κατὰ τὰς ἐν τῷ Α μονάδας — The preposition κατά with the accusative here means 'according to' or 'in proportion to'. Under the definition of multiplication (Def. 15), this indicates that the number of times B measures Γ is equal to the number of units in A (i.e., the value of A).
§7.prop.16ἐμέτρει — The imperfect tense of the verb μετρέω. In Greek mathematical style, the imperfect (or a past tense) is frequently used to refer back to a fact that has already been established earlier in the proof ('the unit E measured the number B the same number of times...').
§7.prop.17ἔστιν ἄρα ὡς ἡ Ζ μονὰς πρὸς τὸν Α ἀριθμόν — The equivalence of the number of times that the unit Z measures A and B measures Δ is directly translated into the equality of ratios (Z : A = B : Δ), based on the definition of proportional numbers (Def. 20).

Cite this passage

Euclid, Elements §7.prop.16-7.prop.18. Humanitext Reader, https://reader.humanitext.ai/en/text/urn:cts:greekLit:tlg1799.tlg001.humanitext-grc2:7.prop.16-7.prop.18

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