Page 23 - Euclid's Elements of Geometry

ST EW aþ.

ELEMENTS BOOK 1

Εἰ γὰρ μή, ἤτοι ἴση ἐστὶν ἡ ΑΓ τῇ ΑΒ ἢ ἐλάσσων· ἴση For if not, AC is certainly either equal to, or less than,
μὲν οὖν οὐκ ἔστιν ἡ ΑΓ τῇ ΑΒ· ἴση γὰρ ἂν ἦν καὶ γωνία ἡ AB. In fact, AC is not equal to AB. For then angle ABC
ὑπὸ ΑΒΓ τῇ ὑπὸ ΑΓΒ· οὐκ ἔστι δέ· οὐκ ἄρα ἴση ἐστὶν ἡ ΑΓ would also have been equal to ACB [Prop. 1.5]. But it
τῇ ΑΒ. οὐδὲ μὴν ἐλάσσων ἐστὶν ἡ ΑΓ τῆς ΑΒ· ἐλάσσων is not. Thus, AC is not equal to AB. Neither, indeed, is
γὰρ ἂν ἦν καὶ γωνία ἡ ὑπὸ ΑΒΓ τῆς ὑπὸ ΑΓΒ· οὐκ ἔστι AC less than AB. For then angle ABC would also have
δέ· οὐκ ἄρα ἐλάσσων ἐστὶν ἡ ΑΓ τῆς ΑΒ. ἐδείχθη δέ, ὅτι been less than ACB [Prop. 1.18]. But it is not. Thus, AC
οὐδὲ ἴση ἐστίν. μείζων ἄρα ἐστὶν ἡ ΑΓ τῆς ΑΒ. is not less than AB. But it was shown that (AC) is not
equal (to AB) either. Thus, AC is greater than AB.
Α A

Β B

kþ Γ C

Παντὸς ἄρα τριγώνου ὑπὸ τὴν μείζονα γωνίαν ἡ μείζων Thus, in any triangle, the greater angle is subtended
πλευρὰ ὑποτείνει· ὅπερ ἔδει δεῖξαι. by the greater side. (Which is) the very thing it was re-
quired to show.

Proposition 20
.
Παντὸς τριγώνου αἱ δύο πλευραὶ τῆς λοιπῆς μείζονές In any triangle, (the sum of) two sides taken to-
εἰσι πάντῃ μεταλαμβανόμεναι. gether in any (possible way) is greater than the remaining
(side).
∆ D

Α A

Β Γ B C
῎Εστω γὰρ τρίγωνον τὸ ΑΒΓ· λέγω, ὅτι τοῦ ΑΒΓ For let ABC be a triangle. I say that in triangle ABC
τριγώνου αἱ δύο πλευραὶ τῆς λοιπῆς μείζονές εἰσι πάντῃ (the sum of) two sides taken together in any (possible
μεταλαμβανόμεναι, αἱ μὲν ΒΑ, ΑΓ τῆς ΒΓ, αἱ δὲ ΑΒ, ΒΓ way) is greater than the remaining (side). (So), (the sum
τῆς ΑΓ, αἱ δὲ ΒΓ, ΓΑ τῆς ΑΒ. of) BA and AC (is greater) than BC, (the sum of) AB

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