Page 372 - Euclid's Elements of Geometry

ST EW iþ.

ELEMENTS BOOK 10

ΔΘ μέσον ἐστίν. καὶ παρὰ ῥητὴν τὴν ΔΙ παράκειται πλάτος AB and BC is medial, and is equal to DE, DE [is] thus
ποιοῦν τὴν ΔΖ· ῥητὴ ἄρα ἐστὶ καὶ ἡ ΔΖ καὶ ἀσύμμετρος medial. And it is applied to the rational (straight-line)
τῇ ΔΙ μήκει. καὶ ἐπεὶ ἀσύμμετρά ἐστι τὰ ἀπὸ τῶν ΑΒ, ΒΓ DI, producing DG as breadth. Thus, DG is rational, and
τῷ δὶς ὑπὸ τῶν ΑΒ, ΒΓ, ἀσύμμετρον ἄρα καὶ τὸ ΔΕ τῷ incommensurable in length with DI [Prop 10.22]. Again,
ΔΘ. ὡς δὲ τὸ ΔΕ πρὸς τὸ ΔΘ, οὕτως ἐστὶ καὶ ἡ ΔΗ πρὸς since twice the (rectangle contained) by AB and BC is
τὴν ΔΖ· ἀσύμμετρος ἄρα ἡ ΔΗ τῇ ΔΖ. καί εἰσιν ἀμφότεραι medial, and is equal to DH, DH is thus medial. And it is
ῥηταί· αἱ ΗΔ, ΔΖ ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι. applied to the rational (straight-line) DI, producing DF
ἀποτομὴ ἄρα ἐστίν ἡ ΖΗ· ῥητὴ δὲ ἡ ΖΘ. τὸ δὲ ὑπὸ ῥητῆς as breadth. Thus, DF is also rational, and incommen-
καὶ ἀποτομῆς περιεχόμενον [ὀρθογώνιον] ἄλογόν ἐστιν, καὶ surable in length with DI [Prop. 10.22]. And since the
ἡ δυναμένη αὐτὸ ἄλογός ἐστιν· καὶ δύναται τὸ ΖΕ ἡ ΑΓ· ἡ (sum of the squares) on AB and BC is incommensurable
ΑΓ ἄρα ἄλογός ἐστιν· καλείσθω δὲ ἡ μετὰ μέσου μέσον τὸ with twice the (rectangle contained) by AB and BC, DE
ὅλον ποιοῦσα. ὅπερ ἔδει δεῖξαι. (is) also incommensurable with DH. And as DE (is) to
DH, so DG also is to DF [Prop. 6.1]. Thus, DG (is) in-
commensurable (in length) with DF [Prop. 10.11]. And
they are both rational. Thus, GD and DF are ratio-
nal (straight-lines which are) commensurable in square
only. Thus, FG is an apotome [Prop. 10.73]. And FH
ojþ and its square-root is irrational. And AC is the square- †
(is) rational. And the [rectangle] contained by a rational
(straight-line) and an apotome is irrational [Prop. 10.20],

root of FE. Thus, AC is irrational. Let it be called
that which makes with a medial (area) a medial whole.
(Which is) the very thing it was required to show.
† See footnote to Prop. 10.41.

Proposition 79
.
Τῇ ἀποτομῇ μία [μόνον] προσαρμόζει εὐθεῖα ῥητὴ [Only] one rational straight-line, which is commensu-
δυνάμει μόνον σύμμετρος οὖσα τῇ ὅλῃ. rable in square only with the whole, can be attached to
an apotome. †
Α Β Γ ∆ A B C D
῎Εστω ἀποτομὴ ἡ ΑΒ, προσαρμόζουσα δὲ αὐτῇ ἡ ΒΓ· αἱ Let AB be an apotome, with BC (so) attached to it.
ΑΓ, ΓΒ ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι· λέγω, ὅτι AC and CB are thus rational (straight-lines which are)
τῇ ΑΒ ἑτέρα οὐ προσαρμόζει ῥητὴ δυνάμει μόνον σύμμετρος commensurable in square only [Prop. 10.73]. I say that
οὖσα τῇ ὅλῇ. another rational (straight-line), which is commensurable
Εἰ γὰρ δυνατόν, προσαρμοζέτω ἡ ΒΔ· καὶ αἱ ΑΔ, in square only with the whole, cannot be attached to AB.
ΔΒ ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι. καὶ ἐπεί, ᾧ For, if possible, let BD be (so) attached (to AB).
ὑπερέχει τὰ ἀπὸ τῶν ΑΔ, ΔΒ τοῦ δὶς ὑπὸ τῶν ΑΔ, ΔΒ, Thus, AD and DB are also rational (straight-lines which
τούτῳ ὑπερέχει καὶ τὰ ἀπὸ τῶν ΑΓ, ΓΒ τοῦ δὶς ὑπὸ τῶν are) commensurable in square only [Prop. 10.73]. And
ΑΓ, ΓΒ· τῷ γὰρ αὐτῷ τῷ ἀπὸ τῆς ΑΒ ἀμφότερα ὑπερέχει· since by whatever (area) the (sum of the squares) on AD
ἐναλλὰξ ἄρα, ᾧ ὑπερέχει τὰ ἀπὸ τῶν ΑΔ, ΔΒ τῶν ἀπὸ τῶν and DB exceeds twice the (rectangle contained) by AD
ΑΓ, ΓΒ, τούτῳ ὑπερέχει [καὶ] τὸ δὶς ὑπὸ τῶν ΑΔ, ΔΒ τοῦ and DB, the (sum of the squares) on AC and CB also ex-
δὶς ὑπὸ τῶν ΑΓ, ΓΒ. τὰ δὲ ἀπὸ τῶν ΑΔ, ΔΒ τῶν ἀπὸ τῶν ceeds twice the (rectangle contained) by AC and CB by
ΑΓ, ΓΒ ὑπερέχει ῥητῷ· ῥητὰ γὰρ ἀμφότερα. καὶ τὸ δὶς ἄρα this (same area). For both exceed by the same (area)—
ὑπὸ τῶν ΑΔ, ΔΒ τοῦ δὶς ὑπὸ τῶν ΑΓ, ΓΒ ὑπερέχει ῥητῷ· (namely), the (square) on AB [Prop. 2.7]. Thus, alter-
ὅπερ ἐστὶν ἀδύνατον· μέσα γὰρ ἀμφότερα, μέσον δὲ μέσου nately, by whatever (area) the (sum of the squares) on
οὐχ ὑπερέχει ῥητῷ. τῇ ἄρα ΑΒ ἑτέρα οὐ προσαρμόζει ῥητὴ AD and DB exceeds the (sum of the squares) on AC
δυνάμει μόνον σύμμετρος οὖσα τῇ ὅλῃ. and CB, twice the (rectangle contained) by AD and DB
Μία ἄρα μόνη τῇ ἀποτομῇ προσαρμόζει ῥητὴ δυνάμει [also] exceeds twice the (rectangle contained) by AC and

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