Page 321 - Euclid's Elements of Geometry

ST EW iþ.

ELEMENTS BOOK 10

sum of the (squares) on AD and DB is incommensurable
with the (rectangle contained) by AD and DB.
Thus, two straight-lines, AD and DB, (which are) in-
lþ contained) by them medial, and, moreover, incommensu-
commensurable in square, have been found, making the
sum of the (squares) on them medial, and the (rectangle
†
rable with the sum of the squares on them. (Which is)
the very thing it was required to show.

′
† AD and DB have lengths k ′1/4 q [1 + k/(1 + k ) ]/2 and k ′1/4 q [1 − k/(1 + k ) ]/2 times that of AB, respectively, where k and k are
2 1/2
2 1/2
defined in the footnote to Prop. 10.32.
.
Proposition 36
᾿Εὰν δύο ῥηταὶ δυνάμει μόνον σύμμετροι συντεθῶσιν, ἡ If two rational (straight-lines which are) commensu-
ὅλη ἄλογός ἐστιν, καλείσθω δὲ ἐκ δύο ὀνομάτων. rable in square only are added together then the whole
(straight-line) is irrational—let it be called a binomial
(straight-line). †

Α Β Γ A B C
Συγκείσθωσαν γὰρ δύο ῥηταὶ δυνάμει μόνον σύμμετροι For let the two rational (straight-lines), AB and BC,
αἱ ΑΒ, ΒΓ· λέγω, ὅτι ὅλη ἡ ΑΓ ἄλογός ἐστιν. (which are) commensurable in square only, be laid down
᾿Επεὶ γὰρ ἀσύμμετρός ἐστιν ἡ ΑΒ τῇ ΒΓ μήκει· δυνάμει together. I say that the whole (straight-line), AC, is irra-
γὰρ μόνον εἰσὶ σύμμετροι· ὡς δὲ ἡ ΑΒ πρὸς τὴν ΒΓ, οὕτως tional. For since AB is incommensurable in length with
τὸ ὑπὸ τῶν ΑΒΓ πρὸς τὸ ἀπὸ τῆς ΒΓ, ἀσύμμετρον ἄρα ἐστὶ BC—for they are commensurable in square only—and as
τὸ ὑπὸ τῶν ΑΒ, ΒΓ τῷ ἀπὸ τῆς ΒΓ. ἀλλὰ τῷ μὲν ὑπὸ τῶν AB (is) to BC, so the (rectangle contained) by ABC (is)
ΑΒ, ΒΓ σύμμετρόν ἐστι τὸ δὶς ὑπὸ τῶν ΑΒ, ΒΓ, τῷ δὲ ἀπὸ to the (square) on BC, the (rectangle contained) by AB
τῆς ΒΓ σύμμετρά ἐστι τὰ ἀπὸ τῶν ΑΒ, ΒΓ· αἱ γὰρ ΑΒ, ΒΓ and BC is thus incommensurable with the (square) on
ῥηταί εἰσι δυνάμει μόνον σύμμετροι· ἀσύμμετρον ἄρα ἐστὶ τὸ BC [Prop. 10.11]. But, twice the (rectangle contained)
δὶς ὑπὸ τῶν ΑΒ, ΒΓ τοῖς ἀπὸ τῶν ΑΒ, ΒΓ. καὶ συνθέντι τὸ by AB and BC is commensurable with the (rectangle
δὶς ὑπὸ τῶν ΑΒ, ΒΓ μετὰ τῶν ἀπὸ τῶν ΑΒ, ΒΓ, τουτέστι contained) by AB and BC [Prop. 10.6]. And (the sum
τὸ ἀπὸ τῆς ΑΓ, ἀσύμμετρόν ἐστι τῷ συγκειμένῳ ἐκ τῶν of) the (squares) on AB and BC is commensurable with
ἀπὸ τῶν ΑΒ, ΒΓ· ῥητὸν δὲ τὸ συγκείμενον ἐκ τῶν ἀπὸ the (square) on BC—for the rational (straight-lines) AB
τῶν ΑΒ, ΒΓ· ἄλογον ἄρα [ἐστὶ] τὸ ἀπὸ τῆς ΑΓ· ὥστε καὶ ἡ and BC are commensurable in square only [Prop. 10.15].
ΑΓ ἄλογός ἐστιν, καλείσθω δὲ ἐκ δύο ὀνομάτων· ὅπερ ἔδει Thus, twice the (rectangle contained) by AB and BC is
δεῖξαι. incommensurable with (the sum of) the (squares) on AB
and BC [Prop. 10.13]. And, via composition, twice the
(rectangle contained) by AB and BC, plus (the sum of)
the (squares) on AB and BC—that is to say, the (square)
on AC [Prop. 2.4]—is incommensurable with the sum of
the (squares) on AB and BC [Prop. 10.16]. And the sum
of the (squares) on AB and BC (is) rational. Thus, the
(square) on AC [is] irrational [Def. 10.4]. Hence, AC
is also irrational [Def. 10.4]—let it be called a binomial
‡
(straight-line). (Which is) the very thing it was required
to show.
† Literally, “from two names”.
‡ Thus, a binomial straight-line has a length expressible as 1 + k 1/2 [or, more generally, ρ (1 + k 1/2 ), where ρ is rational—the same proviso
applies to the definitions in the following propositions]. The binomial and the corresponding apotome, whose length is expressible as 1 − k 1/2

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