Page 166 - Euclid's Elements of Geometry

ST EW þ.

ìri sma then the triangles around the perpendicular are similar

ELEMENTS BOOK 6

to the whole (triangle), and to one another. [(Which is)
the very thing it was required to show.]

.
Corollary
jþ is) the very thing it was required to show.
᾿Εκ δὴ τούτου φανερόν, ὅτι ἐὰν ἐν ὀρθογωνίῳ τριγώνῳ So (it is) clear, from this, that if, in a right-angled tri-
ἀπὸ τῆς ὀρθῆς γωνάις ἐπὶ τὴν βάσις κάθετος ἀχθῇ, ἡ angle, a (straight-line) is drawn from the right-angle per-
ἀχθεῖσα τῶν τῆς βάσεως τμημάτων μέση ἀνάλογόν ἐστιν· pendicular to the base then the (straight-line so) drawn
†
ὅπερ ἔδει δεῖξαι. is in mean proportion to the pieces of the base. (Which

† In other words, the perpendicular is the geometric mean of the pieces.

Proposition 9
.
Τῆς δοθείσης εὐθείας τὸ προσταχθὲν μέρος ἀφελεῖν. To cut off a prescribed part from a given straight-line.

Γ C
Ε E

∆ D

Α Ζ Β A F B
῎Εστω ἡ δοθεῖσα εὐθεῖα ἡ ΑΒ· δεῖ δὴ τῆς ΑΒ τὸ προ- Let AB be the given straight-line. So it is required to
σταχθὲν μέρος ἀφελεῖν. cut off a prescribed part from AB.
᾿Επιτετάχθω δὴ τὸ τρίτον. [καὶ] διήθχω τις ἀπὸ τοῦ Α So let a third (part) have been prescribed. [And] let
εὐθεῖα ἡ ΑΓ γωνίαν περιέχουσα μετὰ τῆς ΑΒ τυχοῦσαν· καὶ some straight-line AC have been drawn from (point) A,
εἰλήφθω τυχὸν σημεῖον ἐπὶ τῆς ΑΓ τὸ Δ, καὶ κείσθωσαν encompassing a random angle with AB. And let a ran-
τῇ ΑΔ ἴσαι αἱ ΔΕ, ΕΓ. καὶ ἐπεζεύχθω ἡ ΒΓ, καὶ διὰ τοῦ Δ dom point D have been taken on AC. And let DE and
παράλληλος αὐτῇ ἤχθω ἡ ΔΖ. EC be made equal to AD [Prop. 1.3]. And let BC have
᾿Επεὶ οὖν τριγώνου τοῦ ΑΒΓ παρὰ μίαν τῶν πλευρῶν been joined. And let DF have been drawn through D
τὴν ΒΓ ἦκται ἡ ΖΔ, ἀνάλογον ἄρα ἐστὶν ὡς ἡ ΓΔ πρὸς τὴν parallel to it [Prop. 1.31].
iþ BA (is) triple AF.
ΔΑ, οὕτως ἡ ΒΖ πρὸς τὴν ΖΑ. διπλῆ δὲ ἡ ΓΔ τῆς ΔΑ· Therefore, since FD has been drawn parallel to one
διπλῆ ἄρα καὶ ἡ ΒΖ τῆς ΖΑ· τριπλῆ ἄρα ἡ ΒΑ τῆς ΑΖ. of the sides, BC, of triangle ABC, then, proportionally,
Τῆς ἄρα δοθείσης εὐθείας τῆς ΑΒ τὸ ἐπιταχθὲν τρίτον as CD is to DA, so BF (is) to FA [Prop. 6.2]. And CD
μέρος ἀφῄρηται τὸ ΑΖ· ὅπερ ἔδει ποιῆσαι. (is) double DA. Thus, BF (is) also double FA. Thus,

Thus, the prescribed third part, AF, has been cut off
from the given straight-line, AB. (Which is) the very
thing it was required to do.

.
Proposition 10
Τὴν δοθεῖσαν εὐθεῖαν ἄτμητον τῇ δοθείσῃ τετμημένῃ To cut a given uncut straight-line similarly to a given
ὁμοίως τεμεῖν. cut (straight-line).

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