Page 528 - Euclid's Elements of Geometry

ST EW igþ.

ELEMENTS BOOK 13

πλευρά. ἔστι δὲ καὶ ἡ ΠΥ πενταγώνου· ἰσόπλευρον ἄρα eral. And side QE is (the side) of a hexagon (inscribed
ἐστὶ τὸ ΠΟΥ τρίγωνον. διὰ τὰ αὐτὰ δὴ καὶ ἕκαστον τῶν in circle EFGHK), and EP (the side) of a decagon, and
ΠΛΡ, ΡΜΣ, ΣΝΤ, ΤΞΥ ἰσόπλευρόν ἐστιν. καὶ ἐπεὶ πεν- (angle) QEP is a right-angle, thus QP is (the side) of a
ταγώνου ἐδείχθη ἑκατέρα τῶν ΠΛ, ΠΟ, ἔστι δὲ καὶ ἡ ΛΟ pentagon (inscribed in the same circle). For the square
πενταγώνου, ἰσόπλευρον ἄρα ἐστὶ τὸ ΠΛΟ τρίγωνον. διὰ on the side of a pentagon is (equal to the sum of) the
τὰ αὐτὰ δὴ καὶ ἕκαστον τῶν ΛΡΜ, ΜΣΝ, ΝΤΞ, ΞΥΟ (squares) on (the sides of) a hexagon and a decagon in-
τριγώνων ἰσόπλευρόν ἐστιν. scribed in the same circle [Prop. 13.10]. So, for the same
(reasons), PU is also the side of a pentagon. And QU
is also (the side) of a pentagon. Thus, triangle QPU is
equilateral. So, for the same (reasons), (triangles) QLR,
RMS, SNT , and T OU are each also equilateral. And
since QL and QP were each shown (to be the sides) of a
pentagon, and LP is also (the side) of a pentagon, trian-
gle QLP is thus equilateral. So, for the same (reasons),
R E triangles LRM, MSN, NT O, and OUP are each also
equilateral.

Q
R
Z a W M F L Z E P

S H Y F X U S G X V a W K U

T N H O

T
Εἰλήφθω τὸ κέντρον τοῦ ΕΖΗΘΚ κύκλου τὸ Φ σημεῖον· Let the center, point V , of circle EFGHK have been
καὶ ἀπὸ τοῦ Φ τῷ τοῦ κύκλου ἐπιπέδῳ πρὸς ὀρθὰς ἀνεστάτω found [Prop. 3.1]. And let V Z have been set up, at
ἡ ΦΩ, καὶ ἐκβεβλήσθω ἐπὶ τὰ ἕτερα μέρη ὡς ἡ ΦΨ, καὶ (point) V , at right-angles to the plane of the circle. And
ἀφῃρήσθω ἑξαγώνου μὲν ἡ ΦΧ, δεκαγώνου δὲ ἑκατέρα τῶν let it have been produced on the other side (of the cir-
ΦΨ, ΧΩ, καὶ ἐπεζεύχθωσαν αἱ ΠΩ, ΠΧ, ΥΩ, ΕΦ, ΛΦ, ΛΨ, cle), like V X. And let V W have been cut off (from XZ
ΨΜ. so as to be equal to the side) of a hexagon, and each of
Καὶ ἐπεὶ ἑκατέρα τῶν ΦΧ, ΠΕ τῷ τοῦ κύκλου ἐπιπέδῳ V X and WZ (so as to be equal to the side) of a decagon.
πρὸς ὀρθάς ἐστιν, παράλληλος ἄρα ἐστὶν ἡ ΦΧ τῇ ΠΕ. εἰσὶ And let QZ, QW, UZ, EV , LV , LX, and XM have been
δὲ καὶ ἴσαι· καὶ αἱ ΕΦ, ΠΧ ἄρα ἴσαι τε καὶ παράλληλοί joined.
εἰσιν. ἑξαγώνου δὲ ἡ ΕΦ· ἑξαγώνου ἄρα καὶ ἡ ΠΧ. καὶ And since V W and QE are each at right-angles
ἐπεὶ ἑξαγώνου μέν ἐστιν ἡ ΠΧ, δεκαγώνου δὲ ἡ ΧΩ, καὶ to the plane of the circle, V W is thus parallel to QE
ὀρθή ἐστιν ἡ ὑπὸ ΠΧΩ γωνία, πενταγώνου ἄρα ἐστὶν ἡ [Prop. 11.6]. And they are also equal. EV and QW are
ΠΩ. διὰ τὰ αὐτὰ δὴ καὶ ἡ ΥΩ πενταγώνου ἐστίν, ἐπειδήπερ, thus equal and parallel (to one another) [Prop. 1.33].

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