Page 503 - Euclid's Elements of Geometry

ST EW ibþ. ìri sma

ELEMENTS BOOK 12

Corollary
.
᾿Εὰν δὲ καὶ εἰς ἑτάραν σφαῖραν τῷ ἐν τῇ ΒΓΔΕ σφαίρᾳ And, also, if a similar polyhedral solid to that in
στερεῷ πολυέδρῳ ὅμοιον στερεὸν πολύεδρον ἐγγραφῇ, τὸ sphere BCDE is inscribed in another sphere then the
ἐν τῇ ΒΓΔΕ σφαίρᾳ στερεὸν πολύεδρον πρὸς τὸ ἐν τῇ polyhedral solid in sphere BCDE has to the polyhedral
ἑτέρᾳ σφαίρᾳ στερεὸν πολύεδρον τριπλασίονα λόγον ἔχει, solid in the other sphere the cubed ratio that the diameter
ἤπερ ἡ τῆς ΒΓΔΕ σφαίρας διάμετρος πρὸς τὴν τῆς ἑτέρας of sphere BCDE has to the diameter of the other sphere.
σφαίρας διάμετρον. διαιρεθέντων γὰρ τῶν στερεῶν εἰς For if the solids are divided into similarly numbered, and
τὰς ὁμοιοπληθεῖς καὶ ὁμοιοταγεῖς πυραμίδας ἔσονται αἱ πυ- similarly situated, pyramids, then the pyramids will be
ραμίδες ὅμοιαι. αἱ δὲ ὅμοιαι πυραμίδες πρὸς ἀλλήλας ἐν similar. And similar pyramids are in the cubed ratio of
τριπλασίονι λόγῳ εἰσὶ τῶν ὁμολόγων πλευρῶν· ἡ ἄρα πυ- corresponding sides [Prop. 12.8 corr.]. Thus, the pyra-
ραμίς, ἧς βάσις μέν ἐστι τὸ ΚΒΟΣ τετράπλευρον, κορυφὴ mid whose base is quadrilateral KBPS, and apex the
δὲ τὸ Α σημεῖον, πρὸς τὴν ἐν τῇ ἑτέρᾳ σφαίρᾳ ὁμοιοταγῆ point A, will have to the similarly situated pyramid in the
πυραμίδα τριπλασίονα λόγον ἔχει, ἤπερ ἡ ὁμόλογος πλευρὰ other sphere the cubed ratio that a corresponding side
πρὸς τὴν ὁμόλογον πλευράν, τουτέστιν ἤπερ ἡ ΑΒ ἐκ τοῦ (has) to a corresponding side. That is to say, that of ra-
κέντρου τῆς σφαίρας τῆς περὶ κέντρον τὸ Α πρὸς τὴν ἐκ τοῦ dius AB of the sphere about center A to the radius of the
κέντρου τῆς ἑτέρας σφαίρας. ὁμοίως καὶ ἑκάστη πυραμὶς other sphere. And, similarly, each pyramid in the sphere
τῶν ἐν τῇ περὶ κέντρον τὸ Α σφαίρᾳ πρὸς ἑκάστην ὁμοταγῆ about center A will have to each similarly situated pyra-
πυραμίδα τῶν ἐν τῇ ἑτέρᾳ σφαίρᾳ τριπλασίονα λόγον ἕξει, mid in the other sphere the cubed ratio that AB (has) to
ἤπερ ἡ ΑΒ πρὸς τὴν ἐκ τοῦ κέντρου τῆς ἑτέρας σφαίρας. the radius of the other sphere. And as one of the leading
καὶ ὡς ἓν τῶν ἡγουμένων πρὸς ἓν τῶν ἑπομένων, οὕτως (magnitudes is) to one of the following (in two sets of
ἅπαντα τὰ ἡγούμενα πρὸς ἅπαντα τὰ ἑπόμενα· ὥστε ὅλον proportional magnitudes), so (the sum of) all the lead-
ihþ is to say, that diameter BD (has) to the diameter of the
τὸ ἐν τῇ περὶ κέντρον τὸ Α σφαίρᾳ στερεὸν πολύεδρον ing (magnitudes is) to (the sum of) all of the following
πρὸς ὅλον τὸ ἐν τῇ ἑτέρᾳ [σφαίρᾳ] στερεὸν πολύεδρον τρι- (magnitudes) [Prop. 5.12]. Hence, the whole polyhedral
πλασίονα λόγον ἕξει, ἤπερ ἡ ΑΒ πρὸς τὴν ἐκ τοῦ κέντρου solid in the sphere about center A will have to the whole
τῆς ἑτέρας σφαίρας, τουτέστιν ἤπερ ἡ ΒΔ διάμετρος πρὸς polyhedral solid in the other [sphere] the cubed ratio that
τὴν τῆς ἑτέρας σφαίρας διάμετρον· ὅπερ ἔδει δεῖξαι. (radius) AB (has) to the radius of the other sphere. That

other sphere. (Which is) the very thing it was required to
show.

A
Proposition 18
.
Αἱ σφαῖραι πρὸς ἀλλήλας ἐν τριπλασίονι λόγῳ εἰσὶ τῶν Spheres are to one another in the cubed ratio of their
ἰδίων διαμέτρων. respective diameters.
B G B C L
A

E H D Z E H G D K M F N

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