Page 499 - Euclid's Elements of Geometry

ST EW ibþ.

ELEMENTS BOOK 12

ἐπὶ τὸ Ν, καὶ ἐπεζεύχθωσαν αἱ ΛΔ, ΔΝ· ἴση ἄρα ἐστὶν ἡ than AD [Prop. 10.1]. Let it have been left, and let it be
ΛΔ τῇ ΔΝ. καὶ ἐπεὶ παράλληλός ἐστιν ἡ ΛΝ τῇ ΑΓ, ἡ δὲ ΑΓ LD. And let LM have been drawn, from L, perpendicu-
ἐφάπτεται τοῦ ΕΖΗΘ κύκλου, ἡ ΛΝ ἄρα οὐκ ἐφάπτεται τοῦ lar to BD, and let it have been drawn through to N. And
ΕΖΗΘ κύκλου· πολλῷ ἄρα αἱ ΛΔ, ΔΝ οὐκ ἐφάπτονται τοῦ let LD and DN have been joined. Thus, LD is equal to
ΕΖΗΘ κύκλου. ἐὰν δὴ τῇ ΛΔ εὐθείᾳ ἴσας κατὰ τὸ συνεχὲς DN [Props. 3.3, 1.4]. And since LN is parallel to AC
ἐναρμόσωμεν εἰς τὸν ΑΒΓΔ κύκλον, ἐγγραφήσεται εἰς τὸν [Prop. 1.28], and AC touches circle EFGH, LN thus
ΑΒΓΔ κύκλον πολύγωνον ἰσόπλευρόν τε καὶ ἀρτιόπλευρον does not touch circle EFGH. Thus, even more so, LD
izþ sided polygon, not touching the lesser circle EFGH, will
μὴ ψαῦον τοῦ ἐλάσσονος κύκλου τοῦ ΕΖΗΘ· ὅπερ ἔδει and DN do not touch circle EFGH. And if we continu-
ποιῆσαι. ously insert (straight-lines) equal to straight-line LD into
circle ABCD [Prop. 4.1] then an equilateral and even-

†
have been inscribed in circle ABCD. (Which is) the very
thing it was required to do.
† Note that the chord of the polygon, LN, does not touch the inner circle either.
E

.
Proposition 17
Δύο σφαιρῶν περὶ τὸ αὐτὸ κέντρον οὐσῶν εἰς τὴν There being two spheres about the same center, to in-

T
μείζονα σφαῖραν στερεὸν πολύεδρον ἐγγράψαι μὴ ψαῦον τῆς scribe a polyhedral solid in the greater sphere, not touch-
ἐλάσσονος σφαίρας κατὰ τὴν ἐπιφάνειαν. ing the lesser sphere on its surface.
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S Z R U X L S T U
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B Y W A F H D B K X Y P V W G Q A F R O D

G H N

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Νενοήσθωσαν δύο σφαῖραι περὶ τὸ αὐτὸ κέντρον τὸ Α· Let two spheres have been conceived about the same
δεῖ δὴ εἰς τὴν μείζονα σφαῖραν στερεὸν πολύεδρον ἐγγράψαι center, A. So, it is necessary to inscribe a polyhedral solid
μὴ ψαῦον τῆς ἐλάσσονος σφαίρας κατὰ τὴν ἐπιφάνειαν. in the greater sphere, not touching the lesser sphere on
Τετμήσθωσαν αἱ σφαῖραι ἐπιπέδῳ τινὶ διὰ τοῦ κέντρου· its surface.
ἔσονται δὴ αἱ τομαὶ κύκλοι, ἐπειδήπερ μενούσης τῆς Let the spheres have been cut by some plane through
διαμέτρου καὶ περιφερομένου τοῦ ἡμικυκλίου ἐγιγνετο ἡ the center. So, the sections will be circles, inasmuch
σφαῖρα· ὥστε καὶ καθ᾿ οἵας ἂν θέσεως ἐπινοήσωμεν τὸ as a sphere is generated by the diameter remaining be-
ἡμικύκλιον, τὸ δι᾿ αὐτοῦ ἐκβαλλόμενον ἐπίπεδον ποιήσει hind, and a semi-circle being carried around [Def. 11.14].
ἐπὶ τῆς ἐπιφανείας τῆς σφαίρας κύκλον. καὶ φανερόν, And, hence, whatever position we conceive (of for) the
ὅτι καὶ μέγιστον, ἐπειδήπερ ἡ διάμετρος τῆς σφαίρας, ἥτις semi-circle, the plane produced through it will make a

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