Page 489 - Euclid's Elements of Geometry

ST EW ibþ.

ELEMENTS BOOK 12

ΑΕ, ΕΒ, ΒΖ, ΖΓ, ΓΗ, ΗΔ, ΔΘ, ΘΑ· καὶ ἕκαστον ἄρα than the cone. For it encompasses it. Thus, the pyramid
τῶν ΑΕΒ, ΒΖΓ, ΓΗΔ, ΔΘΑ τριγώνων μεῖζόν ἐστιν ἢ τὸ whose base is square ABCD, and apex the same as the
ἥμισυ μέρος του καθ᾿ ἑαυτὸ τμήματος τοῦ ΑΒΓΔ κύκλου. cone, is greater than half of the cone. Let the circum-
καὶ ἀνεστάτωσαν ἐφ᾿ ἑκάστου τῶν ΑΕΒ, ΒΖΓ, ΓΗΔ, ΔΘΑ ferences AB, BC, CD, and DA have been cut in half
τριγώνων πυραμίδες τὴν αὐτὴν κορυφὴν ἔχουσαι τῷ κώνῳ· at points E, F, G, and H (respectively). And let AE,
καὶ ἑκάστη ἄρα τῶν ἀνασταθεισῶν πυραμίδων κατὰ τὸν EB, BF, FC, CG, GD, DH, and HA have been joined.
αὐτὸν τρόπον μείζων ἐστὶν ἢ τὸ ἥμισυ μέρος τοῦ καθ᾿ And, thus, each of the triangles AEB, BFC, CGD, and
ἑαυτὴν τμήματος τοῦ κώνου. τέμνοντες δὴ τὰς ὑπολει- DHA is greater than the half part of the segment of cir-
πομένας περιφερείας δίχα καὶ ἐπιζευγνύντες εὐθείας καὶ cle ABCD about it [Prop. 12.2]. And let pyramids having
ἀνιστάντες ἐφ᾿ ἑκάστου τῶν τριγώνων πυραμίδα τὴν αὐτὴν the same apex as the cone have been set up on each of the
κορυφὴν ἔχουσαν τῷ κώνῳ καὶ τοῦτο ἀεὶ ποιοῦτες κα- triangles AEB, BFC, CGD, and DHA. And, thus, in the
ταλείψομέν τινα ἀποτμήματα τοῦ κώνου, ἃ ἔσται ἐλάττονα same way, each of the pyramids set up is more than the
τῆς ὑπεροχῆς, ᾗ ὑπερέχει ὁ κῶνος τοῦ τρίτου μέρους τοῦ half part of the segment of the cone about it. So, (if) the
κυλίνδρου. λελείφθω, καὶ ἔστω τὰ ἐπὶ τῶν ΑΕ, ΕΒ, ΒΖ, ΖΓ, remaining circumferences are cut in half, and straight-
ΓΗ, ΗΔ, ΔΘ, ΘΑ· λοιπὴ ἄρα ἡ πυραμίς, ἧς βάσις μέν ἐστι lines are joined, and pyramids having the same apex as
τὸ ΑΕΒΖΓΗΔΘ πολύγωνον, κορυφὴ δὲ ἡ αὐτὴ τῷ κώνῳ, the cone are set up on each of the triangles, and this is
μείζων ἐστὶν ἢ τρίτον μέρος τοῦ κυλίνδρου. ἀλλ᾿ ἡ πυραμίς, done continually, then we will (eventually) leave some
ἧς βάσις μέν ἐστι τὸ ΑΕΒΖΓΗΔΘ πολύγωνον, κορυφὴ δὲ segments of the cone whose (sum) is less than the excess
ἡ αυτὴ τῷ κώνῳ, τρίτον ἐστὶ μέρος τοῦ πρίσματος, οὗ βάσις by which the cone exceeds the third part of the cylinder
μέν ἐστι τὸ ΑΕΒΖΓΗΔΘ πολύγωνον, ὕψος δὲ τὸ αὐτὸ τῷ [Prop. 10.1]. Let them have been left, and let them be
κυλίνδρῳ· τὸ ἄρα πρίσμα, οὗ βάσις μέν ἐστι τὸ ΑΕΒΖΓΗΔΘ the (segments) on AE, EB, BF, FC, CG, GD, DH, and
πολύγωνον, ὕψος δὲ τὸ αὐτὸ τῷ κυλίνδρῳ, μεῖζόν ἐστι τοῦ HA. Thus, the remaining pyramid whose base is poly-
κυλίνδρου, οὗ βάσις ἐστὶν ὁ ΑΒΓΔ κύκλος. ἀλλὰ καὶ ἔλατ- gon AEBFCGDH, and apex the same as the cone, is
τον· ἐμπεριέχεται γὰρ ὑπ᾿ αὐτοῦ· ὅπερ ἐστὶν ἀδύνατον. οὐκ greater than the third part of the cylinder. But, the pyra-
ἄρα ὁ κύλινδρος τοῦ κώνου ἐλάττων ἐστὶν ἢ τριπλάσιος. mid whose base is polygon AEBFCGDH, and apex the
ἐδείχθη δέ, ὅτι οὐδὲ μείζων ἢ τριπλάσιος· τριπλάσιος ἄρα ὁ same as the cone, is the third part of the prism whose
κύλινδρος τοῦ κώνου· ὥστε ὁ κῶνος τρίτον ἐστὶ μέρος τοῦ base is polygon AEBFCGDH, and height the same as
κυλίνδρου. the cylinder [Prop. 12.7 corr.]. Thus, the prism whose
Πᾶς ἄρα κῶνος κυλίνδρου τρίτον μέρος ἐστὶ τοῦ τὴν base is polygon AEBFCGDH, and height the same as
αὐτὴν βάσιν ἔχοντος αὐτῷ καὶ ὕψος ἴσον· ὅπερ ἔδει δεῖξαι. the cylinder, is greater than the cylinder whose base is
circle ABCD. But, (it is) also less. For it is encompassed
iaþ neither (is it) greater than three times (the cone). Thus,
by it. The very thing is impossible. Thus, the cylinder is
not less than three times the cone. And it was shown that

the cylinder (is) three times the cone. Hence, the cone is
the third part of the cylinder.
Thus, every cone is the third part of the cylinder which
has the same base as it, and an equal height. (Which is)
the very thing it was required to show.

Proposition 11
.
Οἱ ὑπο τὸ αὐτὸ ὕψος ὄντες κῶνοι καὶ κύλινδροι πρὸς Cones and cylinders having the same height are to one
ἀλλήλους εἰσὶν ὡς αἱ βάσεις. another as their bases.
῎Εστωσαν ὑπὸ τὸ αὐτὸ ὕψος κῶνοι καὶ κύλινδροι, ὧν Let there be cones and cylinders of the same height
βάσεις μὲν [εἰσιν] οἱ ΑΒΓΔ, ΕΖΗΘ κύκλοι, ἄξονες δὲ οἱ whose bases [are] the circles ABCD and EFGH, axes
ΚΛ, ΜΝ, διάμετροι δὲ τῶν βάσεων αἱ ΑΓ, ΕΗ· λέγω, ὅτι KL and MN, and diameters of the bases AC and EG (re-
ἐστὶν ὡς ὁ ΑΒΓΔ κύκλος πρὸς τὸν ΕΖΗΘ κύκλον, οὕτως spectively). I say that as circle ABCD is to circle EFGH,
ὁ ΑΛ κῶνος πρὸς τὸν ΕΝ κῶνον. so cone AL (is) to cone EN.

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