Page 485 - Euclid's Elements of Geometry

ST EW ibþ.

ELEMENTS BOOK 12

ὡς [ἡ] ἐν τῇ ἑτέρᾳ μία πυραμὶς τρίγωνον ἔχουσα βάσιν both equal in number, and corresponding, to the wholes
πρὸς τὴν ἐν τῇ ἑτέρᾳ μίαν πυραμίδα τρίγωνον ἔχουσαν [Prop. 6.20]. As one pyramid having a triangular base in
βάσιν, οὕτως καὶ ἅπασαι αἱ ἐν τῇ ἑτέρᾳ πυραμίδι πυραμίδες the former (pyramid having a polygonal base is) to one
τριγώνους ἔχουσαι βάσεις πρὸς τὰς ἐν τῇ ἑτέρᾳ πυραμίδι pyramid having a triangular base in the latter (pyramid
πυραμίδας τριγώνους βάσεις ἐχούσας, τουτέστιν αὐτὴ ἡ having a polygonal base), so (the sum of) all the pyra-
πολύγωνον βάσιν ἔχουσα πυραμὶς πρὸς τὴν πολύγωνον mids having triangular bases in the former pyramid will
βάσιν ἔχουσαν πυραμίδα. ἡ δὲ τρίγωνον βάσιν ἔχουσα πυ- also be to (the sum of) all the pyramids having triangu-
ραμὶς πρὸς τὴν τρίγωνον βάσιν ἔχουσαν ἐν τριπλασίονι λόγῳ lar bases in the latter pyramid [Prop. 5.12]—that is to
jþ sides [Prop. 12.8]. Thus, a (pyramid) having a polygonal
ἐστὶ τῶν ὁμολόγον πλευρῶν· καὶ ἡ πολύγωνον ἄρα βάσιν say, the (former) pyramid itself having a polygonal base
ἔχουσα πρὸς τὴν ὁμοίαν βάσιν ἔχουσαν τριπλασίονα λόγον to the (latter) pyramid having a polygonal base. And a
ἔχει ἤπερ ἡ πλευρὰ πρὸς τὴν πλευράν. pyramid having a triangular base is to a (pyramid) hav-
ing a triangular base in the cubed ratio of corresponding

base also has to to a (pyramid) having a similar base the
cubed ratio of a (corresponding) side to a (correspond-
ing) side.

Proposition 9
.
X
Τῶν ἴσων πυραμίδων καὶ τριγώνους βάσεις ἐχουσῶν The bases of equal pyramids which also have trian-
R H equal. H R O P L
ἀντιπεπόνθασιν αἱ βάσεις τοῖς ὕψεσιν· καὶ ὧν πυραμίδων gular bases are reciprocally proportional to their heights.
τριγώνους βάσεις ἐχουσῶν ἀντιπεπόνθασιν αἱ βάσεις τοῖς And those pyramids which have triangular bases whose
ὕψεσιν, ἴσαι εἰσὶν ἐκεῖναι. bases are reciprocally proportional to their heights are

A

E D Z B G E D G B A C M

Q
F
῎Εστωσαν γὰρ ἴσαι πυραμίδες τριγώνους βάσεις ἔχουσαι For let there be (two) equal pyramids having the tri-
τὰς ΑΒΓ, ΔΕΖ, κορυφὰς δὲ τὰ Η, Θ σημεῖα· λέγω, ὅτι τῶν angular bases ABC and DEF, and apexes the points G
ΑΒΓΗ, ΔΕΖΘ πυραμίδων ἀντιπεπόνθασιν αἱ βάσεις τοῖς and H (respectively). I say that the bases of the pyramids
ὕψεσιν, καί ἐστιν ὡς ἡ ΑΒΓ βάσις πρὸς τὴν ΔΕΖ βάσιν, ABCG and DEFH are reciprocally proportional to their
οὕτως τὸ τῆς ΔΕΖΘ πυραμίδος ὕψος πρὸς τὸ τῆς ΑΒΓΗ heights, and (so) that as base ABC is to base DEF, so
πυραμίδος ὕψος. the height of pyramid DEFH (is) to the height of pyra-
Συμπεπληρώσθω γὰρ τὰ ΒΗΜΛ, ΕΘΠΟ στερεὰ παραλ- mid ABCG.
ληλεπίπεδα. καὶ ἐπεὶ ἴση ἐστὶν ἡ ΑΒΓΗ πυραμὶς τῇ ΔΕΖΘ For let the parallelepiped solids BGML and EHQP
πυραμίδι, καί ἐστι τῆς μὲν ΑΒΓΗ πυραμίδος ἑξαπλάσιον have been completed. And since pyramid ABCG is
τὸ ΒΗΜΛ στερεόν, τῆς δὲ ΔΕΖΘ πυραμίδος ἑξαπλάσιον equal to pyramid DEFH, and solid BGML is six times
τὸ ΕΘΠΟ στερεόν, ἴσον ἄρα ἐστὶ τὸ ΒΗΜΛ στερεὸν τῷ pyramid ABCG (see previous proposition), and solid
ΕΘΠΟ στερεῷ. τῶν δὲ ἴσων στερεῶν παραλληλεπιπώδων EHQP (is) six times pyramid DEFH, solid BGML is

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