Heron of Alexandria, Mensuration

LCL 362: 480-481

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Greek Mathematics

ἤχθωσαν αἱ ΔΗ, ΕΘ. δέδεικται δὲ Διονυσοδώρῳ ἐν τῷ Περὶ τῆς σπείρας ἐπιγραφομένῳ, ὅτι ὃν λόγον ἔχει ὁ ΒΓΔΕ κύκλος πρὸς τὸ ἥμισυ τοῦ ΔΕΗΘ παραλληλογράμμου, τοῦτον ἔχει καὶ ἡ γεννηθεῖσα σπεῖρα ὑπὸ τοῦ ΒΓΔΕ κύκλου πρὸς τὸν κύλινδρον, οὗ ἄξων μέν ἐστιν ὁ ΗΘ, ἡ δὲ ἐκ τοῦ κέντρου τῆς βάσεως ἡ ΕΘ. ἐπεὶ οὖν ἡ ΒΓ μονάδων ι̅β̅ ἐστίν, ἡ ἄρα ΖΓ ἔσται μονάδων ς̅ ἔστι δὲ καὶ ἡ ΑΓ μονάδων η̄· ἔσται ἄρα ἡ ΑΖ μονάδων ι̅δ̅ τουτέστιν ἡ ΕΘ, ἥτις ἐστὶν ἐκ τοῦ κέντρου τῆς βάσεως τοῦ εἰρημένου κυλίνδρου· δοθεὶς ἄρα ἐστὶν ὁ κύκλος· ἀλλὰ καὶ ὁ ἄξων δοθείς· ἔστιν γὰρ μονάδων ι̅β̅ ἐπεὶ καὶ ἡ ΔΕ. ὥστε δοθεὶς καὶ ὁ εἰρημένος κύλινδρος· καὶ ἔστι τὸ ΔΘ παραλληλόγραμμον <δοθέν>1· ὥστε καὶ τὸ ἥμισυ αὐτοῦ. ἀλλὰ καὶ ὁ ΒΓΔΕ κύκλος· δοθεῖσα γὰρ ἡ ΓΒ διάμετρος. λόγος ἄρα τοῦ ΒΓΔΕ κύκλου πρὸς τὸ ΔΘ παραλληλόγραμμον δοθείς· ὥστε καὶ τῆς σπείρας πρὸς τὸν κύλινδρον λόγος ἔστι δοθείς. καὶ ἔστι δοθείς ὁ κύλινδρος· δοθὲν ἄρα καὶ τὸ στερεὸν τῆς σπείρας.

Συντεθήσεται δὴ ἀκολούθως τῇ ἀναλύσει οὕτως. ἄφελε ἀπὸ τῶν κ̄ τὰ ι̅β̅ λοιπὰ η̅. καὶ πρόσθες τὰ κ̅· γίγνεται κ̅η̅ καὶ μέτρησον κύλινδρον, οὗ ἡ μὲν διάμετρος τῆς βάσεώς ἐστι μονάδων κ̅η̅ τὸ δὲ ὕψος ι̅β̅ καὶ γίγνεται τὸ στερεὸν αὐτοῦ ͵̅ζ̅τ̅ς̅β̅ καὶ μέτρησον κύκλον, οὗ διάμετρός ἐστι μονάδων ι̅β·̅ γίγνεται τὸ ἐμβαδὸν αὐτοῦ, καθὼς ἐμάθομεν, ρ̅ι̅γ̅ ζʹ· καὶ λαβὲ τῶν κ̅η̅ τὸ ἥμισυ· γίγνεται ι̅δ̅ ἐπὶ τὸ ἥμισυ τῶν ι̅β̅· γίγνεται π̅δ̅· καὶ πολλαπλασιάσας

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Mensuration: Heron of Alexandria

ΑΒ. Now it is proved by Dionysodorusa in the book which he wrote On the Spire that the circle ΒΓΔΕ bears to half of the parallelogram ΔΕΗΘ the same ratio as the spire generated by the circle ΒΓΔΕ bears to the cylinder having ΗΘ for its axis and ΕΘ for the radius of its base. Now, since ΒΓ is 12, ΖΓ will be 6. But ΑΓ is 8; therefore ΑΖ will be 14, and likewise ΕΘ, which is the radius of the base of the aforesaid cylinder. Therefore the circle is given; but the axis is also given; for it is 12, since this is the length of ΔΕ. Therefore the aforesaid cylinder is also given; and the parallelogram ΔΘ is given, so that its half is also given. But the circle ΒΓΔΕ is also given; for the diameter ΓΒ is given. Therefore the ratio of the circle ΒΓΔΕ to the parallelogram is given; and so the ratio of the spire to the cylinder is given. And the cylinder is given; therefore the volume of the spire is also given.

Following the analysis, the synthesis may thus be done. Take 12 from 20; the remainder is 8. And add 20; the result is 28. Let the measure be taken of the cylinder having for the diameter of its base 28 and for height 12; the resulting volume is 7392. Now let the area be found of a circle having a diameter 12; the resulting area, as we learnt, is 1131. Take the half of 28; the result is 14. Multiply it by the half of 12; the result is 84. Now multiply

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DOI: 10.4159/DLCL.heron_alexandria-mensuration.1941