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Euclid: Elementa

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Euclid: Elementa
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gre X,18
Λῆμμα
Ἐπεὶ δέδεικται, ὅτι αἱ μήκει σύμμετροι πάντως καὶ δυνάμει [εἰσὶ σύμμετροι], αἱ δὲ δυνάμει οὐ πάντως καὶ μήκει, ἀλλὰ δὴ δύνανται μήκει καὶ σύμμετροι εἶναι καὶ ἀσύμμετροι, φανερόν, ὅτι, ἐὰν τῇ ἐκκειμένῃ ῥητῇ σύμμετρός τις ᾖ μήκει, λέγεται ῥητὴ καὶ σύμμετρος αὐτῇ οὐ μόνον μήκει, ἀλλὰ καὶ δυνάμει, ἐπεὶ αἱ μήκει σύμμετροι πάντως καὶ δυνάμει.
Pic2753
eng
LEMMA.
[Since it has been proved that straight lines commensurable in length are always commensurable in square also, while those commensurable in square are not always commensurable in length also, but can of course be either commensurable or incommensurable in length, it is manifest that, if any straight line be commensurable in length with a given rational straight line, it is called rational and commensurable with the other not only in length but in square also, since straight lines commensurable in length are always commensurable in square also.
lat Sic
(Lemma)
Quoniam vero demonstratum est quoniam que longitudine omnino et potentia sunt commensurabiles. Que vero potentia non omnino et longitudine, sed manifeste possunt et longitudine commensurabiles esse et incommensurabiles. Manifeste quoniam si proposite riti commensurabilis aliqua fuerit longitudine, dicetur riti et commensurabilis ipsi non solum longitudine sed et potentia. Que enim longitudine commensurabiles omnino et potentia.
lat Gerard
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lat Adelard
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lat Hermann
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ara Uppsala
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ara Tuṣi
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ara Nairizi
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per Shirazi
.
san
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lat Clavius
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kin 幾何原本
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