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Euclid: Elementa

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gre X,23
Πόρισμα
Ἐκ δὴ τούτου φανερόν, ὅτι τὸ τῷ μέσῳ χωρίῳ σύμμετρον μέσον ἐστίν.
[δύνανται γὰρ αὐτὰ εὐθεῖαι, αἵ εἰσι δυνάμει σύμμετροι, ὧν ἡ ἑτέρα μέση:
ὥστε καὶ ἡ λοιπὴ μέση ἐστίν.]
Ὡσαύτως δὲ τοῖς ἐπὶ τῶν ῥητῶν εἰρημένοις καὶ ἐπὶ τῶν μέσων ἐξακολουθεῖ, τὴν τῇ μέσῃ μήκει σύμμετρον λέγεσθαι μέσην καὶ σύμμετρον αὐτῇ μὴ μόνον μήκει, ἀλλὰ καὶ δυνάμει, ἐπειδήπερ καθόλου αἱ μήκει σύμμετροι πάντως καὶ δυνάμει.
ἐὰν δὲ τῇ μέσῃ σύμμετρός τις ᾖ δυνάμει, εἰ μὲν καὶ μήκει, λέγονται καὶ οὕτως μέσαι καὶ σύμμετροι μήκει καὶ δυνάμει, εἰ δὲ δυνάμει μόνον, λέγονται μέσαι δυνάμει μόνον σύμμετροι.
Pic2759
eng
PORISM.
From this it is manifest that an area commensurable with a medial area is medial.
[And in the same way as was explained in the case of rationals [Lemma following X. 18] it follows, as regards medials, that a straight line commensurable in length with a medial straight line is called medial and commensurable with it not only in length but in square also, since, in general, straight lines commensurable in length are always commensurable in square also.
But, if any straight line be commensurable in square with a medial straight line, then, if it is also commensurable in length with it, the straight lines are called, in this case too, medial and commensurable in length and in square, but, if in square only, they are called medial straight lines commensurable in square only.]
lat Sic
(Porisma)
Ex hoc ergo manifestum quoniam quod medio spatio commensurabile medium est. Possunt enim ipsa recte que sunt potentia commensurabiles, quarum altera media, quare et reliqua media est.
Similiter autem eis que in ritis dicta sunt et in mediis consequitur medie commensurabilem longitudine dici mediam et commensurabilem ipsam non solum longitudine sed et potentia quoniam quod universaliter que longitudine commensurabiles, omnino et potentia.
Si vero medie commensurabilis aliqua fuerit potentia, siquidem et longitudine, dicuntur et ita medie et commensurabiles longitudine et potentia. Si vero potentia solum, dicuntur medie potentia solum commensurabiles.
Sunt vero rursum et alie recte que longitudine quidem incommensurabiles sunt medie, potentia vero solum commensurabiles et dicuntur rursum medie eo quod commensurabiles sint potentia medie, et commensurabiles sibi invicem secundum quod medie, sed commensurabiles ad invicem vel longitudine manifeste et potentia vel potentia solum. Et si que quidem longitudine, dicuntur et ipse medie longitudine commensurabiles consequente quoniam et potentia. Si vero potentia solum sunt commensurabiles, dicuntur et ita medie potentia Solum commensurabiles.
Quoniam vero medie commensurabiles sunt, ita demonstrandum: quoniam que medie medie alicui commensurabiles sunt, que vero eidem commensurabilia, et sibi invicem sunt commensurabilia, que ergo medie commensurabiles sunt.
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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san
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lat Clavius
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kin 幾何原本
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