2/35-pr-14-lemma-2
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2/35-pr-14-lemma-2 |
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| LEMMA. | LEMMA |
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| Sit linea AC maior ipsa DF, et habeat AB ad BC maiorem rationem quam DE ad EF: dico, AB ipsa DE esse maiorem. | Let AC be a line which is longer than DF; and let the ratio of AB to BC be greater than that of DE to EF. Then, I say, AB is greater than DE. |
| Quia enim AB ad BC maiorem rationem habet quam DE ad EF; quam rationem habet AB ad BC, hanc habebit DE ad minorem quam EF. Habeat ad EG: et quia AB ad BC est ut DE ad EG, erit, componendo et per conversionem rationis, ut CA ad AB, ita GD ad DE: est autem CA maior GD: ergo BA ipsa de maior erit. | For, if AB bears to BC a ratio greater than that of DE to EF, then DE will bear to some length shorter than EF, the same ratio which AB bears to BC. Call this length EG; then since AB:BC = DE:EG, it follows, componendo et convertendo, that CA:AB = GD:DE. But since CA is greater than GD, it follows that BA is greater than DE. |
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2/35-pr-14-lemma-2 |
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