Bibliotheca Polyglotta

Euclid: Elementa

eng

Proposition 31. If there be these three quantities, and those three quantities, such as the ratio of the first and the second of these is greater than that between the first and the second of those, while the ratio of the second and the third of these is greater than that of the second and third of those, if arranged in order. then ex equali the ratio between the first and the third of these will also be greater than the ratio between the first and the third of those. If a:b>d:e and b:c>e:f then a:c>d:f.¹

1. Translation of the Chinese by Engelfriet, p. 258.

lat Clavius

No Latin

kin 幾何原本 p. 272-273

第三十一題
此三幾何。彼三幾何。此第一與二之比例。大於彼第一與二之比例。此第二與三之比例。大於彼第二與三之比例。如是序者。以平理推。則此第一與三之比例。亦大於彼第一與三之比例。
解曰。甲、乙、丙。此三幾何。丁、戊、己。彼三幾何。而甲與乙之比例。大於丁與戊。乙與丙之比例。大於戊與己。如是序者。題言以平理推。則甲與丙之比例。亦大於丁與己。
論曰。試作庚與丙之比例。若戊與己。卽乙與丙之比例。大於庚與丙。而乙幾何大於庚本篇　\\　十是甲與小庚之比例。大於甲與大乙矣。本篇　\\　八夫甲與乙之比例。元大於丁與戊。卽(p. 二七三)甲與庚之比例。更大於丁與戊也次作辛與庚之比例若丁與戊卽甲與庚之比例。亦大於辛與庚。而甲幾何大於辛。本篇　\\　十是大甲與丙之比例。大於小辛與丙矣。本篇　\\　八夫辛與丙之比例。以平理推之。若丁與己也。本篇　\\　廿二則甲與丙之比例。大於丁與己也。

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