Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/240

216 and for the same reason whatever multiple the arc EN is of the arc EF the same multiple is the sector EHN of the sector EHF. And if the arc BL be equal to the arc EN the sector BGL is equal to the sector EHN; and if the arc BL be greater than the arc EN the sector BGL is greater than the sector EHN; and if less, less.

Therefore, since there are four magnitudes, the two arcs BC, EF, and the two sectors BGC, EHF;

and that of the arc BC and of the sector BGC have been taken any equimultiples whatever, namely, the arc BL and the sector BGL; and of the arc EF and of the sector EHF have been taken any equimultiples whatever, namely, the arc EN and the sector BGL; and since it has been shewn that if the arc BL be greater than the arc EN, the sector BGL is greater than the sector EHN; and if equal, equal; and if less, less; therefore as the arc BC is to the arc EF, so is the sector BGC to the sector EHF. [V. Definition 5.

Wherefore, in equal circles &c.