214 | EUCLID'S ELEMENTS. |

And if the arc *BL* be equal to the arc EN, the angle BGL is equal to the angle EHN; [III. 27.

and if the arc *BL* be greater than the arc *EN*, the angle *BGL* is greater than the angle *EHN*; and if less, less.

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

and that of the arc *BC* and of the angle *BGC* have been taken any equimultiples whatever, namely, the arc *BL* and the angle *BGL*;

and of the arc *EF* and of the angle *EHF* have been taken any equimultiples whatever, namely, the arc *EN* and the *EHN*;

and since it has been shewn that if the arc *BL* be greater than the arc *EN*, the angle *BGL* is greater than the angle *EHN*; and if equal, equal; and if less, less;

therefore as the arc *BC* is to the arc *EF*, so is the angle *BGC* to the angle *EHF*. [V. Definition 5.

But as the angle *BGC* is to the angle *EHF*, so is the angle *BAC* to the angle *EDF*, [V. 15.

for each is double of each; [III. 20.

therefore, as the arc *BC* is to the arc *EF* so is the angle *BGC* to the angle *EHF*, and the angle *BAC* to the angle *EBF*.

Also as the arc *BC* is to the arc *EF*, so shall the sector *BGC* be to the sector *EHF*.

Join *BC*, *CK*, and in the arcs *BC*, *CK* take any points *X*, *O*, and join *BX*, *XC*, *CO*, *OK*.