*BOOK I*. 3, 4.

*AB* is the greater: it is required to cut off from *AB* the greater, a part equal to C the less.

From the point *A* draw the straight line *AD* equal to *C*; [I. 2.

and from the centre *A*, at the distance *AD*, describe the circle *DEF* meeting *AB* at *E*. [*Postulate* 3.

*AE* shall be equal to *C*.

Because the point *A* is the centre of the circle *DEF*, *AE* is equal to *AD*. [*Definition* 15.

But *C* is equal to *AD*. [*Construction*.

Therefore *AE* and *C* are each of them equal to *AD*.

Therefore *AE* is equal to *C*. [*Axiom* 1.

Wherefore *from AB the greater of two given straight lines a part AE has been cut off equal to C the less.* q.e.f.

PROPOSITION 4. *THEOREM*.

If two triangles have two sides of the one equal to two sides of the other, each to each, and have also the angles contained by those sides equal to one another, they shall also have their bases or third sides equal; and the two triangles shall be equal, and their other angles shall be equal, each to each, namely those to which the equal sides are opposite.

]Let *ABC*, *DEF* be two triangles which have the two sides *AB*, *AC* equal to the two sides *DE*, *DF*, each to each, namely, *AB* to *DE*, and *AC* to *DF*, and the angle *BAC* equal to the angle *EDF*: the base *BC* shall be equal to the base *EF*, and the triangle *ABC* to the triangle *DEF*, and the other angles shall be equal, each to each, to which the equal sides are opposite, namely, the angle *ABC* to the angle *DEF*, and the angle *ACB* to the angle *DFE*.