*EUCLID'S ELEMENTS*.

PROPOSITION 2. *PROBLEM.*

From a given point to draw a straight line equal to a *given straight line.*

Let *A* be the given point, and *BC* the given straight line: it is required to draw from the point *A* a straight line equal to *BC*.

From the point *A* to *B* draw the straight line *AB*;

[*Post*. 1.

and on it describe the equilateral triangle *DAB*, [I. 1.

and produce the straight lines *DA*, *DB* to *E* and *F*. [*Post*. 2.

From the centre *B*, at the distance *BC*, describe the circle *CGH*, meeting *DF* at *G*. [*Post*. 3.

From the centre *D*, at the distance *DG*, describe the circle *GKL*, meeting *DE* at *L.* [*Post*. 3.

*AL* shall be equal to *BC*.

Because the point *B* is the centre of the circle *CGH*, *BC* is equal to *BG*. [*Definition* 15.

And because the point *D* is the centre of the circle *GKL*, *DL* is equal to *DG*; [*Definition* 15.

and *DA*, *DB* parts of them are equal; [*Definition* 24.

therefore the remainder *AL* is equal to the remainder BG. [*Axiom* 3.

But it has been shewn that *BC* is equal to *BG*;

therefore *AL* and *BC* are each of them equal to *BG*.

But things which are equal to the same thing are equal to one another. [*Axiom* 1.

Therefore *AL* is equal to *BC*.

Wherefore *from the given point A a straight line AL has been drawn equal to the given straight line BC*. q.e.f.

PROPOSITION 3. *PROBLEM*.

From the greater of two given straight lines to cut off a part equal to the less.

Let *AB* and *C* be the two given straight lines, of which