If there are two straight lines, and one of them is cut into any number of segments whatever, then the rectangle contained by the two straight lines equals the sum of the rectangles contained by the uncut straight line and each of the segments.

Let *A* and *BC* be two straight lines, and let *BC* be cut at random at the points *D* and *E.*

I say that the rectangle *A* by *BC* equals the sum of the rectangle *A* by *BD,* the rectangle *A* by *DE,* and the rectangle *A* by *EC.*

Draw *BF* from *B* at right angles to *BC.* Make *BG* equal to *A.* Draw *GH* through *G* parallel to *BC.* Through *D, E,* and *C* draw *DK, EL,* and *CH* parallel to *BG.*

Then *BH* equals the sum of *BK, DL,* and *EH.*

Now *BH* is the rectangle *A* by *BC,* for it is contained by *GB* and *BC,* and *BG* equals *A*; *BK* is the rectangle *A* by *BD,* for it is contained by *GB* and *BD,* and *BG* equals *A*; and *DL* is the rectangle *A* by *DE,* for *DK,* that is *BG,* equals *A.* Similarly also *EH* is the rectangle *A* by *EC.*

Therefore the rectangle *A* by *BC* equals the sum of the rectangle *A* by *BD,* the rectangle *A* by *DE,* and the rectangle *A* by *EC.*

Therefore *if there are two straight lines, and one of them is cut into any number of segments whatever, then the rectangle contained by the two straight lines equals the sum of the rectangles contained by the uncut straight line and each of the segments.*

Q.E.D.

As described in II.Def.1, the phrase “the rectangle contained by the two straight lines” means any rectangle constructed with two sides equal to the two given sides. In some sense this is the product of the two lines.
When the sides have names, such as *A* and *BC,* we will refer to that rectangle by “the rectangle *A* by *BC*” since that is a little clearer than Euclid’s terse “the rectangle *A, BC.*”

In this proposition Euclid proves that if

In modern algebraic notation this could be stated as follows:
If *y* = *y*_{1} + *y*_{2} + ... + *y _{n},* then

Here *x* and the various *y _{i}*’s are all lines, and