To cut off a prescribed part from a given straight line.

Let *AB* be the given straight line.

It is required to cut off from *AB* a prescribed part.

Let the third part be that prescribed.

Draw a straight line *AC* through from A containing with *AB* any angle. Take a point *D* random on *AC,* and make *DE* and *EC* equal to *AD.*

Join *CB,* and draw *DF* through *D* parallel to it.

Then, since *DF* is parallel to a side *CB* of the triangle *ABC,* therefore, proportionally, *AD* is to *DC* as *AF* is to *FB.*

But *DC* is double *AD,* therefore *FB* is also double *AF,* therefore *AB* is triple of *AF.*

Therefore from the given straight line *AB* the prescribed third part *AF* has been cut off.

Q.E.F.

Simson complained that proving the general case by using a specific case, the one-third part, “is not at all like Euclid’s manner.” But it is very much Euclid’s manner throughout books V and VI to prove a general numerical statement with a specific numerical value.

Abu’l-Abbas al-Fadl ibn al-Nayrizi (fl. c. 897, d. c. 922) wrote a commentary on the first ten books of the Elements. He gives another construction to divide a line AB into n equal parts. First, construct equal perpendiculars at A and B in opposite directions, mark off n – 1 equal parts on each of them, and connect the points as illustrated. The diagram shows AB divided into five equal parts.
Al-Nayrizi’s construction takes considerably less work than Euclid’s. The proof that this construction is valid is about the same length as that for Euclid’s construction. |