Proposition 37

If a number is measured by any number, then the number which is measured has a part called by the same name as the measuring number.

Let the number A be measured by any number B.

I say that A has a part called by the same name as B.

java applet or image

Let there be as many units in C as the times that B measures A.

Since B measures A according to the units in C, and the unit D also measures the number C according to the units in it, therefore the unit D measures the number C the same number of times as B measures A.

VII.15

Therefore, alternately, the unit D measures the number B the same number of times as C measures A. Therefore, whatever part the unit D is of the number B, the same part is C of A also. But the unit D is a part of the number B called by the same name as it, therefore C is also a part of A called by the same name as B, so that A has a part C which is called by the same name as B.

Therefore, if a number is measured by any number, then the number which is measured has a part called by the same name as the measuring number.

Q.E.D.

Guide

 This proposition says that if b divides a, then a has a one-bth part (namely, a/b). For example, 3 divides 12, therefore 12 has a one-third part (namely, 4 = 12/3).

Use of this proposition.

This proposition is used in the proof of proposition VII.39.