## 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. 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. 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. VII.15 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.
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.

#### Use of this proposition.

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

Next proposition: VII.38

Previous: VII.36

 Select from Book VII Book VII intro VII.Def.1-2 VII.Def.3-5 VII.Def.6-10 VII.Def.11-14 VII.Def.15-19 VII.Def.20 VII.Def.21 VII.Def.22 VII.1 VII.2 VII.3 VII.4 VII.5 VII.6 VII.7 VII.8 VII.9 VII.10 VII.11 VII.12 VII.13 VII.14 VII.15 VII.16 VII.17 VII.18 VII.19 VII.20 VII.21 VII.22 VII.23 VII.24 VII.25 VII.26 VII.27 VII.28 VII.29 VII.30 VII.31 VII.32 VII.33 VII.34 VII.35 VII.36 VII.37 VII.38 VII.39 Select book Book I Book II Book III Book IV Book V Book VI Book VII Book VIII Book IX Book X Book XI Book XII Book XIII Select topic Introduction Table of Contents Geometry applet About the text Euclid Web references A quick trip