## Proposition 38

 If a number has any part whatever, then it is measured by a number called by the same name as the part. Let the number A have any part whatever, B, and let C be a number called by the same name as the part B. I say that C measures A. Since B is a part of A called by the same name as C, and the unit D is also a part of C called by the same name as it, therefore the part B of A is the same part of the unit D of the number C. Therefore the unit D measures the number C the same number of times that B measures A. Therefore, alternately, the unit D measures the number B the same number of times that C measures A. Therefore C measures A. VII.15 Therefore, if a number has any part whatever, then it is measured by a number called by the same name as the part. Q.E.D.
This proposition says that if a has a one-cth part of a, then c divides a. For example, 12 has a one-third part, 3 divides 12. This is a converse of the last proposition.

#### Use of this proposition.

This proposition is used in the proof of the next proposition.

Next proposition: VII.39

Previous: VII.37

 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