## Proposition 7

 If a composite number multiplied by any number makes some number, then the product is solid. Let the composite number A multiplied by any number B make C. I say that C is solid. Since A is composite, it is measured by some number D. Let there be as many units in E as times that D measures A VII.Def.13 Since D measures A according to the units in E, therefore E multiplied by D makes A. And, since A multiplied by B makes C, and A is the product of D and E, therefore the product of D and E multiplied by B makes C. VII.Def.15 Therefore C is solid, and D, E, and B are its sides. Therefore, if a composite number multiplied by any number makes some number, then the product is solid. Q.E.D.
Numbers with at least two factors are plain numbers; those with at least three are solid numbers.

Perhaps Euclid takes extra steps that we would miss because he sees "d measures a a number e times" as saying something different from the product of d and e equals a."

Next proposition: IX.8

Previous: IX.6

 Select from Book IX Book IX intro IX.1 IX.2 IX.3 IX.4 IX.5 IX.6 IX.7 IX.8 IX.9 IX.10 IX.11 IX.12 IX.13 IX.14 IX.15 IX.16 IX.17 IX.18 IX.19 IX.20 IX.21 IX.22 IX.23 IX.24 IX.25 IX.26 IX.27 IX.27 IX.28 IX.29 IX.30 IX.31 IX.32 IX.33 IX.34 IX.35 IX.36 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