## Proposition 24

 If two numbers have to one another the ratio which a square number has to a square number, and the first is square, then the second is also a square. Let the two numbers A and B have to one another the ratio which the square number C has to the square number D, and let A be square. I say that B is also square. Since C and D are square, C and D are similar plane numbers. Therefore one mean proportional number falls between C and D. VIII.18 And C is to D as A is to B, therefore one mean proportional number falls between A and B also. And A is square, therefore B is also square. VIII.18 VIII.22 Therefore, if two numbers have to one another the ratio which a square number has to a square number, and the first is square, then the second is also a square. Q.E.D.
The proof of this proposition is straightforeward.

Next proposition: VIII.25

Previous: VIII.23

 Select from Book VIII Book VIII intro VIII.1 VIII.2 VIII.3 VIII.4 VIII.5 VIII.6 VIII.7 VIII.8 VIII.9 VIII.10 VIII.11 VIII.12 VIII.13 VIII.14 VIII.15 VIII.16 VIII.17 VIII.18 VIII.19 VIII.20 VIII.21 VIII.22 VIII.23 VIII.24 VIII.25 VIII.26 VIII.27 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