## Proposition 14

 If a square measures a square, then the side also measures the side; and, if the side measures the side, then the square also measures the square. Let A and B be square numbers, let C and D be their sides, and let A measure B. I say that C also measures D. Multiply C by D to make E. Then A, E, and B are continuously proportional in the ratio of C to D. as inVIII.11 And, since A, E, and B are continuously proportional, and A measures B, therefore A also measures E. And A is to E as C is to D, therefore C measures D. VIII.7 VII.Def.20 Next, let C measure D. I say that A also measures B. With the same construction, we can in a similar manner prove that A, E, and B are continuously proportional in the ratio of C to D. And since C is to D as A is to E, and C measures D, therefore A also measures E. VII.Def.20 And A, E, and B are continuously proportional, therefore A also measures B. Therefore, if a square measures a square, then the side also measures the side; and, if the side measures the side, then the square also measures the square. Q.E.D.
This proposition is to prove its contrapositive, VIII.16.

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 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