If three numbers be in continued proportion, and the first be square, the third will also be square.
Let A, B, C be three numbers in continued proportion, and let A the first be square; I say that C the third is also square.
For, since between A, C there is one mean proportional number, B, therefore A, C are similar plane numbers. [VIII. 20]
But A is square; therefore C is also square. Q. E. D.
Euclid. Euclid's Elements. Sir Thomas Little Heath. New York. Dover. 1956.
The National Science Foundation provided support for entering this text.
Purchase a copy of this text (not necessarily the same edition) from
An XML version of this text is available for download,
with the additional restriction that you offer Perseus any modifications you make. Perseus provides credit for all accepted
changes, storing new additions in a versioning system.