Euclid, Elements, BOOK VIII., PROPOSITIONS, PROPOSITION 20.

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PROPOSITION 20.

If one mean proportional number fall between two numbers, the numbers will be similar plane numbers.

For let one mean proportional number C fall between the two numbers A, B;
I say that A, B are similar plane numbers.

Let D, E, the least numbers of those which have the same ratio with A, C, be taken; [VII. 33] therefore D measures A the same number of times that E measures C. [VII. 20]

Now, as many times as D measures A, so many units let there be in F; therefore F by multiplying D has made A, so that A is plane, and D, F are its sides.

Again, since D, E are the least of the numbers which have
the same ratio with C, B, therefore D measures C the same number of times that E measures B. [VII. 20]

As many times, then, as E measures B, so many units let there be in G;
therefore E measures B according to the units in G; therefore G by multiplying E has made B.

Therefore B is plane, and E, G are its sides.

Therefore A, B are plane numbers.

I say next that they are also similar.

For, <*> since F by multiplying D has made A, and by multiplying E has made C, therefore, as D is to E, so is A to C, that is, C to B. [VII. 17]

Again, <*> since E by multiplying F, G has made C, B respectively,
therefore, as F is to G, so is C to B. [VII. 17]

But, as C is to B, so is D to E; therefore also, as D is to E, so is F to G.

And alternately, as D is to F, so is E to G. [VII. 13]

Therefore A, B are similar plane numbers; for their sides
are proportional. Q. E. D. ¹

1 25. For, since F......27. C to B. The text has clearly suffered corruption here. It is not necessary to infer from other facts that, as D is to E, so is A to C; for this is part of the hypotheses (ll. 6, 7). Again, there is no explanation of the statement (l. 25) that F by multiplying E has made C. It is the statement and explanation of this latter fact which are alone wanted; after which the proof proceeds as in l. 28. We might therefore substitute for ll. 25-28 the following.

“For, since E measures C the same number of times that D measures A [l. 8], that is, according to the units in F [l. 10], therefore F by multiplying E has made C.

And, since E by multiplying F, G,” etc. etc.

Euclid. Euclid's Elements. Sir Thomas Little Heath. New York. Dover. 1956.

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Euclid, Elements
Thomas L. Heath, Sir Thomas Little Heath, Ed.

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Table of Contents:

PROPOSITION 20.

Euclid, Elements Thomas L. Heath, Sir Thomas Little Heath, Ed.

This text is part of:

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Table of Contents:

PROPOSITION 20.

Euclid, Elements
Thomas L. Heath, Sir Thomas Little Heath, Ed.