Book 8
20
Therefore A, B are similar plane numbers; for their sides are proportional. Q. E. D. 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.
PROPOSITION 21.
21
If two mean proportional numbers fall between two numbers, the numbers are similar solid numbers.
21
For let two mean proportional numbers C, D fall between the two numbers A, B; I say that A, B are similar solid numbers.
21
For let three numbers E, F, G, the least of those which have the same ratio with A, C, D, be taken; [VII. 33 or VIII. 2] therefore the extremes of them E, G are prime to one another. [VIII. 3]
21
Now, since one mean proportional number F has fallen between E, G, therefore E, G are similar plane numbers. [VIII. 20]
21
Let, then, H, K be the sides of E, and L, M of G.
21
Therefore it is manifest from the theorem before this that E, F, G are continuously proportional in the ratio of H to L and that of K to M.
21
Now, since E, F, G are the least of the numbers which have the same ratio with A, C, D, and the multitude of the numbers E, F, G is equal to the multitude of the numbers A, C, D, therefore, ex aequali, as E is to G, so is A to D. [VII. 14]
