Now, since, as the unit C is to the number D, so is D to E,
therefore the unit C measures the number D the same number of times as D measures E. [VII. Def. 20]
But the unit C measures the number D according to the units in D;
therefore the number D also measures E according to the units in D;
therefore D by multiplying itself has made E.
Again, since, as C is to the number D, so is E to A,
therefore the unit C measures the number D the same number of times as E measures A.
But the unit C measures the number D according to the units in D;
therefore E also measures A according to the units in D;
therefore D by multiplying E has made A.
For the same reason also F by multiplying itself has made G, and by multiplying G has made B.
And, since D by multiplying itself has made E and by multiplying F has made H,
therefore, as D is to F, so is E to H. [VII. 17]
For the same reason also, as D is to F, so is H to G. [VII. 18]
Therefore also, as E is to H, so is H to G.
Again, since D by multiplying the numbers E, H has made A, K respectively,
therefore, as E is to H, so is A to K. [VII. 17]
But, as E is to H, so is D to F;
therefore also, as D is to F, so is A to K.
Again, since the numbers D, F by multiplying H have made K, L respectively,
therefore, as D is to F, so is K to L. [VII. 18]
But, as D is to F, so is A to K;
therefore also, as A is to K, so is K to L.
Further, since F by multiplying the numbers H, G has made L, B respectively,
therefore, as H is to G, so is L to B. [VII. 17]
But, as H is to G, so is D to F;
therefore also, as D is to F, so is L to B.
But it was also proved that, as D is to F, so is A to K and K to L;
therefore also, as A is to K, so is K to L and L to B.
Therefore A, K, L, B are in continued proportion.
Therefore, as many numbers as fall between each of the numbers A, B and the unit C in continued proportion,
so many also will fall between A, B in continued proportion.
Q. E. D.
Complete text
Book I