G.307 numbers fall numbers

VIII.10

If numbers fall between each of two numbers and an unit in continued proportion, however many numbers fall between each of them and an unit in continued proportion, so many also will fall between the numbers themselves in continued proportion.

numbers_fall_numbers()

prove 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

assert Let the numbers D, E and F, G fall between the two numbers A, B and the unit C in continued proportion

assert I say that, as many numbers as have fallen between each of the numbers A, B and the unit C in continued proportion, so many numbers will also fall between A, B in continued proportion

assert Let D by multiplying F make H, and let the numbers D, F by multiplying H make K, L

assert Since, as the unit C is to the number D, so is D to E,

∵ numbers_proportional_multiple_defassert the unit C measures the number D the same number of times as D measures E

assert The unit C measures the number D according to the units in D

assert the number D also measures E according to the units in D

assert D by multiplying itself has made E

assert Since, as C is to the number D, so is E to A,

assert the unit C measures the number D the same number of times as E measures A

assert The unit C measures the number D according to the units in D

assert E also measures A according to the units in D

assert D by multiplying E has made A

assert For the same reason also

assert F by multiplying itself has made G, and by multiplying G has made B

assert since D by multiplying itself has made E and by multiplying F has made H,

∵ number_multiplying_numbersassert as D is to F, so is E to H

assert For the same reason also,

∵ numbers_multiplying_numberassert As D is to F, so is H to G

assert also, as E is to H, so is H to G

assert D by multiplying the numbers E, H has made A, K,

∵ number_multiplying_numbersassert as E is to H, so is A to K

assert But, as E is to H, so is D to F

assert also, as D is to F, so is A to K

assert the numbers D, F by multiplying H have made K, L,

∵ numbers_multiplying_numberassert as D is to F, so is K to L

assert But, as D is to F, so is A to K

assert also, as A is to K, so is K to L

assert Further, since F by multiplying the numbers H, G has made L, B,

∵ number_multiplying_numbersassert as H is to G, so is L to B

assert But, as H is to G, so is D to F

assert also, as D is to F, so is L to B

assert It was also proved that, as D is to F, so is A to K and K to L

assert also, as A is to K, so is K to L and L to B

assert A, K, L, B are in continued proportion

∴ 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

Dependency Graph