Kitap 9
19
Next, let A not measure D; I say that it is impossible to find a fourth proportional number to A, B, C.
19
For, if possible, let E have been found; therefore the product of A, E is equal to the product of B, C. [VII. 19]
19
But the product of B, C is D; therefore the product of A, E is also equal to D.
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Therefore A by multiplying E has made D; therefore A measures D according to E, so that A measures D.
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But it also does not measure it: which is absurd.
19
Therefore it is not possible to find a fourth proportional number to A, B, C when A does not measure D.
19
Next, let A, B, C not be in continued proportion, nor the extremes prime to one another.
19
And let B by multiplying C make D.
19
Similarly then it can be proved that, if A measures D, it is possible to find a fourth proportional to them, but, if it does not measure it, impossible. Q. E. D.
PROPOSITION 20.
20
Prime numbers are more than any assigned multitude of prime numbers.
20
Let A, B, C be the assigned prime numbers; I say that there are more prime numbers than A, B, C.
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For let the least number measured by A, B, C be taken, and let it be DE; let the unit DF be added to DE.
20
Then EF is either prime or not.
20
First, let it be prime; then the prime numbers A, B, C, EF have been found which are more than A, B, C.
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Next, let EF not be prime; therefore it is measured by some prime number. [VII. 31]
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Let it be measured by the prime number G.
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I say that G is not the same with any of the numbers A, B, C.
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For, if possible, let it be so.
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Now A, B, C measure DE; therefore G also will measure DE.
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But it also measures EF.
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Therefore G, being a number, will measure the remainder, the unit DF: which is absurd.
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Therefore G is not the same with any one of the numbers A, B, C.
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And by hypothesis it is prime.
20
Therefore the prime numbers A, B, C, G have been found which are more than the assigned multitude of A, B, C. Q. E. D.
PROPOSITION 21.
21
If as many even numbers as we please be added together, the whole is even.
