If as many numbers as we please beginning from an unit be in continued proportion, by however many prime numbers the last is measured, the next to the unit will also be measured by the same.
若从单位开始任意多个数成连比例，则末项被多少质数度量，紧邻单位的项也被同样的质数度量。
从单位起连比例 A、B、C、D；D 被质数 E 量度，设 D = E·F、C = E·G、B = E·H；推得 E 量 A，故 D 的素因子也是 A 的素因子。
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Let there be as many numbers as we please, A, B, C, D, beginning from an unit, and in continued proportion; I say that, by however many prime numbers D is measured, A will also be measured by the same. For let D be measured by any prime number E; I say that E measures A. For suppose it does not; now E is prime, and any prime number is prime to any which it does not measure; [VII. 29] therefore E, A are prime to one another. And, since E measures D, let it measure it according to F, therefore E by multiplying F has made D. Again, since A measures D according to the units in C, [IX. 11 and Por.] therefore A by multiplying C has made D. But, further, E has also by multiplying F made D; therefore the product of A, C is equal to the product of E, F.
设从单位开始的连比例数A、B、C、D，且D被质数E度量。假设E不度量A，则E与A互质。
Therefore, as A is to E, so is F to C. [VII. 19] But A, E are prime, primes are also least, [VII. 21] and the least measure those which have the same ratio the same number of times, the antecedent the antecedent and the consequent the consequent; [VII. 20] therefore E measures C. Let it measure it according to G; therefore E by multiplying G has made C. But, further, by the theorem before this, A has also by multiplying B made C. [IX. 11 and Por.] Therefore the product of A, B is equal to the product of E, G.
由于E度量D，设D=E×F；又A度量D得A×C=D，故A×C=E×F，从而A:E=F:C。因A、E互质且为最小，故E度量C，设C=E×G。
Therefore, as A is to E, so is G to B. [VII. 19] But A, E are prime, primes are also least, [VII. 21] and the least numbers measure those which have the same ratio with them the same number of times, the antecedent the antecedent and the consequent the consequent: [VII. 20] therefore E measures B. Let it measure it according to H; therefore E by multiplying H has made B. But further A has also by multiplying itself made B; [IX. 8] therefore the product of E, H is equal to the square on A. Therefore, as E is to A, so is A to H. [VII. 19] But A, E are prime, primes are also least, [VII. 21] and the least measure those which have the same ratio the same number of times, the antecedent the antecedent and the consequent the consequent; [VII. 20] therefore E measures A, as antecedent antecedent.
由前定理，A×B=C，故A×B=E×G，得A:E=G:B。同理E度量B，设B=E×H，则A×A=B=E×H，故E:A=A:H。
But, again, it also does not measure it: which is impossible. Therefore E, A are not prime to one another. Therefore they are composite to one another. But numbers composite to one another are measured by some number. [VII. Def. 14] And, since E is by hypothesis prime, and the prime is not measured by any number other than itself, therefore E measures A, E, so that E measures A.
因A、E互质且为最小，E度量A，与假设矛盾。故E与A不互质，而为合数。因E为质数，只能度量自身，故E度量A。