| If any number of numbers are proportional, then one of the antecedents is to one of the consequents as the sum of the antecedents is to the sum of the consequents. | ||
| Let A, B, C, and D be as many numbers as we please in proportion, so that A is to B as C is to D.
I say that A is to B as the sum of A and C is to the sum of B and D. |
||
| Since A is to B as C is to D, therefore A is the same part or parts of B as C is of D. Therefore the sum of A and C is the same part or parts of the sum of B and D that A is of B. | VII.Def.20
VII.5 VII.6 | |
| Therefore A is to B as the sum of A and C is to the sum of B and D. | VII.Def.20 | |
| Therefore, if any number of numbers are proportional, then one of the antecedents is to one of the consequents as the sum of the antecedents is to the sum of the consequents. | ||
| Q.E.D. | ||
then each of these ratios also equals the ratio
Euclid takes n to be 2 in his proof.
This proposition is used in VII.15 , VII.20, and IX.35.
Book VII Introduction - Proposition VII.11 - Proposition VII.13.