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Proposition 5

If a number is part of a number, and another is the same part of another, then the sum is also the same part of the sum that the one is of the one.
Let the number A be a part of BC, and another number D be the same part of another number EF that A is of BC. I say that the sum of A and D is also the same part of the sum of BC and EF that A is of BC.
Since, whatever part A is of BC, D is also the same part of EF, therefore, there are as many numbers equal to D in EF as there are in BC equal to A. |
Divide BC into the numbers equal to A, namely BG and GC, and EF into the numbers equal to D, namely EH and HF. Then the multitude of BG and GC equals the multitude of EH and HF. And, since BG equals A, and EH equals D, therefore the sum of BG and EH also equals the sum of A and D. For the same reason the sum of GC and HF also equals the sum of A and D.
Therefore there are as many numbers in BC and EF equal to A and D as there are in BC equal to A. Therefore, the sum of BC and EF is the same multiple of the sum of A and D that BC is of A. Therefore, the sum of A and D is the same part of the sum of BC and EF that A is of BC.
Therefore, if a number is part of a number, and another is the same part of another, then the sum is also the same part of the sum that the one is of the one.
Q.E.D.

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This is the first of four propositions that deal with distributivity of division and multiplication over addition and subtraction. This one says division distributes over addition. Algebraically, if a = b/n and d = e/n, then a + d = (b + e)/n.

This proposition is used in the proofs of five of the next seven propositions.

Book VII Introduction - Proposition VII.4 - Proposition VII.6.

© 1996
D.E.Joyce
Clark University