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G.257 number part number — Heath¶

If a number be a part of a number, and another be the same part of another, the sum will also be the same part of the sum that the one is of the one.

For let the number A be a part of / B C /, and another, D, the same part of another / E F / that A is of / B C /;

I say that the sum of A, D is also the same part of the sum of / B C /, / E F / that A is of / B C /.

For since, whatever part A is of / B C /, D is also the same part of / E F /,

therefore, as many numbers as there are in / B C / equal to A, so many numbers are there also in / E F / equal to D.

Let / B C / be divided into the numbers equal to A, namely / B G /, / G C /, and / E F / into the numbers equal to D, namely / E H /, / H F /;

then the multitude of / B G /, / G C / will be equal to the multitude of / E H /, / H F /.

And, since / B G / is equal to A, and / E H / to D,

therefore / B G /, / E H / are also equal to A, D.

For the same reason

/ G C /, / H F / are also equal to A, D.

Therefore, as many numbers as there are in / B C / equal to A, so many are there also in / B C /, / E F / equal to A, D.

Therefore, whatever multiple / B C / is of A, the same multiple also is the sum of / B C /, / E F / of the sum of A, D.

Therefore, whatever part A is of / B C /, the same part also is the sum of A, D of the sum of / B C /, / E F /.

Q. E. D.

Dependency Graph¶