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Proposition 21

If as many even numbers as we please are added together, then the sum is even.
Let as many even numbers as we please, AB, BC, CD, and DE, be added together. I say that the sum AE is even. |
Since each of the numbers AB, BC, CD, and DE is even, therefore each has a half part, so that the sum AE also has a half part. But an even number is that which is divisible into two equal parts, therefore AE is even. | VII.Def.6
Therefore, if as many even numbers as we please are added together, then the sum is even.
Q.E.D.

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This proposition is used in the next two and in IX.28.

Book IX Introduction - Proposition IX.20 - Proposition IX.22.

© 1996
D.E.Joyce
Clark University