Book 7
2
Therefore CF is a common measure of AB, CD.
2
I say next that it is also the greatest.
2
For, if CF is not the greatest common measure of AB, CD, some number which is greater than CF will measure the numbers AB, CD.
2
Let such a number measure them, and let it be G.
2
Now, since G measures CD, while CD measures BE, G also measures BE.
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But it also measures the whole BA; therefore it will also measure the remainder AE.
2
But AE measures DF; therefore G will also measure DF.
2
But it also measures the whole DC; therefore it will also measure the remainder CF, that is, the greater will measure the less: which is impossible.
2
Therefore no number which is greater than CF will measure the numbers AB, CD; therefore CF is the greatest common measure of AB, CD.
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Porism. From this it is manifest that, if a number measure two numbers, it will also measure their greatest common measure.
2
Q. E. D.
PROPOSITION 3.
3
Given three numbers not prime to one another, to find their greatest common measure.
3
Let A, B, C be the three given numbers not prime to one another; thus it is required to find the greatest common measure of A, B, C.
3
For let the greatest common measure, D, of the two numbers A, B be taken; [VII. 2] then D either measures, or does not measure, C.
3
First, let it measure it.
3
But it measures A, B also; therefore D measures A, B, C; therefore D is a common measure of A, B, C.
3
I say that it is also the greatest.
