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Proposition 39

To find the number which is the least that has given parts.
Let A, B, and C be the given parts. It is required to find the number which is the least that will have the parts A, B, and C.
| Let D, E, and F be numbers called by the same name as the parts A, B, and C. Take G, the least number measured by D, E, and F. | VII.36
Therefore G has parts called by the same name as D, E, and F. | VII.37
But A, B, and C are parts called by the same name as D, E, and F, therefore G has the parts A, B, and C. I say next that it is also the least number that has. If not, there is some number H less than G which has the parts A, B, and C.
Since H has the parts A, B, and C, therefore H is measured by numbers called by the same name as the parts A, B, and C. But D, E, and F are numbers called by the same name as the parts A, B, and C, therefore H is measured by D, E, and F. | VII.38
And it is less than G, which is impossible. Therefore there is no number less than G that has the parts A, B, and C.
Q.E.D.

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(Forthcoming)

Book VII Introduction - Proposition VII.38 - Book VIII Introduction.

© 1996
D.E.Joyce
Clark University