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