The Part Called by the Name of a Number

Theorem 209 (Euclid VII.37). If a number be measured by any number, the number which is measured will have a part called by the same name as the measuring number.

Proof: Let A be measured by B. As many times as B measures A, so many units let there be in C. B measures A according to the units in C, and the unit D measures C according to the units in it. So D measures C the same number of times that B measures A. Alternately, D measures B the same number of times that C measures A (Theorem 55). So whatever part the unit D is of B, the same part is C of A. The unit D is a part of B called by the same name as B. So C is a part of A called by the same name as B.

Theorem 210 (Euclid VII.38). If a number have any part whatever, it will be measured by a number called by the same name as the part.

Proof: Let A have a part B, and let C be the number called by the same name as B. B is a part of A called by the same name as C, and the unit D is a part of C called by the same name as C. So D measures C the same number of times that B measures A. Alternately, D measures B the same number of times that C measures A (Theorem 55). So C measures A.

Theorem 211 (Euclid VII.39). To find the number which is the least that will have given parts.

Proof: Let the given parts be A, B, C, and let D, E, F be the numbers called by the same name as they. Let G be the least number measured by D, E, F (Theorem 75). G has parts called by the same name as D, E, F (Theorem 209), that is, the parts A, B, C. It is the least: if H, less than G, had the parts A, B, C, it would be measured by D, E, F (Theorem 210), though less than G: which is impossible.

Remark. "The part called by the same name as n" is the third part for 3, the fourth part for 4. Taking that part is the specific operation 1/n. These two propositions say exactly when it applies to a number A: when n measures A, and only then. That is the ground in Euclid for "sameness where both apply," and for why 1/n is a derived number and not a number as such.