Any number is either a part or parts of any number, the
less of the greater.

Let *A* and *BC* be two numbers, and let *BC* be the less.

I say that *BC* is either a part, or parts, of *A.*

Either *A* and *BC* are relatively prime or they are not.

First, let *A* and *BC* be relatively prime.

Then, if *BC* is divided into the units in it, then each unit of those in *BC* is some part of *A,* so that *BC* is parts of *A.*

Next let *A* and *BC* not be relatively prime, then *BC* either measures, or does not measure, *A.*

Now if *BC* measures *A,* then *BC* is a part of *A.* But, if not, take the greatest common measure *D* of *A* and *BC,* and divide *BC* into the numbers equal to *D,* namely *BE, EF,* and *FC.*

Now, since *D* measures *A,* therefore *D* is a part of *A.* But *D* equals each of the numbers *BE, EF,* and *FC,* therefore each of the numbers *BE, EF,* and *FC* is also a part of *A,* so that *BC* is parts of *A.*

Therefore, *any number is either a part or parts of any number, the less of the greater.*

Q.E.D.

This proposition says that if *b* is a smaller number than *a,* then *b* is either a part of *a,* that is, *b* is a unit fraction of *a,* or *b* is parts of *a,* that is, a proper fraction, but not a unit fraction, of *a.* For instance, 2 is one part of 6, namely, one third part; but 4 is parts of 6, namely, 2 third parts of 6.

It seems obvious that when one number *b* is less than another *a,* that since the unit *u* is a part of *a* and *b* is a multiple of *u*, then *b* is some multiple of a part of *a*. Yet, the proof of this proposition ignores that possibility, except in the special case when *b* and *a* are relatively prime. In the case of *a* = 4*u* and *b* = 6*u*, the proof will find that *a* is 2 third parts of *b*. Thus, it appears that a satisfactory answer to the question “How mary parts of *a* is *b*?” requires finding the least number of parts.

The proof has three cases.

- If
*b*and*a*are relatively prime, then*b*consists of*b*of the*n*^{th}parts of*a*where*a*=*nu*. - If
*b*divides*a,*then*b*is one part of*a.* - Otherwise they’re not relatively prime, and
*b*does not divide*a.*Let*d*be their greatest common divisor. Then*b*is some multiple*m*of*d*, that is,*b*=*md*, and*a*is some other multiple*n*of*d*, that is,*a*=*nd*Therefore,*b*consists of*m*one-*n*^{th}parts of*a.*

Again, if 1 were considered a number, then the three cases could be consolidated into one.

We'll write the statement such as *b* is one *n*^{th} part of *a* as the equation *b* = *a*/*n*, and a statement such as *b* consists of *m* one-*n*^{th} parts of *a* as the equation *b* = (*m*/*n*)*a*.