If a number is part of a number, and another is the same part of another, then the sum is also the same part of the sum that the one is of the one.

Let the number *A* be a part of *BC,* and another number *D* be the same part of another number *EF* that *A* is of *BC.*

I say that the sum of *A* and *D* is also the same part of the sum of *BC* and *EF* that *A* is of *BC.*

Since, whatever part *A* is of *BC, D* is also the same part of *EF,* therefore, there are as many numbers equal to *D* in *EF* as there are in *BC* equal to *A.*

Divide *BC* into the numbers equal to *A,* namely *BG* and *GC,* and *EF* into the numbers equal to *D,* namely *EH* and *HF.* Then the multitude of *BG* and *GC* equals the multitude of *EH* and *HF.*

And, since *BG* equals *A,* and *EH* equals *D,* therefore the sum of *BG* and *EH* also equals the sum of *A* and *D.* For the same reason the sum of *GC* and *HF* also equals the sum of *A* and *D.*

Therefore there are as many numbers in *BC* and *EF* equal to *A* and *D* as there are in *BC* equal to *A.* Therefore, the sum of *BC* and *EF* is the same multiple of the sum of *A* and *D* that *BC* is of *A.* Therefore, the sum of *A* and *D* is the same part of the sum of *BC* and *EF* that *A* is of *BC.*

Therefore, *if a number is part of a number, and another is the same part of another, then the sum is also the same part of the sum that the one is of the one.*

Q.E.D.

If a number *a* is one *n*^{th} of a number *b*, and if *d* is
one *n*^{th} of *e,* then *a* + *d* is one *n*^{th}
of *b* + *e.*
As a single algebraic equation this says

The sample value taken for 1/*n* in the proof is 1/2.

Although this proposition is only stated for the sum of two numbers, it is used for sums of arbitrary size.

Suppose that *a* is one *n*^{th} of *b* and *d* is one *n*^{th} of *e*. Then *b* is a sum of *n* *a*’s while *e* is a sum of *n* *d*’s. Therefore, *b* + *e* is a sum of *n* (*a* + *d*)’s. Therefore, *a* + *d* is one *n*^{th}
of *b* + *e.*

Note that there are no justifications that we can give for the steps in the proof. Even the definition of number VII.Def.2 is not relevant. The argument is completely based in principles of informal addition, namely unrestricted commutativity and associativity of addition where the number of terms *n* is unrestricted.

In the middle of the proof it is shown that *n*(*a* + *b*) = *na* + *nb*. In particular, when *b* = *a*, that says, *n*(2*a*) = 2(*na*). As the proposition is used for more than 2 terms, and the proof works as well for *m* terms, it can be used to show *n*(*ma*) = *m*(*na*), which is done in VII.16, the proposition preceding the statement for commutativity of multiplication of numbers.

Thus, the foundations of formal number theory in the *Elements* are principles of informal number theory.