Diophantus of Alexandria/Arithmetica/Book 1/Problem 1

Problem

 * To divide a given number into two having a given difference.

Proof
Let $N$ be the given number.

Let $d$ be the given difference.

Let $x$ be the smaller of the two numbers into which $N$ is to be divided.

Thus the greater of the two numbers is $x + d$.

We also have that the greater of the two numbers is $N - x$.

Hence the smaller of the two numbers is found by:

and it follows that the greater of the two numbers is $x + d$.