I've received a number of questions about Axiom III in Theoretical Arithmetic, which states:
Quantities which are both equal to one and the same third are equal to one another.
When we read "one and the same third", all we mean is some other quantity that is used to measure two quantities.
Let's say that you have two trees--one in your front yard and one in your back yard and you want to know if they are equal in height. What you could is take a piece of rope (that is, a third thing) and stretch it to the top of one tree and cut the rope off at the length that measures the tree. Then, you can take that rope to the other tree and stretch it to the top. If both of the trees are equal to the same third (the rope), then they must be equal to one another.