Euclidean algorithm in Euclid's words

Question

When describing the Euclidean algorithm in his book, Elements, Euclid says the following:

When the less of the numbers $a$ and $b$ is continually subtracted from the greater, some number is left which measures the one before it.

So when looking for a GCD of, say, $25$ and $15$, we can subtract $15$ from $25$ and get $10$. So $10$ is the number left which should measure(divide) the one before it. It measures neither $15$ nor $25$.

Can anyone interpret what Euclid wanted to say with that sentence?

Answer 1

You're missing the word "continually".

In your example: $(25,15) \to (10,15) \to (10,5)$ and now $5$ measures $10$ and we're done.

I guess the sentence could also use the addition of "eventually":

When the less of the numbers $a$ and $b$ is continually subtracted from the greater, eventually some number is left which measures the one before it.