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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?

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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.

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  • $\begingroup$ But "continually" seems to mean to subtract as long as that lesser number can go "into" the greater. And the text also makes it seems as if at every step of the subtraction a number will be left that divides the number from the previous step. Do you know what I mean? $\endgroup$ – Michael Munta Feb 13 at 16:27

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