 In Pythagorean mathematics, one quantity measures another if the latter is a whole number multiple of the former. So here the quantity on the right is a measure of the quantity on the left because the quantity on the left is a multiple of the quantity on the right. But this quantity is not a measure because the quantity is not a multiple. Now if we have two quantities, we might have a common measure, something that measures both, not this, but the quantity at the bottom measures both. So it's a common measure. And the greatest common measure is the largest that measures both quantities. To find the greatest common measure of two numbers, we can use the Euclidean algorithm. This was actually discovered by the Pythagoreans and the simplest implementation is as follows. Repeatedly subtract the smaller quantity from the larger and the last amount subtracted is the greatest common measure. This is easiest to do with concrete objects. So let's find the greatest common measure of the two amounts shown. So the amount on the right is the smaller. So we'll subtract it from the larger by separating out and discarding an amount equal to the smaller amount. So we'll separate out, then discard, and again separate, and discard. So now the amount on the left is the smaller, so we'll separate and amount, and discard, and separate, and discard. Now remember what's left was the last amount we discarded, and so that's the greatest common measure. Since this is the greatest common measure, both of our original quantities can be expressed in terms of this measure. So for example, let's say we have two line segments we want to find the greatest common measure. So we'll take the smaller line segment and subtract the smaller from the larger. And now we have a smaller line segment. We'll subtract the smaller from the larger. And again, we'll subtract the smaller from the larger. And since nothing is left, then the last thing we subtracted, which is the remaining line segment, is going to be our greatest common measure. Now there's some evidence that this is how the Pythagoreans actually looked at the Euclidean algorithm. They saw it as a tangible operation performed on objects, and not as a written operation performed on numbers. But we can reduce this to an arithmetic operation. So here we see we have a number, well that's actually the number 15, and another number, the number 6. We subtract the smaller from the larger. 15 minus 6 gets us 9. We subtract the smaller from the larger. 9 minus 6 gets us 3. We subtract the smaller from the larger. 6 minus 3 gets us 3. And one final subtraction, 3 minus 3, gets us 0. And so 3 is the greatest common measure. Or since we've been working with numbers, we also now call this the greatest common divisor of our original numbers, 15 and 9. So we might find the greatest common measure of 45 and 13. So we'll subtract the smaller number from the larger, 45 minus 13. And again, and again. Now we've been subtracting 13 and we now have 6 left. So now 6 is the smaller number, so we'll subtract 6. And again. And we've been subtracting 6, but now we have 1. So 1 is our smaller number, so we'll subtract 1 until we get down to 0. The last amount subtracted is the greatest common measure, or the greatest common divisor. And so we can say the greatest common divisor is 1. While the Euclidean algorithm is typically applied to numbers, it's important to keep in mind that we could apply it to geometric figures as well. And in fact, this is probably how it was first discovered. And this leads to an interesting problem. What if we try to find the greatest common measure of the side and diagonal of a square? So let ABCD be a square with diagonal AC. Now we want to subtract the smaller from the larger, and that's going to be the side. So let CE be equal in length to the side CB. And then we have subtracted the larger from the smaller, leaving AE. Now our next step will be to subtract the smaller AE from the larger, CB. And we'll do that as follows. Since we have a square, CB is the same as AB. And to continue, note that if we take AE equal to EF, where EF is perpendicular to AC, then EF will be FB as well. So that means we can subtract AE from CB by removing FB from AB. Now our next step would be to remove FB from the remainder AF, but this is the side from the diagonal of a smaller square. And this leads to a rather disturbing conclusion, at least for the Pythagoreans. The Euclidean algorithm ends by finding the greatest common measure of two quantities. But if we apply the Euclidean algorithm to find the greatest common measure of the side and diagonal of a square, it doesn't end. And that's because the first time through what we end up with when we subtract the side from the diagonal is the side and diagonal of a smaller square. And what this means is that the side and diagonal of a square have no common measure. We say they are incommensurable. And that means the Pythagorean dictum that all is number can't be applied.