Euclid's Elements, Book X, Proposition 5 (X.5)

Hello! This will be a demonstration of Proposition 5 from Book 10 of Euclid's Elements, which says that magnitudes which are commensurable with one another are to each other as a number is to a number. That is to say, the ratio that two commensurable magnitudes have is the same as a ratio of two numbers. We're given two commensurable magnitudes, A and B. I will be representing these as lines, but this proposition works no matter what your magnitude is. It can be an angle, it can be an area, it can be a volume, it can be just any magnitude. And the only qualification in our given is that A and B are commensurable, they're not incommensurable. And what we have to prove is that the ratio which A has to B is the same as a ratio which some number has to some other number. Now, because A and B are commensurable, we were given that, we know that there will be some magnitude which measures the both of them in common, right? That's just what it means for two magnitudes to be commensurable. So we're gonna lay out that magnitude, call it C. Now this magnitude C will be of the same type as A and B, so all three of these will either be lines, or they'll be angles, or areas, or volumes, or whatever. And what's true about C is that it measures both A and B in common. That there is some number of C's that fit into A, and another number of C's that fit into B. And whatever those numbers are, we will call D and E. That's our next step. We're gonna make the two numbers D and E like this, such that as many C's as there are in A, there are as many units in D, and as many C's are in B, there are as many units in E. An example of this would be, say that there are three C's that fit into A, then our number D would be three, or if there are two C's that fit into B, then there will be two, and that'll be the number of E. E will be two. Now, because the magnitude C measures the magnitude A, the same number of times that the unit measures the magnitude D, or the number D, my bad, that's how we created the number D, that C measures A the same number of times that our unit measures the number D, we can then say by the definition of proportionality, definition 720, that C is to A, just as the unit is to the number D. Now, the clairvoyant among you might already notice that this is kind of a weird step, and we'll get to that later, but for now we'll just take it that C is to A, just as the unit is to the number D. Now, we can also, of course, invert this proportion according to the porism of proposition 5-7, and say that inversely, A is to C, just as the number D is to the unit. Now, in a similar way to how we obtained that C is to A, just as the unit is to the number D, we can likewise say that C is to B, just as the unit is to the number E. This would again follow from the fact that C measures B the same number of times that the unit measures the number E, so definition 720, which is the definition of proportion, tells us that C is to B, just as the unit is to the number E. Now, what you'll notice is that we have two proportions, A to C, just as the number D is to the unit, and C to B, just as the unit is to the number E. These two proportions then allow us to say, ex-i-quali, by proposition 5-22, that A is to B, just as the number D is to the number E, and so what we see is that the ratio of our two given commensurable magnitudes, A and B, is the same as this ratio of the number D to the number E, which is obviously a ratio of two numbers, and this is what we set out to prove, that the ratio of our two commensurable magnitudes is the same as a ratio of two numbers, namely D and E, so having proven that, we are therefore done with the proposition, therefore it's set at QED. I mentioned very briefly, when we were going through the second step, that there is some strangeness going on about how Euclid obtains this proportion, that C is to A, just as the unit is to the number D, and likewise, how he obtains that C is to B, just as the unit is to the number E. You notice what I cited is definition 720, which defines proportionality, but it only defines proportionality for numbers, but in our proportion, that C is to A, just as the unit is to the number D, we have two magnitudes on the one side, and two numbers, the unit isn't a number, but for right now, I'll just call it a number, we have two numbers on the other side. Now, this is already strange enough, given that throughout the entire elements, up until now, we've had magnitudes on the one hand, and numbers on the other hand. These were two distinct categories of things, and we had to prove even the same things for both magnitudes and numbers in two different propositions, as for example, proposition 516, which was the definition, or not the definition, but it said that if four magnitudes are proportional, they are proportional alternately, and likewise, we had to prove that in proposition 711, but for numbers, so I wouldn't blame you, and this is what I think is true as well, is that magnitudes aren't numbers, numbers aren't magnitudes either, which is why we had to prove the same things for them in different propositions, but what we're now seeing in book 10 is this proportion with two magnitudes on the one side, and two numbers on the other side. Now, this wouldn't be the most strange thing, but it is strange that you could invoke the definition, which was exclusive to numbers, when talking about magnitudes, because you remember in book 5, the definition of proportionality there, which is the fifth definition of that book, it's different from the definition of proportionality with numbers. It is very different, in fact, so for you could to use this definition about numbers and apply it to a magnitude, it is a weird move that, at the very least, requires more propositions to explain them. The magnitudes which are proportional in the sense of definition 55 are also proportional in the sense of definition 720, and the converse, that if you have magnitudes which are proportional in the definition 720 sense, they're also proportional in the definition by 5 sense. I'm not going to go through the proofs of those two facts right there, you would be able to find them online, you can look them up in, for example, Sir Thomas Little Heath's commentary on book 5, so these resources are out there for you to look at, or if you want to, write them yourself, which I highly encourage, but for right now, and this isn't the first time we've seen this, but this is the first time I'm really gonna point it out, is this correspondence of proportionality of magnitudes and numbers, because what we're gonna see throughout a lot of book 10, not just in this proposition, but in many more to follow, is the kinship of magnitudes to numbers. Numbers aren't magnitudes, but they also share a lot of properties, and it's not just that they share a lot of properties and propositions, but there is some sort of fundamental relationship that magnitudes bear to numbers. That's all I've got to say about this proposition right here, so I'll end the video here and move on to proposition 10-6.