 The Rhine papyrus begins with the long section known as the recto. That's actually a generic term for any manuscript. There is one side known as the recto, and the other side known as the verso. The recto computes the quotients of the form 2 divided by an odd number for the odd numbers from 3 to 101, and so it gives us a very good idea of how the ancient Egyptians performed division. For example, one of the problems is finding 2 divided by 7. Now, 7 doesn't have any eloquent parts worth talking about, but we can at least take half of it, so we might begin by having, and in fact, Achmos does begin by having. So, when half of 7 is 3.5, we can find a quarter that's half of a half by having each of the individual parts. So, half of 3 is 1.5, then half of a half is a quarter. At this point, Achmos does something rather interesting. He doubles the divisor to get 14, and then doubles it again to get 28, and then he finds that 1.28 of 7 is a quarter. And the thing to notice here is that if we put together the quarter and the 28th, that's 1.5 and 2 quarters, in other words 2. And so we find that 2 divided by 7 is a quarter and a 28th. So what did Achmos do in these last couple of lines? To understand Achmos' procedure, remember the table represents multiples of 7. So when Achmos writes 4.28, what he's really saying is that 4 times 7 is equal to 28. And the next line says that 1.28 times 7 is equal to a quarter. And notice what we've done here. We've actually moved this 28 score to opposite sides, and so we'll call this step an inversion. So let's find 2 divided by 11, and we'll begin by repeated halving. So 1 gets to 11, half, well that's 5.5, a quarter, we can take half of each of these pieces. So half of 5 is 2.5, and half of a half is a quarter. For an 8th, we can have each of the pieces again. So half of 2 is 1, half of a half is a quarter, and half of a quarter is an 8th. And at this point, since we're trying to make 2, we might take stock and see what we have so far. And what we need to figure out is what additional pieces we need to make 2. Now one useful feature about repeating halving is that we have a nice visual picture of what each of these represents. So if we start with the 8th, we have 1, we have a quarter, which is this much of a block, and an 8th is this much of the block. And since we have the quarter and the 8th, what we need is an 8th and a half. And we can get those through an inversion. So again starting with our 111, double to 222, and invert a 22nd is a half, double again to get us 444, one more time to get 888, invert, and selecting the pieces we need, and so 2 divided by 11 is an 8th, a 22nd, and an 88th. Now sometimes Achmos found the quotient by repeated halving, but there's no obvious pattern to how Achmos found 2 divided by n. Most of the time, he took some arbitrary divisor and worked with it. For example, he found 2 divided by 5, and he started by taking 1 3rd. So if we start with 1 5, to find 1 3rd easily, it's worth noting that since 5 is 3 plus 2, then 1 3rd of 5 is a 3rd of 3 and a 3rd of 2. Well, a 3rd of 3 is easy, that's 1, and a 3rd of 2, well, that is the one non-unit fraction the ancient Egyptians routinely worked with, that's 2 3rds. So our first line of the table, 1 3rd is 1 2 3rds. Now notice that at this point we have 1 2 3rds, and so we need an extra 1 3rd to make 2, and we can get that through an inversion. So we'll find 3 5's, that's 15, and if we invert it, that's 1 15th, gives us our 1 3rd, and so we select our pieces, and we find our quotient, a 3rd and a 15th. Now at this point it's worth raising the following question, why 2 3rds? The one non-unit fraction the Egyptians used routinely was 2 3rds, which we've designated as having a 3 with 2 lines over it. The hieroglyphic symbol for 2 3rds is very suggestive. So remember this symbol, this row indicates that we have a unit fraction, a reciprocal if you will, and it looks like this is trying to say it's the reciprocal of 1 and a half. And it's used in the 2 over n problems as very suggestive. And it shows up in problems like 2 divided by 9. The one regularity of the divisions performed by Achmos in the recto of the Rind is that any time you could take 2 3rds and get a whole number he always did so. So when we divide 2 by 9 we can take 2 3rds and we can immediately perform an inversion because that gives us a 6th and the reciprocal of 2 3rds is 1 and a half. The importance of that is that at this point what we need to make 2 is a half and we can get that by another inversion. First we'll double our divisor then invert. Then we'll choose the pieces we need and that gives us a quotient 2 divided by 9 is a 6th and an 18th.