 Very little is known about the life of Diophontus. He most likely lived during the third century, but even that is uncertain. His best-known work is his arithmetic, which contained 13 books, 10 survive. Diophontus is known for introducing a form of notation. However, there is a problem with introducing notation in Greek mathematics. The Greeks used letters of the alphabet to represent numbers. One was alpha, two was beta, three was gamma, and so on. The system is very similar to the Egyptian Hieratic Numeration. But this meant that a letter couldn't be used as a variable. X always meant a specific number. The one exception was the terminal sigma, the form taken by the letter sigma at the end of a word. For example, arithmos. Since terminal sigma did not represent a number, it was the only Greek letter that could be used to represent the unknown quantity. Diophontus used syncopated notation. These are a breath for the nose and the ankh in the ankhon. Syncopated notation differs from true notation in that it records, but it does not direct. For example, when we write 3x, this is true notation. It's a record of what is present, 3 of x, but it also represents an arithmetic operation, multiplication, 3 times x. Now when you're dealing with algebraic quantities, it's necessary to record two things. The type of quantity and the number of that type. So when we write 3x, we are saying that we have 3 of the quantity x. To record this, Diophontus wrote this as terminal sigma gamma, terminal sigma indicating the type unknown value, while gamma gave the number 3. Now there might be a pure number. To represent a pure number like 5 with no unknowns, Diophontus used type indicator m with a circle, and so 5 would be m-circle epsilon. So remember, Diophontus was able to use the terminal sigma to represent the unknown, but do note the slight difference between the terminal sigma representing the unknown and stigma, this value representing 6. Now this really only matters if you intend to read Diophontus in the original Greek. The higher powers were a little easier. The square of the unknown was written this way from the first two letters of the dinamos power, the cube of the unknown, again from the first two letters of the word kubos, the fourth power, dinamo dinamos, the fifth power, dinamo kubos, and so on. For reciprocals of the unknown and its power, Diophontus used a superscripted symbol we usually render as kai. So where we might write 1 over x squared, that would be dinamos kai, and so on. Then Diophontus used syncopated notation, which is a record of what's present, so we can't really talk about addition and subtraction, but when we do have these terms present, the terms will be juxtaposed. So 3x plus 5 means we have a 3x and a 5 present. So we've recorded their presence as, and we might read this as unknown things, 3 units 5. To indicate a subtraction, Diophontus would use a symbol that looked like this, which he described as an inverted capital C truncated. And all of the terms following the truncated C would be subtracted. Or rather you can think about these as terms that are not there anymore because they've already been removed. So we would write 3x minus 5, Diophontus would write, and again we might read that as we have unknowns 3 present, but we used to have 5 ones which have been taken away. So let's try to express in Diophonti notation 3x squared minus 2x plus 5, and we'll use 1 for alpha, 2 for beta, 3 for gamma, 4 for delta, and 5 for epsilon. So the x squared is dinamos, and the 3 would be gamma, so we'd start out dinamos gamma, 3 of the squares. The other quantity that's actually present is the 5. The 5 would be epsilon, and since it's a pure number, it must be preceded with an M-circle. Now we also have to consider this minus 2x, so that's arithmos beta, arithmos 2 unknowns, and since it's going to be subtracted, it'll be preceded by our symbol. To make it a more complicated expression, 3x cubed, that's cubes, and 3 of them, so that's the other term that's present is the 1, so that's 1, a pure number, M-circle, to indicate it's a number, and alpha, the value. Now the subtracted terms are 2x squared and 5x, so 2x squared would be, and 5x would be. And since both of these are being subtracted, they'd be preceded by our subtraction symbol. We could also go backward. Let's interpret this. So here, this represents x to the third, and beta means that there's 2 of the x cubed, so we have a 2x cubed, the M-with-the-circle means there's a number, and epsilon means the number is 5, and we have a couple of terms that follow the subtraction, so we know these are going to be subtracted. This is x squared, and the alpha indicates 1 of them, and finally, terminal sigma represents x and there are gamma equals 3 of them, so we're subtracting x squared and 3x. While Daphontus's notation is interesting, it had no lasting impact, so moving ahead we'll use modern notation, because of greater interest are Daphontus's problems. We'll take a look at those next.