 The big three mathematicians of Great Britain were all professors, John Wallace, civilian professor at Oxford, Isaac Rao, Lucassian professor at Cambridge, and Isaac Newton, also Lucassian professor at Cambridge. In contrast, the big three mathematicians of Northern Europe were all hobbyists, Renee Descartes, philosopher, Pierre Fermat, lawyer, and Blaise Pascal, philosopher. And joining their ranks was Gottfried Wilhelm Leibniz, a diplomat. In the 1660s, Leibniz began considering sequences, consider a sequence of numbers. Leibniz considered the differences between successive terms what we now call the first differences. So here, 0 and 10 have difference 10, 10 and 26 have difference 16, 26 and 60 have difference 34 and so on. But these differences formed another sequence, and you can find the difference between the successive terms of that sequence, what we now call the second differences. So the difference between 10 and 16 is 6, the difference between 16 and 34 is 18 and so on, and lather, rinse, repeat, we can find third differences, fourth differences, and so on. So for example, let's find the first, second, and third differences of the sequence of squares. So we'll set down the sequence of squares. Now we'll find the difference between each term and the preceding term. So the difference between 0 and 1 is 1. The difference between 1 and 4 is 3. The difference between 4 and 9 is 5 and so on. And so our first differences are the sequence 1, 3, 5, 7, 9, and so on. Now find the difference between each of these terms and the preceding terms. So the difference between 1 and 3 is 2. The difference between 5 and 3 is 2 and so on, and our second differences form the sequence 2, 2, 2, 2, and so on. And since these are all the same, the third differences will all be 0. What if our sequence is generated by some formula like n squared plus 3n? So we'll form our sequence starting with n equals 0. Our first differences form the sequence. Our second differences form the sequence. And again, because our second differences are all the same, our third differences are all 0. And after trying out a number of different sequences, Leibniz concluded, if the terms of a sequence are produced by a kth degree polynomial, then the kth differences will be constant, and the k plus first differences will also be 0. Now, Leibniz also knew of what he called the combinatorial numbers, which formed a table that looks like this. And with this idea of differences in mind, he recognized that each row was a different sequence of the row that followed it. So for example, this third row, 1, 3, 6, 10, and so on, these are the first differences of the sequence of numbers in the row that follows it. And what this also means is that every entry was the sum of the entries in the row above it. For example, this entry 35, that's the sum of the entries in the row above it, 1 plus 3 plus 6 plus 10 plus 15. Now, later on Leibniz admits that he realized his work wasn't actually original, but he wasn't a professional mathematician. He pursued it because it was interesting. So he communicated with mathematicians and physicists, and in 1672 Christian Huygens suggested that Leibniz should examine the work of Pascal, probably because many of Leibniz's investigations paralleled what Pascal did. And so Leibniz looked into the work of Pascal and began tackling the problem of serious summations in general. Now, one defect of Pascal's approach is that it required finding formulas for the sums of all lower powers. So if we wanted to find the sum of the tenth powers, we'd need to find a formula for the sum of the ninth powers, which would require finding a formula for the sum of the eighth powers, and so on. And so the question is, could we find these summation formulas directly without having to build up to them? Around 1673 Leibniz had the following idea. Suppose we wanted to find the sum of the first x-square numbers. And we'll start with 0, so we're actually summing 0 squared plus 1 squared plus and so on all the way up to x minus 1 squared. I'd let the sum be z equal to some formula px cubed plus qx squared plus rx plus s. Here Leibniz assumed the formula for the sum of the squares could not require any power greater than the third. This assumption is justified by the work that follows. And alternatively, if you don't believe that, if we had assumed there was an x to the fourth term in the formula, its coefficient would turn out to be 0. Now consider the difference d between successive values of a quantity. dz will be the difference between the successive values of the sum of the squares. Since the difference will just be the next term, and z goes up to x minus 1 squared, then dz is x squared. But the difference between the successive values of a sum is the sum of the differences. So the difference of the px cubed plus qx squared plus rx plus s is the difference of px cubed plus the differences of qx squared plus the differences of rx plus the differences in s. And since p, q, r and s are constants, they will simply multiply the difference. So this is p times the differences in the x-cubes plus q times the differences in the x-squared plus r times the differences in the x's plus s times the difference in the ones. Now dx cubed is the difference between a cube x cubed and the next greater cubed, x plus 1 cubed. Similarly, dx squared is the difference between a square x squared and the next greater square, x plus 1 squared, dx is the difference between x and the next greater x. So that's the difference between x plus 1 and x, d1 is the difference between 1 and the next greater 1. And so that's the difference between 1 and 1. And that gives us a second expression for the difference in the sum of the successive squares. And if we expand, we can rewrite the right-hand side as a polynomial in x. But we have a polynomial on both sides, which means our coefficients have to be the same. So our coefficients of x squared, x, and the constant over on the left are going to be 1, 0, and 0. While on the right, the coefficients of x squared is 3p, the coefficient of x is 3p plus 2q, and the constant is p plus q plus r. And so comparing our coefficients, we can solve for p, q, and r. And that gives us a formula for the sum of the squares of the integers from 0 to x minus 1.