 After seeing Leibniz's mathematical work, Huygens suggested he read the works of Pascal. Huygens also posed this question to Leibniz in 1672, finding the sum of the reciprocals of the triangular numbers. In other words, 1 over 1, plus 1 over 3, plus 1 over 6, plus 1 over 10, and so on. After reading Pascal's treatise on the arithmetic triangle, Leibniz created his own variation known as the harmonic triangle. So let's consider Pascal's triangle. Except for the sides, which are all ones, every entry is the sum of the two entries above it. So for example, this 10 is the sum of 4 and 6. But suppose you formed a triangle whose sides were the harmonic sequence, 1, 1 half, 1 third, and so on. Then fill in the remaining entries as the sum of the two entries below. So this 1 half, it should be 1 third plus something. And so we find that something, that's 1 half minus 1 third, gives us. And 1 sixth plus 1 third is 1 half. This entry 1 third, well that should be 1 quarter plus something. And so we can find that by subtraction. 1 twelfth. 1 sixth, well that's 1 twelfth plus something. Also 1 twelfth. This 1 quarter is a fifth plus something, which works out to be. This 1 twelfth is a twentieth plus something, and we find that. And we can fill out the rest of the entries in the harmonic triangle. Now consider any entry, say 1 half. It's the sum of the two entries below it, 1 third and 1 sixth. But 1 sixth is the sum of the two entries below it, 1 twelfth and 1 twelfth. So 1 half is a third plus a twelfth plus a twelfth. But lather, rinse, repeat. 1 twelfth is the sum of the two entries below it, 1 thirtieth and 1 twentieth. 1 twentieth is the sum of the two entries below it, 1 sixtieth and 1 thirtieth. 1 thirtieth is the sum of the two entries below it and so on. And so we see that this entry, 1 half, is the sum of all of the entries along this diagonal. And we can summarize that in the harmonic triangle. Each entry is the sum of the infinite series of the next diagonal, beginning with the term below and to the left of it. Leibniz also noticed that the denominators along the kth diagonal of the harmonic triangle are k times the entry of the k plus first diagonal of the arithmetic triangle. So if we look at the denominators along this second diagonal, they're two times the entries along the third diagonal of the arithmetic triangle. Or if we take a look at the entries along the third diagonal, they're three times the entries along the fourth diagonal. And what this means is that we can also use the diagonals of the arithmetic triangle to sum the reciprocals of the entries of the arithmetic triangle. So again, that third diagonal of the harmonic triangle tells us that 1 half is the sum, but each entry is three times the entries of the fourth diagonal, and so that'll tell us what the sum of the reciprocals of those entries are. The harmonic triangle allowed Leibniz's sum a number of different series. Leibniz also considered the series where the denominators were products of successive odd numbers. To sum this series, Leibniz noted that this reciprocal of the product of successive odd numbers, 1 over 2n plus 1 times 2n plus 3, could in fact be rewritten as a difference. And so that means 1 third is 1 half, 1 over 1 minus 1 over 3, 1 fifteenth is half of 1 third minus a fifth, 1 thirty-fifth is half of 1 fifth minus a seventh, and so on. And so our series can be rewritten. And in modern terms, this is a telescoping series because all of the terms after the first term collapse, and so our sum is just 1 half. These results convinced Leibniz that any series could be summed. And so the natural question to ask then is, what else can we use these series for? And in particular, he began to apply series to geometric objects. And here's where things get a little bit strange. So given any curve, we can view dy as the difference between successive values of the ordinate. So to Leibniz this meant that at some x value we have a y value, at the next x value we have a different y value, and dy is the difference between the successive values of the ordinate. And if we sum up the y values over some interval, we get the final ordinate. So Leibniz wrote this in the following way, where this symbol is a 17th century s, and it's a shorthand for sum. Moreover, if two consecutive points on a curve, whatever that means, differed by dx and dy, then the ratio dy over dx is what we would now call the slope of the tangent line. And at this point, things should all start to look very familiar to those who've taken calculus. And over the next few years Leibniz developed what we now recognize as the rules of differentiation. So as a single example, we'll consider the product rule. Leibniz explained the product rule. dxy is the difference between the products of successive values of x and y. So that means that dxy, well, it's the difference between the product xy and the product of the next x value with the next y value, x plus dx times y plus dy. We can expand that out and get our final result. And at this point Leibniz says the following. This product, dxdy, is infinitely small in comparison to the others, so it may be neglected giving us dxy is just xdy plus ydx. And this is the product rule we're all familiar with from calculus. And in a 1684 publication, Leibniz gave most of what we now recognize as differential calculus. And unfortunately for Leibniz and really mathematicians everywhere, Leibniz set off a rather contentious debate over who really invented calculus. And this debate had far-reaching consequences, and we'll talk about those later.