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Wallis observed that these were figurate numbers. For example, the second row/column consisted of triangular numbers, the third row/column of pyramidal numbers, and so on. It was already known (though Wallis may have rediscovered this) that these numbers could be expressed as ratios of two products. Thus the numbers in the pth row were given by (q + 1)(q + 2) · · · (q + p) . p! Therefore, if w(p, q) = 1 1 (1 − x 1/p )q dx 0 , then Wallis had w(p, q) = (q + 1)(q + 2) · · · (q + p) . p! 4) Wallis then assumed that the formula continued to hold when q was a half integer.

5 Euler: Series and Continued Fractions The continued fraction a+ α b+ c+ β d+ 39 γ σ e + etc.

M. He also gave a method for obtaining the ci themselves. 7) 20 Sums of Powers of Integers Like Narayana Pandita, Faulhaber also considered sums of sums. To understand Faulhaber’s insight here, consider 2 nm ≡ n j n 3 k m and nm ≡ j =1 k=1 j h km j =1 h=1 k=1 r and so on. Faulhaber saw that the r-fold sum n could be written as a polynomial in n(n + r) times r n2 , while r n2m+1 would be such a polynomial times r n. He explicitly wrote down the 17th degree polynomial equal to 11 n6 . In fact, this expression was verified in 1993 by Donald Knuth.