Wallis's table of characteristic ratios

Area, Ratios
Move the yellow square to see the table entry.
John Wallis, in Arithmetica Infinitorum (1656) John Wallis presented this table. The column number is p, and the row number is q. The entry in the table is the ratio of the area under the curve y=(1-x^(1/q))^p, to the area of the circumscribed unit square; this is called the characteristic ratio of the function. We would now describe this an integral. The values of these areas were known for positive integer values of p and q. The value was also known for p=q=1/2, where the region is a quarter circle, with area pi/4. Wallis split the table, interpolating values for half integers, and found a recursive pattern for these values. This was a critical moment in mathematics: Isaac Newton read Wallis's book as a young man, and generalized the table to invent the binomial series. Newton was also the first to use fractional exponents. Note that the integer values are Pascal's triangle; the p,q entry is C(p+q,p). See if you can find a number pattern for each integer row of the table. Then fill in the same pattern for the columns. Then try to figure out a pattern for the entries where both p and q are half integers. Wallis was a pre-calculus mathematician. He found the pattern purely as a number pattern, not by doing integrals. For details, see Mathematical Intentions at http://www.quadrivium.info.