Analysis — page 5
Where to take the samples
Every numerical integral is a weighted sum of samples of the curve. Spacing the samples evenly gives errors that shrink like a power of their number; choosing where they go, as Gauss did in 1814, makes six samples exact for every polynomial up to degree eleven and turns the power into a geometric rate. The surprise is that the humble trapezoid rule, evenly spaced and exact only for straight lines, does the same thing on any smooth curve that goes once round a circle — and that a single corner defeats every one of them.
The sine of one degree
Every table of sines ever made by hand needed the sine of one degree, and no construction with ruler and compass can produce it: one degree is a third of three, and thirds are what the constructions cannot make. Ptolemy fenced it in with an inequality and got it right to the precision his table carried. Thirteen centuries later Jamshīd al-Kāshī wrote the triple-angle formula as a rule for improving a guess and let it run, gaining almost three decimal places at every turn — a cubic solved without Cardano, by doing the same thing again.