Search Loci: Convergence:
...mathematics is distinguished from all other sciences except only ethics, in standing in no need of ethics. Every other science, even logic, especially in its early stages, is in danger of evaporating into airy nothingness, degenerating, as the Germans say, into an arachnoid film, spun from the stuff that dreams are made of. There is no such danger for pure mathematics; for that is precisely what mathematics ought to be.
"The Essence of Mathematics" in J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Benjamin Banneker's Trigonometry Puzzle
It is clear that Banneker is using the Law of Sines:2 In a triangle, the ratios of the sine of an angle to the length of its opposite side are equal. Where Banneker writes in his proportion, "logarithm base 26," or "logarithm of the hypotenuse," he is anticipating the use of logarithms for the computation. Banneker understood, as his calculations correctly show, that the ratios involved are the sine of an angle to the side opposite, not to the log of the side. In what follows, all angles are expressed in degrees.
I. To find the hypotenuse, Banneker used the Law of Sines: sin C/c = sin B/ b
"Sine complement of the angle at A," is sin 60, the sine of the angle complementary to angle A. If x is the length of the hypotenuse, then sin 60/ 26 = sin 90/ x and x = 26 sin 90/sin 60. . Taking logarithms, we have log x = log 26 + log sin 90 - log sin 60 and, substituting values from a suitable set of tables, log x = 1.41497 +10 – 9.93753 = 1.47744. Notice that, for reasons we shall see later, log sin 90 = Log 1010 = 10. We now find x as the antilogarithm of 1.47744, which is very close to 30. It is unlikely that Banneker would have had access to tables of antilogarithms, a late eighteenth century invention, but would simply have used his table of logarithms in reverse.