CHAPTER XVIII. OBSERVATION.

In other sciences the force of a negative argument will often depend upon the number of possible alternatives which may exist. It was long believed that the quality of a musical sound as distinguished from its pitch, must depend upon the form of the undulation, because no other cause of it had ever been suggested or was apparently possible. The truth of the conclusion was proved by Helmholtz, who applied a microscope to luminous points attached to the strings of various instruments, and thus actually observed the different modes of undulation. In mathematics negative inductive arguments have seldom much force, because the possible forms of expression, or the possible combinations of lines and circles in geometry, are quite unlimited in number. An enormous number of attempts were made to trisect the angle by the ordinary methods of Euclid’s geometry, but their invariable failure did not establish the impossibility of the task. This was shown in a totally different manner, by proving that the problem involves an irreducible cubic equation to which there could be no corresponding plane geometrical solution.‍¹⁸ This is a case of reductio ad absurdum, a form of argument of a totally different character. Similarly no number of failures to obtain a general solution of equations of the fifth degree would establish the impossibility of the task, but in an indirect mode, equivalent to a reductio ad absurdum, the impossibility is considered to be proved.‍¹⁹