Given, for instance, the series of figures in the accompanying diagram, measurement will show that the curved lines approximate to semicircles, and the rectilinear figures to right-angled triangles. These figures may seem to suggest to the mind the general law that angles inscribed in semicircles are right angles; but no number of instances, and no possible accuracy of measurement would really establish the truth of that general law. Availing ourselves of the suggestion furnished by the figures, we can only investigate deductively the consequences which flow from the definition of a circle, until we discover among them the property of containing right angles. Persons have thought that they had discovered a method of trisecting angles by plane geometrical construction, because a certain complex arrangement of lines and circles had appeared to trisect an angle in every case tried by them, and they inferred, by a supposed act of induction, that it would succeed in all other cases. De Morgan has recorded a proposed mode of trisecting the angle which could not be discriminated by the senses from a true general solution, except when it was applied to very obtuse angles.[142] In all such cases, it has always turned out either that the angle was not trisected at all, or that only certain particular angles could be thus trisected. The trisectors were misled by some apparent or special coincidence, and only deductive proof could establish the truth and generality of the result. In this particular case, deductive proof shows that the problem attempted is impossible, and that angles generally cannot be trisected by common geometrical methods. [142] *Budget of Paradoxes*, p. 257. *Geometrical Reasoning.*