A little experience of problems in which it is necessary to divide by sine or cosine of an angle will prepare a boy for the definitions of secant, cosecant and cotangent as the reciprocals of the cosine, sine arid tangent respectively, though, as a matter of fact, he can get on very well without them. At this stage a boy will probably appreciate the fact that since an acute angle is determined if me know any one of the six trigonometrical ratios, it should be possible, if any one of the ratios is given, to find the others. Problems such as this : “Given sin A = ·64, find tan A” should be done (i) by drawing a figure to scale, i.e. constructing the angle A whose sine is ·64, and finding the tangent by measurement (ii) by drawing a right-angled triangle, hypotenuse 100 units, one side 64 units, and calculating the other side by Pythagoras’ theorern; (iii) by taking from the tables the angles whose sine is ·64 and looking up its tangent.