Now that there is a tendency to cut out a good deal of useless matter in arithmetic and algebra, and to spend less time over formal geometry, it is possible, and I think profitable, to introduce boys quite early to a course of numerical trigonometry. For the last fifteen years, I have seen some such course as is here described taught successfully to boys of thirteen and fourteen years of age. The inclusion of certain types of question in arithmetic and algebra courses—such, for instance, as those wonderful creations that you will find in chapters on vulgar fractions—is, I suppose, justified on the ground that they call for patience, orderly arrangement of work, and accuracy. But scope can be found for the exercise of all these qualities in numerical problems which arise in connection with other branches of their work—geometry, mechanics, physics, etc.—and the range of such problems is greatly increased if the boy has an elementary knowledge of trigonometry Moreover, the work involved in solving triangles is often fairly complicated and must be done carefully and arranged in an orderly manner, otherwise it is liable to end in disaster. There is a certain amount of drudgery involved, but it is drudgery with some interest attached to it.