A KEY to a set of geometrical exercises is needed for much the same reasons as render it desirable to supply the teacher with answers to exercises in arithmetic and algebra. To the instructor who has skill in working out such exercises, the doing so is a mere drudgery in a task that has lost the interest of novelty; while to those who find some difficulty in the task, the time and mental exertion required is a serious addition to their necessary burdens. The unusually large number of exercises of various kinds given in the work to which this serves as a key, renders it all the more imperative to lighten the teacher's burden as much as possible. A great proportion of the exercises, such as the questions at the end of each book and the exercises at the foot of the page, are of so easy a character as to require little more than suggestions, though even in these assistance is given proportioned to the difficulty of the case. Numerical answers are given correct to two places of decimals when not expressed as rational numbers. The number of exercises at the end of each book is still so large as to preclude the possibility of giving a full-length demonstration of each, if the book is to be kept within reasonable limits of size and price. It will be found, however, that all needful information is given as concisely as is consistent with clearness, the diagrams, when not given, being easy to draw, according to directions, on a plan explained in the Introduction. In this Introduction will also be found a general discussion of the difficulties to be met in the solution of geometrical exercises, and how best to overcome them. 3 184026 |