PREFACE. So many books have already been written on Practical Geometry, that the appearance of a new work on the subject requires some justification. Hitherto the study of Practical Geometry has been made almost entirely dependent on the exercise of the memory. The theoretical principles which should have formed the basis of the study, and so made the work of the student an intellectual and not a merely mechanical effort, have been left for him to discover as best he might. Instead of being familiarized with these principles, on which most of the practical solutions are founded, he has had to be content with working out each individual problem, as if its construction had nothing in common with other problems. Where classification has been attempted, it has been based on similarity of form, rather than on identity of principle. Here is an illustration in point, where four consecutive problems depend for their solution on four totally distinct principles: 1. To construct an isosceles triangle, given the base and one side. 2. To construct an isosceles triangle, given the base and vertical angle. 3. To construct an isosceles triangle, given the altitude and vertical angle. 4. To construct an isosceles triangle, given the altitude and one of the angles at the base. In the present work each of these four principles would form the subject of a separate lesson, and such problems as best illustrate the principles would be grouped and taught together. Here, for example, are four problems (see page 20) constructed on the same principle, which being once grasped, will help the student to solve every problem in which that principle is involved: PRINCIPLE " The angle in a semicircle is a right angle." PROBLEM-1. To draw a line perpendicular to a given line from a given point. 2. To construct a right-angled triangle, given its hypotenuse and one side. 3. To construct a square on a given line. 4. To construct an oblong, given the diagonal and one side. Though this book contains over 200 problems and exercises, it might have been considerably enlarged, for applications of these principles can be multiplied almost ad infinitum. It has not been deemed necessary to do this, for, independently of adding to the cost, it would have interfered with the plan of the work, which has been designed to encourage the student to exercise thought in his study, and to enable him to solve any problem or exercise in Practical Geometry whose construction is based on the principles here illustrated. ST. JOHN'S WOOD, Oct. 1880. J. C. |