enunciations of Euclid's propositions are given in an Appendix. It is not necessary, and perhaps not desirable, that on his first reading the student should work through every example in each section. He should in each case, however, write out a sufficient number to insure his mastery of the principles involved; the others will be found useful when he comes to revise. Through the kindness of friends the book has been tested, when in proof, by actual work with pupils, and the satisfactory result of this experiment has encouraged the authors to believe that the treatise may be found generally useful. They have to acknowledge valuable suggestions and assistance from Messrs. Butters, Clark, and Walker, Heriot's Hospital School, Edinburgh; Mr. R. F. Davis ; the Rev. W. F. Failes, Westminster School; Mr. Hayward, Harrow School; Mr. Macdonald, Daniel Stewart's College, Edinburgh; Dr. Mackay, Edinburgh Academy; Rev. J. J. Milne; Dr. Muir, Glasgow High School; Professor Raitt, Glasgow Technical College; Mr. Robertson, Edinburgh Ladies' College; Rev. G. Style, Giggleswick School; Mr. Tucker, University College School, and other friends. Additional parts, corresponding to the remaining books of Euclid, are in preparation. is equal in every respect to, is congruent with, or is identically equal to. Congruent figures are such as can be made to coincide by superposition. CHAPTER I. THEOREMS. § 1. (Bookwork, EUCLID, I. 1-4.) 1. Any point on the bisector of the vertical angle of an isosceles triangle is equally distant from the extremities of the base. Let ABC be an isosceles triangle, and let AP bisect its vertical angle; it is required to prove that 2. The straight line which bisects the vertical angle of an isosceles triangle bisects the base, and is at right angles to the base. (A Standard Theorem.) A Д B (2) LADB= LADC=rt. 4. [Def. of rt. < . 3. ABC is an isosceles triangle, and in AB, AC points G, H are taken so that AG = AH. Show that (1) BH = CG, H Use Euc. I. 4 to show that A BAH A CAG. Examine the case in which G and H are on AB and AC produced. A 4. If through the mid-point of a given straight line a perpendicular to the line be drawn, any point in the perpendicular shall be equally distant from the extremities of the given line. (A Standard Theorem.) B Use Euc. I. 4 to show that A ACP A BCP. NOTE.-Standard Theorems may be used in proving other deductions in the same way as Euclid's propositions, but instead of referring to them by number, the student should quote the enunciation, |