mathematical world. Some teachers may not at first recognize such terms as congruent, pencils, anti-parallels, prismatic space, and a few others, but they may be assured that these are old and that their introduction materially simplifies the demonstration of certain familiar theorems. 5. The arrangement of proofs is believed to be the best model for the student. The numbered steps enable the teacher to discuss with ease the difficulties of any particular statement whether upon the printed page or in written work upon the blackboard. 6. The exercises are not all intended for solution by any one class. They are graded so as to offer a selection to the teacher and may be used either with each theorem or at the end of each section or book. The exercises, however, do not uniformly refer to the theorems under which they are printed but afford a continual review of the preceding work. Occasionally they anticipate some subsequent proposition, leading up to it independently, and thus possessing a special propaedeutic value. 7. Methods of attack are suggested early, and at the end of Book III are treated with considerable fullness. 8. Historical notes, designed to increase the interest of the student, are frequently inserted, and a biographical table is printed at the end of the work. 9. A table of etymologies follows the biographical table. 10. It is impossible to make complete acknowledgment of the helps that have been used. The leading European textbooks have been constantly at hand. Special reference, however, should be made to such standard works as Henrici und Treutlein, Lehrbuch der Elementar-Geometrie ; Rouché et de Comberousse, Traité de Géométrie; Petersen, Methods and Theories for the Solution of Geometrical Problems ; Faifofer, Elementi di Geometria ; and the Nixon, Harpur, Mackay, and Hall and Stevens Euclids. 11. The following suggestions may be of value : (a) Make haste slowly at the beginning of plane and of solid geometry. (6) The exercises should be begun early in the course so that the student shall soon come to depend upon himself. (c) The figures should be correctly drawn with compasses and straight edge. Many a simple exercise is made difficult by reason of a poor figure. (d) Impromptu work may be called for in dealing with the easier exercises, the arrangement rendering this possible at the end of any assigned lesson. (e) Written work should be required-with much frequency from the first, thus training the eye, the hand and the logical faculty together. In connection with this the teacher will find the authors' Geometry Tablet (Ginn & Company) of value. (f) While everything is done for the student at the beginning of plane and of solid geometry, he is soon thrown upon his own resources, the frequent word “Why?” calling for the statement of reasons in full. The references by book and number of proposition are not to be given, but, in general, the statement of the proposition should be required. 12. Such merits as the work may possess will be found enhanced, it is believed, by the beauty and accuracy of the figures, and the excellence of the typographical make-up. The authors desire to record here their appreciation of the valuable and timely assistance rendered by various friends, and especially by Professor Clarke Benedict Williams, of Kalamazoo College, in the wearisome task of proof-reading. The authors will gladly welcome any corrections or suggestions for improvement. W. W. BEMAN, ANN ARBOR. OCTOBER, 1895. CONTENTS. 66 3. ANGLES FORMED BY CHORDS, SECANTS, AND TANGENTS 112 . SECTION 1. - FUNDAMENTAL PROPERTIES 3. — A PENCIL OF LINES CUT BY PARALLELS |