Particular attention has been given to the arrangement of the propositions and corollaries in a form for convenient reference. The statement of the corollary has in every case been printed in italics; and in nearly every proposition in which more than one truth is stated, the various parts are distinguished by numerals. Thus, when reference is made to a preceding section, the pupil will readily find the precise statement which is to be quoted. The exercises are upwards of eight hundred in number, and have been selected with great care. In certain exercises which might otherwise present difficulties to the pupil, reference is made to a previous section or exercise which may be used in the solution. The exercises in each Book are numbered consecutively. In the Plane Geometry, the new exercises are largely numerical; but in the Solid Geometry, there is a considerable increase in the number of both numerical exercises and original theorems. A number of the exercises are in the nature of alternative methods of proof for preceding propositions. In the Appendix to the Plane Geometry will be found an additional set of exercises of somewhat greater difficulty than those previously given. The pages have been arranged in such a way as to avoid the necessity, while reading a proof, of turning the page for reference to the figure. The attention of teachers is specially invited to the explanations given in the Introduction, commencing on page vii. The author desires to express his thanks to the many teachers, in all parts of the country, who have furnished him with valuable suggestions and criticisms. WEBSTER WELLS. MASSACHUSETTS INSTITUTE OF TECHNOLOGY, 1894. TO TEACHERS. AMONG the most important objects of the study of Geometry are the development of the reasoning faculties, and the cultivation of the power of clear and accurate expression. To attain these ends, the pupil should be required to state the various parts of a demonstration in concise and logical language, and to give after each statement of a proof the reason in full. Throughout Book I., and in the first sixteen propositions of Book VI., the required authority is printed in each case directly after the statement, in smaller type, enclosed in brackets. In the remaining portions of the work, the formal statement of the reason is omitted, and there is given, in parenthesis, only the number of the section where the authority is to be found. In every such case, the pupil should be held, as in Book I., to the full statement of the reason. The statements of the corollaries are in all cases printed in italics; so that when a previous section is referred to in a proof, the pupil will always find printed in italics the precise statement to be quoted. Thus in Prop. II., Book II., reference is made to § 143; this calls for the following statement: : All radii of a circle are equal. While in general the complete statement of the reference should be insisted on, if the proposition referred to states more than one truth, that portion only need be quoted which applies to the case under consideration. Thus, in the proof of § 29, reference is made to § 28; here the complete reference is not given, but only the portion actually used. In most cases, the various parts of a proposition are indicated by numerals; and when reference is made to a section, the numeral following the number of the section shows which portion of the statement is to be quoted. Thus, in Prop. XXI., Book II., Case I., reference is made to § 83, 1; this calls for the following statement: — An exterior angle of a triangle is equal to the sum of the two opposite interior angles. If a previous case of the same proposition is referred to, the reference given should be the statement of the theorem, followed by the statement of the previous case. 66 Thus, on page 92, the reference "§ 189, Case I." calls for the following: In the same circle, two central angles are in the same ratio as their intercepted arcs, when the arcs are commensurable. Considerable practice should be given in writing demonstrations on the blackboard; the authority for each statement being given in full, just as when the proof is given orally. The symbols given on page 4 should be used whenever possible. The following abbreviations will also be found of use: The author has thought that it would be an aid to teachers to put a few propositions in a form which is recommended for blackboard work. |