INCLUDING PLANE, SOLID, AND SPHERICAL GEOMETRY. BY GEO. W. HULL, M. A., PH.D., PROFESSOR OF MATHEMATICS IN THE FIRST PENNSYLVANIA STATE PHILADELPHIA: E. H. BUTLER & CO. PREFACE. In the preparation of this work the author has kept constantly in view not only the object to be attained by the study of Geometry, which is the power of deductive reasoning, but also the needs of the student in the acquisition of this power. A careful study of the work in the class-room indicates that, if symbols are used too extensively, the language and frequently the logic of the successive steps of a demonstration are impaired; and, on the other hand, if no symbols are used, the subject becomes tedious and the student is discouraged. The endeavor here has been to avoid both of these errors, by the present form and arrangement of the demonstrations. A student beginning the study of Geometry requires a great deal of aid before he can give a clear and logical demonstration; therefore, at first, the reason for each step of the demonstration is given in small type immediately below the statement. But too much aid of this kind has been found injurious, and leads to careless habits of study. These references have therefore been limited to the first nine theorems of Book I. In all other cases the numbers have been given to indicate the references. No student can make any great progress in Geometry without frequent practice in original demonstrations; hence a large number of well-selected and well-graded theorems for original thought are given in each book, and, further to obviate the discouraging difficulties which a young student generally encounters in his endeavor to prove a new truth, the diagrams have been given for a few of these theorems, and suggestive references for many others. Modern mathematical thought is almost unanimously of the opinion that the Theory of Limits is the only rational basis of Geometry. Therefore this treatise is based upon this important principle. Attention is called to the form and arrangement of the demonstrations, and to their brevity and simplicity, due to the use of Continued Proportion. Other commendable features are the concise discussion of the Theory of Limits; the practical solution of the 7 Proposition; the use of a single letter to represent an angle, a line, or a polygon; the diagrams of Solid and Spherical Geometry; and the disposition of the corollaries and converse propositions. The author returns his sincere thanks for the assistance which he has received from experienced teachers. MILLERSVILLE, PA., GEORGE W. HULL. |