GEOMETRY. A COMPLETE COURSE IN THE ELEMENTS OF THE SCIENCE. BY EDWARD BROOKS, A. M., PH. D., LATE PRINCIPAL OF STATE NORMAL SCHOOL, PENNSYLVANIA, AND AUTHOR "LET NO ONE IGNORANT OF GEOMETRY ENTER HERE." PLATO. PHILADELPHIA: CHRISTOPHER SOWER COMPANY, 614 ARCH STREET. PREFACE. ABOUT twenty years ago I published a "shorter course " in geometry entitled the Normal Elementary Geometry. This little book led to the introduction of geometry into many schools that were not prepared for the larger works on the subject then in use. That work is still adapted to the wants of many institutions, but others in the progress of education have outgrown this shorter course and unite with other educational institutions in a request for a more complete work on the subject. To meet this demand the present work has been prepared. It is called a "Complete Course" to distinguish it from the "Elementary Geometry," though it contains only what is known as the elements of plane and solid geometry. The design of the work is to present such a course in geometry as is usually required in academies, colleges, and other higher institutions of learning. The matter is so graded, however, and the treatment so simplified, that schools not requiring the entire course can use the work with convenience and satisfaction. GENERAL TREATMENT.-Great pains have been taken to give clearness and simplicity to the treatment. The methods of demonstration adopted in general are those which experience has shown to be most satisfactory to the science and most readily understood by the pupil. New methods of proof or simplifications of older methods have been introduced only when they were obviously improvements on the methods in ordinary use. The object has been to present the established principles of the science in such a manner that they may be most readily understood and mastered by the student. THE TEXT. To aid the learner in seeing the relation of the dif ferent parts of a demonstration, no method has been found more useful than the breaking up of the demonstration into its constituent parts. This method was advocated by Prof. De Morgan many years ago, and is now adopted by many English authors of geometry. In adopting this plan in the present work, care has been taken to present distinct or closely related statements or arguments in a single line or paragraph, so that they are readily caught by the eye of the student. At the same time equal care has been exercised not to ride this feature as a hobby and carry the plan into such useless details as to confuse the mind by breaking the connection between closely-related parts. In teaching geometry it is found that pupils often experience difficulty in distinguishing between what is "given" and what is "to be proved," and also in seeing just where the argument for the proof begins. This difficulty has been obviated by the use of the side headings-Given, To Prove, and Proof. The classification of the subject matter in the different books, a new and popular feature of the "Elementary Geometry," is also retained in the present work. GEOMETRICAL SYMBOLS.-Within a few years symbols for the words angle, triangle, perpendicular, etc., as presented in the work of Hamblin Smith, have been extensively adopted by teachers of geometry. These are especially useful for written recitations or the examination of a class in geometry. While an excessive use of them in a text-book tends to obscure the presentation and reduce the science to a symbolic form that would diminish its value as a discipline of general thought-power, a judicious use of symbols tends to simplicity and convenience. The aim has been therefore to make such use of these symbols as will render the text the most easily understood by the student. EQUATIONAL THOUGHT..—The tendency of modern methods of instruction in geometry is toward the equational form of thought and expression. This tendency has been stimulated by the prevalence of written recitations and examinations. The method has its advantages in that it often simplifies the argument and enables the pupil to grasp and present the relations with greater ease and certainty than by the ordinary method. To attempt to adopt this method exclusively, however, would be to dispense with that flow of logical thought which is so valuable in its discipline to the mind of the ordinary thinker and speaker. Pains have been taken, therefore, to reduce the argument to equational forms where it really simplifies the demonstration, and to preserve the ordinary logical form where that form seems preferable. The aim has been to combine that culture which is purely mathematical with that general culture of the reasoning powers that prepares the student for the general duties of life. DOCTRINE OF LIMITS.-In treating incommensurable quantities the doctrine of limits seems to command the approval of the best mathematicians of the country. This method has been adopted in the present treatise. The effort has been made to present it in its simplest form; and it is hoped that it will meet the approval of teachers who have been accustomed to the older methods of infinites and the reductio ad absurdum. This PRACTICAL EXERCISES.--A radical defect of former text-books on geometry was the abstract manner of their treatment, giving pupils no idea of the application of the principles they learned. defect was met in the "Elementary Geometry" by the presentation of a collection of " practical examples" at the close of each book, showing the application of the principles of the science. This feature is so valuable that it has been adopted by several recent authors of text-books on geometry. In the present work will also be found a large collection of examples carefully graded, and of a character to show the practical value of the science. UNDEMONSTRATED THEOREMS.-Another defect of the older works was the lack of matter for original thought and the training of the inventive powers of the student. This defect was remedied in the former work by the presentation of a collection of theorems for original thought. This valuable feature is also adopted in the present work. At the close of each book will be found a collection of prop |