623171 Entered according to Act of Congress, in the year 1888, by G. A. WENTWORTH, in the Office of the Librarian of Congress, at Washington. TYPOGRAPHY BY J. S. CUSHING & Co., BOSTON. PRESSWORK BY GINN & Co., BOSTON. PREFACE. M OST persons do not possess, and do not easily acquire, the power of abstraction requisite for apprehending geometrical conceptions, and for keeping in mind the successive steps of a continuous argument. Hence, with a very large proportion of beginners in Geometry, it depends mainly upon the form in which the subject is presented whether they pursue the study with indifference, not to say aversion, or with increasing interest and pleasure. In compiling the present treatise, the author has kept this fact constantly in view. All unnecessary discussions and scholia have been avoided; and such methods have been adopted as experience and attentive observation, combined with repeated trials, have shown to be most readily comprehended. No attempt has been made to render more intelligible the simple notions of position, magnitude, and direction, which every child derives from observation; but it is believed that these notions have been limited and defined with mathematical precision. A few symbols, which stand for words and not for operations, have been used, but these are of so great utility in giving style and perspicuity to the demonstrations that no apology seems necessary for their introduction. Great pains have been taken to make the page attractive. The figures are large and distinct, and are placed in the middle of the page, so that they fall directly under the eye in immediate connection with the corresponding text. The given lines of the figures are full lines, the lines employed as aids in the demonstrations are shortdotted, and the resulting lines are long-dotted. In each proposition a concise statement of what is given is printed in one kind of type, of what is required in another, and the demonstration in still another. The reason for each step is indicated in small type between that step and the one following, thus preventing the necessity of interrupting the process of the argument by referring to a previous section. The number of the section, however, on which the reason depends is placed at the side of the page. The constituent parts of the propositions are carefully marked. Moreover, each distinct assertion in the demonstrations and each particular direction in the construction of the figures, begins a new line; and in no case is it necessary to turn the page in reading a demonstration. This arrangement presents obvious advantages. The pupil perceives at once what is given and what is required, readily refers to the figure at every step, becomes perfectly familiar with the language of Geometry, acquires facility in simple and accurate expression, rapidly learns to reason, and lays a foundation for completely establishing the science. Original exercises have been given, not so difficult as to discourage the beginner, but well adapted to afford an effectual test of the degree in which he is mastering the subjects of his reading. Some of these exercises have been placed in the early part of the work in order that the student may discover, at the outset, that to commit to memory a number of theorems and to reproduce them in an examination is a useless and pernicious labor; but to learn their uses and applications, and to acquire a readiness in exemplifying their utility is to derive the full benefit of that mathematical training which looks not so much to the attainment of information as to the discipline of the mental faculties. PHILLIPS EXETER ACADEMY, 1878. G. A. WENTWORTH. TO THE TEACHER. WHEN the pupil is reading each Book for the first time, it will be well to let him write his proofs on the blackboard in his own language; care being taken that his language be the simplest possible, that the arrangement of work be vertical (without side work), and that the figures be accurately constructed. This method will furnish a valuable exercise as a language lesson, will cultivate the habit of neat and orderly arrangement of work, and will allow a brief interval for deliberating on each step. After a Book has been read in this way, the pupil should review the Book, and should be required to draw the figures free-hand. He should state and prove the propositions orally, using a pointer to indicate on the figure every line and angle named. He should be encouraged, in reviewing each Book, to do the original exercises; to state the converse of propositions; to determine from the statement, if possible, whether the converse be true or false, and if the converse be true to demonstrate it; and also to give well-considered answers to questions which may be asked him on many propositions. The Teacher is strongly advised to illustrate, geometrically and arithmetically, the principles of limits. Thus a rectangle with a constant base b, and a variable altitude x, will afford an obvious illustration of the axiomatic truth that the product of a constant and a variable is also a variable; and that the limit of the product of a constant and a variable is the product of the constant by the limit of the variable. If x increases and approaches the altitude a as a limit, the area of the rectangle increases and approaches the area of the rectangle ab as a limit; if, however, a decreases and approaches zero as a limit, the area of the rectangle decreases and approaches zero for a limit. An arithmetical illustration of this truth may be given by multiplying a constant into the approximate values of any repetend. If, for example, we take the constant 60 and the repetend 0.3333, etc., the approximate values of the repetend will be fo, 1%, 3 3 333 3333 1000, 10000, etc., and these values multiplied by 60 give the series 18, 19.8, 19.98, 19.9998, etc., which evidently approaches 20 as a limit ; but the product of 60 into (the limit of the repetend 0.333, etc.) is also 20. Again, if we multiply 60 into the different values of the decreasing series, 30, 3000, 30800, etc., which approaches zero as a limit, we shall get the decreasing series 2, 1, 3, 5ʊ, etc.; and this series evidently approaches zero as a limit. In this way the pupil may easily be led to a complete comprehension of the subject of limits. The Teacher is likewise advised to give frequent written examinations. These should not be too difficult, and sufficient time should be allowed for accurately constructing the figures, for choosing the best language, and for determining the best arrangement. The time necessary for the reading of examination-books will be diminished by more than one-half, if the use of the symbols employed in this book be allowed. PHILLIPS EXETER ACADEMY, 1879. G. A. W. |