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ELEMENTS OF GEOMETRY

PLANE AND SOLID

BY

JOHN MACNIE, A. M.

AUTHOR OF THEORY OF EQUATIONS"

EDITED BY

EMERSON E. WHITE, A.M., LL.D.

AUTHOR OF WHITE'S SERIES OF MATHEMATICS

NEW YORK.:. CINCINNATI .:. CHICAGO
AMERICAN BOOK COMPANY

623209

White's Series of Mathematics.

ORAL LESSONS IN NUMBER. (For Teachers.)

FIRST BOOK OF ARITHMETIC.

NEW COMPLETE ARITHMETIC.

SCHOOL ALGEBRA. (In Preparation.)

ELEMENTS OF GEOMETRY.

ELEMENTS OF TRIGONOMETRY. (In Preparation.)

COPYRIGHT, 1895, BY AMERICAN BOOK COMPANY.

Printed at
The Eclectic Press

Cincinnati, W. S. A.

PREFACE.

In this treatise, an endeavor is made to present the elements of geometry with a logical strictness approaching that of Euclid, while taking advantage of such improvements in arrangement and notation as are suggested by modern experience. It has been carefully kept in mind that the purpose of such a work is only in a secondary degree the presentation of a system of useful knowledge. A much more important purpose is to afford those who study this subject the only course of strict reasoning with which the great majority of them will ever become closely acquainted. A mind that, by exercise in following and weighing examples of strict logical deduction, has learned to appreciate sound reasoning, and, by practice on suitable exercises, has been trained to reason out a sound logical deduction for itself, has gained what is of far greater importance than mere knowledge; it has gained power. A treatise on rational geometry ought, accordingly, to have for guiding principles those laid down by Pascal as the chief laws of demonstration, substantially as follows: to leave no obscure terms undefined; to assume nothing not perfectly evident; to prove everything at all doubtful, by reference to admitted principles.

In accordance with the first principle, great care has been taken in the wording of the definitions. In the case of some terms, such as straight line and angle, for which no definitions quite free from objection have as yet been proposed, those adopted have been chosen, not as theoretically perfect, but as best suited to the comprehension of the beginner, and most available in deducing the properties of the things defined.

The use of hypothetical constructions has been abandoned for several reasons. To assume them silently, as is now usually done, is unwarrantable in a treatise upon a science supposed, above all others, to consist of a series of rigorous deductions from admitted truths. Why state so carefully that we must assume the possibility

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of drawing or prolonging a straight line, and say nothing in regard to constructions so much less obvious? The author's experience in teaching geometry has convinced him that these stealthy assumptions are decidedly adverse to the acquisition, on the part of the learner, of those habits of strict reasoning which it is the main object of geometrical study to impart. The teacher should have it in his power to inquire, at every step, not only why such a statement is true, but also why such a construction is allowable. On the other hand, the insertion of the few problems really needed as auxiliaries in the demonstration of theorems, imparts to their sequence a logical consistency that cannot obtain where the learner is continually required to perform operations, the possibility of which has neither been proved nor formally assumed.

In regard to the use of circumferences as construction lines in the first book, it may be remarked that, of all lines, the circumference is the easiest to define, to construct, and to conceive, — far more so than is the case with the straight line. No property of the circle, not even its name, is introduced in the first book, but only that property of the circumference which is given in its definition, and some immediate consequences from that property. The employment, at the outset, of the simpler of the two lines treated of in elementary geometry would hardly require apology or explanation but for the force of custom. Yet, strangely enough, in treatises that scrupulously defer the definition of the terms circumference and radius to a subsequent book, there seems to be no scruple against the employment of arcs in illustration of the nature of angles, and we see gravely laid down the postulate: A circumference may be described about any point as center, etc.

The deviation in this work from the usual order of propositions is comparatively slight. In the early part of the first book, so important as the foundation of the science, the properties of triangles are introduced immediately after the discussion of the general properties of angles. This arrangement, especially as regards the different cases of equal triangles, presents several advantages: these propositions are immediately deducible from first principles, or from each other; they are easily grasped by the beginner; above all, they are of the highest utility as aids to further acquisition. It is in itself no slight advantage for the learner to become accustomed, from the first, to the use of these important auxiliaries in demon

stration. From the second book, again, certain propositions that treat of proportional angles have been removed to the place where they belong, after the discussion of ratio and proportion. In the treatment of these subjects, while adhering to the now prevalent method, an endeavor has been made to obviate one frequent source of confusion by making a clear distinction between concrete quantities and their numerical measures.

In regard to propositions and corollaries, the rule observed has been to admit only such as are important in themselves, or have a bearing on subsequent demonstrations and studies. In this era of over-crowded curricula, the aim of an elementary text-book should be to present the necessary rather than the novel or merely interesting. For this reason some subjects have been relegated to an appendix, where they may be studied or omitted according to circumstances.*

The exercises have been carefully selected with a view to their bearing upon important principles, and are, with few exceptions, of such slight difficulty as not to discourage the learner of average ability. In the first sets of exercises, ample assistance is afforded the student by means of references and diagrams, aids that are withheld from the point where the student should have learned to help himself. It is by no means expected that the average class will find time for all the exercises, but enough are given to afford the teacher full opportunity of choice.

SUGGESTIONS. It is earnestly recommended that, before any book work is assigned to a beginning class, a lesson be devoted to the constructions given in Arts. 200–206. The teacher, having shown on the blackboard, for example, how to bisect a straight line, should set the class to doing the same, and require each pupil to bring in, next day, one or more neatly worked examples of the required constructions. The practical familiarity thus gained with the geometrical concepts involved will amply repay the time thus spent. The two great sources of difficulty to the beginner in geometry are the comparative novelty of the subject matter and the unaccustomed clearness of conception and exactness of expres

* The whole of the last part of Book IX., treating of spherical angles and polygons, may as well be omitted by pupils not to take up spherical trigonometry.

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