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AN

INTRODUCTION

TO

GEOMETRY

AND THE

SCIENCE OF FORM.

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PREPARED FROM THE MOST

APPROVED PRUSSIAN TEXT-BOOKS.

Mrs. Anna Cabot (Jackson) well.

STEREOTYPE EDITION, CAREFULLY REVISED.

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✔BOSTON AND CAMBRIDGE:

JAMES MUNROE AND COMPANY.

1857.

Eduz T 148.57.535

Matti 1858. April. 5.

5086514

1405 66

sift of
The Publishers.

Entered according to Act of Congress, in the year 1845, by
JAMES MUNROE AND COMPANY,

In the Clerk's Office of the District Court of the District of Massachusetts

BOUND MAR 1 0 1310

I HAVE carefully examined the manuscript of "An Introduction to Geometry" and think it admirably adapted to supply an important want in education. It is not a mere geometrical logic, but a natural and simple introduction to the Science of Form. By a beautiful and original series of inductive processes, it avoids tedious demonstrations, develops the taste for observation, which is so strong in the quick mind of youth, and leads the pupil to a real and practical knowledge of the truths of Geometry with a rapidity which would not have been anticipated. From these considerations, and from observing the strange neglect into which this science has fallen in our schools, I have strongly urged the publication of this excellent treatise, and think that its study should be insisted upon, as a valuable preliminary to a good education either at college or in business.

BENJAMIN PEIRCE,

Cambridge, April 21, 1843.

}

Perkins Professor of Astronomy ana
Mathematics in Harvard University.

Stereotyped by
GEORGE A. CURTIS;

NEW ENGLAND TYPE AND STEREOTYPE FOUNDRY.

PREFACE.

Do my readers remember the fable of the contest between Nature and Education?" Nature chose a vigorous young pine, the incipient mainmast of a man-of-war. But while she was feeding her pine with a plenty of wholesome juices, Education passed a strong rope round its top, and pulling it downwards with all her force, fastened it to the trunk of a neighboring oak. The pine labored to ascend, but not being able to surmount the obstacle, it pushed out to one side, and presently became bent like a bow. Still, such was its vigor, that its top, after descending as low as its branches, made a new shoot upwards; but its beauty and usefulness were quite destroyed."

Let the pine in this short tragedy represent those childish faculties which long to become acquainted with the actual world, and the cunning senses which wait on these faculties, and we have a true tale of the geometrical non-culture of the young.

The curiosity which speaks in children's busy eyes and hands should be to us the voice of Nature, bidding us make our beginnings early. The infant, who cannot speak, gazes earnestly and thoughtfully at the most common object, returning to it, and glancing from one part to another, as if to learn their connection. When he can walk, he goes round it, handling it, and studying it with all his senses. When he speaks, his questions are of size, form, and distance. If our answers are careless or unsatisfactory, his quick eyes and mind, not blunted by habit, detect our errors. He loves com

parison of objects, and the imaginary multiplication and extension of them; he is pleased with the new and the different, and equally pleased with resemblance and equality to things known before.

But

Happy age of natural geometry; when each look and motion, nay his very games, lead the boy on to the laws which shape the spheres and hold the planets in their course. we put a rope around the pine-and the eye grows dull and careless, and the faculties are starvelings and the youth hears with a stupid indifference what he sought instinctively ten years before. He no longer loves comparison-his conceptive faculty disdains such materials; he cares not for properties of circles and triangles, things long familiar to his eye, and whose closer acquaintance he does not desire.

Feeling this injustice to nature, I sought a work on Geometry, which should connect the instincts of the child with the studies of the youth; which should in a pleasurable manner develop the powers, inform the memory, strengthen, sharpen, and expand the understanding. I found some German works which pleased me; particularly, one by Diesterweg, an author of distinguished reputation, and the Director of a Prussian Normal School, which treats the subject in a masterly manner, though it does not bring it down, as I wished, to the busiest and most questioning age.

From these works, at my suggestion, the "Introduction to Geometry" has been prepared. Many alterations and additions were necessary to adapt them to the modes of instruction usually pursued in our schools; but the original and natural mode of development has been carefully preserved, and must interest students of every age.

All is tangible, orderly, and leading to some conclusion. Solids are first presented, and their properties, particularly symmetry of form, displayed; different solids are compared, and each is minutely described by the pupil, and a skeleton of its surface constructed. He considers the solid as bounded by surfaces; the surfaces by lines; and at last he separates the immaterial form from the material substance.

Then lines are taken up, and the possible number and variety of their intersections are proved geometrically by interesting exercises, and arithmetically by series which might be difficult, were they not presented in so simple and orderly a manner. Angles are treated in the same systematic manner, and experiments are made on them, requiring much ingenuity and accuracy.

Then follow the principal propositions of the Elements of Geometry, which are made more interesting to the student by introducing examples of their practical utility. The book thus recommends itself equally to the practical student, and to him to whom it is the stepping stone to more abstruse knowledge; and it is hoped that it will contribute to make this study, which is nothing less than the study of the laws according to which the universe exists, as popular as it should be.

THE AUTHOR OF THE THEORY OF TEACHING.

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