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Eastern District of Virginia, to wit: Be it remembered, That on the twelfth day of July, in ***** the fifty-third year of the Independence of the

United States of America, FERDINAND R.

HASSLER, of the said District, hath deposi******* ted in this Office the title of a book, the right whereof he claims as author, in the words following, to wit:

Elements of the Geometry of Planes and Solids. With four Plates. By F. R. Hassler, F. A. P. S.

In conformity to the Act of the Congress of the United States, entitled " An act for the encouragement of learning, by securing the copies of maps, charts and books, to the authors and proprietors of such copies, during the times therein mentioned."

R'D. JEFFRIES, Clerk of the

Eastern District of Virginia.

INTRODUCTION.

GEOMETRY, is that branch of Mathematics which treats of the magnitude and relation of figures, in the most general acceptation of the word. Its immediate subject is, therefore, Dimension, which it compasses in any form and magnitude that may be required; in order to represent exactly, or approximate as near as desired, any extension, surface or solid, presented in nature, or imagined with the view of application to nature, and the relations of which it determines. Elementary Geometry treats of these relations only as limited: higher Geometry determines them without reference to limitation.

The first elements of Geometry take their rise in principles so strong and so simple as to be undeniable and self evident, before any scientific form is given to them. It, therefore, begins by a simple statement of these, under the denomination of Axioms; or if not actually stated, they are tacitly supposed as admitted.

Then, under the title of Postulates, it requests the admission, (intellectually,) that a straight line may be drawn, and a circle described, both of which it supposes exact, without entering on the question how near this accuracy may be attainable in the mechanical practice.

Upon these few and simple data the science is built, by the synthetical process of reasoning, to a full system, presenting all the necessary means to apply its results to nature; either by the construction of them, according to its own principles, (which is practical Geometry,) or by delivering its results over to the calculation, as magnitudes, the relations of which it has determined.

While, therefore, the Elements of Arithmetic present us with the principles of exact reasoning, in the analytical form,

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