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Entered according to Act of Congress, in the year Eighteen Hundred and Fifty-one,


In the Clerk's Office of the District Court of the United States for the Southern

District of New York.




Those who are conversant with the preparation of elementary text-books, have experienced the difficulty of adapting them to the various wants which they are intended to supply.

The institutions of education are of all grades, from the college to the district school, and although there is a wide difference between the extremes, the level, in passing from one grade to the other, is scarcely broken.

Each of these classes of seminaries requires text-books adapted to its own peculiar wants; and if each held its proper place in its own class, the task of supplying suitable works would not be difficult.

An indifferent college is generally inferior, in the system and scope of its instruction, to the academy or high school; while the district school is often found to be superior to its neighboring academy.

The Geometry of Legendre, embracing a complete course of Geometrical science, is all that is desired in the colleges and higher seminaries; while the Practical Mathematics for Practical Men, recently published, is designed to meet the wants of those schools which are strictly elementary and practical in their systems of instruction.

But still a large class of seminaries remained unsupplied with a suitable text-book on Elementary Geometry and Trigonometry : viz., those where the pupils are carried beyond the acquisition of facts and mere practical knowledge, but have not time to go through with a ful course of mathematical studies.

It is for sucn, that the following work is designed. I has been the aim of the author to present the striking and important truths of Geometry in a form more simple and concise than could be adopted in a complete treatise, and yet to preserve the exactness of rigorous reasoning.

In this system of Geometry nothing has been taken for granted, and nothing passed over without being fully demonstrated.

The Trigonometry, including the applications to the measurements of heights and distances, has been written upon the same plan and for the same objects: it embraces all the important theorems and all the striking examples.

In order, however, to render the applications of Geometry to the mensuration of surfaces and solids complete in itself, a few rules have been given which are not demonstrated. This forms an exception to the general plan of the work, but being added in the form of an appendix, it does not materially break its unity.

That the work may be useful in advancing the interest of education, is the hope and ardent wish of the author. FISHKILL LANDING,

May, 1851.

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