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DISTRICT OF PENNSYLVANIA, TO WIT:

BE IT REMEMBERED, That on the fourth day of December, L.S. in the thirty-fifth year of the Independence of the United States of America, A. D. 1810,

KIMBER AND CONRAD, of the said district, have deposited in this office the title of a book, the right whereof they claim as proprietors, in the words following, to wit: Trigonometry, Plane and Spherical ; with the Construction and Applica

tion of Logarithms. By Thomas Simpson, F. R. S. With an Appen.

dix on Spherical Projections. 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." And also to the act, entitled, “ An act supplementary to an act, 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,” and extending the benefits thereof to the arts of designing, engraving, and etching historical and other prints."

D. CALDWELL, Clerk of the District of Pennsylvanią.

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PREFACE.

97

THE work, of which an American edition is now offered to public acceptance, needs no laboured encomium, as it is the production of an author whose peculiar facility in combining the elegance of the profound geometer with the perspicuity of the practical teacher, has justly established his character with the mathematical world.

The tract having been originally designed as an appendix to a treatise on geometry published by the same writer, it was thought expedient to adapt the references to a different work, more generally studied in our schools. They are accordingly suited to Playfair's Geometry, but they will mostly apply to Simson's translation of Euclid's Elements. Some cases, which were omitted by T. Simpson, have been supplied; particularly the use and demonstration of Napier's circular parts.

The appendix, now subjoined, it is hoped, will prove acceptable, and render the work more extensively useful. The propositions are principally extracted from the writings of Emerson; but many of the demonstrations are either partly or entirely new.

E. L. Newgarden, 11 mo. 24, 1810.

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PLANE TRIGONOMETRY.

DEFINITIONS.

1. PLANE Trigonometry is the art whereby, having given any three parts of a plane triangle (except the three angles), the rest are determined. In order to which, it is not only requisite that the peripheries of circles, but also certain right lines in and about the circle be supposed divided into some assigned number of equal parts.

2. The periphery of every circle is supposed to be divided into 360 equal parts, called degrees; and each degree into 60 equal parts, called minutes; and each minute into 60 equal parts, called seconds, or second minutes, &c.

3. Any part AB (fig. 1.) of the periphery of the circle is called an arch, and is said to be the measure of the angle ACB at the centre, which it subtends.

Note. The degrees, minutes, seconds, &c. contained in any arch, or angle, are written in this manner, 50°18' 35", which signifies that the given arch, or angle, contains 50 degrees, 18 minutes, and 35 seconds,

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