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The methods of demonstration, in several of the Books, have been entirely changed. By regarding the circle as the limit of the inscribed and circumscribed polygons, the demonstrations in Book V. have been much simplified; and the same principle is made the basis of several im portant demonstrations in Book VIII.

The subjects of Plane and Spherical Trigonometry have been treated in a manner quite different from that employed in the original work. In Plane Trigonometry, especially, important changes have been made. The sepa ration of the part which relates to the computations of the sides and angles of triangles from that which is purely analytical, will, it is hoped, be found to be a decided improvement.

The application of Trigonometry to the measurement of Heights and Distances, embracing the use of the Table of Logarithms, and of Logarithmic Sines; and the application of Geometry to the mensuration of planes and solids, are useful exercises for the Student. Practical examples cannot fail to point out the generality and utility of abstract science.

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