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In the preparation of the present edition of the Geometry of A. M. LEGENDRE, the original has been consulted as a model and guide, but not implicitly followed as a standard. The language employed, and the arrangement of the arguments in many of the demonstrations, will be found to differ essentially from the original, and also from the English translation by DR. BREWSTER.

In the original work, as well as in the translation, the propositions are not enunciated in general terms, but with reference to, and by the aid of, the particular diagrams used for the demonstrations. It is believed that this departure from the method of Euclid has been generally regretted. The propositions of Geometry are general truths, and as such, should be stated in general terms, and without reference to particular figures. The method of enunciating them by the aid of particular diagrams seems to have been adopted to avoid the difficulty which beginners experience in comprehending abstract propositions. But in avoiding this difficulty, and thus lessening, at first, the intellectual labor, the faculty of abstraction, which it is one of the primary objects of the study of Geometry to strengthen, remains, to a certain extent, un improved.

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 important 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 separation 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 ex. amples cannot fail to point out the generality and utility of abstract science. FISHKILL LANDING, I

July, 1851.

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