The Applications of Geometry to Mensuration, given in the eleventh and twelfth books, are designed to show how the theoretical principles of the science are connected with manifold practical results.

The Miscellaneous Exercises, which follow, are calculated to test the thoroughness of the scholar's geometrical knowledge; and sufficient Applications of Algebra to Geometry are given to show the relation existing between these two branches of the mathematics.

The Elements of Plane and Spherical Trigonometry present a complete system, theoretical and practical, fully adapted to the wants of advanced classes.

The trigonometric functions, in this treatise, have been regarded as ratios, since this improved method has not only now superseded the ancient method in English and French works, but has been approved and adopted generally by the best Ameriican mathematicians. Reference, however, is made to whatever is especially valuable in the old method.

In the preparation of this work the author has received valuable suggestions from many eminent teachers, to whom he would here express his sincere thanks. Especially would he acknowledge his great obligations to H. B. Maglathlin, A. M., who for many months has been associated with him in his labors, and to whose experience as a teacher, skill as a mathematician, and ability as a writer, the value of this treatise is largely due.


BRADFORD, Mass., July 25, 1861.


A KEY, comprising the Solutions of the Problems contained in this work, is published, for Teachers only; and the same will be mailed, postpaid, to the address of any Teacher who will forward thirty-six cents in stamps to the Publishers.

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1. GEOMETRY is the science of Position and Extension. The elements of position are direction and distance. The dimensions of extension are length, breadth, and height or thickness.

2. MAGNITUDE, in general, is that which has one or more of the three dimensions of extension.

3. A POINT is that which has position, without magnitude.

4. A LINE is that which has length, without either breadth or thickness.

5. A STRAIGHT LINE, or RIGHT LINE, is one which has the same direction in its whole extent; as the line AB.



The word line is frequently used alone, to designate a straight line.

6. A CURVED LINE is one which continually changes its direction; as the line CD.

The word curve is frequently used to designate a curved




7. A BROKEN LINE is one which is composed of straight lines, not lying in the same direction; as the line EF.


8. A MIXED LINE is one which is composed of straight lines and of curved lines.

9. A SURFACE is that which has length and breadth, without height or thickness.

10. A PLÁNE SURFACE, or simply a PLANE, is one in which any two points being taken, the straight line that joins them will lie wholly in the surface.


11. A CURVED SURFACE is one that is not a plane surface, nor made up of plane surfaces.


12. A SOLID, or VOLUME, is that which has length, breadth, and thickness.



13. A PLANE ANGLE, or simply an ANGLE, is the difference in the direction of two lines, which meet at a point; as the angle A.



The point of meeting, A, is the vertex of the angle, and the lines A B, A C are the sides of the angle. An angle may be designated, not only by the letter at its vertex, as C, but by three letters, particularly when two or more angles have the same vertex; as the angle ACD or DCB, the letter at the vertex always occupying the middle place.




The quantity of an angle does not depend upon the length, but entirely upon the position, of the sides; for the angle remains the same, however the lines containing it be increased or diminished.

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