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made manifest in the practical tests to which the work has been subjected.

In the present Edition, numerous changes have been made, both in the Geometry and in the Trigonometry. The definitions have been carefully revised-the demonstrations have been harmonized, and, in many instances, abbreviated-the principal object being to simplify the subject as much as possible, without departing from the general plan. These changes are due to Professor Peck, of the Department of Pure Mathematics and Astronomy in Columbia College. For his aid, in giving to the work its present permanent form, I tender him my grateful acknowledgments.

The Edition of LEGENDRE, referred to in the last paragraph, will not be altered in form or substance; and yet, Geometry must be made a more practical Science. To attain this object, without deranging a system so long used, and so generally approved, an Appendix has been prepared and added to LEGENDRE, embracing many Problems of Geometrical construction, and many applications of Algebra to Geometry.

It would be unjust to those giving instruction, to add to their daily labors, the additional one, of finding appropriate solutions to so many difficult problems: hence, a Key has been made for the use of Teachers, in which the best methods of construction and solution are fully given.

June, 1875.


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