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THE special object of the present work is to meet the requirements of the Science and Art Department's Examinations in the first three stages of Pure Mathematics as set down in the Syllabus of the Science Directory. This will account for the arrangement of the subject matter. I hope, however, that it will be found not unsuitable as a general classbook in Elementary Mathematics.

In the Arithmetical Section my object has been to deduce the rules from first principles, avoiding as much as possible algebraical considerations.

The Geometry consists simply of the first three books of Euclid's Elements with exercises, the only point calling for remark being the marginal notes. I have found it useful in class teaching to put down upon a blackboard the chief steps of the proposition-just those points, in fact, which it is necessary to retain in the memory; and to encourage the pupil to depend upon himself for supplying the connecting links. This skeleton, as it were, of the proposition is placed in the margin, with the hope that it will be specially appreciated by many students of the industrial classes to whom the language of Euclid is ordinarily an insuperable barrier. Particular reference is here made to such of those classes as have grown up to almost manhood without any mathematical training.

In the Algebraical Section of Stage I., the more difficult examples are set particularly for the exercise of those

students who take up the work of Stage II. without having previously used this book for Stage I.

The amount of Plane Trigonometry included in this volume is small, extending only to the solution of triangles and the simpler cases of heights and distances. There is, however, sufficient to fully cover the requirements of Stage II. of the Government Syllabus. The treatment of the higher parts of Algebra and Plane Trigonometry, as well as that of Spherical Trigonometry, is reserved for a second volume.

E. A.

LEICESTER, November, 1873.

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