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It will be perceived that the rules in general are not systematically detached from the demonstrations; this, the student whose object is real knowledge, will not consider as a defect. in method, because it may frequently prove the means of enforcing the study of principles. A more commodious arrangement might therefore have been adopted for those who wish to acquire the practice of arithmetic only. That examples however, may not be wanting, we have added a great variety in the different rules, beginning with Vulgar Fractions. See from p. 125 to p. 159.

Euclid's Elements of Geometry, in the most concise form, generally make a separate work, and are therefore too extensive to be admitted at length in a volume of this kind. But we have endeavoured to give all the theorems necessary for the two most useful practical branches, Trigonometry and Mensuration the latter however, is supposed to include such figures only as depend on right-lines and the circle. And with a view to facilitate the transition from theory to practice, when ratios or proportions are concerned, we have sometimes abridged the demonstrations by referring to analogous operations in the arithmetic. This may be deemed ungeometrical: but it ought to be remembered, that many who study Euclid do not wholly comprehend the doctrine of proportion as it is laid down in the fifth Book, without tracing the methods of demonstration by means of an arithmetical, or algebraic process.

Under Surveying the reader is not to expect the methods of plotting and measuring estates; but only such trigonometrical problems as are generally applicable to surveying. This part however, with the articles on Heights and Distances, are principally intended as introductory to the construction of military maps and plans. And to complete, or rather to render the Trigonometry independent, a table of logarithms sufficiently extensive for common practice is subjoined.

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The subjects which compose this volume have so frequently been handled at full length in separate publications, that new principles cannot be expected in a work which may be considered as an abridgement, or compilation. What originality it is therefore entitled to, must principally consist in the arrangement. Most of the examples however, in the application of Trigonometry were selected from actual operations during the summer months in the field. And the practical questions and problems in the other parts of the volume, which are adapted to military concerns, have been furnished from the author's manuscript papers that from time to time were drawn up for the use and instruction of the Officers in the Senior Department of the College.

This edition is much more correct than the former: and several improvements and additions will be found in both the Arithmetic and Geometry.

High Wycombe,
May, 1807.

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