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factory to dwell on the properties of the trigonometrical ratios, and to exemplify their use in easy problems, than to bewilder a beginner with an angular system the use of which he cannot appreciate, and which at this stage furnishes nothing but practice in easy Arithmetic.

The subject of Logarithms and their application has been treated very fully, and illustrated by a selection of carefully graduated Examples. It is hoped that the examples worked out in this section may serve

as useful models for the student, and may do something to cure that inaccuracy in logarithmic work which is so often due to clumsy arrangement.

In the experience of most teachers it is found extremely difficult to get boys to handle problems in Heights and Distances with any degree of confidence and skill. Accordingly we have devoted much thought to the exposition of this part of the subject, and by careful classification of the Examples we have endeavoured to make Chapters VI. and xvII. as easy and attractive as possible.

Very little advance can be expected in Trigonometry until the principal formulæ can be quoted readily, but whether it is advisable for learners to have lists of formulæ compiled for them, so as to be easily accessible at all times, is a matter upon which teachers hold different views. In our own opinion it is distinctly mischievous to furnish such lists; it encourages indolent habits, and fosters a spurious confidence which leads to disaster when the student has to rely solely upon his own knowledge.

In the general arrangement and succession of the different parts of the subject we have been mainly guided by our own long experience in the class room; but as the manuscript and proof-sheets have been read by several skilled teachers, and have been frequently tested by pupils in all stages of proficiency, the hope is entertained that our treatment is such as to enable beginners to take an intelligent interest in the subject from the first, and to acquire a sound elementary knowledge of practical Trigonometry before they encounter the more theoretical difficulties. At the same time, as each chapter is, as far as possible, complete in itself, it will be easy for teachers to adopt a different order of treatment if they prefer it; the full Table of Contents will facilitate the selection of a suitable course of reading, besides furnishing a useful aid to students who are rapidly revising the subject.

We are indebted to several friends for valuable criticism and advice; in particular, we have to thank Mr T. D. Davies of Clifton College for many useful hints, and for some ingenious examples and solutions in Chapters xxiv. and xxv.

H. S. HALL.

S. R. KNIGHT.

November, 1893.

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