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THE Series of Mathematical Exercises here offered to the public is collected from those which the author has, from time to time, proposed for solution by his pupils during a long career at the Royal Military Academy: they are, in the main, original; and having well fulfilled the purpose for which they were first framed, it is hoped that they may be made still more widely useful.

The aim in proposing them was not so much to set before the pupil intricate and puzzling questions, as to determine, from the form of solution, whether his mind had fairly grasped the fundamental principles of the particular subject, and was capable of applying those principles: so that a student who finds that he is able to solve the larger portion of these exercises, may consider that he is thoroughly well grounded in the elementary principles of Pure and Mixed Mathematics.

It has not been considered desirable to place the questions strictly in order of presumed difficulty; first, because, on such a point, no two opinions would always agree; and secondly, because a student should be exercised to pass from one style of solution to another with as little effort as his mental capacity will allow.

It has been thought advisable to place the answers at

the end of the volume, in a form which the author hopes will preclude the loss of time which such an arrangement usually entails. For this purpose the numbering of the questions is continuous throughout; and at the head of each page of answers are placed the index numbers of the solutions which commence and terminate the page,

Increasing attention is now being paid to the Method of Synthetic Division and to the Solution of Numerical Equations by Horner's Method, neither of which processes is given, in a form comprehensible by any but advanced students, in any of the treatises of the day on Elementary Algebra. It has therefore been found necessary to give a short and very practical outline of the principles upon which these methods are based; and the author is not without hope that his Second Appendix may lead to the introduction of the general numerical solution of equations in its natural position in every course of Elementary Algebra, immediately after the subject of Quadratic Equations, where it would tend to develope in the mind of the student a knowledge of the nature of algebraic functions, which would be of the utmost service throughout his subsequent course.

For the single purpose of the determination of the numerical roots of an equation, a vast amount of unnecessary and somewhat intricate investigation of the properties of equations is always entered into; and it is a real boon to the learner to clear away all redundancies and reduce the theory of solution to its simplest elements. This has been done in Appendix II. by basing the determination of a root, first upon the law of signs, for hypothetical position, and, ultimately, upon the test-fact of its satisfying the condition implied by the equation; this fact being shown in actual substitution by a process of synthetic division originated by

Horner for this very purpose. The simple logic of solution then becomes this: "If the roots be all real, we have found by the law of signs that one of them lies between a and b ; treating this assumed root by Horner's process of developement, we find that the developed root, a + &c., satisfies the equation more and more nearly according to the extent of its developement. There can be no doubt therefore that this value is a root, whether the remaining roots contain among them imaginary forms or not." By this means the mass of difficulties involved in the various theorems of Newton, Sturm, Fourier and others, regarding the limits of the roots, may be safely ignored until the learner is better prepared to undertake their study.

The method of dealing with approximately equal roots by a system of reciprocal equations, was first published by the author in 1842, and he still believes it to be the most efficient algorithm yet proposed for application to those delicate cases in which two or more roots are identical to more than one or two places of decimals, supposing this fact to be unknown to the worker.

The exercises in Practical Mechanics will, it is hoped, assist in calling attention to this important, though much neglected subject.

To prevent the necessity of reference to other books on mere matters of memory, a few pages of the more useful tables and elementary formulæ in all the subjects comprised in the present work have been given in a form which, while perfectly intelligible to students who have really studied the particular subjects, will probably be of little or no use to those who have merely acquired a smattering.

The collection of Examination Papers, also from original sources, will be useful to Instructors in ascertaining pro

gress from time to time; and, with this view, the answers to them have been omitted.

In selecting and arranging questions from a mass of papers, spread over many years, it is very difficult to avoid the insertion of duplicates: although every effort has been made to do this, the author regrets that he has not been, in all cases, successful.


9 Feb. 1866.


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