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THE main object of this work is to provide a book on modern lines that is suitable for the beginner and can be continued in use for higher forms. In the order adopted and in general treatment the suggestions of the Mathematical Association have been largely followed.

The early chapters are very full in explanation, and are written in simple, even colloquial, language. As far as possible, mathematical terms are not introduced until the processes to which they refer have been rendered familiar by use or are actually going to be used. The usual preliminary list of technical words and their definitions is avoided by placing a general index at the end of the book.

By means of examples, some worked in the text, others set as exercises, the student is led to discover or verify the fundamental laws of Algebra. This experimental method is continued throughout the book. Many examples are set with the intention of leading the student up to a difficulty: thus in some of the early examples such expressions as 3-5 occur before negative quantities have been mentioned. These examples show the necessity for the next step in the subject, and by giving the student the chance of discovering that step for himself tend to develop the faculty of research.

The necessity of rigorous mathematical treatment is kept in view, and the usual formal proofs are given after the rules have been treated experimentally.

In the middle of Part I there is a Revision Chapter which goes over the early portions again, laying stress on the important points, and introducing new matter which was considered too difficult for the early chapters. There are also several cases of intentional repetition. For example, the importance of the formula a2 — b2 = (a + b) (a-b) is insisted upon two or three times. The author thinks that repetition is more effective than mere reference to a preceding page.

In many places suggestions are made as to how the solutions to questions should be thought out, and how the work should be arranged. On such points of detail each teacher probably has his own views. But it is hoped that the insertion of such matter in a textbook will not only be of help to the inexperienced teacher, but will save the time of the experienced teacher in cases where the pupil has forgotten some previous rule or method of work.

Graphs are treated as illustrations of Algebraic principles or scientific observations, rather than as objects of importance in themselves. But to use them with skill and confidence considerable practice is necessary, and ample provision is made for this.

The practical use of Logarithms is so important that a short treatment of Fractional and Negative Indices is given in Part I, thus enabling this Part to conclude with a chapter on the use of Logarithms.

Horner's Synthetic Method of Division is introduced as an alternative to the usual method. It is surprising that this simple and compact process has not been brought into common use sooner.

The problem is regarded as the important type of example, and most of the various Algebraic operations and processes are introduced as being necessary for the solution of some practical problem. But the more mechanical type of example is not neglected: such examples afford practice in accuracy and neatness, and no student can do good work in Higher Mathematics who has not had experience in dealing with long mechanical Algebraic operations.

Many of the examples are original, but the greater number are taken from recent Examination Papers by permission of the following authorities :


The Controller of His Majesty's Stationery Office.

The University of Oxford.

The University of Cambridge.

The University of London.

The Joint Matriculation Board of the Scottish Univer


The Joint Matriculation Board of the Universities of

Leeds, Liverpool, Manchester and Sheffield.

The Intermediate Education Board for Ireland.

The Central Welsh Board.

The Delegacy for Oxford Local Examinations.

The Syndicate for Cambridge Local Examinations.
The College of Preceptors.

The author's thanks are due to his colleagues, Mr. F. W. Crampton, M.A., Mr. L. Green, M.A., and Mr. C. B. Wheeler, M.A., for suggestions and help in reading the proofs.

W. E. P.

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