HARVARD COLLEGE MAY 17 1011 LIBRARY Jarrar fund HENRY FROWDE, M.A. PUBLISHER TO THE UNIVERSITY OF OXFORD LONDON, EDINBURGH NEW YORK AND TORONTO PREFACE THE material for mathematical education may be chosen for the value of the knowledge of the material itself, or for the value of the training got in the course of acquiring the knowledge. The knowledge-value is greater the more closely the thing studied is related to human life and interests, and less the more remote it is from these. As to training-value, the appropriate progress from concrete to abstract is most possible in connexion with things of concrete human interest; so that a selection of material made for its knowledge-value is confirmed by the criterion of training-value. In range this book is in general agreement with the practice of our schools; containing the Geometry, Algebra, and Trigonometry usually read by pupils that do not specialize in mathematics. In detail there are a few differences. Thus the addition theorem in Trigonometry is of no great use for the solution of triangles, nor has the manipulation involved sufficient training-value to justify the inclusion of the theorem. On the other hand, a certain amount of Solid Geometry is included from a belief in its value for both knowledge and training. The more the pupils can develop the subject for themselves and without help from the teacher, the truer is their knowledge of it and the more valuable the training received in the process; but care must be taken that this intensive method does not too greatly restrict the range of knowledge. This book has the form of a summarized discussion between teacher and pupils. In the earlier chapters the discussion is given in some detail. In later chapters it is more condensed, so that towards the end of the book the reply put in the pupil's mouth is the final formal conclusion, which is reached only after a good deal of discussion. The order of the development of the subject must depend to some extent on the suggestions made by the pupils in discussion, and so must vary from class to class. This book shows one possible development; more importance is attached to the method than to the order of development. In order that the discussion may be a real one, each discussion should take place before the pupils look at the corresponding part of the book. Some teachers may for the sake of drill and mechanical dexterity desire an increase in the number of exercises; they can easily multiply them by numerical alteration of the data. But I venture to think that this country pays too much attention to dexterity, and that to work honestly through the problems of the text and the exercises will result in as great a degree of dexterity as any one should require. Occasional reference is made to principles. For a discussion of principles readers are referred to Some Principles of Mathematical Education by Mr. Benchara Branford, which is being issued by the Clarendon Press. The writing of my book has been in part due to reading Mr. Branford's manuscript. Valuable help in suggestions and proof reading has been given by Messrs. Leonard Blaikie, Benchara Branford, J. G. Hamilton, and E. L. Kearney. Most of the exercises are taken from examination papers of the Civil Service Commission, by the permission of the Controller of His Majesty's Stationery Office. Criticisms and suggestions will be gratefully received. I shall be especially glad to hear of pupils' suggestions which (whether right or wrong) have proved fruitful for the development of the subject. DAVID MAIR. BANSTEAD, SURREY, CONTENTS Determination of position of cache. pass; right angle; degree; circle. experiment and from general reasoning. Chord properties of circle. Work is to be done by pupils, CHAPTER II. POSITION OF CHAIR ON FLOOR Position of a point given by distances from walls. Parallels. Loci. Symbols, and their right use. Meaning of word 'Geometry'. Units of area. brackets; precedence of x over +. Suitable degree of approximation. Area of rectangle, of right-angled triangle, Air-space of a room. Volume of a cylinder (gallon CHAPTER V. To COPY A MAP, SIZE UNCHANGED Copy made by measurement of distances; number of measurements needed. Case of three places, other methods. Plane table. To make an angle equal to a given angle; to bisect an angle. Conditions that fix a triangle. Variation of angles of a triangle that has two sides given. Triangle fixed by three sides, by two sides and an angle, by a side and 59 |