General Editor: WILLIAM BRIGGS, LL.D., D.C.L., M.A., B.Sc., PRINCIPAL OF UNIVERSITY CORRESPONDENCE COLLEGE. INTERMEDIATE GEOMETRY. GEOMETRY, THEORETICAL AND PRACTICAL. Part I.-Containing the matter of Euclid, I, III (1-34), IV (1-9). 2s. 6d. Part II.-Containing the matter of Euclid, II, III (35-37), IV (10-16), VI. 2s. Part III.—Containing the matter of Euclid, XI. 1s. 6d. This work is issued in Sections as follows : SECTION I.-Introductory Course. 9d. Sections I-IV are also published in one volume under the title of Matriculation Geometry. Price 3s. 6d. A set of Geometrical Instruments, specially designed for use with Workman and Cracknell's Geometry, Theoretical and Practical, can be obtained from Messrs. Clive & Co., 48 Southampton Row, W.C. Price 2s. 6d. INTERMEDIATE BEING SECTIONS V AND VI OF “GEOMETRY, THEORETICAL AND PRACTICAL” BY W. P. WORKMAN, M.A., B.Sc., HEADMASTER OF KINGSWOOD SCHOOL, BATH, AND A. G. CRACKNELL, M.A., B.Sc., F.C.P., SCIENCE DIRECTOR OF UNIVERSITY CORRESPONDENCE COLLEGE, 66 AUTHOR OF PRACTICAL MATHEMATICS." UNIVERSITY PRESS TUTORIAL Clnißersity Tutorial Press Lo 157 DRURY LANE, W.C. PREFACE. This book contains the subject-matter of Euclid Books VI and XI treated entirely on modern lines, together with such other elementary theorems in modern Geometry as are usually associated with them. In accordance with the accepted modern practice the theory of Proportion and Similar Figures has been based on the arithmetical definition of Ratio, as the Euclidian definition is beyond the comprehension of the average schoolboy. This treatment necessitates the assumption that any two magnitudes of the same kind may be regarded as commensurable. For the sake of logical completeness, however, a simple account of Irrational numbers (omitting Irrational indices) has been given in Chapter XXVI; the treatment is original, and it is hoped that it will prove interesting and instructive to the more advanced pupils. The usual theorems on similar triangles and polygons have been inserted, and an account of the centre and ratio of similitude. A concise explanation of the theory of similar figures as applied to scale drawing has also been included. The difficulty experienced in teaching Euclid's Book XI in schools is largely due to the fact that Euclid's treatment is cursory, and omits certain important concepts and theorems. The student is hurried on too rapidly, without becoming sufficiently experienced in the properties of solid space. For example, the idea of a line parallel to a plane is ignored by Euclid, though |