Euclides THE CONTENTS OF THE FIFTH AND SIXTH BOOKS ARRANGED AND EXPLAINED BY M. J. M. HILL, M.A., D.Sc., F.R.S. PROFESSOR OF MATHEMATICS AT UNIVERSITY COLLEGE, LONDON ; AND FOR THE CIVIL SERVICE OF INDIA. CAMBRIDGE : AT THE UNIVERSITY PRESS. [All Rights reserved.] PREFACE. HE object of this work is to remove the chief difficulties felt by those who THE desire to understand the Sixth Book of Euclid. It contains nothing beyond the capacity of those who have mastered the first four Books, and has been prepared for their use. It is the result of an experience of teaching the subject extending over nearly twenty years. The arrangement here adopted has been used by the Author in teaching for the past three years and has been more readily understood than the methods in ordinary use, which he had previously employed. The Sixth Book depends to a very large extent on the Fifth, but this Fifth Book is so difficult that it is usually entirely omitted with the exception of the Fifth Definition, which is retained not for the purpose of proving all the properties of ratio required in the Sixth Book, but only for demonstrating two important propositions, viz., the 1st and 33rd. The other properties of ratio required in the Sixth Book are usually assumed, or so-called algebraic demonstrations are supplied. The employment side by side of these two methods of dealing with ratio confuses the learner, because, not being equivalent, they do not constitute, when used in this way, a firm basis for the train of reasoning which he is attempting to follow. A better method is sometimes attempted. This is to insist on the mastering of the Fifth Book, expressed in modern form as in the Syllabus of the Association for the Improvement of Geometrical Teaching, before commencing the Sixth Book. But it is far too difficult for all but the best pupils, and even they do not grasp the train of reasoning as a whole, though they readily admit the truth of the propositions singly as consequences of the fundamental definitions, which are (I) The fifth definition, which is the test for the sameness of two ratios. (II) The seventh definition, which is the test for distinguishing the greater of two unequal ratios from the smaller. H. E. b |