OF EUCLID'S ELEMENTS FOR THE USE OF SCHOOLS PARTS I. AND II. CONTAINING BOOKS I.—VI. BY H. S. HALL, M.A. FORMERLY SCHOLAR OF CHRIST'S COLLEGE, CAMBRIDGE; AND 1 F. H. STEVENS, M.A. FORMERLY SCHOLAR OF QUEEN'S COLLEGE, OXFORD; MASTERS OF THE MILITARY AND ENGINEERING SIDE, CLIFTON COLLEGE. PREFACE. THIS volume contains the first Six Books of Euclid's Elements, together with Appendices giving the most important elementary developments of Euclidean Geometry. The text has been carefully revised, and special attention given to those points which experience has shewn to present difficulties to beginners. In the course of this revision the Enunciations have been altered as little as possible; and, except in Book V., very few departures have been made from Euclid's proofs: in each case changes have been adopted only where the old text has been generally found a cause of difficulty; and such changes are for the most part in favour of well-recognised alternatives. For example, the ambiguity has been removed from the Enunciations of Propositions 18 and 19 of Book I.: the fact that Propositions 8 and 26 establish the complete identical equality of the two triangles considered has been strongly urged; and thus the redundant step has been removed from Proposition 34. In Book II. Simson's arrangement of Proposition 13 has been abandoned for a well-known alternative proof. In Book III. Proposition 25 is not given at length, and its place is taken by a simple equivalent. Propositions 35 and 36 have been treated generally, and it has not been thought necessary to do more than call attention in a note to the special cases. Finally, in Book VI. we have adopted an alternative proof of Proposition 7, a theorem which has been too much neglected, owing to the cumbrous form in which it has been usually given. These are the chief deviations from the ordinary text as regards method and arrangement of proof: they are points familiar as difficulties to most teachers, and to name them indicates sufficiently, without further enumeration, the general principles which have guided our revision. A few alternative proofs of difficult propositions are given for the convenience of those teachers who care to use them. With regard to Book V. we have established the principal propositions, both from the algebraical and geometrical definitions of ratio and proportion, and we have endeavoured to bring out clearly the distinction between these two modes of treatment. In compiling the geometrical section of Book V. we have followed the system first advocated by the late Professor De Morgan; and here we derived very material assistance from the exposition of the subject given in the text-book of the Association for the Improvement of Geometrical Teaching. To this source we are indebted for the improved and more precise wording of definitions (as given on pages 286, 288 to 291), as well as for the order and substance of most of the propositions which appear between pages 297 and 306. But as we have not (except in the points above mentioned) adhered verbally to the text of the Association, we are anxious, while expressing in the fullest manner our obligation to their work, to exempt the |