GEOMETRY FOR SCHOOLS. Comprising Books I. and II. of Euclid, WITH SOME ADDITIONS AND NUMEROUS EXERCISES. BY A. J. G. BARCLAY, M.A., MATHEMATICAL Master in GEORGE WATSON'S COLLEGE, EDINBURGH. EDINBURGH: OLIVER AND BOYD, TWEEDDALE COURT. 1883. Price One Shilling. 183. 153. 29 NOV83 OXFORD PREFACE. WHILE engaged in teaching Geometry, I have often been asked by my pupils to write on the black-board, to be copied by them, the proposition that we were working out together for next day's lesson. This led me to think of printing (in the first instance for their use) the first two Books of Euclid in the style that they found, even in their imperfect copy, easier to learn from than the ordinary text. The symbols and contractions so largely used are readily understood and easily acquired; and, as they enable the steps of a proposition to be brought closer, the connexion of the related parts becomes the more obvious. The diagrams and the arrangement of the matter are also designed to render such help to the beginner as mechanical means can give. A section on Analysis and Loci has been added to the First Book. Numerous easy Exercises have been introduced under the Propositions, and some more difficult ones in the continuation at the end. Those numbered in bolder type are important propositions, which form together a fairly complete sequel to this part of Euclid. The numbering of the Propositions has been retained as in Euclid; and, for the sake of those preparing for examinations in which only Euclid is accepted, the Euclidean Propositions of Book I., omitted in the text, are supplied in an Appendix. I beg to tender my thanks for much valuable help to Mr A. Y. Fraser, M.A., George Watson's College, Edinburgh, and to Mr Wm. Raitt, M.A., B.Sc., College of Science and Arts, Glasgow. GEORGE WATSON'S COLLEGE, A. J. G. BARCLAY. Q.E.F. is for "Quod erat faciendum," and indicates that a problem has been effected. Q.E.D. is for "Quod erat demonstrandum," and indicates that a theorem has been proved. In the writing out of propositions the use of the following may also be allowed to the pupils, at the discretion of the teacher. ~for equals in every respect, extr. for exterior. is ΟΥ, congruent with. INTRODUCTION. The Science of Geometry is concerned with lines, surfaces, and solids. A solid is bounded by surfaces, and these again by edges or lines. A solid has three dimensions, Length, Breadth, and Thickness; a surface has two, Length and Breadth; and a line has only one, Length. We may look upon a line as produced by the motion of a point through space; a surface by the motion of a line; and a solid as traced out by a moving surface. The Elements of Plane Geometry include the properties of the Straight Line and the Circle, and of combinations of these on a plane or flat surface. The truths of Geometry are deduced by reasoning, beginning from certain axioms, or fundamental truths, which are self-evident—that is, which can be admitted as true without requiring demonstration; and each truth, when discovered and demonstrated, may be used, when required in future demonstrations, like an axiom. Hence the Science of Geometry is deductive: hence, also, it is imperative on the student of Geometry thoroughly to master each step as he proceeds; and the more thoroughly he does so, the more certain and rapid will be his future progress in Mathematical Science. |