PREFACE. To preserve Euclid's order, to supply omissions, to remove defects, to give brief notes of explanation and simpler methods of proof in cases of acknowledged difficulty-such are the main objects of this Edition of the First and Second Books of the Elements. The work is based on the Greek text, as it is given in the Editions of August and Peyrard. To the suggestions of the late Professor De Morgan, published in the Companion to the British Almanack for 1849, I have paid constant deference. A limited use of symbolic representation, wherein the symbols stand for words and not for operations, is generally regarded as desirable, and I have been assured, by the highest authorities on this point, that the symbols employed in this book are admissible in the Examinations at Oxford and Cambridge *. I have generally followed Euclid's method of proof, but not to the exclusion of other methods recommended by their simplicity, such as the demonstrations by which I propose to replace (at least * I regard this point as completely settled in Cambridge by the following notices prefixed to the papers on Euclid set in the SenateHouse Examinations : I. In the Previous Examination : In answers to these questions any intelligible symbols and abbreviations may be used. II. In the Mathematical Tripos: In answers to the questions on Euclid the symbol must not be used. The only abbreviation admitted for the square on AB is "sq. on AB," and for the rectangle contained by AB and CD, "rect. AB, CD." for a first reading) the difficult Theorems 5 and 7 in the First Book. I have also attempted to render many of the proofs, as, for instance, Propositions 2, 13, and 35 in Book I, and Proposition 13 in Book II, less confusing to the learner. In Propositions 4, 5, 6, 7, and 8 of the Second Book I have ventured to make an important change in Euclid's mode of exposition, by omitting the diagonals from the diagrams and the gnomons from the text. In the Third Book, which I am now preparing, I intend to deviate with even greater boldness from the precise line of Euclid's method. For it is in treating of the properties of the circle that the importance of certain matters, to which reference is made in the Notes of the present volume, is fully brought out. I allude especially to the application of Superposition as a test of equality, to the conception of an Angle as a magnitude capable of unlimited increase, and to the development of the methods connected with Loci and Symmetry. The Exercises have been selected with considerable care, chiefly from the Senate-House Examination Papers. They are intended to be progressive and easy, so that a learner may from the first be induced to work out something for himself. I desire to express my thanks to the friends who have improved this work by their suggestions, and to beg for further help of the same kind. J. HAMBLIN SMITH. CAMBRIDGE, 1871. ELEMENTS OF GEOMETRY. INTRODUCTORY REMARKS. WHEN a block of stone is hewn from the rock, we call it a Solid Body. The stone-cutter shapes it, and brings it into that which we call regularity of form; and then it becomes a Solid Figure. Now suppose the figure to be such that the block has six flat sides, each the exact counterpart of the others; so that, to one who stands facing a corner of the block, the three sides which are visible present the appearance represented in this diagram. Each side of the figure is called a Surface; and when smoothed and polished, it is called a Plane Surface. The sharp and well-defined edges, in which each pair of sides meets, are called Lines. The place, at which any three of the edges meet, is called a Point. A Magnitude is any thing which is made up of parts in any way like itself. Thus, a line is a magnitude; because we may regard it as made up of parts which are themselves lines. The properties Length, Breadth (or Width), and Thickness (or Depth or Height) of a body are called its Dimensions. We make the following distinction between Solids, Surfaces, Lines, and Points: A Solid has three dimensions, Length, Breadth, Thickness. A Surface has two dimensions, Length, Breadth. A Line has one dimension, Length. A Point has no dimensions. S. E. 1 EUCLID. BOOK I. DEFINITIONS. I. A POINT is that which has no parts. This is equivalent to saying that a Point has no magnitude, since we define it as that which cannot be divided into smaller parts. II. A LINE is length without breadth. We cannot conceive a visible line without breadth; but we can reason about lines as if they had no breadth, and this is what Euclid requires us to do. III. The EXTREMITIES of finite LINES are points. A Point marks position, as for instance, the place where a line begins or ends, or meets or crosses another line. IV. A STRAIGHT LINE is one which lies in the same direction with regard to its points. V. A SURFACE is that which has length and breadth only. VI. The EXTREMITIES of a SURFACE are lines. VII. A PLANE SURFACE is one in which, if any two points be taken, the straight line between them lies wholly in that surface. Thus the ends of an uncut cedar-pencil are plane surfaces; but the rest of the surface of the pencil is not a plane surface, since two points may be taken in it such that the straight line joining them will not lie on the surface of the pencil. In our introductory remarks we gave examples of a Surface, a Line, and a Point, as we know them through the evidence of the senses. |