INTRODUCTION TO THE ELEMENTS OF EUCLID. He lived at Alexandria, EUCLID was born 300 years B.C. and wrote or compiled, in fact formed into one uniform whole, the treatise on geometry, known as The Elements of Euclid. It was written by him in Greek. The work consists of thirteen books. The first six, with twenty-one propositions of the eleventh, and two from the twelfth, form the Euclid commonly read in English schools. Euclid is looked upon as the father of geometrical reasoning, and so identified is his name with what he wrote about, that the word 'Euclid' in English schools is almost synonymous with 'geometry.' The first six books of Euclid form a treatise on plane geometry. That is to say, all the lines you see on opening the pages of any Euclid, to the close of the sixth book, are supposed to be drawn on a plane or flat surface like the face of the table you write upon. The eleventh and twelfth books are about solid geometry, and refer to lines which might be drawn not only on the face of the table before you, but in any direction, either towards the ground or the ceiling or the walls of the room. B On Points, Lines, and Surfaces. At present you have nothing to do with solid geometry, and yet it may help you the better to understand the definitions which Euclid gives of certain words which he is about to use, if we begin by considering a solid. Every object which can be held in the hand has three dimensions, commonly called length, breadth, and thickness; and that which has length, breadth, and thickness, is called a solid. It matters not whether the object has a regular form like a brick, or an irregular form like a chance stone that might be picked up. A D Let us now take in hand and consider a solid. Here is a brick, we will say a wooden brick. Its length measured from A to B is so many inches, that is one dimension. Its breadth measured from A to C is so many inches, that is a second dimension. Its thick B ness measured from A to D is so many inches, that is a third dimension. Suppose now that its thickness be taken gradually away (planed away for instance) then the brick will become thinner and thinner; and when the thickness has been all taken away, there will remain only two dimensions, length and breadth (that is, in fact, the face of the brick). This is Euclid's superficies or surface. After this, if the breadth of this surface be gradually taken away, the surface will become narrower and narrower, and when the breadth is all gone, one dimension only remains, that of length (the edge, in fact, of the brick). This is what Euclid calls a line. It has no thickness, no breadth, only length. Lastly, suppose this line to shrivel up from one end; in other words suppose its length to be gradually taken away, the line will become shorter and shorter, till, when all the length is gone, there will remain no dimension at all; only a dot, as it were, standing at the corner of the brick. Which dot has neither thickness, nor breadth, nor length. All the dimensions of the solid are gone; one after another they have been taken away. Now this dot without length, breadth, or thickness, is Euclid's point; it is called by him onpeior. And we are now prepared for his Definition 1.-A point is that which has no parts, or which has no magnitude. Some commentators on Euclid add, but position. Observe, we began with a solid brick, and taking it to pieces, resolving it,' that is, taking away successively its thickness, breadth, and length, we came at last to a Point. Now Euclid takes the opposite course: he sets out with a point, and he seems to have before him the idea of the point moving. And as a pencil point moving along paper and leaving a trace behind, forms or generates a visible line, so his point, having no dimensions, no parts, no magnitude, by moving along, will form or generate a mathematical or Euclidian line, which he thus describes in Definition 2.-A line is length without breadth. He adds, what is suggested by the idea of the motion of a point generating a line, Definition 3.-The extremities of a line are points. He might have added, the intersection of two lines (that is, where they cross one another) is a point. Now the motion of the point may be curved or crooked. The point may, in its motion, swerve first to one side and then to another, still the moving point (having itself no parts, no magnitude) will generate a line, that is, length without breadth. But if, in its motion, the point do not swerve one way or another, Euclid calls such a line a straight line, for he gives as 1 Resolve, ava-λów, hence analysis. Definition 4.-A straight line is that which lies evenly between its extreme points. This much Euclid says of lines, and then he proceeds to the next magnitude of which he speaks; namely, that which has length and breadth. Here he seems to have before him the idea of the line, which has length only and no breadth, being drawn along in a direction different to its length (as the end of a rake is drawn by its handle), and so generating a surface which he thus defines : Definition 5.-A superficies (or surface) has only length and breadth. He adds, what is suggested by this idea of a surface being generated by the motion of a line in a direction different to its length, Definition 6.-The extremities of a superficies (or surface) are lines. In the seventh definition there is given a neat and satisfactory test of a flat surface. It is one naturally used by the polishers of marble surfaces. They take what they call a straight edge, and lay it in different directions on the surface which they are polishing. If the straight edge, wherever it is placed, exactly lies along the polished surface, so that no daylight is visible between the straight edge and the polished surface, then they know that the surface is truly flat. The actual words of the definition or test of a flat surface are these: Definition 7.-A plane superficies (in other words, a flat surface) is that in which any two points being taken, the straight line between them lies wholly in that superficies (or surface). Of course if a flat surface were pushed in any direction except that of its length or breadth, it would generate a solid; in other words, the space through which the surface was pushed, if immediately crystallised, would be a solid having the three dimensions of length, breadth, and thick ness. 1 Thus beginning with a point, by the method of putting together, that is, by adding successively length, breadth, and thickness, we get to a solid. Euclid, however, stops at surface, because the present portion of his treatise is on plane geometry. We may now, then, suppose that we have a flat surface, defined as above, before us, on which to place points and to draw straight, or other lines. We must here observe that the points which we place on the flat surface must have some magnitude, and the lines which we draw on it must have some thickness, else we should not see them. But it is not difficult to conceive the point to become finer and finer till at last it has lost all its magnitude. In like manner, we may conceive the line to become thinner and thinner, till it has lost all its thickness, and has only the dimension of length remaining. Thus the physical points and lines, for so they are called when they have magnitude and thickness-become mathematical points and lines according to Euclid's Definitions. On Angles. it a little way. The Take up your compass, amount of opening between the legs of the compass is an angle. As you open the compass more, the angle becomes greater, until the legs are drawn so far asunder that they are in a straight line; then there ceases to be (according to Euclid's definition) an angle between them. His definition of an angle being as follows: Definition 9.2—A plane rectilineal angle is the inclination of two straight lines which meet together, but are not in the same straight line. The legs of the compass represent the two straight 1 Putting together, ovv-Tienμi, hence synthesis. 2 Definition 8 is omitted; it refers to angles contained by curved lines, and is not required at present. |