OUTLINES OF GEOMETRY. SECTION I. CHAPTER I. THE INTEREST OF SCIENTIFIC INQUIRY. WHO that has ever felt it, cannot recall the sensation which his first view of the sea or of distant mountains gave him? The first glow of delight once past, what was the feeling that succeeded but a longing to reach them, and enjoy the glorious prospect which they must afford? Just such, we may imagine, were the feelings of those who first thought of scientific inquiry. The universe lay stretching before them in all its beauty,—sun, moon, and stars moving in a wonderful order, seed-time and harvest, summer and winter following each other in unbroken succession. Nor in smaller matters was order less prevalent: in the cell of the bee, for example, that affords the greatest possible room in the honeycomb; and in the spider's web, that so cunningly entangles its prey. But such a desire as this could not be satisfied by gazing; and by degrees the impulse became irresistible to penetrate to their utmost the secrets of nature, and arrive at the knowledge which must at first have appeared far beyond their reach. We need not follow them in their labours, but any one who has made his way B across country to some distant hill may have a faint idea of what their toil must have been. He will remember how he looked about for the best starting point, how every kind of impediment rose up in his way, how he fell on the stones,-how he got into a wood and lost himself. Then, as difficulties thickened, how the distant objects lost all their interest; until at last, looking round thoroughly weary and dispirited, he may have found himself scarcely a step in advance of his first position. But yet he is advanced a step; and this thought imparts new vigour, and of itself seems to overcome the obstacles, until another and another height is gained, and he stands at last upon the very summit of the range he saw at first. Every difficulty and repulse is forgotten in that moment, as he beholds the former scene no longer in confused and tangled beauty, but each object in the place exactly fitted for it, and performing the part for which it is specially designed. Such a reward was that of Copernicus, when breaking through the old conception of the earth as the central point of the system in which we are placed, he took his stand in thought upon the sun, and beheld planet after planet sweeping in its appointed orbit round him, each having its size and its path proportioned to its distance from the central sun. How Here then lies the interest which the pursuits of science, and more especially of Mathematics, afforded to former generations. How is it that to us they have become so hard, lifeless, and insipid? The answer is obvious, that the subject has grown stale. eager are the inquiries about a land never before traversed-how dull the history of some well-trodden every-day route. With what pleasure do we read "Captain Speke,"-with what reluctance approach a "Bradshaw." Let us plunge headlong in medias res, is the feeling of every modern traveller, for the beginning is insupportable. And so it has come to pass, that the student is set down at once amidst Theorems and Axioms, Definitions, Points, Problems, Postu lates, and Propositions, until his brain is wearied out with all the ideas which seem so new and strange, though they are indeed so familiar to him. In reading, therefore, the few pages of this little treatise, he may consider himself to be listening to the tale of one who has visited some of the smaller hills, and who wishes to recount each step of the path by which he arrived at them. On the part of the student, therefore, no mathematical knowledge whatever will be requisite, but only those powers of body and mind which the Creator has given to all who are not in some especial manner deficient. ONE of our earliest actions, performed probably quite without our being aware of it, was to "use our eyes." We saw and noticed trees, houses, dogs, horses, men. Let us suppose, then, that some two of these objects, say two trees, were to attract our notice particularly, we might very naturally wish to draw them-that is, to represent them on paper. And the attempt at this would require no knowledge of mathematics, as may be proved by the various figures with which young gentlemen delight to cover their books long before they have comprehended even the definitions of Euclid. We have then our two trees thus (1.) 1. One remark we may make at once-namely, that in our drawing we should place them not close together, but at some little distance apart; and this because the real trees were so situated. Suppose now, that in addition to the two objects we have already described, we draw something from one to the other, then evidently we have three things on the paper, of two of which we know the meaning, while the third is as yet unexplained (2.) 2. What is this then that is drawn from one object to the other? "A line," is the natural answer, which is in truth no answer at all; for what we wanted to know was not the name but the nature of what we have drawn, and about this we know no more than we did before. We will however call it a line. Now we must go on to examine what" a line" must be. Then, by Placing ourselves in our former position, suppose we wish to measure how far it is from one tree to the other. fastening a measuring-tape by one end to one of them, and carrying it along to the other, we could easily find out the distance; and if we were to draw the trees and the measuring-string on the paper we should have this result (3):— 3. That is one which corresponds exactly with what we had above in figure 2. Here therefore is an explanation of "a line," namely, that it represents the distance between two objects. For the present let this be sufficient about it, and let us examine the method by which we arrived at the statement we have just made. CHAPTER DEFINITION. 66 III. FIRST, there were upon paper two objects and something between them, of which we had to find out the meaning. This was a "Problem," which is the finding out of something new from facts which we understood before. Then we went on to give this unknown idea a name, i.e., a line;" and lastly, having examined what it would represent, we attached that meaning to the name we had given it. In other words, we "defined" it. This then is the course we must pursue with the various new ideas we shall meet with as we proceed. First, to give them a name; secondly, examine what they represent; thirdly, attach that meaning to the name given. It is for this reason that Euclid begins with “Definitions,” because all the ideas about which he speaks are new to the student. We have arrived, therefore, at the following mathematical ideas:1. "A Problem," that is something to be done. 66 2. " Definition," or the statement of the meaning of a word declaring what it represents. 3. "A Line," which is the representation of distance. THE NAMES OF OBJECTS IN GEOMETRY. It will be well now before tracing out any further ideas to obtain if possible some method by which the frequent repetition of the word "objects" may be avoided. In doing this, it must be borne in mind that the two objects that we first chose, i.e. two trees, were merely two of innumerable instances that might have been chosen-two men, two carts, a signpost and a milestone, a house and a wall-all would have answered equally well; in fact, all that was wanted was something and something else. A simple plan |