very fact, which thus proves by its mere existence that the distance from the sun to the earth, when compared with the surface of the burning-glass, may be considered infinite. END OF PART I. CONDITIONS SUFFICIENT FOR THE ARREST OF A MOVING POINT. HITHERTO We have examined the laws which regulate the motion of any point in general, and of one moving in a straight line or circle in particular. But to complete our subject there still remain for us to consider the circumstances under which the point ceases to move, i.e. is arrested or fixed in any particular position. Thus we have seen already what conditions were required to bring a point back to its original line of motion after once leaving it: viz. that the sum of its inclinations should be altogether equal to the angle in a straight line. We have seen also in a simple case the conditions necessary for the arrest of a point in a certain position— namely, when a straight line was to be drawn from one given point to another given point. For this there were required: 1st. A starting point. 2nd. A point through which the straight line should pass. But these two latter points coincided in this particular case. Such a coincidence however is of course by no means necessary, and instances where each of these "determining" points is distinct have already occurred in the examples given of circular motion. Hence for the fixing of the position of any point P, three points are required:The first, To fix the place of starting. Here then it may be at once observed are three requirements, to one of which at least that namely of direction-an equivalent may be found by means of a method already discussed, that is, by angles instead of points. Thus in considering the conditions of arrest, we have in effect three points, the distances and directions of which mutually affect each other and have varying relations between themselves. To examine these, let us take any three points A,B,C (1,) and join them by straight lines. have a triangle ABC (2.) 1. A Then we B CB 2. 3. The problem now appears under the following form. Every triangle has six parts, three sides and three angles, and may assume an infinite variety of shapes, according as the number of degrees in each of its angles varies, or of sizes according to the variation of the lengths of its sides. These six parts however cannot all vary independently of each other. If certain parts are fixed, then certain others are fixed also; and what the parts are, which, when fixed, determine the others also, is our present question. From what has just preceded, it is evident that two parts alone are not sufficient. Such a condition would only determine the direction of each of the points, and not the distance from the starting point. Three therefore is the first number to be tried, and accordingly Euclid tries no fewer. Nor is it sufficient that any three parts should be fixed: for three angles would determine only the shape of the triangle, and not the size (3.) The reason of this is evident. For we saw above that the angles of every triangle must be equal to 180°. Given, therefore, the number of degrees in any two angles of a triangle, the number in the third is easily reckoned. This third condition therefore is valueless, as it might be obtained from the other two: and thus only two conditions are really given. Triangles which are thus formed, i.e. which have their angles equal, respectively, to those of another, are called "similar," and have certain important properties, some of which Euclid examines in his sixth book. But every other set of three parts is sufficient, and Euclid accordingly examines each of them, proving them to be so, and adding at the same time one other case, in which the requisite conditions are fulfilled for the determination of the size of the triangle, though not of the shape. CHAPTER XXIV. THE COMPARISON OF TRIANGLES. 66 WHAT, then, is the method which Euclid adopts of answering the question just raised? It is as follows:-A triangle will evidently be definite in all its parts if each part is equal to the corresponding part of another triangle which is definite. He therefore conceives two triangles, one with all its parts, the other with only certain of them definite or given;" and shows that in certain cases the remaining parts of the latter must be equal respectively to those of the former, and therefore themselves definite and calculable." The comparison of triangles, then, is the subject with which we are at present concerned, but before enumerating the results which Euclid obtains from the process, it will be well to say a few words-1, as to his general style and kind of arguments; and 2, as to the instru ment by which he effects the comparison, namely, Equality. a It would perhaps be more correct to say that the results obtained by Euclid may be applied in this manner, than to say that he so applies them. E CHAPTER XXV. ON EXISTENCE AND RELATION. A FABLE runs that an aged ass, seeing her little grandchild running towards her at a frantic pace, cried out, "Whither so fast, my child? What do you tremble at?" "O, mother," replied the poor little thing breathlessly, "I asked a great animal what a very tempting looking plant was, and he said such dreadful words that I ran away as hard as I could go." "And what were the dreadful words he used?" 66 Eryngion multifoliense, or some such fearful thing,” replied the little donkey, trembling. "You are indeed an ass, my child," replied the fond mother; "he did but answer your question, and had you stayed to taste you would have found those dreadful words meant a most delicious thistle." Two such thistles are now presented to the student. They are "Existence" and "Relation." They are indeed prickly in appearance but pleasant in taste, and not only do they prove strengthening when digested, but there is scarcely an intellectual dish worth tasting of which they do not form a principal ingredient. These two ideas are so simple and general that it is needless to define them any further. Suffice it to say, that everything which is conceived must be conceived as existing, or having existed; and that everything which exists must be conceived to have some relation to every other existing thing. Now of "Existences" we have had three principal kinds—Magnitudes, Angles, and Figures, and of these we have had various subordinate classes. Of "Relations" we have had those of Loci, Limits, Determination of Position, Equality, and Simultaneous Variation (i.e. that which takes place when two quantities vary together, either directly, as in the case of an angle and the arc it subtends; or inversely, as in the case of the distance of a receding point and the angle at it.) |