CHAPTER XXI. PARALLELISM-SECOND CONDITION OF CONTINUITY OBTAINED. A QUESTION arises from the considerations we have just made. Given any two straight lines, is the point they have in common at a measurable or immeasurable distance? The fact above proved furnishes an easy test. For if there be any two straight lines, AP, BP, having in common a point P, then taking in each of them re- P spectively, a point (C and D) and D 1. A -B joining CD, we have a triangle, CDP, the angles of which must altogether contain 180° (1.) And we have seen that if P is at an infinite dis 2. D B tance the angle at P is zero, and consequently the angles at C and D will together contain 180° (2.) Conversely, if the angles at C and D together contain 180° the point which the two lines have in common is at an infinite distance. Lines which have this relation to each other are called Parallel, and if the opposite sides of any four-sided figures be parallel, the figure is called a Parallelogram. Of such lines instances are abundant. The furrows of a plough, the ruts of a cart, the trees in an avenue, are but a few, and these not the commonest examples of this most common relation. We have now this test as to whether two straight lines are parallel or not. If one straight line falling upon two other straight lines makes the two interior angles on the same side less than two rightangles, these lines will meet if produced. But an easier test is furnished by a little further consideration. For if AB and CD be two parallel straight lines, and EF meets them in H and G, then by virtue of what has been said 3. Hence parallel straight lines may also be defined to be, such as make equal angles with the same given straight line. And a straight line may be considered to be a circle which has its centre at an immeasurably great distance from the circumference; its radii, parallel straight lines; and the angle at the centre between any two radii zero. Thus a circle is the limiting form of a polygon, and the straight line the limiting form of the circle, and therefore of the polygon also. And this is what we should expect when we consider that a Straight line is of infinite length, and yet is made up of infinitesimals, so that it must be composed of an infinite number, not of infinitesimals, for that would only make up a finite line, but of infinities of infinitesimals, and therefore is an infinity of infinities," or an infinity of the second order." 66 It must be remarked further that from the above process we obtain a test whether the second condition necessary for the enclosure of space, be fulfilled or not. That is, since each of the boundary points of the figure must lie in two of its sides, an easy test whether the point be at a finite or an infinite distance has been obtained. And if no point of the figure is at an infinite distance, the space is enclosed. CHAPTER XXII. EXAMPLES OF CONTINUITY. SEVERAL curious consequences have their origin in these apparent paradoxes. First, it follows that any one moving in the same direction in a straight line will after an incalculable time return to the point from which he set out; and this result will appear less extraordinary if we consider that any one starting from England and passing eastward through China would inevitably meet another man travelling on the same line of latitude westward through America; yet to each it would seem as though he were continually going in a straight line further and further from home. So again, a comet coming in a hyperbolic orbit (a) from an infinite distance till it arrives at some certain distance from the sun passes away (b) down another infinite branch of the f curve, and yet reappears again from a diametrically opposite direction. (c) b An illustration also of the latter test, as to whether two straight lines meet at a finite 1 2 distance or not (i.e. whether they make equal angles with another stright line) is found in the distance of the fixed stars. The angle between the axis of the earth's orbit and the line of distance of one of these is measured twice at six months' interval, the earth being at the second time a hundred and ninety millions of miles distant from her position the first time. And in each case the angle is found to be the same. Consequently, supposing the star to be the vertex of a triangle, of which the base line is a hundred and ninety millions of miles, the sides of this triangle will still be parallel. 190.000.000 MILES E In other words, a hundred and sixty millions of miles bear no relation whatever to the distance between the earth and the nearest fixed star. Of course this cannot be properly represented on paper. There is one still more curious, though rather complicated instance, which practically proves that a distance, even when measurable, may be so great as to have, when compared with another, all the properties of infinity in regard to it. There is a certain curve already mentioned, which is called a Parabola, and which may be formed in the following manner :— 1. Suppose there is a fixed point, S Y (called the Focus;) a straight line, OX, directrix (parallel to the axis) is always o equal to a straight line drawn from it to the focus (2.) This curve proved that the angle LPT equals the angle TPS (3.) Consequently, if there be a glass in the shape of a parabola, and if a ray of light coming along LP, (i.e., a line parallel with the axis of the parabola,) strike it at any point P, then the ray strikes the curve at an angle LPT, (chap. xv.) and by a well-known law of optics is refracted through the glass at an angle with the curve equal to the angle LPT, i.e. at an angle TPS. That is to say a ray of light coming in the direction of LP, and striking the glass at P will after refraction travel along the line PS (4.) 4. And as this is the case at any point of the curve; it follows that a number of rays, travelling along different lines, all parallel to the axis of the glass, and striking the glass at different points, will all be refracted to the one point S. But for this to happen the rays must be parallel to the axis. Now the source of light need not of course be in the directrix, but may be at any distance from the glass. Suppose, then, there is a candle at C, a short distance from the glass. Then the rays will emanate from the source in all directions, and if they fall upon the glass (which is parabolic in form) they will strike it in various directions, and consequently be refracted to various 5. S points in the axis, some even outside the curve (5.) But the further the source of light, the less will the divergency of the rays be felt; until, if the distance be infinite, the rays will move in parallel lines. Such a distance is that of the sun when compared with a very small surface of glass; and its rays will therefore, if our principle be true, be all collected 6. into one point, namely, the focus of the parabola (6.) Upon this principle the "burning-glass is constructed, the name of focus indeed, as that of the French "foyer," being derived from this |