Miquel's theorem, a circle belonging to it. These six circles meet in the same point, and so on for ever. Any even number (2n) of straight lines determines a point as the intersection of the same number of circles. It we take one line more, this odd number (2n+1) determines as many sets of 2n lines, and to each of these sets belongs a point; these 2n +1 points lie on a circle. § 8. The Conic Sections. The shadow of a circle cast on a flat surface by a luminous point may have three different shapes. These are three curves of great historic interest, and of the utmost importance in geometry and its applications. The lines we have so far treated, viz. the straight line and circle, are special cases of these curves; and we may naturally at this point investigate a few of the properties of the more general forms. If a circular disc be held in any position so that it is altogether below the flame of a candle, and its shadow be allowed to fall on the table, this shadow will be of an oval form, except in two extreme cases, in one of which it also is a circle, and in the other is a straight line. The former of these cases happens when the disc is held parallel to the table, and the latter when the disc is held edgewise to the candle; or, in other words, is so placed that the plane in which it lies passes through the luminous point. The oval form which, with these two exceptions, the shadow presents is called an ellipse (i). The paths pursued by the planets round the sun are of this form. If the circular disc be now held so that its highest point is just on a level with the flame of the candle, the shadow will as before be oval at the end near the candle; G but instead of closing up into another oval end as we move away from the candle, the two sides of it will continue to open out without any limit, tending however to become more and more parallel. This form of the shadow is called a parabola (ii). It is very nearly the orbit of many comets, and is also nearly represented by the path of a stone thrown up obliquely. If there were no atmosphere to retard the motion of the stone it would exactly describe a parabola. (iii) (ii) FIG. 26. (iv) If we now hold the circular disc higher up still, so that a horizontal plane at the level of the candle flame divides it into two parts, only one of these parts will cast any shadow at all, and that will be a curve such as is shown in the figure, the two sides of which diverge in quite different directions, and do not, as in the case of the parabola, tend to become parallel (iii). But although for physical purposes this curve is the whole of the shadow, yet for geometrical purposes it is not the whole. We may suppose that instead of being a shadow our curve was formed by joining the luminous point by straight lines to points round the edge of the disc, and producing these straight lines until they meet the table. This geometrical mode of construction will equally apply to the part of the circle which is above the candle flame, although that does not cast any shadow. If we join these points of the circle to the candle flame, and prolong the joining lines beyond it, they will meet the table on the other side of the candle, and will trace out a curve there which is exactly similar and equal to the physical shadow (iv). We may call this the anti-shadow or geometrical shadow of the circle. It is found that for geometrical purposes these two branches must be considered as forming only one curve, which is called an hyperbola. There are two straight lines to which the curve gets nearer and nearer the further away it goes from their point of intersection, but which it never actually meets. For this reason they are called asymptotes, from a Greek word meaning 'not falling together.' These lines are parallel to the two straight lines which join the candle flame to the two points of the circle which are level with it. We saw some time ago that a surface was formed by the motion of a line. Now if a right line in its motion always passes through one fixed point, the surface which it traces out is called a cone, and the fixed point is called its vertex. And thus the three curves which we have just described are called conic sections, for they may be made by cutting a cone by a plane. In fact, it is in this way that the shadow of the circle is formed; for if we consider the straight lines which join the candle flame to all parts of the edge of the circle we see that they form a cone whose vertex is the candle flame and whose base is the circle. We must suppose these lines not to end at the flame but to be prolonged through it, and we shall so get what would commonly be called two cones with their points together, but what in geometry is called one conical surface having two sheets. The section of this conical surface by the horizontal plane of the table is the shadow of the circle; the sheet in which the circle lies gives us the ordinary physical shadow, the other sheet (if the plane of section meets it) gives what we have called the geometrical shadow. The consideration of the shadows of curves is a method much used for finding out their properties, for there are certain geometrical properties which are always common to a figure and its shadow. For example, if we draw on a sheet of glass two curves which cut one another, then the shadows of the two curves cast through the sheet of glass on the table will also cut one another. The shadow of a straight line is always a straight line, for all the rays of light from the flame through various points of a straight line lie in a plane, and this plane meets the plane surface of the table in a straight line which is the shadow. Consequently if any curve is cut by a straight line in a certain number of points, the shadow of the curve will be cut by the shadow of the straight line in the same number of points. Since a circle is cut by a straight line in two points or in none at all, it follows that any shadow of a circle must be cut by a straight line in two points or in none at all. When a straight line touches a circle the two points of intersection coalesce into one point. We see then that this must also be the case with any shadow of the circle. Again, from a point outside the circle it is possible to draw two lines which touch the circle; so from a point outside either of the three curves which we have just described, it is possible to draw two lines to touch the curve. From a point inside the circle no tangent can be drawn to it, and accordingly no tangent can be drawn to any conic section from a point inside it. This method of deriving the properties of one curve from those of another of which it is the shadow, is called the method of projection. The particular case of it which is of the greatest use is that in which we suppose the luminous point by which the shadow is cast to be ever so far away. Suppose, for example, that the shadow of a circle held obliquely is cast on the table by a star situated directly overhead, and at an indefinitely great distance. The lines joining the star to all the points of the circle will then be vertical lines, and they will no longer form a cone but a cylinder. One of the chief advantages of this kind of projection is that the shadows of two parallel lines will remain parallel, which is not generally the case in the other kind of projection. The shadow of the circle which we obtain now is always an ellipse; and we are able to find out in this way some very important properties of the curve, the corresponding properties of the circle being for the most part evident at a glance on account of the symmetry of the figure. For instance, let us suppose that the circle whose shadow we are examining is vertical, and let us take a vertical diameter of it, so that the tangents at its ends. are horizontal. It will be clear from the symmetry of the figure that all horizontal lines in it are divided into two equal parts by the vertical diameter, or we may say that the diameter of the circle bisects all chords parallel to the tangents at its extremities. When the shadow of this figure is cast by an infinitely distant star (which |