800 PROJECTION same involution on the line at infinity. The latter is therefore called the circular involution on the line at infinity; and the involution which a circle determines at its centre is called the ciicular involution at that point. All circles determine thus on the line at infinity the same involution ; in other words, they have the same two invisible points in common with the line at inanity. THEOREM. All circles may be considered as passing through the same two points at infinity. These points a< - e called the circular points at infinity, and by Prof. Ciiyley the absolute in the plane. They- are the foci of the circular involution in the line at infinity. Conversely Every conic which passes through the circular points is a circle ; because the involution at its centre is circular, hence conjugate diameters are at right angles, and this property only circles possess. We now see why we can draw always one and only one circle through any three points : these three points together with the circular points at infinity are five points through which one conic only can be drawn. Any two circles are similar and similarly situated because they have the same points at infinity ( 21). Any two concentric circles may be considered as having double contact at infinity, because the lines joining the common centre to the circular points at infinity are tangents to both circles at the circular points, as the line at infinity is the polar of the centre. Any two lines at right angles to one another are liarmonic conjugates with regard to the rays joining their intersection to the circular points, because these rays are the focal rays of the circular involution at the intersection of the given lines. To bisect an angle with the vertex A means (G. 23) to find two rays through A which are harmonic conjugates with regard to the limits of the angle and perpendicular to each other. These rays are therefore, harmonic with regard to the limits of the given angle and with regard to the rays through the circular points. Thus perpendicularity and bisection of an angle have been stated in a projective form. It must not be forgotten that the circular points do not exist at all ; but to introduce them gives us a short way of making a state ment which would otherwise be long and cumbrous. We can now generalize any theorem relating to metrical pro perties. For instance, the simple fact that the chord of a circle is touched by a concentric circle at its mid point proves the theorem: If two conies have double contact, then the points where any tangent to one of them cuts the other are harmonic with regard, to the point of contact and the point where the tangent cuts the chord of contact. DESCRIPTIVE GEOMETRY. For many, especially technical, purposes it is of the utmost importance to represent solids and other figures in three dimensions by a drawing in one plane. A variety of methods have been introduced for this purpose. The most important is that which towards the end of the last century was invented by Monge under the name of "descriptive geometry." We give the elements of his method. It is based on parallel projections to a plane by rays perpendicular to the plane. Such a pro jection is called orthographic ( 18). If the plane is horizontal the projection is called the plan of the figure, and if the plane is vertical the elevation. In Monge s method a figure is represented by its plan and elevation. It is therefore often called drawing in plan and elevation, and sometimes simply orthographic projection. 28. We suppose then that we have two planes, one horizontal, the other vertical, and these we call the planes of plan and of eleva tion respectively, or the horizontal and the vertical plane, and denote them by the letters ir l and ir 2 . Their line of intersection is called the axis, and shall be denoted by the letter x. If the surface of the drawing paper is taken as the plane of the plan, then the vertical plane will be the plane perpendicular to it through the axis x. To bring this also into the plane of the drawing paper we turn it about the axis till it coincides with the horizontal plane. This process of turning one plane down till it coincides with another is called rabatting one to the other. Of course there is no necessity to have one of the two planes hori zontal, but even when this is not the case it is convenient to retain the above names. The whole arrangement will be better understood by a glance at Cg. 10. A point A in space is there projected by the perpendicular AAj and AA 2 to the planes ir l and ir 2 , so that Aj and A,, are the horizontal and vertical projections of A. If we remember that a line is perpendicular to a plane that is perpendicular to every line in the plane if only it is perpen dicular to any two intersecting lines in the plane, 1 we sec that the axis which is perpendicular both to AA! and to AA 2 is also per pendicular to A]A and to A 2 A because these four lines are all in the same plane. Hence, if the plane ir., be turned about the A, R -f-- * Fig. 10. Fig 11. axis till it coincides with the plane TTJ, then A._,A n will be the con tinuation oi AjA . This position of the planes is represented in fig. 11, in which the line A^ is perpendicular to the axis x. Conversely any two points A 1; A 2 in a line perpendicular to the axis will be the projections of some point in space when the plane 7r 2 is turned about the axis till it is perpendicular to the plane TTJ, because in this position the two perpendiculars to the planes 7r x and in, through the points A x and A a will be in a plane and there fore meet at some point A. REPRESENTATION OF POINTS. We have thus the following method of representing in a single plane the position of points in space : we take in the plane a line x as the axis, and then any pair of poiiits A 1; A 2 in the plane on a line perpendicular to the axis represent a point A in space. If the line AjA.^ cuts the axis at A , and if at A t a perpendicular be erected to the plane, then the point A will be in it at a height A a A = A A 2 above the plane. This gives the position of the point A relative to the plane TTJ. In the same way, if in a perpendicular to ir. 2 through A 2 a point A be taken such that A 2 A = A Aj, then this will give the point A relative to the plane TT,. 29. The two planes TTJ, 7r 2 in their original position divide space into four parts. These are called the four quadrants. We suppose that the plane -rr. 2 is turned as indicated in fig. 10, so that the point P comes to Q and R to S, then the quadrant in which the point A lies is called the first, and we say that in the first quadrant a point lies above the horizontal and in front of the vertical plane. Now we go round the axis in the sense in which the plane TT, is turned and come in succession to the second, third, and fourth quadrant. In the second a point lies above the plane of the plan and behind the plane, of elevation, and so on. In fig. 12, which represents a side view of the planes in fig. 10 the quadrants are marked, and in each a point with its projection is taken. Fig. 11 shows how these are repre sented when the plane ir 2 is turned down. We see that A point lies in the first quadrant if the plan lies below, the elevation Q~~ above the axis; in the second if plan and elevation both lie above; in the third if the plan lies above, the eleva tion below ; in the fourth if plan and elevation both lie below the axis. If a point lies in the horizontal plane, its elevation lies in the axis B. 111 ET Fig. 12. and the plan coincides with the point itself. If a point lies in the vertical plane, its plan lies in the axis and the elevation coincides with the point itself. If a point lies in the axis, both its plan and elevation lie in the axis and coincide with it. Of each of these propositions, which will easily be seen to be true, the converse holds also. 30. REPRESENTATION OF A PLANE. As we are thus enabled to represent points in a plane, we can represent any finite figure by representing its separate points. It is, however, not possible to represent a plane in this way, for the projections of its points com pletely cover the planes ir l and IT.,, and no plane would appear different from any other. But any plane a cuts each of the planes TT,, IT., in a line. These are called the traces of the plane. They cut each other in the axis at the point where the latter cuts the plane a. A plane is determined by its two traces, which are two lines that meet on the axis, and, conversely, any two lines ichich meet on the axis determine a plane. 1 It is very convenient here to make use of the modern extension of the mean ing of an angle according to which we take as tho angle between two non-inter secting lines the angle between two intersecting lines parallel respectively to the given ones. If this angle is a right angle, the lines are called perpendiculars. Euclid s definition (XI. def. 3), and theorem (XI. 4) may then be stated as in the
text. Compare also aitiele GEOMETRY (EUCLIDIAN), 75, vol. x. p. 38C.