GEOM is very careful never to admit anything but finite quan- tities. The treatment of the infinite is in fact another fundamental difference between the two methods. Euclid avoids it. In modern geometry it is systematically i11tr0- dueed, for only thus is generality obtained. ()f the different modern methods of geometry, we sh:1ll treat principally of the methods of projection and corre- spondence which have proved to be the most powerful. ’l'he.<e have become independent of Euclidian Geonietry, especially through the 0'c0melrz'c dcr Lagc of V. Staudt, and the A/¢sr7r'7z)z1()2gs7c71)'e of Grassmann. .l"or the sake of brevity we shall presuppose a know- ledge of Euclid’s I*.'len1cnf.~', although we shall use only :1 few of his propositions. § 1. 'e consider space as filled with points, lines, and planes, an-l these we call the elements out of which our figures are to be flintcd, calling any combination of these elements a “ figure." ’;y :1 line we mean :1 straight line in its entirety, extending both w.1ys to infinity; and by a plane, a plane surface, extending in all directions to infinity. 'e suppose That throu_=_;h any two points in space one and. only one line may he dr:1vn (Eucl. I., Def. 4, Ax. 10, Post. 2); Th-.1t through any three points which a.rc not in a line, one and only one plane may be placed (compare p. 386, 5' 73, above); 'l'hat tl1e intersection of two planes is a line (Eu:-l. XI. 13); That a line which has two points in common with a plane lies in the plane (liucl. I., Def. 7), hence that the intersection of :1 line and a plane is a single point; and That three planes which do not meet in a line have one single point in common. These results may be stated differently in the following form :— . A plane is determined-— A point is determined-— 1. By three points which do 1. By three planes which do not lie in a line; not pass through :1 line ; . lly two intersecting lines; 2. ’»_v two intersecting lines; . lly a line and a point which 3. By a line and a pl:1ne which does not lie in it. does not pass through it. II. A line is determincd— 1. lly two points ; 2. lly two planes. The reader will observe that not only are planes determined by points, but also points by planes‘; that therefore we have a right to consider the planes as elements, like points ; and also that in any one of the above statements we may interchange the words point and plane, and we obtain again a correct st:1tement, provided that these statements themselves are true. As they stand, we ought, in several cases, to add “if they are not parallel,” or some such words, parallel lines and planes being evidently left altogether out of consideration. To correct this we have to reconsider the theory of parallels. _ § ‘.2. Let 11s take in a plane a line 3; (fig. 1), a point S not in this line, and a line q drawn through S. Then this line q will meet the line 1) in a point A. If we turn the line q about S to- wards q’, its point of intersection with 1) will move along 1) towards I’), passing, on con- tinued turning, to a S greater and greater dis- tance, until it is moved out of our reach. If we turn (1 still farther, its continuation will meet. p, but now at the 1, other side of A. The point of intersection has disappeared to the right and reappeared to the left. There is one inter- mediate position where ‘Z q is parallel to 12-that Fin. 1 is where it does not cut °' ' p. In every other position it cuts 12 in some finite point. If, on the othcr hand, we move the point A to an infinite distance in p, then the line q which passes through A will be a line which does not cut 1) at any finite point. 'I‘lu1s we are led to say: Every line through S which joins it to any point at an infinite distance in p is parallel to 1). But by E1tclid’s 12th axiom there is but one line parallel to 1) through S. The difficulty in which we. are thus involved is d11e to the fact that we try to reason about infinit_v as if we, with our finite ea iabilitics, could compre- hend the infinite. To overcome this difficulty, we may say that all points at infinity in a line appear to us as one, and may be replaced by a single “idcal" point, just as all points in :1 fixed star—whieh is not at an infinite, only at a great distauee—c:1n- not be distinguished by us and to beings on the earth count as PROJECTIVE] >—< CS [5 A B (ll ETRY 339 one. Ve may therefore now give the following definitions and ax1om:— De/£11il1'on.-—Lines which meet at infinity are called parallel. A.1'iom.—All points at an infinite distance in a line may be con- sidered as one single point. ‘ Dr_fin1'tio71.—'l‘l1is ideal point is called the point at infinity in the line. _ The axiom is equivalent to Euclid’s Axiom 12, for it follows from either that through any point only one line may be drawn parallel to a given line. This point at infinity in a line is reached whether we move a point in the one or in the opposite direction of a line to infinity. A line thus appears closed by this point, and we speak as if we could move a point along the line from one position A to another I} in two ways, either through the point at infinity or through finite points only. It must never be forgotten that this point at infinity is ideal, that the results based on this assumption are true for that finite region of space which is within our reach, and that beyond this region they may or may not be true, —we do not know. The advantage of this view of parallels will become apparent at every step as we go on. § 3. Having thus arrived at the notion of replacing all points at infinity in :1 line by one ideal point, there is no difficulty in re- placing all points at infinity in a plane by one ideal line. To make this clear, let us suppose that a line 1), which c11ts two fixed lines (6 aml b in the points A and B, moves parallel to itself to a greater and greater distance. It will at last cut both a and b at their points at infinity, so that a line which joins the two points at infinity in two intersecting lines lies altogether at infinity. Every other line in the plane will meet it therefore at infinity, and thus it contains all points at infinity in the plane. All points at itlfillily in (6 plane lie in a line, ichich is called the line at z'n_fi)12'l_I/ in the plant‘. It follows that parallel planes must be considered as planes having :1 common line at infinity, for any other pl:1nc cuts them in parallel lines (Eucl. XI. 16), which have a point at infinity in common. If we next take two intersecting planes, then the point at infinity in their line of intersection lies in both planes, so that their lines at infinity meet. Hence every line at infinity meets every other line at infinity, aml they are therefore all in one plane. All points nt2'n_fi7z1'l_I/ in space may be co)1si(lc7'cd as I_z/1'n_q in one 1'd«'al plane, 1!‘/tic/L is called the plane at 2'7:_/iizitg/. § 4. Ve have now the following definitions :— Parallel lines are lines which meet at infinity ; I’:u-allel planes are planes which meet at infinity; A line is parallel to a ilane if it meets it at infinity. Theorems like this— incs (or planes) which are parallel to a third are parallel to each othcr—-follow at once. This view of parallels leads therefore to no contradiction of Euelid's .EIcmcnl..s‘. As immediate consequences we get the propositions :— I-lvery line meets a plane in one point, or it lies in it; Every plane meets every other plane in :1 line ; Any two lines in the same plane meet. § 5. We have called points, lines, and planes the elements of geometrical figures. We also say that an element of one kind coatrL1'ns one of the other if it lies in it or passes through it. All the elements of one kind which are contained in one or two elements of a different kind form aggregates which have to be enumerated. They are the following :— 1. Of one dimension. 1. The row, or range, of points formed by all points in a line, which is called its base. 2. The flat pencil formed by all the lines through a point in
- 1 plane. Its base is the point in the plane.
3. The axial pencil formed by all planes through a line which is called its base or axis. 11. Of two dimensions. 1. The field of points and lincs—that is, a plane with all its points and all its lines. 2. The pencil of lines and planes— that is, a point in space with all lines and all planes through it. III. Of three dimensions. The space of points—that is, all points in space. The space of planes—that is, all planes in space. IV. Of four dimensions. The space of lines, or all lines in space. § 6. The word dimension in the above nccds explanation. If in a plane we take a row )9 and a pencil with centre Q, then through every point in 1) one line in the pencil will pass, and every ray in Q will cut 1) in one point, so that we are entitled to say a row contains as many points as a flat pencil lines, and, we may add, as an axial pencil planes, because an axial pencil is cut by a plane in a flat pencil.
The number of elements in the row, in the flat pencil, and in the