The propositions of this geometry are, to a large extent, the same as those of the projective geometry of reals. Two points determine one and only one line ; two lines meet in a point ; five points, no three collinear, determine a conic ; and so on. But a great many propositions assume a simpler and more symmetric form than they do in the real geometry. For example, in the real geometry a line which is not tangent to a conic may meet it either in two points or not at all; in the complex geometry a line which is not tangent always meets the conic in two points. This is be cause the problem of finding the points of intersection of a straight line and a conic reduces to the solution of a quadratic equation. In general any geometric problem depending on the solution of an algebraic equation of higher than the first degree will assume a simpler form in the complex than in the real domain.
Among the complex points there is a sub-class consisting of those for which the ratios and x2/x3 (or x1/x2, if = o) are real. The co-ordinates of these points can always be taken to be real, so that these points can be identified with the ordinary real points. Hence the real projective plane is habitually thought of by mathematicians as immersed in a complex projective plane. Every real straight line or real conic is thought of as containing not only all its real points but also a collection of complex points. More over, free use is made of imaginary straight lines, conics, and so on. For example, if we were to limit attention to real figures we should have to say that from certain points of the plane it is im possible to draw lines tangent to a given conic. From the point of view of complex geometry we say that there are two tangents through any point not on the conic ; if the point is exterior to the conic the two tangents are real lines, but if the point is interior they are conjugate imaginary lines. No attempt is made to visual ize the "imaginary" complex elements. We simply make such inferences about them as follow by the rules of logic from the definitions adopted. The results so obtained are bound to be as
self-consistent as any theorems about numbers.
Geometry of N Dimensions.—Three-dimensional projective geometry can be approached from the point of view of elementary geometry much as we have approached one and two-dimensional projective geometry in the discussion above; or it may be derived by a purely logical process on the basis of its own axioms ; or it may be taken up analytically in the manner explained above for the two-dimensional complex geometry. Points are then defined in terms of sets of homogeneous co-ordinates If these co-ordinates are restricted to real numbers we obtain the real projective geometry of three dimensions. If they are allowed to be any complex numbers we obtain the complex projective geometry of three dimensions.
From the algebraic point of view, there is no reason why the number of homogeneous co-ordinates should be restricted to four, and no such restriction is made by mathematicians. A point in a projective space of n dimensions is defined by homogeneous co-ordinates (xl,x2, . . , xn.0) and projective transformations are defined by equations of the form If the co-ordinates and coefficients are real we arrive at the real projective geometry of n dimensions, if complex, at the complex projective geometry of n As the number n increases the geometry becomes more com plicated because the number of primary figures to consider be comes greater, but it does not become more difficult in principle. It is no longer possible to visualize our results as we do in the one-, two- and three-dimensional cases, but the logical processes by which they are proved remain the same. In order to explain the situation rapidly we have based the conception of an n-dimensional space on the notion of co-ordinates, but this is not necessary. It can also be developed without the use of co-ordinates from a purely descriptive set of axioms.