PHILLIPS AND MOORE. — ALGEBRA OF PROJECTIVE GEOMETRY. 739 



does not properly exist. There is, however, a number associated with a 

 point and line which has some of the properties of distance from point 

 to line. We define an area that has the usual sum properties and in 

 fact becomes identical (after proper choice of unit) with Euclidean area 

 when the singular line is thrown to infinity. 



In the course of the work we use the notations AB and ABC for seg- 

 ment and area respectively. Interpreting these as products we find that 

 they have the properties of Grassmann's products. They are definable 

 in terms of our distance, angle and area, in the same way that outer 

 products are expressible in metrical concepts of Euclidean geometry. 



The method used in this paper is not postulational. In fact, we are 

 more interested in the results than in the method of obtaining them. 

 In some cases our definitions have not been as simple as possible. In 

 accordance with our primary aim we have interpreted quantities neces- 

 sarily existing in the algebra instead of introducing notions that were 

 not required. 



4. It is our purpose to use this scheme in the solution of problems 

 in projective geometry. Two ways of doing this are suggested. In the 

 first place the above scheme of distance and angle gives us a great va- 

 riety of coordinate systems. We may, for example, represent a point 

 by its distance from two fixed points (singular point not on singular 

 line) and a line by the angles it makes with two fixed lines. The equa- 

 tions of point and line are then of first degree and the incidence relation 

 bilinear. The distance between two points is a bilinear function of their 

 coordinates. Similarly for the angle between two lines. The differen- 

 tial of arc is of the form 



xdy — ydx. 



A kind of curvature is easily defined and thus we build up a differen- 

 tial projective geometry of plane curves. 



In the second place we may start with the theory of the triangle. In 

 terms of our distance, angle and area, we can then express descriptive 

 relations such as perspectivity, inscribability in a conic, etc., and by pro- 

 cesses similar to those in Euclidean geometry determine fi-om these their 

 projective consequences. This is especially easy since the expressions 

 determining one part of a triangle in terms of three others are all ra- 

 tional. It is our intention to develop these applications in a later 

 paper. 



§ 1. Addition. 



5. The addition of two points A, B naturally divides itself into two 

 cases depending on the significance given to AA. In metric geometry 

 these two additions have been defined in terms of metric concepts. If 



