On Triple Ratios: 4 of 6

          The Lines at the Infinities

          Consider the “one-infinities”; ratios of the form (0;b;c). You can also write them as (a:b;c)* ∞₁, where  ∞₁ equals (0;1;1). This applies:

          (x;y;z)+₁(0;b;c)             =       (0; b; c)      if x is not zero

                                                          (0;0;0)        if x is zero

          So a 1-infinity plus any 1-finite ratio (i.e. with nonzero first term) absorbs it; and any two 1-infinities add to the indefinite ratio.

          (0;y;z)+₂(0;b;c)             =       (0; by; cy+bz)     

                                                =       (0; 1; (c/b)+(z/y))

                                                =       (0; (b/c)[+](y/z); 1)

          (0;y;z)+₃(0;b;c)             =       (0; bz+cy; cz)     

                                                =       (0; (b/c)+(y/z); 1)

                                                =       (0; 1; (c/b)[+](z/y))

          So +₂ and +₃ are addition and reduction on the line at one-infinity. Similarly, +₃ and +₁ are addition and reduction on the line at two-infinity, and +₁ and +₂ are addition and reduction on the line at three-infinity.