62 
MR, A. B. EEMPE ON THE THEORY OF MATHEMATICAL FORM. 
Geometry in General. 
350. In most geometrical investigations the units constitute a system of a high 
order of multiplicity; we have points, straight lines, conics, cubics, &c., unified 
collections of two, three, or more of these, operators such as quarternions, &c., &c., &c. 
It will, however, be sufficient for the purpose of illustration to refer briefly to some 
comparatively simple systems. 
351. It is to be understood that points and the line at infinity are not regarded 
as distinguished from other points and lines, and consequently parallelism is looked 
upon merely as intersection on a particular line. 
System of Coplanar Points and Straight Lines. 
352. In a general consideration of coplanar points and straight lines the points 
compose a single system, being undistinguished from each other, and the same 
thing is the case with the straight lines. The connecting pairs of the two 
systems are of two sorts, for a point may either lie on a line or off it; and, as 
there is no other special relation between a point and a line, the system of con¬ 
necting pairs is a double one. i 
353. We may graphically represent the lines by the graphical units O O O O, 
and the points by the graphical units •••••. If a point lies on a line we may 
connect the corresponding graphical units by a link thus O-•, no link being 
drawn in the other case. 
354. The number of points on each line is infinite, and an infinite number of lines 
pass through each point. Every two points lie on one line and one only, so that if a 
and h are any two points, the line P which is linked to both is unique with respect to 
a, h (fig. 65). Pairs of points are accordingly all undistinguished from each other. 
Fig- 65. Fig. 60. 
355. Every two lines pass through one common point, and one only; so that if 
there is a point a linked to two lines P and Q, a is unique with respect to PQ 
(fig. 66). 
356. Every form which component collections of lines and points possess is also 
possessed by component collections of points and lines; the two component single 
systems being of precisely the same form. 
357. We may take as fundamental laws defining the distribution of the links, and 
therefore the form of the double system the two well-known theorems : 
