586 A SIXTH MEMOIR UPON QU ANTICS. [l58 
It is hardly necessary to remark, that in the present case the notion of the quadrantal 
relation of two points has altogether disappeared, and that the unit of distance 
is arbitrary. 
214. Passing now to geometry of two dimensions, we have here to consider a 
certain conic, which I call the Absolute. Any line whatever determines with the 
Absolute (cuts it in) two points which are the Absolute in regard to such line con 
sidered as a space of one dimension, or locus in quo of a range of points, and in 
like manner any point whatever determines with the Absolute (has for tangents of 
the Absolute through the point) two lines which are the Absolute in regard to such 
point considered as a space of one dimension, or locus in quo of a pencil of lines. 
The foregoing theory for geometry of one dimension establishes the notion of distance 
as regards each of these ranges and pencils considered apart by itself; in order to 
bring the different ranges and pencils into relation with each other, it is necessary to 
assume that the quadrant which is the unit of distance for these several systems 
respectively, is one and the same distance for each system (of course, when, as in the 
analytical theory, we actually represent the quadrant by the ordinary symbol r, the 
above assumption is tacitly made; but substituting the thing signified for the 
definition, and looking at the quadrant merely as the distance between two points, 
or as the case may be, lines, harmonically related to the point-pair, or as the case 
may be, line-pair, constituting the Absolute, the assumption is at once seen to be an 
assumption, and it needs to be made explicitly). But the assumption being made, the 
foregoing theory of distance in geometry of one dimension enables the comparison not 
only of the distances of points upon different lines, or of lines through different points, 
but of the distances of points on a line and of lines through a point. The pole of 
any line in relation to the Absolute may be termed simply the pole, and in like 
manner the polar of any line in relation to the Absolute may be termed simply the 
polar, and we have the theorem that the distance of two points or lines is equal to 
the distance of their polars or poles, or what is the same thing, that the distance 
of two poles and the distance of the two corresponding polars are equal. And we 
may, as a definition, establish the notion of the distance of a point from a line, viz. 
it is the complement of the distance of the polar of the point from the line, or 
what is the same thing, the complement of the distance of the point from the pole 
of the line. The distance of a pole and polar is therefore the complement of zero, 
that is, it is the quadrant. 
215. It has, by means of the preceding assumption as to the quadrant, been 
possible to establish the notion of distance, without the assistance of the circle, but 
this figure must now be considered. A conic inscribed in the Absolute is termed a 
circle; the centre of inscription (or point of intersection of the common tangents) and 
the axis of inscription (or line of junction of the common ineunts) are the centre and 
axis of the circle. All the points of a circle are equidistant from the centre; all 
the tangents are equidistant from the axis, and this distance is the complement of 
the former distance.