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61. 
ON GEOMETRICAL RECIPROCITY. 
[From the Cambridge and Dublin Mathematical Journal, vol. in. (1848), pp. 173—179.] 
The fundamental theorem of reciprocity in plane geometry may be thus stated. 
“ The points and lines of a plane P may be considered as corresponding to the 
lines and points of a plane P' in such a manner that to a set of points in a line 
in the first figure, there corresponds a set of lines through a point in the second 
figure, (namely through the point corresponding to the line); and to a set of lines 
through a point in the first figure, there corresponds a set of points in a line in the 
second figure, (namely in the line corresponding to the point).” 
And from this theorem, without its being in any respect necessary further to 
particularize the nature of the correspondence, or to consider in any manner the relative 
position of the two planes, an endless variety of propositions and theories may be 
deduced, as, for instance, the duality of all theorems which relate to the purely 
descriptive properties of figures, the theory of the singular points and tangents of 
curves, &c. 
Suppose, however, that the two planes coincide, so that a point may be considered 
indifferently as belonging to the first or to the second figure: an entirely independent 
series of propositions (which, properly speaking, form no part of the general theory of 
reciprocity) result from this particularization. In general, the line in the second figure 
which corresponds to a point considered as belonging to the first figure, and the line 
in the first figure which corresponds to the same point considered as belonging to the 
second figure, will not be identical; neither will the point in the second figure which 
corresponds with a line considered as belonging to the first figure, and the point in 
the first figure which corresponds to the same line considered as belonging to the 
second figure, be identical, 
c. 
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