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IV. A Sixth Memoir vfon Quantics. By Aethur Cayley, Esq., F.B.S. 
r)! r. ' 
Received November 18, 1858, — Read Januarj 6, 1859. 
I PROPOSE in the present memoii- to consider the geometrical theory : I have alluded to 
this part of the subject in the articles Nos. 3 and 4 of the Introductory Memoir. The 
present memoir relates to the geometry of one dimension and the geometry of two 
dimensions, corresponding respectively to the analytical theories of binary and ternary 
quantics. But the theory of binary quantics is considered for its own sake ; the geometry 
of one dimension is so immediate an interpretation of the theory of binary quantics, that 
for its own sake there is no necessity to consider it at all ; it is considered with a view 
to the geometry of two dimensions. A chief object of the present memoir is the esta- 
bhshment, upon pui-ely descriptive principles, of the notion of distance. I had intended 
in this introductory paragraph to give an outline of the theory, but I find that in order 
to be intelligible it would be necessary for me to repeat a great part of the contents of 
the memoir in relation to this subject, and I therefore abstain from entering upon it. 
The paragraphs of the memoir are numbered consecutively with those of my former 
Memoirs on Quantics. 
147. It will be seen that in the present memoir, the geometry of one dimension is 
treated of as a geometry of points in a line, and the geometry of two dimensions as a 
geometry of points and lines in a plane. It is, however, to be throughout borne in 
mind, that, in accordance with the remarks No. 4 of the Introductory Memoir, the terms 
employed are not (unless this is done expressly or by the context) restricted to their 
ordinary significations. In using the geometry of one dimension in reference to geometry 
of two dimensions considered as a geometry of points and lines in a plane, it is necessary 
to consider, — 1°, that the word point may mean point and the word line mean line ; 
2°, that the word point may mean line and the word hne mean point. It is, I say, 
necessary to do this, for in such geometry of two dimensions we have systems of points 
in a line and of lines through a point, and each of these systems is in fact a system 
belonging to, and which can by such extended signification of the terms be included in, 
the geometry of one dimension. And precisely because we can by such extension com- 
prise the correlative theorems under a common enunciation, it is not in the geometry of 
one dimension necessary to enunciate them separately ; it may be and very frequently is 
necessary and proper in the geometry of two dimensions, where we are concerned with 
systems of each kind, to enunciate such correlative theorems separately. It may, by 
way of further illustration, be remarked, that in using the geometry of one dimension 
in reference to geometry of three dimensions considered as a geometry of points, lines, 
MDCCCLIX. K 
