[157 
158] 
561 
¿c 2 , and 
fiay be 
h 3 
- 6 
- 3 
- 12 
- 9 
- 6 
-24 
- 18 
- 6 
-24 
- 6 
-18 
-48 
-12 
- 9 
-24 
-24 
- 6 
-48 
158. 
A SIXTH MEMOIR UPON QUANTICS. 
[From the Philosophical Transactions of the Royal Society of London, vol. cxlix. for 
the year 1859, pp. 61-—90. Received November 18, 1858,—Read January 6, 1859.] 
I propose in the present memoir 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, [139]. 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 qualities. 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 establishment, upon purely 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 
C. II. 71