59 
in the Cartesian system, as the analogue of the unit of the 
algebraical equation, is a unit length of a mathematical 
straight line, with a convention as to signs which we need 
not describe. For a reason that will presently appear, we 
here make the otherwise simple observation, that if we sub- 
stitute — in the Cartesian method — the idea of pairs of 
coincident ordinates and take the product of their values 
instead of the square of a single ordinate, we only add a 
convention that leaves the analogy between the algebra and 
the geometry in question untouched. We further remark 
that this analogy will still be perfect, if we conceive of the 
two coincident ordinates as tubes instead of mathematical 
straight lines, providing the tubes be of infinitesimal trans- 
verse section, which may be of circular form and (say) of 
radius ft, and providing further that the numerical values 
of the tubes are the quantities of curved surface which they 
possess, the convention as to signs being unaltered: the 
unit of curved surface must obviously be 27 rR. Instead, 
however, of the curve being described by the motion of the 
central point of the extremity of the ordinate, let it be de- 
scribed by a diameter, through that point, perpendicular to 
the plane of the coordinates : it will still be of infinitesimal 
depth. Further let us take, as the fundamental ideal of the 
tubular ordinate, the one whose height is R. It may be 
varied in value by elongation in a positive or negative direc- 
tion, its diameter remaining constant ; or it may be varied 
by dilatation into larger and larger circles, its height 
remaining constant : in the latter case the ordinate, which 
may be called circular to distinguish it from the tubular 
ordinate, will be of positive sign if it be on the positive side 
of the coordinate plane, which is perpendicular to the tubu- 
lar ordinate, and of negative sign if on the opposite 
side of that plane. Elongation to unit length of the tube, 
and dilatation to unit radius of the circle, alike make the 
quantity of curved surface 27 rR, which is the unit of this 
