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magnitude. Adopting this latter unit as the analogue of 
the algebraical unit, we shall thus have two distinct but 
connected systems of variation of the magnitude— that of 
elongation yielding tubular ordinates analagous to and co- 
extensive with the ordinates of the Cartesian geometry; 
and that of dilatation yielding the circular ordinates, the 
nature of which we now proceed to consider. 
Just as in our modified Cartesian system, the ordinate is 
erected with the axis of the tube at the extremity of the 
corresponding coordinate, and with a length determined by 
the equation ; so here the circular ordinate must be described 
(on the plane perpendicular to the tubular ordinate) with 
its centre at the extremity of the corresponding coordinate, 
and its radius the function of the latter coordinate given 
by the equation. Also as the curve, in the Cartesian 
system, separates the region of greater ordinates than its 
own at any point from the region of lesser ordinates, so here 
the curve must be such as will do the same ; in other words, 
it must be the envelope of the series of circular ordinates. 
The reason will now appear why we substituted the pro- 
duct of a pair of coincident tubular ordinates for the square 
of a single ordinate : one of the pair we suppose to be pro- 
duced by the elongation of a fundamental tube, of infini- 
tesimal height and radius, on the positive side of the 
perpendicular coordinate plane, and the other by elongation 
of a like tube on the negative side of that plane. The 
equal dilation of these tubes will produce, not coincident, 
but adjacent circular ordinates, and these of opposite sign, 
and of which the product will therefore be a negative square. 
The curve will be described by the line of intersection of 
contiguous circular ordinates, that line being of infinitesimal 
length 2R, half on the positive side and half on the negative 
side of the plane of the circles. 
Once again, we introduce an arbitrary convention that is 
only additional to, not in contravention of, those of the 
