61 
Cartesian geometry. In whichever manner the pair of 
fundamental tubes enlarges, whether by elongation or dila- 
tation, we regard the product of the pair of ordinates pro- 
duced as the counterpart of the square of the corresponding 
variable in the algebraical equation. Thus, if y be the 
magnitude, regardless of sign, of the geometrical ordinate, 
then, when y 2 =a?, we may for the algebraical y 2 substitute 
( + r)( + 7) or (-y)(- y), and we obtain the result y=±a; 
and, when y 2 — -a 2 , we obtain (d-y)( - y)= - a 2 , whence 
y = a. In this way we avoid altogether the introduction of 
the algebraical symbol s/~\ as affecting the geometrical 
representation of the equation. Although in the case of 
circular ordinates the value y = a is single, it is obvious that 
two points in the curve— that is, in the envelope of a series 
of circles whose centres are on a straight line— have the 
same ordinate and abseissa, one point, namely, on each side 
of the straight line. 
The writer is prepared to have what he has advanced so 
far regarded as rather curious than instructive, unless it can 
be shown that the system he describes leads to interesting 
results. In what follows it will be shown that such results 
are not wanting. 
We now write down a series of equations and construct 
a figure to represent them ; and we then proceed to comment 
on the equations and curves, and on their points of interest 
in connection with what has gone before. 
x l + f-l 
a 2 5 2 
& ,t - _i 
a 2 6 2 
X 2 Z 2 -t 
a"-b 2 b 2 
... (fi) 
SC 2 Z 2 _ -| 
d 2 - 6 2 6 2 
-■m 
f 2 2 1 
'if i 
0 7 0 """ ■!*•#♦«» 
a - b* a 
