1 16 Mr, Woodhouse on the Independence of the 
multiple arc may be found in terms of the chord of the simple 
nz 
arc ; for* chord n% == { v/^i } e 2 
XXII. In the above demonstrations* no formulas are borrowed 
from geometry ; and the general law of the coefficients is clearly 
expressed ; it is, I think, most conveniently expressed by means 
of the symbol or note of derivation d. The operation which 
this symbol indicates is as' certain as any other operation, 
whether arithmetical or algebraical. 
XXIII. The demonstrations and method of deduction given 
in this paper shew, I think, with sufficient evidence, the intro- 
duction of geometrical expressions and formulas into analytical 
investigation to be perfectly unnecessary. It has appeared like- 
wise, that such introduction embarrasses investigation, and 
causes ambiguity, by concealing the true derivation of expres- 
sions ; and, although I do not wish to give importance to my 
own observations, by supposing a greater confusion of notion 
to exist than really does, yet, I think, In what has been written 
and said, there may be detected a lurking opinion, that the 
value of certain expressions essentially demand the existence of 
geometrical curves and figures, and the investigation of their 
properties. 
XXIV. In the Appendix to the Arithmetica Universalis , 
p. 200. 219. &c. Newton, with great clearness and force of 
argument, has shewn the distinction to be made between the 
order of classing curves, analytically considered, that is, defined 
but the demonstration of the latter theorem is not, it appears to me, to be reckoned in 
the number of strict demonstrations. The only objection against the demonstration of 
the very learned and ingenious author of the Calcul des Derivations, is, that it is rather 
indirect, and blended with geometrical expressions and formulas. 
