108 Mr. Woodhouse on the Independence of the 
That analytical science was advanced by the discovery of this 
theorem, is indeed true; but the circle and its lines were no farther 
useful or necessary, than as they afforded a mode of expressing, 
in geometrical language, an analytical truth. What is analyti- 
cally expressed, may be analytically combined and resolved; 
and, if Cotes, by the properties of figures, has expressed his 
discovery, it is because the mathematicians of the time in which 
he lived, were more skilful and dexterous with the geometrical 
method than with the analytical. 
In order to demonstrate Cotes's property of the circle, consi- 
dered, as such, one of two different methods must be pursued. 
Either let the demonstration be strictly geometrical, according 
to the method of the ancients, or as completely analytical as pos- 
sible ; that is, let the demonstration be effected by the analytical 
method, from as few fundamental principles as possible. I know 
not on what grounds of perspicuity and rigour, the propriety of 
a demonstration half geometrical, half algebraical, can be estab- 
lished; for, besides the want of symmetry in such a demon- 
stration, in strictness of reasoning, a separate discussion is 
necessary, to shew the propriety and justness of the application 
of analysis to certain properties of extension demonstrated 
geometrically. 
It is beside my present purpose, to inquire whether Cotes's 
theorem can be demonstrated strictly after the method of the 
ancients: hitherto it has not been so demonstrated. To demon- 
strate it analytically, in the most simple and direct manner, we 
must proceed from as few fundamental principles as possible ; * 
and give to the quantities concerned, their true and natural 
* For the analytical demonstration, all that is necessary to be known, is what is 
proved in the 47th of the Elements. 
