Review of the Principia of Newton. 245 
Prop. 8, Prob. 3, is but a particular case of a general 
problem, which our author has solved according to the mo- 
dern analysis, in the scholium of the 13th section of the first 
book ; viz. by a development of the function of the ordinate ; 
and here we have the origin, and indeed the whole sub- 
stance of 'Taylor’s celebrated theorem, which shows the rela- 
tion between the increment of a function and its different 
orders of fluxions. ‘The second term of the development is 
proportional to the first differential ; the third term will be 
paseernone! to the second differential, &c. This is demon- 
strated in Newton’s quadrature of curves, by Stewart, and in 
other comments on his works. Now, as the second fluxion 
of the ordinate is & known: to be proportional to the Geweyet 
gen 
is peo to a is made by Newton the expression of 
the force. If, for Samp te the curve be a parabola, and the 
force be supposed to act in the are of the ordinate, we 
; 
shall lotmayena' ce. 3h, Soe "haa aiaeea) or as 
1 1 
y °, or the force, as y *. Ifthe abscissa be assumed as the 
2 
us and « oc y and 
ordinate to a co-ordinate axis, then 2= 
a(x 2yy) xc 2®Wy, andx 2 y, or the force is constant, as 
we know also from other principles. Any of the conic sections 
will require the same law of force, acting according to the di- 
rection of its ordinate. If it be ashigher curve of the same 
class, the force may easily be found in the same manner ; 
ewhich solution, together with the innumerable fl fluxional 
lems solved in the Principia, as well as in the earlier produc- 
tions of Newton, long before the forms and rales of the flux- 
ional or differential _— were published, prove that We 
noble science owes its existence to our great t author. 
further add, that the eae and third sections now - 
consideration, have originated a new science, viz. the» ap 
sophy of the — saci saaa ae ae ta 
hysics, or cele echanics. 
= oak an not considered as owiele. of that sub- 
comprising 
lime wales, as the author himself has treated it, and 
