by a Method of Differences . 383 
Cor. If in these we either put qr — p or \qr in the place of 
p, we shall get theorems for the rectangles of sine and co- 
sines, rectangles of cosines and the rectangles of sines similar 
to those of Cor. 11. and hi. (respectively) Example 2. Theorem II. 
for the simple sines and cosines. 
General Scholium. 
It is necessary to obeserve, that there may be particular 
cases, in the summation of which these methods fail, and 
which, if not properly considered, may lead to great error, 
especially when new series are derived from those containing 
failing cases, by multiplying by fluxions, and finding the 
fluents of the expressions thence arising. For if the cor- 
rection should happen to be sought from any of the failing 
cases, the summation of the new series might not only fail 
in the failing case of the primary expression, but in every 
other. 
From Example 1. Theorem I. we have sine of pz -f- sine of 
P + % + sine ofp + stq.z &c. = - - s ‘ ° f - ~ ; this when 
z = o, will be sine of o 4 ” sine of o + sine of o & c. or o + 0 
+ 0 &c - = r£ e4rs = that is the sum of a ^ries 
of noughts infinite, which is absurd. Again, in Example 2. 
Theorem I. Cor. 1. cos. of nz -f- cos. of %nz -}- cos. of 571% 
&c. = o, therefore if z be taken == o it will be 1 -j- 1 -{- 1 
&c. = o which ought to be infinite, and in Cor. 11. z being = o 
we have 1 1 -f- 1 1 & c . = — i. 
In order to explain the reason of these absurdities, and to 
prevent the errors they may produce, it is necessary to con- 
