of certain differential Expressions, Sec. 
221 
It is on the grounds of convenience of calculation, and of 
systematic arrangement, that differential expressions, such as 
have been just exhibited, are resolved into a series of terms 
P dx -f P' dx + P "dx + Sec. -j- Q ^ i-\- ^ where P dx -f- P 'dx, 
P "dx are integrable; for, remove those grounds, and it will be 
In the application of the differential calculus to curve lines, 
after making certain arbitrary assumptions, it appears that hy- 
perbolic areas, and arcs of circles, may be computed from the 
integrals of the expressions ; the integrals of which 
are in fact afforded by the several methods that relate to the 
quadratures of the circle and hyperbola; and mathematicians, 
either for the sake of embodying in some degree their specula- 
tions, or from a notion of a necessary connexion subsisting be- 
tween circles, hyperbolas, and the integrals of , 4r — have 
expressed the integrals by the arcs and areas of those figures. 
Although the computation of the integrals, is totally inde- 
pendent of the existence of the figures, and of their properties, 
yet it is curious, that the simplest transcendental expressions 
of analysis, should express parts of the simplest figures in 
geometry. 
* This series arises from expanding and from integrating each term mul- 
tiplied into x lm , dx. 
dx 
-f- &c. is not an integral of equally exact as 
