484 
Stethen whose sum Sede + dy + 7.4% is the complete 
Guxion of w. Our limits, however, will not permit us 
to enter farther into this branch of the : besides, 
it is easy to extend what has been ly shewn, to 
functions of any number of quantities. 
Application of Infinitesimals to Fluxions. 
107. The colshated Lelhadts Gatuten hie thecry of 
the differential calculus upon the doctrine of infinitel 4 
little quantities, To this. method it has been obj 
that the notion of infinitely goer bee ie tint fy) 
to form the foundation of a mathematica theory, and 
on this account it has been laid aside by late writers, in 
establishing the principles of the calculus. It must be 
confessed, however, that this view of the subject gives 
readily all the rules for the calculus, and affords a great ed. 
facility in its applications to geometry and mechanics, 
particularly in questions of an intricate nature. On 
this account it is 2 and, besiiene we a: 8 
that in the developements of the mee a Tine selve 
tion, all the Tomeglected, that were hi 
to find ime 
pei mca having deel the proc 
MORSE SIRS Ral 
ction xy, 
‘E eg en rept of tev 
nitely less any_ 
then dad y tonat be lntinitely less than dy or dx, 
== then (See By substituting now 2°, 
,.md 22d 2x instead of dy, in the formula’ 
d(2 (ey) fedy ty de, we get d( (2) = 25 Free 
inakseny; e fl vfluxions. of the high 
With regard to fluxions, of her orders, his, 
theory required that-fluxions, ‘or 
second 
respect to those of the first order ; and therefore as ho- 
with the squares of these last. Hence,, to 
Lag gp deri his her differentials, it was only ne- 
we to find iginiageee be 
» other any sinligaea telat 
* resolved, is far Sion, The ary 
quantity represented by 1, 
tials of the | 
, should be considered as infinitely little in _. 
indicate thé pian aie Fact 
FLUXIONS. 
Thas, $Y will manifestly be the trigonometrical tangent 
of the angle which a tangent to Whe’ curve makes with 
the axis, and consequently § 
Also, y dx will be the i increment, or dif- 
tere of the sheet ad, ie re 
led hav Vide )= ee 
an, » we e = 
ap Sy for the differential e ae ” 
110. Al Lalbnite’s view of the oi led to 
correct results, dil ot scze the tr oA 
method. He coghe be here vena beter 
rejecting certain quantities, 
definitely small, when com i 
retain The trath was, Cupiic Wine eee 
jected, STs (Riet ay Catton! ifan hog On 
small 
Mra Ce i ae 
SECTION Ill. 
Or rue Inverse Metuop or Fivxions. 
11. Tue I Method of malt aun Inverse 
a ic anes ap 
fluent of 
;in« 
he 
cena 
—— are itis carr to 
’ lation ofthe 
to the sittin 
con ee 
ramet amy 744 ubeta a 
ee the.direct ames. quantities are aseaee 
=e 
of any flaxion, we | employ character 
” the i et ee ge ee toe 
to consi differentials as new, variable Pe aid? by 5 
whi had themselves, differentials. opted \generally by foreign, British 
= 3 ofa snare ar ai te, meri d ates | a. 
terms “were of an to iy wi jon a2” wis ~ cals 
Raieiecierae led.a fluent, is: “tree the fg, 
tity is i inate omnis it 
aed ca from which it is to des 
vo dg 
dz; then, cor, ving pig 2 ae | 
ay Be and, z | ee Oey called Ji 
nimnanl PH kirk 
hy ) that “in?a 
such as are cons, 
5° . 
esares Pines ah Ne ad 
~ sey ahd asa tile wih? 
wilt * ve ye 
11s. We 
function ¢ 
