= 
J 
+ 
* 
PLUXIONS. 
Pea the number B is the Napierean logarithm of 10, so that 
"principles e"=10, na aad log. e = com. log. 10=1; and 
: hence com. log. ex qy = 434294481903, By inspect- 
Cy AB = age The Pee yee 
ing a table of logarithms, it appears tt that ——— 
3 its more accurate ap 
to be e=>2.718281828459. Let us now aol 
mil rents (F) 
2*1.(b). 
= it is to be observed, that in Napier’s system I. (4) 
= From’ this formula, and the series which we have 
, — tage 
pads ar 
that here gi 
pap ee. 
ils estigate the general 
eeeere tes 
Rie is, for the 
of the basis-ot” the system), we 
have, by formula a (F, 
x 
ar ) 
y {(e+a—it ; 
B(2-h)* 
Ba 
and therefore, ; 
Bu zt=n (<*"—1), 
7 1 
Bu’ (z+hy=n {+i} 3 
and taking the difference, and dividing by h, 
pfu(egp—ue} aflegnpoe ‘} 
h ee h s 
"by formula (B), (Art. 7.) the second member of 
AO Tes bale 
Pret ad ¥(S—') Gy being a quantity between 
0&1). Therefore, 
B fears ALL; 
Stes ico uc aera eae 
please, and and always of i 
magnitude between 0 and} iC: 0 dappled $6 be 
may be each reckoned =0, 
fet} of G1) 
very great, — an =, ana + 
oe a 
and then (2-+)*, 2%, 2", and {rth are each to. 
be accounted =1 ; our formula becomes:now simply, 
BW) eel i 
and hence, pertieke for u’ and u their values, 
log. (t-4+h)—log. « i _ ©) 
h 1l— (24-h)* 
From this formula we see, that the function which 
expresses the ratio of the increments of log. «, and z, 
has the’ ies which we have shewn in 
Art. 7. to belong to the function which expresses the: 
ratio of the increments of " and x; but that the ex- 
pression for its boundaries has a different form. If we 
make z’=0, we get — for one boundary ; .and if we 
make 2’=1, we haves Fy fo the other boundary; 
and between these, i 4 ar of the function express- 
ing the ratio is always contained. Or we may indicate 
1 
both boundaries at once, by this eS aay 
where i’ is some quantity cm, bart 0, and re than 
h; and, again, as a yee 
where H is mb ah uantity eG the a of 
baie yo = 0; we cas also 
log. aesaihent wr ’ 
h =sasri=mte (@) 
14. Next let the function be a”, where a denotes a 
constant’ positive quantity, and x any variable quan- 
tity. When «x changes to «4h, then a” changes to 
a* th — g* q* therefore a* +4 a*= a (isn 1). 
We have found (art. 9, and art. 12.) that if a=)” y 
then u= 
_—! 
#(1) n being any number, and 2 a cera 
B® . 
tain quantity between 0 and 1, and B=1(b). In this 
formula put 5" instead of its representative x, and it 
u 
becomes wax 2(F—1). To adapt this expression to our 
BE 
present purpose, change wu into h, also 4 into.a, and 
B into A, (that is 1 (2) into 1(a)) and it becomes 
sh 
pl) , or putting for the present A’=Aa*, we 
Aor 
ahat=(1 + p. 
Now we have found (art. 7.) that 
(ath)? —z amas h; 
397) 
Fundamen. ° 
tal 
Principt 
———— 
