FUNCTION. 
Therefore, in general, if y or @ x = fin. : 
a cof = 55% = A’, (eee ea ee 
"d x 
* Making thefe fubftitutions in the expanded feries for 
Q(x + Ax) 
; APAxX’? 
fin, (x + Aw) = fin.x + A. Ax. cofe— ——— fin, x 
Ax 
et vp One fin. x + &e. 
In the fame manner, if yor e@x=cof. x 
d dy dy .. 
i= A fin, 23 = es AOE ak = Ahn 3 
d 
oa aren = A‘*cof. x, &e. 
d xt 
And thefe fubftitutions will give 
. A? is *) 00 
cof.x + Ax =cof.x« —A. (Ax) fin. « — —-——+ 
+ Se) 5 ellos cof. x — &c., 
AY Aas) 
go 
co A.Ax~—— 
gar MOY 
4 
A on 
At Aa 
2+3+4 
er aa cae 
—Piin. x 
=Q. 
“Ae io 
+ 
2.3+4.5.6 
&e. 
Then ary : i ee 
co An) = 
eer Wee P, aaeoe — 
Thus, whatever be the angle Ax 
ALY 
fin. Ax =z A. Ax — aslas} + A’- 
2.3 2. 
alae) 4 
ZN 5 
el aad &e. 
A 
cofAxn= 1 — a Gg 
It is evident that ae feriesare re a en convergent 
aehibngs the angle Ax is fuch that A. Ax be lefs or equal 
© unity ; 3 in that cafe 
Ax 
fn.Ax ZA.Awxwand 7A.Ax — 
A}. 
ae 
for the terms ai alternate figns and fia een diminifh- 
ing, the fum of the fecond and third of the fourth and fifth, 
&g. will be all n negative, and on the contrary, the fumso 
the third and fourth, the fifth and fixth, &c. will be all po- 
fitive. 
Now by the theorem of Archimedes, we may affum 
that = frei is ers lefs than the arc, and the tangent always 
greater. Hen 
~-Axw 
in. Age Kaae — = Ae 
a 
but olAge Poeeereres 
fin, A 
therefore a > Awys fin? Aw > x)? 
Vp—fintaAwge - | ; . 
- (1 — fin? A x) 
hence fin. Aw >= 
ace 3) 
And from the property of circle 
I 
fn. Aw < Ax, sa ies =a 
1+ (Axzy 
Ef the angle A wx be taken lefs than 90°,, and fo finall that 
A. Ax be lefs than unity, then 
fin. Ae <A. Ax, and > ——— 
ram + =S5 =" 
and confequently A. A xr > —21- and A > 
VIF Ox 
3 
rere ; ikewile Go: Avr >A.Ag— ane 
a I + aa ° 
3 
and < Aw, 7 confequently A A + — aan = oll ~CAw 
At (Az) A @ oy 
and A — <AworA 14 ——— 
As thefe seul ibas mutt take place owes fmall A a, it 
refults that - cannot be lefs than unity, for if A -— I wé 
fhould have = 4 > 13; but the condition A os 
v3 4 E+ Tes; wo)” 
gives x ee ee + (Ax)? 3 therefore if 
by ever a {mall a quantity, it will ei always poffible to take: 
furpafles unity 
Ax fuch, that /1 + (Aa)? < — = “<-> whereas this quantity” 
ft ‘ : . . 
mutt always be > A 
I ears from the fecond condition that A cannot be 
greater than unity, for if A furpafles unity by ever fo fmall 
a a it will always be poffible to take A x fo fmall, 
sl Aw) 
thatr + —~— < A, whereas we fhould always have 
Ther hie — - can oe be lefs or greater thaw 
unity, it follows t 
Ther efore the des ived fandtion of the eae co-efficient. 
of fin. w is fim ply = cof. x 
and that of ae c= . at; wv denoting any angle eat 
ever, that is, an arc re of a circle whofe rad. = 1. Ther 
fore for any angle A 
inAgeae OO, Go, (as? 
24 De FrAs§ 2.304.5-6.7 
+ &e. 
Avy? 4 3 
cof, Av = 1 — coe Seed, jap . 
2 ooo 2.324.520 
+ &e. 
which well known formule were difcovered by Newton. 
In the fame manner as the fine and co-fine are functions of the 
angle; the angle itfelf may be yaa a6 a. fundtion of the 
fine or co-fine an and its differentia 
x ~= angle of fine v; geese fin. 6 we = 5 
- x become wv a A , and fuppofe 
De 
a OM + dae 
= n3 then fin, rea e+n) met Aw= fin. (¢ x) cof. x: 
+ cof. (2 #). fin. 2; yas and fim (gu) = 43. col. (9 x). =. 
V1 — fin Oe = 
Moreover by . formule 
Ae = (A wy3 + &e.. 
fn aman — St &c, and col war — = + Kee 
Making: 
