510 
FLUXIONS. 
Ex. io. 
duhim in an extremely fmall arch of a circle: or ii/anv 
arrh 11 Y 
other arch. 
131. Let the length of the Dendulum 
the time of vibration in a very fmall arch, is to the time 
in the arch, the chord of whole half is c, as i to i 
CB = r, cord 
-A + 
y-cc 
•B + 
5 2 cc 
;C + 
7 2 cc 
■D 4 - &c. 
AB=c, BF=x, arch BE=w, 'Ee—v\ F f=x, then BD = 
r-i ~ cc 
LV_ 
\F 
- 
2 r 
BG — —, DG 
2 r 
c 
— X 
2 C 
r 
EG — ycx — 
4 rr ’ 
D Alfo let t— time of a body’s 
defeending or afeending 
through the cord AB, z = 
time of defeending or af. 
vending through the arch 
BE, then %t=z time of de- 
feribing AB with the velo- 
the circle v— 
r~ 
2 y/cx- 
ccxx 
4 rr 
The times of deferibing any 
fpaces uniformly are as the fpaces dire&ly and the velo¬ 
cities reciprocally; but fince the pendulum falls from 
A and is fuppofed to deferibe the arch AB in defcendin<c, 
or BA in afeending, therefore the velocity in B and E 
areasi/DBand^/DG,oras^ABand ^"aF; therefore %t; 
2 2 -4 rr 1 42.40- 1 6 2 .4cr * S 2 .4cc 
Cor. Hence if the pendulum r meafures time by vi¬ 
brating in a very fmall arch ; and if c be the chord of 
the arch ay then the feconds loft in twenty-four hours by 
, , 86400 xcc 
vibrating in the arch 2 a, will be nearly - or 
i6rr 
54O0CC oocc 
•-, and the minutes loft-. Or, if time be mea- 
rr rr 
fured by vibrating in the arch 2a, then the feconds loft 
by vibrating in the arch 2A (C being the chord of A) 
CAOO - 
will be-x OC— cc. nearly. 
rr 
Ex. 11. To find the meridional parts for any latitude. 
132. Let radius CA—r, the given arch of latitude 
AB=it/, line DB —y, meridional parts of AB=z. By 
conftrudtion of Mercator’s chart, as q 
cofine of the latitude (Vrr — yy) : ra- 
II • 
dins (r) :: (v : z ::) v : z =- 
TV 
C 
V c 
y c- 
y/ c — X ’ 
whence z— 
tv 
trx 
fmall) 
r " 4 rr ; 
— cc 
_t 
tx 2 X 
4 err x-J-cxx 
2 y/ cc—ex 
(rejedling exx—eex as extremely 
y/rr—yy 
but by the nature of the circle v= C 
ry . rry 
— - - ■ ; whence z — - 
y/ rr -—yy 
whence, by form the 6th, z-z 
rr —yy ’ 
2.3025S r 
x log.^r! = 2.30258c x 
r—y 
log. 
X r—y 
But in the tri. 
and the fluent is z— t x arch of 
4 V c —x 2 r 
this circle whofe fine is r^-, and when xxc, 
t t 
4 hen z— — X quadrant BH=-x 3.1416; and 2Z or 
2 r 4 J T 
angle EBF, as EB (y/rr — yy) : rad. (r) :: EF (r-f-j)V 
tangent of the angle B = 
rX.r+y fr+V 
““ r -'l—y = 
cotan- 
ik 
y/ rr—yy 1 1 f 
gent of half the complement of the latitude A B, whence 
z=z2.3025rx log. cotangent of half the complement of 
the latitude. And by correction z — 2. 302585c x log. 
cct. of half the co. lat. — 2.30258c X log. rad. and this. 
is 
