Encyclopaedia Britannica > Volume 7, ETM-GOA

FLUXIONS.
equal to ATX AG, the product o fthe meafures of the
two former abfciffas.
Firft, in order to determine the equation of the curve
(which muft be known before the area can be found), let
the ordinates GD and NR move parallel to themfeives
towards HF ; and then having put GD=j, NR=2,
AT=fl, AG=j, and AN=«, the fluxion of the area
CDGB will be rcprefented by ys, and that of the area
CRNB by zu: which two expreflions muft, by the
nature of the problem, be equal to each other; becaufe
the latter area CRNB exceeds the former CDGB by
the area CSTB, which is here confidered as a conftant
quantity : and it is evident, that two expreflions, that
differ only by a conftant quantity, muff always have
equal fluxions.
Since, therefore,yj is^z//, and u—as, by hypothefis,
it follows, that u=as, and that the firft equation (by
fubftituting for w) will become jj-rnzzr, or y — az, or
laftly ys = zas, that is, GDxAG = NRXAN : there¬
fore, GD : NR :: AN : AG; whence it appears, that
every ordinate of the curve is reciprocally as its corre-
fponding abfciffa.
Now, to find the area of the curve fo determined,
put AB—1, BC—and BG=.v.* then, fince AG (i+x)
: AB (1) : : BC (6) : GD (y) we have ^=7^,
• • lx ...
confequently « ( =yx)=z — iXx—xx -j-x*x—
n3x-j-xx4—&c. Whence, B G D C, the area itfelf
x* x3 x4 xs
will be —IXx——4-——— + —» &c. which was
2 3 4 5
to be found.
Hence it appears, that as thefe areas have the fame
properties as logarithms, this feries gives an eafy me¬
thod of computing logarithms; and the fluent may be
found by means of a table of logarithms, without the
trouble of an infinite feries: and every fluxion whofe
fluent agrees with any known logarithmic expreflion,
may be found the fame way. Hence the fluents of
fluxions of the following forms are deduced.
The fluent of
-~hyp. log. of x-\-*/xzz±za* >
of -~7==^rr=~hyp.log.aX.x4-<v/a^x-j-x*;
V ZtfX+XX
of ^ hyp. log. of 2±2,
7-hyp- log-
a — a,/a^z+zx7'
a + A/a'z+zx*
Pros. 2. To determine the length of curves.
Fig. 5. Becaufe Cde is a right-angle triangle, Cd4
nCe!,4-f/e*; wherefore the fluxions of the abfcifla and
ordinate being taken in the fame terms and fquared,
their fum gives the fquare of the fluxion of the curve ;
whofe root being extra&ed, and the fluent taken, gives
the length of the curve.
Examp. To find the length of a circle from its tan¬
gent. Make the radius AO (fig. 5.) ~a, the tangent
of AC=/, and its fecant sr, the curve 2=2, and its
fluxion =2; becaufe the triangles OTC, O C S,
are fimilar, O T : O C :: O C :: O S ; whence O S
= —, and SA=a =: a — == ; whofe
s s yV-j-r*
fluxion is 7--^ . 11 ; and becaufe the triangles OTC,
dCe are fimilar, TC (~t) : TO ( = ya*4-rs) :: Ce
(a2tt V alt
■=—) : CJ=— fluxion of the curve.
a'+t^T'1 a+t
Now by converting this into aft infinite feries we have the
fluxion of the cuive =:t——LL, See. and con-
,3 tS /7 #9
fequently 2 = — -—1 —-f , &c. = A R.
_ 5fl4 7«6 9«8
Where, if (for example’s fake) AR be fuppofed an arch
of 30 degrees, and AO (to render the operation more
eafy) be put =_unity, we ihall have/ezyG-ez .5773502
(becaufe OVi = M (±) :: OA (1) : AT (/)=yi)
Whence,
t3 (=/X^=/Xf) = .1924500
ts 0641500
*7 = .0213833
t9 (z=.t',Xti=.^Sj — .oo,]i2']7
t" [ = t9Xt2-~j=z.002^59
t'3 ^/-‘X/1^) = -0007919
^=/,3X/’=—j = .0002639
&c.
And therefore AR = -5773502—ll?34700^
.0641500 .0213833 .0091277 .00237
7 +’ 9 — 11 ^
5
.0007919
r3
.0000097
> .0002639 t .0000879 .0000293
*5 +' 17 ~ ^9 "F
,000003 2
2I ~ ^ = -523598? : for the length of
an arch of 30 degrees, which multiplied by 6 gives
3-I4I_592+ f°r the length of the femi-periphery of
the circle whofe radius is unity.
Other feries may be deduced from the verfed fine
and fecant; and thefe are of ufe for finding fluents
which cannot be exprefled in finite terms.
V' 2«W—
Var-
W4/W*—
Verfed fine
Right fine
Tangent
Secant
Rr 2
is —, and
Radius Unity.
Pro®.