FRACTIONS. 
for a farther account of thefe conti nved fraGions to Dr. p — fp) be 
Ope = 
.Wallis’s Arithinet. ee “Prop. 1gl. 
mat. Planetar. in 0 er 
p- 173, feq. edit. ; — particulatly to Mr. 
Evnuler’s Analyf. Infinit. vol. i. cap p- 295> feq. who 
shas fhewn the ufe and + ication ce he one in many 
idnitances 
Lord Disaucker feems to have been the rey who canfi- 
dered continued fraGtions, or at leaft who firft applied them 
The hint fen ufeful, but 
n pretty much neglected, Sue approxima- 
tions to Me actions or ratios exprefled in great numbers. See 
the article Rarie. His feries for the quadcature of the 
circle, is 
in his and Dr. Lia s rotations aie amounts to the 
fame thing as 1 
Butt S42 as 
2+, &c. 
in the notation of Huygens and Euler 
The fymbol 0 denotes as eae the ratio of the {quare 
of the diameter to the area of the c 
Fracr » Vanifli ings 3 are thoi in which the a 
and ae cemiaice vanith, me equal to o, at the fam 
time. Thus, when | quai N is expreffed by a 
fration-—~, if P and Q vanifh at the fame time, we are not 
thence to conclude, that N is o. Suppofe N = 
@a—- ax ~. 3 . 
= ; when x = ay-the numerator and denominator 
a—~a 
of N vanith together; but if we reduce the value of N to 
amore fimple form, by dividing the numerator and denomi- 
nator by their cc common | divifor va — x, we Shall find 
2va 
= wicaeed ax 
( ) / a 
N=a x ———- 
a. The idea of fuch fractions originated, as Mon- 
tucla has informed us ee Math. 
teft among fome h mathematicians, in arig= 
_ non and Rolle were ae principal combatants, in seer eli to 
the diferential calculus, then introduced, which Rolle 
Among other arguments againit it 
a tan 
mathematician particularly by 
a (Al gebta, p. 212.apud Works, 
d De Me (Mifcell, Anal. p. 165. Wea is 
iar eis were the occafion of a fharp controverly bee 
Powell and Waring in their competition for the ie ee 
-at Cambridge. Waring maintained that the fra€&tion — ae 
“became 4, when p was = 1. Powell aia to ce con- 
aaa as abfurd, alleging that when p = 1, 
5 
+ 
vol. ii.) ina aie con." 
864.) Saunderfon (Algebra, . } 
he fraétion.. 
— I 
oe ee 
tof 
a | 
si cand loft the profefforfhip. Waring 
replied that meg + p+ p + pi by common 
Givifion = 1 i-+-1+ 1 4, when p is 
Mathematicians have pr opofed * ae tod of fin ding 
the — of thefe fra€tions. ‘The one is by confidering the 
terms of the fra€tion as two eet te quantities, continually 
joealue till they both vanifh together; or finding the 
ltimate value of the ratio denoted by the fra&tion. is - 
confidered, it appears that, as the terms of the fraction are 
oe ofed to decreafe till they vanifh, or become only equal 
o their Auxions or increments, the value of the fra&tion, in 
ee {tate, will be equal to the fluxion or increment of the 
numerator divided by that of the denominator, ¢.g. take 
—_— 
13 the fluxion of the numerator is = 
e — 5x* x, and ae the denominator — +; eee 
I 
3 xt 
— x — 1 I =3 = 
. ew! 
the value of the fraction — ———, whenx = 13. Or thus,. 
x 
x ss : 
becaufe x = 1, — will be = — ; then the flux- 
x 
ion of the numerator, — 4 oa x, divided by the an of 
the denominator, — x, gives 4 x* or 4, as be for 
The other method is by edgcne the given-e enetiones 
another, in a more fimple form,.and then aan the : 
values of the letters. In the former example, <r 
— x 
I — xt a 
eee, when x = 1, divide the numerator i the deno-- 
x 
minator, and it becomes:-1 -+ x + x? + whic 
eing = 1, becomes 4 for the le of the panies as - 
av ax ax — xXx 
before. Again, to find the value of 2, when 
a= 
x is = a, in which cafe both the numerator and denomi- 
nator become = 0; divide the numerator by the denomi- 
. — x 
nator, and the quotient. / ax +x +x J —, when, 
x being = a, becomes a+ a = 34, forthe gd of” 
the fraction in that ftate of i "As ton? 3 Math. 
For a farther view of this iabje, and particularly of Mr. 
oodhoufe’s icles ions againit vanifhing fraGtions; fee 
Analytical Funct 
FRACTION, Repetend of a decimal. 
Fraction, Logarithm of a. See 
Fractions, /umming of infinite. 
sbi 
See REPETEND. 
LoGaritTHn. 
See Cateutus, and 
ae) ¥3 
e 
a 
their origin, o 
the concords, the only intervals marked {tro 
ses each other, and thefe diffe 
sie may be called the aoa interva n ob 
tained, as the fame are exhibited in a ta e, plate V. 
in vol, xxviii, of the Philofophiical Magazine, See 
INTERVAL+ 
