aha dei 
mator is 3 bd. And thus 4 and fs or — a —, become 
wr and “ae : 
e 
But where the denominators have a.commen the, 
it is fuicient to- multiply them alternately by the quo- 
tien its. 
Thus the fragtions —— — and qm reduced to thefe 
a and - 7 < by erm aa by the quoti- 
ents ¢ a dy: arifing by aa divifion of the denominators by 
the common divifor 6. other uate las refpect 
to conben: are eafily aed to algebraic quantitie 
, and fubtraé& ion of jraétions in Jfpecie ;.—-The 
Addition 
ee is, in all mapect> the lame in {pecies as in aanben: 
fe 
—_— 
“Suppole it be required to add the fraCtions + and 
d e 
‘ oe aad 
Ihefe, when reduced to the fame denomination, will be Fa 
be ad+be 
and : ; io confequently their fum is —-—77-- 
So, if the fraétion — were io be f{ubtraéted from “5 ; 
b 
af 2 ts : ad be 
" having reduced them they. will be a and ia? as before. 
—a qd. 
Their difference therefore is —— “75 
MM ultiplication « sped divifion of frattions in 0 hes -~Here, 
too, the procefs is perfe€tly the fame as in vulgar arithmetic. 
Thus, e gr- fu) pre a the faCtors, or fraéti ~~ to be m 
: the produ& will be rec 
tiplied, + —_ aca 
a 
Or fappofe the fra€tions required to be. divided, < and 
; 3 the quotient will be i x = = = ae = > 
Hence as 4 = =; the produé of a into Ss that is, 
of an integral quantity into a fradtion; — x <= fbi 
d 
Whence it appears, that the numerator of the fbi. is 
20 be multiplied by the integer 
Hence, alfo, the asourle of 7 < by a, that is, of the 
7 I. @¢@ 
| a. ada 
_ Befide the common notion of a fraGtion,’ there is another 
ard to has underftood. Thu 
broken quantity divided by the whole one, 
us, 
thi “epee iifkead of 
pa 
e integers as the 
(viz. 4, the ee a then: 
i geometers and. sigebraft, he re - ak a ai 
vided by 8 
15'S. . oP 
t 
RACTION, continued, i is ufed for a one aie denomi- 
10 number with a fraétion, the 
ch is again a whole number and a 
fraction, and fo on, siceer this affection - continued ad 
infinitum, or whether the feries breaks off after a aad 
number of terms. ‘Fhus, 
I 1 
aes or-—~ 2 
2+-— 2+ ; 
Ste ae 
5 +, ke, 5 
-, &c. aré continued (ee iaie. 
£ we make ufe of letters inftead of numbers, we fhall 
have general expreffions of thefe fraétions, thus, 
I 
a+ ree ss ; t 
€ | +- a 
and 7 +, &e. 
‘+ F450 
c#"- 3 
F +3&& 
The reduction of thefe continued fractions to thofe of a 
mon form is not difficult by the ufual rules of arith- 
metic and - ebra, ‘Thus to give an lich ca 
at the fam 
fuppofe the ena fractions, - 
oor 
rs — i! 
1+ => ; 
+s ¢ whic 
eeonel the excumiee ee of a circle, otic “diameter f is 
one 5 a we ftop at = we fhall have 3 + >= ae 
=. If we ftop at 2. we fhall have 
7 3 
pth iss. 1 = SEES = 8 
7 I 106 06 
15 
But if we ftop at 4, which is awa on account o 
the’ fmall fraétion st added to the laft denominator 1, 
we fhall then find, 3+— 1 = 22. Thefirg 
7 + Te i5 += 113 . ; 
. the reduétions give the ca ‘of Archimedes, and 
he laft that of Adrian Met 
But as beginning at the ia denominator of the conti- 
nued fra&tion makes the computation aeaipere tedious, 
fous methods is ve been contrived for the reduction. of 
thefe fractions, and. for a continued approximation to their 
nd Mr. Cotes’s method for the reduction of 
continued fraction are each units, th inators will 
be the quotients arifing from the continued divifions in Mr. 
Cotes’s method's or in the common, for g fractions. 
e i in 
a lower di 
Euclid’s method for findin 
of two magnitudes, libs x. prop. 3 
things weuld Jead us too: far, .we. therefore -refer the ae 
for 
