A L G 
in y, « times, then ma—x and na—y, therefore x±y—ma 
dzna=?ndtn.a, i. e. a is contained in x±y, m±n times, or 
it meafures x±y by the units in m±n. 
Now it appears thata— pb—c, and b — qc—d\ every quan¬ 
tity therefore which meafures a and b, meafures pb, and 
a—pb or c •, hence alfo it meafures qc, and b— qc or d ; that 
is, every common meafure of a and b meafures d. ft ap¬ 
pears alfo from the divifion, that a—pb-±-c, b=iqc-\-d, c— 
rd, therefore d meafures c', and qc, and qc-Vd or b, hence 
it meafures pb and pb-\-c or a. Every common meafure 
then of a and b meafures d, and d meafures a and b, there¬ 
fore d is their greatefl common meafure. 
Ex. To find the greatefl; common meafure of a* — x * 
— x* 
to its 
E B R A. 
2S5 
and a 3 — a 2 x — ax'-\-x‘, and reduce 
loweft terms. 
-cdx —ax'-J-x 3 
a° — ax- 
-ax 2 -\-x 3 ) ad —v 4 ( a- j-x 
ad — a 3 x — rdx^f-ax 3 
a 3 x-\-aV — ax 3 —x 4 
adx — a*x z — ax 3 -\- x 4 
leaving out ex', which is found in each term of the remain¬ 
der, the next divifor is ad — x*. 
a 2 — x % ) a 3 — adx —cx'-j-x 3 ( a.—x 
’ a 3 — ax * 
—a'x-J-x 3 
— a 2 x-\-x 3 
c*— x 2 is therefore the greatefl common meafure of the two 
quantities, and, if they be refpeCtively divided by it, the 
fraction is reduced to - its lowed terms. The 
a —x 
quantity, 2X 2 , found in every term of one of the divifors, 
2 adx 2 —2X% but not in every term of the dividend, a 3 — 
adx — ax 2J r x 3 , mud be left out; otherwife the quotient will 
be. fractional, which in the proof of the rule is fuppofed 
to be integral; and, by omitting this part, 2x% no common 
meafure of the divifor and dividend is left out, becaufe, 
by the fuppofition, no part of zx 2 is found in all the terms 
of the dividend. If zx° be retained in the operation, and 
a 4 —x 4 , or a 3 — a’x — ax 2 - j-x 3 be divided by the lad divifor 
thus obtained, the quotient will be fractional. 
To find the greatefl common meafure of three quantities, 
a, b, c, take the greatefl common meafure d, of a and b, 
and alfo the greatefl meafure e, of b and c; then the greatefl 
meafure of d and e is the greatefl common meafure re¬ 
quired. Becaufe every common meafure of a, b, and c, 
meafures d and e; and every meafure of d and e meafures 
a, b, and c ; therefore the greatefl common meafure of d 
and e muft be the greatefl: common meafure of a, b, andc. 
If one number be divided by another, and the preceding 
divifor by the remainder, the remainder w ill at length be 
lefs than any quantity that can be afligned. Fora— pb-\~c; 
and b, and confequently pb, is greater than c, therefore pb 
; a . 
4-r or a is greater than 2 c, and - is greater than c; there- 
2 
fore from a a quantity greater than its half has been taken; 
in the fame manner, when c is the dividend, more than its 
half is taken away, and fo on : but if from any quantity 
there be taken more than its half, and from the remainder 
more than its half, and fo on, there will at length remain 
a quantity lefs than any that can be afligned. 
If—be a fraftion in its loweft terms, and - alfo in its 
c n ab . c 
loweft terms, — is in its loweft terms. If pofllble, let ab 
«nd c have a common meafure m, and let abz=.mr, and c= 
m, then a whole number, and fince - is in its low- 
eft terms, — and ms have no common meafure, there. 
Vol. I. No. iS. 
— is a fraction in its loweft terms. 
c -,3 
fore the faCtor m, in — is taken out by b, or m and b have 
a common meafure, hence ms and b, that is, c and b have a 
common meafure, which is contrary to the fuppofition; 
therefore ab and c have no common meafure. 
If b be fuppofed equal to a , it follows that dd and chave 
no common meafure, or 
In the fame manner likevvife —, —, &c. are in their low- 
c c 
eft terms. 
Def. When two numbers have no common meafure 
but unity, they are laid to be prime to each other. 
Fractions are changed to others of equal value, with a 
common denominator, by multiplying each numerator by 
every denominator except its own, for the new numera¬ 
tor ; and all the denominators together for the common 
denominator. 
ace a jf 
■Let -j, -, y, be the propofed fractions; then ~~ t 
edb n •* 
are fractions of the fame value with the for- 
C JL 
bdf bdf 
mer, having the common denominator bdf: for 
cbf c edb e ydf b 
an d TT ?—the numerator and denominator of 
bdf d bdf f 
each fraction having been multiplied by the fame quanti¬ 
ty, viz. the product of the denominators of all the other 
fractions. 
On the Addition and Subtraction of Fractions. 
If the fractions to be added have a common denomina¬ 
tor, their film is found by adding the numerators toge¬ 
ther and retaining the common denominator; thus - 4 -— 
_«+c b b 
b 
If the fractions have not a common denominator, they 
muft be transformed to others of the fame value which 
have a common denominator, and then the’addition may 
. , , , r a c ad be ad-V-bc 
take place as before. Ex. T -1———1 —-— 
r b ' d bd' bd bd. 
If two fractions have a common denominator, their 
difference is found by taking the difference of the nume¬ 
rators and retaining the common denominator. Thus, 
a c a—c 
b b b 
If they have not a common denominator, they muft be 
transformed to others of the fame Value which have a 
common denominator, and then the fubtraCtion may take 
, , - „ a c ad be ad—be 
place as before. Ex. 
b d bd bd' 
bd 
On the Multiplication and Divifion of Fractions. 
A fraCtion is multiplied by any quantity, by multiply¬ 
ing the numerator by that quantity and retaining the deno¬ 
minator. Thus, yX c —~r> for, if the quantity to be di- 
b b 
vided be c times as great as before, and the divifor the 
fame, the quotient muft be c times as great. 
a ab . 
Cor. 1. r y,b———a. That is, if a fraction be multi- 
b b 
plied by its denominator, the produCt is the numerator. 
Cor. 2. The refult is the fame whether the numerator 
be multiplied by a given quantity, or the denominator di¬ 
vided by it. Let the fraCfion be—, and let its numerator 
be multiplied by c, the reiult then is — , or—, being the 
quantity which arifes from dividing its denominator by c. 
The produCt of two fractions is found b.y multiplying 
4 D tim 
