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N. B. The foregoing Dividend may be writ in the following manner,

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dd In which the Members of the second Term, as also they of the third are United, by adding in each Term the Factors of the Letter (a), in respect of which the Terms of the Dividend were plac'd.

But if ddd be made the firft Term of the Dividend, then the DiviGon will stand thus,

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n-aa+ba fca-bc for He Divis

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If the Divisor be not an Aliquor part of the Dividend, the Quotient may; in some Cases, be continued to an infinite Series. But if, after you have plac'd as many Terms in the Quotient as you think proper, you have a mind to have the exact Quotient, place the last Remainder as a Numerator over your Divisor, which Fraction annex to the before found Quotient, with its proper Sign + or

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P A R T - II.

Of Fractional or Broken Duantities.

CHA.P.-J. -;Rotation of Frađional Quantities. FRaftional Quantities are express’d or ser down like Vulgar Fractions, in common Arithmetick.

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2bc Numerators,
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Denominators. How tbey came to be lo, may be seen by the general Rule in the beginning of Division.

These Fractional Quantities are managd in Algebra, as Broken Numbers in Arithmetick.

Thus {

СНА Р. ІІ.
Hedution of fracional Quantities.

Sect. 1. To reduce Frations having different Denos

minatozs, to fragions of the same Value, that fhall have a common Denominatoz.

Hule. Multiply, all the Denominators continually by each other,

(i.e. Multiply the first Denominator by the second, and thar Product by the third, and so on) and reserve their Product for a new and common Denominator ; then Multiply any of the Numerators by all the Denominators bur its own, continually, and the last Product put for a Numerator over the said common Denominator ; fo this Fraction is equal to that, whole Numeratorycy Multiplied into all obe Denominators, but its own continually. Do fo with the rest of the Numerators, and you'll have your desire,

Examples,

ac

a

is

Examples. 1. Let it be required to bring and

Te

to one Denomination, First, b'xc=bc, is the common Denominator. Secondly, ax c = ac;

therefore =

be b bd

d Thirdly, bxd=bd; therefore

bc Consequently and are che rwo Fractions required.

be

bc
cdf
and

be all brought to a common fd Denominator.

Firft, ex fd x cmd=efdc edd, is the common Denomi

is

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3. Let

e

nator.

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Therefore ccddff

ofdd

Secondly, cdf x fd x cmd=ccddff-cdddf; ab xexced

abec - abed; and a dxe xf = aefd + ddef, are the Numerators,

cdddff

abec abed efdc

efdd efdc And aefd + ddef

are the Fractions required. efdc

d 3. If

and be brought to one Denominacion, a + b

%
bb ut bc - bd - do

dc they will be

ad - 40. + bd be and

relbat bb da • bd

bat bb bd pectively.

efdd

b toc

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Se&t. 2. To reduce a whole Quantitp into an Æquivalent fraction of a given Denomination.

Aule. Moltiply the whole Quantity by the given Denominator, under which Product place the said Denominator with a Line between them, and you will have the Fraction required.

Examples: i. Let it be required to bring á +b into a Fraction, whose Denominator (hall be d. First, tbx st a=da fudb

da + db Then

ba
is the Fraction required.

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2; b being reduc'd to a Fraction, whose Denominator will

er is =

be q

bq -br

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Note, when whole Quantities are to be set down Fra&tion-wise ; Subscribe an Unit for the Denominator; :bus ab is =

ab 446 +b=

and as

Lemma to Se&t. 3. How to find the greatest common Divisoz (or common

Measure) of cwo given whole Quantities.

a

Hule. Divide the Greater given Quantity by the Less, and if any thing remains, Divide your Divisor thereby, and if any thing yet remains, Divide your last Divisor thereby; and thus proceed 'till norhing remains (as in Vulgar Fractions), if the first Term of each Remainder be a Sub-Multiple of the first Term of the next foregoing Divisor: But here Note, that every Remainder whose firft Term is not a Sub-Multiple of the first Term of the next foregoing Divisor, must be Divided by such of its Sub-Multiples as is Prime to that Divisor, and will give the first Term of the Quotient' a Sub-Multiple of the ficft Term of that Divisor: by which Quotient thar Divisor must be Divided, and if there be a Remainder, the faid Quotient will be your next Dividend to be Divided by this Remainder, or &c. And the last Divisor is the greatest common one required.

Suppose the Greater of the two given * Note, this Divifion Quantities to be=b, and the Less=0; is not necesary nor pro

and fuppose c to be * Divided by a Sub. per, if the first Term of Multiple thereof af Prime to b, and

= cbe a Sub-Multiple of giving the first Term of the Quorient ibe firf Term of b.

a Sub. Multiple of the first Term of b

f (where Note, that f will be = 1, if the first Term of c be a SubAultiple of the first Term of b) : then suppose b to be Divided by is and the Remainder=r. Again, suppose r to be Divided

by a Sub Multiple thereof =s Prime to j, and giving the first

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