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Sect. 6. Multiplication of Fractional Quantities. FIRST prepare mixed Quantities (if there be any) by making them improper Fractions, and whole Quantities by fubfcribing an Unit under them; as per Sect. 3. Then,

RULE.

Multiply the Numerators together for a new Numerator, and the Denominators together for a new Denominator; as in Vulgar Fractions.

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36+4c. These prepared for the Work (per Sect. 3.) will

itand

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N. B. Any Fraction is multiplied with it's Denominator by cafting off, or taking the Denominator away. Thus

b

x a gives

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TH

Sect. 7. Division of Fractional Quantities.

HE Fractional Quantities being prepared, as directed in the laft Section.

Then,

RULE.

Multiply the Numerator of the Dividend, into the Denominator of the Divifor, for a new Numerator; and multiply the other two toge ther for a new Denominator; as in Vulgar Fractions.

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Suppofe it were required to divide aa+

a+b aaa+4aab+3abb
aaa + 4 a ab + 3 abb (aaa+ t

@+46

=

aaa+ 4aab+3abb aa+36

aa+56a+406 a+46

3 abb 9+46

by a + b.

The Work will ftand thus,

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aa + 4a ab+3abb

But

aa+56a+466

(per Sect. 4.)

minators, and divide the Numerators. Thus, if

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it will be bb) a b3 (ab the Quotient required.

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When Fractions are of one Denomination, caft off the Deno

ab3 ab3

ab c

For) (c. But a 8 (per Sect. 4.)

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b b

bbc

Again, fuppofe it were required to divide

a3 — abb
d

by

aa+2ab+bb Cafting off c-d in both, it will be a a+

C

d

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Nvolve the Number into itself for a new Numerator, and the Denominator into itself for a new Denominator; each as often as the Power requires.

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13bc

a

2 ad

bb

122

a a

a- C

166+2bd+ dd

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4 a add

b b b 2 7 b b b c c c b
b b b c c c b b b + 3 b b d + 3 b d d + d d d

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Sect. 9. Evolution of Fractional Quantities.

F the Numerator and Denominator of the Fraction have each of them fuch a Root as is required (which very rarely happens) then evolve them; and their refpective Roots will be the Nume rator and Denominator of the new Fraction required.

Thus 1944bb
| 9 a abba a +2ab+bb

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27 a a a b b b ja aa+3aab +3 a b b + b b b

8 ddd

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aaa

3aab+3abb — b b b

Sometimes it fo falls out, that the Numerator may have such a Root as is required, when the Denominator hath not; or the Deno

minator

minator may have fuch a Root, when the Numerator hath not, In those Cafes the Operations may be fet down.

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But when neither the Numerator, nor the Denominator have juft fuch a Root as is required, prefix the radical Sign of the Root to the Fraction; and then it becomes a Surd; as in the laft Step, which brings me to the Bufinefs of managing Surds.

THE

CHAP IV.

Of Surd Duantities.

HE whole Doctrine of Surds (as they call it) were it fully handled, would require a very large Explanation (to render it but tolerably intelligible); even enough to fill a Treatife itfelf, if all the various Explanations that may be of Ufe to make it eafy should be inferted; without which it is very intricate and troublesome for a Learner to understand. But now thefe tedious Reductions of Surds, which were heretofore thought ufeful to fit Equations for fuch a Solution, as was then understood, are wholly laid afide as ufelefs: Since the new Methods of refolving all Sorts of Equations render their Solutions equally eafy, although their Powers are never fo high. Nay, even fince the true Ufe of Decimal Arithmetick hath been well understood, the Bufinefs of Surd Numbers has been managed that Way; as appears by feveral Inftances of that Kind in Dr Wallis's Hiftory of Algebra, from Page 23, to 29.

I fhall therefore, for Brevity Sake, pass over those tedious Re ductions, and only fhew the young Algebraift how to deal with fuch Surd Quantities as may arife in the Solution of hard Questions.

Sect. 1. Addition and Subtraction of Surd Quantities.

Cafe 1. WHEN the Surd Quantities are Homogeneal, (viz. are alike) add, or fubtract the rational Part, if they

Z 2

are

are joined to any, and to their Sum, or Difference, adjoin the irrational or Surd.

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Examples in addition.

5√ be 6b√ acl b√ aa+cc
7bc4b√ ac 3 b va a +cc

1+2 312 √ be 10 b√ a c 4 by aa+cc

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Cafe 2. When the Surd Quantities are Heterogeneal, (viz. their Indices are unlike) they are only to be added, or fubtracted by their Signs, viz. + or - And from thence will arife Surds

either Binomial, or Residual.

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1+213│√/bc:+√ba|4d√a:+3b/ac|3√ac—ba: +√ac+ba

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