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duce the same with the given Power; consequently 50+40-6 is the Cube Root required.

But if the new Power, raised from the supposed Root (being involved to it's due Height) should not prove the same with the given Power, viz. if it hath either more or fewer Terms in it, &c. then you may conclude the given Power to be a Surd, which must bave it's proper Sign prefixed to it, and cannot be otherwise expressed, until it come to be involved in Numbers.

Example 5. Suppose it were required to extract the Cube Root of 27 aa a +54 baa +8bbb. Here are two diftinct and perfect Cubes, viz. 27 a a a, and 8 bbb, whose Cube Roots are 3 a and 2b. Wherefore one may suppose the Root sought to be 3 a +2b, which being involved to the third Power, is 27 a a a +54 baa + 36bba +8bb. Now this new raised Power hath one Term (viz. 36b ba) more in it than the given Power bath; but this being a perfect Cube, one may therefore conclude the given Power is not so, viz. it is a Surd, and hath not such a Root as was required, but must be expressed, or set down,

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Thus V 27 aa a +54baa+86bb. If these Examples be well understood, the Learner will find it very easy by this Method of proceeding, to discover the true Root of any given Power whatsoever.


Of Algebzaick Fradions, or Broken Duantities.

Sect. 1. Notation of Fractional Quantities.

Fra&tional Quantities are expressed or fet down like Vulgar

Fractions in common Arithmetick.

2 bc 56-42 Numerators.

Thus {osmos 2756 Denominators.

How they come to be so, see Cafe 4, in the last Chapter of Division. These Fractional Quantities are managed in all respects like Vulgar Fractions in Common Arithmetick.


Y 2

Sect. 2. To alter or Change different fractions into

one Denomination, retaining the same Value.


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ULTIPLY all the Denominators into each other for e

new Denominator, and each Numerator into all the Densminators but it's own for new Numerators.



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d Let it be required to bring and into one Denomination.

b First a xc, and d xb, will be the Numerators, and bxc will

bd be the common Denominator, viz. and

the two bo b c

bd d Fractions required : that is,

and bo

b c buto

d Again, let

and be brought in one Denomination, ato

6 d

bbtbc-bd-do ad-act-bd-bc and they will be


&c. ba+66-da-

bat bb-darbd


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Sect. 3. To Bring whole Duantities into Frations

of a given Denomination.

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TULTIPLY the whole Quantities into the given Deno-

nominator for a Numerator, under which subscribe tbe given Denominator, and you will have the Fraction required.

EX AMPLES. Let it be required to bring a + b into a Fraction, whose Denominator is d - a. Firft a +bxda is datbd-aa-ba:

da tubimaa- ba Then

is the Fraction required.

aa - da Again B +ä will be And a will be

d a atbb Allo a tot will be


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2 da

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When whole Quantities are to be set down Fraction-wise,


And subscribe an Unit for the Denominator. Thus a b is

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аас аа
- is

is a


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Sect. 4. To abbreviate,

To abbreviate, or Reduce Fra£tional Quantities into their lowest Denomination.


Ivide both the Numerator and Denominator by their greatest

common Divisor, viz. by such Quantities as are found in both; and their Quotients-will be the Fraction in it's lowejt Term. abbb bb

b dc Thus

And a +

=a+d. de In such fingle Fractions as there, the common Divisors (if there be any) are easily discovered by Inspection only; but in compound Fractions it often proves very troublesome, and must be done either by dividing the Numerator by the Denominator, until nothing remains, when that can be done : or else finding their common Measure, by dividing the Denominator by the Numerator, and the Numerator by the Remainder, and so on, as in Vulgar Fractions (Sect. 4. Page 51.)

E X A M P L E S. aac-aad Suppose

were to be reduced lower. ¿ d - dd

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aaa а

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In this Example it so happens that the Numerator is divided just off by the Denominator; but in the next it is other wise, and requires a double Division to find out the common Measure, viz.

- abb Let it be required to reduce

to it's loweft Terms.

aa + 2ab + bb First aa + 2ab + bb) a aa-abb (a

aaa + 2 a abtabb

- 2 aab-2abb the Remainder. Then - 2aab- 2 abb) aat2 abtbb

I aatab


2 ab+bb abtbb



Hence ië appears that -- 2a ab- 2 abb is the common Meafure; by which a aa-abb being divided.

Viz. - 2a a 6 2 abb) aaa-abb

a a a fudab

-aab abb -aab • abb

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a is the new Denominator. But

2a + 2b

+ 2a



2 a
the Numerator; and


the Denominator. Let both be multiplied with 2ba, 2 ba

aa tab the Numerator, and you will have

Or changing

b the Denominator, the Signs of all the Quantities, it will be

the new Frac

a+b aa

ааа. abb tion required. That is,

at-b 2.2 + 2ab +66

dd - bb Again, let it be required to reduce

ddd bbb The common Measure of this Fraction will be the easiest found (as appears from Trials) by dividing the Denominator by the Numerator, &c. Thus,

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ab 3

ddbb) d d d -b bb (d

+-bbd-bbb) ddb, bb

+bd-bbbbdb3 (6


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Hence it appears that bdbb is the common Measure that will divide both the Numerator and the Denominator.


Confequently bod-abos + 1, is the new Numerator.




o And bd-65) ddd - bbb dd dddddb

+d+b the new Denominatora

+ ddb bbb


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Let both be multiplied with b, and then you will have d+b the Numerator,

. dd+bd +66 the Denominator,

But if after all Means used (as above) there cannot be found one common Measure to both the Numerator and Denominator; then is that Fraction in it's leaft Terms already.

Note, These Operations will be understood by a Learner after he hath passed thro' Multiplication, and Division of Fractions.

Sect. 5.

5. addition and Subtraction of Fractional

Quantities. THE

HE given Fractions being of one Denomination, or if they are not, make them fo, per Sect. 4.

4. Then,

RUL E. Add or subtract their Numerators, as Occasion requires, and to their Sum of Difference, subscribe the common Denominator s as in l'ulgar Fractions.

Examples in addition. bb

-bte d

data 2a-to 26

atbd to

dta 66. 3a+ + d dy

dta Examples

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2 a

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2 a

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