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Examples in Subtraction.
20 to atbd d+c
atd I-23 d to d
Sect. 6. gultiplication of Fractional Quantities. FIRST prepare
mixed Quantities (if there be any) by making them improper Fractions, and whole Quantities by subscribing an Unit under them; as per Sect. 3. Then,
Multiply the Numerators together for a new Numerator, and the Denominators together for a new Denominator ; as in Vulgar Fractions. Thus
ab 132 — 26
40 + 26
2 a 6 - 466 1 x 2 3
2 ddt de
b Suppose it were required to multiply 20+ 36+46. These prepared for the Work (per Sect. 3.) will itand
| 2 actb-25
6bac+365-7500+80cc+46c-100CC 2 x 2 3
N. B. Any Fraction is multiplied with it’s Denominator by cafting off, or taking the Denominator away. Thus xa gives
Sect. 7. Division of Fractional Quantities. THE
HE Fractional Quantities being prepared, as directed in the lait Section. Then,
R U L E. Multiply the Numerator of the Dividend, into the Denominator of the Divifor, for a new Numerator ; and muliiply the other two together for a new Denominator ; as in Vulgar Fractions.
be divided by the Work may stand thusg of ab'd rabdo
per Sect. 4. of labe
bbb Or thus i
Suppose it were required to divide a at by a + b.
a + 4b The Work will ffand thus, aaa + 4 a ab+3abb paaa+4 a a
2a a +4 a abt 3abb
But a +46
aatsbat 466 aa a + 4 a ab +3abb a a +36 a á + 5 bat 406
(per Sect. 4.)
a +46 When Fractions are of one Denomination, caft off the Denominators, and divide the Numerators. Thus, if were to be
by bol it will be ! b) a b) (ab the Quotient required.
For 83 za b c For
But =ab(per Sect. 4.)
abo Again, suppose it were required to divide
6-8 aa + 2ab +66
Casting off c d in both, it will be eat cd
ba 2 ab +66) a aarabb
Sect. 8. Involution of Fractional Quantities. À 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.
3bc 6td Thus
4 Q add
66 19bbcc 166+2bd + dd
2 ar too 66
bbbccc1868 +366d+36dd todd 3 a a a18 aa add dla a a
заас+3асс — ссс
Sect. 9. Evolution of Fractional Quantities. IF F the Numerator and Denominator of the Fraction have each
of them such a Root as is required (which very rarely happens) then evolve them; and their respective Roots will be the Nume rator and Denominator of the new Fraction required.
9 a abblaa + 2ab +66 Thus
6 27 a a abbbja a a +32ab +3abb+bbb Again I
8 d d d la a a — 3 aab +3 abb-bob
Sometimes it so falls out, that the Numerator may have such a Root as is required, when the Denominator hath not; or the Deno
minator may have such a Root, when the Numerator hath not, In those Cases the Operations may be set down,
But when neither the Numerator, nor the Denominator have just such a Root as is required, prefix the radical Sign of the Root to the Fraction; and then it becomes a Surd; as in the last Step, which brings me to the Business of managing Surds.
CHA P. IV..
Of Surd Duantities. THE 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 Treatise itfell, if all the various Explanations that may be of Use to make it easy thould be inserted; without which it is very intricate and troublesome for a Learner to understand. But now these tedious* Reductions of Surds, which were heretofore thought useful to fit Equations for such a Solution, as was then understood, are wholly laid afide as useless : Since the new Methods of resolving all Sorts of Equations render their Solutions equally easy, although their
Powers are never so high. Nay, even fince the true Use of · Decimal Arithmetick hath been well understood, the Business of
Surd Numbers has been managed that Way; as appears by several Inftances of that Kind in Dr Wallis's History of Algebra, from Page 23, to 29.
I shall therefore, for Brevity Sake, pass over those tedious Rex ductions, and only thew the young Ålgebraift how to deal with fuch Surd Quantities as may arise in the Solution of hard Questions.
Sect. 1. Addition and Subtraction of Surd Quantities. Case1. WHEN the Surd Quantities are Homogeneal, (virgo
are alike) add, or subtract the rational Part, if they
are joined to any, and to their Sum, or Difference, adjoin the
Examples in addition.
7 , bcl abraczba aatio 1+213112 V bclicóva0|4hvaa toc
Cafe 2. When the Surd Quantities are Heterogeneal, (viz, their Indices are unlike) they are only to be added, or subtracted by their Sigos, viz. + or -- And from thence will arise Surds cither Binomial, or Refidual.
Examples ire additioii.
Examples in Subtraction.
d-2avba+dd 1---213 1vbe-bab-dvāa a toca ;-d+2avbd + dd