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is

aa + 2ab by Seat. 3. Chap. ILA

Suppose it were required to divide da izbba

aa | by ait bi

a +46 The Work (when prepard as before directed) will stand thus •+5)

aaa + 4aab t Babb ana + 42ab + 3abb 4 + 46

44 tsba + 466 And

ada + 4aab + 3abb

na + sba + 466 a + 4b When Fractions are of one Denomination, Caft off the Deno. minators, and divide one Numerator by the other. abbb

bb Thus, If

were to be divided by it will be bb) abbb (ab the Quotient required.

bb abbb abbbc For

Cobe

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Again suppose it was required to Divide

ada abb

Co na + 2ab + bb

d

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By Cafting off c d from both, it will Átand thus

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PART III.

Jnvolution.

CH A P.. I.

Involution of whole Quantities. IN Nvolution is the Railing or Producing of Powers, from any

propos d Root; and is perform'd in all respects like Multiplication; fave only in this ; Multiplication admits of any different Factors, but Involution Atill Retains the fame.

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Examples.

I the Root, or fingle Power. iO 2 2 jaa Itaa | Square, or second Power. IG 33 10a 1-asa | Cube, or third Power, IC 414 daaa I aaaa | Biquadrat, or 4ch Power. 10 sis taaaaa 1-aaaaa, Sarsolid, or. 50h Power, &c.

Note, The Figures plac'd in the Margent, after the Sign (0) of Involution; thew to what height the Root is Involved ; and are called Indices of the Power; and are usually placed over the Involved Quantities, in order to 'contract the Work, especially when the Powers are any thing high.

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If the Quantities have Coefficients, the Coefficients must be Invölyed along with the Quantities. As in thele.

Tbus

Thus i
24
34

is be
iO 22 444

t gaa

25bbcc
1633
8aan -27444

125bbbccc
1644

164444 +814aaa 623bbbbcecc 16 515 32aaaaa-243aaaaa 3125bs c. &c.

(

2

Involution of Compound Quantities is performed in the same manner, due regard being bad to their Signs and Coefficients.

As for Instance, Suppole a + b were given to be Multiplyed to
the sth Power.
Thus I a. tb called a Binomial Root.

a-t-b
IX a 2laa tab
IX6 3 tab-+-bb

-
2 + 3, or 1 2 4a tab+bb the Square of a + b

a-t-b
4x4 5aaa fazaab + abb

4 xbo aab+zabb-t-bbb
576--
5+6,or 1 & 3 7a+3aab+3abb-+bbb the Cube of a +b.

. 4 + b 7x4 844 +3431 +32°b2+ abs 8

-t 7x9

43b13a*bs +3ab1 tbt 8 + 9, or 1 41024+44b-6a2b2 +42b3-4b4 the Biquadrat lato

(of atb 10 Xust4496toa:bit4ao b3 +

ab4
10 x 612 aub +42b2 +6a2b3 + 4abu+bs

十4
25-+5a9b 10a3b+104?b3 + sabi tbs the

Surfolid or sth Powr of 4 +b required. &c. Again, Let & -b a Residual Root, be given to be involved. Then 14-b

-b ixa 2 aa-ab IX-b3

31

-ab-tbb
16 2 449--sab+bb the Square of ?

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.X 3113_2ab + abb

,
4X- 64 - abtzabi_63
1 G 3711-34*b+sabi-b3 the Cube of 1-6

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7 % al 84-313b +34'b? - abs 7X-9 a3b + 3a2b2_31b3f6+ 1 G 4101-4436 6a?bi-49b3 tbm the 4th Power of b

(=b 10 x aulas-4946+6a3b2-4a9b3 + abe TO X-bu a4b-4a3b2-613b3 + 4abc-bs IG 5 1 3 as-Sab+1023l:2-104-b3 + 5abe_bs the 5th

Power of a-b, &c. By comparing these two Exainples together, you may make the following Observations. 1. That the Powers rais'd from a Residual Roor, (viz. the Dif.

a ference of the two Quantities) are the same with their like Powers rais'd from a Binomial Root, (or the Sum of two Quantities) save only in their Signs, viz. the Bioomial Powers bave the Sigor + to every Term; but the Refidual Powers have the Signs † and - interchangeably to every other Term.

2. The Indices of the Powers of the Leading Quantity ( a ) continually decrease in 'Arithmetical Progression ; viz. in the Square it is a', a! ; in the Cube a', , at; in the Biquadrat 4*, al, a', 4'; &c.

3. The Ipdices of the other Quantity b, do continually increase in Arithmetical Progreffion, viz. in the Square it is Bt, ba; in the Cube b, b, b3 ; in the Biquadrat b, b, b, bt; &c.

4. The firft and last Terms, are always pure Powers of the fingle Quantities, and are both of the same height.

5. The Sum of the Indices of any two Letters join'd together, in the intermediate Terms are always equal to the Index of the higheft Power, viz. of the first or latt Term.

These Observations being duly congder'd, it will be easy to conceive, how the Terms of any propos’d Power, rais'd from a Binomial or Relidual Root, mult stand without their Uncie, .ot Numera! Figures, or Coefficients,

For Instance, suppose it were required to raiie the Binomial Root a + b; to the 7th Power ; then the Terms of that Power will stand without their Uncie in this Order Viza? tabta! b: t **b: -t a b + ab + ab + b1

And

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And because the Uncia (not only of any single Letter, but also) of every single Power, how high' soever it be, is an Unit or i (which neither Multiplies nor Divides) and all the Powers of any Binomial of Residual Root are naturally raised by Multiplying of the Precedent Power into its original Root, which is done by only joining each Letter in the Root to the Precedent Power with its Uncia, and then removing the faid' Power, when it is fo join'd to the second Letter, one place forwards (either to the Left, or Right Hand), it must needs follow.

That the Unciæ of the Second Term (iņ any such Power) will always be the Sum of so many Units Added together more one, as there bath been Multiplications of the first Root, which will always be determined by the Index of the first Term in the Power.

And because the Uncie of all the intermediate Terms are only removed along with their Letters, it also follows; ihar if they are Added together, their respective Sums will produce the true Uncie of the intermediate Terms in the new raised Power. As doth plainly appear from the following Numbers so removed without their Letters; which both thews and Demonstrates an easic way of producing the Uncie of any ordinary Power (viz. of one not very bigh) raised from either a Binomial, or Readual Root.

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: 3 3

The Uncie of the Cube.
3 3
4

6 4. The Cincie of the 4th Power,
4

6
5 .10 .10 s

Uncia of the 5th Power.
I
5.10.10

s 6.15.20 .15

6

Uncia of the 6th Power, 6.15 .20 15.

6 7 .21 .35 -35 .21 hos Uncie of the 7th Power,

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And so on in this manner ad infinitum.

Now

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