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s. Take one Example of a mixt Involution of Numbers, the Converle of which is called the Numeral Exegesis.

If y = 123, what is the Value of ty?
Suppose a+b=y.
Then a3 + 3a*b+ zab3 + b3 = y3.
And 4 +
Therefore ai † 342b + 2ab + b3 ) Is the Canon for Invol-
ta + b

3 ving And 123 +100 + 20+ 3,

Operation.

b=y

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Nore, that most Men do not make use of the foregoing Cyphers; but, if they are not Written, they must be understood as Written.

CHA P. III.

Involution of Fracions, THE "He Rule for Involving Fractional Quantities or Numbers, is

this; viz. Involve the Numerator into it self for a new Numerator, and the Denominator into it self for a new Denominator ; each so often as the Power requires,

Thu

G 2

.

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62 + abd -t dd

2ac + cc
b3

27b3c3 b3 + 3bed + 3bd2 + di
8 ayd

3a'c + 340

10 2,2

9bc

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Evolution of whole Quantities. that is, it is the converse Work to that of Involution ; and in

: single Quantities it is easy, if the given Power have such a Root as is required, which may be thus known,

If the given Power have no Numbers prefixt to it, and its Index can be Divided by the Number Denominating the Root required, the Quotient will be the Index of the Roor roughr.

Thus, If the Cube Root of aaaaaa, viz. As were required, (the Number that Denominates the Cube. Roor is 3) then 3) 6 (2; that is 3/46 = a’ the Root required. And fuch Operations are usually let down. Thusu a I 4966 abocs

az a?biz | a3b3c3 I w313 a? | apbe) : a*b*cz 3 w214 | ab láb c

6

I w 2 2

Nore, The Figures plac'd in the Margin after the Sign (w) of Evolution, denote the Number Denominating the Root to be Extra&ted.

If the given Powers have Coefficients (viz. Numbers prefixed to them,) then you must Extract their respective Roore, Thus 8114 | 1296 2*6* | 20736 a*b*c*

36 a+b+
34
6 a?l?

12 a b c
2 w 214 34
6 aabb ]

But

I w2

2

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144 a bac?

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12 ab

But if the Root required cannot be truly Extracted out of both the Coefficients and Indices of the given Power, then it is a Surd, and must have the Sign of the Root required prefixt to it (or its Index over it.) See Part V.

I w22

Thus as 1 67444 | 216 bida

Vas 1 Jógaan 1 216 bid} = 216 bid?. I w313 3 Vas 1 3167024 | 6bd

Evolution of Compound Quantities may be perform’d by the following

Hule. First Involve any Binomial, as x +%, to such a Power as the Root to be Extracted requires ; that is, to the second Power, if it be the Square Root; to the third Power, if it be the Cube-Root; &c. that is to be Extracted. And the Power thus produc'd, is your Canon for Evolving, which is to be us'd in the following Manner, vize

First find ihe Root of the first or greateft Term of the given Compound Quantity, and call it x, then Substract the said Term from the laid Compound Quantity; the Remainder call your Resolvend; and the Value of the Coefficient of y in the second Term, of your Canon, call your Divisor. By this Divide the Refolvend, and the firft Term of the Quotient is = y. But bere Nore, thar in the continuance of the Operation, you may make more of the first Terms of the Quotient=. Then find the Sum of the VAlues of all the Terms of your Canon but the first; and call it your Aviativum, which Substract from your Resolvend; and call the Remainder, if there be any, your second Refolvend. The Sum of the Values of the next foregoing x and , thus found, call x, li.c.

x your next x); and proceed with this Value of x, in order to find that of the next y as before is taught; and so on.

Example, 1. Suppose it was required to Extract the Square-Roor of

- on +99* +2%

6nx + **

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11.x+y** + y = xx + 2xy + x is your Canon for Extracting the Square-Root. +

2dly

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6n

2dly. The Square-Root of aa, (the first Term of the given Compound Quantity) is a ; and therefore a = x(i.e. 1£. x); and 4=* which taken from the said Compound Quantity, leaves

tonn

a 6nx for a Resolvend. The Coefficients of , in the for + 2x

+ XX cond Term of the Canon, is 2x, and its Value 24 is your Divifor; by which the Resolvend being Divided, the firft Term of the

- 6n Quotient is

3M ; wherefore 249 + 3 = 7

A + x

Onx is the +2x

+ xx Ablativum, which raken from the faid Resolvend, leaves o: consc quently, a

- 3" is the Root sotight.
+x

Operation:

tann 6n

6NX + 2x

t **

tonn

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(*T

3n Root.

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Or the same Root may be Extracted in an eafier, but more tedious manner; thus ai – 6an + onn + 24x - 6nx + xx (4 - 3n -+-** Root

xx ,

XX

6nx to xx Resolvend.

6an fon fo 2ax Divisor 22 2x

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