Page images
PDF
EPUB

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

If y

123, what is the Value of y†y?

Suppole a+b=y.

Then a +3a2b+3ab2 + b2=y3.

And a + b=y.

Therefore a +34b+3ab+b Is the Canon for Invol

3a2b+3ab2

tot b

And 123 100 +20 +3,

Operation.

100 = a
20=

1000000=

100 = A

Sving.

600000=3a2b
20 b

120000 =3ab2
8000= bs

1728120 = 43+ a.
1296003aab
3=6
32403ab2
27= b3

Here a 120, And b 3.

1860990=y3+y Answer.

Note, that moft 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 Rule for Involving Fractional Quantities or Numbers, is this; viz.

Involve the Numerator into it felf for a new Numerator, and the Denominator into it self for a new Denominator; each fo often as the Power requires.

G 2

Thu

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][ocr errors][merged small][ocr errors][ocr errors][ocr errors][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][graphic][ocr errors][merged small]
[blocks in formation]

Evolution of whole Quantities.

Evolution is the Extracting of Roots from any given Power; that is, it is the converfe Work to that of Involution; and in fingle Quantities it is eafy, if the given Power have fuch a Root as is required, which may be thus known,

1

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 Root fought.

Thus, If the Cube Root of aaaaaa, viz. 4° were required, (the Number that Denominates the Cube-Root is 3) then 3) 6 (2; that is 34642 the Root required. And fuch Operations are ufually fet down.

[merged small][ocr errors][ocr errors][subsumed][merged small][ocr errors][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small]

Note, 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 muft Extract their refpective Roots.

[merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][ocr errors][merged small][merged small][ocr errors][merged small][ocr errors][merged small][merged small][merged small][merged small]

Part IV. 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.

[merged small][merged small][ocr errors][ocr errors][merged small][merged small][ocr errors][merged small][merged small][merged small][ocr errors][merged small][merged small]

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

Hule.

First Involve any Binomial, as x+y, to fuch a Power as the Root to be Extracted requires; that is, to the fecond 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, viz.

Firft find the Root of the firft or greatest Term of the given Compound Quantity, and call it x, then Subftract the faid Term from the laid Compound Quantity; the Remainder call your Refolvend; and the Value of the Coefficient of y in the fecond Term, of your Canon, call your Divifor. By this Divide the Refolvend, and the firft Term of the Quotient isy. But here Note, that in the continuance of the Operation, you may make more of the firft Terms of the Quotienty. Then find the Sum of the Values of all the Terms of your Canon but the firft; and call it your Abiativum, which Subftract from your Refolvend; and call the Remainder, if there be any, your fecond Refolvend. The Sum of the Values of the next foregoing x and y thus found, call x, fi.e. your next x); and proceed with this Value of x, in order to find that of the nexty as before is taught; and fo on.

Example.

r. Suppofe it was required to Extract the Square-Root of

[blocks in formation]
[ocr errors]

6nx + xx

i ft. x + y × x + y = xx+2xy +, is your Canon for Extracting the Square-Root.

[ocr errors]

2dly?

2dly. The Square-Root of aa, (the first Term of the given Compound Quantity) is a; and therefore ax (i.e. ft. x); and 4x2 which taken from the said Compound Quantity, leaves +9nn

6n 4-6nx for a Refolvend. The Coefficients of y in the fe+ 2x

+ xx

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

[merged small][ocr errors]

+9nn

[ocr errors]

3n

wherefore 2xy +7= a 6nx is the +2x + xx

[ocr errors]

6n

Ablativum, which taken from the faid Refolvend, leaves o: confe

[merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][ocr errors][merged small][ocr errors][merged small][merged small][merged small][ocr errors][merged small][merged small]

Or the fame Root may be Extracted in an eafier, but more tedious manner; thus

a2 — 6an + 9nn + 24x — 6nxxx (a-3n+xx Root!

xx

— 6an +9n1 ́+2ax — 6nx+xx Refolvend.

[merged small][ocr errors]
« PreviousContinue »