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243
405 ) 1213 (-
33554432 524^880) 2874588,0 ( 5
14, TO2- 13824, would come out too great to be fubtrafled from the Figures 13312 that precede the fecond Point, there mufl only 3 be writ in the Quotient : Then the Quo- tient 23 being in a feparate Place multiplied by 23, "ives the Square 529, which again multiplied by 23, gives the Cube 12167, and this taken from 15312, will leave 1145- which augmented by the next Figure of the Rcfolvcnd o, and divided by the triple Square of the Quotient 23, viz,. by feeking how many Times 3 x 529, or 1587, is contained in 1145a, it gives 7, lor the third Figure of the Quotient. Then the Quotient 237, multiplied by 237, gives the Square 5S169, which again multiplied by 237, gives the Cube 13312053, and this taken from the Refolvend, leaves 0. Whence it is evident, that the Root fought is 237.
So, to Extras the Quadrate Cubical Root of 315-450820, it muft be pointed over every fifth Figure, and the Figure 3, whofe Quadrate Cube, or fifth Power 243, is the next leaft to 564, viz. to the firft Point, mud be writ in the Quotient. Then the Quadrate Cube 243, being fubtrafled from ;o"4, there remains 121, which augmented by the next Figure of the Rcfolvcnd, ' 36430820 ( 32, 5 v> z - 3. and divided by
. five Times the Biquadrate
of the Quotient, -viz. by feeking how many Times 5 x8i, or 405, is conrain'd in 1213, ir gives 2 for the fecond Figure. That Quo- tient 32, being thrice mul- tiplied by it fclf, makes the Biquadrate 1048571?; and this again multiplied by 32, makes the Quadrate Cube 33554432, which being fubtractcd from the Refolvend, leaves 28711388. Therefore 32 is the integer Part of the Root, but not the true Root; wherefore if you have a Mind to profecutc the Work in Decimals, the Remainder, augmented by a Cy- pher, muft be divided by five Times the aforefaid Biqua- drate of the Quotient, by feeking how many Times 5 x 1048575, or 5242880, is contain'd in 2876388, o, and there will come out the third Figure, or the firll Decimal 5. And fo by fubtrafling the Quadrate Cube of the Quotient 32, 5 from the Refolvend, and dividing the Remainder by five Times its Biquadrate, the fourth Figure may be obtained. And fo on in iufmimm.
In fome Cafes, 'tis convenient only to indicate the Ex- traBion of a Root; efpecially where it cannot be had exaflly. Now, the Sign, or Character, whereby Roots are tfenuted, is y : To which is added the Exponent of the Power, if it be above a Square. E. gr. */' denotes the Square Root. •/* the Cube Root, iSc.
When a Biquadratick Root is to be ExtraBei, you may ExtraS twice the Square Root, becaufe ■/* is as much as •/* 1/-. And when the Cubo Cubick Root is to be ExtraBei, you may firft ExtraB the Cube Root, and then the Square Root of that Cube Root, becaufe thci/» is the fame as -/' ^*5 whence fome have called thefe Roots, not Cubo Cubick ones, but Quadrato Cubes. And the fame is to be obferved in other Roots, whofe Indexes are not prime Numbers.
Ito frove the ExtraBion of Roots.
1°. For a Square Root: Multiply the Root found by it felf, and to the Product add the Remainder, if there were any : If the Sum be equal to the Number given, the Operation is juft.
2°. For a Cube Root: Multiply the Root found by it felf ; and the Product, again, by the fame Root. To the laft Product, add the Remainder if there were any. _ If the Sum come out the Number firft given, the Work is juft.
After the like Manner may the ExtraBion of other Roots be proved.
yi? Extract the Roots of Equations, or Algebraic gHtantities.
The ExtraBion of Roots out of Ample Algebraic Quantities, is evident, even from the Nature, or Marks of Notation it felf, as — . that /aa is a, and that /aacc is a c, and that •/oaacc is jac; and that y' 49 a 4 « «
is 7 a ax. And alfo that •/ — , or — — ; s — , and that
v £hh ; s a_, b and tlmt v gztzz . s 3^ and that cc c 2 jbb 5b
V% is f , and that </ is and that i/'aabb
. 27 a 3a'
js ■/ a b. Moreover, that b i/aicc, or b into ■/ a a c c,
is b into ac, or a b c. And that 3 c V 9 ** z ? is
v 25 b b
3 c X Si; or 211? And ,w a + 3 * . , *b±£!
5 b . 5 b
a -r?# z b x so
I m 1
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o. -f- 2 a b -|- b b + 2ab+bb
And that
c zabxx-j-sbx*
propofed Quantities are produced, by multiplying , the Root, into themfelvcs (as a a f rom ' a x a, a a cYfrern! ac
St"' ?i a VY 1 3 £ Cmt ° * a <=.«**•) But when Quan- tities cc.nf.fi of feveral Terms, the Bufincfs is performed as m Numbers. r
Thus to extraB the Square Root out of a a + 2 a b + b b, in the firft Place, write the Root of the firft Icrm a a, viz. a in the Quotient, and havi n? f u b- tracted its Square a X a, „ „ , , . , , °
there will remain 2 ab + bb aa+2ab+bb ( a -f b to find the Remainder of .*_*_ the Root by. Say therefore, How many Times is the double of the Quotient, or » a, contained in the firft
Term of the Remainder
2 a b ? I anfwer b, therefore write b in the Quotient and havin g fubtrafled the Product of b into 2 a"+b, or
\ ab J~, bb ' there wil1 remain nothing. Which fhews that the Work is finifhed, the Root coming out a + b.
And thus to extraB the Root out of a»-f 6a 3 b + 5 a * b , b ~ " JJ b! + 4 b* firft fet in the Quotient the Root of the firft lerm a", viz. a a, and having fubtractcd its Square a a x a a, or a*, there will remain tfa'b + 5 a a bb — 12 a b' -f 4 b* to find the Remainder of the Root. Say therefore, How many Times is 2 a a contained in 6 at b ? Anfwer 3 ab ; wherefore write 3 a b in the Quotient, and having fubtrafled the Produfl of 3 a b, into 2 a a-+- 3 a b. or 6 a' b -(- 9 a a b b, there will yet remain — 4 a a b b
— 12 ab ! +4b« to carry on the Work. Therefore, fay again, How many Times is the double of the Quotient, yik. 2 aa-f- 6 ab contained in — 4 aa bb — 12 a b', or which is the fame Thing, fay, How many Times is the doable of the firft Term of the Quotient, or 2 a a, contained in the Erfl Term of the Remainder — 4 a a b b ? Anfwer —2b b. Then having writ — 2 b b in the Quotient, and fubtrafled
the Product — 2 b b into 2 a a -j- 6 a b 2 bb or — .
4aabb — 12 a b ! -f 4 b 4 , there will remain nothing. Whence it follows, that the Root is a a -f- 3 a b 20 b.
a*-ffia'b-i-5aabb — i2ab ! +4M(;aa-j-3ab — 2 bb a*
6 a 5 b-]- 5 aabb — 12 ab*-}-4b*
o -f- tfa 1 b-f-9 a abb
— 4 a a b b — 12 a b* -- 4 M
— 4aabb — 12 ab 1 -f- 4 b 4
And thus the Root of the Quantity x x — ■ a a? -X i aa is x — \ a ; and the Root of the Quantity y 4 -f- 4 yi — 8 y + 4 is y y -f 2 y — 2 ; and the Root of the Quan- tity 16 a 4 — 24 a a x x -f- 9 x* -\- 12 b b * x — is a a b b -f- 4 b 4 is 3 x x — 4 a a -- 2 b b, as may appear under- neath.
xx *— aa-f + aa (^ — i'u x x
o — a x -f- \ a a
-j- 16 a* i
ox* ~" 24 ? a x- — 16 aab - - ( ii'T* + "bb , bt \ ] +2
[.a a ■ bti
+ 16 a 4 + vbb + 4fa ,
y 4 + 4 y' _8y + 4Cyy+sy~a'
4y ! +4yy
o — 4yy
— 4 y y — 8y + 4
If you would extraB the Cube Root of &• + 3 a a Ij ~\- 3 a b b -j- b J , the Operation is performed thus : a ! -(-3 aa b + 3a.bb~|-b , ( a _|_b.
r '.
V 5 I s
81 aa
I fay thefe are
3aa)o + 3aab(b
•j " ?ac
all evident, becaufe it will appear at firft Sight, that the
' + 3aab + 3abb-fb s
o o 9
ExtraB
