EXT
[ 374 1
EXT
f* The Operation being repeated according to the third and fourth Steps, ./. e. the Remainder being ftitt divided by the double of the Root as far as extratted, and trom the Remainder, the Square of the Figure that laft came out, with the Duple of that forefaid Divifor augmented thereby, being fubtracted ; you will have the Root re- quired.
E. gr. To extraB the Root of 99856", point it after the following Manner, 9'Ai^> * en feek a Number, whole Square ihaS equal the foil Figure 9, wa 3, and write it in the Quotient ; then having fubftracted from 9, 3 X " or 9, there will remain ; to which fet down 3 ' 3 ' ' the Figures as far as the next Point,
- : : via. 98 for the following Operation.
9985a ( 3 lS Then, taking no Notice of the laft Fi- 9 gure 8, fay, How many Times is the
' T double of 3, or 6, contained in the firft
°j* Figure 9 ? Anfwcr 1. Wherefore hav-
ing wrote 1 in the Quotient, fubtract „ ^ . the Product, of I X 61, or 61 from 98,
\L\$ and there will remain 37, to which
, " connect the laft Figures 56, and you will
have the Number 37 5G", in which the
Work is next to be carried on. Wherefore alfo neglecting the laft Figure of this, viz. 6, fay, How many Times is the double of 31, or 62, contain'd in 375, (which is to be guefs'd at from the initial Figures 6" and 37, by taking Notice how many Times 6 is con- tained in 37 ?) Anfwer 6 ; and writing 6 in the Quotient, fubtract 6x626, or 3755, and there will remain 05 whence it appears, that the Bufincls is done, the Root coming out 316".
Otherwife, with the Divifors fet down, it will ftand thus :
99855 ( 31S 9
6) 98
362)115(59 &c. And then having wrote
ei) 3756 37 ; 6
6-17 609
88791
8468 r
o And fo in others.
Again, if you were to JSm-aff the Root out of 22178791: firft, having pointed it, feek a Number, whofe Square (if it cannot be exactly cquall'd) lhall be the next left Square, (or neareft) to 22, the Figures to the firft Point, and you will find it to be 4. For 5 X 5, or 25, is greater than 225 and 4x4, or 16, is left 5 wherefore 4 will be the firft Fi- gure of the Root. This
'q-~1 r ,-~~ .~r~- ccf- therefore being writ in 2^78791 ( 47°?. 43« 3 /. Vt, thc Quotient> B f, om 22;
take the Square 4X4, or 16 ; and to the Remain- der 6, adjoin the next Figures 17, and you'll have 617 ; from whofe Divifion, by the double of 4, you are to obtain 41 1000 the fecond Figure of the
2*6736" Root, viz,, neglecting the
, 1 1 laft Figure 7, fay, How
3426400 many Times 8 is contain-
3825649 ed in 61? Anfwer, 75
1 wherefore write 7 in the
©"0075100 Quotient, and from 617,
56513196 take the Product of 7
' into 87, or 609, and there
356190400 will remain 8, to which
282566169 join the two next Figures
' 87, and you will have
7362423! g 87; by the Divifion
whereof by the double of 47, or 94, you are to obtain the third Figure ; in order to which fay, How many Times is 94 contained in 88 ? Anfwer, o ; wherefore write o in the Quotient, and adjoin the two laft Figures 91, and you will have 88791, by whofe Divifion by the double of 470, or 940, you are to obtain the laft Figure, viz. fay, How many Times 940 in 8879 ? Anfwer 9 ; wherefore write 9 in the Quotient, and you will have the Root 4709. But fince the Product 9x9409, or 84681, fubtrafted from 88791 leaves 4110, the Number 4709 is not the Root of the Number 22178791 precifely, but a little lefs.
If then it be required ro have the Root approach nearer; carry on the Operation in Decimals, by adding to the Remainder two Cyphers in each Operation. Thus the Remainder 4110, having two Cyphers added to
it, becomes 41 1000 ; by the Divifion whereof, by the double of 4709, or 9418, you will have the firft Decimal Figure 4. Then having writ 4 in the Quotient, fubtract 4X94184, or 376736, from 411CO0, and there will rc _ main 34264. And fo having added two more Cyphers the Work may be carried on at Plcafure, the Root at length coming out 4709, 43637, Cfo
But when the Root is carried on half Way or above, the reft of the Figures may be obtain'd by Divifion alone; As in this Example if you had a Mind to extratl the B.oot to nine Figures, after the five former 4709,4 are extracted, the four latter may be had, by dividing the Remainder by the double of 4709,4.
Thus if the Root of 32976, were to be Extra-tied to five Places, in Numbers ; after the Figures are pointed, write 1 in the Quotient, as being the Figure whofe Square I x I, or 1, is the greater! that is contain'd in
- , thc Figure to the firft Point ; and having taken the
Square of 1 from 3, there will _ _ _ remain 2 : Then having fet thc two 32976 ( 181, 59
next Figures, viz. 29 to it, (viz.
to 2) feek how many Times the z
double of 1, viz. 2, is contain'd
in 22, and you will find indeed that it is contained more than to Times ; but you are never to take your Divifor 10 Times, no, nor 9 Times in this Cafe ; becaufe thc Product of 9 x 29, or 261, is greater than 229, from which it would be to be taken, or fub- tracted ; Wherelore write only 8.
8 in the Quotient, and fubtracted 8 x 28, or 224, there will remain 5 ; and having fet down to. this the Figures 76, feck how many Times rhe double of 18, or 36, is contained in 57, and you will find 1, and fo write 1 in the Quotient ; and having fubtraaed 1 x 361, or 361, trom 576, there will remain 215. Laftly, to obtain the remaining Figures, divide this Number 215, by the double of 181, viz. 362, and you will have the Figures 59, which being writ in the Quotient, give the Root 18 1, •y').
After the fame manner are Roots Extracted out of Decimal Numbers. —Thus the Root of 329, 76 is 18, 159;^ and the Root of 3,2976 is I, 8159; and thc Root ot 0,032976, is 0,18159, and fo on. But the Root of 3297,6 is 57,4247; and the Root of 32,976 is 5, 74*47' And thus the Root of 9, 9856 is 3, its.
to Extract the Cube, or other higher Root, out of a given Number.
The Extraction of the Cubic Root, and of all other Roots may be comprehended under one general Rule ; via. Every third Figure, beginning from Unity, is firft to be pointed, if the Root to be Extn:c'ied be a Cubic one; or every fifth, if it be a Quadrato Cubic, (or of the fifth Power) and then fuch a Figure is :o b writ in the Quo- tient, whofe greateft Power (that is, whofe Cube, if it be a Cubick Power, or whofe Quadrato-Cube, if it be the fifth Power, &c.) (hall either be equal to the Figure, or Figures, before the firft Poinr, or next lefs under them 5 and then having fubtracted that Power, the next Figure will be found by dividing the Remainder augmented by the next Figure of the Refoivend, by the next leaft Power of the Quotient multiplied by the Index of thc Power to be ExtraSed, that is, by the triple Square, if the Root be a Cubick one; or by the Quintuple Biquadrate (that is, five Times the Biquadrate) if the Root be of the fifth Power, (3c, And having again fubtracted the Power of thc whole Quotient from the firft Refoivend, the third Figure will be found by dividing that Remainder, aug- mented by the next Figure of the Refoivend, by the next leaft Power of the whole Quotient, multiplied by the Index of the Power to be Extracted. .
Thus, to Extratl the Cube Root of 13312053, the Num- ber is firft to be pointed after this manner, viz. 13312053 , then you are to write the Figure 2, whofe Cube is 8, in the firft Place of the Quotient, as, that which is the next leaft Cube to the Figures 1 3, (which is not a perfect Cube Number) or as far as thc firft Point; . ...
and having fubtracted 13312053(237
that Cube, there will re- Subtract the Cubs 8
main 5 ; which being ,
augmented by the next 12) rem: 53(4 or 3
Figure of the Refoivend 3, and divided by the triple Square of the Quo- tient 2, by feeking how many Times 3 X 4, or 12, 13312053
is contained in 53, it gives Remains o
4 for the fecond Figure
of the Quotient. But fince the Cube of the Quotient ^- 24,
Subtrail Cube I587)rem:
2167
15° C 7
