R O O
», Divifion may be performed by the fame rule. Thus, if it were required to divide i byrf-f-f. Then ^r~ =
i+e — d+e . Hence by the rule, P — d, Q_ = -j
ROS
'AQ.= - l J Eft
' 4- Eft.
- I J A = P* = «' ' = <* =i Br=
, v 1 X -"==- and C ='11 n -
That
2 + '
See AWt. Ep'ijt ad Oldenburgh. Tom. 3. Oper. Wallis, ■Analyf. per quart. Series, &c. Lond. 17 11. p. 23. feq. The method of extracting the roots of numbers, mentioned hi the Cyclopedia under the head Extraction, is the feme as that given long ago by Vieta, and is to be met with in almoft all books of arithmetic and algebra. Monf* de Lagni and Dr. Hullcy have given us much more expedi- tious methods. Dr. Halley's rules for the fquare and cube are inferted in the Cyclopaedia, under Approximation j only one of the formulae for the cube is erroneously printed. We (hall here infer t the doctor's rules which extend to the feventh power, and may eafily be continued at pleafure, the law of the continuation being obvious.
„ a b
jaa-)rb=L/aa-¥b, or a ± — —
> + £:
- +£:
■f
y±aa.
or, (7 +
2 J. /' L *
■■-" +\/ -aa±i '
3 9 ~ baa
,« +
$a>±b _ab
v /«»±*=^-+ y~ ,aa + ^r or,+-^i_ V - 4 ' v ,6 — Ion 3 , s« +
y^^ ^\a + y /^n^P_±
See Phil. Tranf. N" us Vol. 1. c. i. Soft. xx.
S" s + zb
- /T7T +1 / x . * j " b
V - 6 +V 36 ««± al ,, j0 r e ±— j-^
For the application of thefe formula or rules, obferve, that a denotes the neareft root to that of the propofed number, whe- ther greater or lefs than the true ; and that b is the difference between this net, involved to the proper power, and the propofed number. Thus, if it were propofed, to extraft the cube -mt (of 231, the neareft cube to it is 216, the™* of which is 6 = „, and the difference between 231 and 16
is is = 1. The root required will therefore be = -a + _ . 2 '
y\ a a + h = 3 + y* + 1 =3 + v/» 8333
= 6.1358. But this is only a firft approximation. For a fecond fuppofe 6.1358 = „. tnen its cube will be = 231.000853894712, which is greater than the truth, and the difference 0.000 853894712 r= b. There- fore the root :=
o-\- y Laa V 4
3"
^ = 3.0679
966195897 true to the eighteenth figure, which Dr. Halley aflures us he computed within an hour. By this example the method of working may be fufficiently clear, and the ufe of the double fign + is obvious. When a is lefs than the truth, then <j 3 -f- b reprefents the given number ; but when a is greater than the truth, the propofed number will be reprefented by a' — b. In the firft cafe
the root i:
1 „_L /' J*
— fl -r * / — a a *r —
2 V a — ia.
3",
or o-f-
ab
3« 3 ±/',
and
in the fecond cafe it is ab
« +
/ 1 . i
3"' — *•
Dr. Halley juftly obferves, that the irrational formula;.
+ y~<"> ± — is
V A 3"»
preferable to the rational,
a — „ al . -. j becaufe the extraction of the fquare root
is much eafier than a divifion by fo great a divifor, as 3"' + *.
Thefe rules are at leaft triple the figures of the root at every new operation. Thus, in the foregoing example, by the hra operation, we had five figures, viz. 6.1358 for the root ; and at the fecond, we had the root true to eighteen places. °
Monf. de Lagni publifhed a rule for the furfolid root, or , ! po *T'> whlch quintuplicates the figures of the roots, at leaft. The rule is this,
aa 4 517 4
or Lowthorp's Abridgment, p. 81. Mac Laurin's Algebra, p. 242. Simfon's Algebra, p. 155. Where various ge- neral theorems for approximation to the roots of pSre powers are given. See alfo Equation. For common ufe, thefe extractions may be moft commo- dioully performed by logarithms : dividing the logarithm of the power by the index of the required root, that is two for the fquare, three for the cube, Eft. the quotient is the logarithm ot the net ; which may therefore be found bv the tables. '
In fome books we have tables of fquare and cube numbers, by which the neareft fquare or cube to any propofed num- ber within the limits of the table, may be found by in- fpexion. Pre/let has given a table of this, kind in his Elt- mens des Mathematiques. For the roots of adfected equations. See Equation.
W»«/Root. Seethe article Seminal.
CttbeS.00 t, in geometry. Obferve, that the mbiroot of any quantity A' may be A, or ~ - 1 + > / ~^ xA, or ~ '~ •/— 3 X A .
2 2
This follows from what was fhewn under the head Unity,
that the cube roots of unity or 1 are I, ' X i£ h
2
J * Hence the cube roots of A' or of 1 x A'
will be as above mentioned. Mac Laurin's Algebr. p. 226, where he fhews the ufe of fuch expreflions. In general, every power has as many roots, real and imagi- nary, as there are units in the exponent of the power . Mac Laurin Ibid. p. 128. Impofftble Root is not only the fquare root of a negative quantity, but any other root denominated by any even number. Thus
\/~ l > \/~ 1 > \/ — I, or in general ^/— 1 or
y/ — x are all impoflible roots, or quantities.
Some call them imaginary roots or quantities. ROPALON, in botany, a name given by fome authors to the nympbxa, or water tilly, and alfo to the faba agyptia of the river Nile. ROPE {Cycl)— Standing Ropes, in a (hip, the fhrowds and ftays are fo called, becaufe they are not removed, unlcfs to be eafed or fet taught. ROQUET* in zoology, the name of a fpecies of American lizard, of fmall fize, and of a reddifh brown colour, varie- gated with black and yellow fpots. Its fore legs are remark- ably long for a creature of this kind ; its eyes are particu- larly vivid and fparkling, and its head is carried conti- nually erect ; and the creature is almoft eternally in mo- tion, hopping about like a bird, and it ufually carries its tail bent into a femi-circle over the back. It is far from be- ing my or timorous, and is delighted at the fight of men ; when tired with play or with running, it will open its mouth and pant, and loll out its tongue as the dogs do. Rochefort, Hift. Antill. ROS {Cycl. — Ros mayalis, maydew, the dews that fall in the beginning of fummer, particularly in this month, have great virtues attributed to them by many authors, when newly gathered and clean filtered. It is not of a watery appear- ance, or colourlcfs, but is yellowifh, and much refemblcs the urine of a healthful perlbn. The common methods of putrifying water alfo have no effei5f on dew. Several quan- tities of it have been placed in digeftion with different de- grees of heat, from that of dung to that of a fand furfacc, but no putrefaction brought on it, though continued two months together ; the heat, in whatever degree, ferving to purify it, not to hurt it. The fummer fun, fur all the hot mouths together, has no more effect on it. It only fends out a green vegetable matter of the conferva kind which floats on the furface, and adheres to the fides of the veflel, and the whole quantity of the liquor remains as pure as before. Phil. Tranf. N« 3.
Though thefe common methods of putrefaction do not take place in this fluid, a much more fimple procefs will. If a quantity of it be put into an open wooden veflel, and fet in a cool fhady place, with a canvas ftretched over it to keep out duft and infects, it will in three or four weeks putrefy and ftink, and depofite a large quantity of a black and thick mud. The manner of the formation of this Pedi- ment is this : a fmall quantity of a foul matter firft feparates itfelf from the liquor, and floats at the top in form of a ikin or film; this foon after precipitates itfelf to the bottom, and another fucceeds in its place at the top, which is foon after precipitated in the fame manner. Thus the mud is a congeries of thefe films one on another. If a quantity of dew be put into a long and narrow glafs veflel, thefe films, inftead of finking, as foon as formed, will combine one with another at the top, and form a thick fenm, which may be taken off with a fpoon, and is very different from the black mud, feparated to the bottom in the former experiment,
being
