EQJJ
E CtU
will the confequent term N, divided by the antecedent M, be nearly equal to the required root.-
For inftance, let us take one of the foregoing Equation!, the ieaft root of which was found to be — T W* i aIul let tnc Equation be thus difpofed :
1-4 = 4*3 — S*« + 2* — ••
Then form this feries,
i.i . i.i.o.— 4- — 'S- — 4 1 ' — 97- — 20 9- And ^ will be nearly equal to the greateft root of the pro- pofed Equation.
in the application of the rule for finding the leaft root, the author obferves, that fome difficulties may arife in two cafes. Firft, when the leaft root of the Equation may be taken ei- ther affirmatively, or negatively. Secondly, when the leaft root is imaginary, as if the Equation had thefe roots, -|L v /_ 4 , _ v '_4, and 5; of which the laft is to be confidered as greateft, that is, according to his definition, as fartheft diftant from nothing.
In the firft cafe, the alternate terms only arc to be confidered, and if they tend towards a conftant ratio, while the contiguous terms deviate, it will argue an equality between the affirma- tive and negative root. And, in this cafe, a term in the feries mult be divided by the alternate term following ; and the fquare root of the quotient will be the root of the E- quation: For inftance, if we had, I = — y + 4>'J , + 4V 5 5 then forming a feries according to the firft rule, we fhall have i . i . o . — 4 . 36 . — 20 . 148 . — 84 . 596 . — 3+°' 5.388, faV. And the tenth term 596, divided by the alternate term fol
lowing, or the twelfth, 2388, gives 4?eh
I*', nearly
equal to the fquare of the required root, which will therefore be, nearly, + \/ Uh T his example (hews that the rule approximates fufficiently faft; for the number found i*rrs differs from the truth only by „'„, the true root being + i_. The inconvenience, here mentioned, might alfo be remedied, by taking * =y -j- a ; and then finding the root of the trans- formed Equation by the firft rule.
In the fecond cafe, when the leaft root is imaginary or im- poffible, it is to be confidered whether the root will be affir- mative or negative ; if the former, put x =y + a, if the latter, x=y — a. And then the value of y may always be found by the rule ; providing that a be affirmed greater than x. But though we fhall never be difappointed by taking any greater number for a ; yet it is to be obferved, that the lefs a — x is, the more eafily and quickly will the feries tend to the required root. Some circumfpe&ion is therefore requi- lite.
For inftance, let 1 = .
2 If we form a feries without any previous examination, fuch as
! j t I.:.I._J-._i-._Ii._i2--, Vc.
■248 16 32 64 128 we fhall never find any root, as the terms of the feries do not tend to any conftant ratio. But fuppofing x = y — 2 (for it is vifible that the negative root cannot be great) wc fhall have
_ 1 ;_y — W +>' — 8
or, to avoid fractions, fuppofe, y = 82:, that is, x—%z — 2. Then we have
I = 15Z — 5622; -f- 642;', From this Equation the following feries arifes :
I.I.I .23.353.4071.42769.436151 . &c.
Hence 2; = -—!■ — ; and x = —
436151 4361 51
For a fecond example, take this Equation,
X XX -- X
from whence this feries arifes,
e —11
8' 16'
37 —9
'^-. CSV.
4 a 10 32 64 128 Which gives no root ; but taking ^=^ + 3 (as it appears that x muft be affirmative) this Equation will arife, 1 = — 20; — $yy — j*
- 3
-, or rather taking 31 = 132, that is, *• =
13^+3, we fhall have i= — 20a — 10422 — 1692: 1 . This laft Equation gives the fallowing feries : o . . 1 . — 20 . 296 . — 4009 . 52776 . — 688608
Whence z = — J' 7 ? 5 1 zndx = ^§2l. = 2,0036 near-
688608 688608 °
ly ; x being exactly = 2.
As to the cafe before mentioned, of the equality of two roots, it is alfo to be obferved, that a like rule may be applied, where more than two roots arc equal, providing they be real. But in cafe fome of thefe equal roots fhould be impoflible, and the reft real ; as if x* — i = o s where #=1, x=. — 1,
- = */ — i, and* = — v' — i, all which roots, by our
author's definition, are to be confidered as equal, or equally diftant from nothing. We may obviate this inconvenience, by fuppofing x = y -f- a ; which is therefore an univerlal re- medy.
This will lead us to a method of extracting the roots of powers. For inftance, if we wanted the cube root of 2. ■Svpfi. Vol. I.
Then xi = 2 or 1 = - &
2 The feries, formed according to this Equation^
1 1 1 1 1 1 1 1 1 , . ,f.
i.i.i.---'- — - • &c. only (hews s
222444888 J
that *■! = 2. But taking x — y-$-i, the propofed Equation
will be transformed into the following, 1 = 3^ + 3/7 + y* ?
From whence this feries may be derived,
l.I.i .7 .25. 97 .373. 1435. 5521.
Thercfore y = -22Z - and x = ~^— =1.2 <;q7 nearly.
5521 5C21 ■"' J
Again* let it be required to extract the biquadrate root of 20; then „t4 ss 20, and fuppofing x~y + 2, we (hall have
1 sss 8y -j- 6yy -• 2'y + -y* ; from whence this feries is ob-
4
tained, 0.0.0.1.8-70. 610. 5316 -.46332. TM.* 2126:/ C3i6x'\ , 4 2126;
1 his gives y = i I z= 2J — t I and x = 2 — -.
& J 185328V 4633V 185328
Which root approaches very near to the truth ; fo that it is much to be doubted whether the fame could be obtained fo quickly, and fo accurately, by any other method. Suppofe again, that the fquare root of 26 were required. Then xx = 26. Let .v == y -\- 5. And 1 — I oy + yy. Hence we have the feries o . 1 . 10 . 101 . 1020. 10301 . 104030.
Therefore y = — - — ;
104030
arid x = »/ 26 = 5
104030 5 . 09901951360, differing only by unit from 5. 09901951359 found by the common method. In the cafe where the greateft root of the Equation is re- quired, the like obfervations may be made, as when we feek the leaft root, viz. that we may obviate all inconveniencies by fuppofing x=y^a. Both methods often take place without any preparation ; fometimes only one, and fome- times neither; but this rarely. Both will fucceed by aproperiiib- flitution of x—y + tf, providing the Equation have real roots. To avoid fractions in both cafes, fuppofe, firft, the general Equation a~ bx-\-cxx-\- dxi -• csV. where fl, b, c, d 9 &c. are integers; to which form all Equations may be reduced. Then we avoid fractions by fuppofing -v = ay. In the fecond method, we have the general Equation ax m — bx*-~* t +
cA-m— 2 _j_ ^ where we may put x = - ; but this fubftl-
tution may often be well omitted.
It is to be obferved, that this method may be fometimes ufe- fully extended to literal Equation:. Thus in the general cubic Equation, 1 = ax -\-bxx--cx*, form the feries o . o . o . 1 . rt . aa . aa + b . a"> -j- 2fli + c . a4 + Z aa ^ 4" 2ac + bb . a* + 4^1 b + %aac -J- yibb + %U 9 &c. then will the ,- _ a* 4- -laab 4- zac 4- bb
proximate value of x = — ; rft* T — TT^I — T*
r fl s -J- 4tf Ji -\- $ a ac -f- labb -\- zbc
The learned author of thefe methods afterwards applied them to the refolution of infinite Equations ; for which we muft refer the reader to the Acta Petropol. Tom. 5. p. 63—82. The ingenious Mr. Simpfon, gives us the following method for the folution of Equations in numbers, when only one Equation is given, and one quantity (.v). to be determined. Take the fluxion of the given Equation, fuppofing x to be the variable quantity ; and having divided the whole by x, let the quotient be reprefented by A. Eftimate the value of x pretty near the truth, fubftituting the fame in the Equation, as alfo in the value of A, and let the error, or refulting num- ber, in the former, be divided by this numerical value of A, and the quotient be fubtracted from the faid former value of *; and from thence will arife a new value of that quantity, much nearer the truth than the former, wherewith proceeding as before, another new value may be had, &c. till we arrive to any degree of accuracy defired.
For inftance, fuppofe 300* — # 3 — 1000 = 0. To find x, take the fluxion of the given Equation, and expunging x, we have, 300 — 3**= A. It appearing, by infpection 5 that 300* — * 3 , when x = 3, will be lefs, and when *= 4, will be greater than rooo, eftimate x at 3 . 5, and fubfti- tute it inftead thereof both in the Equation, and in the value of A. The error in the former ==17.125. and the value of the latter = 263 . 25 : Wherefore taking
= 0.027 from 3.5, there remains 3.473 for a
263.25
new value of x. With this proceeding as before, the next error, and the next value of A will be 0.00962518 and 263.815 refpectively. Hence the third value of* = 3. 47296351. which is true to 7 or 8 places. The propofed Equation need not always be delivered^from radicals, though this be fometimes convenient, yet it is alfo fometimes infupport ably tediou s. Thus, if we had the Equa- tion v/i — x + s/ 1 — 2xx + s/ 1 — 3* J — 2= T 0; the clearing it from furds would be very tedious ; but Mr. Simp- fon's method gives the root without any previous reduction. The like is to be obferved, when two Equations are given, and as many unknown quantities {x and y) are to be determined. For all which we refer to the author himfelf, in his effays printed 1740. p. 82. feq.
jo K The
