Isaac Newton.

The method of fluxions and infinite series : with its application to the geometry of curve-lines online

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it may be obferved, that towards the end of the Operation I neg-
lect all thofe Terms, whofe Dimenfions would exceed the Dimenfions
of the laft Term, to which I intend only to continue the Root,

fuppofe to *' ,2.


iz. Alfo the Order of the Terms may be inverted in this man-
ner xx +- aa, in which cafe the Root will be found to be

a a

10 A* iz A-

13. Thus the Root of aa xx is ^ -Jj - ^7

14. The Root of x xx is #'" i** 4-.v* T ' r **, 8cc.

. . AT A.' A' b*X* g

15. Of -+- f ## is a -f- - ^ , Sec.

. i + <z * A- . i 4- '- * * i a * A- 4 + ,'_ n 3 x- 6 . &c- j

1 6. And v/r^rr, .Ii* - .4-. ;,,. . c . and more -

over by adually dividing, it becomes

i -|- -i/^r + |^^4 -+- ^frx 6 , &c.
-4- T^ -f- T^ H- rV^ x

17. But thefe Operations, by due preparation, may very often
be abbreviated; as in the foregoing Example to find \/;_***'
if the Form of the Numerator and Denominator had not been the
fame, I might have multiply'd each by </ 1 bxx, which would

y^i -f-rt* 1 - ab x *

have produced & and the reft of the work might

I b x x

have been performed by extracting the Root of the Numerator only,
and then dividing by the Denominator.

1 8. From hence I imagine it will fufficiently appear, by what
means any other Roots may be extracted, and how any compound
Quantities, however entangled with Radicals or Denominators, (fuch

Vx \fi xx V x i! 2x t x i v

as x"> -}- ; _. j may be reduced to

^/axx -\- A- 3 * x-{-xx " 2X x.1 '

infinite Series confifting of iimple Terms.

Of the ReduStion of offered Equations.

19. As to aftedled Equations, we mufl be fomething more par-
ticular in explaining how their Roots are to be reduced to fuch Se-
ries as thefe ; becaufe their Doctrine in Numbers, as hitherto de-
liver'd by Mathematicians, is very perplexed, and incumber'd with
fuperfluous Operations, fo as not to afford proper Specimens for per-
forming the Work in Species. I fhall therefore firfl (hew how the


Method of FLUXIONS,

Refolutidn of affected Equations may be compendioufly perform'd
in Numbers, and then I fhall apply the fame to Species.

20. Let this Equation _y l zy 5 = be propofed to be re-
folved, and let 2 be a Number (any how found) which differs from
the true Root lefs than by a tenth part of itfelf. Then I make
2 -\-p =y, and fubftitute 2 4-/> for y in the given Equation, by
which is produced a new Equation p> 4- 6p l 4- iop i =o,
whofe Root is to be fought for, that it may be added to the Quote.
Thus rejecting />> 4- 6// 1 becaufe of its fmallnefs, the remaining
Equation io/> i = o, or/>=o,i, will approach very near to
the truth. Therefore I write this in the Quote, and fuppofe
o, i 4- ^ =/>, and fubftitute this fictitious Value of p as before,
which produces q* 4- 6,3^ 4- 1 1,23? 4- 0,06 1 =o. And fince
1 1,23^ 4- 0,06 1 =o is near the truth, or ^= 0,0054 nearly,
(that is, dividing 0,06 1 by 11,23, ^ * many Figures arife as
there are places between the firft Figures of this, and of the prin-
cipal QmDte exclufively, as here there are two places between 2 and
0,005) I write 0,0054 in the lower part of the Quote, as being
negative; and fuppofing 0,0054 4- r=sg, I fubftitute this as
before. And thus I continue the Operation as far as I pleafe, in the
manner of the following Diagram :

y~' zy 5 =o


+ 2,09455148, &c. =y

Z+p=J>. + 7 *


4 zp

The Sum

-i + iop+6p* + p->

+ i/

+ o 3 ooi+ 0,035 +o, 5 5 2 + 2*
+ o, 06 + i 3 2 + 6,

+ 1, + 10,

1 he 6um

o, 061 -|- 1 1) 23 i + 6, 3 q * + 2*

o,oo54 + r= q. <ji

+ II,2 ??

+ 0,06 1

o, oooooo i f74^+ o,ooo0#7-4&V 0, 0tfai +)'

+ 0,00018370^ 0,06804: +^;?
0,060642 +11,23

+ o, 061

The Sum

+ 0,0005416 +II,l62r

0,000048^2 + * = r.



21. But the Work may be much abbreviated towards the end by
this Method, efpecially in Equations of many Dimenfions. Having
firft determin'd how far you intend to extract the Root, count fo
many places after the firft Figure of the Coefficient of the laft Term
but one, of the Equations that refult on the right fide of the Dia-
gram, as there remain places to be fill'd up in the Quote, and reject
the Decimals that follow. But in the laft Term the Decimals may
be neglected, after fo many more places as are the decimal places
that are fill'd up in the Quote. And in the antepenultimate Term
reject all that are after fo many fewer places. And fo on, by pro-
ceeding Arithmetically, according to that Interval of places: Or,
which is the fame thing, you may cut off every where fo many
Figures as in the penultimate Term, fo that their loweft places may
be in Arithmetical Progreffion, according to the Series of the Terms,
or are to be fuppos'd to be fupply'd with Cyphers, when it happens
otherwife. Thus in the prefent Example, if I defired to continue
the Quote no farther than to the eighth place of Decimals, when
I fubftituted 0,0054 -f- r for q, where four decimal places are
compleated in the Quote, and as many remain to be compleated, I
might have omitted the Figures in the five inferior places, which
therefore I have mark'd or cancell'd by little Lines drawn through
them ; and indeed I might alfo have omitted the firft Term r J ,
although its Coefficient be 0,99999, Thofe Figures therefore
being expunged, for the following Operation there arifes the Sum
0,0005416 -f- 1 1,1 62?% which by Divifion, continued as far as
the Term prefcribed, gives 0,00004852 for r, which compleats
the Quote to the Period required. Then fubtracting the negative
part of the Quote from the affirmative part, there arifes 2,09455148
for the Root of the propofed Equation.

22. It may likewife be obferved, that at the beginning of the
Work, if I had doubted whether o, i -f-/> was a fufficient Ap-
proximation to the Root, inftead of iof> i = o, I might have
fuppos'd that o/** -f- i op i = o, and fo have wrote the firft
Figure of its Root in the Quote, as being nearer to nothing. And
in this manner it may be convenient to find the fecond, or even
the third Figure of the Quote, when in the fecondarjr Equation,
about which you are converfant, the Square of the Coefficient of
the penultimate Term is not ten times greater than the Product of
the laft Term multiply'd into the Coefficient of the antepenulti-
mate Term. And indeed you will often fave fome pains, efpecially
in Equations of many Dimensions, if you feek for all the Figures


8 Tie Method of FLUXION'S,

to be added to the Quote in this manner ; that is, if you extract the
lefier Root out of the three lafl Terms of its fecondary Equation :
For thus you will obtain, at every time, as many Figures again
in the Quote.

23. And now from the Refolution of numeral Equations, I mall
proceed to explain the like Operations in Species; concerning which,
it is neceflary to obferve what follows.

24. Firft, that fome one of the fpecious or literal Coefficients, if
there are more than one, fliould be diftinguifh'd from the reft, which
either is, or may be fuppos'd to be, much the leaft or greateft of
all, or neareft to a given Quantity. The reafon of which is, that
becaufe of its Dimeniions continually increafing in the Numerators,
or the Denominators of the Terms of the Quote, thofe Terms may
grow lefs and lefs, and therefore the Qtipte may conftantly approach
to the Root required ; as may appear from what is faid before of
the Species x, in the Examples of Reduction by Divifion and Ex-
traction of Roots. And for this Species, in what follows, I mall
generally make ufe of A: or z ; as alfo I fliall ufe y, p, q, r, s, &c.
for the Radical Species to be extracted.

25. Secondly, when any complex Fractions, or furd Quantities,
happen to occur in the propofed Equation, or to arife afterwards
in the Procefs, they ought to be removed by fuch Methods as
are fufficiently known to Analyfts. As if we mould have

y* -+- j 1 >' 1 x"= = o,. multiply by b x, and from the Pro-
duct by* Kyi'-l-fry* bx^ -+ x*-= o extract the Root y. Or

we might fuppofe y x b x=v, and then writing ^~ x for y t
we mould have i; J -+- &*v* fax* -\- 3/5*** ^hx' -+. x 6 = o,.
whence extracting the Root v r we might divide the Quote by b x,,
in order to obtain y. Affo if the Equation j 3 xy* -f- x$ = o
were propofed, we might put y?= v, and x j = z, and fo wri-
ting vv for y, and z* for x, there will arife v 6 z=v -f- z* = o ;
which Equation being refolved, y and x may be reftored. For the
Root will befound^=2-f-s3_|_5~s 5 5cc.andrei1:onngjyandA;, we have
y* = x^ -f- x -+- 6x^ &c. dien fquaring, y =x^-+- 2X J ~f- 13*", &c..

26. After the fame manner if there mould be found negative Di-
menfions ofx and jy, they may be removed by multiplying by the fame
x andjy. As if we had the Equation x*-}-T > x*-y~ I '2.x~ I i6y- 3 =o,
multiply by x and j 3 , and there would arife x*y* -+- 3# 3 jy 1 2_v 5

A J -r 1 -r-v aa 2ai i 1 a 4

O. And U tjie Equation were x = ~ + ?r

y \.


by multiplying into jy } there would arife xy*-=.a' i y*
And fo of others.

27. Thirdly, when the Equation is thus prepared, the work be^
gins by finding the firfr. Term of the Quote ; concerning which, as
alfo for finding the following Terms, we have this general Rule,
when the indefinite Species (x or 2) is fuppofed to be fmall ; to
which Caie the other two Cafes are reducible.

28. Of all the Terms, in which the Radical Species (y,/>, q, or
r, &c.) is not found, chufe the loweft in refpect of the Dimenlions
of the indefinite Species (x or z, &c.) then chufe another Term in
which that Radical Species is found, fuch as that the Progreflion of
the Dimenfions of each of the fore-mentioned Species, being con-
tinued from the Term fir ft afTumed to this Term, may defcend as
much as may be, or afcend as little as may be. And if there
are any other Terms, whofe Dimenfions may fall in with this
Progreflion continued at pleafure, they muft be taken in 1 ike-
wife. Laftly, from thefe Terms thus felected, and made equal to
nothing, find the Value of the faid Radical Species, and write it in
the Quote.

29. But that this Rule may be more clearly apprehended, I fhall
explain it farther by help of the following Diagram. Making a
right Angle BAC, divide its fides AB, AC, into equal parts, and
raifing Perpendiculars, diftribute the Angular Space into equal Squares
or Parallelograms, which you may conceive to be denominated from
the Dimenfions of the Species x and y,

as they are here infcribed. Then, when

any Equation is propofed, mark fuch of

the Parallelograms as correfpond to all

its Terms, and let a Ruler be apply'd

to two, or perhaps more, of the Paralle-

lograms fo mark'd, of which let one

be the loweft in the left-hand Column at AB, the other touching

the Ruler towards the right-hand ; and let all the reft, not touching

the Ruler, lie above it. Then felecl: thofe Terms of the Equation

which are reprefented by the Parallelograms that touch the Ruler,

and from them find the Quantity to be put in the Quote.

30. Thus to extract the Root y out of the Equation y 6 5xy s -+-

)'* ja*x 1 y 1 +6a i x*-\-& 1 x4=o, I mark the Parallelograms belong-



A 4








A? 3

A 5 4




**. 3




A - ;





s 1




The Method of FLUXIONS,





ing to the Terms of this Equation
with the Mark #, as you fee here
done. Then I apply the Ruler
DE to the lower of the Parallelo-
grams mark'd in the left-hand
Column, and I make it turn round
towards the right-hand from the
lower to the upper, till it begins
in like manner to touch another,
or perhaps more, of the Parallelograms that are mark'd ; and I fee
that the places fo touch'd belong to x 3 , x*-y* y and_y 5 . Therefore
from the Terms y 6 7a z x*-y <L -}-6a*x*, as if equal to nothing, (and
moreover, if you pleafe, reduced to v 6 7^*4- 6= o, by making
$=rv'\fitx t ) I feek the Value of y, and find it to be four- fold,
-\-</ax, </ax, -+-</2ax, and ^/2ax, of which I may take
any one for the initial Term of the Quote, according as I defign to
extract this or that Root of the given Equation.

31. Thus having the Equation y* 6y*-i-()&x* x 3 =o, I chufe
the Terms by- -\-gbx*-, and thence I obtain 4-3* for the initial
Term of the Quote.

32. And having y">-i-axy-{-aay x* 2rt 3 =o, I make choice of
y'-i-a^y 2<2 3 , and its Root -\-a I write in the Quote.

33. Alfo having x*y s ^c^xy 1 c I .v a 4- 7 =o, I felect vV i y f 4-<r 7 J

which gives ^/ c for the firft Term of the Quote. And the

like of others.

34. But when this Term is found, if its Power fhould happen
to be negative, I deprefs the Equation by the fame Power of the
indefinite Species, that there may be no need of depreffing it in the
Refolution ; and befides, that the Rule hereafter delivei'd, for the
fuppreffion of fuperfluous Terms, may be conveniently apply'd.
Thus the Equation 8z; 6 _) i3 4-^2 5 >' a 27^5=0 being propofed, whofe

Root is to begin by the Term ^ I deprefs by s% that it may be-
come Sz+yt-^azy 2ja !> z~ 1 =o, before I attempt the Refolu-

3 5. The fubfequent Terms of the Quotes are derived by the fame
Method, in the Progrefs of the Work, from their feveral fecondary
Equations, but commonly with lefs trouble. For the whole affair
is perform'd by dividing the loweft of the Terms affected with the
indefinitely fmall Species, (x, x 1 , x 3 , &c.) without the Radical Spe-
(/>, q, r } &c.) by the Quantity with which that radical Species

i of


of one Dimenfion only is affected, without the other indefinite Spe-
cies, and by writing the Refult in the Quote. So in the following

Example, the Terms -> ~ } - ~> &c. are produced by dividing

a l x, TrW", TTT-v 3 , &c. by ^aa.

36. Thefe things being premifed, it remains now to exhibit the
Praxis of Refolution. Therefore let the Equation y*-{-a z y-\-axy
za* x z =o be propofed to be refolved. And from its Terms
y=-\-a*y 2 3 =o, being a fictitious Equation, by the third of the
foregoing Premifes, I obtain y a=o, and jtherefore I write -{-a in
the Quote. Then becaufe -\~a is not the compleat Value ofy, I put
a+p=y, and inftead of y, in the Terms of the Equation written
in the Margin, I fubftitute a-\-p, and the Terms refulting (/> 3 -{-
3rf/ 1 -f-,?,v/>, &c.) I again write in the Margin ; from which again,
according to the third of the Premifes, I felect the Terms -+-^p
-H2 l .v=o for a fictitious Equation, which giving p= ^x, I
write ~x in the Quote. Then becaufe ^.v is not the accurate
Value of p, I put x-\-q=p, and in the marginal Terms for p
I fubftitute ^x-t-q, and the refulting Terms (j 3 -^x^+^a^, &c.)
I again write in the Margin, out of which, according to the fore-
going Rule, I again feledl the Terms 4^ _ I 3 -drx*=o for a ficti-
tious Equation, which giving =^> I write -^ in the Quote.
Again, fince ^ is not the accurate Value of g, I make -^ - {-r=q t
and inftead of a I fubftitute ~ - \-r in the marginal Terms. And

&4 '

thus I continue the Procefs at pleafare, as the following Diagram
exhibits to view.


Method of FLUXIONS,

X 3






- X*

T '




'31** 509*4

37. If it were required to continue the Quote only to a certain
Period, that x, for inilance, in the laft Term {hould not afcend
beyond a given Dimenfion ; as I fubftitute the Terms, I omit fuch as
I forefee will be of no ufe. For which this is the Rule, that after
the firft Term refulting in the collateral Margin from every Quan-
tity, fo many Terms are to be added to the right-hand, as the In-
dex of the higheft Power required in the Quote exceeds the Index
of that firft refulting Term.

38. As in the prefent Example, if I defired that the Quote, (or
the Species .v in the Quote,) mould afcend no higher than to four
Dimenfions, I omit all the Terms after A-*, and put only one after x=.



Therefore the Terms after the Mark * are to be conceived to be
expunged. And thus the Work being continued till at laft we come

to the Terms -^ -^ - H-rfV axr,'m which />, q, r, or

reprefenting the Supplement of the Root to be extracted, are only
of one Dimenfion ; we may find fo many Terms by Divifion,

131*3 _, 509*4 \ as we fl^n e want j n g to compleat the Quote.

16384(13 /




... XX 13 1.*' kuyAT _

So that at laft we {hall have y=a 7*-f"6^-t-^l~*- r^I; icc -

39. For the fake of farther Illustration, I mail propofe another
Example to be refolved. From the Equation -L_y< .Ly4_f_iy3 iy=.
_^_y z=o, let the Quote be found only to the fifth Dimenfion,
and the fuperfluous Terms be rejected after the Mark,

_!_ 5j &c.

+ ^ 5 , &c.

-L;S 4 Z'p, &C.



s , &c.
% &c.

40. And thus if we propofe the Equation T 4-rjrJ' '+TT|-T )'' +
-rT T ;' 7 -t-TW' J -i-r.)' 3 +y =o, to be refolved only to the ninth Di-
menfion of the Quote ; before the Work begins we may reject the
Term -^^y" ; then as we operate we may reject all the Terms
beyond 2', beyond s 7 we may admit but one, and two only after

Y4 The Method of FLUXIONS,

z f ; becaufe we may obferve, that the Quote ought always to afcerrd
by the Interval of two Units, in this manner, z, .s j , z s , &c. Then
at laft we fliall have ;'=c fs3_j__|_. s _ T _5__ 2; ^_ J _^_' T ^_. 3 9 ) & C .
41. And hence an Artifice is difcover'd, by which Equations,
tho' affected hi injinitum, and confiding of an infinite number of
Terms, may however be refolved. And that is, before the Work
begins all the Terms are to be rejected, in which the Dimenfion of
the indefinitely fmall Species, not affected by the radical Species,
exceeds the greateft Dimenfion required in the Quote ; or from,
which, by fubftituting inftead of the radical Species, the firfl Term,
of the Quote found by the Parallelogram as before, none but fuch
exceeding Terms can arife. Thus in the laft Example I mould have
omitted all the Terms beyond y>, though they went on ad injini-
tum. And fo in this Equation

8 -f-3 1 4S 4 -f-92 lS l6 8 , &C.

) j' 1 in z* s 4 -}- z 6 z*y &c.

that the Cubick Root may be extracted only to four Dimenfions of z,
I omit all the Terms in infinitum beyond -f-j 5 in z, 1 J.-4_|_.L 2 >
and all beyond y- in z 1 a 4 -(-.c 6 , and all beyond -+-y in .c 1 2z 4 ,
and beyond S-}-;s tt 42 4 . And therefore I aflurr.e this Equation
only to be refolved, -^z 6 y* z*y* -{-?* ;> s 6 ^ 1 -}-^ 4 ^ 1 z^y* 2z*y
-i-z'-y 4s4_j_ s i 8=0. Becaufe?. ',(*''- ~^ { ' Term of the Quote,)
being fubflituted inflead of y in the reft of the Equation deprefs'd
by z^y gives every where more than four Dimenfions.

42. What I have faid of higher Equations may alib be apply'd to
Qi\adraticks. As if I defired the Root of this Equation


.r 1 A* A 4 -

h-r-f - ; &c.

as far as the Period x f , I omit all the Terms in infinititm., beyond
y in <?_[-*+ ' and affume only this Equation, j* ay xy

2" \ 4

-y+ =0. This I refolve either in the ufual manner, by making

& 4-*-*


j-^; or more expedition fly by
the Method of affected Equations deliver'd before, by which we fhall
have _}'= 3 #> where the laft Term required vanifhes, or

becomes equal to nothing.

43. Now after that Roots are extracted to a convenient Period,
they may fometimes be continued at pleafure, only by oblerving the
Analogy of the Series. So you may for ever continue this z-t-i-z*
^_^.25_j__'_ 2; 4_{_ T i_2;s j &c. (which is the Root of the infinite Equa-
tion 5r==)'-f-^ i _j_^5_|_y4 j foe.) by dividing the laft Term by thefe
Numbers in order 2, 3, 4, 5, 6, &c. And this, z f^-H-rlo-^' '
yj l TB .27-f_ TrT ' TTy2 ;9 j &c. may be continued by dividing by thefe Num-

bers 2x3, 4x5, 6x7, 8x9, &c. Again, the Series

"-'g , &c. may be continued at pleafure, by multiplying the Terms
refpectively by thefe Fractions, f } 7, , -, T V, &c> And
fo of others.

44. But in difcovering the firft Term of the Quote, and fome-
times of the fecond or third, there may ftill remain a difficulty
to be overcome. For its Value, fought for as before, may happen
to be furd, or the inextricable Root of an high affected Equation.
Which when it happens, provided it be not alfo impoffible, you
may reprefent it by fome Letter, and then proceed as if it were
known. As in the Example y*-\-axy-{-ii*-y x 3 2a>=o : If the
Root of this Equation y^^-a'-y 2 5 =o, had been furd, or un-
known, I mould have put any Letter b for it, and then have per-
form'd the Refolution as follows, fuppofe the Quote found only to
the third Dimenfion.


fbe Method of FLUXIONS,

y s -\-aay-\rtxy 2 a 3 ;

, tf^A- 4jC ft

^=0. Make a - \-T,b 1 =c 2 , then

ii | (v*r*

rTTv* .8 ,8 ,10 .

AT 3

\-b~i -f-?^ i ^-j-2^/ :1 -f-/ )J

~" w :;

5;'3 A 3

' j &C.

A' 3

6<? 1 A.-^ C43.V 1

3i3,;S /,.4iA* ~X* 3 3 .%3

i z / 1 4

~* + t ( , h ^ r 8

45. Here writing in the Quote, I fuppofe b - p=y, and then
for y I fubftitute as you fee. Whence proceeds p'^-^bp 1 , &c. re-
jecting the Terms b'-^a'-b 2tf 3 , as being equal to nothing : For b
is fuppos'd to be a Root of this Equation jy3_j_ fl *y 2<? 3 =o. Then

the Terms ^p-^-a^p-^-abx give '/^V* : 1 to be fet in the Quote,, and

to be fubflituted for p.

46. But for brevity's fake I write a- for aa-^-^l>l>, yet with this
caution, that aa-\-^bb may be reflored, whenever I perceive that
the Terms may be abbreviated by it. When the Work is finim'd,
I aflume fome Number for a, and refolve this Equation y*-\-?.'-\'
2^ ; =o, as is fhewn above concerning Numeral Equations ; and I
fubftitute for b any one of its Roots, if it has three Roots. Or
rather, I deliver fuch Equations from Species, as far as I can, efpe-
cially from the indefinite Species, and that after the manner before
insinuated. And for the reft only, if any remain that cannot be
expunged, I put Numbers. Thus y'-^-a^y 2^ 5 =o will be freed
from a, by dividing the Root by a, and it will become y*+)' 2=0,
whofe Root being found, and multiply'd by a, muft be fubftituted

inftead of b.



47. Hitherto I have fuppos'd the indefinite Species to be little.
But if it be fuppos'd to approach nearly to a given Quantity, for
that indefinitely fmall difference I put fome Species, and that being
fubftituted, I folve the Equation as before. Thus in the Equation
f}-' ^y* -+- ^y l y* -t-y -\-a x = o, it being known or fup-
pos'd that x is nearly of the fame Quantity as a, I fuppofe z to be
their difference; and then writing a-\-z or a z for x, there will
arife y y* -f- jj 5 y* -{-y + z=o, which is to be folved
as before.

48. But if that Species be fuppos'd to be indefinitely great, for
its Reciprocal, which will therefore be indefinitely little, I put fome
Species, which being fubflituted, I proceed in the Refolution as
before. Thus having y* -+-\ l -f-jv x> =o, where x is known
or fuppos'd to be very great, for the reciprocally little Quantity

- I put z, and fobflituting - for .v, there will arife y> -f-.)' 1 + y
~ =o, whofe Root is .y = ^ - ^z + z* -f- ^2', &c. where
x being reflored. if you pleafe, it will be y=:x - H- H

J * 3 9* 8 i**

&c '

49. If it fhould happen that none of thefe Expedients mould
fucceed to your defire, you may have recourfe to another. Thus
in the Equation y* x^y 1 -+- xy* -f- Z) 1 2y -+- i = o, whereas
the firft Term ought to be obtain'd from the Suppofition that
jy-4_j_2yt 2 y + 1 = 0, which yet admits of no poffible Root;
you may try what can be done another way. As you may fuppofe
that x is but little different from + 2, or that 2-{-z-=x. Then
fubftituting 2-{-z inftead of A*, there will arife y* z'-y* -\zy*
2y -f- 1 = 0, and the Quote will begin from -j- i. Or if you

fuppole x to be indefinitely great, or l - = z, you will have ^ 4

>* y 1

- { - -+-2y* 2y H- i = o, and -f- z for the initial Term of
the Quote. ,

50. And thus by proceeding according to feveral Suppofitions,
you may extract and exprefs Roots after various ways.

51. If you mould delire to find after how many ways this
may be done, you mufl try what Quantities, when fubfHtuted for
the indefinite Species in the propofed Equation, will make it divifible
by_y, -f-or fome Quantity, or by^ alone. Which, for Example

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