Isaac Newton.

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

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and therefore iox= i, or ,v = o, i nearly. This added tog or 2,
makes a new = 2,1, and x being ftill the Supplement, 'tis y =
2,1 +x, which being fubftituted in the original Equation _y 5 zy
= 5, produces 11,23^-4- 6,3** + x 3 = 0,6 1, to determine the,
new Supplement x. He rejects the Powers of x, and thence derives
^___oj __ 0,0.054, and confequently y = 2,0946, which

I 1 ,25

not being exaft, becaufe the Powers of x were rejected, he makes
the Supplement again to be x, fo that y= 2,0946 -f- x, which be-
ing fubftituted in the Original Equation, gives 11,162^-+- &c. =
0,00054155. Therefore to find the third Supplement x, he has

.v =" ,' 1 6 5 2 4 ' 5S = 0,00004852, fo that y =.2,0946 + *=
2,09455148, &c. and To on.



By this Procefs we may fee how nearly thefe two Methods agree,
and wherein they differ. For the difference is only this, that our
Author conftantly profecutes the Refidual or Supplemental Equations,
to find the firft, fecond, third, &c. Supplements to the Root : But
Mr. Raphjbn continually corrects the Root itfelf from the fame fup-
plementaf Equations, which are formed by fubftituting the corrected
Roots in the Original Equation. And the Rate of Convergency will 1
be the fame in both.

In imitation of thefe Methods, we may thus profecute this In-
quiry after a very general manner. Let the given Equation to be
refolved be in this form ay m -+- by"-* -4- cy-* -J- dy m ~* , &c. = o, in
which fuppofe P to be any near Approximation to the Root y, and
the little Supplement to be p. Then is y = P -4-/>. Now from
what is (hewn before, concerning the raifing of Powers and extrac-
ting Roots, it will follow that y m = P -h/> I m = P* -f- wP m -'/>, &c.
or that thefe will be the two firft Terms of y m ; and all the reft, ,
being multiply'd into the Powers of />, may be rejected. And for
the fame reafon y m ~ l = P m ~ I -h m iP m ~ l p, &c. y m ~ l = P m ~- -+
m 2 P"-=p, &c. and fo of all the reft. Therefore thefe being fub-
ftituted into the Equation, it will be
a ]>> .4- niaP n - l p , &c."l
~ -, &c.

m 2c n ~*p, &c. >= o ; Or dividing by P" ,
m 7 JP "-"</>, &c.


'- -j-^/P-s , &c.

-\-m ^dP~*p, &c. = o. From whence taking the Value of />,
we mail have/ = - + *P-' + cP- + rfp-* . ar,. _ and

ma?- 1 + lbV~' -{.m z^~3 + m J^P-4 , Jjff .

confequently r=

^ J


To reduce this to a more commodious form, make Pi= - , whence

P=A-'B, P- I =A- i B% &c. which being fubftituted, and
alfo multiplying the Numerator and Denominator by A" 7 , it will be

~ ~ ~ 'B +" A."-"-B*+ =4rfA"-?Bi. ^c-. will

be a nearer Approach to the Rootjy, than jp or P, and fo much



77je Method of FLUXIONS,

the nearer as ' is near the Root. And hence we may derive a very

convenient and general Theorem for the Extraction of the Roots of
Numeral Equations, whether pure or affected, which will be this.
Let th,e general Equation ay m -^- by" 1 1 -+ cy m ~~- -f- d) m ~ s , &c.

=: o be propofed to be folved ; if the Fraction - be affumed
as near the Root y as conveniently may be, the Fraction

. iAA m 1 3 1 +7 zcA m -S F 3 4. z 3n/A 4B4,'fef f .

nearer Approximation to the .Root. And this Fraction, when com-
puted, may,be,ufed inflead of the Fraction - , by which means a

Bearer Approximation may again -be had ; and fo on, till we ap-
proach as near the true Root as we pleafe.

This general Theorem may be conveniently refolved into as many
particular Theorems as we pleafe. Thus in the Quadratick Equa-

A 1 -4- rB z

tion y 1 + by === c, it will be y = , ,"7 D p , fere. In the Cubick

if if * f 2t\ J DO X D

. . .... 2 A

Equation y* + ty + cy = d, it will be y == 3<i .
y^r^. In the Biquadratick Equation y* -{- by* -\- cy 1 -+-dy=ze, it

irt* -+- 2/)AB-4- iB~ x A 1 -f- rB4 _ . 111-1 c i l_

be ^ == >/^' And the llke of hl S her


; For an 'Example of the Solution of a Quadratick Equation, let propofed to extract the Square-root of 12, or let us find the
value of_y in this Equation y 1 #= 1.2. Then by comparing with
the general formula, we fliall have b =. o, and <: = 12. And


taking 3 for the firft approach to the Root, or making g =T>
that is, Az=3 and B;= i, we fliall have by Substitution y ^==.
^~ =4-, fora nearer Approximation. Again, making A = 7

and B = 2, we fliall have y = 12l == || for a nearer Approxi-

14 X 2

mation. Again, making A = 97 and B = 28, we fliall have _y=
97j + i:! x 28i . __ lil7 fo r a nearer Approximation. Aeain,

'94* 2 S45 2 _ _ _

making A= 18817 and B =543 2 > we ^ a11 have y= '

7o8ic8o77 /- A i -r i

== 1|^ ior a nearer Approximation. And if we go on in the

fame method, we may find as near an Approximation to the Root as
y/e pleafe,



This Approximation will be exhibited in a vulgar Fraction, which,
if it be always kept to its loweft Terms, will give the Root of the
Equation in the fhorteft and fimpleft manner. That is, it will al-
ways be nearer the true Root than any other Fraction whatever^
whofe Numerator and Denominator are not much larger Numbers
than its own. If by Divifion we reduce this laft Fraction to a De-
cimal, we mall have 3,46410161513775459 for the Square-root
of 12, which exceeds the truth by lefs than an Unit in the lall place.-

For an Example of a Cubick Equation, we will take that of our
Author _y j * 2? = 5, and therefore by Companion b = o,
=. 2, and d==. 5. And taking 2 for the firft Approach to the

Root, or making ^- = 4., that is, A = 2 and B=i, we mall

have by Subftitution y ==- = 44 f r a nearer Approach to

the Root. Again, make A = 21 and B = 10, and then we
mall have y = 9- 1 + 2500 __ Hj-L f or a nearer Approximation.

6615 1000 5615

Again, make A= 11761 and 6 = 5615, and we mall have

~ .

y = =

3x11761 1 ^561 5 2x5615 1 3 J 9759573 16 495

proximation. And fo we might proceed to find as near an Approxi'
mation as we think fit. And when we have computed the Root
near enough in a Vulgar Fraction, we may then (if we pleafe) re-
duce it to a Decimal by Divifion. Thus in the prefent Example we
fhall have ^ = 2,094551481701, &c. And after the fame manner
we may find the Roots of all other numeral affected Equations, of
whatever degree they may be.

SECT. IV. The Refolution of Specious Equations by infinite
Series ; and firft for determining the forms of the
Series^ and their initial Approximations.

23, 24. TTT^ROM the Refolution of numeral affected Equations,
J/ our Author proceeds to find the Roots of Literal, Spe-
cious, or Algebraical Equations alfo, which Roots are to be exhibited
by an infinite converging Series, confiding of fimple Terms. Or
they are to be exprefs'd by Numbers belonging to a general Arithme-
tical Scale, as has been explain'd before, of which the Root is de-
noted by .v or z. The affigning or chufing this Root is what he
means here, by diftinguiming one of the literal Coefficients from the
reft, if there are feverul. And this is done by ordering or difpofing


Method of FLUXIONS,

the Terms of the given Equation, according to the Dimenfions of
that Letter or Coefficient. It is therefore convenient to chufe fuch a
Root of the Scale, (when choice is allow'd,) as that the Series may
converge as faft as may be. If it be the leaft, or a Fraction lefs
than Unity, its afcending Powers muft be in the Numerators of the
Terms. If it be the greateft quantity, then its afcending Powers
muft be in the Denominators, to make the Series duly converge.
If it be very near a given quantity, then that quantity may be con-
veniently made the firft Approximation, and that fmall difference,
or Supplement, may be made the Root of the Scale, or the con-
verging quantity. The Examples will make this plain.

25, 26. The Equation to be refolved, for conveniency-fake, iliould
always be reduced to the fimpleft form it can be, before its Refo-
Jution be attempted ; for this will always give the leaft trouble. But
all the Reductions mention'd by the Author, and of which he gives
us Examples, are not always neceflary, tho' they may be often con-
venient. The Method is general, and will find the Roots of Equa-
tions involving fractional or negative Powers, as well as cf other
Equations, as will plainly appear hereafter.

27, 28. When a literal Equation is given to be refolved, in diftin-
guifhing or affigning a proper quantity, by which its Root is to con-
verge, the Author before has made three cafes or varieties ; all which,
for the fake of uniformity, he here reduces to one. For becaufe
the Series mull neceffarily converge, that quantity muft be as fmall
-,as poffible, in refpect of the other -quantities, that its afcending
Powers may continually diminim. If it be thought proper to chufe
the greateil quantity, inftead of that its Reciprocal muft be intro-
duced, which will bring it to the foregoing cafe. And if it approach
near to a given quantity, then their fmall difference may be intro-
duced into the Equation, which again will bring it to the firft cafe.
So that we need only purfue that cale, becaufe the Equation is al-
ways fuppos'd to be reduced to it.

But before we can conveniently explain our Author's Rule, for
finding the firft Term of the Series in any Equation, we muft con-
fider the .nature of thofe Numbers, or Expreffions, to which thefe
literal Equations are reduced, whofe Roots are required ; and in this
Inquiry we ihall be much aiTifted by what has been already difcourfed
of Arithmetical Scales. In affected Equations that were purely nume-
ral, the Solution of which was juft now taught, the feveral Powers
of the Root were orderly difpoied, according to a fingle or limple
Arithmetical Scale, which proceeded only in longum, and was there



fafficient for their Solution. But we muft enlarge our views in thefe
literal affected Equations, in which are found, not only the Powers
of the Root to be extracted, but alfo the Powers of the Root of the
Scale, or of the converging quantity, by which the Series for the
Root of the Equation is to be form'd ; on account of each of which
circumftances the Terms of the Equation are to be regularly difpofed,
and therefore are to conftitute a double or combined Arithmetical
Scale, which muft proceed both ways, in latum as well as in longum,
as it were in a Table. For the Powers of the Root to be extraded,
fuppofe y, are to be difpofed in longum, fo as that their Indices may
conftitute an Arithmetical Progreffion, and the vacancies, if any,
may be fupply'd by the Mark #. Alfo the Indices of the Powers
pf the Root, by which the Series is to converge, fuppofe x, are to
be difpofed in latum, fo as to conftitute an Arithmetical Progreffion,
and the vacancies may likewife be fill'd up by the fame Mark *,
when it hall be thought neceffary. And both thefe together will
make a combined or double Arithmetical Scale. Thus if the Equa-
tion y s $xy 4- i!y4 7* #/ 4- -6a* x* 4- fax* =a=-o, were given,
to find the Root y, the Terms may be thus difpofed :

y 6 y* V4 yS y* yl yo

= 0;

Alfo the Equation v f by* 4- gbx\ x* =o fhould be thus dif-
jpofed, in order to its Solution :

y' * * by* *


y *




# *


* -5*7*




* *


* *


* "~"7


* *

** 4



* *



* +6a*x*


* *


* 4~^*# 4

: 1-

4- tfx* r

And the Equation y* 4- axy -J- a*y # 2<z 3 = o thus :

jK J * + a*y za


* f = >
*' J

And the Equation x'y* y*xy* c'x* 4-^ = thus :

* *

* * * y*xy* * * ^ sss 0.

x*y' * *
And the like of all other Equations.

C c When

Method of FLUXIONS,

When the Terms of the Equation are thus regularly difpos'd, ft
is then ready for Solution ; to which the following Speculation will
be a farther preparation.

29. This ingenious contrivance of out' Author, (which we may
call Tabulating the Equation,) for finding the firft Term of the
Root, (which may indeed be extended to the finding all the Terms,
or the form of the Series, or of all the Series that may be derived
from the given Equation,) cannot be too much admired, or too care-
fully inquired into : The reafon and foundation of which may be
thus generally explained from the following Table, of which the
Construction is thus.



za b

za zb





a b




3 a

3 a zb




50 b

; 3






70+ zJ


In a Pfor.e draw any number of Lines, parallel and equidiftant, and
cthers_at; right Angles to them, fo as to divide the whole Space, as
far as is neceffary, into little equal Parallelograms. Aflume any one
of thefe,- in which write the Term o, and the Terms a, za, 30, 4.a,
&c. in-the fuceeeding Parallelograms to the right hand, as alfo the
Terms -*-^ 2a, 3^7, &c. to the left hand. Over the Term
o, in. the fame Column, write the Terms ^, zb, 3^, 4^, &c. fuc-
ceffively",' and the Terms b, zb, 3^, &c. underneath. And
thefe Ave ma^f call primary Terms. Now to infert its proper. Term
in any other afitgiVd. Parallelogram, add the two primary 'Terms
together.., that' ftand over-againft if each- way, and write the Sum
in the given Parallelogram. And-* thus all the Parallelograms be-
ing fill's, as-far as there is oecafion every way, the whole. Space



-will become a Table, which may be called a combined Arithmetical
ProgreJJion in piano, compofed of the two general Numbers a and t\
of which thefe following will be the chief properties.

Any Row of Terms, parallel to the primary Series o, a, za, ^a,
&c. will be an Arithmetical Progreflion, whofe common Difference
is a ; and it may be any fuch Progreflion at pleafure. Any Row or
Column parallel to the primary Series o, , zb, 3^, &c. will be an
Arithmetical Progreflion, whofe common difference is ^j and it may
be any fuch Progreflion. If a ftr-ait Ruler be laid on the Table,
the Edge of which mall pafs thro' the Centers of any two Parallelo-
grams whatever ; all the Terms of the Parallelograms, whofe Cen-
ters mail at the fame time touch the Edge of the Ruler, will conftitute
an Arithmetical Progreflion, whofe common difference will coniiit of
two parts, the firfl of which will be fome Multiple of a, and the other
a Multiple of b. If this Progreflion be fuppos'd to proceed injeriora.
verjus, or from the upper Term or Parallelogram towards the lower ;
each part of the common difference may be feparately found, by fub-
tracling the primary Term belonging to the lower, from the primary
Term belonging to the upper Parallelogram. If this common diffe-
rence, when found, be made equal to nothing, and thereby the Re-
lation of a and b be determined ; the Progreflion degenerates into a
Hank of Equals, or (if you pleafe) it becomes an Arithmetical Progref-
fion, whofe common difference is infinitely little. In which cafe, if
the Ruler be moved by a parallel motion, all the Terms of the Parallelo-
grams, whofe Centers mall at the fame time be found to touch the Edge
of the Ruler, fhall be equal to each other. And if the motion of
the Ruler be continued, fuch Terms as at equal diftances from the
firfl: fituation are fuccerTively found to touch the Ruler, fliall form
an Arithmetical Progreflion. Laftly, to come nearer to the cafe in
hand, if any number of thefe Parallelograms be mark'd out and di-
flinguifh'd from. the reft, or aflign'd promifcuoufly and at pleafure,
through whofe Centers, as before, the Edge of the Ruler ihall fuc-
ceflively pafs in its parallel motion, beginning from any two (or more)
initial or external Parallelograms, :whofe Terms are made equal ; an
Arithmetical Progreflion may be found, which ihall comprehend and
take in all thofe promifcuous Terms, without any regard had to the
Terms that are to be omitted. Thefe are fome of the properties of
this Table, or of a combined Arithmetical Progreflion in piano >, by
. which we may eafily underfland our Author's expedient, of Tabu-
lating the given Equation, and may derive the neceflary Confequen-
~es from it.

C c 2 For

196 The Method of FLUXIONS,

For when the Root y is to be extracted out of a given Equation,
confifting of the Powers of y and x any how combined together-
promifcuoufly, with other known quantities, of which x is to be
the Root of the Scale, (or Series,) as explain'd before ; fuch a value
of y is to be found, as when fubftituted in the Equation inftead of
y, the whole (hall be deftroy'd, and become equal to nothing. And
firft the initial Term of the Series^or the firfl Approximation, is to be
found, wtyich in all cafes may be Analytically reprefented by Ax* 1 ;
or we may always put y = Ax m , &c. So that we mail have y 1 =
A*x, &c. _>' 3 = A*x* m , 6cc. _}> 4 = A 4 * 4 *, &c. And fo of other
Powers or Roots. Thefe when fubftituted in the Equation, and by
that means compounded with the feveral Powers of x (or z} already
found there, will form fuch a combined Arithmetical Progreffion in
flano as is above defcribed, or which may be reduced to fuch, by
making a=m and = I. Thefe Terms therefore, according to
the nature of the Equation, will be promifcuoufly difperfed in the
Table j but the vacancies may always be conceived to be fupply'd,
and then it will have the properties before mention'd. That is, the
Ruler being apply'd to two (or perhaps more) initial or external
Terms, (for if they were not external, they could not be at the be-
ginning of an Arithmetical Progreffion, as is neceflarily required,)
and thofe Terms being made equal, the general Index m will thereby
be determined, and the general Coefficient A will alfo be known.
If the external Terms made choice of are the loweft in the Table,
which is the cafe our Author purfues, the Powers of x will proceed
by increafing. But the higheft may be chofen, and then a Series
will be found, in which the Powers of x will proceed by decreafing.
And there may be other cafes of external Terms, each of which will
eommonly afford a Series. The initial Index being thus found, the
other compound Indices belonging to the Equation will be known
alfo, and an Arithmetical Progreffion may be found', in which they
are all comprehended, and confequently the form of the Series wifll
be known.

Or inftead of Tabulating the Indices of the Equation, as above,
it will be the fame thing in effedt, if we reduce the Terms themfelves
to the form of a combined Arithmetical Progreffion, as was fhewn
before. But then due care mufl be taken, that the Terms may be
rightly placed at equal diftances j otherwife the Ruler cannot be ac-
tually apply'd, to difcover the Progreffions of the Indices, as may
be done in the Parallelogram.




For the fake of greater perfpicuity, we will reduce our general
Table or combined Arithmetical Progreffion in piano, to the parti-
cular cafe, in which a-=.m and b=. i -, which will th.n appear
thus :

- 2M+0

zm-if 3

2m 3

_,, +5


- AO-f- 2

m-\- 1

m 3

+ 5

+ 4
+ 3







< 3



2OT+ 1


> 3

4'"+ 4



5 M-6



601 3


7 m +4

Now the chief properties of this Table, fubfervient to the prefent
purpofe, will be thefe. If any Parallelogram be feledted, and an-
other any how below it towards the right hand, and if their included
Numbers be made equal, by determining the general Number m,
which in this cafe will always be affirmative ; alfo if the Edge of the
Ruler be apply 'd to the Centers of thefe two Parallelograms ; all the
Numbers of the other Parallelograms, whofe Centers at the fame time
touch the Ruler, will likewife be equal to each other. Thus if the
Parallelogram denoted by m -+- 4 be feleded, as alfo the Parallelo-
gram 377* -f- 2 ; and if we make m -t- 4 = ^m -H 2, we mall have
m=i. Alfo the Parallelograms ;;; -h 6, m -f- 4, 3^7 -|- 2, $m,
j m 2 , &c. will at the fame time be found to touch the Edge of
the Ruler, every one of which will make 5, when m= i.

And the fame things will obtain if any Parallelogram be felecled,
and another any how below it towards the left-hand, if their in-
cluded Numbers be made equal, by determining the general Number
m, which in this cafe will be always negative. Thus if the Parallelo-
gram denoted by 5/w-i-4be felecled, as alfo the Parallelogram 402 -f- 2;
and if we make ^m-\-^.-=^.m -t-2, we fliall have>= 2. Alib
the Parallelograms 6w+6, 5^4-4, 4^ + 2, 3?;;, zm 2, 6cc.


198 7%e Method of FLUXIONS,

will be found at the fame time to touch the Ruler, every one of
which will make 6, when m = 2.

The fame things remaining as before, if from the firft fituation of
the Ruler it (hall move towards the right-hand by a parallel motion,
it will continually arrive at greater and greater Numbers, which at
equal diftances will form an afcending Arithmetical Progeffion. Thus
if the two firft felected Parallelograms be zm 1 = 5;;; 3, whence
m=., the Numbers in all the correfponding Parallelograms will
be -j. Then if the Ruler moves towards the right-hand, into the
parallel fituation %m-\- i, 6m I, &c. thefe Numbers will each be
3. If it moves forwards to the fame diftance, it will arrive at
4/7; -{-3, 7/ +- i, &c. which will each be 5^. If it moves forward
again to the fame diftance, it will arrive at yn -f- 5, %m -f- 3, &c.
which will each be 8f. And fo on. But the Numbers f, 3, 52.,
8y, &c. are in an Arithmetical Progreffion whofe common diffe-
rence is 2-i. And the like, mutatis mutandis, in other circum-

And hence it will follow <? contra, that if from the firft fituation
of the Ruler, it moves towards the left-hand by a parallel motion,
it will continually arrive at lefler and leifer Numbers, which at equal
diftances will form a decreafing Arithmetical Progreffion.

But in the other fituation of the Ruler, in which it inclines down-
wards towards the left-hand, if it be moved towards the right-hand
by a parallel motion, it will continually arrive at greater and greater
Numbers, which at equal diftances will form an increafing Arith-
metical Progreffion. Thus if the two firft feleded Numbers or Pa-
rallelograms be 8m + i = $m i, whence m = ~ ~ } and the
Numbers in all the correfponding Parallelograms will be 4!.. If
the Ruler moves upwards into the parallel fituation 5^-4-2, 2;;;, 8fc.
thefe Numbers will each be i f. If it move on at the fame diftance,
it will arrive at 2m + 3, m-+- i, 6cc. which will each be i-i. If it
move forward again to the fame diftance, it will arrive at m -f- 4,
4/;z -+- 2, &c. which will each be 4^. And fo on. But the Num-
bers 4,1, i|, i-i, 4.1, &c. or .Ll, i, |, -L, &c. are in an in-
creafing Arithmetical Progreffion, whofe common difference is , or 3.

And hence it will follow alfo, if in this laft fituation of the Ruler
it moves the contrary way, or towards the left-hand, it will conti-
nually arrive at lefler and lefler Numbers, which at equal diftances
will form a decreafing Arithmetical Progreflion.

Now if out of this Table we fhould take promifcuoufly any num-
ber of Parallelograms, in their proper places, with their refpeclive




Numbers included, neglecYmg all the reft ; we mould form fome cer-
tain Figure, fuch as this, of which thefe would be the properties.



5;;;+ 1

The Ruler being apply'd to any two (or perhaps more) of the
Parallelograms which are in the Ambit or Perimeter of the Figure,
that is, to two of the external Parallelograms, and their Numbers
being made equal, by determining the general Number m ; if the
Ruler paffes over all the reft of the Parallelograms by a parallel mo-
tion, thofe Numbers which at the fame time come to the Edge of the
Ruler will be equal, and thofe that come to it fuccefllvely will form
an Arithmetical Progreffion, if the Terms mould lie at equal diftan-
ces ; or atleaft-they may be reduced to fuch, by fupplyingany Terms
that may happen to be wanting.

Thus if the Ruler fhould be apply'd to the two uppermoft and
external Parallelograms, which include the Numbers 3/w-f-^ and
^m ~}_ 5, and if they be made equal, we mall have m = o, fo that
each of thefe Numbers will be 5. The next Numbers that the Ruler
will arrive at will be m -f- 3, 4;;; +3, 6/ -f- 3, of which each will

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