Isaac Newton.

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

. (page 21 of 30)
Online LibraryIsaac NewtonThe method of fluxions and infinite series : with its application to the geometry of curve-lines → online text (page 21 of 30)
Font size

Quantity fhould not be taken fo fmall, as to make the Series con-
verge very faft, yet it would however converge to the true Root,
tho' by more fteps and flower degrees. And this would obtain in
proportion, even if it were taken never fo large, provided we do
not exceed the due Limits of the Roots, which may be difcover'd,
either from the given Equation, or from the Root when exhibited
by a Series, or may be farther deduced and illuftrated by fome
Geometrical Figure, to which the Equation is accommodated.

So if the given Equation were yy =. ax xx, it is eafy to ob-
ferve, that neither^ nor x can be infinite, but they are both liable to

H h flv.rul

The Method of FLUXIONS,

ieveral Limitations. For if x be fuppos'd infinite, the Term ax
would vaniih in refpedt of xx, which would give the Value ofjyy
impoffible on this Supposition. Nor can x be negative; for then the
Value of yy would be negative, and therefore the Value of_y would
again become impoffible. If x = o, then is^ = o allb ; which is
one Limitation of both quantities. As yy is the difference between
ax and xx, when that difference is greateft, then will yy, and con-
fequently^, be greateft alfo. But this happens when x = a, as
alfo y = ftf, as may appear from the following Prob. 3. And in
general, when y is exprefs'd by any number of Terms, whether
finite or infinite, it will then come to its Limit when the difference
is greateft between the affirmative and negative Terms j as may ap-
pear from the fame Problem. This laft will be a Limitation for y t
but not for x. Laftly, when x = a, then_y = o; which will limit
both x and y. For if we fuppofe x to be greater than a, the ne-
gative Term will prevail over the affirmative, and give the Value of
yy negative, which will make the Value of y impoffible. So that
upon the whole, the Limitations of x in this Equation will be thefe,
that it cannot be lefs than o, nor greater than a, but may be of any
intermediate magnitude between thofe Limits.

Now if we refolve this Equation, and find the Value of y in an
infinite Series, we may ftill difcover the fame Limitations from
thence. For from the Equation yy = ax xx, by extracting the

3. _5.

fquare-root, as before, we fhall have y =. a^ L ~

za* Sa 1

X 1 ' i X ** * 3 O TT

- , c. that is, y == d*x* into i - , &c. Here

i6a z

x cannot be negative ; for then x? would be an impoffible quantity.
Nor can x be greater than a ; for then the converging quantity ~

or the Root of the Scale by which the Series is exprefs'd, would be
greater than Unity, and confequently the Series would diverge,, and
not converge as it ought to do. The Limit between converging and
diverging will be found, by putting x=a, and therefore y = o ;
in which cafe we fhall have the identical Numeral Series i = i
_l_ ^ -if. _' r , &c. of the fame nature with fome of thofe, which we
have elfewhere taken notice of. So that we may take x of any
intermediate Value between o and a, in order to have a converging
Series. But the nearer it is taken to the Limit o, fo much fafter
the Series will converge to the true Root ; and the nearer it is taken
to the Limit a, it will converge fo much the flower. But it will

however

4

and INFINITE SERIES. 235

'however converge, if A: be taken never fo little lefs than a. And by
Analogy, a like Judgment is to be made in all other cafes.

The Limits and other affe&ions of y are likewife difcoverable from
this Series. When x = o, then y = o. When x is a nafcent quan-
tity, or but juft beginning to be pofitive, all the Terms but the rirft
may be negledted, and y will be a mean proportional between a and x.
Alfo y = o, when the affirmative Term is equal to all the negative

Terms.or when i= - -f- - u- -? , &c. that is, when x = a.

z 8a* ib3 '

For then i = -f. 4. _f_ _ f rj &c. as above. Laftly, y will be a
Maximum when the difference between the affirmative Term and all
the negative Terms is greateft, which by Prob. 3. will be found
when x = ^a.

Now the Figure or Curve that may be adapted to this Equation,
and to this Series, and which will have the fame Limitations that
they have, is the Circle ACD, whofe Diameter is AD = a, its Ab-
/cifs AB = x, and its perpendicular Ordinate BC =.}' For as the
Ordinate BC=^ is a mean proportional
between the Segments of the Diameter
AB rrn x and BD = a x, it will be
yy ==. ax xx. And therefore the Ordi-
nate BC = _y will be exprefs'd by the fore-
going Series. But it is plain from the na-
ture of the Circle, that the Abfcifs AB cannot be extended back-
wards, fo as to become negative ; neither can it be continued for-
wards beyond the end of the Diameter D. And that at A and D,
where the Diameter begins and ends, the Ordinate is nothing. And
the greateft Ordinate is at the Center, or when AB = ^

SECT. VI. 'Trqnfitton fo the Method of Fluxions.

55. ' | "^HE learned and fagacious Author having thus accom-
plifh'd one part of his deiign, which was, to teach the
Method of converting all kinds of Algebraic Quantities into fimplc
Terms, by reducing them to infinite Series : He now goes on to
fhew the ufe and application of this Reduction, or of thefe Series,
in the Method of Fluxions, which is indeed the principal defign of
this Treadle. For this Method has fo near a connexion with, and
dependence upon the foregoing, that it would be very lame and
defective without it. He lays down the fundamental Principles of

H h 2 this

The Method of FLUXIONS,

this Method in a very general and fcientiflck manner, deducing
them from the received and known laws of local Motion. Nor is
this inverting the natural order of Science, as Ibme have pretended,
by introducing the Doctrine of Motion into pure Geometrical Spe-
culations. For Geometrical and. Analytical Quantities are belt con-
ceived as generated by local Motion; and their properties may as
well be derived from them while they are generating, as when their
generation is fuppos'd to be already accomplifh'd, in any other way.
A right line, or a curve line, is defcribed by the motion of a point,
a fmface by the motion of a line, a folid by the motion of a fur-
face, an angle by the rotation of a radius ; all which motions we
may conceive to be performed according to any ftated law, as occa-
fion (hall require. Thefe generations of quantities we daily fee to
obtain in rerum naturd, and is the manner the ancient Geometricians
had often recourfe to, in confidering their production, and then de-
ducing their properties from fuch adhial defcriptions. And by ana-
logy, all other quantities, as well as thefe continued geometrical
quantities, may be conceived as generated by a kind of motion or
progrefs of the Mind.

The Method of Fluxions then fuppofes quantities to be generated
by local Motion, or fomething analogous thereto, tho' fuch gene-
rations indeed may not be eflentially neceflary to the nature of the
thing fo generated. They might have an exiftence independent of
thefe motions, and may be conceived as produced many other ways,
and yet will be endued with the fame properties. But this concep-
tion, of their being now generated by local Motion, is a very fertile
notion, and an exceeding ufeful artifice for discovering their pro-
perties, and a great help to the Mind for a clear, diftincl:, and me-
thodical perception of them. For local Motion fuppofes a notion
of time, and time implies a fucceffion of Ideas. We eafily diflin-
guifh it into what was, what is, and what will be, in thefe ge-
nerations of quantities ; and fo we commodioufly confider thofe
things by parts, which would be too much for our faculties, and ex-
tream difficult for the Mind to take in the whole together, without
fuch artificial partitions and distributions.

Our Author therefore makes this eafy fuppofition, that a Line
may be conceived as now defcribing by a Point, which moves either
equably or inequably, either with an uniform motion, or elfe accor-
ding to any rate of continual Acceleration or Retardation. Velocity
is a Mathematical Quantity, and like all fuch, it is fufceptible of
infinite gradations, may be intended or remitted, may be increafed

or

and INFIN ITE SERIES. 237

or dlminifhfd in different parts of the fpace delcribed, according to
an infinite variety of fluted Laws. Now it is plain, that the fpace
thus defcribed, and the law of acceleration or retardation, (that is,
the velocity at every point of time,) mufl have a mutual relation
to each other, and muft mutually determine each other ; fo that
one of them being affign'd, the other by neceflary inference may be
derived from it. And therefore this is ftrictly a Geometrical Pro-
blem, and capable of a full Determination. And all Geometrical
Propoluions once demonftrated, 1 or duly investigated, may be fafely
made ufe of, to derive other Proportions from them. This will
divide the prefent Problem into two Cafes, according as either the
Space or Velocity is affign'd, at any given time, in order to find the
other. Arid this has given occasion to that diftin<5lion which has
lince obtain'd, of the dirctt and irrcerje Method of Fluxions, each of
which we fhall now confider apart.

56. In the direct Method the Problem is thus abftractedly pro-
pofed. From the Space defer i bed, being continually given, or affumed,
or being known at any point of Time ajjigrid ; to find the Velocity of the
Motion at that Time. Now in equable Motions it is well known,
that the Space defcribed is always as the Velocity and the Time of
defcription conjunclly ; or the Velocity is directly as the Spice de-
fcribed, and reciprocally as the Time of defcription. And even in
inequable Motions, or fuch as are continually accelerated or retarded,
according to fome ftated Law, if we take the Spaces and Times very
fmall, they will make a near approach to the nature of equable Mo-
tions ; and flill the nearer, the fmaller thole are taken. But if we
may fuppofe the Times and Spaces to be indefinitely fmall, or if
they are nafcent or evanefcent quantities, then we fhall have the Ve-
locity in any infinitely little Space, as that Space directly, and as the
tempufculum inverlely. This property therefore of all inequable Mo-
tions being thus deduced, will afford us a medium for folving the
prefent Problem, as will be fhewn afterwards. So that the Space
defcribed being thus continually given, and the whole time of its
defcription, the Velocity at the end of that time will be thence de-
terminable.

57. The general abflract Mechanical Problem, which amounts to
the lame as what is call'd the inverfe Method of Fluxions, will be
this. From the Velocity of the Motion being continually given, to de-
termine the Space defcribed, at any point of Time affign'd. For the
Solution of which we fhall have the afTiflance of this Mechanical
Theorem, that in inequable Motions, or when a Point defcribes a

Line

2 <>8 *!} Method of FLUXIONS,

Line according to any rate of acceleration or retardation, the indefi-
nitely little Spare defcribed in any indefinitely little Time, will be in
a compound ratio of the Time and the Velocity ; or thejpafiolum will
be as the velocity and the tempiijculum conjunctly. This being the
Law of all equable Motions, when the Space and Time are any finite
quantities, it will obtain allb in all inequable Motions, when the
Space and Time are diminiih'd in infinitum. For by this means all
inequable Motions are reduced, as it were, to equability. Hence the
Time and the Velocity being continually known, the Space delcribed
may be known alfo ; as will more fully appear from what follows.
ThisTroblem, in all its cafes, will be capable of a juft determina-
tion ; tho' taking it in its full extent, we mult acknowledge it to
be a very difficult and operofe Problem. So that our Author had
good reafon for calling it moleftijfimum & omnium difficilltmum pro-
blema.

58. To fix the Ideas of his Reader, our Author illuftrates his
general Problems by a particular Example. If two Spaces x and y
are defcribed by two points in fuch manner, that the Space x being
uniformly increafed, in the nature of Time, and its equable velocity
being reprefented by the Symbol x ; and if the Space y increafes in-
equably, but after fuch a rate, as that the Equation y =. xx ihall
always determine the relation between thofe Spaces j (or x being
continually given, y will be thence known ;) then the velocity of
the increafe of y fhall always be reprefented by 2xx. That is, if the
fymbol y be put to reprefent the velocity of the increafe of y, then
will the Equation y =. zxx always obtain, as will be (hewn hereafter.
Now from the given Equation y = xx, or from the relation of the
Spaces y and x, (that is, the Space and Time, or its representative,)
being continually given, the relation of the Velocities y=.2xx is
found, or the relation of the Velocity y, by which the Space increafes,
to the Velocity x, by which the reprefentative of the Time increales.
And this is an inftance of the Solution of the firft general Problem,
or of a particular Queftion in the direct Method of Fluxions. But
-.vice versa, if the kit Equation y = 2xx were given, or if the Ve-
locity y, by which the Space y is defcribed, were continually known
from the Time x being given, and its Velocity x , and if from thence.
we ihould derive the Equation y = xx, or the relation of the Space
and Time : This would be an inftance of the Solution of the fecond
.general Problem, or of a particular Queftion of the inverfe Method
of Fluxions. And in analogy to this defcription of Spaces by mov-
ing points, our Author confiders all other quantities whatever as ge-
nerated

and INFINITE SERIES. 239

nerated and produced by continual augmentation, or by the perpe-
tual acceffion and accretion of new particles of the fame kind.

59. In fettling the Laws of his Calculus of Fluxions, our Author
very fkilfully and judicioufly difengages himfelf from all confidera-
tion of Time, as being a thing of too Phyfkal or Metaphyfical a
nature to be admitted here, efpecially when there was no abfolute
neceffity for it. For tho' all Motions, and Velocities of Motion,
when they come to be compared or meafured, may feem neceflarily
to include a notion of Time; yet Time, like all other quantities,
may be reprefented by Lines and Symbols, as in the foregoing ex-
ample, efpecially when we conceive them to increafe uniformly.
And thefe reprefentatives or proxies of Time, which in fomc mea-
fiire may be made the objects of Senfe, will anfwer the prefent pur-
pofe as well as the thing itfclf. So that Time, in fome fenle, may
be laid to be eliminated and excluded out of the inquiry. By this
means the Problem is no longer Phyfical, but becomes much more
fimple and Geometrical, as being wholly confined to the defcription
of Lines and Spaces, with their comparative Velocities of increafe
and decreafe. Now from the equable Flux of Time, which we
conceive to be generated by the continual acceflion of new particles,
or Moments, our Author has thought fit to call his Calculus the
Method of Fluxions.

60, 6 1. Here the Author premifes fome Definitions, and other
neceflary preliminaries to his Method. Thus Quantities, which in
any Problem or Equation are fuppos'd to be fufceptible of continual
increafe or decreafe, he calls Fluents, or flowing Quantities ; which
are fometimes call'd variable or indeterminate quantities, becaufe they
are capable of receiving an infinite number of particular values, in
a regular order of fucceilion. The Velocities of the increafe or de-
creafe of fuch quantities are call'd their Fluxions ; and quantities in
the fame Problem, not liable to increafe or decreafe, or whofe Fluxions
are nothing, are call'd conftant, given, invariable, and determinate
quantities. This diftindlion of quantities, when once made, is care-
fully obferved through the whole Problem, and infinuated by proper
Symbols. For the firft Letters of the Alphabet are generally appro-
priated for denoting conftant quantities, and the laffc Letters com-
monly lignify variable quantities, and the fame Letters, being pointed,
repreient the Fluxions of thofe variable quantities or Fluents refpec-
tivcly. This diftinction between thefe quantities is not altogether
arbitrary, but has fome foundation in the nature of the thing, at
leafl during the Solution of the prefent Problem. For the flowing

or

24-O 7#* Method of FLUXIONS.

or variable quantities may be conceived as now generating by Motion,
and the conftant or invariable quantities as fome how o other al-
.ready generated. Thus in any given Circle or Parabola, the Diame-
ter or Parameter are conftant lines, or already generated ; but the
Abfcifs, Ordinate, Area, Curve-line, &c. are flowing and variable
quantities, becaufe they are to be underftood as now defcribing by
local Motion, while their properties are derived. Another diftinc-
tion of thefe quantities may be this. A conftant or given Irne in any
Problem is tinea qtitzdam^ but an indeterminate line is line a qua-vis
vel qutzcunque, becaufe it may admit of infinite values. Or laftiy,
conftant quantities in a Problem are thofe, whole ratio to a common
Unit, of their own kind, is fuppos'd to be known ; but in variable
quantities that ratio cannot be known, becaufe it is varying perpe-
tually. This diftinction of quantities however, into determinate and
indeterminate, fubfifts no longer than the prefent Calculation requires;
for as it is a diftinftion form'd by the Imagination only, for its own
conveniency, it has a power of abolifhing it, and of converting de-
terminate quantities into indeterminate, and vice versa, as occaiion
may require ; of which we fhall fee Inftances in what follows. In
a Problem, or Equation, theie may be any number of conftant quan-
tities, but there muft be at leaft two that are flowing and indeter-
minate ; for one cannot increafe or diminifh, while all the reft con-
tinue the fame. If there are more than two variable quantities in
a Problem, their relation ought to be exhibited by more than one
Equation.

ANNO-

( 241 )

ANNOTATIONS on Prob.i,

O R,

The relation of the flowing Quantities being given,
to determine the relation of their Fluxions.

SECT. I. Concerning Fluxions of the firft orcler^ and t(f
Jlnd their Equations.

HE Author having thus propofed his fundamental Pro-'
blems s in an abftra<ft and general manner, and gradually
brought them down to the form mod convenient for*
his Method ; he now proceeds to deliver the Precepts
of Solution, which he illuftrates by a fufficient variety of Examples,!
referving the Demonftration to be given afterwards, when his Rea-
ders will be better prepared to apprehend the force of it, and when
their notions will be better fettled and confirm'd. Theie Precepts
of Solution, or the Rules for finding the Fluxions of any given'
Equation, are very fliort, elegant, and compreheniive ; and appeal-
to have but little affinity with the Rules ufually given for this pur-
pofe : But that is owing to their great degree of univerfality. We
are to form, as it were, fo many different Tables for the Equation,
as there are flowing quantities in it, by difpofing the Terms accor-
ding to the Powers of each quantity, fo as that their Indices may'
form an Arithmetical Progreflion. Then the Terms are to be mul-
tiply'd in each cafe, either by the Progreflion of the Indices, or by '
the Terms of any other Arithmetical Progreflion, (which yet mould
.have the fame common difference with the Progreffion of the Indices ;) '

I i as

242 Tfo Method of FLUXIONS.

as alfo by the Fluxion of that Fluent, and then to be divided by
the Fluent itfelf. La ft of all, thefe Terms are to be collected, accor-
ding to their proper Signs, and to be made equal to nothing; which
will be a new Equation, exhibiting the relation of the Fluxions.
This procefs indeed is not fo fhort as the Method for taking Fluxions,
(to be given p relent ly v ) which he el fe where delivers, and which is
commonly follow' d ; but it makes fufficient amends by the univer-
lality of it, and by the great variety of Solutions which it will afford.
For we may derive as many different Fluxional Equations from the
lame given Equation, as we .(hall think fit to affume different Arith-
metical Progreffions. .Yet all thefe Equations will agree in the main,
and tho' differing in form, yet each will truly give the relation of
the Fluxions, as will appear from the following Examples.

2. In the firft Example we are to take the Fluxions of the Equa-
tion x> ax 1 -{- axy y"> = o, where the Terms are always
brought over to one fide. Thefe Terms being difpofed according
to the powers of the Fluent x, or being conlider'd as a Number ex-
prefs'd by the Scale whofe Root is x, will iland thus x> - ax 1 -f-
ayx* y>x = o; and affuming the Arithmetical Progrefiion 3, 2,
], o, which is here that of the Indices of x, and multiplying each
Term by each refpedlively, we fhall have the Terms jx 3 zax-
H- ayx * j which again multiply'd by i , or xx~ l , according to
the Rule, will make ^xx 1 2axx -f- ayx. Then in the fame Equa-
tion making the other Fluent/ the Root of the Scale, it will ftand
thus, _y 5 -f- oy*-i- axy 1 ax*y = o ; and affuming the Arith-

- >

metical Progreffion 3, 2, I, o, which alfo is the Progreffion of the
Indices of y, and multiplying as before, we fhall have the Terms

3_)' ; * -+- axy *, which multiply'd by - , or yy~*, will make

3i>' a -+ ax J- Tlien colle( ^i n g the Terms, the Equation yxx 1
zaxx + ayx tyy* -f- axj = o will give the required relation of the
Fluxions. For if we refolve this Equation into an Analogy, we fhall
have x : y : : 3>' 2 ax i^x 1 zax -h ay -, which, in all the values that
x and y can affume, will give the ratio of their Fluxions, or the
comparative velocity of their increafe or decreafe, when they flow
according to the given Equation.

Or to find this ratio of the Fluxions more immediately, or the
value of the Fraction 4' by fewer fteps, we may proceed thus. Write
down the Fraction ? with the note of equality after it, and in the

Numerator

and INFINITE SERIES. 243

Numerator of the equivalent Fraction write the Terms of the Equa-
tion, difpos'd according to x, with their refpective figns ; each be-
ing multiply'd by the Index of x in that Term, (increafed or di-
minifh'd, if you pleafe, by any common Number,) as alib divided
by .v. In the Denominator do the fame by the Terms, when dii-
pofed according to y, only changing the figns. Thus in the pre-
fent Equation x"' ax 1 -f- axy ;' 3 = o, we (lull have at once

y i,x*2ax-\-av +

* ~ J>* * ax *

Let us now apply the Solution another way. The Equation x ;

ax* -f- axy y* = o being order'd according to x as before,
will be x 1 ax* -(- ayx 1 y*x =. o ; and fuppofing the Indices
of x to be increas'd by an unit, or aifuming the Arithmetical Pro-
greffion -j- , -~~ t ^ , ~ , and multiplying the Terms refpectively,

we fhall have thefe Terms ^.xx* ^axx -}- zayx y-xx- 1 . Then
ordering the Terms according to /, they will become _)' 3 -f- oy 1
-\-axy f -i- x*y =.0; and fuppofing the Indices ofy to be diminifli'd
ax*

by an unit, or afluming the Arithmetical Progreffion ^ , L Si iJ,

.> y ' y ' y
and multiplying the Terms refpecYively, we mall have thefe Terms

2yy* * * x*yy- 1 + ax*yy~*. So that collecting the Terms,
we lhall have 4.v.v* -^axx +- 2 ayx y>xx~ l zyy* x'>yy-' -+-
ax*yy~* = o, for the Fluxional Equation required. Or the ratio

c ^1 T>1 -11 i y 4 X * T a -f-f- 2ay v'v * . . ,

of the Fluxions will be - = , -. 3_J : _ . w hich ratio

x Z) 2 * * -f-As, J ax l \< l

may be found immediately by applying the foregoing Rule.

Or contrary-wife, if we multiply the Equation in the fir ft form
by the Progreffion ~ , ? } ~ , ^ v , we flinll have the Terms zxx 1

axx * -\-ytx\- 1 . And if we multiply the Equation in the fc-
cond form by - , ^ , l y 5 y - , we fiiall have the Terms 4^* *,

H- zcixy -+- x=j}~ ! cx-yy~'. Therefore collecting 'tis a.v.v 1 ^v. v
+ rxx~> ^v}*+ 2axy-i-x>j}- > ~fix 1 y}-'~o. Or the ratio
of the Fluxions will be | = ^ ^ ~^:^^.,-r , which might
l.avc been found at once by the foregoing Rule.

And in general, if the Equation x"> - -ax % - axy y* o, in
the form x- a\- -f- <?.yv - > ; ,v = o, be multiply'd by the Terms
of this Arithmetical Progreffion " ;+ 3 v "L+J. ; r w ;n

O ) -v, .v JL \> 11

produce the Terms m -\-y.\-~ m-+-2n>:x-{- m -\- icxt mj'xx-'-,

I i 2 and

244 e ^ )e Method of FLUXIONS,

and if the fame Equation, reduced to the form y*-\-

_f- K\y= o, b; multiply 'd by the Terms of this Arithmetical Pro-

Online LibraryIsaac NewtonThe method of fluxions and infinite series : with its application to the geometry of curve-lines → online text (page 21 of 30)