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which it is adapted, and the Angle VAE
being a right Angle. Now any right Line
AKD being drawn through A, cutting the
Circle in K, and the Quadratrix in D, and
the perpendiculars KG, DB being let fall
to AE } call AV =.a, AG = c;, VK = x, and BD = y, and it

T 2 will

The Method of FLUXIONS.

will be as in the foregoing Example, x =.z 4- -r~ 4- j; 4- -
&c. Extract the root js, and there will arife z= x ^ 4-

, * 7 , &c. whofe Square fubtract from AKq. or a l s and the

s 4 " a 4

root of the remainder # 4- -^ , ; , &c. will be GK.

2^j 9Aa9 *7?r\/j9 *

Now whereas by the nature of the Quadratrix 'tis AB = VR = x,
and fince it is AG : GK :: AB : BD (y), divide AB x GK by AG,

and there will arife y = a ^ ^ -^ - , , &c. And thence,
(by Prob. i.) y = - ^ ^. ^ , &c. to the fquare of
which add i , and the root of the fum will be i 4- ^ -f-

'-J il il

_6o4^ &c ^ __ \vhence (by Prob. 2.) / may be obtain'd,

1Z /S~S U

or the Arch of the Quadratrix ; viz. YD = x 4- ^j -f

6 4 ' v7 &c.

895025

THE

THE

METHOD of FLUXIONS

AND

INFINITE SERIES;

O R,

A PERPETUAL COMMENT upon
the foregoing TREATISE,

u

.

;

THE

METHOD of FLUXIONS

AND

INFINITE SERIES.

ANNOTATIONS on the Introduction :

OR,

The Refolution of Equations by INFINITE SERIES.

S E c T. I. Of the Nature and ConftruElion of Infinite

or Converging Series.

great Author of the foregoing Work begins
it with a fhort Preface, in which he lays down
his main defign very concifely. He is not to be
here underftood, as if he would reproach the mo-
dern Geometricians with deferting the Ancients,
or with abandoning their Synthetical Method of
Demonftration, much lefs that he intended to difparage the Analy-
tical Art ; for on the contrary he has very nauch improved both
Methods, and particularly in this Treatife he wholly applies himfelf
to cultivate Analyticks, in which he has fucceeded to univerial ap-
plaufe and admiration. Not but that we mail find here fome ex-
amples of the Synthetical Method likewife, which are very mafterly
and elegant. Almoft all that remains of the ancient Geometry is
indeed Synthetical, and proceeds by way of demonftrating truths
already known, by mewing their dependence upon the Axioms, and

other

144 : -tb e Method of FLUXIONS,

other fir ft Principles, either mediately or immediately. But the
hiiinefs of Analyticks is to invcftiga'te fuch Mathematical Truths as
really are, or may be fuppos'd at leaft to be unknown. It afiumes
thofe Truths as granted, and argues from them in a general man-
ner, till after a .fcries of argumentation, in which the -feveral fteps
have a. neceftary. connexion wjth each other, it arrives at the know-
ledge of the propofition required, by comparing it with fomething
really known or given. This therefore being the Art of Invention,
it certainly deferves to be cultivated with the utmoft induftry. Many
of our modern Geometricians have been perfuaded, by confidering
the intricate and labour'd Demonftrations of the Ancients, that they
.were Mailers of an Analyfis purely Geometrical, which they ftudi-
ouily conceal'd, and by the help of which they deduced, in a direct
and fcientifical manner, thofe abftrufe Proportions we fo much ad-
mire in tome of their writings, and which they afterwards demon-
ftrated Synthetically. But however this may be, the lofs of that
Analyfis, if any fuch there were, is amply compenfated, I think,
by our prefent Arithmetical or Algebraical Analyfis, especially as it
is now improved, I might fay perfected, by our fagacious Author in
the Method before us. It is not only render 'd vaftly more univerfal,
and exterriive than that other in all probability could ever be, but is
likewife a moft compendious Analyiis for the more abftrufe Geome-
trical Speculations, and for deriving Conftructions and Synthetical
Demonftrations from thence ; as may abundantly appear from the
enfuing Treatife.

2. The conformity or correfpondence, which our Author takes
notice of here, between his new-invented Doctrine of infinite Series,
and the commonly received Decimal Arithmetick, is a matter of con-
fiderable importance, and well deferves, I think, to be let in 3. fuller
Light, for the mutual illuftration of both ; which therefore I fhall
here attempt to perform. For Novices in .this Doctrine, tJho' they
inay already be well acquainted with the Vulgar Arithmetick, and
with the Rudiments of the common Algebra, yet are apt to appre-
hend fomething abftrufe and difficult in infinite Series ; whereas in-
deed they have the fame general foundation as Decimal Arithmetick,
efpecially Decimal Fractions, and the fame Notion or Notation is only
tarry'd ftill farther, and rendered more univerfal. But to mew this
in fome kind of order, I muft inquire into thefe following particulars.
Firft I muft (hew what is the true Nature, and what are the genuine
Principles, of our common Scale of Decimal Arithmetick. Secondly
what is the nature of other particular Scales, which have been, or

may

and INFINITE SERIES. 145

may be, occasionally introduced. Thirdly, what is the nature of a
general Scale, which lays the foundation for the Doctrine of infinite
Series. Laftly, I ihall add a word or two concerning that Scale ot
Arithmetick in which the Root is unknown, and thcrefoi-e propofcd
to be found ; which gives occafion to the Doctrine of Affected Equa-
tions.

Firft then as to the common Scale of Decimal Arithmetick, it is
that ingenious Artifice of expreffing, in a regular manner, all con-
ceivable Numbers, whether Integers or Fractions, Rational or Surd,
by the feveral Powers of the number Ttv/, and their Reciprocals;
with the affiftance of other fmall Integer Numbers, not exceeding
Nine, which are the Coefficients of thofe Powers. So that Ten is
here the Root of the Scale, which if we denote by the Character X,
as in the Roman Notation and its feveral Powers by the help of this
Root and Numeral Indexes, (X 1 = 10, X 1 = ico, X 3 = 1000,
X 4 = 10000, &c.) as is ufual ; then by ailuming the Coefficients
o, i, 2, 3, 4, 5, 6, 7, 8, 9, as occafion (hall require, we may form or
exprefs any Number in this Scale. Thus for inflance 5X 4 -f- jX 3 -f-
4X 1 + 8X 1 -rf- 3X will be a particular Number exprefs'd by this
Scale, and is the fame as 57483 in the common way of Notation.
Where we may obferve, that this laft differs from the other way of
Notation only in this, that here the feveral Powers of X (or Ten)
are fupprefs'd, together with the Sign of Addition -f-, and are left
to be fupply'd by the Underftanding. For as thofe Powers afcend
regularly from the place of Units, (in which is always X, or i,
muhiply'd by its Coefficient, which here is 3,) the feveral Powers
will ealily be understood, and may therefore be omitted, and the
Coefficients only need to be fet down in their proper order. Thus
the Number 7906538 will (land for yX 6 -+- gX 5 -f- oX* -+-6X 3 -f-
^X* -f-3X' -f-3X, when you fupply all that is underftood. And
the Number 1736 (by fuppreffing what may be ealiiy -underftood,)
will be equivalent to X 3 -+- 7X 1 -f- 3X -f- 6 ; and the like of all other
Integer Numbers whatever, exprefs'd by this Scale, or with this
Root X, or Ten.

The fame Artifice is uniformly carry'd on, for the expreffing of
all Decimal Fractions, by means of the Reciprocals of the ll-vcral

Powers of Ten, fuch as ^ = o, i ; 5^1 = 0,0 1 ; ^ = 0,001 ; c.:c.
which Reciprocals may be intimated by negative Indices. Thus the
Decimal Fraction 0,3172 (lands for 3X~'-j- iX~~ : -f-7X -{- 2\~~ 4 i
and the mixt Number 526,384 (by {applying what is underfl ;

U becomes

Method <?/* FLUXIONS,

becomes 5 X 4 + 2X> -f- 6X -f- 3 X~' -f- 8X" 1 -f- 4 X- ; and the
infinite or interminate Decimal Fraction 0,9999999, &c. ftands for
9 X^' -f- gX- 1 -4- 9X~ 3 H- 9X~ 4 -f- 9 X~ 5 -+- yX~ & , &c. which infi-
nite Series is equivalent to Unity. So that by this Decimal Scale, (or
by the feveral Powers of Ten and their Reciprocals, together with
their Coefficients, which are all the whole Numbers below Ten,) all
conceivable Numbers may be exprefs'd, whether they are integer or
fracled, rational or irrational ; at leaft by admitting of a continual

And the like may be done by any other Scale, as well as the Deci-
mal Scale, or by admitting any other Number, befides Ten, to be
the Root of our Arithmetick. For the Root Ten was an arbitrary
Number, and was at firft aflumed by chance, without any previous
confideration of the nature of the thing. Other Numbers perhaps
may be affign'd, which would have been more convenient, and which
have a better elaim for being the Root of the Vulgar Scale of Arith-
metick. But however this may prevail in common affairs, Mathe-
maticians make frequent life of other Scales ; and therefore in the
fecond place I (hall mention fome other particular Scales, which
have been occafionally introduced into Computations.

The moft remarkable of thefe is the Sexagenary or Sexagefimal Scale
of Arithmetick, of frequent ufe among Aflronomers, which expreffes
all poffible Numbers, Integers or Fractions, Rational or Surd, by the
Powers of Sixty, and certain numeral Coefficients not exceeding fifty-
nine. Thefe Coefficients, for want of peculiar Characters to repre-
fent them, muit be exprefs'd in the ordinary Decimal Scale. Thus
if ftands for 60, as in the Greek Notation, then one of the/e Num-
bers will be 53^ -f- 9^' -+- 34!, or in the Sexagenary Scale 53", 9*,
34, which is equivalent to 191374 in the Decimal Scale. Again,
the Sexagefimal Fraclion 53, 9', 34", will be the fame as 53^= -f-
9|f+ 34~ z , which in Decimal Numbers will be 53,159444, &c.
aa infinitum. Whence it appears by the way, that fome Numbers
may be exprefs'd by a finite number of Terms in one Scale, which
in another cannot be exprefs'd but by approximation, or by a pro-
greffion of Terms in infinitum.

Another particular Scale that has been confider'd, and in fome
meafure has been admitted into practice, is the Duodecimal Scale,
which exprefles all Numbers by the Powers of Twelve. So in com-
mon affairs we fay a Dozen, a Dozen of Dozens or a Grofs, a Dozen
of GrofTes or a great Grofs, Off. And this perhaps would have been
the mod convenient Root of all otherSj by the Powers of which

to

and IN FINITE SERIES. 147

to conftruct the popular Scale of Arithmetick ; as not being fo lig
but that its Multiples, and all below it, might be eafily committed
to memory ; and it admits of a greater variety of Divifors than any
Number not much greater than itfelf. Befides, it is not fo fmall,
'but that Numbers exprefs'd hereby would fufficiently converge, or
by a few figures would arrive near enough to the Number required;
the contrary of which is an inconvenience, that muft neceflarily
attend the taking too fmall a Number for the Root. And to admit
this Scale into practice, only two fingle Characters would be wanting,
to denote the Coefficients Ten and Eleven.

Some have confider'd the Binary Arithmetick, or that Scale in
which TIDO is the Root, and have pretended to make Computations
by it, and to find considerable advantages in it. But this can never
be a convenient Scale to manage and exprefs large Numbers by, be-
caufe the Root, and confequently its Powers, are fo very fmall, that
they make no difpatch in Computations, or converge exceeding flowly.
The only Coefficients that are here necelTary are o and i. Thus
i x 2 5 -f- i x 2* -h o x2 3 + i x2* -f- i x 2' -f- 0x2 is one of thefe
Numbers, (or compendioufly 110110,) which in the common No-
tation is no more than 54. Mr. Leibnits imngin'd he had found
great Myfteries in this Scale. See the Memoirs of the Royal Academy
of Paris, Anno 1703.

In common affairs we have frequent recourfe, though tacitly, to
Millenary Arithmetick, and other Scales, whofe Roots are certain
Powers of Ten. As when a large Number, for the convenience of read-
ing, is diftinguifli'd into Periods of three figures: As 382,735,628,490.
Here 382, and 735, &c. may be confider'd as Coefficients, and the
Root of the Scale is 1000. So when we reckon by Millions, Billions,
Trillions, &c. a Million may be conceived as the Root of our Arith-
metick. Alfo when we divide a Number into pairs of figures, for
the Extraction of the Square-root ; into ternaries of figures for the
Extraction of the Cube-root ; &c. we take new Scales in effect, whofe
Roots are 100, 1000, &c.

Any Number whatever, whether Integer or Fraction, may be made
the Root of a particular Scale, and all conceivable Numbers may be
exprefs'd or computed by that Scale, admitting only of integral and
affirmative Coefficients, whofe number (including the Cypher c)
need not be greater than the Root. Thus in (Quinary Arithmetick,
in which the Scale is compofed of the Powers of the Root 5, the
Coefficients need be only the five Numbers o, i, 2, 3, 4, and yet all
Numbers whatever are expreffible by this Scale, at leaft by approxi-

U 2 mation,

j^B 77oe Method of FLUXIONS,

mation, to v/hat accu-racy we pleafe. Thus the common Number
2827,92 in this Arithmetick would be 4 x 5 4 -+- 2 x 5' -|- 3 x 5* -\~
ox5 I H-2x5-f-4x5~ I H-3x 5~ s ; or if we may fupply the feveral
Powers of 5 by the Imagination only, as we do thofe of Ten in the
common Scale, this Number will be 42302,43 in Quinary Arithme-
tick.

All vulgar Fractions and mixt Numbers are, in fome meafure, the
expreffing of Numbers by a particular Scale, or making the Deno-
minator of the Fraction to be the Root of a new Scale. Thus is
in effect o x 3 + 2 x^" 1 ; and 8-f- is the fame as 8 x 5 '-f- 3 x j-'j
and 25-5- reduced to this Notation will be 25x9 + 4x 9' , or ra-
ther 2x9' -4- 7x9 -4-4X9"" 1 . And fo of all other Fractions and
mixt Numbers.

A Number computed by any one of thefe Scales is eafily reduced
to any other Scale affign'd, by fubftituting inftead of the Root in one
Scale, what is equivalent to it exprefs'd by the Root of the other
Scale. Thus to reduce Sexagenary Numbers to Decimals, becaufe
60 = 6x10, or|=6X, and therefore | s = 3 6X 1 , ^=2i6X 3 ,
&c. by the fubilitution of thefe you will eafily find the equivalent
Decimal Number. And the like in all other Scales.

The Coefficients in thefe Scales are not neceflarily confin'd to be
affirmative integer Numbers lefs than the Root, (tho' they mould be
fuch if we would have the Scale to be regular,) but as occafion may
require they may be any Numbers whatever, affirmative or negative,
integers or fractions. And indeed they generally come out promif-
cuoully in the Solution of Problems. Nor is it neceflary that the
Indices of the Powers mould be always integral Numbers, but may
be any regular Arithmetical Progreffion whatever, and the Powers
themielves either rational or irrational. And thus (thirdly) we are
come by degrees to the Notion of what is call'd an univerfal Series,
or an indefinite or infinite Series. For fuppofing the Root of the
Scale to be indefinite, or a general Number, which may therefore
be reprefcnted by x, or y, &c. and affuming the general Coefficients
a, b, c, d, &c. which are Integers or Fractions, affirmative or nega-
tive, as it may happen ; we may form fuch a Series as this, ax* -f-
lx* _j_ ex* -f- dx l -f- ex, which will reprefent fome certain Number,
exprefs'd by the Scale whofe Root is x. If fuch a Number pro-
ceeds in hfif.itum, then it is truly and properly call'd an Infinite
Series, or a Converging Series, x being then fuppos'd greater than
Unity. Such for example is x + \x~ '-\-^.x' - + ^*~ 3 , &c. where
the reft of the Terms are underftood ad in/initum, and are iniinuated

and INFINITE SERIES. 149

bv, oV. And it may have any dcfcending Arithmetical Progreffion
for its Indices, as x m \x m ~ l -+- ^v* 1 -+-"*.. \s, Gfc.

And thus we have been led by proper gradations, (that is, by
arguing from what is well known and commonly received, to what
before appear'd to be difficult and obfcure,) to the knowledge of
infinite Series, of which the Learner will find frequent Examples
in the lequel of this Treatife. And from hence it will be eafy to
make the following general Inferences, and others of a like nature,
which will be of good ufe in the farther knowledge and practice of
t-hefe Series ; viz. That the firft Term of every regular Series is al-
ways the mo ft coniiderable, or that which approaches nearer to the
Number intended, (denoted by the Aggregate of the Series,) than
any other lingle Term : That the fecond is next in value, and fo on :
That therefore the Terms of the Series ought always to be difpoled
in this regular defcending order, as is often inculcated by our Author :
That when there is a Progreflion of fuch Terms-/;? infinitum, a few of
the firft Terms, or thofe at the beginning of the Series, are or fhould
be a fufficient Approximation to the whole ; and that thefe may
come as near to the truth as you pleafe, by taking in ftill more
Terms : That the fame Number in which one Scale may be exprefs'd
by a finite number of Terms, in another cannot be exprefs'd but by
an infinite Series, or by approximation only, and vice versei : That
the bigger the Root of the Scale is, by fo much the fafter, cafen'.i
paribus, the Series will converge ; for then the Reciprocals of the
Powers will be fo much the lefs, and therefore may the more fafely
be neglected : That if a Series coir e T os by increafing Powers, fuch
as ax -^ bx* -+- ex* -|-</.v 4 , &c. the Root x of the Scale mull be un-
derftood to be a proper Fraction, the lefler the better. Yet when-
ever a Series can be made to conveige by the Reciprocals of Ten,
or its Compounds, it will be more convenient than a Series that
converges fafter j becaufe it will more eafily acquire the form of the
Decimal Scale, to which, in particular Cafes, all Series are to be ul-
timately reduced. LafHy, from fuch general Series as thefe, which
are commonly the refill t in the higher Problems, we muft pafs (by
fubftitution) to particular Scales c; Series, and thofe are finally to be
reduced to the Decimal Scale. And the Art of finding fuch general
Series, and then their Reduction to -particular Scales, and laft all
to the common Scale of Decimal Numbers, is ulmoll the whole of

j abrtiull-r pares of Amly ticks, as may be fecn in a good meaiiire'by
the prefent TrcuUic.

I

Method of FLUXIONS,

I took notice in the fourth place, that this Doctrine of Scales, and
Series, gives us an eafy notion of the nature of affected Equations,
or fhews us how they ftand related to fuch Scales of Numbers. In
the other Inflances of particular Scales, and even of general ones,
the Root of the Scale, the Coefficients, and the Indices, are all fiip-
pos'd to be given, or known, in order to find the Aggregate of the
Series, which is here the thing required. But in affected Equations, on
the contrary, the Aggregate and the reft are known, and the Re ot of
the Scale, by which the Number is computed, is unknown and re-
quired. Thus in the affected Equation \$x* -j- 3*2 -f- ox* -+- 7*-
53070, the Aggregate of the Series is given, viz. the Number
53070, to find x the Root of the Scale. This is eafily difcern'd to
be 10, or to be a Number exprefs'd by the common Decimal Scale,
efpecially if we fupply the feveral Powers of 10, where they are un-
derftood in the Aggregate, thus 5X 4 -+- 3X 3 -f-oX 1 +7X' -4-oX
= 53070. Whence by companion 'tis x = X=io. But this
will not be fo eafily perceived in other instances. As if I had the
Equation 4^+4- ax 3 -f- 3** -f-ox" -f- 2x -f- ^x~ f -f- ^x~ 1 = 2827,92
I Ihould not fo eafily perceive that the Root x was 5, or that this is
a Number exprefs'd by Quinary Arithmetick, except I could reduce
it to this form, 4x5* -+- 2x \$ 3 + 3*5* + 0x5' -f- 2 x 5 H- 4x5 *
-+- 3 x 5~~ ; = 2827,92, when by comparifon it would preiently ap-
pear, that the Root fought muft be 5. So that finding the Root of
an affected Equation is nothing elfe, but finding what Scale in Arith-
jnetick that Number is computed by, whofe Refult or Aggregate is
given in the common Scale ; which is a Problem of great ufe and
extent in all parts of the Mathematicks. How this is to be done,
either in Numeral, Algebraical, or Fluxional Equations, our Author
will inflruct us in its due place.

Before I difmiis this copious and ufeful Subject of Arithmetical
Scales, I fhall here make this farther Observation ; that as all con-
ceivable Numbers whatever may be exprefs'd by any one of theie
Scales, or by help of an Aggregate or Scries of Powers derived frcm
any Root ; fo likewife any Number whatever may be exprefs'd by
fome fingle Power of the fame Root, by affuming a proper Index,
integer or fracted, affirmative or negative, as occafion fhall require.
Thus in the Decimal Scale, the Root of which is 10, or X, not
only the Numbers i, 10, 100, 1000, &c. or i, o.i, o.oi, o.ooi, &c.
that is, the feveral integral Powers of 10 and their Reciprocals, may
be exprefs'd by the fingle Powers of X or 10, viz. X , X' , X 1 , X s ,
or X, X- 1 , X~% X - % &c. refpectively, but alfo all the inter-
mediate

and INFINITE SERIES. 151

mediate Numbers, as 2, 3, 4, Gff. u, 12, 13, Gfr. may be exprefs'd
by fuch fingle Powers of X or 10, if we aflame proper Indices.

Thus 2 = X' JOI03> &C- , 3 = X '477",&c. 4 = __ Xo/o-.o, &e. g^ Qr jj

_.X''4'3!>. &C - i 2=== X'>7i" 8 ' &e> 456 = X*.s89s,&c. And the like of
all other Numbers. Thefe Indices are ufually call'd the Logarithms of
the Numbers (or Powers) to which they belong, and are fo many
Ordinal Numbers, declaring what Power (in order or fucceflion) any
given Number is, of any Root aflign'd : And different Scales of Lo-
garithms will be form'd, by afluming different Roots of thofe Scales.
But how thefe Indices, Logarithms, or Ordinal Numbers may be
conveniently found, our Author will likewife inform us hereafter.
All that I intended here was to give a general Notion of them, and
to mew their dependance on, and connexion with, the feveral Arith-
metical Scales before defcribed.

It is eafy to obferve from the Arenariiu of Archimedes, that he
had fully confider'd and difcufs'd this Subject of Arithmetical Scales,
in a particular Treatife which he there quotes, by the name of his
a'^^tl, or Principles ; in which (as it there appears) he had laid the
foundation of an Arithmetick of a like nature, and of as large an
extent, as any of the Scales now in ufe, even the moft univerlal. It
appears likewife, that he had acquired a very general notion of the
Dodtrine and Ufe of Indices alfo. But how far he had accommo-
dated an Algorithm, or Method of Operation, to thofe his Princi-
ples, muft remain uncertain till that Book can be recover'd, which
is a thing more to be wim'd than expedled. However it may be
fairly concluded from his great Genius and Capacity, that fince he
thought fit to treat on this Subject, the progrefs he had made in it
was very confiderable.

But before we proceed to explain cur Author's methods of Ope-
ration with infinite Series, it may be expedient to enlarge a little
farther upon their nature and formation, and to make fome general
Reflexions on their Convergency, and other circumftances. Now
their formation will be beft explain'd by continual Multiplication
after the following manner.

Let the quantity a -+- bx -{-ex 1 -+- <A' 3 -+- ex 4 , 6cc. be aflumed as
a Multiplier, confming either of a finite or an infinite number of

Terms ; and let alfo - -+- x = o be fuch a Multiplier, as will give
the Root x= - . If thefe two are multiply'd together, they

will produce 3 + 2Xf?* + 2f_V + "1^5^ + *i V, &c.

* a a a n

152 The Method of FLUXIONS,

. o ; and if inftead of x we here fubflitute its value - , the Series

ap fy+"<! f tp+bq f- dp + cq /3 ' ef+t/f p*

wi 1 become - TTT - x - -f- x -*- x -. -f- -^-^- ? x - >

q q q if 11* 9 j*

&c. = o ; or if we divide by -, and tranfpofe, it will be "*" aq .

tp + bg p dj> + eg /* ep + Jq t* ....

x y + j x ^ x - , &c. = ,7 : which Series,

thus derived, may give us a good infight into the nature of infinite
Series in general. For it is plain that this Series, (even though it
were continued to infinity,) mufl always be equal to a, whatever
may be fuppofed to be the values of p, q, a, b y c, d } &c. For

- , the firft part of the firflTerm, will always be removed or deflroy'd
by its equal with a contrary Sign, in the fecond part of the feeond

Term. And x- , the firfl part of the fecond Term, will be re-

i i
moved by its equal with a contrary Sign, in the fecond part of -the

Online LibraryIsaac NewtonThe method of fluxions and infinite series : with its application to the geometry of curve-lines → online text (page 13 of 30)