Isaac Newton.

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

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Online LibraryIsaac NewtonThe method of fluxions and infinite series : with its application to the geometry of curve-lines → online text (page 28 of 30)
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310 7%e Method 0/* FLUXIONS,

teger affirmative Number, the Series will break off, and become
finite, at a number of Terms denominated by m. But in all other
cafes it will be an infinite Series, which will converge when x is
lefs than a.

Indeed it can hardly be faid, that this, or any other that is de-
rived from the Method of Fluxions, is a ftridl Inveftigation of this
Theorem. Becaufe that Method itfelf is originally derived from the
Method of railing Powers, at leaft integral Powers, and previoufly
fuppofes the knowledge of the Unci<z, or the numeral Coefficients.
However it may anfwer the intention, of being a proper Example
of this Method of Extraction, which is all that is neceffary here.

There is another Theorem for this purpofe, which I found many
years ago, and then communicated it to my ingenious Friend Mr.
A. dc Moivre, who liked it fo well as to infert it in a Mathematical
Treadle he was then publiming. I mall here give the Reader its
Inveftigation, in the fame manner it was found.

Let us fuppofe a -f- x \ m = a m -f- />, and that a -f- x = z, and
therefore z = x = i. Now becaufe z m =a m -i-f, it will be

m

p = niz"' 1 = ^- ; where for z m writing its value a m -+- f, we

fH fnf, M

ma . m P . . ma x

fhall have/ = H 1 . Now if we make/>= 7- -J- q, it

m m

will be p = '- "^r -+- q> And comparing thefe two values

ttn* 1 ?%P

of p, we mall have q = ^ -f- ~ ; where if for p we write its

ma m x m 1 a m x m 1

value as above, it will be q = -^- + ^ + ~ , or q = m x

_ m . _i_ w j

m ~\- i x ~ + ~l ; make q = m x " 2 - x ^~ -f- r; therefore

- m -a

- -I- r. From which

m
a x

= m x m -+- i x m x m -+- i x

m 4 OT _

two values of <7 we fhall have r = m -x. m -}- i x ^- ~1~ 7" . And

2 2,3 -

for y fubftituting its value, it will be r = m x m -h i x
+ ' ..- OT * a

mr

i .

Or r = m

AT 1

Make r-=.m

4- j ; then, &c. So that we

mall

and INFINITE SERIES. 311

* "'+' *'"*= ,

/lull have a + x I ra = a" -f- m x ^. -f- w x -7- ^7 ~

=^*=fX;0i.^

Now this Series will Hop of its own accord, at a finite number
of Terms, when m is any integer and negative Number ; that is,
when the Reciprocal of any Power of a Binomial is to be found.
But in all other cafes we mall have an infinite converging Series for
the Power or Root required, which will always converge when a
and x have the fame Sign ; becaufe the Root of the Scale, or the

converging quantity, is ^ , which is always lefs than Unity.

By comparing thefe two Series together, or by collecting from

, : m _ a i

each the common quantity - ^ - , we fliall have the two

_ . TO x

equivalent Series - -+- -j- x - -f- -j-x x - , &c. =
, 11 x x + 11 x "L_ 2 x == ^=- , &c. from whence we

a + x |* 2 3 a-i-x I 5

might derive an infinite number of Numeral Converging Series, not
inelegant, which would be proper to explain and illuftrate the na-
ture of Convergency in general, as has been attempted in the for-
mer part of this work. For if we aflume fuch a value of i as
will make either of the Series become finite, the other Series will
exhibit the quantity that arifes by an Approximation ad infinitum.
And then a and A; may be afterwards determined at pleafure.

As another Example of this Method, we fliall fhew (according
to promife) how to derive Mr. de Moivre's elegant Theorem ; for
raifing an Infinitinomial to any indeterminate Power, or for extract-
ing any Root of the fame. The way how it was derived from the
abftract coniideration of the nature and genefis of Powers, (which
indeed is the only legitimate method of Inveftigation in the prefent
cafe,) and the Law of Continuation, have been long ago communi-
cated and demonftrated by the Author, in the Philofbphicai Tranf-
actions, N 230. Yet for the dignity of the Problem, and the bet-
ter to illuftrate the prefent Method of Extraction of Roots, I lhall
deduce it here as follows.

Let us aflume the Equation a -+- b~ -+- cz* -f- itz* -4- ez* t &.c. | *
= )', where the value of y is to be found by an infinite Series, of
which the firfl: Term is already known to be a m , or it is y a",
&c. Make v = a -f- l>z -f- cz, 1 -J- dz> -f- ez*, &c. and putting
z = i, and taking the Fluxions, we mail have -y r^zi b -+- 2cz -+-

, I2 Tie Method of FLUXIONS,

7</jz -f- 4s*. &c - Then becaufe y = u", it is ^ = mviT-', where
if we make -y = <*, &c. and V = t>, &c. we fliall have _y =
ma'-'b, &c. and taking the Fluents, it will be _> = * maF*ks t

For another Operation, becaufe _y = mvv"- 1 , it is _y =

_l_ ;, ; x ; ii>i;"'~ z . And becaufe <u = 2f -f- 6<fe + 12^2;*, &c. for

y, v, and -yfubftituting their values a, &c. ^, &c. and 2c, 6cc. refpec-
tively, we fliall have jy = zmca" 1 - 1 -+- m x OT i6*a m - z , &c. and
taking the Fluents 7 = * 2fnca m - J z -\-m-x. m i*tf*-**, 6cc. and
taking the Fluents again, y = * * mca m - l z* -\-m~x. ^^^a m - 2 z 1 )

&c. ..

For another Operation, becaufe/ = mw m - 1 -+- mxm iv 1 v m ~ 1 t

'tis y m i v"j m ~ l + yn x / iv m ~ z -vv-{-m\m ixw 2 t v m *v">.
And becaufe -u = 6^+24^2;, &c. for v, v, v, v, fubftituting a,

6cc. b, &c. 2c, &c. 6^/, &c. we fliall have )"=, 6mda m ~ l -f- 6m x

w \bca m ~' i -\- m x. m i x m 2l> 3 a m ~*, &c. And taking the

Fluents it will be y = * 6mda m ~ l z -+- 6m x m \bca m -*z -+- m x
w i Km 2foa m ~*z, 8cc. l/ = * *

x ^^ x w 2foa m -iz 1 , &c. and/ = * * * mda m - l z* -+ m x

m x

. And fo on z' /-

finitum. We fliall therefore have tf

And if the whole be multiply'd by s", and continued to a due
length, it will have the form of Mr. de Moivres Theorem.

The Roots of all Algebraical or Fluential Equations may be ex-
tradled by this Method. For an Example let us take the Cubick
Equation y* -\~axy -\-a*y x"' 2 3 =o, fo often before refolved,
in which y = a, &c. Then taking the Fluxions, and making
x = i, we fliall have ^yy- -+- ay +- axy -f- a*-y 3x z = o. Here
if for y we fubftitute a, &c. we fliall have ^y -\- a 1 4- axy 3**,

&c. =o, or>= -j:+^" = ^' &c - =. - &C ' A " d
taking the Fluents, v = * -i-x, &c. Then taking the Fluxions

j again

and INFINITE SERIES. 313

again of the lafl Equation, we fliall have 37}* -f- 6y 1 y 4- 2 ay 4- axy
4- a l y 6x= o. Where if we make 7 = a, &c. and 7 = ^.,
&c. we fliall have y= ZZif+j''-^: __ _!_ &c. and therefore

' 43% CSV. 32a '

AT 1 '""

y = * 4 , &c. and y= * * -h z ' & c - Again, ?yy* -{- iSyy'y

* "\2.ii (3 -I.'? ^

_j_ 675 -f- 3^7 -f- axy -\- a*-j 6 = 0. Make 7 = a, &c. 7 =
^ &c. and 7 = , 6cc. then 7 ^ + ^~ ^- 4 " 6 5j; C- __

' 3 2a 4 a

^|j, , &c. and therefore 7 = * ^^ , &c. 7 ^ * * = ^ , 6cc.

and 7 = * * * - ^ 5 &c. Again, 377* -1-24777-1- 18^*74- 3677*
4- 4^7 4- ^7x7 4- a*y = o. Make 7 = a, &c. 7 = ^, &c./ =
- , &c. and 7 = - |^ , &c. then 7 =

:, &c. and7=# '- ^-. , &c. 7 = * *

, &c. and y = * * * * r68a ' ^ C ' ^ nc ^ ^ on as ^ ar as we
Therefore the Root is y = ai x + f- H-H^ + -|2|fl &c.

643 12^- ' i68' J

The Series for the Root, when found by this Method, muft al-
ways have its Powers afcending ; but if we defire likewife to find
a Series with defcending Powers, it may be done by this eafy arti-
fice. As in the prefent Equation y* -\- axy -f- a*y x"' 2a* = o,
we may conceive x to be a conftant quantity, and a to be a flowing
quantity ; or rather, to prevent a confufion of Ideas, we may change
a into x, and x into a, and then the Equation will be _y 5 -f- axy -+
x*y rt 3 ax 3 = o. In this we mall have y = a, &c. and ta-
king the Fluxions, 'tis 3j/y* -4- ay -f- axy -f- 2xy +- x*y 6x* = o,

ay 2X],"-r- 6-V z n . i r o > "~" #& e>

or y= 7-1 H r . But becauie y=a, &c. tis y= - , ficc.
y +<**+* r ^ 3

== i, &c. and therefore/ = * -i-.v,&c. Again taking the Fluxions
'tis 3_)^ 4 -f- 6)/ 1 / 4- zay +- axy -\-2y-\- ^.xy -+- x*y I2x = o, or
- -yj>-^-'j-4y+"* -6^-2^-2, Qr

/ yJ+flX-f-* 1 y*

king _y = a, &c. and _y = |, &c. 'tis y == ~^ + f~ 2a , &c.

3^

=s y rt , &c. and _y = ^ , &c. and ;'=** ^ , &c.

Again it is 3vy* -t- i8;7/ -f- 6y 5 -f- 3^} 4- axy -+. 67 H- 6x/ H- x*_y

S f 12

The Method of FLUXIONS,

_ 12 = O, Or y = -rv- ? -

v = a t &c. y = 4-' &c - and y = -, &c.)~ 4 + v -T 2 + 2 + 12 ,
/ y 3> / 3**

&c. = -^ , &c. Then taking the Fluents, 7 = * , &c.

y = * * ^T , &c. and y = * * |^ , c. And fo on. There-

fore we mall have y = <z . x 7^ + 7^7 , &c. Or now we
may again change x into #, and a into x ; then it will be y = x

_ itf _ -. _i_ ii^j , &c. for the Root of the given Equation, as

3*

was found before, pag. 2 16, &c.

Alfo in the Solution of Fluxional Equations, we may proceed in
the fame manner. As if the given Equation were ay a*x -{- xy
- o,. (in which, if the Radius of a Circle be reprefented by a, and
if y be any Arch of the fame, the corresponding Tangent will be
reprefented by x 3) let it be required to extradl the Root y out of
this Equation, or to exprefs it by a Series compofed of the Powers
of a and x. Make x = i, then the Equation will be ay a 1 -J-

a

x a -yr=o. Here becaufe_y' = _, = i, &c. taking the Fluents
it will be y = * x, &c. Then taking the Fluxions of this Equa-
tion, we (hall have ay -f- 2xy -+- xy = o, or y= -^T 1 . But
becaufe we are to have a conftant quandty for the firft Term of y,
we may fuppofe y='~^^ i = o, &c. Then taking the Fluents
'tisj/= * o, &c. and y =. * * o, &c. Then taking the Fluxions

again, 'tis a'y -f- 27 + 4*7 -{- xy = o, or y = "J'^.'T* Here
if for y and_y we write their values i, &c. and o, &c. we mall have

}'= ^ , &c. whence y * il , & c . y = * ^ , c .
and 7 == * * * ,71 > ^ c - Taking the Fluxions again, 'tis

ay +6y + 6xy +xy = o, 01-7 ~*?~^ J = o, <5cc. There-
fore y = * o, &c. _y= * * o., &c. j.'= * * * o, &c. y =. **#
Q, &-C. Again, ay -+- I2y 4- 8^,7 -f- A' 4 j = o, or y = ~ l 2 ^!.'' 1 -

and INFINITE SERIES.

= + , &c. Then Jr^*^^, &c. ;< = **4- ^ , &c.
_>>=:***-{- , &c. jy = * * * * 4- , &c. and _}=*****

45

- - - } &c. Again, a*y-\- zoy -f- loxy -f- xy = o, whence y=s

5 "' r56 7?

***** o, &c. Again, a*j+ 307 -f- i2*y -+- A,"^ = o, or y=.

y~^=:- 30 x 2 4 rf- 9 , &c. Then > =- . _ !^2f , &c .

12x30** 4x301-3

j ==- * * - ^ , &c. y = * * * - ^ , &c. ; = *** *

30*4 6*5 o * 6

- -, &c. _y = # * * * * -^ , c^c. j)/=******

A' 7

c. and _/ = ******* , &c. And fo on. So that we

have here y = * # -H ox- 1 ^ + OA ' 4 + ^; > &c - that is, _y =
fl , . *! & c

* ' P + 5^ /' '

This Example is only to {hew the universality of this Method,
and how we are to proceed in other like cafes ; for as to the Equa-
tion itfelf, it might have been refolved much more fimply and ex-

peditioufly, in the following manner. Becaufe y = -^- - } by
Divifion it will be y = i ~ + ^ *- 6 -f- *L , & c . And ta-

king the Fluents, y = x ^ -+- ; - 6 4- ^ > &c.

In the fame Equation ^ a*x 4- x 1 ^ = o, if it were requir'd
to exprefs x by y, (the Tangent by the Arch,) or if x were made
the Relate, and y the Correlate, we might proceed thus. Make

jc x

y = i, then a* a?x +- x* = o, or si = 14 - = i 5c c

J G,

Then x = * y, &e. And taking the Fluxions, 'tis x = ^ =s

~-^ , &c. = 04-^, &c. whence A- = * o, &c. and x = * * o,

&c. So that the Terms of this Series will be alternately deficient,
and therefore we need not compute them. Taking the Fluxions

. , * Zxx 2 n i-i->i r 2V

again, tis x = -j- =: , &c. Therefore x = * -^ , &c.

A- = * * j-, , &c. and ^== # * * ^ , &c. Again, x=z -^ -l-^r >

S f 2 and

Method of FLUXIONS,

2XX

>X xx XX c ,n_. tf ,2

and again, A- = -f- -^ h . Subitituting i, &c. and >-

&c. for # and x } and alfo o, 6cc. for # and #, it will be x =
16 c t. i6y c gy*

, &c. whence x =. * - , occ. * = ** , &c. x = * # *

<j4 ** * '

8v' c 2 >' 4 C J 21*

_, , &c. A- = * * * * ^ , &c. and . x ==-***#, &c..

6 ' " J 7 ' :: 5 6

. 20xx+ iQAr- 4" 2A ' ;xr J 2n*- I -4 - ;o.;x4-i2.v.v-(-2.vA-

Again, x = - - , and again, x = - - -
Here for x, x, and x writing i, &c. ^ , &c. and l , &c. re-
fpeaively, 'tis x = 8 ^U. 6 , & c . ^ , Sec. Then x =
^i v , &c. J = ..i4\ &c. x=***i^- J , &c. 2==.....

fi 3^

ig 4 , &c. ^= * * * * * 7^ , &c. x= ****** JJpr, &c. and
*, &c. That is, X =y+.^ + -l. +

7.1 Cfl 6

For another Example,, let us take the Equation ^*jj* .y l j> ,

/z*x* o, (in which, if the Radius of a Circle be denoted by a,
and if y be any Arch of the fame,, then the correfponding right Sine
will be denoted by x j) from which we are to extract the Root y.

Make x=i, then it will be a*y*~ x^y 1 = a*, or ;}* = -^^
M J} &c. or j/= i, &c. and therefore y=. * x, &c. Taking
the Fluxions we fhall have za*yy zxy* 2X 1 yy = o ) or a*y>
X y __ x y = 0, or "y = ^r~7i = ' ^^ ^ nc ^ ta ^ing. the Fluxions

again, 'tis a*y j/ 3* ^ a /= o, or ^ = __. . y

Therefore } = * J } &c. ^ = * * , &c. and _y = * * * 1- ,
&c. Then a ;y 4^ 5^' x*y = o, and again <7 a j/ 9 _y ,
x ^=o, orj = i^=,&c. = J,& C . There-

fore y := * , &c, y ' * * ^ > &c. ^ = * * * 4 , &c. ^5^

* * *

INFINITE SERIES, 317

* * , . j^- 4 , 6cc. and r = ***#* -IL , &c. Taking the

6 :: S 6 7

Fluxions again, 'tis n l y i6y g.y x*y = o, and again, a*jr

2^Y + i i xv c

z\$y 1 1*7 x*y = o, or ;< = ~ -^ = -/, &c. =
^ , &c. Therefore j = * ^.v, &c. v = * * ^V, &c.

' J a* J 2a 6

y = * * * -2-3 x r t &c. _)' = * * * * gJT x4 > ^ c> y === * * * * *

^-|x r , &c. y = ****** r 6 , &c. and y = *****#*
-^-. . &c. Or v = x -i- ~ 4- -^- 4- -^ , &c.

TI2a 6 J * t><i l 40^.4 II2*

If we were required to extraft the Root x out of the fame Equa-
tion, a l y* x 1 )/ 1 rt 1 * 1 = o, (or to exprefs the Sine by the:
Arch,) put y = i, then a 1 x 1 a** 1 = o, or x* = i

-*, and therefore x = i, &c. and x = * y, &c. Taking the-

Fluxions 'tis axx 2a*xx = o, or x= =: o, &c,-
Therefore x = * o, &c. x =: * * o, &c. Taking the Fluxions

again, 'tis x = ^ = ^ , &c. Thence x= * ^, &c.
x = , &c. andx=* * * - , &c. Again, x = ^>

2^1 Oi fi

and x = ~ = -+- ^ , &c. Therefore x = ^ , &c. x =

t 1 ) J j 4

-^ , occ. x = * * * 7 , &c. x = * * * - , occ. and x

v* x x

=== * * - , &c. Again, x = , and x =

i ?r/24 * O * a fl

i 6 v ? y 1

- 6 , 6cc. Therefore x = - ^j , x = * ~ *

&c. x * -^-j , 6cc. x * * - 1 g , &c. x = * * *

j.J V* J

_ -^- t , &c. x = .. - f , &c. and x= **

I 2Ofl* 72Ofl

*__ - -^ - 6 , 6cc. And therefore x= < y> ^ -f-

&c.

If it were required to extraft the Root y out of this Equation,
*y . .x*y* + m*y* w 1 ^ = o, (where x =s i,) we might pro-
ceed

3i 8 The Method of FLUXIONS,

ceed thus. Becaufe y~ ==. '" ^ ~^ - v - = ;*, &c. 'tis ;/ = ;;;, See.

an d_y= -# w*> &c. Taking the Fluxions, we fhall have 2a*yy
2xy* 2x*yy -bJzr\$*yy = o, or ay xy xy -+- my o, or
r= p ^r= j ^ c - Therefore taking the Fluxions again, 'tis

3xy x 1 / = o, or y = -I ^^_ v jf^l; and making y = m, &c.
'tis y= OTX ' ~ OT &c. and therefore y = * '" x ' "~ ; "'x, &c. y =

^ X /2 A -^

i^LV, &c. and y = * * * OT x ' ~^\-', &c. Taking the

za'' * zx. ya

Fluxions again, 'tis a*y -+- m 1 4 x_y \$xy xy = o j and again,
cfy {- in* 9 x y 7xy x 2 i/ o, or y Q "'' x ->' ~t~ " vv ___

, - x 9 -^ &c> Therefore y= * ; " x ' ~ g ' lx 9 ~ CT \y & c v

4 4 ' ^

This Series is equivalent to a Theorem of our Author's, which (in
another place) he gives us for Angular Sections, For if A- be the
Sine of any given Arch, to Radius a ; then will y be the Sine of an-
other Arch, which is to the firft Arch in the given Ratio of m to
i. Here if m be any odd Number, the Series will become finite j
and in other cafes it will be a converging Series.

And thefe Examples may be lufficient to explain this Method of
Extraction of Roots ; which, tho' it carries its own Demonftration
along with it, yet for greater evidence may be thus farther illustrated.
In Equations whofe Roots (for example) may be reprefented by the
general Series y = A -+- Ex -f- Cx 4 + Dx 3 , 6cc. (which by due Re-
duction may be all Equations whatever,) the firfc Term A of the
Root will be a given quantity, or perhaps = o, which is to be
known from the circumftances of the Queilion, or from the given

Equation,

and INFINITE SERIES.

319

Equation, by Methods that have been abi ^antly explain'd already.
Then making x= i, we flrall have have y = B -f- 2C.v -+- 30**,
&c. where B likewife is a conftant quantity, or perhaps = o, and
reprefents the firft Term of the Series y. This therefore is to be
derived from the firft Fluxional Equation, either given or elfe to
be found ; and then, becaufe it is y = B, &c. by taking the Fluents
it will be y = * Ex, ccc. whence the fecond Term of the Root
will be known. Then becaufe it is_y= zC -f- 6D.v, &c. or becaufe
the conftant quantity zC will reprefent the firft Term of y ; this is
to be derived from the fecond Fluxional Equation, either given or
to be found. And then, becaufe it is y = zC, &c. by taking the
Fluents it will be y = * zCx, &c. and again y = * * Cx 1 , cc. by
which the third Term of the Root will be known. Then becaufe

it is_y=6D, &c. or becaufe the conftant quantity 6D will repre-
fent the firft Term of the Series y ; this is to be derived from the

third Fluxional Equation. And then, becaufe it is y = 6D, &c.
by taking the Fluents it will be v = * 6Dx, &c. y = * * 3D*'-,
See. and _)'==.* * * D* 3 , &c. by which the fourth Term of the
Root will be known. And fo for all the fubfequent Terms. And
hence it will not be difficult to obferve the compofition of the Co-
efficients in moft cafes, and thereby difcover the Law of Continua-
tion, in fuch Series as are notable and of general ufe.

If you ihould defire to know how the foregoing Trigonometri-
cal Equations are derived from the Circle, it may be fhewn thus : on
the Center A, with Radius AB = a t let the Quadrantal Arch BC be
defcribed, and draw the Radius AC. Draw the Tangent BK, and
through any point of the Circum-
ference D, draw the Secant ADK,
meeting the Tangent in K. At any
other point d of the Circumference,
but as near to D as may be, draw
the Secant A.tte, meeting BKin/ ; on
Center A, with Radius AK, defcribe
the Arch K/, meeting A in /,
Then fuppofing the point d con-
tinually to approach towards D, till
it finally c<- : .ncides with it, theTri-
lineum K//6 will continually approach to a right-lined Triangle,
and to funilitude w/ith the Triangle ABK : So that when Dd is a

Moment

c

320 The Method of FLUXIONS,

Moment of the Circumference, it will be K-! = ^4 x =

Da &.1 L)J ~ AB

x ^ . Make AB = a, the Tangent BK = x, and the Arch
BD=y ; and inflead of the Moments Kk and Dd, fubftitute the
proportional Fluxions x and y, and it will be - = " - + *- , or a 2 v

J y a* J

+ x*y a*x = o.

From D to AB and de let fall the Perpendiculars DE and Dg-,
which Dg meets de, parallel to DE, in g. Then the ultimate form
of the Trilineum Ddg will be that of a right-lined Triangle fimi-
lar to DAE. Whence "Dd : dg :: AD : AE = v / /iJJ\$ F D&q.
Make AD = a, BD=_>', and DE=x; and for the Moments
Dd, dg, fubftitute their proportional Fluxions y and x, and it will
be y : x : : a : \/ a 1 AT*. Or^ 1 : x 1 :: a* : a 3 - x 1 , or a^y 1
- x^y* a'-x 1 - = o.

Hence the Fluxion of an Arch, whofe right Sine is x, being

exprefs'd by f^_^ ; and likewife the Fluxion of an Arch, whofe

right Sine is y, being exprefs'd by i?_ ,. ; if thefe Arches are to
each other as i to m, their Fluxions will be in the fame proportion,
and vice versa. Therefore " x , : v x : : i : ;, or . x .

J a x */ a y da x

= -T37i } or putting #= i, 'tis a*y l

== o ; the fame Equation as before refolved.
' We might derive other Fluxional Equations, of a like nature with
thefe, which would be accommodated to Trigonometrical ufes. As
if/ were the Circular Arch, and x its verfed Sine, we mould have
the Equation zaxy* x'-y'- a^x* = o. Or if y were the Arch,
and x the correfponding Secant, it would be x^y* a*-x*y l # 4 x*
= o. Or inftead of the natural, we might derive Equations for
the artificial Sines, Tangents, Secants, &c. But I fhall leave thefe
Difquifitions, and many fuch others that might be propofed, to ex-
ercile the Induftry and Sagacity of the Learner.

SECT,

and INFINITE SERIES. 321

'SECT. VI. An Analytical Appendix ', explaining fome
Terms and Expreffions in the foregoing work.

BEcaufe mention has been frequently made of given Equations,
and others a framed ad libitum, and the like ; I mall take oc-
calion from hence, by way of Appendix, to attempt fome kind of
explanation of this Mathematical Language, or of the Terms giver/,
afligjfd, affiimed, and required Quantities or Equations, which may
give light to fome things that may otherwife feem obfcure, and
may remove fome doubts and fcruples, which are apt to arife in
the Mind of a Learner. Now the origin of fuch kind of ExpreiTions
in all probability feems to be this. The whole affair of purfuing
Mathematical Inquiries, or of refolving Problems, is fuppofed (tho'
tacitely) to be tranfacled between two Perfons, or Parties, the Pro-
pofer and the Refolver of the Problem, or (if you pleafe) between the
Mafter (or Inftruclor) and his Scholar. Hence this, and fuch like
Phrafes, datam reffam, vel datum angtthim, in iniperata rations Je-
care. As Examples inftrudl better than Precepts, or perhaps when
both are join'd together they inftrucl beft, the Mafter is fuppos'd to
propofe a Queftion or Problem to his Scholar, and to chufe fuch
Terms and Conditions as he thinks fit ; and the Scholar is obliged
to folve the Problem with thofe limitations and reftriclions, with
thofe Terms and Conditions, and no other. Indeed it is required
on the part of the Mafter, that the Conditions he propofes may be
confident with one another ; for if they involve any inconfiftency
or contradiction, the Problem will be unfair, or will become ab-
furd and impomble, as the Solution will afterwards difcover. Now
thefe Conditions, thefe Points, Lines, Angles, Numbers, Equations,
Gfr. that at firft enter the ftate of the Queftion, or are fuppofed to
be chofen or given by the Mafter, are the data of the Problem, and
the Anfwers he expects to receive are the qii(?/ita. As it may fometimes
happen, that the data may be more than are neceffary for determining
the^Qiu ft'.on , and lo perhaps may interfere with one another, and the
Problem (as now propofed) may become impolTible ; fo they may be
fewer than are neceffary, and the Problem thence will be indetermin'd,
and may require other Conditions to be given, in order to a compleat De-
termination, or perfectly to fulfil the quafita. In this cafe the Scholar is
to fupply what is wanting, and at his difcretioa miy a (Jit me fuch and fo
many otherTerms and Conditions, Equations and Limitations, as he finds

T t will

322 7&? Method of FLUXIONS,

will be neceffary to his purpofe, and will befl conduce to the fim-
pleft, the eafieft, and neateft Solution that may be had, and yet in
the moft general manner. For it is convenient the Problem fhouM
be propofed as particular as may be, the better to fix the Imagina-
tion ; and .yet the Solution mould be made as general as poffible,
that it may be the more inftrucHve, and extend to all cafes of a
like nature.

Indeed the word datum is often ufed in a fenfe which is fome-
thing different from this, but which ultimately centers in it. As
that is call'd a datum, when one quantity is not immediately given,
but however is neceffirily infer'd from another,, which other perhaps
is neceffarily infer'd from a third, and fo on in a continued Series,
till it is neceffarily infer'd from a quantity, which is known or given
in the fenfe before explained. This is the Notion of Euclid'?, data,

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