1000000030

60 7 )

LU:CroNS,

a metltod of eakulation whieh great ly

t:t:dit.Hes eorr.p"tlt'ons in lhe

I,,~hcr

pms oi'

011'

Ihc01Jlics. Sir Ir.I..,c N(WIOnand Mr Lribnilz coJltend ·

ed fur the Itnnour of inveming it

rroba~¡(

they

¡,Id both maJe progref, in the lame d,CcoYtry. unknown

lO' ?eh olher, before there \Vas

~ny

publication on the

fubje[t,

In th i, branch of mathematics magnitudes of every

kinJ

fup?ofed genermd by motion: a I,ne by the

moti~n

of a point, a furf,cc by the motion of a line, "nd

fol,J by the mOtioDof a fu,fAce. And fome pan of

ligure is fuppofed gencrmd by an uniform motion; in

ronfequl nc~

of which the olher pans m"y increale uni·

formly l'r \Vith .n

a~cekrated

or warded motion, or lIlay

r.medfe in

a~y

of th,Ce ways; and the compu tations are

marlé by traciog lhe comparative veloeities with \Vhieh

t!::

pam

flo.·.

Fig. t. If lhepmlldogram AlleD be generated by

3n uoifo, m motion of lhe line AB tOI'.'..d CD whtlé

mOl'eS from

lowardsj(, \Vhik thelinefl Freeeiyes thein·

crement Ff, and the fig"re \ViII be ioereafed by the parolle.

logram

F,;

the line FE io this cafeundergoes 00 varialioD.

The Jluxioo of aoy m"goitude at aoyPOiDt is the inere·

meot that it 1V0uld receiye in anygiven time, Cuppofing it

toinmaCe uniformly fromtha! point ; and as the mea·

fure, ..ill be the fame, whatever the time be, \Ve are al

liberty to fuppoCe it

I<C¡

thanan)' afligned time.

Tt!e firn leners in the alphdbet are ufed to reprefent

,nvan,ble quantities; the lener,

vari.ble quanti.

ties; and the fame Imers with points over them ;,

reprefent their

ftuxio~.

Thtrtfore if

3=11,

an¿ EF=x ;

Ff,

the fl uxioo of

BF,

wiU be= ; , and F" the Rnxion of AF,

=11;.

I ~

the re.'langle be fuppoféd geomted by the onlform

mouon of fe

tO..

~rds

eD,

at the fame time He moyes

unifo;mly

tOwarJ.

AD, thé point

Iteeping always on

thedtagooal, ,he lines fe

IVill fiolV unifo,mly,; for

"'h,le B/ ,mires the inCrentént

and HIJ , the incre·

~e,nt

IIK, FG

\ViII receiye the incremem

and HG the

lnerement

hl(.

and they wdl receive equal iocrements in

e~ual

fueeelhe times. But the paralldogram will fiolV

W'th ao aeceler:ued motÍon; for while F flows tO

and

it i, increafeel by the gnomoo

Ke/:

but while

30d

fio\\'

throu~h

the equal fpaces

K1,

it is io·

~reared

by the

~OOO1en

LglII

gremr thdO Ke( ; confe–

queotl~

IVhen Ru.ioos of the fi des of a p",tllelogram

are uOlform, the flux,on of the pJr.Ile1ogram

,ncrcal~s

t.Onunu:<lIy.

The

fic~ion

(lf the pmllrlogmn BHeF is the tIVO

pmUdograms

and

Gf;

fur th, u;:h the pu nolcte,

reeCilrs ao increment of the I!nOnt1'l1

KG!.

IVI"le its

~dlS

fiolV to

and

the pm

IS ou',ng to the addi-

1I00:,j

vellllity whe'ewlth

1111

r"'dl~dl)gr;:m ilow~

clur:ng

that,IImr; ano thc"f",c i, no I,;r' nf the meaf"r. of the

JX'OO, wlnd,

mlla

CUtllputCU

r"l'l'of:ng the rar,t·

meter to flow ttn:fo rn,ly as it oiJ at the beginning, witl.·

OUt auy acceler.tioo.

ThLfefure if the fiJ,s of

paraJ!c1ogram be

and

théir fluxions wiUbe

):; and the flu xion of the pa·

t.J1elogfdm

xj+J;:

and if

x=)',

that ¡s, if the figure be a

fq u,tre, the Bux,on of

will be

2X;.

FiC'

Let the ttiangle ABC be dcrctibcd by the

unifurm mo:ioo of

from

tOlVards

the point

moying in the line DF, fo dS dIVays to toueh the lines

AC, CH; while D moves from Ato F, DE is uniform–

Iy increafed, aod the inereafe of the triangle is uniform–

Iyaec/bated. When

is in the polition FC, it is

a maximum. As D moy/s from F

the line

decreares, and the triaeg!e ,ocréafes, but with a motion

unifonnly retardcd.

Fig

the femicircle AFB be geoeraleo by the

uniform motionof CD f,om

toward.

while Cmoves

fro",

¡\

tO G, the line CD wiJl increare, but \Vitn are·

tarded motion: the circumfertnce alfo incre,fes with a

ret~rrlcd

motion, aod tbt circular fpace increafes

~'ith

accdeldted motlon, but not unifo,dy, the degtee. of

acrelcrationgro.,ing Itfs as CD approache, tO the pofitioo

GF. Wheo e moyes from

(J ,

it decreafes \Vitl¡

a motion cominually aecclerated, the circumfmnce in–

creafes IVilh a motion continually aecelerated, aod the

area increafes \Vith a motion cominually rrtarded, and

quic~ly

retarded as CD approaches to B.

The fluxion of a quaotity which decre, fes

tObe con-

fidereJ as negatiye.

When a quantity does not fio\V uniformly, its fl uxioD

may bereprefented by avariable quantity, or a line of

."riable léog'o; the Ruxion of fu h a I,ne is eaJled the

ftcond fiux,on of the quantity " hofe fiuxion tbat line is:

and if it be variable, a third flnxion m"y be dedueed from

it,.and higher order! from th& in the fdme manae': the

Ctcond flux ion is rep" rcnr"d by t\Vo poiots, as ; .

The increment a quanttty receives by flowing for any

gi"en time, contaios meafurtl of all tht ddrérent order!

01' flux,ons ; for if it incré,les unifotmly, the whole in–

errmen! is the hrfl fiuxion; and it h"s no fecond IIltx:oo.

If it incre" frs with a motion u",fonnly ,ccebateel, the

palt oi' the ,ncrement oceafi"ned by th, (,rll motion mea·

fUlcl tile r,rn lIuxion, and the par! occafioo,d by the ac·

rder.tion mcafurcs thc I.c,,"d fl,Ixion.

[1'

the mvtion

not ollly accdératrd, but thé degree of aCederation

continu. lly increafrd, the't\Vo lira fiU XlllOS are me"lur,tI

as betore; aoJ rhe par! of the inen

ment

occafioncd hy

tite additillnal degrce uf accdrratioo me"furc) the th"d ;

aotl

Thde rr.cafi,rt! rcqll" e tO be (lI"céted, antl

Jre only mention,d hae tO tllullrm thefubj<éL

D I R E CT

O D,

AnJ JI,..I

¡,\~

q/ldr.titJ b,'ing git:n,

pntl itl JI:/xi ,n.

To finu tite fiuxiun of any pOll'cr of a qUJn.

titi