DIFFERENTIAL 
It is called a differential, or differential quantity, becaufe 
frequently confidered as the difference o 
and, as aa it is the foundation of the differential eal 
eae Sir Ifaac Newton, and the Englifh. cail it a mo- 
ment, as sing ‘confidered as the momentary increafe or de- 
creafe of a variable Aah ity. Mr. Leibntz, and others, 
call it alfo an ane mal. See [nrinirestmAL. 
DIFFERENTIAL of tei eon i3¢. degree. 
FERENTIO- —— FRENTI F 
DirFsrentiat cale a or method, is a method of dif- 
Recut quantities ; that is, of finding a d-ffcrential, or in- 
fisitely {mall quantity, ane, taxén an inhnite number of 
times, is equal to a given slacks See Carcutus, Dir. 
FERENTIAL methed, and Fruxio 
Dirrrgenrio-DiFFERS NTIAL "callus is a method of 
aifferencins die penta qvantit 
As the fizn of a differ ae is ie teter d prefixed to the 
quantity, dwis the diff: an ial of xs that of a es 
i d the differential of ddx isdddx, & 
fimilar to the Auxiona x, %. #, &e. 
nus we have degrees of di eee 
e differential of an ordinary sail : called a dif- 
ferential of the firft order or degree, as dx; that of the fe- 
cond degree, is an Hees of a diffe cal quantity of 
the firft “decree, asddx of the third degree, is an 
infinitefimal of a : ferenal ane of the fecond degree, 
x, and fo 
The powers ot ‘d fferentials are differenced after the fame 
manner as the powers of ordin : and, a 
as compound differentials either multiply or divide each 
or are powers of differentials of the firlt degree ; dif- 
ferentials are differenced after the fam 8 ordinar 
quantities ; and, therefore, the differ pee differential cal- 
culus is the ral in effet, with the differential, or the 
ee of flux 
ENTIAL, in e Doftrine of Logarithms. aa 
calls el ogarithms of gents aiferetils whic 
ufually call artificial tangents. e Lecantrun, ee 
RivtHm, and T'AnGENT 
DIFFERENTIAL eguation d by fome mathemati- 
cians for an equation involving infinitefimal differences, or 
fluxions. Tus the svete 3° dx —2ax 
—39 dy+ axdy = the foreign raga or 3x 
—24 - sy +axy= o in th 
tion, is called a differential equa ation 
See Dir- 
have a the term differential 
enfe, to certain equations defining the 
See ake 
n Ma thematics an appellation 
given toa ‘aaa . cade: quantities by means of their 
fucceffive differe 
This peed ig cable of very a ara and 
ufe, in the conftruGion of tables, fummation of feriefes, &c. 
It was firft ufed, and the rules of it laid down, Briggs, 
in his conftruction of logarithms numbers, much 
in his ‘* Conftru 
Briggs’s Arithmetica a cap. 12, 13. and his 
« Trigonometria Britan 
is method is ren in Sansa form by fir [aac New- 
ton in the fifth lemma of the third book of his Principia. . 
the 
He treats of it asa method of defcribing a curve of 
parabolic kind, through any given number of points: And 
he diftinguifhes two cafes of this problem; the fuft, when 
the ordinates, drawn from the given a - any line gives 
n, are et equal diltances from ea see r; and the 
ld 
cafes, but ne demon- 
{tration in that place, which has Grce heen fupplied by him. 
felf and others. See his Methodus Di abot publiihed 
with other traéts of the fame author, by Mr. Jones, Lon. 
don, 17113 and Surling’ 3 pet - the Newtonian 
N° 
differential method, in the Phil. Trasf. 362; Cotes, De 
Methodo a ential poate niana, in his works publithed 
by Dr. Sm Herman, Phoronomia; and Le Sear and 
ee in ree Comment on fir Ifa4ac Newton’s Prine 
cipia 
It is to be obferved, that the methods there demonflrated, 
by fome of thefe authors, extend to the defeription of any 
algebraic curve through a given number of points, which 
fir Ifaac, writing to Mr, Leibnitz, mentions as a problem 
of the ky ua 
enmed hate 
a therefore. ae ame-~ 
Aan Le Newton 
thod for interpolations. 
Stirling? s Method. 
Method. Differ. prop. 5. 
Any curvilinear figure may alfo be fquared ieaily. of 
which fome ordinates may be found. Newt. Meth. Diff, 
Simpfon’s Math. D: . p. 115. 
e extended to the conitruction of 
mathematical tables by ae: Newt. Meth. Diff. 
=? 
The re ae differences of the ordinates of parabclic 
curves, becoming u'tima nl equal, and the intermediate or- 
dinate required, bei 0 
rules, by thefe fee ‘of the 
this method being called the differential method. 
ittle more particular. 
he cafe of fir I{aac’s problem amounts to this: a 
feries of einer placed at equal intervals, being given, to 
find any intermediate number of that ferics when its interval 
from the firll term of the feries is given. 
Subtra& every term of the feries from th 
ing, and let the remainders be called firlt differences ; then 
fubtraét each difference from the next following, and let 
thefe remainders be called fecond differences; again Jet each 
fecond difference be fubtra&ted from the next paged oe ins 
let ey remainders be called third differences, and fo 
then i 
To 
e next follow- 
tw of ie ies and any ae fought, 
that is, let the number of terms fro » both inclu- 
five, be = x +1, then will the term fought, 
H.M—T wd = 
E=A+4-* f4e st = df RS gp RET 
1.2.3 I 
m—a~2 65 
anes 3 gy 4, &e. which feries differs from the 
2. 
3: d" al 
evens in thia, that the quantities rar eae oa 
; 22.3 
nie 
pn inane here ufed, fignify the fame with d’, @”, ufed 
<a 
- Gir Ifaac Newton. 
ant - the differenees of any order become equal, that 
s, if a e quantities d”, d”, d"", become = 0, we 
thall cae a cane expreffion for E, the term fought; it be- 
4K 2 ing 
