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# Title: Lexicon Technicum: Or, An Universal English Dictionary Of Arts And Sciences (shelfmarks Leibn. Marg. 71; ZEN Leibn. Marg. 71)
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M A X
M a L
ment Petits, Sect. 3, (90. In the Acta Frud. Lips
1. D. 1683. P, 122. there is allo a Method o
determinins Maxima C Minima, bY one D. 3
chiesr applicable to the Dramins oF Pangents
to Curres: And in che lame B0Ok, and For che
Xear 1684. P. 467. Vou hare a Method oF che
FaMOUS MIE. LeibnitZ, FoOr che Came Purpole, 2c-
cordins ro his Calculus Differentialis. Sec alle
ihe Serench BOOk OF MI. De La Fires Conick
Sections, in Lacin; and Chap. 7. OF NieumentiitS
Anabsiis Insinitorum. There is also princed in
the First Volume oF Descartes Geometry, 2 MC.
chod oF Fruddes, For Aindins che Maxima (5 MI
mima, P. 107.
The Eollomins Acconne oF this Method, Wa
communicare d ro me by MIr. FTumphry Dit
ton a Peron very Skiltul in chele MaTeIS.
LR OBL F M.
To determine any Flomins Quantity in an F. quation
proposd, to an Extream Value.
Tho chere are varions excellent Mechods Fo
the doins of chis, vet there is nochins chat seems
(0 clear and natural, and is really (0 general, (0
quick and casie, 25 thac wrhich che Doctrine of
Fluxions Furnictres us wwichal.
ToTolre rhe Problem in any Case ThaE Can BC
proposd, is onty co make a just Application of
this General Rule, viz.
FTaving put the E quarion into Fluxions, let the
Fluxion os that Ouantity (mhose Fxtream Value
is souche) be supposed =0; by mhich means all
those Members of the Equation in which it v
found vill vanist, and ehe remainins ones mill
sive the Determination of the Maximum or Mi-
nimum desired.
DENIONSTRATION.
gr = 2rx, vod x = 2; which Value oF X
cubsticured in ihe Equarion ot ihe Curre, sives
=ry, and (0y = 1:  = rhe Semi con-
Iugace.
Again, Suppose yy u = d y 2 = c cd - 0
Wrhere y, Z. un are all Florvins Ouanriries; and
y is ro be determind eo an Lxtremum. Nou in
all such Cales, Wvhere che Equation will involve
theFluxions oF (0 many disterent Flowins Ouan-
ciries, e must endearour by convenient substi
curions ro expunse fome ot chem, chac ve may
hare no more than one TOrr OF Fluxions letrro
deal wirh; ot which che Lquation may be
clear d br ordinarr Diriion. And I chink chis
Mechod, which PIl proxose in the Arst place, c0
be a very easie and general one in order eo chis
End. The Equation is y y u - dy Z- ccd =0,
Crom wotence 2y y u -1 yy u = dy Z = dZy
o and purrins y = 0, yy i = dy 2 =0
Nou ro chron oux chele Fluxions in che last F.
quacion, we necd onty substiture che Value ot
cirher ot chem trom che decond Equation, wohere
ve sindt it (ex. er.) = dy 2. F d 7y = 297u
erxo yyu = dy Z = dy 2 ] d 2y = 27y u
dy Z = dy Z - 2 y y 0 =0, er∞0 d Z = 2yu
o Which is che Equarion delired
Suppole y y x = Zy X = m = 0, and y 10
Be decermined c0 an EXtremum. THeN 2yy X
F yy X = 2y X = Zy X = Zy X = 0, ane
matins y = 0, yy X = Z y X = Zy X = 0,
buc  = 271 x + 77 r = 27 x = 2) X, ereo
Substimrins ehis in che room oF Z in the FOrmer
Every Maximum or Minium is in ils orn Na. Equarion, Wve tave yy x - 2 y y X = y y X
eure a Stable Ouantity: F0 determine cheretore
any Florvins Ouantier to a Maximum OF Mini-
mum, is eo makre ic (instead ot a Florwing) a
Permanent one; buE the Fluxion OF a PerMA
nent duanties is equal ro Noching. From whence
che Realon ot che Rule is Susticieny clear.
Lerus illustrace chis by Come P. amples. F. X. 81
Suxpose BB x - y y X -] - cy x - d3 = 0
wvhere y and x are Florvins Ouancities, and
 2y ∞ - 2y ∞ = 2y x = 0; That 15, Zy X
= 2 yy x = 0, 2nd Z X = 2 y X-
BHE NIT. Niementiit Furnistres an Expedient for
this Purpose al(o in che 7ch Chap. OF his Analysis
Infinitorum: Ile considers che Flovins Ouanti
ries in the Rquarion, a5 che Ordinares of (0 ma-
ny several Curres, wrhich hare one common
abscissa; chen substicurins che Values oF chC
Fluxions of chese Ordinates (sorren by che sc
neral Properer oF all Curres) he brings aIl chc
is eo be determined t0 an Fxtream Valuc. Then Fluxions EO one Expreltion. LX. Sr. In THCF
Bb X - 2y y X = y y ∞ -) - c y x - (= c X y = 0, Quacion abOvC, y y u = d y Z = ccd = -0, Wvhere
and makins y = 0, bb x - yy x- L cy x = c
and y y = cy = bb, trom wtich Ouadrarick
Equation y mas be derermind.
we had yy i = dy 2 = 0; let y, u, Z BC Con-
ceirdro bethe Ordinares oFchree lereral Curres
wrhose common Abscisse ler = x, and ler the
Subcansent belongins 10 ube = n, and chat for
Equation ot an Fllipte, Wohere che L arus Rectum (Z be = 1; chem v =  = and Z = 2 , ereo
=r, che Latus Fransversum == 9; the Abscise
x, and che Ordinare = y; then ve have
13uX d9Z2
=0, chat 15 17 u = d Zn = 0.
2yy = 0, Hom irhence che Egnarion For y decermined e0 an Fxtremum.
Fis rrue ehis Equarion appeaes somectiins disse-
La 1 X = - = - 97 = 0, wotich is the