FUNCTION. 
whofe equation yg -as= 0, fubftituting gt 2S for y> 
and # + ¢ for x, the equation becomes 
y+ ae fre —axex—ae=o. 
= 0; therefore rejeCting thefe terms, and 
dividing others »y % 
ar + = —-a=o 
rejeCting the fécond term, ¢ being now fuppofed = 0, 
i= ay, =24. 
We may here likewife trace the analogy between the 
thod of Fermat and the differential calculus; for the in- 
determinate quantity e, by which » is augmented, anf{wers 
to the differential da ; and oe the correfponding sugmentae it 
tion of y, anfwers to its oe d 
This idea of — feems to have been original, ‘at 
confifts in aera into an equa- 
s afterwards fup- 
of the problem 
e whole equation has 
This idea ae 7” confidered as 
the new dope by which fuch rapid advances 
the 
BE: 
have been made in geom d anics. But 
obfcurity of its origin fe ll remains a blemifh in the 
elementary principles, as delivered mo riters on 
this fubje&t. They are all liable to the objeGtion, urged-by 
Berkly, of a /hifting 
eftablifhed upon the 
actual exiftence as a quantity, and confequences 
eftablifhed which a ee to hold good, aoe the 
oe is enieely changed by fel ae the fymbol equa 
of a fymbo 
ero. 
The cotemporaries of Fermat did not enter into the {pirit 
-of this mode of pede el confidered it as.only a ea 
ticular artifice, applica a few éafes, and fubjeét to 
ewe difficu ae rh the ‘third volume. of as sade of 
length Barrow ea: to fubftitute for thefe 
vanifhiag quantities of Fermat others fuppofed infinitely 
fmall. In 1674, he Lins his method of tangents, whic 
only a geometrical co ruction ofthe method of Fermat, by 
means of a triangle infinitely imall, formed by the fides e 
and -” < and by an infinitely fmall fide of the curve fup- 
pofed .a polygon. ‘With 
Infinitefimal quantities, from which arofe the differential 
calculus 
But t to yale has the fabje ect. patented relating to the 
ae and minima of - itities were not unknown to the 
One entire book © 
that can be see from a given Dont to the ar 
metho 
the proof, ce t every other ftraight lise drawn to the conic 
fe€tion is lefs in caie of the ma aa and pees in that 
thé minimum, than that which h 3 determined, and 
£ tha e 
this method has fince been generally followed by all tho fe 
tio 
of Aypothefis ; that 18, equations are 
upp ai on 
ae an 
the 
him originated the fyftem of 
who = endeavoured to {dlve thefe: problems ty fimple 
geom 
Fermet 
is to be a maximum or minimum, equal to zero. 
gave rife to 
But there is a {pecies of ae related to thefe, of a 
which, by revolving on its 
oppote d the leaft poffible ane 
direction of its axis. Thou ugh fome of thefe Pr scape had 
‘been oceafionally folved in particular inftan yet no 
general rule had been found epee ri them, ‘ill ‘Lagrang e 
dev dene the method known by the name of calculus varia- 
a fluid moving in the 
The firft methods attempted — the folution ‘of 
thefe problems was, according to the differential notion, 
by dividing the curve into an ane nite number of polygons, 
then determining the pofition pel two adjacent fides, fo that 
oan Propo ofed becomes a ma 
nly a 
was aor rare 7 
at {cent was fuch, 
fine of the angle which one of its infinitely fmall files make 
with the vertical, muft always be proportional to the velo- © 
pel which is as the {quare root of the height from oleh 
body has defcended ; and this proportions reduced to a 
ai ae ntial equation, gives the cyc method was 
afterwards applied to Cobian fall more complica ia 
ticularly to thofe called i/operimetrical, in which i 
eae to find among all 
gr or leaft om the difficulty of thefe 
problems, joined to the as which they acquired by 
the iboun of Euler and the feprbahagp: the general cad 
of ifoperimetrical was given to them, even in the cafe w 
the condition of equality in fhe length was not a 
n all the ears ape Sar relative to maxima ay 
ie, when z has been fuppofed a fundtion of «x, 
have been f{uppofed i have fome conftant relation i an 4 
other during the courfe of the problem, but in rO- 
blems, particularly in thofe under confideration, thele rela. 
tions are w pente peer el to change. Ina 
x initance, + a a and dy havea relation to each 
other defined by he nature a the curve, but in the method 
of variations, the ordinates y are no longer bounded by the 
original curve, may pafs into another, having no deters 
aminate relation with the former. 
o 
Let 
