History. 
—— ions ; and-one of the 
a 
Nicholas 
and Waniel 
Bernoulli. 
The sei 
FLUXIONS. 
binomial, leads to many beautiful applications of flux- 
Giaticebenathere ticiienhi iain 
late L , has made it the foundation 
of his theory of the calculus. f 
received considerable improvement from 
the mathematicians in Germany, particularly in that 
branch which relates to the fluents of fluxions, contain- 
ing several variable tities. The two Nicolas Ber- 
noullis, one a:son of James and the other a son of John 
Bernoulli, and Daniel Bernoulli, another son of John, 
/ i SORT woe 3% 
of profound disquisitions relating to the 
calculus; and to these may be»added the labours:of 
improvement of the calculus, by 
as challenges 
by name, to find the-path of a projectile, moving in a 
oa ee 
the velocity. » i quickly resolv “problem, 
not only in that particular case, but also when the re- 
sistance was as any a whatever of the velocity. He 
then required that should ce his own solu- 
tion ; but Keill had not resolv 
lor 
particular 
scodiegeatldigeteaienda tiine ipnaedivctesl 
to gui i resolve Taylor's 
problem, and more that he would Seaphearen 
which T: should not resolve, but whi 
could resolve himself.. Taylor did not accept this chal- 
pre mg solution of Taylor’s problem 
in the Leipsie Acts. 
The new calculi excited a con of a different 
kind, respecting the accuracy of their principles. These 
were attacked on the continent Niewentiit, and 
Rolle; and defended by Leibnitz, Varignon. 
rin. In England, Dr Berkeley, Bishop of Cloyne, called 
in ¢ ly the accuracy of the rea- 
employed toestablish the theory of fluxions, but 
aglish. As John Bernoulli 
aimed at, he offered 
also the faith of mathematicians in general, in regard to 
matters of at He began the in his 
work entitled the Minute P; But the principal 
addressed to an Infid , (understood to be 
Dr Halley,) wherein it is examined whether the object, 
principles, and of the modern analysis are more 
inferences 
listime tl ied than relict ix end! wed 
of faith: One of the best answers to the Bishop came 
from the of Benjamin Robins, in A discourse con- 
Berkeley, however, had some reason for his objecti 
The very: Guriéiss om 
manner in which the great inventor 
387 
had promulgated his discovery, might leave room for a History. 
ispute about the accuracy of the terms. Instead of de- 
Setitgsaheon it was better to adopt a mode of expla~ 
nation more intelligible, and consonant to the common 
methods of mathematical reasoning. This was done 
by Maclaurin, in his Treatise of Fluxions, (1742.) He 
has there placed the principles of the method upon the 
firm basis of geometrical demonstration; but his de- 
monstrations are tedious, so that we fear few’ have pa~ 
paves en to study the subject, as delivered in the 
ort-of hi 
ject.is considered in the usual manner, and algebraic 
characters are employed, is very valuable, and indeed 
the whole work abounds with original views of the 
theories connected with fluxions, and it proves the au- 
thor to belong to the highest class of mathematicians. 
Before the publication of Maclaurin’s treatise, Mr 
Thomas Simpson had given the first edition of his New 
Treatise of Fluxions, (1787.) He new modelled the 
work, published it in 1750. This was a 
very valuable work at the time it appeared, and, as far 
as it goes, is at the present time one of the best intro- 
ductions to the bnetbiod of fluxions in the English lan« 
ar ane a Doctrine of Fluxions came out in 1743. 
This has also been always much esteemed in England. 
It contains a great number of applications ; but ‘as it 
oo oo less within the reach of a beginner, 
impson’s is, we believe, more — 
- It is to the celebrated Euler that the calculus is in- 
debted for its greatest improvements. Indeed these are 
far too rumerous to find a place in the brief view 
which our limits allow us to give of the progress of the 
science ; even the titles of his various memoirs would 
fill several of our pages: his more remarkable works 
will be given in the list of books relating to the sub- 
ject in the conclusion. That branch of the calculus 
which treats of the higher class of problems, De mazi- 
mis et minimis, such as the solid of least resistance, the 
carve of swiftest descent, &c. was first reduced by him 
to the form of a distinct , in his Methodus invenia 
endi Lineas Curvas Maximi Minimive proprietate gau- 
dentes, Sive solutio Problematis Isoperimetrici lalisst» 
mo sensu accepti, (1744.) This was improved 
and new modelled by Lagrange, and denominated the 
Method of Variations. \t is aremarkable instance of 
Euler’s candour, that he took up the subject a second 
time, and laying aside his own theory, treated it accord+ 
ing to La ’s views, employing also the same 
notation. Euler's writings on the analysis of infinites, 
and the differential and integral calculus, are a trea- 
sure of analytical know 
fore produced by the labours of an individual. 
Adi 
Marquis Fagnano, or Fagnani, has contributed con- 
cident to the i ent of a branch of the flux« 
ional He found that it is always: possible to 
assign two arcs of an ellipse, reckoned 
mity of each axis, such, t cree, my egy be exe 
algebraic quantities ; a t any hyper- 
which has led to some remarkable transformations of 
fluxional formula, appears to have been but little known 
in Britain, as we do not recollect to have seen it in any 
of the mathematical works published in this country 
until it was also found by our ingenious coun nD, 
Mr Landen, who added to it another remarkable dis- 
covery, namely, that any hyperbolic are may always 
Maclaurin. 
work. The second, in which the subs 
Simpson. 
Emerson. 
Euler. 
ledge, richer than was ever be« | 
from one extre«" 
made by an Italian mathematician, the Fagnano. 
\ 
