8838 
Neil and Van-Heuraet gave the first example of a History. 
curve that may be rectified, one of the cubic parabo- ““v——* 
las.) Van-Heuraet’s od reduced the problem of Van-Heu- 
rectifications to that of quadratures. Brouncker and tet 
Mercator, proceeding in the path of Wallis’ discoveries, Brouncker. 
FLUXIONS. 
ner of the ancients, he gave them with that clearness 
pe er os la ought always to characterize the. 
style of warks-on science: ~~ = eal Ti 
earlier than Descartes a method of 
"Fermat 
“Rat he-only published it after Descaites had - 
History. 
‘ ov Ne 
Fermat. 
Slussius, 
de 
8. Vincent. 
Wallis. 
ee nistmosn, and ‘he joined to it a method de mazi- 
nis et minimis. "These are more simple than Descartes” 
methods, but their author, fir from imitating the f¥ank= 
mazimis et minimis, his analysis, and the mode of de- , 
monstration. By a multitude-of discoveries, several of- 
y his communicative character, and the simple man- 
ner in which he has presented his researches. arvel 
Huygens first’ demonstrated: Fermat's two: rules. 
Slussius afterwards found:a simple method of drawing 
tangents, which at bottom was but the enunciation of. 
the caleulus required by: Fermat’s method ; i 
ccontvived? lie-chiarpelicristic tritmgle)whicl in faetrige 
the same as the triangle that measures: the fluxions of 
the abscissa, the and curve; and’ thus the 
of tangents: 
plication. of al 
was infinite, the. last term. may be reckoned as 
. Considering, then, surfaces as formed ‘of a 
series-of lines, the'terms of which follow a certain law, 
he found the ion. for the surface by summing. 
the series; The area ofa triangle, for example; was 
na 
of ny 
iw 
te: J 
found the first series known for the rectification of the 
circle, and h bola. Brouncker also first. noticed con- 
tinued fractions ; and he shewed that the fundamental 
~ principle employed by Neil in the rectification of: 
curves, and by which Mercator squared the hy- 
were to be feund in the works’ of Wallis. 
‘Mercator published his Logarithmotechnia in Septem- Mercator. 
ber 1668, which contained his quadrature of the hy- 
perbola ; and soon after the book came out, Mr Collins, 
to the Royal Society, sent'a copy to Barrow, 
at Cambridge, who put it into the hands of Sir Isaac, y.wton,. 
then Mr Newton, and a fellow of Trinity College. 
Presently afterwards, viz. in July 1669, Barrow wrote 
to Collins, that«a friend of :his*(Newton,) who had ‘an 
excellent —_ to these things, had brought him some 
papers, wherein he had set down methods of calcula-- 
ting the dimensions of ‘magnitudes, like:that of Mr~ 
Mercator for the:hyperbola, but very general; as also 
of resolving equations: Barrow afterwards: sent these- 
papers'to Collins,-saying, that: he presumed he would. 
be much’pleased: with them, and “requesting him to- 
shew: them ‘to Lord’ Brouncker.. Their 'title;was: De 
analysi per equationes numero terminorum-infinilas.. In 
this manuscript, the -method- of flaxions« was first indi- 
cated, and rules deduced from it given forthe quadra-: 
* ture of curves, to-which it was observed, their-rectifica-: 
tion, and the determinatiow of the quantity, and ‘the su- 
perficies of solids, and of the‘centre of gravity, may be. 
all:reduced: moreover, the author ‘there asserted, that» 
he knew no problems relating to the quadrature or rec-: 
tification of curves, to which his method would:not ap-: 
ply; and-that by means of -it,.he:could:draw tangents’: 
to mechanical curves; so, there can: be no doubt: butr 
that. then, Newton: ‘the method ‘of fluxions,, 
and therefore he must. be reckoned: the. first» inventor. 
Indeed it a: that although his discovery was»pro« 
mulgated forthe first time, he ‘had been ‘inpos« 
session of it from about’ the year 1666; which wasitwo 
ears before Mercator published hisquadrature of the. 
bola. And although the MS: memoir De analysi 
 ccthne omer ey &e. professes-to' explain ‘the method 
? 
rather than to:demonstrate “it. accurately, yet 
there was enough to shew, that:the author-was aware 
of its great i as-an instrument. of investiga- 
tion, and that he had reduced it'in-some measure'to the 
form of an analytical theory: 
Barrow, Collins, and Oldenburg, . (another 
_ to the Royal Society, ) disseminated: the analytical dis- 
| coveries of Newton by. their: 
ce, and: coms 
municated them to several geometers on the-continent, 
_ stch as Slussius, and Borelli.. 
In‘the-year 1672, the celebrated Leibnitz, who: af-' 
terwards also claimed the honour of the-diseovery of the 
method of fluxions, appeared for the first time upon’ the’ 
scene. »to be in London, he communicated: 
first method was icable, deserves particular atten- to some members of the Royal Society, certain re-' 
tion, because it was the germ of Newton’s most beauti- searches relating to the differences of numbers; but he. 
fal discoveries, and is at present the most i ant part. was given to understand, that this subject had been al- 
of the theory of series. This method led ‘him to.are- ready treated by Mouton, an astronomer? of Lyons: 
markable expression for ‘the area ofa circle. Wallis upon this, he turned his attention to the doctrine of in< 
must be allowed to have contributed ‘tothe pro- finite series, which, at that time,\ engaged ‘all the ma- 
of sis, er his oan rd and ‘his thematicians ; and, in 1674) he announced: to Olden- 
i e doctrine of series, which led to’ burg, that he possessed important theorems relative. to 
all the great discoveries of that period. © ' the quadrature of the circle by series; and that be-had 
Leibnitz. 
