F L U 
FLUXTBI'LITY, f. The (late or quality of being 
fiuxible. Scott. 
FLUX'I BLE, adj. Capable of being fluxed. Not much 
itfed. 
fjLUXTBLENESS,/ The (late of being fluxible. Scott. 
FLUXI'LITY, f. [fluxus, Lat.] Eadnefs of reparation 
of parts; poflibili’ty of liquefaction.—Experiments feem 
to teach, that the fuppofed averfation of nature to vacuum 
is but accidental, or in confequence, partly of the weight 
and fluidity, or at leaftfluxility, of the bodies here below. 
Boyle. 
FLUX'ION, f. [ fluxio, from fluo, Lat. ] The aft or 
movement of a flowing body ; the matter that flows. In 
mathematics, it is applied to the analyfis of variable quan¬ 
tities ; or to the method of finding a quantity from its 
rates of increafe or decreafe. Mod foreign writers define 
this as the method of differences, or differentials, being 
the analyfis of indefinitely frnall quantities, which taken 
an infinite number of times, make a finite quantity. But 
our Englifli mathematicians confider all quantities as ge¬ 
nerated by motion ; as a line by the flux or motion of a 
point; or a furface generated by the flux of a line. Ac¬ 
cordingly, the variable quantities are called fluents, or 
flowing quantities; and the method of finding either the 
fluxion, or the fluent, the method of fluxions. Leibnitz 
conliders the fame infinitely fmall quantities as the diffe¬ 
rences, or differentials, of quantities; and the method of 
finding thefe differences, he calls the differential calculus-, 
whence foreign geometricians have adopted thefe terms: 
the two methods, however, are effentiaily the fame. 
The method of fluxions is unqueffionably one of the 
fublimetl difeoveries of the feventeenth century ; and it 
owes its invention to the immortal Newton. That honour, 
however, was long difputed by the celebrated Leibnitz; 
and the partifans on each fide became embroiled in a vio¬ 
lent controverfy on the fubjcdl, till at length an applica¬ 
tion was made by Leibnitz himfelf to the Royal Society 
of London to decide the claim. The prefident, on this 
important oecafion, 1 appointed a committee of its mem¬ 
bers to inveftigate all the letters, papers, and documents, 
that related to the point; and, after a very ffriCl exami¬ 
nation and enquiry, they gave in their report as follows : 
4 ‘ That Mr. Leibnitz was in London in 1673, and kept a 
correfpondence with Mr. Collins by means of Mr. Olden- 
burgh, till September 1676, when he returned from Paris 
to Hanover by way of London and Amfferdam: that it 
did not appear that Mr. Leibnitz knew any thing of the 
differential calculus before his letter of the 21ft of June 
1677, which was a year after a copy of a letter, written 
by Newton in 1672, had been fent to Paris to be commu¬ 
nicated to him, and above four years after Mr. Collins 
.began to communicate that letter to his correfpondents ; 
in which the method of fluxions was fufnciently explained, 
to let a man of his fagacity into the whole matter: and 
that fir Ifaac Newton had even invented his method be¬ 
fore the year 1669, and confequently fifteen years before 
M. Leibnitz-had given any thing on the fubjeCt in the 
Leiptic A6ls, or in any other publication.”—Hence they 
unanimoutly concluded, that the fluxionary calculus owed 
its difeovery to fir Ifaac Newton. 
In the following Treatife we do not propofe to alter fir 
Ifaac Newton’s 'notion of a fluxion, but to explain and 
demonflrate his method by deducing it from a few felf- 
evident truths ; and, in treating of it, to abftradl from all 
principles and poftulates that may require the imagining 
any other quantities but fuch as may be eafily conceived 
to have a real exiftence. We (hall not confider any part 
of fpace or time as indivifible, or infinitely little; but we 
(hall confider a point as a term or limit of a line, and 
a moment as a term or limit of time : nor (hall we re- 
folve curve lines, or curvilinea! (paces, into redtilineal 
elements of any kind. In delivering the principles of 
this method, <ve apprehend it is better to avoid fuch 
fuppofitions : but after thefe are demonftrated, (hort and 
concife ways of fpeaking, though lefs accurate, may be 
FLU 47-5 
permitted, when there is no hazard of our introducing 
any uncertainty or obfeurity into the fcience from the ufe 
of them, or of involving it in difputes. In the doilrine 
of fluxions which we propofe to explain, we have recourfe 
to the genelis of quantities, and either deduce their rela¬ 
tions, by comparing the powers which are conceived to 
generate them; or, by comparing the quantities that are 
generated, we difeover the relations of thefe powers, and 
of any quantities that are fuppofed to be reprefented by 
them. The power by which magnitudes are conceived 
to be generated in geometry, is motion .• and therefore we 
mud begin with fome account of it. 
No quantities are more clearly conceived by us than 
the limited parts of fpace and time. They confitt indeed 
always of parts; but of fuch as are perfectly uniform and 
fimilar. Thofe of fpace exifl together ; thofe of time flow 
continually: but by motion they become the meafures of 
each other reciprocally. The parts of fpace are perma¬ 
nent ; but, being deferibed fucceflively by motion, the 
fpace may be conceived to flow as the time. The time 
is ever perifhing; but an image or reprefentation of it is 
preferved and prefented to us at once in the fpace de¬ 
feribed by the motion. Time is conceived to flow always 
in an uniform courfe, that ferves to meafure the changes 
of all things. When the fpace deferibed by motion flows 
as the time, fo that equal parts of fpace are deferibed in 
any equal parts of the time, the motion is uniform ;. and 
the velocity is meafured by the fpace that is deferibed in 
any given time. As this fpace may be conceived to be 
greater or lefs, and to be fufceptible of all degrees of 
affignable magnitude ; fo may the velocity of the motion 
by which we fuppofe the fpace to be always deferibed in 
a given time. The velocity of an uniform motion is the 
fame at any term of the time during which it continues ; 
but motion is fufceptible of the fame variations with other 
quantities, and the velocity in other inflances may increafe 
or decreafe while the time increafes. In thefe cafes, how¬ 
ever, the velocity at any term of the time is accurately 
meafured by the fpace that would be deferibed in a given 
time, if the motion,were to be continued uniformly from 
that term. 
Any fpace and time being given, a velocity is deter¬ 
mined by which that fpace may be deferibed in that given., 
time; and, converfeiy, a velocity being given, the fpace 
which would he deferibed by it in any given time is alfo 
determined. This being evident, it does not feem to be- 
neceffary, in pure geometry, to enquire further what is 
the nature of this power, affedlion, or mode, which is 
called velocity, and is commonly aferibed to the body that 
is fuppofed to move. It feems to be fufticient for our 
purpoie, that, while a body is fuppofed in motion, it mud 
be conceived to have fome velocity or other at any term 
of the time during which it moves ; and that we can de- 
monftrate accurately what are the meafures of this velo¬ 
city at any term, in the enquiries that belong to this 
dodlrine. 
There are two fundamental principles of this method. 
The firil is, that, when the quantities which are gene¬ 
rated are always equal to each other, the generating 
motions muff be always equal. The fecond is the cbn- 
verfe of the firil, that, when the generating motions 
are always equal to each other, the quantities that are 
generated in the fame time muff be always equal. The 
firff: is the foundation of the direB method of fluxions; 
the fecond of the inverfe method. When any quantity is 
propofed, all others of the fame kind may be conceived 
to be generated from it; fuch as are greater than it, by 
fuppofing it to be increafed ; fuch as are lefs, by fuppo¬ 
fing it to be diminilhed. In the common arithmetic, in¬ 
teger numbers are conceived to be produced by adding a 
given quantity or unit to itfelf continually ; and fractions 
are produced by fuppofing it to bedivide4 into Inch parts 
as by a like addition would generate the given quantity 
itfelf. But in geometry, that all degrees of magnitude 
may be produced, and in fuch a way as may found a gene¬ 
ral 
