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two kinds, the vitreous and the faline. The vitreous are thofe which either naturally have a gloffy form, or readily affume one in the fire ; the principal of thefe are glafs of lead, glafs of antimony and borax.
By the faline fluxes, authors generally mean thofe which are compofed of falts, whether tartar, nitre, fixed alkali or the like, as the black jfojir, fandiver, pot afbes, and the like. The vitreous fluxes feem deftined more immediately to act upon the irony or vitrefcible matter, wherewith the Irubborn ores are frequently mixed, and the faline kind to act more im- mediately upon the matter of the ore itfelf, for the due exclu- fion or feparation of the metal.
The more kindly ores require no Fluxes at all, and often they contain in themfelves, the neceflary matter for that yJaxr; thus there are copper ores, which being only ground to powder, and put into a common wind furnace, will yield as much pure metal at the full operation, as can be procured from them, by all the fluxes. Shaw's Lectures, p. 256. FLUXION (Cy*/.)— The elements of the do&rine ofyWnr, have been delivered by its great author, in (o concifc a manner, as to give occafion to one of the moft ingenious writers of this age, to reprefent it, as founded on inconceivable principles, and full of falfe reafonings. This author in a letter, under the title of the Analyfl, publifhed in the year 1734, has been at great pains to convince his readers, that the object, principles, and infe- rences of the modern analyfis by fluxions, are not more dis- tinctly conceived, or more evidently deduced, than religious myfteries and points of faith. He fays he does not controvert the truth of the conclufions, but only the logic and method of mathematicians. He asks how they demonftrate, what objects they are converfant with, and whether they conceive them clearly ; what principles they proceed upon, how found they may be, and how they apply them ; declaring himfelf not concerned about the truth of the theorems, but only about the way of coming at them, whether it be legitimate or ille- gitimate, clear or obfcure, fcientific or tentative. He con- fiders the conclufions not in themfelves, but in their premi- fes ; not as true or falfe, ufeful or inlignificant, but as deri- ved from fuch principles, and by fuch inferences. And for as much as it may feem an unaccountable paradox, that mathe- maticians {hould deduce true propofitions from falfe principles, be right in the conclufion, and yet err in the premifes, he en- deavours particularly to explain, how this may come to pafs, and fhew how error may bring forth truth, tho' it cannot bring forth fcience. His folution of the paradox is, that in the application of the method of infinitcfimals and Fluxions, two errors are committed which being equal and contrary, deftroy each other. We cannot enter into a detail of all he fays on thefe and many other heads, nor of all that has been faid on the other fide, in defence of the method of Fluxions, and of its inventor Sir Ifaac Newton. An Anfwer to the Analyfl, ap- peared very early under the name of ' Philalethes Cantabrigienfis, fuppofed to be Dr. Jurin ; a fecond by the fame hand, in de- fence of the firft; a difcourfe of Fluxions by Mr. Robins; a treatife of Sir Ifaac Newton's,- with a commentary by Mr. Colfon ; and feveral other pieces were published on this fub- je£t ; particularly a very full and excellent treatife of Fluxions, by Mr. Madaurin, late profeflbr of mathematics in the uni verfity of Edinburgh, containing not only a moft diftinct ac- count of the principles of Fluxions, but alio of the chief difco- veries in geometry, and mathematical philofophy of this age. The curious may find an elegant account of this treatife in the Philofophical Tranfactions, N". 468, 469. We prefume that Mr. Maclaurin's demonftrations are fuffi- cient to fatisfy the moft fcrupulous ; it would exceed the bounds of our defign to infert them at length here; but we cannot omit mentioning what feems neceflary to explain and illuftrate the notion of Fluxions ; and the principles on which this me- thod of computation is founded.
In the doctrine of Fluxions, magnitudes are conceived to be ge- nerated by motion, and the velocity of the generating motion ■ is the Fluxion of the magnitude. Lines are fuppofed to be gene- rated by the motion of points. The velocity of the point that defcribes the line, is Its Fluxion, and meafures the rate of its increafe or decreafe. When the motion of a point is uni- form, its velocity is conftant, and is meafured by the fpace defcribed by it in a given time. When the motion varies, the velocity at any term of the time is meafured by the fpace which would be defcribed in a given time, if the motion was to be continued uniformly, from that term, without any va- riation. And this is analogous to the general doctrine of pow- ers, or may be confidered as a particular application of it. As a power which acts continually and uniformly is meafured by the effect that is produced by it in a given time, fo the ve- locity of an uniform motion, is meafured by the fpace that is defcribed in a given time. If the action of the power vary, then its exertion at any term of the time is not meafured by the effect that is produced after that term in a given time, but by the effect that would have been produced if its adtion had -continued uniform from that term; and in the fame manner, the velocity of a variable motion at any given term of time, is not to be meafured by the fpace that is actually defcribed
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after that term in a given time, but by the fpace that would have been defcribed, if the motion had continued uniformlv from that term. If the action of a variable power, or the velocity of a variable motion, may not be meafured in this manner, they mull not be fufcc-ptible of any menfuration at all. When it is fuppofed that a body has fome velocity or other at any term of the time, during which it moves ; it is not therefore fuppofed, that there can be any motion in a term, limit or moment of time, or in an indivifible point of fpace ; and as velocity is always meafured by the fpace that would be delcribtd by it, continued uniformly for fome given finite time, it cannot furcly be faid, that geometers pietend to conceive motion or velocity without regard to fpace or time, as the author of the Analyfl often fuggefts. This is a fhort view of the nature and tendency of the dorTtrine of Fluxions, which we fhall now proceed to explain more particularly. We have already faid that lines may be conceived as generated by the motion of points; in like manner furfaces may be con- ceived as generated by the motion of lines; folids by the mo- tion of furfaces ; angles, by the rotation of their fides ; the flux of time being fuppofed to be always uniform. The velo- city with which a line flows, is the fame as that of the point which is fuppofed to defcribe or generate it. The velocity with which a furface flows, is the fame as the velocity of a given right line ; that, by moving parallel to itfelf, is fuppofed to generate a reflangle which is always equal to the furface. The velocity with which a folid flows, is the fame as the ve- locity of a given plain furface, that, by moving parallel to it- felf, is fuppofed to generate an ereS prifm or cylinder that is always equal to the folid. The velocity with which an angle flows, is meafured by the velocity of a point, that is fuppoled to defcribe the arch of a given circle, which always fubtends the angle, and meafures it. In general, all quantities of the fame kind (when we confider their magnitude only, and ab- flraa from their pofition, figure, and other affeflions) may be reprefented by right lines, that are fuppoled to be always in the fame proportion to each other as thefe quantities. They are reprefented by right lines in this manner in Euclid's'Ele- ments, in the general doclrine of proportion, and by right lines and figures in the data of that accurate geometer, "in this method likewife, quantities of the fame kind may be re- prefented by right lines, and the velocities of the motions by which they are fuppofed to be generated, by the velocities of points moving in right lines. All the velocities we have men- tioned are meafured, at any term of the time of the motion, by the fpaces which would be defcribed in a given time, by thefe points, lines or furfaces, with their motions continued uniformly from that term.
A Fluxion being the velocity with which a quantity flows, at any term of the time while it is fuppofed to be generated, is therefore always meafured by the increment or decrement that would be generated in a given time by this motion, if it was continued uniformly from that term without any acceleration or retardation : or it may be meafured by the quantity that is generated in a given time by an uniform motion, which is equal to the generating motion at that term. Time is reprefented by a right line that flows uniformly, or is defcribed by an uniform motion ; and a moment or termination of time repiefented by a point or termination of that line. A given velocity is reprefented by a given line, the fame which would be defcribed by it in a given time. A velocity that is accelerated or retarded, is reprefented by a line that in- creafes or decreafes in the fame proportion. The time of any motion being reprefented by the bale of a figure, anil any part of the time by the correfponding part of the bafe; if the or- dinate at any point of the bale be equal to the fpace that would be defcribed, in a given time, by the velocity at the corre- fponding term of the time continued uniformly, then any ve- locity will be reprefented by the correfponding'ordinate. The Fluxions of quantities are reprefented by the increments or de- crements, defcribed in the foregoing paragraph, which meafure them ; and inftead of the proportion of the Fluxions themfelves, we may always fubftitute the proportion of their meafures. When a motion is uniform, the fpaces that are defcribed by it in any equal times are always equal. When a motion is perpetually accelerated, the fpaces afcribed by it in any equal times that fucceed after one another, perpetually increafe. When a motion is perpetually retarded, the fpaces that are defcribed by it in any equal times that fucceed after one an- other, perpetually decreafe.
It is manifeft, converfely, that if the fpaces defcribed in any equal times are always equal, then the motion is uniform. If the fpaces defcribed in any equal times that fucceed after one another perpetually increafe, the motion is perpetually accele- rated : For it is plain, that if the motion was uniform for any time, the fpaces defcribed in any equal parts of this time would be equal ; and if it was retarded for any time, the fpaces de- fcribed in equal parts of this time that fucceed after one an- other would decreafe: both of which are againft the fuppofi- tion. In like manner it is evident, that a motion is perpe- tually retarded, when the fpaces that are defcribed in any equal times that fucceed after one another perpetually decreafe. The
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