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FLUXIONS.

(4.) About the year 1699, Mr. Facio, in a Treatise on the Lineof Swiftest Descent, declared that he was obliged to own Newton as the first inventor of the differential calculus, and the first bymany years; and that he left the world to judge whether Leib-nitz, the second inventor, had taken anv thing from him. Thisprecise distinction between the first and second inventor, withthe suspicion it insinuated, raised a controversy between M. Leib-nitz, supported by the editors of the Leipsic Acts, and the English mathematicians, who declared for Newton ; but Sir Isaac himselfnever appeared on the scene. Writing succeeded each other, atfirst but slowly on either side, probablv on account of the distanceof places, but the controversy grew still hotter and hotter, till atlength, in the year 1711, Mr. Leibnitz complained to the Royal-Society , that Dr. Keil had accused him of publishing the methodof fluxions invented by Sir Isaac Newton , under other names andcharacters. He insisted, that nobody knew better than Sir Isaac himself, that he had stolen nothing from him : and required thatDr. Kefl should disavow the ill construction that might be putupon his words.

(5.) The society referred the matter to a committee, who, afterdue consideration, gave in their report as follows : “ That M.Leibnitz was in London in 1673, ami kept a correspondence withMr. Collins, by means of Mr. Oldenburg/), till Sept. 1676, whenhe returned from Paris to Hanover, by way of London and Am-sterdam : that it did not appear that M. Leibnitz knew any thingof the differential calculus before his letter of the 21st June, 1077,'which was a year after a copy of a letter written by Newton in1672, had been sent to Paris to be communicated to him, andabove four years after Mr. Collins began to communicate that let-ter to his correspondents ; in which the method of fluxions wassufficiently explained, to let a man of his sagacity into the wholematter ; and that Sir Isaac Newton had even invented his methodbefore the year 1669, and consequently 15 years before M. Leib-nitz had given any thing oh the subject in the Leipsic Acts.”From-which they concluded, that Dr. Keil had not at all injuredM. Leibnitz in what he said.

(6.) It seems beyond doubt, that Newton was the first inventor,but some have been of opinion, that notwithstanding the decisiongiven, by the Royal Society , there ar£ strong presumptions of M.Leibnitz having found the method for himself, without being pre-viously acquainted with Newton ’s invention.

(7.) As the method' of fluxions was invented by Sir Isaac New-ton , so it was carried by him to a considerable degree of perfec-tion, as may be seen in his Fluxions, and in his Quadrature ofCurves. The same has been since farther prosecuted and improv-ed by Maclaurin in his Treatise of Fluxions; by D’Alembert inthe MemoireS de 1’ Acad, de Berlin ; by Euler in bis variouswritings, as well as his Institutio Calculi Integralis ; by Cotes inhis Harmonia Mensurarum ; byWalmesley in his Analyse des Me-sures des Rapportes et des Angles, &c. and bv different authors,both of this and other countries, as Emerson, Simpson, Landen,Waring, Vince, &c. also, L’Hopital, the Bernoullis, Herman,Riccalt, La Grange, Cousin, and many others.

Definitions, Principles, and Notation of Fluxions.

(8.) In the method of fluxions, magnitudes or quantities of everykind, are considered as generated by continued motion, by meansof. which they increase or decrease ; as a line by the motion of apoint, a surface by the motion of a line, a solid by the motion of asurface, and an angle by the rotation of one ot the lines whichcontain it: the flux of time being supposed to be always uniform ;and since all quantities, when we consider their magnitude only,without regarding their position, figure, and other affections, maybe represented by lines, it follows, that quantities of every kindmay be considered as generated by motion.

(9.) Any variable quantity thus generated, is called a Fluent orFlowing Quantity ; and the rate, or proportion, according to whichany flowing quantity increases at any position or instant, is thefluxion of that quantity, at that position or instant. This rate ofincrease is proportional to the magnitude by which the flowingquantity would be uniformly increased in a given time with the ge-nerating celerity continued during that time, which magnitudemay therefore be taken as a measure of the fluxion of the variablequantity.

(10.) To illustrate these definitions, suppose a point m, Plate

LXXII, fig. 1, be conceived to move from the position A, andto generate a line AP, by a motion any how- regulated ; and sup-pose the celerity of the point In, at any position P, to be such, aswould, if from thence it should become, or continue uniform, besufficient to cause the poiht to pass uniformly over, the distancePp in the time allowed for the fluxion, then will the said line P prepresent the fluxion of the fluent or flowing line AP at that po-sition.

(11.) Again, suppose the right line inn, fig. 2, to move, fromthe position AB, continually parallel to itself with any continuedmotion, so as to generate the fluent or flowing rectangle ABQP,whilst the point m describes the line AP ; also let the distance P phe taken, as before, to represent the fluxion of the line, or baseAP, and complete the rectangle PQ q p ; then, like as P p is thefluxion of the line AP, so is P q the fluxion or flowing parallelo-gram AQ ; for if the line P p lie supposed to be generated withan uniform celerity, in a given time, the parallelogram P q willalso he generated with an uniform celerity in the same time.

, (12.) In like manner, if the solid AEUP, fig. 3, be conceivedto lie generated by the plane PQR moving, from the positionABE, always parallel to itself, along the line AD, and if P p de-note the fluxion of the line AP: then, like as the rectangle PQ qpexpresses the fluxion of the flowing rectangle ABQP, so also shallthe fluxion of the variable solid, or prism AfiERQP, be expressedby the prism PQR r qp. And in noth these last two cases, it ap-pears that the fluxion of the generated rectangle, or prism, is equalto the product of the generating line, or plane, drawn into thefluxion of the line along which it moves.

(13.) Hitherto the generating line or plane has been consideredas of a constant or invariable magnitude; in which case the fluent,or quantity generated, is a rectangle or a prism, the former beingdescribed by the motion of a line, and the latter by the motion ofa plane. So in like manner are other figures, whether pTane orsolid, conceived to be described by the motion of a variable mag-nitude, whether it be a line or a plane. Thus, let a variable linePQ, fig. 4, 5, and 6, be carried with a parallel motion along dieline AP, so that while the point P generates the tine AP, the linePQ may generate the area Al’Q ; by this means the point Q willgenerate the line AQ, which will be a straight line when PQ aswell as AP, flows uniformly, otherwise it will be a curve. Heretherefore, there are several fluents or flowing quantities, namely,,the absciss or base AP, the perpendicular or ordinate PQ, thearea APQ, and the straight line or curve AQ.

(14.) In order to exhibit the fluxions of these quantities: letPp, as before, he the fluxion of the base, complete the rectanglePQ r p, this rectangle will be the measure of the fluxion of thecurvilinear arcs APQ. For if the generating line were supposedto become invariable at the position PQ, it is evident that whilethe line Pp,\vas described by t|ie point P with an uniform celerity,the parallelogram P r would also he generated by the line PQuniformly, and with the very celerity with which the area APQwas increasing at the position PQ. Next suppose (hat the variableline PQ increases uniformly, after leaving the position PQ, withthe very degree of celerity ot'increase it had when in that position;it is evident that the point Q will now describe a straight lineQ q, which, as in tig. 5 and 6, will be a tangent to the curve at Q»this line will also be generated with an uniform celerity, viz. thevery celerity with which the generating point was moving in thecurve at the position Q. Hence it appears, that as P pis themeasure of the fluxion of the base, or absciss AP, and P r of thefluxion of the Area APQ, so is the line r q, the measure of thefluxion of the ordinate, or generating line PQ, and Q q themeasure of the fluxion of the curve line A Q.

(15.) In the doctrine of fluxions, the initial letters of the alpha-bet, a, b, c, d, See. are commonly used to denote constant or in-variable quantities ; and the last letters z, y, x, iv, Sc c. to denotevariable or flowing quantities. Thus, the variable line AP, fig*3, may be represented by x, and the constant line PQ by a ; also,in fig. 5 and 6, the variable absciss AP may be represented by X,the ordinate PQ by y, and the curve line A Q by z.

(16.) The fluxion of a variable quantity denoted by a single let-ter is represented by the same letter with a point over it. Thus,the fluxion of x is represented by x, the fluxion of y by y,therefore in fig. 5. the line Pp or Qrwill he represented by *>the line nq by y, and Qq by a. For these fluxions are as their

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