into black scoriæ, neither by itself nor with which appears of a violet by refraction, and fluxes does it exhibit a regulus. It grows red this colour is very fixed in the fire. Cobalt is on roasting: precipitated upon iron from the blue globule, Tin easily melts before the blow-pipe, and but not upon copper. When cals of iron is is calcined. The fluxes dissolve the calx mixed with that of cobalt in a Aux, the former sparingly; and when saturated, contract a is dissolved. This seinimetal takes up about milky opacity. Some small particles of this one-third of its weight of sulphur in fusion, metal dissolved in any flux may be distinctly after which it can hardly be melted again. It precipitated upon iron. Crystallized ore of is precipitated by iron, copper, and several iin, urged by fire upon the charcoal, yields its other metals. The common ore yields an immetal in a reguline state. pure regulus by roasting. The green cobalt, Bismuth presents nearly the same appear- examined by our author, tinges the microcosances of lead; the calx is reduced on the coal, nic salt blue; but at the same time shows red and fused in the spoon. The calx, dissolved spots indicating copper. in microcosmic sali, yields a brownish yellow Zinc exposed to ihe blow-pipe melis, takes globule, which grows inore pale upon cooling, fire, sending forth a beautiful bluish-green at the same time losing some of its transpa- flame, which, however, is soon extinguished jency. Too much calx renders the matter by a lanuginous calx : but if the reguline nuperfectly opaque. Borax produces a similar cleus included in this lanuginous matter (conmass in the spoon, but on the coal a grey one, monly called flowers of zinc) is urged by the which can scarcely be freed from bubbles. On flame, it will be now and then inflamed, and, fusion the glass mokes, and forms a cloud as it were, explode and Ry about. With borax about it. Bismuth is easily precipitated by it froths, and at first tinges the flame. It concopper and iron. Sulphurated Lismuth is easily tinually diminishes, and the Aux spreads upon fused, exhibiting a blue flame and sulphureous the charcoal; but in fused microcosmic salt, smell. Cobalt, when added, by means of sul- it not only froths, but sends forth flashes with phur, enters the globule; but the scoriæ soon a crackling noise. Too great beat makes it swells into distinct partitions; which, when explode with the emission of ignited particles. further urgerl by fire, throw out globules of The white calx, or flowers, exposed to the bismuth. Sulphurated bismuth, by the addi- flame on charcoal, becomes yellowisli, and has tion of borax, may be distinctly precipitated a kind of splendour which vanishes when the by iron or manganese. fame ceases. It remains fixed, and cannot be Regulus of nickel when melted is calcined, melted. The Auxes are scarcely tinged, but hut more slowly than other metals. The calx when saturated by fusion, they grow opaque imparis an hyacinthine colour to fluxes, which and white on cooling. Clouds are formed grows yellow on cooling, and by long-conti- round the globules, of a nature similar to those nued fire may be destroyed. If the calx of of the metallic calx. Dissolved zinc is not nickel is contaminated by ochre of iron, the precipitated by any other metal. When minelatter is first dissolved. Nickel dissolved is ralized by carbonic acid gass, it has the same precipitated on iron, or even on copper; an properties as calcined zinc. In the pseudoevident proof that it does not originate from galena sulphur and iron are present. These either of these metals. Sulphurated nickel is generally, on the charcoal, smell of sulphur, no where found without iron and arsenic; the melt and unge the flame more or less, depositreguus is obtained by roasting, and susing with ing a cloud all around. Those which have no borax, though it still remains mixed with some matrix are tinged by those which contain iron, other metals. and acquire by saturation a white opaque coRegulus of arsenic takes fire by a sudden lour, verging to brown or black, according to heat, and not only deposits a white smoke on the variety of composition. charcoal, but diffuses the same all around. Regulus of antimony, fused and ignited on The calx smokes with a smell of garlic, but the charcoal, affords a beautiful object; for if does not bum. The fluxes grow yellow with- the blast of air be suddenly stopped, a thick nui growing opaque, on adding a proper quan- white smoke rises perpendicularly, while the tity of calx, which is dispelled by a long lower part round the globule is condensed into continuance of the heat. This semimetal is crystalline spiculæ, similar to those called arprecipitated in a metallic form by iron and gentine flowers . The calx tinges fluxes of an copper, but not by gold. Yellow arsenic li- hyacinthine colour ; but on fusion smokes, and quesies, smokes, and totally evaporates : when is easily dissipated, especially on the charcoal, hcated by the external Aame, so as nei- though it also deposits a cloud on it. The disther to liquefy nor smoke, it grows red, and solved metal may be precipitated by iron ant yellow again upon cooling. When it only copper, but not by gold. Crude antimony libegins to meli , it acquires a red colour, which quefies on the charcoal, spreads, smokes, peneremains after cooling. Realgar liquefies more trates it, and at last disappears entirely, except easily, and is besides totally dissipated. a ring which it leaves behind. Regulus of cobalt melts, and may partly be Regulus of manganese scarcely yields to the depurated by borax, as the iron is first calcined Aane. The black calx tinges the fluxes of a and taken up. The smallest portion of the bluish colour; borax, unless saturated, comcaix tinges the flux of a deep-blue colour, municates more of a yellow colour. The co lour may be gradually dissolved altogether by quantities; and the method of finding those difthe interior flame, and again reproduced by a ferences, he calls the differential calculus. smali particle of nitre, or the exterior flame Besides this difference in the name, there is alone.' Coinbined with carbonic acid, it is of another in the notation. Newton expresses the a white colour, which changes by ignition to fluxion of a quantity, as of I, by a dot placed over black. In other respects it shows the same ex- it, thus x; while Leibnitz expresses his differential periments as the black cals. of the same 1, by prefixing the initial letter d, as Flux, in medicine, a disease generally de- dı. But, setting aside these circumstances, the scribed under the term dysentery, and in the two methods are just alike; though the principles nosology of Cullen denominated specifically on which they are established are different: fiusdysenteria sanguinea. See DYSENTERIA. ions being referred to the doctrine of motion; 'Flux. a. (fluxus, Latin.) Unconstant; differentials to such a kind of augmentation as is more directly referable to analysis. not durable; maintained by a constant succes- The method of fluxions is one of the greatest, sion of parts. most subtle, and sublime discoveries of perhaps TO FLUX. v. a. 1. To melt. 2. To sali- any age: it opens a new world to our view, and Fate ; to evacuate by spitring (South). extends our knowledge, as it were, to infinity; FLUXILITY.s. (furus, Latin.) Easiness carrying us beyond the bounds that seemed to of separation of parts (Boyle); have been prescribed to the human mind, at least FLUSION, s. (fluxio, Latin.) 1. The intinitely beyond those to which the ancient act of flowing. 2. The matter that flows geometry was confined. (Wiseman). The history of this important discovery, recent FLUXION, in the Newtonian analysis, de as it is, is a little dark, and embroiled. Two of notes the velocity with which a flowing quan- claimed the invention, sir Isaac Newton, and M. the greatest men of the last age have both of them tity is increased by its generative motion: by Leibnitz; and nothing can be more glorious for which it stands contradistinguished from fluent the method itself, than the zeal with which the or the flowing quantity, which is gradually partizans of either side have asserted their title. and indefinitely increasing, after the manner The two great authors themselves, without any of a space which a body in inotion describes, seeming concern, or dispute, as to the property of Or, a fluxion inay be niore accurately defin- the invention, enjoyed the prospect of the proed, as, the magnitude by which any flowing gresses continually making under their auspices, quantity would be uniformly increased in a till the year 1699, when the peace began to be given portion of time, with the generating ce disturbed. Terity at any proposed position, or instant, sup- Descent, declared, that he was obliged to own M. Facio, in a Treatise on the Line of Swiftest posing it from thence to continue invariable. From this definition it appears, that the calculus, and the first by many years; and that Newton as the first inventor of the differential Auxions of quantities are, always, as the cele. he left the world to judge, whether Leibnitz, the rities by which the quantities themselves in second inventor, had taken any thing from him. crease in magnitude. This precise distinction between first and second Mr. Simpson observes, that there is an ad- inventor, with the suspicion it insinuated, raised vantage in considering Auxions, not as mere a controversy between M. Leibnitz, supported by velocities, but as the magnitudes which these the editors of the Leipsic Acts, and the English velocities would, uniformly, generate in a given mathematicians, who declared for Newton.°sir finite time: the imagination is not here con Isaac himself never appeared on the scene; his fined to a single poini, and the higher orders glory was become that of the nation; and his adof Auxions are rendered much more easy and herents, warm in the cause of their country, needintelligible. And though sir Isaac Newton ed not his assistance to animate them. defines Ausions to be the velocities of motions, either side; probably on account of the distance Writings succeeded each other but slowly, on yet he hath recourse to the increments or mo of places; but the controversy grew still hotter ments, generated in equal particles of time, in and hotter: till at length M. Leibnitz, in the year order to determine those velocities, which he 1711, complained to the Royal Society, that Dr. afterwards teaches us to expound by finite Keil had accused him of publishing the Method magnitudes of other kinds. of Fluxions invented by sir I. Newton,under other Method of Fluxions, is the algorithm and names and characters. He insisted that nobody analysis of fluxions, and fluents or flowing quanti- knew better than sir Isaac himself, that he had ties. stolen nothing from him; and required that Dr. Most foreigners define this as the method of Keil should disavow the ill construction which differences or differentials, being the analysis of might be put upon his words. indefinitely small quantities. But Newton, and The society, thus appealed to as a judge, apother English authors, call these infinitely small pointed a committee to examine all the old letquantities, moments; considering them as the ters, papers, and documents, that had passed momentary increments of variable quantities; as among the several mathematicians, relating to the of a line considered as generated by the flux or mo- point; who after a strict examination of all the tion of a point, or of a surface generated by the evidence that could be procured, gave in their reflux of a line. Accordingly, the variable quanti- port as follows: “That Mr. Leibnitz was in Lonties are called fluents, or flowing quantities; and don in 1673, and kept a correspondence with Mr. the method of finding either the fluxion or the Collins by means of Mr. Oldenburgh, till Septemfluent, the method of fuxions. ber 1676, when he returned from Paris to HanoM. Leibnitz considers the same infinitely small ver, by way of London and Amsterdam: that it quantities as the differences, or differentials of did not appear that Mr, Leibnitz knew any thing of the differential calculus before his letter of the casioned by Collins's communication of the letter 2ist of June, 1677, which was a year after a copy of 1672; and though we, instead of positive, have of a letter, written by Newton in the year 1672, only presumptive proof, we are decidedly of opihad been sent to Paris to be communicated to nion that Leibnitz saw, in Collins's possession, him, and above four years after Mr. Collins began papers which acquainted him with Newton's disto communicate that letter to his correspondents; covery. We request that the reader will compare in which the Method of Fluxions was sufficiently with Leibnitz's acknowledgment the following explained, to let a man of his sagacity into the relation, for the truth of every part of which we whole matter; and that sir I. Newton had even hold ourselves responsible. invented his method before the year 1669, and In the year 1669, amongst other series by sir consequently 15 years before M. Leibnitz had Isaac Newton, one for finding the arc of a circle given any thing on the subject in the Leipsic from the sine, and, in 1671, another by Mr. GreActs." From which they concluded that Dr. gory, for finding the arc from the tangent, were Keil had not at all injured M. Leibnitz in what sent to Mr. Collins, who was very free in comhe had said. municating these and other discoveries. ln 1674 The society printed this their determination, Leibnitz mentions in a letter to Oldenburgh bis together with all the pieces and materials relating being possessed of the first series; and the next to it, under the title of Commercium Epistolicum year those of both Newton and Gregory were sent de Analysi Promota, 8vo. Lon. 1712. This book by Oldenburgh to Leibnitz. But in 1676 Leib. was carefully distributed through Europe, to nitz dropped his pretensions to the first series, vindicate the title of the English nation to the not being able to demonstrate it, and sent to discovery; for Newton himself, as already hinted, Oldenburgh, as his own, that of Gregory, with a never appeared in the affair: whether it was that demonstration. Both Newton and Gregory adhe trusted his honour with his compatriots, who mitted that Leibnitz found out this series; for were zealous enough in the cause; or whether he they knew nothing of Oldenburgh's letter, the felt himself even superior to the glory of it. copy of which lay buried for more than 30 years M. Leibnitz and his friends however could not among the papers of the Royal Society: so that at shew the same indifference: he was accused of a length, though not till 1713, Leibnitz was comtheft; and the whole Commercium Epistolicum pelled to acknowledge Gregory as the original either expresses it in terms, or insinuates it. Soon author. Nay, from the whole tenour of this after the publication therefore, a loose sheet was gentleman's conduct, he may be justly suspected printed at Paris, in behalf of M. Leibnitz, then at of having often learned by information what he Vienna. It is written with great zeal and spirit; affirmed to have invented: for he pretended to and it boldly maintains that the Method of Flux- Mouton's differential method; to a property of a ions had not preceded the Method of Differ- series that had been discovered by Pascal; to ences; and even insinuates that it might have four or five different series invented by ewton; arisen from it. The detail of the proofs however, to a method of progression; to the differential on each side, would be too long, and could not be analysis, when it is certain he was ignorant of it; understood without a large comment, which must and lastly, to some of the principal propositions in enter into the deepest geometry. the Principia. Newton's grand work was first M. Leibnitz had begun to work upon a Com- published in 1686: it was criticised at Leipsic by mercium Epistolicum, in opposition to that of the Leibnitz, in a review managed by himself, in 1987; Royal Society; but he died before it was com- and, two years afterwards, he pretended to have pleted. invented some propositions contained in the PrinA second edition of the Commercium Epistoli. cipia, relative to the motion of the planets in cum was printed at London in 1722; when New- ellipses. Well might this gentleman be characton, in the preface, account, and annotations, terised as having “ a vast and devouring gevius!" which w re added to that edition, particularly for he was determined to devour every choice answered all the objections which M. Leibnitz and morsel that fell in his way. We attempt pot to Bernouili were able to make since the Commer- depreciate his talents: but that he was a playiary cium first appeared in 1712; and from the last by regular habit there can be no reasonable doubi; edition of the Commercium, with the various ori- and that he should abstain from appropriating to ginal papers contained in it, it evidently appears himself unjustly the greatest mathematical inven. that Newton had discovered his Method of Flux- tion of any age, when he seized greedily every ions many years before the pretensions of Leibnitz. smaller discovery, is contrary to all the laws of See also Raphson's History of Fluxions, and the human thought and all the rules of human action. valuable account of the Commercium Epistolicum,' Direct method of Fluxions. All tinite magnitudes given in vol. 29 of the Philosophical Transactions, are here conceived to be resolved into infinitely or New Abridgement, vol. 6. pp. 116–153. small ones, supposed to be generated by motion, There are however, according to the opinion as a line by the motion of a point, a superficies by of some, strong presumptions in favour of Leib. a line, and a solid by a superficies; and they are nitz; i. e. that he was no plagiary: for that New- the elements, moments, or differences, thereof. ton was at least the first inventor, is past all dis. The art of finding these infinitely small quantipute; his glory is secure; the reasonable part, ties, or the velocities by which they are generated, even among the foreigners, allow it: and the and of working on them, and discovering other question is only, whether Leibnitz took it from infinite quantities, by their means, makes the dihim, or fell upon the same thing with him; yet rect method of fluxions. Leibnitz himself acknowledges, that in 1676, being What renders the knowledge of infinitely small in England, he staid some days in London, where quantities of such great use and extent is, that he became acquainted with Collins, who shewed they have relations to each other, which the finite him several letters from Gregory, Newton, and magnitudes, whereof they are the infinitesimals, other mathematicians, which turned chiefly on have not. series. This visit to England was probably oc- Thus e. gr. in a curve, of any kind whatever, Flux y=y the infinitely small differences of the ordinate and Thus, the Aluxion of ry, is xy + 2y. For, let: absciss have the ratio to each other, not of the two right lines, DE and FG, move parallel to ordinate and absciss, but of the ordinate and sub- themselves from two other right lines, BA and tangent; and, of consequence, the absciss and BC, (Plate 68. fig. 10.) and generate the rectangle ordinate alone being known, give the subtangent DF. Let them always intersect each other in the unknown; or, which amounts to the same, the tan curve BHR, and let Dd (i) and Ff (y) be the flusgent itself. The method of notation in fluxions, introduced ions of the sides BD (?) and BF (y); and draw by the iorentor, sir I. Newton, is thus: dm and sn parallel to DH and FH. The fluxion of The variable, or flowing quantity, to be uni. the area BDH is Dm or yx, and that of the area formly augmented, as suppose the absciss of a BFH is Fr or ay, and therefore the fluxion of the curve, he denotes by the final letters 0.77:; whole rectangle EF= ry=BDH+ BFG will be and their fiuxioas by the same letters with dots xy + ry. The fluxion of yzu is yzu + yzu + yzu; for placed over them, thus, v ry s. And the it be put =zu, then gzu will be =yx, and its initial letters a, b, s, d, &c. are used to express invariable quantities. fluxion = yi + xy: but = being =zu, and x= Again, if the fiuxions themselves are also vari. zu + uz, yi+ ry, by substitution, will be =yzu + able quantities, and are continually increasing, or And the fluxion of suyz, is suyz + decreasing, he considers the velocities with which yuz + yzu. the increase or decrease, as the fluxions of the wuyz + xuyz + xuyz; and the fluxion of a+xxbformer fluxions, or second fluxions; which are (the common product being ab+b1- --ya-sy) win denoted by two dots over them, thus, y I. z. bebx-ja-iz-xy. After the same manner one may consider the The fluxion of the square of a variable quanaugments. and diminutions of these, as their city being settled upon sound and unexceptionfurions also ; and thus proceed to the third, able principles, that of the product of two varifourth, &c. fluxions, which will be noted, thus, able quantities might be proved thus; without 112, &c. We may observe in general, the consideration of quantities indetinitely less that the fluxions of all kinds and orders whatever Flux. *= are contemporaneous, or such as may be generated together, with their respective celerities, in one and the same time. Lastly, if the flowing quantity be a surd, as Flux. (1+y)=x+y Flux. (x+y)*= 2 x (x + y) x(x+y) vo-b; he notes its Auxion(V2-6); if a That is, Flui. **+ 2xy + yo=218 + 2xy+ 2y2 + 2yy fraction day or, 2x2 + 2yy +f. 2xy=2x2 + 2xy + 2y3 +2ys: The chief scope and business of fluxions is, from Taking away the common quantities from both the flowing quantity given, to find the fluxion : members of the equation, leaves for this we shall lay down one general rule, as Flux. 2xy=2xy + 2yx stared by Dr. Wallis, and afterwards apply and Czemplify it in the several cases. “Multiply each or, 2 Flux. xy=2(*y+y*) term of the equation separately by the several indices of the powers of all the nowing quantities or, Flux. qy=sý+yi. contained in that term; and in each multiplica Otherwise, thus: tion, change one root or letter of the power into Make x+y=s. its proper flusion: the aggregate of all the pro Then 20 + 2xy + yo=; ducts connected together by their proper signs, whence sy=1,2-13– 4yo will be the fuxion of the equation desired.” Flux, wy=ss-**-YY. The application of this rule will be contained in tbe following cases: But, since s=x+y, s=rty 1. Te express the fiuxions of simple variable quanti- Flux. 3y=(1+y)(3+y)-13-yi ties, es already mentioned.-Put the letter, or letters, which express them, with a dot over them: thus, =x*+ xy + yx + yy-11-yy fluxion of x is x and the fluxion of y is y, and the =xy + yx, as before. III. To find the fluxion of a fraction.-Multiply the fiution of I+y+9+2, is x+y+v+2, &c. flusion of the numerator by the denominator, Note. For the fluxion of permanent quantities, and from this product subtract the fluxion of the when any such are in the equation, imagine 0, or denominator drawn into the numerator; and this a cypher'; for such quantities can have no Aux- will be the numerator, and the equare of the deion, properly speaking, because they are without nominator will be the denominator of the fraction motion, or invariable. Again, the fluxion of a expressing the fluxion of the given fraction. quantity, which decreases, instead of increasing, is to be considered as negative. Thus the fluxion of is *y-xy II. 1o find the fiuxions of the products of two or more y Уу variable or fisuing quantities.- Multiply the fluxion of each simple quantity by the factors of the pro For suppose = %, then will x = yz, which y ducts, or the product of all the rest; and connect equal quantities must have equal Auxions; therethe last products by their proper signs: then the sum, or aggregate, is the fuxion sought. fore x=ys +ży and 4–zy=zy; and dividing all he notes it, ( 3 ) m • -*X 12 מ - מנוי 12 n у 1 =y܀ 11 n become 2 x 1*-1- y ty-13 ху у : wherefore this last is the yy Þ**–*. Or, more briefly, according to the rule, fluxion of the fraction -=&; because ing= will be xxxx , z will be equal to the fluxion of y &c. 72 1 mir And the fluxion of will be ; for the V. To find the flusions of surd quantities.-Suppose permanent quantity a having no fusion, there it required to find the fluxion of v 211–xx, or can be no product of the fluxion of the numerator 27x -xx)}. Suppose 2rx- 1.t -1.71}= =z; then is 2rxinto the denominator, as there would have been, *x=zz; and consequently rx-r5=zz; and, by had a been r, s, or any other variable quantity. ringi IV. To find the fiuxion of a power.-Multiply the division, (by substitution) fluxion of the root by the exponent of the power, and the product by that power of the same root, whose exponent is less by unit than the given ex = to the fluxion of v 211 – 18. ponent; and likewise by the invariable quantity and co-efficient, if there be any. Or, by the preceding rule, the fiuxion of Thus, the fluxion of xx will be 2x*; for ri= * Xx; but the fluxion of xxx=*x + xr=201, &c. 2rx-x* { will be 1 X 271-2xxx ?ra-221-'. and the fluxion of 63 will be 3xxx. That of ra ri-x* - 1. will be 8x7i, &c. that of 5x" will be 25x47; that of 3art will be 12 ax; i. If it be required to find the fluxion of ay-11}; Or, if m express the index of any power, as for ay-2/ } put z; then ay-rx=z; and ay-2xx = zz- }; and multiplying by 3, 3ay— 6x3 = suppose r"; its fuxion will be mr' If the power be produced froin a binomial, &c. zz – į; aud, consequently, Sazij-62{ri=;; as suppose 2 + y'?, or 4x + 2xy + yy, its fluxion will equal (substituting ay--xxl2 = =1) 323y2y — 6ao x2 be 2 xx+y x+s, or 2*x + 2rý + 2ry + 2yy. 1°yy + 3axıy – Ma’ya ** + 12 ayx31 – 61°i = to the fuxion of ay-ra1. If the exponent be negative, as suppose * To find the fluxion of 3 + bx + cord + d13. Put its fluxion will be Þa+61 +629 + dx3 =2. Then (zł).= = 1-1. -1; br+ 2x + 3dxas Or, if it be done by way of fraction, by restitution, &c. 312 34 a + bx + cx?dx3 (for the square of im is xam) = =-xm- 1-3m3= – mx – 171- 1x as before; =, removing **-1 to By a similar process the fluxion of the denominator, by changing the sign of the ex- Du+bx + cao + dx3 + ex*+ &c. to mx" is found to (6+ 2cx + 3d12 +4213+ &c. to (m-1)/19— 1) ponent, be,- If the power be imperfect, i. e. if its exponent be a fraction, as suppose am; or in the other rational and surd quantities. - Let it be required to VI. To find the fluxion of quantities compounded of dotation 1 *, suppose x = 2: then if each find the fluxion of bx? + ca I + ea? x V x1 + aa=z. member be elevated to the power of n, it will stand Put bx' + cax + cao=P, and xx + aa=2. Then thus, * "=z"; the fluxion of which will be, by the given quantity is pa=z, and the fluxion therethis general rule, mr m-=nz*-*. xx Wherefore 2 will be of is pg+qp=s: but ġ is = v** + aa, and p is (by dividing =26xx + car; therefore, in the equation på top =ż, if in the place of p, q, P:9, we restore the both parts by nz" — 1) and quantities they represent, we shall have +2b18 x V pata + caxx X*; or -*: *** -"; putting instead of zo –', V x taa its value o and the above expression will V1? + a? ==. Which being reduced to one do nomination, gives |