into black scoriæ, neither by itself nor with fluxes does it exhibit a regulus. It grows red on roasting. Tin easily melts before the blow-pipe, and is calcined. The fluxes dissolve the calx sparingly; and when saturated, contract a milky opacity. Some small particles of this metal dissolved in any flux may be distinctly precipitated upon iron. Crystallized ore of in, urged by fire upon the charcoal, yields its metal in a reguline state. Bismuth presents nearly the same appearances of lead; the calx is reduced on the coal, and fused in the spoon. The calx, dissolved in microcosmic salt, yields a brownish yellow globule, which grows more pale upon cooling, at the same time losing some of its transparency. Too much calx renders the matter perfectly opaque. Borax produces a similar mass in the spoon, but on the coal a grey one, which can scarcely be freed from bubbles. On fusion the glass smokes, and forms a cloud about it. Bismuth is easily precipitated by copper and iron. Sulphurated bismuth is easily fused, exhibiting a blue flame and sulphureous smell. Cobalt, when added, by means of sulphur, enters the globule; but the scoriæ soon swells into distinct partitions; which, when further urged by fire, throw out globules of bismuth. Sulphurated bismuth, by the addition of borax, may be distinctly precipitated by iron or manganese. Regulus of nickel when melted is calcined, but more slowly than other metals. The calx imparts an hyacinthine colour to fluxes, which grows yellow on cooling, and by long-continued fire may be destroyed. If the calx of nickel is contaminated by ochre of iron, the latter is first dissolved. Nickel dissolved is precipitated on iron, or even on copper; an evident proof that it does not originate from either of these metals. Sulphurated nickel is no where found without iron and arsenic; the regulus is obtained by roasting, and fusing with borax, though it still remains mixed with some other metals. Regulus of arsenic takes fire by a sudden heat, and not only deposits a white smoke on charcoal, but diffuses the same all around. 'The calx smokes with a smell of garlic, but does not burn. The fluxes grow yellow without growing opaque, on adding a proper quantity of calx, which is dispelled by a long continuance of the heat. This semimetal is precipitated in a metallic form by iron and copper, but not by gold. Yellow arsenic liquefies, smokes, and totally evaporates: when heated by the external flame, so as neither to liquefy nor smoke, it grows red, and yellow again upon cooling. When it only begins to melt, it acquires a red colour, which remains after cooling. Realgar liquefies more easily, and is besides totally dissipated. Regulus of cobalt melts, and may partly be depurated by borax, as the iron is first calcined and taken up. The smallest portion of the calx tinges the flux of a deep-blue colour, which appears of a violet by refraction, and this colour is very fixed in the fire. Cobalt is precipitated upon iron from the blue globule, but not upon copper. When calx of iron is mixed with that of cobalt in a flux, the former is dissolved. This semimetal takes up about one-third of its weight of sulphur in fusion, after which it can hardly be melted again. It is precipitated by iron, copper, and several other metals. The common ore yields an impure regulus by roasting. The green cobalt, examined by our author, tinges the microcosmic salt blue; but at the same time shows red spots indicating copper. Zinc exposed to the blow-pipe melts, takes fire, sending forth a beautiful bluish-green flame, which, however, is soon extinguished by a lanuginous calx: but if the reguline nucleus included in this lanuginous matter (conimonly called flowers of zinc) is urged by the flame, it will be now and then inflamed, and, as it were, explode and fly about. With borax it froths, and at first tinges the flame. It continually diminishes, and the flux spreads upon the charcoal; but in fused microcosmic salt, it not only froths, but sends forth flashes with a crackling noise. Too great heat makes it explode with the emission of ignited particles. The white calx, or flowers, exposed to the flame on charcoal, becomes yellowish, and has a kind of splendour which vanishes when the Alame ceases. It remains fixed, and cannot be melted. The fluxes are scarcely tinged, but when saturated by fusion, they grow opaque and white on cooling. Clouds are formed round the globules, of a nature similar to those of the metallic calx. Dissolved zinc is not precipitated by any other metal. When mineralized by carbonic acid gass, it has the same properties as calcined zinc. In the pseudogalena sulphur and iron are present. These generally, on the charcoal, smell of sulphur, melt and tinge the flame more or less, depositing a cloud all around. Those which have no matrix are tinged by those which contain iron, and acquire by saturation a white opaque colour, verging to brown or black, according to the variety of composition. Regulus of antimony, fused and ignited on the charcoal, affords a beautiful object; for if the blast of air be suddenly stopped, a thick white smoke rises perpendicularly, while the lower part round the globule is condensed into crystalline spicule, similar to those called argentine flowers. The calx tinges fluxes of an hyacinthine colour; but on fusion smokes, and is easily dissipated, especially on the charcoal, though it also deposits a cloud on it. The dissolved metal may be precipitated by iron and copper, but not by gold. Crude antimony liquefies on the charcoal, spreads, smokes, penetrates it, and at last disappears entirely, except a ring which it leaves behind. Regulus of manganese scarcely yields to the flame. The black calx tinges the fluxes of a bluish colour; borax, unless saturated, communicates more of a yellow colour. The co lour may be gradually dissolved altogether by the interior flame, and again reproduced by a small particle of nitre, or the exterior flame alone. Combined with carbonic acid, it is of a white colour, which changes by ignition to black. In other respects it shows the same experiments as the black calx. FLUX, in medicine, a disease generally described under the term dysentery, and in the nosology of Cullen denominated specifically dysenteria sanguinea. See DYSENTERIA. FLUX. a. fluxus, Latin.) Unconstant; not durable; maintained by a constant succession of parts. To FLUX. v. a. 1. To melt. 2. To salivate; to evacuate by spitting (South). FLUXILITY. s. (fluxus, Latin.) Easiness of separation of parts (Boyle). FLUXION.' s. (fluxio, Latin.) 1. The act of flowing. 2. The matter that flows (Wiseman). FLUXION, in the Newtonian analysis, denotes the velocity with which a flowing quantity is increased by its generative motion: by which it stands contradistinguished from fluent or the flowing quantity, which is gradually and indefinitely increasing, after the manner of a space which a body in motion describes. Or, a fluxion may be more accurately defined, as, the magnitude by which any flowing quantity would be uniformly increased in a given portion of time, with the generating celerity at any proposed position, or instant, supposing it from thence to continue invariable." From this definition it appears, that the fluxions of quantities are, always, as the cele rities by which the quantities themselves increase in magnitude. Mr. Simpson observes, that there is an advantage in considering fluxions, not as mere velocities, but as the magnitudes which these velocities would, uniformly, generate in a given finite time: the imagination is not here confined to a single point, and the higher orders of Aluxions are rendered much more easy and intelligible. And though sir Isaac Newton defines fluxions to be the velocities of motions, yet he hath recourse to the increments or moments, generated in equal particles of time, in order to determine those velocities, which he afterwards teaches us to expound by finite magnitudes of other kinds. Method of Fluxions, is the algorithm and analysis of fluxions, and fluents or flowing quanti ties. Most foreigners define this as the method of differences or differentials, being the analysis of indefinitely small quantities. But Newton, and other English authors, call these infinitely small quantities, moments; considering them as the momentary increments of variable quantities; as of a line considered as generated by the flux or motion of a point, or of a surface generated by the flux of a line. Accordingly, the variable quantities are called fluents, or flowing quantities; and the method of finding either the fluxion or the fluent, the method of fluxions. M. Leibnitz considers the same infinitely small quantities as the differences, or differentials of quantities; and the method of finding those differences, he calls the differential calculus. Besides this difference in the name, there is another in the notation. Newton expresses the fluxion of a quantity, as of 1, by a dot placed over it, thus x; while Leibnitz expresses his differential of the same x, by prefixing the initial letter d, as dr. But, setting aside these circumstances, the two methods are just alike; though the principles on which they are established are different: fluxions being referred to the doctrine of motion; differentials to such a kind of augmentation as is more directly referable to analysis. The method of fluxions is one of the greatest, most subtle, and sublime discoveries of perhaps any age: it opens a new world to our view, and extends our knowledge, as it were, to infinity; carrying us beyond the bounds that seemed to have been prescribed to the human mind, at least infinitely beyond those to which the ancient geometry was confined. The history of this important discovery, recent as it is, is a little dark, and embroiled. Two of claimed the invention, sir Isaac Newton, and M. the greatest men of the last age have both of them Leibnitz; and nothing can be more glorious for the method itself, than the zeal with which the partizans of either side have asserted their title. The two great authors themselves, without any seeming concern, or dispute, as to the property of the invention, enjoyed the prospect of the progresses continually making under their auspices, till the year 1699, when the peace began to be disturbed. Descent, declared, that he was obliged to own calculus, and the first by many years; and that Newton as the first inventor of the differential Writings succeeded each other but slowly, on either side; probably on account of the distance of places; but the controversy grew still hotter and hotter: till at length M. Leibnitz, in the year 1711, complained to the Royal Society, that Dr. Keil had accused him of publishing the Method of Fluxions invented by sir I. Newton,under other names and characters. He insisted that nobody knew better than sir Isaac himself, that he had stolen nothing from him; and required that Dr. Keil should disavow the ill construction which might be put upon his words. The society, thus appealed to as a judge, appointed a committee to examine all the old letters, papers, and documents, that had passed among the several mathematicians, relating to the point; who after a strict examination of all the evidence that could be procured, gave in their report as follows: "That Mr. Leibnitz was in London in 1673, and kept a correspondence with Mr. Collins by means of Mr. Oldenburgh, till September 1676, when he returned from Paris to Hanover, by way of London and Amsterdam: that it did not appear that Mr. Leibnitz knew any thing of the differential calculus before his letter of the 21st of June, 1677, which was a year after a copy of a letter, written by Newton in the year 1672, had been sent to Paris to be communicated to him, and above four years after Mr. Collins began to communicate that letter to his correspondents; in which the Method of Fluxions was sufficiently explained, to let a man of his sagacity into the whole matter; and hat sir I. Newton had even invented his method before the year 1669, and consequently 15 years before M. Leibnitz had given any thing on the subject in the Leipsic Acts." From which they concluded that Dr. Keil had not at all injured M. Leibnitz in what he had said. The society printed this their determination, together with all the pieces and materials relating to it, under the title of Commercium Epistolicum de Analysi Promota, 8vo. Lon. 1712. This book was carefully distributed through Europe, to vindicate the title of the English nation to the discovery; for Newton himself, as already hinted, never appeared in the affair: whether it was that he trusted his honour with his compatriots, who were zealous enough in the cause; or whether he felt himself even superior to the glory of it. M. Leibnitz and his friends however could not shew the same indifference: he was accused of a theft; and the whole Commercium Epistolicum either expresses it in terms, or insinuates it. Soon after the publication therefore, a loose sheet was printed at Paris, in behalf of M. Leibnitz, then at Vienna. It is written with great zeal and spirit; and it boldly maintains that the Method of Fluxions had not preceded the Method of Differences; and even insinuates that it might have arisen from it. The detail of the proofs however, on each side, would be too long, and could not be understood without a large comment, which must enter into the deepest geometry. M. Leibnitz had begun to work upon a Commercium Epistolicum, in opposition to that of the Royal Society; but he died before it was completed. A second edition of the Commercium Epistolicum was printed at London in 1722; when Newton, in the preface, account, and annotations, which were added to that edition, particularly answered all the objections which M. Leibnitz and Bernoulli were able to make since the Commercium first appeared in 1712; and from the last edition of the Commercium, with the various original papers contained in it, it evidently appears that Newton had discovered his Method of Fluxions many years before the pretensions of Leibnitz. See also Raphson's History of Fluxions, and the valuable account of the Commercium Epistolicum,' given in vol. 29 of the Philosophical Transactions, or New Abridgement, vol. 6. PP. 116-153. There are however, according to the opinion of some, strong presumptions in favour of Leibnitz; i. e. that he was no plagiary: for that Newton was at least the first inventor, is past all dispute; his glory is secure; the reasonable part, even among the foreigners, allow it: and the question is only, whether Leibnitz took it from him, or fell upon the same thing with him; yet Leibnitz himself acknowledges, that in 1676,being in England, he staid some days in London, where he became acquainted with Collins, who shewed him several letters from Gregory, Newton, and other mathematicians, which turned chiefly on series. This visit to England was probably oc casioned by Collins's communication of the letter of 1672; and though we, instead of positive, have only presumptive proof, we are decidedly of opinion that Leibnitz saw, in Collins's possession, papers which acquainted him with Newton's discovery. We request that the reader will compare with Leibnitz's acknowledgment the following relation, for the truth of every part of which we hold ourselves responsible. In the year 1669, amongst other series by sir Isaac Newton, one for finding the arc of a circle from the sine, and, in 1671, another by Mr. Gregory, for finding the arc from the tangent, were sent to Mr. Collins, who was very free in communicating these and other discoveries. In 1674 Leibnitz mentions in a letter to Oldenburgh his being possessed of the first series; and the next year those of both Newton and Gregory were sent by Oldenburgh to Leibnitz. But in 1676 Leibnitz dropped his pretensions to the first series, not being able to demonstrate it, and sent to Oldenburgh, as his own, that of Gregory, with a demonstration. Both Newton and Gregory admitted that Leibnitz found out this series; for they knew nothing of Oldenburgh's letter, the copy of which lay buried for more than 30 years among the papers of the Royal Society: so that at length, though not till 1713, Leibnitz was compelled to acknowledge Gregory as the original author. Nay, from the whole tenour of this gentleman's conduct, he may be justly suspected of having often learned by information what he affirmed to have invented: for he pretended to Mouton's differential method; to a property of a series that had been discovered by Pascal; to four or five different series invented by ewton; to a method of progression; to the differential analysis, when it is certain he was ignorant of it; and lastly, to some of the principal propositions in the Principia. Newton's grand work was first published in 1686: it was criticised at Leipsic by Leibnitz, in a review managed by himself, in 1687; and, two years afterwards, he pretended to have invented some propositions contained in the Principia, relative to the motion of the planets in ellipses. Well might this gentleman be characterised as having a vast and devouring genius!" for he was determined to devour every choice morsel that fell in his way. We attempt not to depreciate his talents: but that he was a plagiary by regular habit there can be no reasonable doubí; and that he should abstain from appropriating to himself unjustly the greatest mathematical invention of any age, when he seized greedily every smaller discovery, is contrary to all the laws of human thought and all the rules of human action. Direct method of Fluxions.-All finite magnitudes are here conceived to be resolved into infinitely small ones, supposed to be generated by motion, as a line by the motion of a point, a superficies by a line, and a solid by a superficies; and they are the elements, moments, or differences, thereof. The art of finding these infinitely small quantities, or the velocities by which they are generated, and of working on them, and discovering other infinite quantities, by their means, makes the direct method of fluxions. What renders the knowledge of infinitely small quantities of such great use and extent is, that they have relations to each other, which the finite magnitudes, whereof they are the infinitesimals, have not. Thus e. gr. in a curve, of any kind whatever, the infinitely small differences of the ordinate and absciss have the ratio to each other, not of the ordinate and absciss, but of the ordinate and subtangent; and, of consequence, the absciss and ordinate alone being known, give the subtangent unknown; or, which amounts to the same, the tangent itself. The method of notation in fluxions, introduced by the inventor, sir 1. Newton, is thus: The variable, or flowing quantity, to be uniformly augmented, as suppose the absciss of a curve, he denotes by the final letters,,y; and their fluxions by the same letters with dots placed over them, thus, vi y z. And the initial letters a, b, c, d, &c. are used to express invariable quantities. Again, if the fluxions themselves are also variable quantities, and are continually increasing, or decreasing, he considers the velocities with which they increase or decrease, as the fluxions of the former fluxions, or second fluxions; which are denoted by two dots over them, thus, y r z. After the same manner one may consider the augments and diminutions of these, as their fluxions also; and thus proceed to the third, fourth, &c. fluxions, which will be noted, thus, 3 ==: 3 x z, &c. We may observe in general, that the fluxions of all kinds and orders whatever 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 √7-6; he notes its fluxion (√.—b); if a : The chief scope and business of fluxions is, from the flowing quantity given, to find the fluxion: for this we shall lay down one general rule, as stated by Dr. Wallis, and afterwards apply and exemplify it in the several cases. Multiply each term of the equation separately by the several indices of the powers of all the flowing quantities contained in that term; and in each multiplication, change one root or letter of the power into its proper fluxion: the aggregate of all the products connected together by their proper signs, will be the fluxion of the equation desired." The application of this rule will be contained in the following cases: 1. To express the fluxions of simple variable quantities, as already mentioned.-Put the letter, or letters, which express them, with a dot over them: thus, fluxion of x is x and the fluxion of y is y, and the fluxion of 1+y+9+ z, is x + y + v + z, &c. Note. For the fluxion of permanent quantities, when any such are in the equation, imagine 0, or a cypher; for such quantities can have no fluxion, properly speaking, because they are without motion, or invariable. Again, the fluxion of a quantity, which decreases, instead of increasing, is to be considered as negative. II. To find the fluxions of the products of two or more variable or flowing quantities.-Multiply the fluxion of each simple quantity by the factors of the products, or the product of all the rest; and connect the last products by their proper signs: then the sum, or aggregate, is the fluxion sought. Thus, the fluxion of ry, is ry + zy. For, let two right lines, DE and FG, move parallel to themselves from two other right lines, BA and BC, (Plate 68. fig. 10.) and generate the rectangle DF. Let them always intersect each other in the curve BHR, and let Dd (x) and Ff (y) be the fluxions of the sides BD (1) and BF (y); and draw dm and sn parallel to DH and FH. The fluxion of the area BDH is Dm or yx, and that of the area BFH is Fn or ay, and therefore the fluxion of the whole rectangle EF=ry=BDH+BFG will be xy+ry. The fluxion of yzu is yzu + yzu+yzu; for if x be put zu, then yzu will be yx, and its fluxion = y + xy: but being =zu, and x = zu+uz, ya+ry, by substitution, will be yzu + And the fluxion of ruyz, is suyz + yuz+yzu. xyz+xuyz+xuya; and the fluxion of a +x × b— (the common product being ab+bx-ya-sy) will be bx-ya-iy-xy. The fluxion of the square of a variable quantity being settled upon sound and unexceptionable principles, that of the product of two vari able quantities might be proved thus; without the consideration of quantities indefinitely less than others. permanent quantity a having no fluxion, there can be no product of the fluxion of the numerator into the denominator, as there would have been, had a been r, z, or any other variable quantity. IV. To find the fluxion of a power-Multiply the 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 exponent; and likewise by the invariable quantity and co-efficient, if there be any. Thus, the fluxion of xx will be 2xx; for rx= Xx; but the fluxion of x x x=xx + xx=2x1, &c. and the fluxion of x3 will be 3xxx. That of r will be 8x7x, &c. that of 5x5 will be 25x+x; that of 3art will be 12 ax3 i. Or, if m express the index of any power, as 772 &c. > n 21x-xx V. To find the fluxions of surd quantities.—Suppose it required to find the fluxion of 2r1-xx, or -xx. Suppose 2rx-1=2; then is 2rx— xx=zz; and consequently rx-x=zz; and, by (by substitution) division, x= to the fluxion of√2rx — 15. Or, by the preceding rule, the fluxion of 2rx-xx will be × 2x-2xxx 2ra—xx {−1 rx-xx + 2rx-xx for ri-xx 2rx-xx If it be required to find the fluxion of ay—11 ay -ax put z; then ay-xx=z and ay-2xx =zz; and multiplying by 3, 3ay—6xx = zz-3; and, consequently, Sazy-6 x} 1x=2; equal (substituting ay-xx2 = ≈}) 3a3y2y — 6a2x2 a2yy+3ax4y — 6a2y2 xx + 12 ayx3x − 6x3 x = to the fluxion of ay-xa\}. To find the fluxion of a + bx + cx3 + da3. Put -1. ✔a + bx + ca2 + dx3=z3 Then (2) |