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lour may be gradually dissolved altogether by the interior Blame, and again reproduced by a small panicle 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 posulogy 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 quanuity 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 ce-rty 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 celenies 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 fluxions are rendered much more easy and intelligible. And though sir Isaac Newton defines fluxions to be the velocities of motions, yet he has recourse to the increments or moRents, generated in equal particles of time, in order to determine those velocities, which he anerwards teaches us to expound by finite magnitudes of other kinds.
Method of Fluxions, is the algorithm and analysis of fluxions, and fluents or flowing quantities.
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 x, by a dot placed over it, thus ; while Leibnitz expresses his differential of the same x, by prefixing the initial letter d, as dx. But, setting aside these circumstances, the two methods are just alike; though the principles ions being referred to the doctrine of motion; difon which they are established are different: fluxferentials 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.
Most foreigners define this as the method of differences or differentials, being the analysis of ir definitely small quantities. But Newton, and other English authors, call these infinitely small quantities, moments; considering them as the mentary increments of variable quantities; as ef a line considered as generated by the flux or motion of a point, or of a surface generated by the fux of a line. Accordingly, the variable quantities are called fluents, or flowing quantities; and the method of finding either the fluxion or the Bent, the method of fluxions.
M. Leibnitz considers the same infinitely small antitics as the differences, or differentials of
M. Facio, in a Treatise on the Line of Swiftest
Descent, declared, that he was obliged to own
Newton as the first inventor of the differential
calculus, and the first by many years; and that second inventor, had taken any thing from him. he left the world to judge, whether Leibnitz, the This precise distinction between first and second inventor, with the suspicion it insinuated, raised a controversy between M. Leibnitz, supported by the editors of the Leipsic Acts, and the English mathematicians, who declared for Newton. Sir Isaac himself never appeared on the scene; his glory was become that of the nation; and his adherents, warm in the cause of their country, need
ed not his assistance to animate them.
either side; probably on account of the distance Writings succeeded each other but slowly, on of places; but the controversy grew still hotter 1711, complained to the Royal Society, that Dr. and hotter: till at length M. Leibnitz, in the year Keil had accused him of publishing the Method of Fluxions invented by sir 1. 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 Loudon in 1673, and kept a correspondence with Mr. Collins by means of Mr. Oldenburgh, till September 1676, when he returned from Paris to Handver, by way of London and Amsterdam: that it did not appear that Mr. Leibnitz knew any thing
of the deferential calculus before his letter of the 21st of June, 1677, which as 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 that 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. kel had not at all injured M. Leibnitz in what be 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 Auslysi Promota, Svo. Lon 1712. This book was carefully distribut d 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 inaintains that the Method of Fluxions had not preceded the Method of Differ. ences; 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 Fluxious 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 Leibuitz 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 hin 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 Newton; 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 doubt; 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,
Thus e. gr. in a curve, of any kind whatever,
the infinitely small differences of the ordinate and absciss bave the ratio to each other, not of the ordinate and abseiss, but of the ordinate and subtangent; and, of consequence, the absciss and erdinate 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 I. 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 v, r, y, s; and their fluxions by the same letters with dots placed over them, thus, v zy z. And the initial letters a, b, c, d, &c. are used to express invariable quantities.
Again, if the fluxions themselves are also varisble quantities, and are continually increasing, or decreasing, be considers the velocities with which they increase or decrease, as the fluxions of the former fluxions, or second fluxions; which are decoted by two dots over them thus, y x 2.
After the same manner one may consider the augments, and diminutions of these, as their fusions also; and thus proceed to the third, fourth, &c. fluxious, which will be noted, thus, yzz: y z 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 if a
-; he notes its fluxion
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,"
he notes it,
The application of this rule will be contained in the following cases:
Thus, the fluxion of ry, is iy+xy. 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 (*) and Ff (y) be the fluxions of the sides BD (x) and BF (y); and draw dm and sn parallel to DH and FH. The fluxion of the area BDH is Dm or yż, and that of the area BFH is Fn or ry, and therefore the fluxion of the whole rectangle EF ry BDH+ BFG will be y+xy. The fluxion of yzu is ÿzu + yzu+yzu; for if z be putzu, then yzu will be yx, and its fluxion yxxy: but x being zu, and i zù+uz, yx+xy, by substitution, will be ➡yzu+ yuz+yzu. And the fluxion of zuyz, is xuyż + ruyz + ruyz + xyz; and the fluxion of a + xx by (the common product being ab + bx—ya—xy) will bebx-ya-xy-ry.
The fluxion of the square of a variable quantity being settled upon sound and unexceptionable principles, that of the product of two variable quantities might be proved thus; without the consideration of quantities indefinitely less than others.
Flux. (x+y)=2 × (x + y) x (x + y)
Flux. x2+2xy + y2 = 2xx + 2×ý+2yx + 2yỷ or, 2xx+2yÿ÷f. 2xy=2xx + 2xÿ + Qyż + Qyÿ. Taking away the common quantities from both members of the equation, leaves
Flux. 2xy=2xy + Qyż or, 2 Flux. ry=2(xỷ+2yx) or, Flux. ry=xy+yx. Otherwise, thus:
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 x, ≈, or any other variable quantity.
: wherefore this last is the
both parts by n≈7-1) and
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 21; for xx = xxx; but the fluxion of x ×x=ix+xi=2ix, &c. and the fluxion of 13 will be 3xxi. That of 19 will be 8x7r, &c.; that of 5a3 will be 25x^; that of 3ax will be 12aa.
Or, if m express the index of any power, as its fluxion will be mx" -1 c. suppose r ; If the power be produced from a binomial, &c. as suppose x + y)2, or xx + 2xy+ yy, its fluxion will bc 2 × x + y × x + ý, or 2rx 4 2xý + 2xy + 2yÿ. If the exponent be negative, as suppose x
its fluxion will be - Mxx
Or, if it be done by way of fraction, (for the square of xm is x) -mx-l as before;=, removing x-1 to the denominator, by changing the sign of the ex-mi
If the power be imperfect, i. e. if its exponent be a fraction, as suppose √xm x; or in the other
z: then if each
member be elevated to the power of n, it will stand thus, "; the fluxion of which will be, by this general rule, ma1ina" −1i.
because z being
; for the
X i; or -:a"-"; putting instead of 2′′−1,
and the above expression will
"-". Or, more briefly, according to the rule,
Or, by the preceding rule, the fluxion of 2rx-xx will be 2ra-2x4 × 2rx-xx)}}−1=
ri―ii + 2rx-xx
If it be required to find the fluxion of ay-xx for ay- put z; then ay-x=2} and y-2x i =zz ̄} ; and multiplying by 3, Say-6xx ; and, consequently, 3az tý−6 z}xi = 5; equal (substituting ay—xa ?
or x2yy+3ax1y—6a2y2xx + 12 ayı3x −6x'x = to the
To find the fluxion of
by restitution, &c..
V. To find the fiuxions of surd quantities.—Suppose it required to find the fluxion of 21-xa, or 2rx-xx. Suppose 2x- =2; then is 2rxxx=zz; and consequently ri-ai=2; and, by Ti-ir
will be XiXI
to the fluxion of 2rx-ax.
a + bx + cx2 + dx3. Then (
by + 2xx + 3dx2x
3 a + bx + cxdx3
By a similar process the fluxion of √u + bx + ca2 + dx3 + ex1 + &c. to mx" is found to
(by dividing of is pq+qp=¿: but q is xx+ua, and p is =2bxi+cai; therefore, in the equation pg+yp
z, if in the place of p, q, p, q, we restore the 2- quantities they represent, we shall have bx3+ cax2 + ea2x × i
+ 2xi × √ u2 + a2 + cai ×
√x2 + aa +2=2. Which being reduced to one denominator, gives
(b + 2cx + 3dx2 + 4ex3 + &c. to (m − 1 ) lx” — 1) ¿ n (a + bx + cx2 + dx2 + ex1 + &c. to mx") 2-1
VI. To find the fluxion of quantities compounded of rational and surd quantities.-Let it be required to find the fluxion of bx2 + cax +ta2 × √x xx+ua=2. Put bx2+ cax+ ea2=p, and
the given quantity is pq=3, and the fluxion therexx+aa=q. Then
Is the former case, put the proposed exponentale, a single variable quantity; then take (cotan 2)= the logarithm of each, so shall log. of z=rx log.
de; take the fluxions of these, so shall
b. of e; bencei-zix log. of e=exix log. of e," the flaxion of the proposed exponential ex; and which therefore is equal to the said proposed Çatity, drawn into the fluxion of the exponent, ad also into the log. of the root.
we have sin, and cos 1; and therefore sin (z+)=sin 2+ cos z; whence sin (3)— sin z, or (sin x) = cos z: viz. the fluxion of the sine of an arc whose radius is unity, is equal to the product of the fluxion of the angle into the cusine of the same arc.
In like manner the fluxion of cos z, or cos (z+¿) — cos z = cos z cos - sin z sin - COS 2, since (art. SINE) COS (2+) = cos z cosi - sin z sin: therefore, because sin =, and cos i=1, we have (cos x)cos x-sin-cos=-ż sin z: that is, the fluxion of the cosine of an arc, radius being 1, is found by multiplying the fluxion of the arc (taken with a contrary sign) by the sine of the same arc.
By means of these two formulæ, many other fluxional expressions may be found. As that (cos mz)'= —mi sîn mx. (sin ms)'= + m2 cos mz. sin % COS z. it Sin 2%
IX. To find the fluxion of a rectangle, when one ade x increases, and the other y decreases.-In this case the fluxion of the decreasing quantity is nega tive with respect to that of the 'reasing quantty (see the beginning of this article), and therefore, the sign of the term affected with it ought to be changed; e. gr. the fluxion of the rectangle 2 in these circumstances will be expressed by zy-zv.
(sec z) =
; sin z
cos % sin 2%
(sin x)=m sin - cos z.
XI. To find the second, third, &c. fluxion of a flowing quantity. These fluxions differ in nothing, except their order and notation, from first fluxions, being actually such to the quantities from which they are immediately derived; and therefore, they may be found, in the same manner, by the general rules already delivered.
Thus, by the 4th rule, the first fluxion of x3 is 3x2; and if he supposed constant, or if the root x be generated with an equable celerity, the fluxion of 31a†, or 34 × â2, will be 34 × 2x¿=6xx2, which is the second fluxion of a3; and 63 will be its third fluxiou: but if the celerity with which is generated be variable, either increasing or decreasing, then being variable, will have its fluxion denoted by x, &c. In this case the fluxion of 3x2 x i will be, by the 21 and 4th rules, 6x¿ × i + 3x2xx=6xx2+3x2%, the second fluxion of a3. And the third fluxion of x3 obtained in like man
(cos z-sin 2%)
ner from the last, will be 6ix. à2 + 6x × 2xx +
X. To find the fluxions of sines, cosines, &c.—
n-1 then y n; and if 2
the sine of the angle or arc denoted by s, we must sappose that by a motion of one of the legs inading the angle, it becomes z+*, then sin (z+) -sine is the fluxion of sin z. But according to yx, the formula for the sines of sums of arcs (see Siz and TRIGONOMETRY), we have sin (z+) = If the function proposed were ar" we should #a z cos i + sin cos z the radius being assumed find (axn) nax : the factors na and & being equal to unity. But the sine of an arc indefinitely small does not differ sensibly from that are itself, regarded as constant in the first fluxion nas nor's cosine differ perceptibly from radius; hence, to obtain the second fluxion it will suffice to
= n x n 1 x x ፰ + nx
ży, then 2ÿ +