« ElőzőTovább »
which naturally involve the mind in metaphyfical difficulties ; but as the magnitudes they uniformly generate in a given finite time, fupposing the fluent or space to be described by an uniform motion. And if the motion by which any magnitude is generated be not uniform, but accelerated or retarded, the idea of a fluxion will still be the same: For though we cannot express the fluxion by any fpace actually generated in a given time, as in uniform motion; yet we can readily aflign the magnitude, or (as it is commonly called) the cotemporary increinent, that would be uniformly generated, if the acceleration or retardation were to stop at any point in which the fluxion is required to be investigated.
Now, as our ideas of magnitude arise from a comparison of the proposed object with some other of determinate dimensions : so, in the method of fluxions, we fix on a given magnitude, which is fupposed to have been uniformly generated in a given time by the motion of a point, line, or plane, as a 1tandard, wherewith to compare any other magnitude, which is supposed to have been generated in the same time, by an accelerated or retarded motion. Thus (for the sake of illuftration) suppose a ball to roll on an horizontal plane, in a straight direction, at the uniform rate of 20 feet in a minute: and also another ball to move uniformly in the same direction, at the rate of 40 feet in the same time; here then it will be plain, that the magnitudes generated in any given time must be in the ratio of 2 to 1; and therefore the fluxion of the latter will be double that of the former. And from hence it appears, that if the fusion of x be x, that of 2x will be 23, 34 will be 36, &c. and generally, that of nx will be nx. But if, while one ball moves along with an uniform velocity, the other is fupe posed to move with an accelerated motion, and that the law of accele. ration is such, that the space described by the latter, from the commencement of motion, is always some power of that described by the former, suppose the square of it; then, if the magnitude by which the space that is uniformly described is increased in a given time be denoted by x, that magnitude which the acoelerated motion would uniformly generate in the fame time, and commencing from the fame instant, will be expressed by 2x%. Thus, in the cale proposed, if the first ball has uniformly described a Ipace of 10 poles, the other must have run 100 poles ; but the former ball moves uniformly at the rate of 20 feet in a minute, therefore the magnitude or space, which the accelerated ball would uniformly describe from the same instant in one minute, will be 400 feet. The Auxions will be therefore at that point in the ratio of 400 to 20; or of 20 to 1."
Of this our author gives a demonstration; adding
“ Hence it appears, that we have the moit rational notion of fluxions from the consideration of time in the generation of the increment or decrement, and that the fluxion of any variable quantity may be truly defined, The magnitude by which any flowing quantity would be incriafed in a given time with the generating velocity at a given inftant, fuppofing it from thence to proceed uniformly or invariably. And with regard to the higher orders of fluxions, how much more obscure are our notions without the idea of time in the operation of the fluent B 2
generating the increment; fince by having recourse to the first ratio of the nascent increment, or the last ratio of the evanescent incre. ment, even to obtain only the first fluxion ot a variable quantity, we unavoidably fall into this absurdity, That a velocity which continues for no time at all actually describes a space. How then can we form any conception not only of fuch a space or increment, but also of an infinite variety of magnitudes of it, generated into one and the same point and instant of time, in which it is well known all the orders of fluxions are confidered, when nothing, I think, can be more evident than that the magnitude or increment imagined to be generated must in such a case be purum putum nihil, or strictly and abfolutely nothing. If the doubt of the existence of an increment under such circumstances be deemed incredulity and a species of infidelity", I am afraid I shall be ftigmatized with those appellations ; for 1 confess it is past my comprehension how a mere point can contain in itself an infinite variety of magnitudes, and which are all at the same time equal to one another. The unnecessary quibbles, and metaphysical niceties, by which some have attempted to explain the principles of fluxions, have not only rendered them quite obscure to the learner, but also exposed them to the ridicule and severe criticisms, it is probable, were not intended to invalidate the method of fluxions (which it is evident may be ftri&tly mathematically demonftrated) but to shew the futility of the method they had taken to elucidate the principles ; in which light it is well known the incomparable inventor never intended they Thould be viewed.
From what has been faid we may draw these practical observations. - j. That the common rule for finding the fuxion of a flowing quantity, viz. Multiply the fuxion of the root by the exponent of the power and the affixed coefficient, and the produt by that power of the fame root of which the exponent is less by unity than the given exponent, is general, and without exceptions, being applicable to any expression whatever consisting of one variable quantity with a constant exponent. 2. If the expression be a compound one, that is a binomial, trinomial, or any multinomial, the fluxion of each term must be found separately, and connected with their respective resulting figns; the sum arifing by such addition is the Auxion of the compound expression. 3. If the expreffion consists of the product of two or more variable quantities, each quantity must flow separately, while the others are supposed to be constant, or as coefficients to that variable quantity; the fum of these ftuxions will be that of the given expreffion. This follows from the general expression nx".'.;. Thus, let the fluxion of y% be proposed to be investigated. Put y+x=v, then will y2 + 2y
v2-02-22 +z=v2; hence yz = =v-}y2-12. And, from what has been before shewn, the fluxion of this will be wijzż; but v=y+z, and v=j +ż, .. by fubftitution, the fluxion of yz is yż + zj. And in the lame manner will the fluxion of xyz be found to be xyz + x2*; and that of uxyz = uxyx + uxzy + uxzi + ayzi; &c."
* See Colson's Newton's Flux. p. 18, Preface.
Our author proceeds to illustrate these observations, by examples; but we have alreany extended this article to a greater length, than we usually allow to elementary tracts of this kind.
The English Garden : a Poem. Book the First and Second,
By W. Mafon, M. A. 4to. gs. Cadell.
As it were superfluous, if not impertinent, to inake any reflections on the poetical abilities of a writer, so well known to the public as Mr. Mason, we shall confine ourselves, on the present occafion. to the giving our readers a specimen or two of his intimacy with the didačtic mufe: in whose placid province his talents appear well adapted to display themselves to advantage. In doing this, however, we hope not to incur his indignation so far as to induce him to commence à prosecution against us for piracy; as we have no intention in the least to injure' his property by preventing the sale of his poem.
The terrible threat, indeed, which he hath hung up in terrorem, by way of advertisement, opposite the title page of the second book, might måke us hesitate a little in our choice of such specimen, could we be persuaded Mr. M. is so litigious or avaricious, 'as he has been represented by a certain profecuted bookseller * Looking upon the quotation we should make, as matter of property, we should, in order to do the least injustice to him poffible, select some of the very word passages in his poem, as being of the leaft value. And yet, if we did this, it is ten to one if he did not complain of our doing both him and his poem injustice. On the other hand, if we should select the best passages, should we not do him greater injustice by robbing him, as he might call it, of the most valuable part of his poem ?-The'matter is intricate ; and we would advise Mr. M. to let his counsel make a case of it, and submit it to higher opinion. In the mean time noftro periculo we lhall proceed impartially to select neither the best nor the worst parts of his poem, to do as inuch justice and as little injustice both to Mr. M. and our readers as poffible.
The poet has thought no other apology for the choice of his subject, necessary, than the following passage from Lord 'Verulam.
" A garden is the purest of human pleasures, it is the greatest refreshment to the spirits of man : without which, buildings and palaces
* See Mr. Murray's letter to Mr. Mason, in our last Review.
are but gross handy-works. And a man shall ever see, that when
66 To thee, divine Simplicity ! to thee,
Thy flowery works with charm perennial please." The poet then proceeds to apostrophize the Muses and declare the motive of his verse; pathetically lamenting the cause.
“ Ye too, ye filter Powers ! that, at my birth,
The last farewell hung trembling, and bespoke This poem was begun in the year 1969, not long after the death of the amiable person here mentioned.
A wish to linger here, and bless the arms
Be dressid (Ah meek MARIA !) in thy charms."
“ O how unlike the scene my fancy forms,
“ Long was the night of error, nor dispelled
“ So taught the Sage, taught a degenerate reign