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ftells, -which they find in vaft abundance on their fca fhores. This was alfo a practice in the time of Diofcorides ; but the lime thus made is not fit for making lime water. The water made from it does not keep long, and is lefs ftyptic, and fweetifh to the tafte, and is greatly inferior to the water made with lime burnt from ftones. The newer the lime is, the lefs it has been expofed to the air, and the drier it has been kept, and finally, the more it has held together with- out crumbling, or mouldering to powder, the better it is for the making lime water. Hot water has alfo been found to make it better than cold, and rain water much better than any other kind. The fame lime will ferve twice to make the medicinal water ; but if it be tried a third time, the water will be found to be quite infipid : if after twice nfing it be calcined again, the fire will give it the fame qua- lities, and it will again make the lime water as ftrong as the firft. It has this inconvenience, however made, that it will not mix with any other liquor. It ferments with all kinds of fyrups, and will not bear the lean: mixture of any thing in the leaft acid without becoming turbid, and de- pofiting a fediment. Mem, Acad. Par. 1700. Experience has fhewn lime water to be an excellent medi- cine in many cafes. In the gravel and ftone, particularly And it has alfo been found very ferviceable in the gout, in habitual relaxations of the bowels, and in other cafes of relaxation. In fome kinds of the fcurvy likewife it is of ufe ; and is often applied, with fuccefs, externally, to ul- cers, &c.
Fabricius ab Aquapendente allures us, he cured a fchirrhus fpleen, and the dropfy, by a continued ufe of fponges dipped in common lime water, and placed near the part affected Boyle's Works Abr. Vol. 1. p. 80.
LIMITS, (Cycl.) in mathematics, is fometimes ufed, in gene- ral, for quantities, one of which is greater, and the other lefs than another quantity. Thus, in the quantities, a, x, I?, if a be lefs than x and b be greater than x, a and b are faid to be limits of x. The word occurs in this fenfe, when we fpeak of the limits of the roots of equations. Sometimes a quantity is faid to be a limit between two others, when it is greater than the one and lefs than the other. So ratio is faid to be a limit between two other ratios, when it is greater than the one and lefs than the other. But limit is often ufed in a more reftrictcd fenfe ; thus, when a variable quantity approaches continually to fome determinate quantity, and may come nearer to it than to have any given difference, but can never go beyond it; then is the determinate quantity faid to be the limit of the variable quantity. Hence, the circle may be faid to be the limit of its circum- fcribed, and inferibed polygons ; becaufe thefe, by increafing the number of their fides, can be made to differ from the circle lefs than by any fpace that can be propofed, how fmall foever. The limit of a variable ratio, is fome determinate ratio, to
■ which the variable ratio, may continually approach, and come nearer to it than to have any given difference, but can never go beyond it. Hence, the ratio of the ordinate to the fub-tangent of a curve, is faid to be the limit of the variable ratio of the differences of the ordinates, to the dif- ferences of the abfeiffe.
The word limit, in this fenfe, fignifies the fame as what Sir Ifaac Newton calls a firft, or prime, and a laft, or ulti- mate ratio.
There are two cafes of a variable quantity, or variable ratio, tending to fuch a limit, as we have been defcribifig. In the firft cafe, the variable quantity, or ratio, will not only approach to its limit nearer than any given difference, but will actually arrive at its limit. In the fecond cafe, the variable quantity, or latio, will only approach its limit within lefs than any given difference, but will never actually arrive at it. Sir Ifaac Newton, to avoid the harfhnefs of the hypothecs, of indivifibles, and the tedioufnefs of demonftxations, ac- cording to the method of the antients, by deductions ad abfurdum, has prcmifed feveral lemmata, in the firft fee tion of the firft book of his principles, relating to the firft and laft fums, and ratio's of nafcent and evanefcent quantities ; that is, to the limits of fums, and ratio's. This doctrine chiefly depends on the firft of thofe lemmata ; the words of which are, .^uantitates ut & quantitatum rationes, qua ad tsqualitatem tempore quovis finito conjlanter tendunt, & ante fine?n temporis illius propius ad inviccm accedunt quam pro data ai/avis differentia, fiunt ultimo aquales.
The learned gentlemen, who have written in defence of Sir Ifaac, againft the author of the Analyit, are not agreed among themfelves, as to the prccife meaning of this lemma. One of thefe gentlemen a fays, that the genuine meaning of this propofition is, that thofe quantities are to be efteemed ultimately equal, and thofe ratio's ultimately the fame, which are perpetually to each other, in fuch a manner, that any difference, how minute foever, being given, a finite time may be afligned, before the end of which, the difference of thofe quantities, or ratio's, fhall become lefs than that given difference. [ — a See Pref. State of the rep. of Letters for Oct. 1735, and for Oct. 1736. — j
What Sir Ifaac Newton intends, we mould underftand by the ultimate equality of magnitudes, and the ultimate iden- tity of ratio's propofed in this lemma, will be beft known from the demonftration annexed to it. By that it appears, Sir Ifaac Newton did not mean, that any point of time was aflignable, wherein thefe varying magnitudes would become actually equal, or the ratios really the fame ; but only that no difference whatever could be named, which they fhould not pafs. The ordinate of any diameter of an hyperbola, is always lefs than the fame continued to the afymptote ; yet the demonftration of this lemma can be applied, with- out changing a fmglc word, to prove their ultimate equa- lity. The fame is evident from the lemma immedi- ately following, where parallelograms are inferibed, and others circumfcribed to a curvilinear fpace. Here the firft: lemma is applied to prove, that by multiplying the number, and diminifhing the breadth of thefe parallelograms in infi- nitumi that is, perpetually and without end, the inferibed and circumfcribed figures become ultimately equal to the curvi- linear fpace, and to each other ; whereas, it is evident, that no point of time can be afligned, wherein they are ac- tually equal j to fuppofe this were to afTert, that the varia- tion, afcribed to thefe figures, though endlefs, could be brought to a period, and be perfectly accomplifhed ; and thus we mould return to the unintelligible language of indi- viftbles. The excellence of this method coniifts in making the fame advantage of this endlefs approximation towards equality, as by the ufe of indivifibles, without being in- volved in the abfurdities of that doctrine. In fhort, the dif- ference between thc{e two may be thus explained. There are but three ways in nature of comparing fpaces : one is by fhewing them to confift of fuch, as by impofition on each other will appear to occupy the fame place : ano- ther is by fhewing their proportion to fome third j and this method can only be directly applied to the like fpaces as the former ; for this proportion muft be finally determined by fhewing, when the multiples of fuch fpaces are equal, and when they differ : the third method to be ufed, where thefe other two fail, is by defcribing upon the fpaces in queftion fuch figures, as may be compared by the former methods ; and thence deducing the relation between thofe fpaces, by that indirect manner of proof, commonly called dedufiio ad abfurdum ; and this is as conclufive a demonftration, as any other, it being indubitable, that thofe things are equal, which have no difference. Thus Euclid and Archimedes de- monftrate all they have written, concerning the comparifon and menfuration of curvilinear fpaces. The method advanced by Sir Ifaac Newton for the fame purpofe, differs from theirs, only by applying this indirect form of proof to fome general propofitions, and from thencededucing the reft by a direct form of reafoning. Whoever compares the fourth of Sir Ifaac Newton's lemmas with the firit, will fee, that the proof of the curvilinear fpaces, there confidered, having the propor- tion named, depends wholly upon this, that if otherwife the figure inferibed within one of them, could not approach, by fome certain diftance, to the magnitude of that fpace; and this is precifely the form of reafoning, whereby Euclid proves the proportion between different circles. As this method of reafoning is very diffufely fet out in the writings of the antients ; and Sir Ifaac Newton has here expreffed himfclf with that brevity, that the turn of his argument may poflibly efcape the unwary j the reading the antients, muft be the beft introduction to the knowledge of his me- thod. The impoflible attempt of comparing curvilinear fpaces, without having any recourfe to the fore- mentioned indirea method of arguing, produced the abfurdity of indi- vifibles.
As the magnitudes, called in this lemma, ultimately equal, may never abfolutely exift under that equality ; fo the vary- ing magnitudes holding to each other the variable ratios, here confidered, may never exift under that, which is here called their ultimate ratio. Of this Sir Ifaac Newton gives an inftance, from lines increafing together by equal additions, and having from the firft a given difference. For the ulti- mate ratio of thefe lines, in the fenfe of this lemma, as Sir Ifaac Newton himfelf obferves, .will be the ratio of equality, though thefe lines can never have this ratio j fmce no point of time can be afligned, when one does not exceed the other. '
In like manner, the quantities called by Sir Ifaac Newton vanifhing, may never fubfift under that proportion here efteemed their ultimate.
In the cafe of drawing tangents to curves, where the ordinate bears the fame proportion to the fubtangent, as that where- with the difference of the ordinates, to the difference of abfciflas, vanifh ; thefe lines muft not be conceived, by the name of an evanefcent, or any other appellation, ever to fubfift under that proportion : for fhould we conceive thefe lines, in any manner, to fubfift under this proportion, though at the inflant of their vanifhing, we fhall fall into the unintelligible notion of indivifibles, by endeavouring to reprefent, to the imagination, fome inconceivable kind of exillence of thefe lines between their having a real mag- nitude, and becoming abfolutely nothing. Sir Ifaac New- ton
