L I M
L I M
ton was himfelf apprehenfive, that this miftake mightbemade; for as he thought fit (in compliance with the bad tafte which then prevailed) to continue the ufe of fome loofe and indiftinct expreflions refembling thofe of indivifibles, for which he has himfelf apologized, he exprefsly cautions us againft mifinterpreting him in this manner, when he fays : Si quando dixero quanthates quam minimis, vel evanefcentes, vel ultimas, cave intelligas quanthates magnitudine determina- tas, fed cogha femper diminuendas Jine ihnite. Thus exprefsly has he declared to us, that vanifhing quantities, or whatever ■ other lefs accurate appellation he names them by, are to be confidered as indeterminate quantities bearing to each other under their different magnitudes, different proportions; which the quantities themfelves can never obtain, and the limit of thefe proportions, is that, for the fake of which thefe quantities are confidered : infomuch, that fmce thefe quantities have different proportions, while they obtain the name of vanifh- ing quantities, the term ultimate is neceffarily added to de- note that proportion, which is the limit of an endlefs num- ber of varying ones. The like remark is neceffary, when thefe quantities are confidered in the other light, as arifing before the imagination : for then the proportion intended muft be fpecified, by calling it the firft, or prime proportion of thefe quantities. And as this additional epithet is necef- fery to exprefs the proportion intended, fo it is abfurd to apply it to the quantities themfelves ; as Sir Ifaac Newton fays, there are rationes prima quatititatum nafcentium, but not quanthates prima: nafcentes. Phil. Tranf. N" 342. p. 205. So that according to the author we have been quoting, all the examples given by Sir Ifaac in the before-mentioned fection, are to be underftood of fuch limits or ultimate ratios, as are never attained to by the quantities and ratios limited, but to which thefe may approach indefinitely, that is, fo as to differ lefs than by a given quantity.
On the other hand, a learned gentleman, who affumed the name of Philalethes Cantabrigienjis a , thinks, that Sir Ifaac means, by the words of the lemma, and proves, in his demonstration, not that the quantities or ratios, are barely to be confidered as ultimately, becoming equal, or are to be eftecmed as ultimately equal; though, in reality, they can never have that proportion to each other ; but that they do at laft become actually, perfectly, and abfolutely equal. [ — a Pref. State of the Rep. of Letters for Nov. 1735.
P- 37 1 *— 1 . • [
He alfo diftinguifhes, as above, between quantities and ra- tios which arrive at their limits, and thofe which do not. And it is infifted on, that every one of the examples given. in the lemmata of this firft fedtion of the firft book of Sir Ifaac's principles, are of fuch quantities and ratios as actually arrive at their refpedtive limits ; nor is there an inftance there given of a quantity, or ratio, which never arrives at its limit, except one at the latter end of the fcholium of this fection (and that by way of Uluftration of a particular ob- jection only) of two quantities, having a given difference, and being equally increafed, ad infinitum, and whofe ratio, it is admitted, never arrives at its limit. But decreafing quantities may really, and, in fadt, be diminifhed ad infi- nitum •' for they may vanifh and come to nothing. The ratio therefore of thefe, fays he, may arrive at its limit, though that of the others cannot.
Neither are thefe learned gentlemen agreed as to the fenfe of the word vanijhing or evanefcent, in the fcholium of this firft fection of Sir Ifaac's principles.
The queftion is, whether the quantities that vanifh, are underftood to fpend fome finite time in vanifhing, or to va- nifh in an inftant, or point of time ; and confequently, whether they bear to one another an infinite number of dif- ferent fucceflive ratios during the vanifhing, or one ratio only, at the point, or inftant, of their evanefcence. This laft is the fenfe in which Philalethes takes the word evanefcent or vanifhing ; arid the difpute, 011 this head, as he obferves a , is of no other confequence, than to deter- mine, whether the fenfe in which he ufes the word be.a- greeable to Sir Ifaac Newton's. For, if the quantities va- nifh in an inftant, arid I take the only ratio with which they vanifh ; or they fpend a finite time in vanifhing, and I take the laft of the ratios, which they fucceffively bear to one another during that time ; flill the ratio, taken in either of thefe cafes, will be one and the fame. [ — a Pref. State of Rep. of Lett, for Nov. 1735, p. 383, 384.—] We cannot pretend to give the whole detail of this con- troverfy : but muft refer the curious to the prefent ftate of the republic of letters for 1735. We fhall only obferve, that this difquifition is partly critical and partly fcientifical. The critical enquiry is into the fenfe of Sir Ifaac, fo far as it may be determined from his own words, and here we cannot help thinking that this is fomewhat doubtful. The other enquiry is about the true or fcientifical notion, upon which this aodtrine ought to be founded. With refpect to which we fhall only afk two queftions, which every rea- der may refolve for himfelf, to wit, whether the conception or notion he has of the ratio or proportion of evanefcent quantities, at the point or inftance of their evanefcence, be more clear and diftindt than the notion of infinitefimals I
And whether the notion of inferibed or circa mfcribed poly* gons to any curve, attaining their laft form, and thereby coinciding with their curvilinear limit, be more clear and diftindt than the notion of polygons of an infinite number of fides, in the method of infinitefimals ? Before we leave this fubject, it may be proper to give the fentiments of an eminent mathematician c about the doc- trine of limits, or of prime and ultimate ratios, and to fhew the connection of this doctrine with that of fluxions. [— c Mr. Maclawin, in his Treat, of Fluxions, Art. 502. — ] Sir Ifaac Newton confiders the fimultaneous increments of flowing quantities as finite, and then inveftigates the ratio which is the limit of the various proportions which thofe in- crements bear to each other, while he fuppofes them to dc- creafe together till they vanifh ; which ratio is the fame with the ratio of the fluxions. In order to difcover this limit, he firft determines the ratio of the increments in general, and re- duces it to the moft fimpleterms, fo as that (generally fpeaking) a part at leaft of each term may be independent of the value of the increments themfelves ; then by fuppofing the increments to decreafe till they vanifh, the limit readily appears.
For example, let a be an invariable quantity,, x a flowing quantity, and any increment of x ; then the fimulttneous increments of xx and ax will be 1x0 + 00 ar.d go, which are in the fame ratio to each other, as 2x-\~o is to a. This ratio of ix-\-o to a, continually decreafes while decreafes, and is always greater than the ratio of 2 x to a, wlula ia any real increment; but it is manifeft, that it continually approaches to the ratio of 2 x to a as its limit \ whence i't follows, that the fluxion of xx is to the fluxion of a x as 2,vistofl'. If a- be fuppofed to flow uniformly, ax Will likewife flow uniformly, but xx with a motion continually accelerated : the motion with which a x flows, may be meafured by ao; but the motion with which x x flows, is not to be meafured by its increment %xo-\-oo, but by the part 7.XQ only, which is generated in confequence of that motion 5 and the part 00 is t@ be rejected, becaufe it is generated in confequence only of the acceleration of the motion with which the variable fquare flows, while the increment of its fide is generated ; and the ratio of ixo to ao is that of ix to a, which was found to be die Ihnh of die ratio of the incre- ments ixo-\- 00 and ao. See the article Fluxion. It is objected againft Sir Ifaac Newton's method of invefti- gating this limit, that he firft fuppofes that there are incre- ments, that when it is faid let the increments vanifn, the former fuppofition is deftroyed, and yet a confequence of this fuppofition, i. e. anexpreffion got by virtue thereof, is re- tained. But the fuppofitions that are made in this method of inveftigating the Ihnh, are not fo contradictory as this objection Teems to import. He firft fuppofes that there are Increments generated, and reprefents their ratio by that of two quantities, one of which is given fo as not to vary With the increments. If he had afterwards fuppofed that no increments had been generated, this indeed had been a fup- pofition directly contradictory to the former. But when he fuppofes thofe increments to be diminifhed till they vanifh, this fuppofition finely cannot be faid to be fo contradictory to the former, as to hinder us from knowing what was die ratio of thofe increments, at any term of the time while they had a real exiftence ; how this ratio varied, and to what limit it approached, while the increments were con- tinually diminifhed. On the contrary, this is a very con- cife and juft method of difcovering the limit which is re- quired.
It is to'be obferved, that the limiting, prime, or ultimate ratio of increments, ftrictly fpeaking, is not the ratio of any real in- crements whatfoever. But as the tangent of an arch is the right
j line that limits the pofition of all the fecants that can pafs through the point of contact, though ftrictly fpeaking it be
j no fecant ; fo a ratio may limit the variable ratios of the
I increments, though it cannot be faid to be the ratio of any real increments. The ratio of the generating motions may be likewife faid to be the laft, or ultimate ratio of the incre- ments, while they are fuppofed to be diminifhed till they vanifh, for a like reafon.
LIMITANEI, among the Romans, an appellation given to the foldiers who were ftationcd on the frontiers of the em- pire. Phifc. Lex. Ant. in voc<
LIMITOTROPHI, among the Romans, the fame with £* mitanei. See Limitanei, fupr.
LIMMA, Ttyppa', refuluum, in the antient mufic, is the dif- ference of the diateflarou and the ditonus. It is exprefled by -m.
Euclid demonftrates it to be lefs than the hemitone «. Boe* thius calls the Hmma, hemitonium minus b . — [" IVallis's Append. Ptolem. Harm. p. 169, 170. b Ibid. p. 170. — ] The limma here defined is that of Ptolemy. Modern mufi- cians have applied this name to feveral other intervals, arif- ing from the fubftraction of femi-tones from the tone ma- jor. Thus the difference between the femi-tone minor and the tone major is % : 2 -| — \\ is by Mr. Euler called limma majus ; and the difference between the femi-tone major and tone major he calls limma minus a . This laft- is expreffed
