A S Y 



rrUnd, particularljr, they were eiitialy aboIifiieJ. See Sakc- 



TUARV. 



ASYMMETRY, derived from the privative a, <rvv, ivhli, 

 and /i»irf«y, mtnfiire, cj. d. without miafure, a want of propor- 

 tion, or corrcfpondence between the parts of a thing. See 

 Symmetry. 



In Mathematics, the word is more particularly ufed for 

 whnt wc more ulually call i.iconr::nenfurabilitv ; wliich is 

 when between two quantities tlitie is no common meafiire : 

 as betsveen the fide and dia'jonal of a fquare. In numbers, 

 fiird ro^ts, as ^ 2, &c. are iucommenfurable to rational 

 numbers. 



ASYMPTOTE, m-Gnw^try, a line which continually 

 approaches nearer and nearer to another; yet will never 

 meet with it, though indefinitely produced. 



The word is compound>.d of the privative a, snv, tv'ilh, 

 and •:^'k-j>, from •^trrlx; I fall ; q. d. iiicoindJeiit, or which 

 never meet. Some Latin authors call thcfe lines intaSa. 



Bertinus enumerates divers forts of afymptotes ; fome 

 ftraiglit, others curve ; fome concave, olhevs convex, &c. 

 a:id farther, propofes an inl'ru:nent for dcfcribing them. 

 Tlfough, in Ih-ictnefs, the term afymptotes feems appro- 

 p.iatcd to right liny*. A fyniptotes, then, are properly right 

 lines, which approach nearer and nearer to fome curve, of 

 which they are faid to be the afymptotes ; but which, 

 though they and their curve were indefinitely continued, 

 would never meet : confeo^i'.ently afvniptotes may be con- 

 ceived as tangents to their curves at an infinite diiiance. 

 Two curves are alfo faid to be afymptotical, when they thus 

 continually approach, without a poffibility of meeting. Thus 

 two parabolas, whofe axes are in the fame right line, are 

 afymptotical to one another. 



Ot lines of the fecond kind, or curves of thefirft kind, that 

 is, the conic feftions, only the hyperbola has afymptotes, 

 whicli are two in number, the properties of which ha've been 

 long ago demonllrated by Apollonius Pergaius. 



All curves of the fecond kind have at leaft one afymptote ; 

 but they may have three ; and all curves of the fourth kind 

 may have four afj-mptotes. 



The conchoid, ciffoid, and logarithmic curve, though not 

 reputed geometrical curves, have each alfo cue afymptote. 



The nature of afymptotes v.-ill be eafily conceived from 

 the inftance of the afymptote of a conchoid. Suppofe 

 MM AM, &c. {Plale Analysis, /o-. I.) to be a part of a 

 conchoid, C its pole, and the right line B D, fo drawn that 

 the parts Q^, E A, O M, &c. of riglit lines drawn from 

 the pole C, are equal to each other ; then will the line BD 

 be an afymptote of the curve ; becaufe the perpendicular 

 MI, &c. is (horter than MO, and MR than M(^&c. fo 

 that the two lines continually approach ; yet the pomts M, 

 &c. and R, &c. can never coincide, fince there is ftill a por- 

 tion of a line to keep them afunder ; which portion of a 

 line is infinitely divifible, and confequently mult be dimi- 

 nilhcd infinitely before it becomes nothing. 



Asymptotes of the hyperbola are thus defcribed. 

 Suppofe a right lineDE {Plats I. Cos\cs, fg. 20.) drawn 

 through the vertex A of the hyperbola, parallel to the or- 

 dinate M;n, and equal to the conjugate axis, viz. the part 

 DA, or AE, equal to the fcmi-axis : then, two right lines 

 drawn from the centre C of the hyperbola through the points 

 D and E, viz. the right hues C F and C G, are afymptotes of 

 the curve. The parts of any right lire, lying between the 

 <;ui-ve of the common hyperbola and its alymptotes, are 

 ■equal to one another on both fides, that is rm ^ M R. 

 Thus alfo, in hyperbolas of the fecond kind, if a right line 

 be drawn, interfccting the curve and its three afymptotes in 

 <hree points, the fum of the two parts of that right line es- 



A S Y 



tended in the fame direftion from any two of the afymptotes 

 to two points of the curve, is equal to the third part v.hicii 

 extends in the contraiy direction from the thiid afymptote 

 to the third point of the cur\e. 



If the hyperbola GMR {fig. 12. '^"2.) be of any kind 

 whofe nature with regard to the curve, and its afymptc^e, 

 is exprcfTe^ by this general equation, x"' y =. a" "^- ; and 

 the right line PM be drawn any where parallel to the afymp- 

 tote CS, and the paralltlo'yi-am PCOM be completed: 

 this parallelogram is to the hyperbolic fpace PMGB, con- 

 tained under the determinate lir;e PM, the Curve of the 

 hyperbola GM indefinitely continned towards G, and the 

 part PB of the afym.ptote indefinitely continued the fame 

 way, as m — n is to n ; and fo if n; be greater than n, the 

 fa-d fpace is finite and quadrable ; but when m = r, as it 

 will be in the common hyperbola, the ratio of the foregoing- 

 parallelogram to that fpace is as o to 2 : that is irfiiiitely 

 greater than the parallelogram, and fo cannot be obtained ; 

 and when m is lefs than n, m — ii will be negative, and the 

 parallelogram will be to the fpace as a n.gative number to 

 a pofitivc one, -and the faid fpace is called by Dr. WaiUi 

 more than infinite. See Hyplrbola. 



Asymptote of a Logarithmic Cvrnie. If MS (f^- 33-) 

 be the logarithmic curve, PR an afymptote, PT the fiib- 

 tangent, and MP an ordinate; then will the indeterminate 

 fpace RPMS =PM X PT; and the folid, generated by 

 the rotation of this curve about the afymptote V'P, will be 

 half of a cylinder whofe altitude is equal to the length of 

 the fubtangent, and t'le femidiamerer of the bafe equal to 

 the ordinate Qjy. See Logarithmic. 

 , Asymptotes, are by fome dillinguifhed into various 

 orders. An afymptote is faid to be of liie firft order, when 

 it coincides with the bafe of the curvilinear figure : of the 

 fecond order, when it is a right line parallel to the bafe : of 

 the third order, when it is a right line oblique to the bafe : 

 of the fourth order, when it is a common parabola, that has 

 its axis perpendicular tb the bafe : and, in general, of the 

 order « +2, when it is a parabola, thoordinate of which is 

 always as a power of the bafe, whofe exponent is n. The 

 alyinptote is oblique to the bafe, when the ratio of the firlt 

 fluxion of the ordinate to the fluxian of the bafe, approaches 

 to an adignable ratio, as its limit ; but it is parallel to the 

 bafe, or coincides with it, when this limit is not aflignable. 



The determination of the afymptotes of curves, is a curious 

 part of the higher geometry. M. de Fontenelle has given ftve- 

 ral theorems relating to this fubject, m his " Georr.ctrie dc 

 rinfini." See alfo Stirling's "LinextertiiOrdiris," Prop. vi. 

 where the fubjccl •f afymptotes is elaborately difcufied ; ami 

 Cramer, " IntroduiSlion a I'Aiialyfe dcs Lignes courbcs,"art 

 147, &c. in which is given an excellent theory of geometrical 

 curves and their branche'S. This fubjeft is alfo treated accu- 

 rately by Mr. Maclaurhi, in his Fluxions, book i. chap. to. 

 where he has been careful to avoid the modem panni' \e; 

 concerning infinites and infinicefimals. The areas bou .del 

 by curves, and their afymptotes, tlio' indefnitily extended, 

 fometimes have limits to which they may approach, fo as 

 to differ lefs from thofe limits than by any given quantity. 

 This happens in hyperbolas of all kinds, except the firft, or 

 i^poUonian. The fame is alfo true of the area, comprifed 

 between the logarithmic curve and its afymptote. SceLoc a- 

 RiTHMic curve. Thofe who do not fcruple to fuppofe the 

 curve and its afymptote to be infinitely produced, lay, that 

 the infinitely extended area becomes, equal to its limit. 



The afymptotical ar^ in the common or Apollonian hy- 

 perbola, and in many other curves, has no limit ; and it i»' 

 ufual to fay, thefe a'eas are infinitely great ; by which, 

 however, uo more is meant, than that the curve, and its 



aiymptotc, 



3 



