681 



A'SYMPTOTE. 



ATABEKS. 



632 



tance between them diminiahea without limit, or they may be brought 

 to any degree of nearness, without ever actually meeting. 



This appears a paradox to beginners in geometry, who are generally 

 unable to imagine it possible that two lines should continue to 

 approach one another for ever, without absolute contact. But this 

 arises from their confounding the thing called a straight line in 

 practice (which is not a straight line, but a thin stroke of black lead or 

 ink, as the case may be) with the straight line of geometry, which has 

 neither breadth nor thickness, but only length. And they also 

 imagine that if two lines might be asymptotic, the fact might be made 

 visible, which is impossible, unless the eye could be made to distin- 



guish any distance, however small. But if the unassisted eye cannot 

 detect a white space between two black lines, unless that space be a 

 thousandth of an inch in breadth, which is about the truth, it is 

 evident that two geometrical surfaces with asymptotic boundaries, 

 such as A B c, D E c, would appear to coincide from the point where the 

 distance between them is about the thousandth part of an inch. The 

 idea of a geometrical asymptote is therefore an effort of pure reason, 

 and the possibility of it must be made manifest to the mind, not to 

 the senses. ALHBCD is a vessel of water, of which the sides and 



bottom are extended indefinitely towards o and B ; the end A L is fixed, 

 but the end BM is moveable parallel to its first position, so as always 

 to form a Water-tight obstacle ; by which means the length of the 

 vessel may be increased to any extent, while its breadth and height 



n tin; K.-IIMO. Let the water be a perfect fluid, without any 

 adhesion to the sides of the vessel (which is mathematically possible, 



ii not physically), and let the bottom of the vessel be geome- 

 trically horizontal. Then, as the end M B changes its position and 

 moves towards OB, it is manifest that the vessel will grow larger, 

 and the level of the water will fall. Suppose the side L K to be of 

 glass. Thus, when the vessel ends at E p, the water may stand at s v ; 

 when the end is at F Q, the water may stand at T w, and so on. But 

 the level of the water never can fall absolutely to the bottom c K ; for 

 so long as the preceding mathematical suppositions hold good, and 

 there is gome water in the vessel, it must stand at some determinate 

 height above the bottom. As the end B M moves to the right, let the 

 curve M V w, &c., mark out the positions of the level upon the edge of 

 the moving end, as is done in the diagram. Then for the reason 

 above given, this curve never can meet the line c K, though obviously 

 in a state of continual approach towards it. Hence the curve M v w 

 and the line c K are asymptotes. 



At another illustration, let there be two parallel lines A B, c D, the 

 perpendicular distance of which is A c ; and from A, with different 



radii, describe arcs of circles pi, Q<?, Ttr, s, &c. From A B on all 

 these circles measure arcs equal in length to the straight line A c ; that 

 is, let p 1, <J 2, R 3, . . . . w 7, &c., be all equal to A c. Now it is plain 

 that the arcs q q, B r, &c., are all greater than A c, and will continue 

 so, however great the radius may be ; for A c is the shortest distance 

 which can be drawn from one parallel to the other. But as the radius 

 is extended, the arcs T t, v r, Ac., become more upright, as a person 

 unused to geometrical phraseology would say, that is, more and more 

 nearly coincident with a perpendicular drawn from A B ; they also 

 become more and more nearly equal to A C. Hence the points 5, 6, 7, 

 &c., come nearer and nearer to c D, with which they would actually 

 cqjncide, if it were possible that one of the arcs could become equal to 

 A c. Hence the curve, 1, 2, 3, &e., is an asymptote to c D. 



The mathematical theory of asymptotes will be found in all works 

 on the theory of curves, and in most on the differential calculus. The 

 following are the most general notions which it will be within our 

 limits to give, and will be understood by a moderately well-informed 

 mathematician. If the equation of a curve be y^<t> (x), and if the 

 function <f> (x) can be separated into two others, say 1/1 (;c) and x (*)> of 



which x (*') diminishes without limit when x is increased without 

 limit ; then the curve whose equation is y = $/ (x) is an asymptote to 

 the curve whose equation is y = </> (x) or i|> (x) + x (*) For the differ- 

 ence of the ordinates of the two curves (to a common value of x} 

 is x (*')> which diminishes without limit. For instance, let the first 

 curve have the equation 



x-a 



since is b + , of which - decreases without limit when 



x a, xa xa 



x is increased without limit, it follows that the straight line having 

 the equation y = b is an asymptote to the curve. If the preceding 

 equation be reversed and put under the form 



ay 

 x=-^L 



y-b 



similar reasoning will show that the straight line whose equation 

 is .) = a is also an asymptote. If the first expression be developed in 

 inverse powers of x, giving 



y = 6+*fL +** + * + Ac. 



X Z 2 X s 



the equations of curves which are asymptotes to the preceding may be 

 found by taking any of the preceding terms for y, provided b be always 

 one. Such are 



, ba 



y=b + 



x 



x- x 3 

 or generally, any curve whose equation is 



where x (*) diminishes without limit, when x is increased without 

 limit, is an asymptote to the preceding. Observe that a curve may 

 first cut another, then recede from it, and afterwards become an 

 asymptote to it. 



The following is a mere sketch of the most general method of 

 finding asymptotes to algebraical curves. The first part of the method 

 detects the number and direction of the rectilinear asymptotes, those 

 only excepted which are parallel to either axis of co-ordinates, which 

 will easily admit of a separate determination. 



Clear the equation of all radicals. Suppose it then of the second 

 degree, though the same reasoning applies to all degrees. Its form 

 will then be (putting all the highest terms on one side) 



The following theorem can then be demonstrated. If the equation 

 ay' + b ay + c.( 3 =Q (A) 



be possible, then it is the collective equation of two lines passing 

 through the origin of co-ordinates, which two lines are parallel to two 

 asymptotes of the curve and the curve can have no others. It is a 

 well-known theorem that any algebraical equation between .T and y, 

 which is homogeneous with respect to these letters, is not the equation 

 of a curve, but of a collection of straight lines passing through the 

 origin. Thus the asymptotes of tlie curve of the third degree will be 

 determined by the solution of an equation of the form 



a x 3 + b 3?ij + c xy 1 + rf^= 



which may belong either to one or three straight lines. 



If y = k x + l be the equation of an asymptote, the value of k may be 



any one of the values of -L determined from the equation (A). ' To 



x 



find 7, remember that any homogeneous algebraical expression of the 

 mth degree, containing x and y, may be expressed by the form 



(I) 



and let the equation of the curve, when its various sets of homogeneous 

 terms have been collected, be 



Then if <f>' (it) represent the differential coefficient of <t> (t), the equation 

 of the asymptote is 



r -i- OL> 



<? (k) 



when the highest dimension in the equation exceeds the next highest 

 by more than one, all the asymptotes must pass through the origin of 

 co-ordinates. 



The term asymptote is first found in the Conic Sections of Apol- 

 lonius ; and the properties of the hyperbolic asymptote are found in 

 the second book of his Conic Sections. 



A'TABEKS are the rulers of several of the small principalities into 

 which the empire of the Seljuk Turks, soon after its establishment, 

 became divided, during the eleventh, twelfth, and thirteenth centuries. 



