TANGENT. 



To determine the tangents of curves, fuppofed to be de- 

 fcribed by the iiiterfeftions of right lines revolving about 

 given poles, fee Mr. Maclaurin's Fluxions, art. 210, fcq. 

 In finding the tangents of curves by tli- method of iiilinite- 

 fimal differences, it has been objeftcd that the conclufion is 

 found by a double error, i. By taking the curve for a 

 polygon of an infinite number of fides. 2. By the falfo 

 rule for taking the differentia] of a power. But there is no 

 need of fuch fuppofitions in the method of fluxions, for it 

 may be geometrically demonftratcd, tliat the fluxions of the 

 bafe, ordinate, and curve, are in the fame proportion to 

 each other, as the fides of a triangle refpeftively parallel to 

 the bafe, ordinate, and tangent. When the bafe is fup- 

 pofed to flow uniformly, if the curve be convex towards 

 the bafe, the ordinate and curve increafe with accelerated 

 motions ; but their fluxions at any term are the fame as if 

 the point which defcribes the curve had proceeded uniformly 

 from that term in the tangent. Any farther increment 

 which the ordinate or curve acquires, is to be imputed to 

 the acceleration of the motions with whicli they flow. See 

 Maclaurin's Fluxions, book i. chap. vii. and viii. 



Any two arcs of curve lines touch together, when the 

 fame right line is the tangent of both at the fame point. 

 But when they are applied to each other in this manner, 

 they never perfeftly coincide, unlefs they be fimilar arcs of 

 Cmilar and equal figures. 



In the Philofophical Tranfaftions, we have the following 

 method of drawing tangents to all geometrical curves, with- 

 out any labour or calculation, by M. Slufius. 



Suppofe a curve, as D Q {Plate XV. Anal. jig. 3.) 

 whofe points are all refcrrible to any right line given, as 

 E A B, whether that right line be the diameter or not ; or 

 whether there be more given right lines than one, provided 

 their powers do but come into the equation. In all his 

 equations, he puts -o for the line D A, y for B A ; and for 

 E B, and the other given lines, he puts b, d, Sec. that is, 

 always confonants only. 



Then, fuppofing D C to be drawn touching the curve in 

 D, and meeting with E B produced in C, he calls the fought 

 line, C A, by the name of a. 



To find which, he gives this general method. I. Rejeft 

 out of the equation all members which have not either 

 ■V or y in them ; then put all thofe that have j on one fide, 

 and all thofe which have v on the other ; with their figns + 

 or — ; and the latter, for diftinftion and eafe fake, he calls 

 the right, the former the left fide. 2. On the right fide, let 

 there be prefixed to each member the exponent of the power, 

 which V hath there ; or, which is the fame thing, let that 

 exponent be multiphed into all the members. 3. Let the 

 fame be done alfo on the left fide, multiplying each member 

 there by the power of the exponent of y ; adding this more- 

 over, that one y mufl:, in each part, be changed into a. 

 This done, the equation thus reformed will (hew the method 

 of drawing the required tangent to the point D ; for, that 

 being given, as alfo y, v, and the other quantities exprefled 

 by confonants, a cannot be unknown. Suppofe an equa- 

 tion iy — yy = v v, in which E B is called b ; ^ A — y, 

 D A = 1), and let a, or A C, be required fo as to find the 

 point C, from whence C D being drawn, (hall be a true 

 tangent to that curve Q D in D. In this example, nothing 

 is to be rejefted out of the equation, becaufejr or v are in 

 each member : it is alfo difpofed, as required by the rule I ; 

 to each part, therefore, there mud be prefixed the expo- 

 nent of the powers of y or i>, as in the rule 2 ; and on the 

 left fide, let one y be changed into a, and then the equa- 

 tion will be in this form, ba - 2ja = zvv, which equa- 

 JO 



2VV 



tion reduced, gives cafily the value of a = ' " '" = A C. 



6 — Zy 



And fo the point C is found, from wliich the tangent DC 

 may be drawn. 



To determine wliich way the tangent is to be drawn, whe- 

 ther towards B or E, he direcis to confidcr the numerator 

 and denominator of the fratlion. For, i. If in bolii paru 

 of the fraftion all the figns are affirmative ; or if the affirm- 

 ative ones are more in number ; then the tangent is to run 

 towards B. 2. If the affirmative quantities are greater than 

 the negative in the numerator, but equal to thole in the de- 

 nominator, the right line drawn through D, and touching 

 the curve in that point, will be parallel to A B ; for in thi» 

 cafe a is of an infinite length. 3. If in both parts of the 

 fraftion the affirmative quantities are lefs tlian the negative, 

 changing all the figns, the tangent mufl: be drawn now alfo* 

 towards B ; for tliis cafe, after the change, comes to be 

 the fame as the firll. 4. If the affirmative quantities are 

 greater than the negative in the denominator, but in the 

 numerator are lefs, or -vice verfi, then changing the figns in 

 that part of the fraftion where they are lefs, the tangent 

 mud be drawn a contrary way ; that is, A C mud be taken 

 towards E. 5. But whenever the affirmative and negative 

 quantities are equal in the numerator, let them be how they 

 will in the denominator, a will vanifh into notiiing : and, 

 confequently, the tangent is either A D itfelf, or E A, or 

 parallel to it ; as will eafily be found by the data. This he 

 gives plain examples of, in reference to the circle, thus : let 

 there be a femicirclc, whofe diameter is E B ; in which there 

 is given any point, as D (Jig. 4.), from which the per- 

 pendicular D A is let fall to the diameter. Let D A = -u, 

 B A =^', B E = i : then the equation will be by — yy -=1 

 ■V V, and drawing the tangent D C, we have AC, or a = 



2 7) 1; 



2y 



Now, if 6 be greater than zy, the tangent rnxiH 



be dravvn towards B j if lefs, towards E ; if it be equal to 

 it, it will be parallel to E B, as was faid in the firil, Ucond, 

 and fourth rules. 



Let there be another femicircle inverted, as N D D 

 i/s- 5-)» the points of whofe periphery are referred to the 

 right line B E, parallel and equal to the diameter. Let N B 

 be called d, and all things elfe as before ; then the equa- 

 tion will be by — yy z= dd ^ -vv — idv; which being ma- 

 naged according to his rules, you have a = '"'" ~ ^"'"^ 



b — 2y 



Now, fince V here is fuppofed to be always lefs than d ; 

 if b be greater than 2y, then the tangent muft be drawn 

 towards E ; if equal, it will be parallel to B E ; if lefs, 

 changing all the figns, the tangent muft be drawn towards 

 B, as by rules fourth, fifth, and third. But there could be 

 no tangent drawn, or at leaft E B would be it, if N B liad 

 been taken equal to the diameter. Let there be another 

 femicircle, whofe diameter N B {fg. 6. ) is perpendieular 

 to E B, and to which its points arc fuppofed to be referred. 



Let N B be called b, and all things elfe as above ; the 



•11 L I J bv — 2W ,, 



equation will he yy = b-u — vv, and a = . If 



zy 



now, b be greater than 2 v, the tangent muft be drawn to- 

 wards B ; if lefs, towards E ; if equal, D A will be the 

 tangent, as appears by rules fourth and fifth. 



Tancjents, Inverfe Method of, is a method of finding the 

 equation, or the couftruAion, of any curve ; from the tan- 

 gent 



