TANGENT. 



fluxions of the abfcifs A P, and tJie ordinate PM- «nd 

 becaufe the triangles M m R and T M P arc fimilar, wc have 

 R m : M R :: P M : P T. Let, therefore, the ahfcifs A P 

 be put = X, and the ordinate P M =^, and we (hall hare 



y : X .: y : —. = p Y. By means of this general eKpref. 



fion for the fub-tangcnt, and the equation of the curve es- 

 prHl.ng the relation between x and y, the ratio of the 

 fluxions * and > will be found, and from thence the length 

 ot the fub-tangcnt ; whence the tangent itfelf may be cafilT 

 determined and drawn. This we fhaU iUuftrate in the fol- 

 lowing examples : 



I. The equation defining a circle hax — Kx=y^- and 

 by taking the fluxions of thefe quantities, ax -2xiz= 



2yy 



confequently — = 



y a — 2 X 



i> J' 



and, molti* 



plying both fides by y, we have ^ = — i!_ = the fub- 



y ifl — .V 

 fuppofmg P/. the increafe of the abfcifs = e, tangent PT {ice fig. 20.) ; whence U a — «) or AC — 



r_„_.:._ : c. „. .u - _..: ^ P, i.e. C P : (^) P M :: (j,) P M?P T ; a property of 



the circle deduced from the principles of common eeo- 

 metry. ° 



II. The equation defining the common parabola is ok 

 = >', a being the parameter, .-c the abfcifs, and y the or- 



^/px; 



dinate ; hence 



ax = 2 yj, 



and — = — ; 



confequently. 



y x 

 y 



a 



2 ax 



As to the methods propofed by Hudde, Roberval, 

 Huygens, &c. they differ fromthofc given above, only in the 

 fame manner as in their methods of maxima and minima ; it 

 would therefore be ufelcfs to defcribe them in this place. 



Barroiu's Method of Tangents — It is obvious from what is 

 faid above, and what has been ftatcd under the article Maxi- 

 ma et Minima, that both the method of tangents, and that 

 for the greateft and leaft ordinates, were very nearly related to 

 the prefent fluxional way of treating the fame fubjcfts ; hut 

 with regard to tangents, a flill nearer approach was made by 

 Dr. Barrow. 



This accurate geometer confidered the little triangle 

 formed by the difference of the two ordinates, their diilance 

 from each other, and the indefinitely fmall part of the curve, 

 as fimilar to that which is formed by the ordinate, the tangent, 

 and fub-tangent. He then fought by the equation of the 

 curve, the ratio of the two fides ba, BiJ, (fg. 18.) of the 

 triangle B 3 a, when the difference of the ordinates is infi- 

 nitely little ; and then faid, 3& ba : B a :: ordinate B P : the 

 fub-tangent T P. 



In the cafe of the parabola, for example, whofe equation 

 hf=px '.."__"./■ 



and b a the correfponding increafe of the ordinate y z= a% 

 then the equation for the ordinate p b becomes 



(jr -f ay =/. (x + e), or 



y^ + zay + a' — px + pi. 



^ubtrafting from both fides j^^ =^ p x, there remains 

 2 ay -ir ci' ^= p e. 



Alfo a being itfelf infinitely fmall, its fquai-e «' may be 

 entirely neglefted, and there refults z ay = pe\ therefore 

 a : e ■.: p : 2y ; but a ■=■ b a, and e =.^ a, alfo y =z 

 therefore, from the propofition ftated above, •uiz. 



ab : aB :: ordinate : fubtangent, 

 we ha»e 



p : 2 ^ px :: ^f p x ; 2 x, the fubtangent required. 



Such were the principles employed in the folution of this 



interefting problem prior to the brilliant difcovery of the 



fluxional calculus, which from its generality fupplanted them 



all, and they are now therefore merely matters of hiftorical 



curiofity ; but as they exhibit the flow and progreflive ad- 

 vances of genius and fcience towards an ultimate ftate of per- 



feftion, they are highly deferving of the attention of the 



mathematician, who will find in them much to admire ; they 



will at the fame time enable him duly to appreciate the tran- 



fcendant talents of that great philofophcr, who formed out 



of them one general and comprehenfive principle of folution, 



which will apply with equal facility to algebraical curves of 



every order. 



The Method of Tangents accordingto the DoSrim of Fluxions. — 



Its ufe is very great in Geometry ; becaufe in determining the 



tangents of curves, we determine at the fame time the quad- 

 rature of the curvilinear fpaces : on which account it well 



deferves to be here particularly infilled on. 



To find the Sub-tangent in any algebraic Curve Let the pro- 

 pofed curve be A M O ( Plate XIV. Anal.jig. 19.), and the 

 right line T M Q a tangent to it at the point M ; let the 

 femiordinate pm \>e infinitely near another P M, and M R 

 parallel to A H ; then the relative celerities of the point M, 

 moving along the curve from A towards O, in the dircftions 



M R and PM, with which A P and P M increafe in this po- _ _ „— _ _ 



fltion, will be truly expreffed by M R and R m ; but the " '^ 3 " •* " /v 2 " " a " 



celerities by which quantities increafe are as the fluxions of T being given, through which the tangent muft pafs, the 

 thofe quantities ; therefore (Mm being the fluxion of the tangent itfelf may be drawn. 



curve line A^) MR and Rw are the correfpondipg V. Becaufe the equation, exhibiting the nnture of all 



kind» 



V ; therefore the fub-tangcnt P T 



{fs- 19- ) is the double of its correfponding abfcifs A P ; 

 which is a well-known property of the parabola. 



III. The general equation for parabolas of any kind 



being a"' k" = ji" + " ; we have n a" x"- ' k 



any 

 m + n X 



' ' i ; and, therefore. 



X _ m-l-nxj''" 



-; whence 



y X m + n X y" 

 y ~ na" xT- 



m-^-ny.a'" x' 



na" x' 



{heczuiey"*' = a"*") 



m -\- n 



X jc = the true value of the fub-tangcnt ; which, 



therefore, is to the abfcifs in the conftant ratio of m -f n to n. 

 IV. The equation defining an cUipfis is b^ x ax — x* 

 = a' f, A P (fg. 21.) being = x, MP=y, AB=a, 

 and the leffer axis = b ; for by the property of the el- 

 lipfis, we have a' : b' :: a x - x'' { A P X P B) : >' (M P') ; 



whence 3' x 



and, therefore, b' x 



ax— 2xx = 



za'yy. 



and — = 



a- y; 

 2a^y 



X a 



2.V 



and, con- 



fequently, the fub-tangent P T p— ) = — - —/I— = 



b'y 



X a — 2x 



whence the point 



