TANGENT. 



Takges'T, Co, or Tjnj^ent ef the Complement, is the tan- 

 «cnt of an arc, whicli is the complement of another arc to a 

 quadrant. 



Thus a tangent of the arc A H, is the co-tangent of the 

 arc A E, or the tangent of the complement of the arc A E. 



Tofnd the length of the tangent of any arc, the fine of the 

 arc being given : fuppofe the arc A E, the given fine A D, 

 and the tangent required E F. Since both the fine and 

 tangent are perpendicular to the radius E C, they are parallel 

 to each other. Wherefore as the cofine D C is to the fine 

 A D, fo is the whole fine to the tangent E F. See Sike. 



Hence, a canon of fines being had, a canon of tangents is 

 eafily conftruAed from it. 



Tangents, jirtificial, are the logarithms of the tangents 

 of arcs. 



Tangents, Line of, is a line ufually placed on the feftor, 

 and Guntcr's fcale ; the dcfcription and ufes of which, fee 

 under Sector. 



Tangent of a Conic SeSion, as of a parabola, is a right 

 line, which only touches or meets the curve in one point, 

 and does not cut or enter within the curve. See Conic 

 SeHioni. 



Tangents, Method of, is a method of drawing tangents 

 to any algebraical curve, or of determining the magnitude of 

 the tangent and fub-tangent, the equation to the curve being 

 given. .' 



The method of tangents is nearly related to that of maxima 

 et minima ; and the fame authors, who in the early ftate of 

 algebra attempted one of thofe cafes, never failed of touching 

 alfo on the other. Hence we have the methods of Def- 

 cartes, Fermat, Roberval, Hudde, &c. We have already 

 explained under the article Maxima et Minima, the feveral 

 methods of thefe authors relating to the latter fubjeft ; and 

 as their methods of tangents differ in no refpeft from this, we 

 fhall not repeat them again in this place, but merely explain 

 the principle which led to fo intimate a conneftion between 

 the two problems. 



Defcartes' Method of Tangents It has been (hewn under 



the article above referred to, that Defcartes' method of 

 maxima and minima, depended upon his making two roots 

 of his equation equal to each other, and the fame principle 

 led liim alfo to his problem of tangents. 



Let us conceive, for example, a curve A B i, ( Plate XIV. 

 Analyft!, fig. l6. ) defcribed on an axis A C ; and from any 

 point in this axis, C, as a centre, let there be defcribed a 

 circle, which fhall cut the curve at leafl in two points, as 

 B, b ; from thefe draw two ordinates, which will ncceffarily 

 be common both to the circle and curve : let us now imagine 

 the radius of this circle to decreafe, while its centre remains 

 fixed ; and it is obvious that thus the two points of inter- 

 feftion will approach each other, and finally coincide, in 

 which cafe the circle will touch the curve at the point E, 

 and the tangent at that point will be common to both, and 

 perpendicular to the radius of the circle at that point. 

 Thus the problem of determining the tangent to a curve, is 

 reduced to finding the pofition of a perpendicular to the 

 curve, drawn from any point in its axis. In order to effeA 

 this, Defcartes fought, in a general manner, the points of 

 ijiterfeftion in the curve made by a circle defcribed with a 

 given radius from a given point in the axis. He thus ar- 

 rived at an equation, which, in the cafe of two interfeftions, 

 ought to contain two unequal roots, expreffing the diflance 

 •f the two ordinates from the vertex of the curve. But 

 when the two points of interfeftion are united in one, as in 

 the cafe of the circle touching the curve, then the tvi'o roots 

 of the equation are neceffarily equal to each other. His 

 objed, therefore, was, in the equation firfl obtaineii, and of 



which the co-efficients were indeterminate, to 

 values, that the two roots fhoiild be equal ; 

 pofe, he compared the propofed equation w 

 of the fame degree, having two equal roots ; 

 equating the co-efficients, obtained the value 

 firft equation. 



In order to illuflrate this, let A B i {fig. 

 bola, and B 4 a circle. Make C A — a, 

 radius C B = r, then CD =; a — k ; and 

 nate B D belongs to the circle, we have 



give them fiicli 

 tor which pur- 

 ith an equation 

 and hence, by 

 of thofe in his 



1 6.) be a para- 



A D = x, the 



fince the ordi- 



CD" = r'- (a- x)-= r'-a'-}- 



;a.■l: — X . 



But the fame ordinate belonging alfo to the parabola, we 

 have from the known property of that curve, y' = px,jf> 

 being the parameter ; therefore 



r^ — a^ -(- 2a.x — .v' = p.x, or 

 s,^ + (J>— 2a) X + {a^ — r') = o, 



which, being an equation of the fecond degree, mull necef- 

 farily have two roots, or values, of x, anfwering to the two 

 abfciffes AD, Ad: for we Ihould arrive at the fame con- 

 clufion, if our equation had been deduced with reference to 

 the point 6 ; and it is obvious that thefe roots depend entirely 

 upon the relation of the co-efficients {/> — 2 a) and 

 (a' — r"), or upon the ratio of the quantities a, p, and r, 

 to each other ; and, confequently, fuch values may be 

 given to thefe quantities, that the two values of x may be 

 equal. 



In order to find this ratio, Defcartes formed an equation 

 of the fecond degree, having two equal roots, as .\- ° — 2e s. 

 -t- e^ =: o, TOz. \x — e) (x — e) =. o; and comparing this 

 with that found above, he obtained the equation .v — a = 

 CD = \p, which (hews that in the parabola, the fub- 

 normal is equal to half the parameter ; whence it alfo fol- 

 lows, that the fub-tangent is equal to double the abfcifs, 

 which is the known property of the curve. 



Defcartes had alfo another method for tangents, a little 

 different from the above in praftice, although it was th^ 

 fame in principle ; thus he conceived a right line to revolve 

 about a fixed point in the axis of the curve produced, 

 which at firft fhould cut the curve in a certain number of 

 points, but by its revolution, thefe points of interfeftion ap- 

 proaching each other would finally coincide, and thus the 

 revolving line become a tangent to the curve. For this 

 purpofe he alfo firft obtained the general equation, which he 

 afterwards equated with another having two equal roots, and 

 thus determined the feveral relations of his indeterminate 

 co-efficients, exaftly as in the cafe above given. 



Fermat' s Method of Tangents. — It will be found by com- 

 paring the above method of tangents of Defcartes, with 

 that of his maxima and minima, that the two ultimately de- 

 pend upon the fame principle, vi%. of making two roots of 

 an equation equal to each other ; and the coincidence of 

 Fermat's methods for thefe two problems is ftill more ob- 

 vious ; in faft, he fcarcely treats of them as diftinft cafes, 

 but refers immediately for the folution of the cafe of tangents 

 to that of his maxima and minima. In order, fays this 

 author, that a line may be a tangent to a curve, as for ex- 

 ample to the parabola A Bi, at the point h, {fig. 17.) it is 

 evident that every ordinate, except B C, will meet that tangent 

 beyond the curve, as in C. Thus the ratio of B C' : ce', 

 which is the fame as C D' : cDSwill be lefsthan that of C B': 

 c l>', or than that of C A to ir A ; but if we fuppofe thefe 

 ratios to be the fame, and confequently the diftance f C to 

 vanifh, the points B, i, will coincide, and we fhall have an 

 equation, which, treated in the fame manner as in his method 

 de tnaximii ct minimis, will give the ratio of C D ; C A. 



A» 



