CONIC SECTIONS. 



163 



Sections. 



Areas of adding in Fig. 100, and subtracting in Fig. 101, the 

 the Conic triangles CAG, CBG are equal : In the same way it 

 appears, that the triangles DCL, BCL are equal. 

 Again, because the semidiameter CB bisects the chord 

 which joins P and Q, it will bisect GL, the tangent 

 which is parallel to that chord ; and therefore the tri- 

 angles GCB, LCB are equal. Honce it follows, that 

 the triangles ACG, GCB, BCL, LCD, are all equal ; 

 and this will be true, whatever be their number. 



Let AK, a tangent at the vertex A, meet the semi- 

 diameters CP, CB, CD in G, H, K : Put the letter a 

 to denote the semitransverse axis CA ; and let the tan- 

 gents AK, AH, AG, &c. which correspond to the whole 

 sector, its half, its fourth, &c. be denoted by /, V, t", 

 &c. respectively : And because (by Prop. 2.) 2a: t:: 

 a t =f=t' 1 : at' : (where the upper sign applies to the cir- 



■a 1 a 1 

 cle, and the lower to the hyperbola) we have- 



I 2V 



V a* 



—; but as similarly = 



2t"' 



t" 

 ''2' 



we have, by 



a 1 a 1 It' l"\ 

 substitution, = --,=+=(— -J- —J, and hence 



a} 



4 at" 



a* I V I" \ 

 zz — =±=[ (- —J. Now the expression &at" or 4CA 



X AG (see Fig. 100. and 101.) is eight times the tri- 

 angle ACG, that is double the polygon CAGBLD 

 which circumscribes the circular, or is inscribed in the 

 hyperbolic sector; therefore putting s' to denote the 

 polygon, we have 



2~?~ t \ 2" "*"¥/' 

 If we carry on the process of bisecting the sector 

 ACD indefinitely, so as to divide it next into 8, then 

 into 1 6, then into 32 equal parts, and so on, and put t'" 

 for the tangent corresponding to its 8th part, and f iv 

 for that corresponding to its l6th part, &c. we shall 

 have in like manner, 



» 3 «*_,__ /^' t" t" t iv . \ 



2V=T— U + X+ T+I6+' &c ) 



f 



4, ' 8 



Now the polygon CAGBLD manifestly approaches 



continually to the area of the sector, as the number of 



its sides is increased, and at last differs from it by less 



than any assignable quantity ; therefore, putting now 



s for the area of the sector, if we substitute * in the 



above expression instead of s', and conceive the series 



V t" 



of terms — , — , 



2' 4' 



&c. to be continued ad infinitum, we 



have the quantity — expressed 

 from which we readily find 

 2 s = <L 3 ' 



by an infinite series, 



t ft' I" V" „ \ 



a -'(2 + T+ir+> &c -) 



the formula to be investigated, and m which it must 

 be observed, that the upper part of the sign =±= ap- 

 plies to the circle, and the lower to the hyperbola. 



As to the successive quantities t', t u , f", &c. they may 

 be determined from each other, and from t, by the se- 

 ries of formulas 2a i t'=t (a 1 :^:*' 2 ), 2d 1 1" =r I' («' — 

 t" 1 ) &c. for it will be found by resolving a quadrate 

 equation, that in the case of the circle, 



but in the case of the hyperbola, 



"=°{W:;-'}; 



and in the latter case the least root is taken, because V 

 ought to be less than /. The value of t", t'", &c. are 

 found each from that before it exactly as V is found 

 from t. 



Scholium. 



The formula for the determination of the tangents I', 

 t", ["', &c each from that before it, in the case of the 

 hyperbola, is entirely similar to that for the circle, dif- 

 fering from it only in the signs of the terms In the 

 hyperbola, however, these tangents have a property 

 which does not belong to them in the circle, by which 

 their calculation may be facilitated : For since in the 

 hyperbola 



a: I:: a*+V*'. 2a V, (by Prop. 2.) 



by mixing, 



a + t:a—t : : a* + 2 a t' + t'? : at — 2a t'+t' 2 4 



that is, a + t : a — t : : (a +f )*: (a — t')* ; 

 and because, similarly, 



a + t': a—i' : : (a + l'y : (a-4"y, 



and therefore 



ia + t')i : (a— ff : : (a+t")* : (a—t")* ; 



therefore a+t : a—t : :(a + t"y : (a — t")* ; 

 and so on : Hence, 



a +t>_,a + tU 

 a—*t' — \a—t) ' 



a + t" _ (a + ('\i 

 a—t"~\a—t'J ' 



a + t' 



a — /' 



_/a + t"\i 

 ~(a^V'J ' 



&c. 



T a + t , a-Lt' i 



Let us now put — —~v, then — '- — — v 

 a — t a — V 



*+<''! 



. = V 



a—t" 

 &c. therefore, resolving these equations in respect of t', 



i", &c. we find t'—a — , 



^ + 1 v z +\ v'+l 



&c. This series of fractions, by which the tangents V, 

 t", &c. is expressed, being formed in a very simple 

 manner from the square, the fourth, the eighth, &c. root 



of the fraction v = — — , they may be easily computed, 



and thence the values of t'., t", &c. and sector s, found. 



The tangents V, t", &c. may also be readily found 

 from the trigonometrical tables, for each is related to 

 that which follows it, exactly as the sine of an angle to 

 the tangent of half that angle. This may be proved 

 as follows : 



Put u and z to denote any two angles, then it ap- 

 peal's, from the third and fourth formulas of Table D, 

 Art. 12, Arithmetic of Sines, that 



Sin.M-f Sin . z_ S in. ^ j u+z) Cos. \ Cu — z) 

 Sin. u — Sin. z~ Cos. ^ (u+z) Sin. ^ (« — z)' 



Tan. ^ (u + z) 



Tan. ^ (« — z)' 



Now, by Art. 26, 



Tan.K«+-)= f ^i^ 

 ' 1 — Tan.i u 



Tan. i z 



Tan. 



