VOL. XL.3 PHILOSOPHICAL TRANSACTIONS. 1'J'J 



describe, or conceive to be described a circle, tiie radius of which is v'/ra, in 

 which take any arc a, the cosine of which is —r\ and let c be the whole cir- 

 cumference. To the same radius take the cosines of the arcs 



AC — AC+a2C — a2C+a3C — ASC+A. -hi 1 r ^ 



-, , , , , , , &c. till the number or them 



be equal to n. Then all these cosines will be so many values of x ; and the 

 quantity y will always be m — xx. 



Prob. 4. — Having given any Equation, of the Kind of those above described; 

 to know whether its Solution is to be referred to the Hyperbola or to the Circle. 



Let n denote the highest dimension of the equation : divide the coefficient of 

 the second term by 2""' X n, calling the quotient m : then see whether the 

 square aa be greater or less than m"; if it be greater, the equation is to be re- 

 ferred to the hyperbola ; but if less, to the circle. 



Let there be given the equation l6,r* — 40:1^ + 20x = 7, where n := 5 ; 

 therefore 2""' X « = 20: divide 40 by 20, the quotient is 2 = ?«; hence m" =. 

 32, and aa := 40; and as this is greater than the power 32, the equation is to 

 be referred to the hyperbola. But since in the hyperbolical case there was put 

 i/ aa — b = m, it follows that aa — b = m^ = 32, and therefore b = aa — 32 

 = 49 — 32 = 17. Now the root of the equation in this case is ^v'7 -)_^ jy 

 ^7 _^ 17 : but v^l? = 4.123105 nearly; therefore 7 +/17 = 11.123105, 

 and 7—^17 = 2.876895 ; also the 5th root of the former number is J.6221, 

 and the 5th root of the latter 1.2353, the sum of which roots is 2.8574, and 

 the half sum 1.4287 is the value of x in the given equation. 



Again, let the equation 16a;* — 40a^ + 20^ = 5 be given ; in which m is 

 \ still = 2, but a = 5, and the square aa is less than 2^ or 32 ; therefore the 

 value of X cannot be obtained without the quinquisection of an angle ; and that 

 is performed by our general theorem, by taking, to the radius i/2, the arc 

 whose cosine is ^ = ^ = -, which is the arc of 27° 55' nearly, the 5th part 

 of which is 5° 35'. Now the log. cosine of that arc, to the radius l, is 

 9.9979347 ; but since our radius is V2, to that log. add the log. of ^2, that 

 is 0.1515150, the sum will be 10.1484497; from which taking away the 10, 

 the remainder 0.1484497 will be the log. of 1.4075 nearly, the number sought. 

 And in like manner the other four roots may be found. 



It may be further remarked, that if the equation be of the hyperbolic kind, 

 and n be an odd number, there will be only one possible root ; but if n be an 

 even number, there will be only one value of the square xx, the rest being 

 impossible. 



If the equation be of the circular kind, all the roots will be possible. 



To know how many of the roots will be affirmative, and how many negative, 



