VOL. XVI.] PHILOSOPHICAL TRANSACTIONS. 403 



But these roots are readily enough found by means of the logarithmetical 

 table of versed sines; viz. if the coefficients be either surds or fractions, and the 

 roots not to be expressed in numbers, as most commonly is the case. And this 

 is the rule; in the first and second formulas, if -J-6i be less than p, let ^p — 

 \bb = d, and putting, for the radius, the difference between -^hp and -^b^ + 

 ■^q (that is, HR) in the first formula, and the difference between -^ijb + ^q 

 and ^^b^ in the second; and find the angle whose tangent is dV d. Then as 

 the co-sine of that angle is to its versed sine, so the difference, which is taken 

 for radius, to a fourth term ; the cube root of which is had by taking 4- of its 

 logarithm. Then dividing 4^p —' ^bb by this cube root, let the divisor be sub- 

 tracted from the quotient, the remainder will be the quantity Y & ; the sum of 

 this remainder and -^b will be the root sought, if the centre fall on the right 

 hand of the axis; otherwise their difference will be the root. But if ^bb be 

 greater than jb, making HR radius, let dy/d, or the distance of the paraboloid 

 from the axis, be the sine of some arch, and let its versed sine be multiplied 

 into the radius, or ^bp — -^b^ + ^q, and taking 4- of the logarithm of the 

 product, its cube root will be obtained, by which let -^bb — -fjb be divided; the 

 sum of the quotient and divisor, after the same manner added to or subtracted 

 from ^b, will give the root sought. And the like holds in the third and fourth 

 formulas, except that ^yi^ + -^bp +_ ^q is to be taken for radius, and -^bb -\- 

 ■^p into ^ -^bb + -^p, or d\/d, for the sine. But these rules will perhaps be 

 better understood by examples. 



Suppose the cubic equation z' — 172^ -|- 54z — 350 = 0, and let the root 

 z be sought. Here -^bb is greater than p, but q is greater than the cube of -i^b, 



and therefore it is explicable by one affirmative root only : now — ~ h d, 



127 127 4913 



and -Q-v^-Tj- 's to be taken for the sine, to the radius — — |- 175 — 153, 



that is, -^=-i and the arch answering thereto is 15° 3' 4g", the log. of its 



yersed sine is 8.5362376, which added to the log. of the radius 2.30959J3» 

 makes 0.8457889, the third part of which 0.281 9276 is the log. of the cube 



127 

 root 1.91394, by which, as a divisor, dividing — -, or d, the quotient is 7.37281 ; 



the sum of the quotient and divisor, increased by the addition of -J-i, is the 

 root sought, viz. 14.9534, &c. 



Having thus dispatched cubic equations, let us proceed to biquadratics. Now 

 these have always either none, or 2, or 4 true roots, the determination of 

 which depends partly on the coefficients, and partly on the sign and magnitude 

 of the absolute number given. In the construction of the equation z* — bz^ + 



3 f2 



