276 PHILOSOPHICAL TRANSACTIONS. [aNNO J 738. 



root of which is 21, which put = m, which makes 3mx = 63x; therefore the 

 equation to be resolved will be 4a;' — 63x = 81, which being compared with 

 the equation for the cosines, viz. 4x^ — 3rrx = rrc, gives rr = 21, hence 



r a 81 27 



r = ^1\, therefore c = — = — = -y. 



27 

 To find then the circular arc to the radius \/1\, and cosme — ; put the 



whole circumference = c, and take the arcs |, ^^, ^-~^, which will easily 

 be known by a trigonometrical calculation, especially by using logarithms; then 

 the cosines of the arcs to the radius \/21, will be three roots of the quantity x; 

 and since y = m — xx, there will therefore be as many values oiy, and thence 

 a triple value of the cube root of the binomial 81 + •/ -r- 2700 ; which must 

 now be accommodated to numbers. 



Make then as y/1\ : V - so is the tabular radius : to the cosine of an arc a, 

 which will be nearly 32° 42' ; hence the arc c — a will be 327° 18', and c + a 

 392° 42', of which the 3d parts will be 10° 54', and 109° 6', and 130° 54'. 

 But now as the first of these is less than a quadrant, its cosine, that is, the sine 

 of 79° 6', ought to be considered as positive ; and both the other two being 

 greater than a quadrant, their cosines, that is, the sines of the arcs 19° 6' and 

 40° 54', must be considered as negative. Now by trigonometrical calculation 

 it appears, that these sines, to radius \/21, will be 4.O4999, and — I.4999, 

 and 3.0000, or f, and —^, and —3. Hence there will be as many values of 

 the quantity y, viz. all those represented by ot — xx, viz. 21 — V> and 21 — -«-, 

 and 21—9, that is, 4-, VS '2, the square roots of which are i\^3, 4-/ 3, 

 2^3 ; therefore the three val ues of .y/ — 3/, wi ll be -l-/ — 3, ■|-v' — 3, 2./ — 3 ; 

 hence the three values of ^81 -\- V — 2700 are 4 + 4.^—3, and —4+4 



^ 3j and — 3 + iy/ — 3. And by proceeding in the same manner, there 



will be found the three values of v^81 —V— 270O, which are 4.-4.^—3, 

 and f — 4/— 3, and — 3 — -i-v^ — 3. 



There have been several authors, and among them Dr. Wallis, who have 

 thought that those cubic equations, which are referred to the circle, may be 

 solved by the extraction of the cube root of an imaginary quantity, as of 81 + 

 V^— 2700, without any regard to the table of sines : but that is a mere fiction; 

 and a begging of the question ; for on attempting it, the result always recurs 

 back again to the same equation as that first proposed. And the thing cannot 

 be done directly, without the help of the table of sines, especially when the 

 roots are irrational ; as has been observed by many others. 



Prob. 3. — To extract the nth Root of the Impossible Binomial a + \/ — b. 



Let that root he x -^ V — y; then making V aa -{■ b = m, and — — =/>, 



