CUBIC EQUATIONS. 589 



desirous of knowing what degree of generality it had. With this view I made 

 the following trial of it : — Since in the first century of numbers there are twenty- 

 seven which can be values of q, and in the second century twenty-eight, the total 

 is fifty-five. But since three of these are found to occur twice, I add them, which 

 gives in all fifty-eight cases. I found by trial, that in fifty of these cases the 

 method was successful, and that it only failed in eight. It may therefore be 

 recommended, so long as the coefficient q is of moderate magnitude. 



But what is to be done, in case the number obtained proves not to be a root ? 

 (which merely shows that there are no " approximate" roots in the equation.) 

 Are we to abandon the solution of the equation by this method ? We need not 

 do so, for I find that the number obtained, if not the root, is generally a good 

 approximation to the root, and that by a kind of easy supplementary process the 

 true root is generally obtainable. This will be best shown by taking a numerical 

 example. 



Rule of Second Approximation. — Suppose the equation x 3 — 1533 x — r — 



to be proposed for solution. Here —q— 1533 .-. — — ^ = 2044, the root of 



o 



which is 45 + decimals. In order to try whether — 45 is the root, take from 

 — \q — 6132, three times 2025, or 45 2 . The remainder is 57, which is not a 

 square number, and therefore — 45 is not the root. Now, to find the root, 

 assume that — 45 is an approximation to it, and proceed as follows. The differ- 

 ences between the successive squares 45 2 , 44 2 , 4L 2 , &c. &c, are 244 + 1, 2-43 -l- 1, 

 &c. &c, or 89, 87, 85, 83, &c. &c. Multiply these by three, and we get the series 

 267, 261, 255, &c, in which each successive term diminishes by six. 



Then taking the number 57, which was the first remainder, when — 45 was 

 tried for the root, add to it the terms of the foregoing series one by one, till a 

 square arises, which (if the numbers are high) may give the computer the trouble 

 of consulting " a table of squares." Thus : — 



45 gives 57 

 2b7 



44 „ 324 

 261 



43 ., 585 

 255 



42 ., 840 

 249 



41 „ 1089 = 33 2 



Therefore, — 41 is the root, and 33 is the difference of the two other roots x 

 and y. Therefore, x + y — 41, and x — y = 33, whence x = 37, y = 4, therefore 

 the roots are 4, 37, — 41 . 



VOL. XXIV. PART III. 7 U 



