“Such gentlemen as may feel desirous of improving their minds by engaging in establishments of a nature similar to this, but who, on account of their residing at a distance from any large town, have not hitherto had the opportunity, will, it is hoped, be induced by the regulations respecting corresponding members, to join the society; and they may depend upon meeting with every attention, whenever the Committee shall be favoured with their communications.”
APPENDIX C.
CUBE ROOTS.
The mode of extracting the roots of exact cubes which I taught the boys, and which was probably that adopted by Zerah Colbourn, will be best shown by an example. Suppose the question to be, What is the cube root of 596,947,688? This looks like a formidable array of figures, and a schoolboy, resorting to the usual mode of extracting the root, would fill his slate with figures, and perhaps occupy an hour in the process. Zerah Colbourn or my class would have solved the question in a minute, and without making any figures at all. My class would have proceeded as follows: They would first fix in their memories the number of millions (596) and the last figure of the cube (8), disregarding all other figures. Then, knowing the cubes of all numbers from 1 to 12 inclusive, they would at once see that the first or left-hand figure of the root must be 8; and deducting the cube of 8 (512) from 596, they would obtain a remainder of 84. This they would compare with the difference between the cube of 8 (512) and the cube of 9 (729), that is to say, with 217; and seeing that it was nearly four-tenths of such difference, they would conclude that the second figure of the root was 4. The third or last figure of the root would require no calculation, the terminal figure of an exact cube always indicating the terminal figure of its root—thus 8 gives 2. The cube root, therefore, is 842. In this process there is some risk of error as regards the second figure of the root, especially when the third figure is large; but with practice an expert calculator is able to pay due regard to that and certain other qualifications which I could not explain without making this note unduly long. As already stated, Zerah Colbourn did occasionally blunder in the second figure; and this circumstance assisted me in discovering the above process, which I have little doubt is the one he followed. If, instead of an exact cube, another number of nine figures be taken, the determination of the third figure of the root, instead of being the easiest, becomes by far the most difficult part of the calculation.
[This part of the explanation was written by Sir Rowland Hill, as a note to the Prefatory Memoir, before the year 1871. What follows was added in 1875.]
Rule for extracting the roots of imperfect cubes divisible into three periods:—
1. Find first and second figures as described above.