2. Deduct cube of first figure from the first period (of the number whose root is to be extracted), modified, if necessary, as hereafter described.
3. Then multiply the number (expressed by both figures) by each figure in succession, and by 3.
4. Deduct the product (or the significant figures thereof—see example), from the remainder obtained as above. (See 2).
5. Divide the remainder now obtained by the square of the number expressed by both figures (see 3), multiplied by 3—dropping insignificant figures (see example),—and the quotient will be the last figure (or 3rd figure) of the root.
I can confidently affirm from experience that there is nothing in the above calculations too difficult for those who, possessing a natural aptitude, are thoroughly well practised in mental arithmetic. I doubt, however, whether the mode just described be exactly that which we followed; our actual mode, looking at the results as described above (which is in exact accordance with my Journal), must, I think, have been more facile; but as it is fully fifty years since I gave any thought to the subject, and as, in the eightieth year of my age, I find my brain unequal to further investigation, I must be contented with the result at which I have arrived.
It must be remarked, however, that cases will arise when some modification of the process will be necessary. As, for instance, when the first period of the cube is comparatively light, it may be necessary to include therein one or more figures of the second period treated as decimals; indeed, if the first period consist of a single figure, it will be better to incorporate it with the second period, and treat both together as one period,[377] relative magnitude in the first period dealt with being important as a means of securing accuracy in the last figure of the root. But expert calculators soon learn to adopt necessary modifications, and by the “give-and-take” process to bring out the correct result. Indeed, I find it recorded in my Journal that “small errors will sometimes arise which, under unfavourable circumstances, will occasionally amount to a unit.” These observations it must be understood to apply only to the extraction of the roots of imperfect cubes, which Zerah Colbourn invariably refused to attempt. When the cube is perfect, the last figure of the root, as shown in the text, requires no calculation at all.
Example.
What is the cube root of 596,947,687?
[Note.—This is the number treated above, except that in the unit’s place 7 is substituted for 8, in order to render the number an imperfect cube; so slight a change, however—though rendering it necessary to calculate the last figure of the root,—will still leave the root as before.]
Following the rule, we find the first and second figures of the root in the manner described above. They are 8 and 4.