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always one step below the trial-divisor, which suggests it. The several steps of the work are of the simplest kind : having found the first figure 8 of the root, the 1 is multiplied by it, the product added to the 0, the result multiplied by the same 8, and the product subtracted from the first period : the 1 is again multiplied by the 8, the product, as before, carried to the divisor-column, and thus one step of the work is completed. The trial-divisor, 16, now suggests the next root-figure, 7, which, as before, is a multiplier of the 1; the product carried to the divisor-column gives the true divisor, and the product of this by the 7, carried to the dividendcolumn, and subtracted, gives the next dividend (2); the 1 is again multiplied by the 7, and the product carried to the divisor-column, completes the second step; and so on, all the steps being uniformly the same.
(101.) The operation for the cube-root of a number is merely an extension of this easy kind of work. In this case, the row at first written consists of 1, 0, 0, and the given number; this latter being now divided into periods of three figures each, the cube-root of the first period forms the first root-figure; this, as before, is used as a constant multiplier throughout the first step or stage of the operation, the products by it contributing to form two columns of work under the O's, and a third column under the given number: thus, the root-figure is applied, as a multiplier, to the 1, and the product is added to the first 0; the same multiplier is applied to the sum, and the product added to the second 0; the same multiplier is applied to this next sum, and the product subtracted from the first period. To complete the step, we return to the 1, applying still the same multiplier, adding the product as before to the first column, applying the same multiplier to the sum, and adding the product to the second column, at which we now stop: we then return again to the 1; apply the multiplier, and add the product to the first column, carrying the process no further : the first step is now completed ; the number last found, in the second column, is the trial-divisor for finding the next root-figure; with which root-figure we proceed through the same course of multiplications, &c., as at first, and thus get, under the trial, the true divisor, and thence the next dividend.
The following example exhibits the steps at length, the numbers in the three columns, which appear at the end of a step, being marked (1), (2), &c., as in the square-root form.
EXTRACTION OF THE CUBE-ROOT.
Ex. 1. Extract the cube-root of 411001037875. 1
411,001,037,875 (7435 7 49
You of course see why the figures below the (1), (2), &c. in the first column are put each one place further to the right; it is because the root-figures which produce them are each one place further to the right, the local value diminishing at a ten-fold rate. The corresponding numbers in the second column are pushed two places to the right, because the multiplicand and the multiplier from which each is produced have both of them advanced one place to the right; and the figures are pushed three places to the right in the next column, because the multiplicand has advanced two places and the multiplier one. *
(102.) The foregoing process may be extended indefinitely. It may be applied to the finding of the fourth, fifth, sixth, or any higher root of a number: for the fourth root there will be four columns of work, for the fifth five columns, and so on.t But this is not the place to prosecute the subject further; for additional information you must consult the works referred to at the close of the foot-note below, as also for an explanation of the reason of the operation above, which cannot be made sufficiently intelligible without the aid of algebra. I shall give you one other example in the cuberoot, in order that you may see how the work is to be contracted when the proposed number ends in a decimal not to be depended upon; or when only a prescribed number of decimals is required in the root. You will observe, by examining what follows, that the contractions are so managed as to exclude every decimal that would extend beyond the proposed limitation.
* Should any reader of this little work be already acquainted with arithmetic, but acquainted with it only as it is taught in the common school.books, he will be agreeably surprised to find that a subject which must have occasioned him so much perplexity as the extraction of the cube-root, resolves itself into the above simple process.
I claim no merit for it myself; it is due to the late Mr. Horner, of Bath, as a particular application of his general method of solving numerical equations. One or two recent writers on arithmetic bave had the discernment to see its superiority over the common rule, but not the generosity to mention, in connection with it, the name of its indefatigable author,-a man who has done more for the practical advancement of that part of calculation to which it belongs than any other mathematician since Newton. In order to prevent the possibility of this reproof being improperly applied, I must add,-though the addition is quite superfluous to those who know anything of his personal character, -that Professor De Morgan, who uses Horner's method in all his arithmetical writings, is never chargeable with this disingenuous oversight.
Students who may wish to know more about Horner's method, may consult my recent “ Introduction to Algebra, and to the Solution of Numerical Equations ;' “ The Analysis and Solution of Cubic and Biquadratic Equations ;” and “The Theory and Solution of Equations of the Higher Orders." This latter work, which, with the separately published “ Appendix,” costs 188., is fit only for the advanced student.
+ The fourth root may be obtained by first finding the square-root, and then the square-root of the result.
2. What is the cube-root of 98375.112? * 1 0
34375 (1) 21136164 4 32
18465245 8 48
1039.112 (2) 2769787 4 756 636.181
27698 12 5556 402.931 (3)
1385 6 792 383.036
5 126 6348 19.895 (4)
1 6 13.81
2131.03300 132 6361.81
852413200 138 6375.63
127861980 •1 8.30
1278620 138.1 638309,3
63931 •1 8.3
213 138.2 6392.2
4 •1 •4
Proof...98375.111 1,38,3 22.214.171.124 To the above work I have annexed the reverse process of finding the cube of the root 46.163112, which, you see, may be depended upon as true, up to the sixth decimal place. In the first multiplication, which gives the square of the root, I have contracted the result to four places : in the next multiplication, which gives the cube, the factors are also arranged for four places of decimals; but, agreeably to the recommendation at page 126, only three of these places are preserved, in adding up the partial products. The result, you see, fully verifies the operation by which the root has been found; for, as the final remainder in that operation is 1, the number
* The decimal points in the several steps of the work may be omitted ; but I think it safer to preserve them.
actually exhausted by the process is 98375.111. And you may, in general, consider the last decimal of a root deiermined in this manner as true to the nearest unit.*
(103.) I shall conclude with a few examples, for exercise, recommending you to use the contracted method, for decimals, as soon as you are familiar with the process in its uncontracted form. And now having conducted you to this point, I think I may presume that you are in full possession of every important principle in the elements of Arithmetic. The remaining few pages will be occupied with certain business-calculations, of too easy and obvious a character to render it necessary, with your present knowledge, that I should accompany then with the same minute details, and lengthened explanations, that I have furnished to you in what has preceded. If you have only made yourself completely master of what has now been taught, you may take up any more extensive treatise on Arithmetic, with the fullest confidence, that you will meet with no difficulty beyond your powers of successfully contending with. And, what is better, you will, I think, have acquired a deeper insight into the true principles of arithmetical calculation, than the majority of such treatises can afford you, inasmuch as you will have been habituated to think for yourself, to look for reasons as well as rules, and so have been fitted, by the proper mental preparation, to enter safely upon any department of science you may hereafter feel disposed to cultivate.
Exercises. Find the cube-root of each of the following numbers. 1. 912673
7. 115.29736 2. 52734375
8. 822650 3. 21024576
9. 78314.6 4. 80677568161
10. 12345.678 5. 411001037875
11. 123.456789 6. 7835.8748
12. 9, to nine decimals.
* In marking off the figures to be dismissed in the contracted portion of the operation, it is of course matter of indifference whether dots be employed, as at page 130, or dashes, as in the example above. In working with the pen, the latter is the more convenient way; though it cannot be used in print without so separating the figures as to drive them out of the proper vertical columns. In print, therefore, dots would seem preferable; but when figures fall below those thus dotted, the dots might be mistaken for decimal points connected with the latter ; so that, in the extraction of roots, it is better to tick off the figures as above, both in printing and in writing.