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Ex. 2. Extract the cubic root of the number 9.
This may be considered as a cubic equation of the form x=9, or x3+0x2+ -9=0.
The following operation was executed by a young Gentleman, who was afterwards a pupil of the Author's.
The following lines were communicated to the Author by the young
Gentleman here alluded to, in regard to the operation now shown. He
"The above work was done above three months before I saw the Analytical and Arithmetical Essays, having had the perusal of a Specimen of the aforesaid works for one evening only.
May 1, 1821."
The Specimen which Mr. Brown alludes to, is the following Letter, published by the Author, showing the application of the rule to the cube root of a number.
62402514690 .... ..1298024613211888598700
62402514691548 129802461327666240894902808300 624025146915570
London, Nov. 24, 1820.
Having now published a Work, entitled " ANALYTICAL AND ARITHMETICAL ESSAYS," consisting of Continued FractionsFigurate Numbers-an entire new Method of the Transformation of Equations, determination of their Limits, the Extraction of their Roots, and their consequent Depression, by an easier Method than that hitherto practised; together with the Decomposition of Algebraic Quantities into others of a given Form; the whole of the Rules resulting from the Theory being such, that the operations performed by them are completely divested of the algebraic form, and thereby rendered purely arithmetical; to which are also added, Essays on the Method of Finite Differences-Indeterminate Equations-and the Combinatorial Analysis; the Method of treating the whole of these Subjects being either new or rendered more obvious by a clearer Explanation; should you be interested in the progress of the Numerical Science, I have not the smallest doubt but that the perusal of the Work will give you full satisfaction; or in the event of your own time being fully occupied, I should feel greatly obliged to you to recommend the Work to those who have both leisure and inclination to pursue such studies, or to such as are likely to promote its object. As a Specimen, I have given the Method of extracting the Cube Root from the Work now recommended.
I am, Sir,
P.S.-The Work may be had of the Author, No. 5, Claremont Place, Judd Street, Brunswick Square.
To extract the cube root of any given number.
Divide the number into as many periods of three figures each, from right to left, as possible; find the nearest cube to the remaining figure or figures on the left, and subtract it from the number formed by these figures; then the root of the cube is the first figure of the root to be extracted.
Call the triple root now found the first co-efficient, the triple square of it the second co-efficient, and the difference between the cube and the number to be extracted the absolute number; then write these three numbers separately, in one line.
1. Divide the absolute number by the second co-efficient, without the last figure, so as only to have one figure in the quotient.
2. Under the first co-efficient construct a column of three numbers, so that the right-hand figure may advance one figure to the right-hand of the unit's place of the co-efficient; under the second co-efficient construct a column of two numbers, so that each may advance two figures to the right of the unit's place of the second co-efficient; and under the remainder construct a column of one number, so as to advance three figures to the right of the absolute number, or to the unit's place of the next period; then, to find each of these numbers,
3. Annex the quotient figure to the first co-efficient, and the sum will be the first number under the said co-efficient: each of the two remaining numbers will be found by increasing the number above .t by the quotient figure.
4. Multiply each of the two upper members of the first column, in succession, by the quotient figure, and the opposite number in the second column will be found by adding the product to the number above it.
5. Multiply the first member of the second column by the quotient figure, and the opposite number in the third column will be found by subtracting it from the absolute number.
Then, if the product be less than the absolute number, the quotient figure is the second figure of the root to be extracted; but if not, the work must be repeated.
Now, considering the third number in the first column as a first co-efficient, the second number in the second column as a second coefficient, and the remainder as an absolute number, the next or third figure of the root will be found in the same manner as the second; and so on.
Ex. Extract the cube root of the number 13.
Here the nearest cube to 13 is 8, the root of which is 2; therefore the co-efficients of the first step are 6 and 12, and the remainder, or absolute number is 5. Now 5 will be found to contain 1, which is the second co-efficient without the last figure five times; but this will be found not to succeed; therefore try 3 in the operation.
Proceed with the above columns of numbers according to the 2d, d, 4th, and 5th parts of the rule.
Then, since 3 succeeds, divide 883 by 158, which is the co-efficient without the last figure, and the quotient 5 is the next figure of the root, which must now succeed; therefore proceed with the next step.
Again, divide 22125 by 16567, and the quotient 1 is the next figure of the root; therefore proceed with the next step.
This process being sufficiently understood, the learner may then work the whole of the steps in one continued operation: thus,
In this operation we may observe, that the multiplications and additions, as also the multiplications and subtractions, are performed in one line, as may be found in some of our best systems of arithmetic.
It may here be remarked, that wherever we stop in the operation, as many figures, except one, may be found as the number of figures in the root already obtained, by dividing the last remainder by the second co-efficient, wanting as many of its last figures as the number of figures to be found: thus, in the present instance, 5550449 divided by 16581 gives 334, which, annexed to the part 2:351 of the root already found, gives 2:351334, which is true to the last figure.
Method of Proof.
This operation will admit of a proof at every step, which may be done by the following
Consider the co-efficients and remainder from which the step to be proved is found as whole numbers, and the figure of the root as a decimal in the place of tenths; then add into one sum the cube of the new figure, the product of the first co-efficient, and the square of the new figure the product of the second co-efficient, and the new figure itself together with the last remainder; then, if the work is right, the sum will be equal to the preceding remainder or absolute number, Ex. The co-efficients and absolute number by which the third figure of the root 5, in the example given, are 69,1587 and 833 considered as whole numbers; then
69x (5) 17:25
1587 × (5) =793.5
OF RATIOS, OR PROPORTION.
(96.) Ratio, is the relation which one quantity bears to another, with respect to magnitude. This relation exists only between quantities of a similar kind; thus, a number must be
compared with a number; a line with a line; &c. &c. for it would be absurd to compare a certain number of feet with a certain number of pounds; &c. &c.
(97.) The magnitude of quantities may be compared in two ways. First, with regard to their difference; and then the question asked, is, "How much one quantity is greater or less than another." This relation of quantities to each other, is called their Arithmetical ratio. The second way in which they may be compared, is, by inquiring "How often one quantity is contained in the other." This relation between quantities is called their Geometrical ratio.
Obs. The term ratio, when simply applied, is generally understood in the latter sense; and it is in this sense that the word will be made use of in the present Article.
(98.) In examining how often one quantity is contained in another, the natural process is to divide the one by the other.
Thus, in comparing the number 12 with the numbers 4 and 3, we find that 4 is contained in 12 three times, and that 3 is contained in 12 four times; from which we infer, the ratio of 12 : 3 is greater than the ratio of 12 to 4, the magnitude of a ratio being measured by the number of times one quantity is contained in another. For the same reason, we may say, that the ratio of 11 7 is less than the ratio of 11: 5. When a ratio is thus expressed, the first term of it is called the antecedent, the last term the consequent, of that
Note. In expressing the ratio of two quantities, the word "to" is generally supplied by two dots thus, the ratio of “a to b” is expressed by “a:b.'
COROL. It appears, from this mode of estimating the magnitude of a ratio, that when the consequent of a ratio is not an aliquot part of the antecedent, the value of the ratio must be expressed by a fraction, the numerator of which is the antecedent, and denominator the consequent of that ratio.
Thus, the magnitude of the ratio of 15: 7 is expressed by the
fraction and that of the ratio 4: 13 by the fraction
(99.) When the antecedent of a ratio is greater than the consequent, it is called a ratio of greater inequality; when the antecedent is less than the consequent, a ratio of lesser inequality; and if the two terms of a ratio be the same, then it is said to be a ratio of equality.
(100.) But the foregoing definitions apply to those instances only, in which the consequent of a ratio is contained a certain number of times in the antecedent, or in which the magnitude of the ratio may be expressed by some definite fraction.