of the latter, we have the triple (= 108) and triple the square (= 3888) of 36 in L and м respectively. The trial divisor 3888 is found to be contained 4 times in 15725, which is annexed to the former figures of the root making 364, and to the column L making 1084. This multiplied by 4 and added to м gives 393136; and this last multiplied by 4 gives 1572544; which being equal to the resolvend, can be subtracted. No remainder occurring, the work is hence terminated at this step, and the root is 364. 2. Find the cube root of 9 to fifteen places of decimals. The decimal points are marked in this solution, but they are unnecessary, as the arrangement of the work itself will lead to a correct disposition of the figures. The first vertical bar in the root marks where the contraction begins to give its figures to the root, and the second where the pure division commences. The root is true in the last figure. This example contains specimens of the case where ciphers occur in the root; and is also a pattern for the method of contraction. Ex. 3. Extract the cube root of 571482.19. Ex. 4. Extract the cube root of 1628 1582. Ex. 5. Extract the cube root of 1332. Ex. 6. Extract the cube root of 3 to six places; and also the cube root of 6: and show how the cube root of 2, of 12, and of 18, may be obtained from these. Ex. 7. Find how many more figures are required to be written in extracting the cube root of 3 to six places of decimals, than in finding it to three places. Ex. 8. Find the cube root of 60610998517, and subtract it from the cube root of 5. Ex. 9. Extract the cube root of 009009009; and thence obtain its sixth and its ninth roots to six and to four places of decimals respectively. Ex. 10. Find the square of the cube root of 1000, and likewise of ·001: and the cube of the square root of 100 and '01; and determine the product of these four results. TO EXTRACT ANY ROOT WHATEVER *. LET P be the given number, ʼn the index of the root to be extracted, R the true root, a the nearest approximation (either greater or less than R) that has been made by trial or otherwise; and let a" = A, and R = a + x. Then, Add or subtract x, according as P is greater or less than a, which gives a new and nearer approximation to R. With this new value of a, find a new value of x, and hence a new approximation to R. Continue this series of approximations till the required degree of accuracy is obtained. The number of figures which may be depended on in each successive value of x, is generally equal to the number of accurate places in a; so that each operation doubles the number of figures already obtained. *This rule is a modification and extension of Dr. Halley's, and was first given in its present. form by Dr. Hutton. See his Tracts, vol. i. p. 213. The following is essentially Dr. Hutton's investigation : = a + p + q n . p-q an x2+ p + q sa ... Now since a is an approximation to/P, a is small in comparison with it, and hence its powers above the second may be rejected, as of values too small to materially affect the calculation within the prescribed limits. Then equating the co-efficients of the homologous powers which remain, we have From the second and third of these equations we get the same relation between p and q: viz. Inserting this value of in the assumed fraction for a + a we obtain (n−1) p + (n+1) a 2 a {P-A} a + x = (n + 1) + (n−1) Aa; or x= *(n+1) p + (n−1) ▲ Α which is the working formula in the text, and perhaps the best form which the correction admits of. In another place I have given an improvement upon this approximation from a memorandum of the late Mr. W. G. Horner, but to which I cannot make more special reference here. Note. It will always be well to reduce the index into factors as far as possible, and extract successively the several roots in the manner directed at the head of the rule. Thus the thirtieth root is the fifth root of the third root of the square root, since 5 × 3 × 2 = 30. It will also be always least laborious to commence with the lowest roots, as with the second before the third, and the third before the fifth, in the case just cited. EXAMPLE. Extract the tenth root of 442504881 64. Here extracting the square root of the given number, we shall have to extract the fifth root of 21035 8. A few trials will show that the root lies between 7.3 and 7.4. Taking a = 73, we have A a5 = 20730 71593. Also P = 21035·8, and n = 5. Hence, 2. 21035 8 + 3. 20730 71593 =0·021360, and hence √21035·8 = 7·321360 nearly. This result is true in the last figure; but it is seldom that the approximation proceeds so far correctly; and even here it would not have been safe to assume its correctness without verification. For a simple and ingenious method of constructing tables of square and cube roots, and the reciprocals of numbers, see Dr. Hutton's Tracts on Mathematical and Philosophical Subjects, vol. i. Tract 24, p. 459. By means of the method there laid down, the tables at the end of the volume were computed. A method adapted to the square root, in which the root is exhibited as a simple vulgar fraction, is also given in the same volume, which is extremely convenient for the extraction of the roots of isolated numbers; but where the roots of a series of consecutive numbers are required, the one above referred to is the most convenient and rapid one ever discovered. The following is the method for the square root. n Let N denote the given number, and the fraction to which its square root is approximately equal; then if n be a near integer or fractional value of the root, the fraction n2+Nd2 will denote one still nearer. Or, for continuing the process, the following is still more convenient. Let Example.-Extract the square root of 920. Here N = 920, and n = 30, d = 1. 30-33150183, which differs from the truth only by 6 in the tenth place of figures, the true value being 30-33150177.-Hutton's Tracts, vol. i. p. 457-549. n2 + Nd2 Then from the 2nd 2n2 2 91 3 OF RATIOS, PROPORTIONS, AND PROGRESSIONS OF NUMBERS. NUMBERS are compared to each other in two different ways: the one comparison considers the difference of the two numbers, and is named Arithmetical Relation; and the difference sometimes the Arithmetical Ratio: the other considers their quotient, which is called Geometrical Relation; and the quotient expresses the Geometrical Ratio. So, of these two numbers, 6 and 3, the difference, or arithmetical ratio is 6 3 or 3, but the geometrical ratio is § or 2. There must be two numbers to form a comparison: the number which is compared, being placed first, is called the Antecedent; and that to which it is compared, the Consequent. So, in the two numbers above, 6 is the antecedent, and 3 the consequent. If two or more couplets of numbers have equal ratios, or equal differences, the equality is named Proportion, and the terms of the ratios Proportionals. So, the two couplets, 4, 2 and 8, 6, are arithmetical proportionals, because 4 - 2 8 6 = 2; and the two couplets, 4, 2 and 6, 3, are geometrical proportions, because = =2, the same ratio. To denote numbers as being geometrically proportional, a colon is set between the terms of each couplet, to denote their ratio; and a double colon, or else a mark of equality, between the couplets or ratios. So, the four proportionals, 4, 2, 6, 3, are set thus, 4: 2:6:3, which means, that 4 is to 2, as 6 is to 3; or thus, 4: 2 6 3, or thus, §, both which mean, that the ratio of 4 to 2, is equal to the ratio of 6 to 3 *. = * The test of equal ratios in arithmetic is that the quotients of the antecedents and consequents in the two alleged ratios is a fraction of the same value. In reference to geometry, however, the test does not involve the idea of quotients or fractions, nor indeed of numbers at all. See Def. v. book v. Elements of Euclid. The difficulty felt in treating proportion by means of arithmetical ideas, and of the arithmetical definition of ratio, arises from the incommensurability of numbers; but this difficulty is not encountered in Euclid's method of treating Proportion is distinguished into Continued and Discontinued. When the difference or ratio of the consequent of one couplet, and the antecedent of the next couplet, is not the same as the common difference or ratio of the couplets, the proportion is discontinued. So, 4, 2, 8, 6, are in discontinued arithmetical proportion, because 4 — 2 = 8 — 6 = 2, whereas 2 8=- - 6: and 4, 2, 6, 3, are in discontinued geometrical proportion, because 2, but, which is not the same. = = But when the difference or ratio of every two succeeding terms is the same quantity, the proportion is said to be Continued, and the numbers themselves make a series of Continued Proportionals, or a progression. So, 2, 4, 6, 8, form an arithmetical progression, because 4 − 2 = 6 −4 = 86 = 2, all the same common difference; and 2, 4, 8, 16, a geometrical progression, because == = 162, all the same ratio. When the successive terms of progression increase, or exceed each other, it is called an Ascending Progression, or Series; but when the terms decrease, it is a Descending one. .... So, 0, 1, 2, 3, 4,... is an ascending arithmetical progression, but 9, 7, 5, 3, 1, is a descending arithmetical progression. Also 1, 2, 4, 8, 16, .... is an ascending geometrical progression, and 16, 8, 4, 2, 1, is a descending geometrical progression. .... ARITHMETICAL PROPORTION AND PROGRESSION. IN Arithmetical Progression, the numbers or terms have all the same common difference. The first and last terms of a Progression are called the Extremes ; and the other terms, lying between them, the Means. The most useful part of arithmetical proportion is contained in the following theorems : THEOREM 1. When four quantities are in arithmetical proportion, the sum of the two extremes is equal to the sum of the two means. Thus, with regard to the four, 2, 4, 6, 8, we have 2 + 8 = 4 + 6 = 10. THEOREM 2. In any continued arithmetical progression, the sum of the two extremes is equal to the sum of any two means that are equally distant from them, or equal to double the middle term when there is an odd number of terms. Thus, in the terms 1, 3, 5, it is 1 + 5 = 3 + 3 = 6. +14= 4 + 12 = 6 + 10 = the subject. It is, however, necessary to remark, that quantities (for numbers belong to the predicate quantity) which fulfil the arithmetical test (or definition, as most writers term it) of proportionality, can be readily subjected to Euclid's test: but the proposition is not convertible. Hence, therefore, though we cannot build a system of proportion adapted to geometry upon the arithmetical basis, we can establish upon grounds equally valid and convincing with those of geometry, all the properties of proportional quantities as expressed by numbers, to whatever branch of pure or applied mathematics they may refer. In reading the fifth book of Euclid, the intelligent teacher will avail himself of the opportunity of doing this. From the very early stage of the student's progress, the common definition and test is necessarily employed in this place: but by no means with a view to supersede the more logical and satisfactory investigations just referred to. |