EXAMPLES. 1st. What is the cube root of 20346417? 20346417(273 2x2x2=8 root of the 1ft. period, or 1ft. Subtra. 23 = 8 = 1st. Subtrah. 2×2=4(=next infericur power,) and, 22X3=12)123=Dividend. 4XS (the index of the given pow.)=12 1ft. Divs. 27X27X27=19683=2d. Subtra. 27X27=729 (next inferiour power) and, 273 = 19683=2d.Subtra. 729×$ (=index of the given pow.)=2187=2d.Ds 272x3=2187)6634=2d. Div. 278X273X273=27346417=3d. Subtra. 2733 = 20346417=3d. Subtra. 2d. What is the biquadrate root of 34827998976? Ans. 431·9+. 3d. Extract the sursolid, or fifth root of 281950621875? Ans 195. 4th. Extract the square cubed, or sixth root of 1178420166015625? Ans. 325. A GENERAL* RULE FOR EXTRACTING ROOTS BY APPROXIMATION. 1. Subtract one from the exponent of the root required, and multiply half of the remainder by that exponent, and this product by that power of the assumed root, whose exponent is two less than that of the root required. 2. Divide The general theorem for the extraction of all roots, by approximation, from whence the rule was taken, and the Theorems deducible from it, as high as the twelfth power. Let G=refolvend whose root is to be extracted. §r+e=root required; r being affumed as near the true root, and m-exponent of the power-then the equation will stand thus. S By this Theorem the fraction is obtained in numbers to the lowest terms in all the odd powers; and in the even powers only by having the numerator and denominator found by this equation 2. Divide the given number by the last product; and from the quotient subtract a fraction, whose numerator is obtained by subtracting two from the exponent, and multiplying the remainder by the square of the assumed root; and whose denominator is found by subtracting one from the exponent and multiplying the square of the remainder by the exponent. 3. After this subtraction is made, extract the square root of the remainder. 4. From the exponent subtract two, and place the remainder as a numerator; then subtract one from the exponent, and place the remainder under the numerator for a denominator. 5. Multiply this fraction by the assumed root; add the product to the square root, before found, and the sum will be the root required, or an approximation to it. EXAMPLE. What is the square cubed root of 1178420166015625? Exponent of the required root is 6. Then,x6-15. 2 3004-8100000000 and this multiplied by 15-121500000000. 1178420166015625÷121500000000=9698-9314, from this And the sum is the approximated root= For the 2d. operation, let 325.43 = assumed root. 240 325.43 ANOTHER ANOTHER METHOD BY APPROXIMATION.* RULE. 1. Having assumed the root in the usual way, involve it to that power denoted by the exponent less 1. 2. Multiply this power by the exponent. 3. Divide the resolvend by this product, and reserve the quotient, 4. Divide the exponent of the given power, less 1, by the exponent, and multiply the assumed root by the quotient. 5. Add this product to the reserved quotient, and the sum will be the true root, or an approximation, 6. For every succeeding operation, let the root last found, be the assumed root. EXAMPLE. What is the square cubed root of 1178420166015625 ? Then * A rational formula for extracting the root of any pure power by approxima tion. Let the refolvend be called G, and let re be the required root, being affumed in the usual way. 1 Let G be required; then re=— G Then, 3005x6=14580000000000. 14580000000000)1178420166015625(80-824. Add x 300-250 330-824-approximated root. For the next operation, let 330-824 be the assumed root. OF PROPORTION IN GENERAL. NUMBERS are compared together to discover the relations they have to each other. There must be two numbers to form a comparison: the number, which is compared, being written first, is called the antecedent; and that, to which it is compared, the consequent. Numbers are compared with each other two different ways: The one comparison considers the difference of the two numbers, and is called arithmetical relation, the difference being sometimes named the arithmetical ratio; and the other considers their quotient, which is termed geometrical relation, and the quotient, the geometrical ratio. Thus, of the numbers 12 and 4, the difference or arithmetical ratio, is 12 12-4-8; and the geometrical ratio is = 3.* 4. If two, or more, couplets of numbers have equal ratios, or differences, the equality is termed proportion; and their terms, similarly posited, that is, either all the greater, or all the less taken as antecedents, and the rest as consequents, are called proportionals. So the two couplets 2, 4, and 6, 8, taken thus, 2, 4, 6, 8, or thus, 4, 2, 8, 6, are arithmetical proportionals; and the two couplets, 2, 4, and 8, 16, taken thus, 2, 4, 8, 16, or thus, 4, 2, 16, 8, are geometrical proportionals.† Proportion * Ratios are, here, always confidered as the refult of the greater term of compar ifon diminished, or divided, by the lefs; not regarding which of them be the antecedent. To denote numbers as being geometrically proportional, the couplets are feparated by a double colon, and a colon is written between the terms of each couplet; we may, alfo, denote arithmetical proportionals by feparating the couplets by a double colon, and writing a colon turned horizontally between the terms of each couplet. So the above arithmeticals may be written thus, 24 :: 6. 8, and 4 .. 2 86; where the first antecedent is lefs or greater than its confequent by just fo much as the fecond antecedent is lefs or greater than its confequent: And the geometricals thus, 2 : 4 :: 8 : 16, and 4 : 2 :: 16 : 8; where the first antecedent is contained in, or contains its confequent, juft fo often, as the fecond is contained in, or contains its confequent. Four numbers are faid to be reciprocally or inverfely proportional, when the fourth is lefs than the fecond, by as many times, as the third is greater than the first, or when the first is to the third, as the fourth to the fecond, and vice verfa. Thus 2, 9,6 and 3, are reciprocal proportionals. Note. Proportion is distinguished into continued and discontinued. If, of several couplets of proportionals, written down in a series, the dif erence or ratio of each consequent, and the antecedent of the next following couplet, be the same as the common difference or ratio of the couplets, the proportion is said to be continued, and the numbers themselves, a series of continued arithmetical or geometrical proportionals. So 2, 4, 6, 8, form an arithmetical progression; for 4-2= 6-4-8-6-2; and 2, 4, 8, 16, a geometrical progressión; for= 8 =16=2 ठ But, if the difference or ratio of the consequent of one couplet, and the antecedent of the next couplet be not the same as the common difference or ratio of the couplets, the proportion is said to be discontinued. So 4, 2, 8, 6, are in discontinued arithmetical proportion; Note. It is common to read the geometricals 2 : 4 :: 8: 16, thus, 2 is to 4 as 8 to 16, or, As 2 to 4 fo is 8 to 16. Harmonical Proportion is that, which is between those numbers which assign the lengths of mufical intervals, or the lengths of ftrings founding mufical notes; and of three numbers it is, when the firft is to the third, as the difference between the first and fecond is to the difference between the second and third, as the numbers 3, 4, 6. Thus, if the lengths of strings be as these numbers, they will found an octave 3 to 6, a fifth 2 to 3, and a fourth 3 to 4. Again, between 4 numbers, when the firft is to the fourth, as the difference between the firft and fecond is to the difference between the third and fourth, as in the numbers 5, 6, 8, 10; for ftrings of fuch lengths will found an octave 5 to 10; a fixth greater, 6 to 10; a third greater 8 to 10; a third less 5 to 6; a fixth lefs 5 to 8 ; and a fourth 6 to 8. A feries of numbers in harmonical proportion is, reciprocally, as another feries in arithmetical proportion. SHarmonical 10 · 12 · 15 · 20 · 30-602 As Arithmetical 6 5 4.. 3. 2. 1 for here 10: 12: 5: 6; and 12 : 15 :: 4 : 5, and fo of all the reft. Whence thofe feries have an obvious relation to, and dependence on, each other. 1. Let a, b, c, be the three numbers in musical proportion. Then, because we have a ca-b: b-c; therefore, ab-ac=ac-bc; whence, if any two of the three · be given, the other may be found by the following Theorems. For Example. Suppofe you would find a mufical mean proportional between the monochord 50=a, and the octave 25=c; then, by Theor. II. which is the length of that chord, called a fifth. 2ac --4 a+c 2500 =b=----=33-3, 75 2. If there be four numbers in musical proportion, as a, b, c, d ; then, fince it is ad: a-b: c-d, we have ac-ad-ad-db. From which equation we have the following Theorems. Hence, when any three of thofe numbers are given, the fourth may be found. Thus, let 10, 8, 6 be given to find a fourth harmonical proportion. This harmonical theory may be carried much farther. See Martin's Newtonian Philofophy, Vol II. page 123. |