SQUARES, CUBES, AND ROOTS. Number. Square. Cube. Square Root. Cube Root. 733870808 30:0333148 9.662040 736314327 30-0499584 9.665609 738763264 30-0665928 9.669176 905 819025 741217625 30-0832179 9.672740 743677416 30-0998339 9-676302 746142643 30-1164407 9.679860 908 824464 909 826281 910 828100 911 829921 912 831744 748613312 30.1330383 9.683416 751089429 30.1496269 9.686970 753571000 30-1662063 9-690521 756058031 758550528 30-1827765 9.694069 30.1993377 761048497 9.697615 30.2158899 9-701158 914 835396 763551944 30.2324329 9.704699 766060875 30.2489669 9.708237 916 839056 768575296 771095213 842724 773620632 30.2654919 9.711772 30.2820079 9.715305 30-2985148 9.718835 776151559 30.3150128 9.722363 778688000 30-3315018 9.725888 781229961 30-3479818 9-729411 922 850084 783777448 30-3644529 9-732931 786330467 30.3809151 9.736448 924 853776 925 855625 926 857476 788889024 30-3973683 9.739963 791453125 30-4138127 9.743476 794022776 30-4302481 9.746986 927 931 932 859329 928 861184 929 863041 930 864900 866761 868624 933 870489 812166237 934 872356 814780504 935 874225 936 876096 937 877969 938 879844 796597983 30-4466747 9.750493 799178752 30-4630924 9.753998 801765089 30-4795013 9-757500 804357000 30.4959014 9.761000 806954491 30-5122926 9.764497 809557568 30.5286750 9-767992 30.5450487 9.771484 817400375 30.5614136 9-774974 30.5777697 9-778462 820025856 30.5941171 9-782946 822656953 30.6104557 9.785429 825293672 30-6267857 9.788909 30.6431069 9.792386 940 883600 830584000 30.6594194 9.795861 941 947 948 898704 851971392 885481 833237621 835896888 942 887364 943 889249 838561807 944 891136 841232384 945 893025 843908625 946 894916 846590536 896809 849278123 30.6757233 9.799334 30.6920185 9.802804 30-7083051 9.806271 30-7245830 9.809736 9.813199 30.7408523 30.7571130 9.816659 30-7733651 9.820117 30.7896086 9.$23572 949 900601 854670349 30.8058436 9.827025 950 902500 857375000 30.8220700 9.830476 OF RATIOS, PROPORTIONS, AND PRO. 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 is 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 -62; 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, 4:26:3, or thus,, both which mean, that the ratio of 4 to 2, is equal to the ratio of 6 to 3. 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 8 26: and 4, 2, 6, 3 are in discontinued geometrical proportion, because == 2, but = 3, 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 Con tinued Proportionals, or a progression. So 2, 4, 6, 8 form an arithmetical progression, because 426-4=862, all the same common difference; and 2, 4, 8, 16, a geometrical progression, because == y=2, all the same ratio. When the following terms of a 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, &c. is an ascending arithmetical progression, but 9, 7, 5, 3, 1, &c. is a descending arithmetical progression. Also 1,2, 4, 8, 16, &c. is an ascending geometrical progression, and 16, 8, 4, 2, 1, &c. is a descending geometrical progression. ARITHMETICAL PROPORTION AND PROGRESSION. IN Arithmetical Progression, the numbers or terms have all the same common difference. Also, 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, of the four 2, 4, 6, 8, here 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 uneven number of terms. Thus, in the terms 1, 3, 5, it is 1 + 5 = 3 + 3 = 6. And in the series 2, 4, 6, 8, 10, 12, 14, it is 2+ 14 = 4 +12=6+10=8+8=16. THEOREM 3. The difference between the extreme terms of an arithmetical progression, is equal to the common difference of the series multiplied by one less than the number of the terms. So, of the ten terms, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, the common difference is 2, and one less than the number of terms 9; then the difference of the extremes is 20218, and 2 X 9 = 18 also. Consequently the greatest term is equal to the least term added to the product of the common difference multiplied by 1 less than the number of terms. THEOREM 4. The sum of all the terms, of any arithmetical progression, is equal to the sum of the two extremes multiplied by the number of terms, and divided by 2; or the sum of the two extremes multiplied by the number of the terms, gives double the sum of all the terms in the series. This is made evident by setting the terms of the series in an inverted order, under the same series in a direct order, and adding the corresponding terms together in that order. Thus, in the series 1, 3, 5, 7, 9, 13, ditto inverted the sums are 11, 15; 15, 13, 11, 9, 7, 5, 3, 1; 16+16 +16 +16+ 16 + 16 + 16 + 16, which must be double the sum of the single series, and is equal to the sum of the extremes repeated as often as are the number of the terms. From these theorems may readily be found any one of these five parts; the two extremes, the number of terms, the common difference, and the sum of all the terms, when any three of them are given; as in the following problems: PROBLEM I. Given the Extremes, and the Number of Terms, to find the Sum of all the Terms. ADD the extremes together, multiply the sum by the number of terms, and divide by 2. EXAMPLES. 1. The extremes being 3 and 19, and the number of terms 9; required the sum of the terms? 2. It is required to find the number of all the strokes a common clock strikes in one whole revolution of the index, or in 12 hours. Ans. 78. |