Numb. Square. Cube. 851 724201 616295051 852 725904 618470208 853 727 609 620650477 854 729316 622835864 855 731025 625026375 856 732736 627222016 857 734449 629422793 858 736164 631628712 859 737881 633839779 860 739600 636056000 861 741321 638277381 862 743044 640503928 863 744769 642735647 864 746496 644972544 865 748225 6472 14625 866 749956 649461896 867 751689 651714363 868 753424 653972032 869 755161 656234909 870 756900 658503000 871 7 58641 660776311 872 760384 663054848 873 782129 665338617 874 387 667627624 875 765625 669921865 876 767 376 672221376 877 769129 67 4526133 878 770884 676836152 879 772641 679151439 880 774400 681472000 881 716161683797841 882 777924 686128968 883 779689 688465387 884 781456 690807104 885 783225 693154125 886 784996 695506456 887 786769 698764103 888 788544 700227072 889 790321 702595369 890 792100 704969000 891 793881 707347971 892 795664 709732288 893 797449 712121957 894 799236 714516984 895 801025 716917375 896 802816 719323136 897 804609 721734273 898 806404 724150792 899 808201 726572699 900 810000 729000000 Square Root, Cube Root. 9.476395 9.480106 9.483813 9.487518 9.491219 9.494918 9.498614 9.502307 9.505998 9.509685 9.513369 9'517051 9'5207 30 9'524406 9.528079 9.531749 9.535417 9.539081 9:542748 9.546402 9.550058 9 553712 9557363 9:561010 9564655 9.568297 9.571937 9.575574 9.579208 9.582839 9.586468 9.590093 9.593716 9 597337 9. 600954 9.604569 9.608181 9.611791 9.615397 9.619001 9 622603 9.626201 9.629797 9.633390 9.636981 9.640569 9.644154 9.647736 9.651316 9.654893 Numb. Square. Cube. Square Root. Cube Root, 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 904401 906304 908209 910116 912025 913936 915849 917764 919681 921600 923521 925444 927369 929296 931225 933156 935089 937024 938961 940900 942841 944784 946729 948676 950625 952576 954529 956484 958441 960400 962361 964324 966289 968256 970225 972196 974169 976144 978121 980100 982081 984064 986049 988036 990025 992016 994009 996004 998001 860085351 862801408 865523177 868250664 870983875 873722816 876467 493 879217912 881974079 884736000 887 503681 890277128 893056347 895841344 893632125 901428696 904231063 907039232 909853209 912673000 915498611 918330048 921167317 924010424 926859375 929714176 932574833 935441352 938313739 941192001 944076141 946966168 949862087 952763904 955671625 958585256 961504803 964430272 967361669 970299000 973242271 97619 1488 979 146657 982107784 985074875 988047936 991026973 994011992 997002999 30-8382879 30.8544972 30.8706981 30.8868904 30.9030743 30.9192497 30.9354166 30.9515751 30.9677251 30.9838668 31.0000000 31.0161248 31.0322413 3180483494 31 0644491 31.0805405 31.0966236 31:1126984 311287648 31'1448230 31:1608729 31.1769 145 31:1929479 31.2089731 31.2249900 31.2409987 31'2569992 31.2729915 31.2889757 31.3049517 31.3209195 31.3368792 31:3528308 31.3687743 31.3847097 31.4006369 31.4165561 31.4324673 31.4483704 31.4642654 31.4801525 31.4960315 31.5119025 31.5277655 31:5436206 31:5594677 31.5753068 31.5911380 31•3069613 9.833923 9.837369 9.840812 9.844253 9.847692 9.851128 9.854561 9.857992 9.861421 9.861848 9.868272 9.871694 9.875113 9.878530 9.881945 9.885357 9.888767 9.892174 9.895580 9.898983 9.902383 9.905781 9.909177 9.912571 9.915962 9.919351 9.922738 9.926122 9.929 504 9.932883 9.936261 9.939636 9.943009 9.946379 9.949747 9.953113 9.956477 9.959839 9.963198 9.966554 9.969909 9.973262 9.976612 9.979959 9.983304 9.986648 9.989990 9.993328 9.996665 OF RATIOS, PROPORTIONS, AND PROGRESSIONS. 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 consi. ders their quotient, which is called Geometrical Relation n; 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 for 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 Proporcionals. 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 geometri. cal proportionals, 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, s, 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 — 2 = 6: and 4, 2, 6, 3 are in discontinued geometrical proportion, because 을 = 2, but if = 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 Continued Proportionals, Proportionals, or a progression. So 2, 4, 6, 8 form an arithmetical progression, because 4–2 = 6-4 = 8-6 = 2, all the same common difference; and 2, 4, 8, 16 a geometrical progression, because = 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 arithnetical 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 proportions, 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, 12, 14, it is 2 + 14 de + 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 20 - 2 = 18, and 2 X 9 - 18 also. Consequently, |