Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

Numb

Cube.

801 S02 S03 €01 805 800 807 808 809 $10 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850

Square. 641601 643204 644809 046416 645025 619036 651249 652864 654481 656100 657721 65934+ 660969 662596 664225 665856 607489 669124 670701 672400 671041 675684 677329 676976 680625 682270 683929 685584 687241 688600 600561 692224 693889 695556 697225 698896 700509 702244 703021 705000 707281 708961 710049 712336 714025 715716 717469 719104 720301 722500)

513922401 515849603 517781027 519718464 521600125 523606616 525557943 527514112 529475129 531441000 533411731 535367328 537366797 539353144 541343375 543338496 545338513 547343432 549353259 551368000 553387061 555412248 557441707 559476224 561515625 563559976 565509283 567663552 569722789 571787000 573856191 -575930368 578009537 580093704 582182875 58+277056 586370253 53&450472 590589719 592704000 594823321 596647688 599077107 60121158+ 603351/25 605495736 607045423 609E00192 611960019 614125000

Square Root., Cube Rooi.
25:3019134 | 9287044
28:3190045 9:200907
28.837254039 204767
28.3548938 9:248623
28-3725219 9 302-77
29-3901391 g 300327
28.4077454 9:310175
28.4253408 | G•314019
28.4426253 9'317859
28.400498999321697
28.4780017 | 9:325532
28-4956137 9329303
29-5131549 9'333191
28.5306852 4:337016
28:5482048 9:340838
26.5657137 9.344657
28:5832119 | 9:348473
28.0000933 9:352285
28.6181760 9356095
28.6356421 9'359901
28.6530976 9:363704
28.67054249.367505
28.68797669-371302
28-7054002 9 375096
28.7228132 9:378887
28.7402157 9:382675
287576077 9386460
18.7749891 9.390241
28.7923601 9391020
28.80372069*397796
28.82707069401509
28.8444102 9:405338
28.8617394 9:409105
28.87905829-412869
28.89036669416630
28.91366-69.420387
28.0309523 9:424141
28.94822979427893
28-96546,67 9•431642
28.9827535 9435388
29-0000000 9:439130
29.0172363 94-12870
29.0344623 9:446607
29.0510781 4:450341
29.06888:37 9:454071
29.0800791 9:457799
24.1032044 9:401524
29.1204396 | 9:465247
29.13700469 468966
29:154756:5 9:4720182

Numb.

Cube.

651 852 853 854 855 856 857 858 859 860 861 862 303 864 865 806 867 868 80g 870

71 872 873 874 875 870 877 878 879 890 881 882 883 884, 385 836 887 889 859 890 891 842 893 894 895 846 897 898 893 900

Square. 724201 725904 727609 729316 731025 732736 731449 736164 737991 739000 741321 743014 744769 746490 748225 749956 751089 753124 755161 756900 758641 760384 762129 763976 765625 767370 709129 770884 772641 774400 776161 777924 779689 781456 783225 784996 786769 788544 790321 792100 793881 795664 797449 799236 801025 802816 801609 806404 808201 810000

616095051 618-170208 620630477 622835804 625026375 627222016 629422793 631628712 633839779 6300500CO 638277381 610503928 6+2735647 644972544 6.+7214625 649461896 651714363 053972032 656234909 658503000 600770311 603054848 665335617 607627624 609921875 672221376 674526133 076336152 679151439 081472000 683797541 686128909 683.105387 690807104 693151125 665506456 097804103 700227072 702595369 701969000 707347971 709732283 712121957 714516981 716917375 719323136 72 734273 724150792 726572699 729000000

Square Root. Cabe Root. 29-17190-13 9470345 29:1890300 09:480106 29.2001037 9.483913 29-22327819 +87518 29:2403830 9:431919 29:2574777 9*494918 29:2745023 9+99014 292916370 9'502307 29'3057018 9:505998 29-3257566 9'509085 29-3428015 9:513369 29-3598365 9*517051 29-3708010 G6520730 29:3938769 9.524406 29*4108823 9:528079 29-42787799-531749 294-148637 9.535417 29:4618397 | 9:5.9081 29-4788059 9.5+2743 29.4957024 9.516 102 29.5127091 9*550058 29.5296401 9:553712 29.5465734 | 9:557363 29:56:34910 9:561010 29*5803989 9:56 1655 29.5972972 9:568297 29.6141858 | 9:571937 29:6310648 9'575574 29.0479325 4:579208 23.0647939 9•582839 29'08161429-585468 23.698-1848 9:590093 29.71.53159 9:593716 29.7321375 9.597337 297-1904'36 9.600954 29-7057521 9-004569 29.7825452 9.608181 2907993259 | 9•611791 29.8101030 9:615397 29.5328078 9.619001 29.8490231 9:622003 29.80630909.626201 29.8831056 9.629797 29 8998328 9.633390 29.91055069.636981 29 9332591 y 640569 29.9499583 9•644154 29.9566481 9:647736 29.9533287 9.651316 30.0000000 9:054393

|

Numb. Square.

Cube. Square Root.Cube Root 811801 731432701 30-0166620 | 9•659468 813604 733870908 30*0333148 9•662040 815409 736314327 30 0499584 1 9.605609 817210 738703204 30*0005928 | 9*069170 810025 74121762. 30:08321791 9:672740 820836 743077416 30-099-3399070301 822049

7-46142613 30 1104407 9:079800 824-164 748013312 30:1330383 9683416 826281 751089429 30-1496269 9 086970 828100 753571000 30*1632003 9.690521 $29921 755058931 30*1827765 9091059 831744 758550528 30'1993377 | 9:097015 533509 761018497 302158899 | 90701159 835396

703551644. 30.2324329 9.704698 837225 760000875 30:2489669 9.703236 839056 768575290 30-2654919 9711772 840389 771095213 30°2820079 9715305 842724 773620032 30-29851489718835 844551 776151559 30•3150128 90722363 846-100 778688000 30•3315018 | 9°725888 8-18241 781229,61 30:3479818 9.729+10 S5008+ 783777448 30:3644529 | 9732930 $51929 786330107 30'3809151 9736448

788989024 30-3973683 97393 855025 791453125 30:4138127 90743-75 857476 704022776 30•4302181 9746985 859329 796597983 30-4466747 9.750493 861184 799178752 30"-1630924 9.753998 863041 801765059 30-4795013 907057500 864900 804357000 3049590!4 | 9761000 860761 800954491 30-5122926 99704497 868624 800557505 30-52807509767992 870 189 812160237 30•5450487 / 98771434

814780504 30:5614136 | 9774974 87225 817400375 30:5777697 | 99778-161 870096 820025856 30:5411171 9782940

822056953 30•6101557 19785423 877841 825293672 30·0267857 9•788903 881721 827931019 30-5431069 9792386 883005 &30584000 30-6594194 9795361 895481 833237621 30-67572339799333

835890898 30-6920185 9'802903 889249 838561807 30:7083051 9.800271

841232384 30 7245830 9.809736 843908625 30-74095239813198

8-10590536 9-7 890809 849278123 30.7733051 9*820117

898704 85.971392 30-7890086 9'823572 919 G00001 854670349 30*8058136 | 9827025 950 992510 85737500) 30*8220700 1.4.830,475

901
9C2
903
904
905
906
907
gos.
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924

: 853776
925
926
927
928
929
930
931
932
933
934

872:59
935
930
937 877909
938
939
940
941
942

887364
943
94-4 891136
445

893025
894916

[ocr errors]

946

0948

Cube.

Nunb. Square.
951 904101
952

905301
953

908209 954

910116 965 912025 956

913936 957 915849 958 917704 959 99581 960 921600 951

923521 962

925444 963

9273019 964

929296 905

931225 906 933156 967 935089 969

937024 909 . 938961 970 910900 971 942841 972

944784 973

946729
974

9-18676
975 950625
976 952576
*977 954529
978 956494
979 958141
980 960400
981 962361
982 G61324
93 966289
9814 968 256
985 970225
986

972196
987 974109
988 976144
959 978121
990 980100

991 982081 1992

98406+ 993 986049 994

988036 945 990025

992010 997 994009 998

990004 999 998001

SÓC035351 862801108 805523177 808250664 870983875 873722816 876407493 879217912 831974079 984736000 887503681 890277123 893050347 895841344 698082125 9014281196 904231013 907039232 909853209 912673000 915498611 918330048 921167317 924010424 920859375 929714176 932574833 935441352 938313739 941192001 944070141 946966168 919862087 952763904 955071625 958595256 951504803 964430272 967361669 970299000 973242271 976191488 979146657 98210778+ 985074875 998017936 991026973 994011992 997002999

Square Root Cabe Root 30-8382379 9.833923 30.8544972 | 9:837369 30.8700ysi 9*4410312 30'88689049.844253 30•4030743 | 9•617 692 30°(192497 9-851128 30-9334106 9.954561 30 9515751 9'857992 30-9677251 99851421 30:9838068 | 9.861818 31•0C03000 9:808272 31.0161248 9•87100+ 31•0322113 9:875113 31•0483494 | 4.878530 31-06-14491 9.881915 31.0805405 9.895357 31.0966236 9.889707 31•1126984. 9:832174 31:1287649 | 9-895580 31:1418230 | 9.898983 31:16J8729 9:902383 31:1769145 | 9.905781 31:1929479 9.999177 31.2089731 | 9.912571 31:2249900 9915902 31.2409987 9.919351 31.2509942 9:922739 31.2729715 9:926122, 31.2889757 | 992050+ 31:3014517 | 9*932883 31•3209145 9.936291 31•3308792 9:13:3636 31•3528:08 903059 31:3687743 | 9*940379 31:38+7097 9:949747 31:1000309 9953113 31:41055016956477 31.432+673 9°:959839 31°4483704 9.903198 31.4642054 9930554 31.4801525 9*969999 31.4963315 9*973202 31.5110025 9 976612 31:5277355 9*979959 31:5-30200 | 9*983304 31.559 +677 9'956)+3 31.5753068 y'999090 31-5911380 9*993328 31.6069613 9990665

996

OF RATICS, 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 dif. ference sometimes the Arithinetical 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

2; and the two couplets 4, 2 and 6, 3, are geometrical 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 Discontis nued." 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, §, 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 4 = ?, 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 Continued

Proportionals,

1

« ΠροηγούμενηΣυνέχεια »