No. Square. Cube. Square Root. Cube Root. No. Square. Cube. Square Root. Cube Root. 631 3981 61 251 239591 25-1197134 8 577152 694 481636 334255384 26·34387978-853598 632 399424 252435968 25 13961028-581680 695 483025 335702375 26.3628527 8.857849 633 400689 2536361 37 25-15949138 586204 696 484416 337153536 26 381 8119 8-862095 634 401956 254840104 25 1793566 8.590723 697 485809 338608873 26 4007576 8.866337 635 403225 256047875 25·1992063 8 595238 698 487204 840068392 26·41 96896 8.870575 636 404496 257259456 25·21 90404 8 599747 699 488601 341 532099 26-4386081 8.874809 637 405769 258474853 25-2388589 8-604252 700 490000 343000000 26 4575131 8.879040 658 407044 259694072 25.2586619 8.608752 701 491 401 344472101 26 4764046 8.883266| 639 408321 260917119 25-2784493 8-613248 702 492804 345948408 26 4952826 8.887488 640 409600 2621 44000 25.2982213 8-617738 703 494209 347428927 26 51 41 472 8-891706 641 410881 263374721 25 3179778 8-622224 704 495616 34891 3664 26·5329983 8.895920 642 412164 264609288 25 33771898-626706 705 497025 350402625 26 551 8361 8.900130 643 413449 265847707 25 3574447 8-631183 706 498436 351895816 26 5706605 8.904336 644 414736 267089984 25-3771551 8.635655 707 499849 353393243 26 5894716 8.908538| 645,41 6025 268336125 25-3968502 86401 22 708 501264 354894912 26 6082694 8.912736 646 417316 269586136 25 4165301 8 644585 709 502681 356400829 26-6270539 8 91 6931 647 418609 270840023 25 4361947 8-649043 710 504100 357911000 26 6458252 8-121121 648 419904 272097792 25 4558441 8 653497 711 505521 359495431 26-6645833 8.925307 649 421201 273359449 25-4754784 8 657946 712 506944 360944128 26-6833281 8-929490 650 422500 274625000 25-4950076 8.662301 713 508369 362467097 26 7020598 8.933668 651 423801 275894451 25-5147016 8 666831 714 509796 363994344 26-7207784 8-937843 652 425104 277167808 25 5342907 8-671266 715 511225 365525875 26.7394839 8 94201 4| 653 426409 278445077 25 5538647 8-675697 716 512656 367061696 26 7581763 8.9461 80 654 427716 279726264 25 57342378-680123 717 51 4089 368601 81 3 26 7768557 8.950343 655 429025 281011375 25 5929678 8-684545 718 51 5524 3701 46232 26 7955220 8 954502 656 430336 282300416 25 6124969 8 688963 719 516961 371694959 26 8141754 8.958658 657 431649283593393 25 6320112 8-693376 720 518400 373248000 26-8328157 8-962809 658 432964 284890312 25-6515107 8-697784 721 51 9841 374805361 26 851 4432 8.966957 659434281 286191179 25-6709953 8.702188 722 521284 376367048 26 8700577 8.971100 660 435600 287496000 25 6904652 8.706587 723 522729 377933067 26 8886593 8.975240 661 436921 288804781 25 7099203 8 710982 724 524176 379503424 26 9072481 8-979376 662 438244 290117528 25-7203607 8.715373 725 525625 381078125 26·9258240 8.983508] 663 439569 291434247 25 7487864 8-719759 726 527076 382657176 26.9443872 8-987637 664 440896 292754944 25-7681975 8.724141 727 528529 384240583 26 9629375 8.991762 665 442225 294079625 25-7875939 8-728518 728 529984 385828352 26.981 4751 8-995883 666 443556295408296 25.80697588-782891 729 531441 387420489 27-0 9-0 667 444889296740963 25.8263431 8.737260 730 532900 38901 7000 27 01 85122 9004113] 668 446224 298077632 25.84569608-741 624 731 534361 390617891 27-0370117 9-008222 669 447561 299418309 25 8650343 8.745984 732 535824 392223168 27·0554985 9 012328 670 448900 300763000 25-8843582 8-750340 733 537289 393832837 27 0739727 9 016430 671 450241 302111711 25.9036677 8-754691 734 538756 395446904 270924344 9 020529 672 451 584 303464448 25-9229628 8-759038 735 540225 397065375 27·1108834 9 024623 673 452929 304821217 25-94224358-763380 736 541 696 398688256 27-12931 99 9 028714 674 454276 306182024 25.961 5100 8.767719 737 543169 40031 5553 27-14774399-032802 675 455625 307546875 25 9807621 8.772053 738 544644 401 947272 27-1661 554 9 036885 676 456976 30891 5776 26-0 8.776382 739 546121 403583419 27 1845544 9 040965 677 458329 310288733 26 01922378-780708 740 547600 405224000 27-2029410 9 045041 678 459684 311665752 26-0384331 8·785029 741 549081 406869021 27-221 31 52 9 049114 679 461041 31 3046839 26 0576284 8.789346 742 550564 408518488 27-2396769 9 053183 680 462400 314432000 26 0768096 8.793659 743 552049 410172407 27-2580263 9 057248 681 463761 31 5821241 26 0959767 8.797967 744 553536 411830784 27-2763634 9 061309 682 465124 317214568 26-1151297 8-802272 745 555025 41 3493625 27-2946881 9-065367 683 466489 318611987 26 1342687 8 806572 746 556516 415160936 27 31 30006 9 069422 684 467856 32001 3504 26 1 533937 8.810868 747 558009 41 6832723 27 3313007 9 073472 685 469225 321 419125 26·1725047 8-815159 748 559504 418508992 27 3495887 9-077519 686 470596 322828856 26·191 601 7 8 819417 749 561001 4201 89749 27 3678644 9 081 563 687 471969 324242703 26-2106848 8.823730 750 562500 421875000 27-3861279 9.085603 688 473344 325660672 26-2297541 8.828009 751 504001 423564751 27 4043792 8-089639 689 474721 327082769 26.2488095 8-832285 752 565504 425259008 27-4226184 9-093672 690 476100328509000 26-2678511 8.836556 753 567009 426957777 27·4408455 9.097701 691 477481 329939371 26-2868789 8.840822 754 568516 428661064 27-4590604 9-101726 692 478864 331373888 26-3058929 8.845085 755 570025 430368875 27-4772633 9.105748 693 480249 332812557 26.3248932 8.849344 756 571536 432081216 27-4954542 9·109766| No. Square. Cube. Square Root. Cube Root. No. Square. Cube. Square Root. Cube Root. 757 573049 433798093 27.5136330 9.113781 820 672400551368000 28 6356421 9.359901 758 574564 43551 9512 27.5317998 9.117793 821 674041 553387661 28 6530976 9.363704 759 576081 437245479 27·5499546 9.121801 822 675684 555412248 28 6705424 9.367505 760 577600 438976000 27.5680975 9.125805 823 677329 557441767 28 6879766 9.371302 761 579121 440711081 27-5862284 9-129806 824 678976 559476224 28-7054002 9.375096 762 580644 442450728 27·6043475 9.133803 825 680625 561 51 5625 28-7228132 9-378887 763 582169 444194947 27·6224546 9.137797 826 682276 563559976 28-74021 57 9-382675 764 583696 445943744 27·6405499 9·141788 827 683929 565609283 28 75760779-386460 765 585225 447697125 27 ·6586334 9·145774 828 685584 567663552 28 7749891 9-390241 766 586756 449455096 27 6767050 9-149757 829 687241 569722789 28-7923601 9-394020 767 588289 451217663 27·6947648 9.153737 830 688900 571787000 28 8097206 9.397796 768 589824 452984832 27 7128129 9.157713 831 690561 573856191 28 82707069-401 569 769 591361 454756609 27 7308492 9.161686 832 692224 575930368 28 8444102 9.405338 770 592900 456533000 27-7488739 9.165656 833 693889 578009537 28-8617394 9-409105 771 594441 458314011 27 7668868 9·169622 834 695556 580093704 28-8790582 9-412869 772 595984 460099648 27 7848880 9.173585 835 697225 582182875 28 8963666 9-416630 773 597529 461 889917 27 8028775 9·177544 836 698896 584277056 28 91 36646 9-420387 774 599076 463684824 27-8208555 9-181500 837 700569 586376253 28-9309523.9-424141 775 600625 465484375 27 8388218 9.185452 838 702244 588480472 28-9482297 9-427893 776 602176 467288576 27·8567766 9.189401 839 703921 590589719 28 9654967 9.431642 777 603729 469097433 27 8747197 9.193347 840 705600 592704000 28 9827535 9-455388 778 605284 470910952 27 8926514 9·197289 841 707281 594823321 29-0 9-439130 779 606841 4727291 39 27 9105715 9.201228 842 708964 596947688 29 0172363 9-442870 780 608400 474552000 27·9284801 9-205164 843 710649 599077107 29 0344623,9-446607 781 609961 476379541 27.9463772 9.209096 844 712336 601211584 29 0516781 9-450341 782 611524 478211768 27·9642629 9.213025 845 714025 603351125 29 0688837 9-454071 783 613089 480048687 27.9821372 9.216950 846 715717 605495736 29 08607919-457799 784 614656 481 890304 28.0 9.220872 847 71 7409 607645423 29·1032644 9.461 524 785 616225 483736025 28-0178515 9-224791 848 719104 609800192 29 12043969-465247 786 617796 485587656 28 0356915 9-228706 849 720801 611960049 29·13760469-468966 787 61 9369 487443403 28-0535203 9-232618 850 722500 614125000 29 15475959-472682 788 620944 489303872 28 0713377 9-237527 851 724201 616295051 29 1719043 9-476395 789 622521 491169069 28-0891 438 9-240433 852 725904 61 8470208 29-1890390 9-480106 790 624100 493039000 28 1069386 9-244335 853 727609 620650477 29-2061637 9.483813 791 625681 494913671 28·12472229-248234 854 729316 622835864 29-22327849-487518 792 627264 496793088 28 1424946 9-252130 855 731025 625026375 29-2403830 9.491219 793 628849 498677257 28·1602557 9-256022 856 732736 627222016 29-2574777 9-494918 794 630436 500566184 28 1780056 9-259911 857 734449 629422793 29-2745623 9-498614 795 632025 502459875 28-1957444 9-263797 858 736164 631 628712 29-2916370 9.502307 796 633616 504358336 28 2134720 9.267679 859 737881 633839779 29 30870189.505998 797 635209 506261573 28 2311884 9-271559 860 739600 636056000 29-3257566 9.509685| 798 636804 508169592 28-2488938 9-275435 861 741 321 638277381 29-3428015 9.513369 799 638401 510082399 28-2665881 9.279308 862 743044 640503928 29 3598365 9.517051| 800 640000 51 2000000 28-28427129.283177 863 744769 642735647 29-3768616 9.520730 801 641601 513922401 28-301 9434 9.287044 864 746496 644972544 29 3938769 9-524406 802 643204 51 5849608 28-3196045 9-290907 865 748225 647214625 29 4108823 9-528079 803 644809 517781627 28-3372546 9-294767 866 749956 649461896 29-4278779 9-531749| 804 646416 519718464 28-3548938 9-298623 867 751689 651714363 29-4448637 9-535417| 805 648025 5216601 25 28-37252199-302477 868 753424 653972032 29-4618397 9-539081 806 649636 523606616 28-3901 391 9.306327 869 755161 656234909 29 4788059 9-542743 807 651 249 525557943 28·4077454 9.310175 870 756900 658503000 29-4957624 9-546402| 808 652864 527514112 28-4253408 9.314019 871 758641 660776311 29-5127091 9.550058' 809 654481 529475129 28.4429253 9.317859 872 760384 663054848 29-5296461 9-553712 810 656100 531441000 28-4604989 9-321697 873 762129 665338617 29-5465784 9.557363 811 657721 533411731 28 4780617 9-325532 874 763876 667627624 29-5634910 9-561010 812 659344 535387328 28-4956137 9-329363 875 765625 669921875 29.5803989 9:564655 813 660969 537367797 28-5131549 9.333191 876 767376,672221376 29-5972972.9.568297| 814 662596 539353144 28 5306852 9.337016 877 769129 674526133 29 6141858 9.571937 815 664225 541313375 28-5482048 9-340838 878 770884 6768361 52 29-6310648 9.575574 816 665856 543338496 28.5657137 9.344657 879 772641 679151439 29·6479325 9:579208 817 667489 545338513 28 5832119 9.348473 880 774400 681 472000 29-6647939 9.582839 818 669124 547343432 28-6006993 9.352285 881 776161 683797841 29-6816442 9-586468| 819 670761 549353259 28-6181760 9-956095 882 777924 686128968 29-6984848 9.590099 No. Square. Cube. Square Root. Cube Root. No. Square. Cube. Square Root. Cube Root. 9.854561 9.857992 883 779689 688465387 29-71531 59 9.593716 942 887364 835896888 30 6920185 9.802803 884 781 456 690807104 29-7321375 9.597337 943 889249 838561807 30.7083051 9.806271 885 783225 6931 54125 29.7489496 9 600954 944 891136 841232384 30 7245830 9.809736 886 784996 695506456 29-7657521 9.604569 945 893025 843908625 30.7408523 9.813198 887 786769 697864103 29-7825452 9.608181 946 894916 846590536 30 7571130 9.816659 888 788544 700227072 29-7993289 9.611791 947 896809 849278123 30 7733651 9.820117 889 790321 702595369 29.8161030 9.615397 948 898704 851971392 30 7896086 9.823572 890 792100 704969000 29-8328678 9.61 9001 949 900601 854670349 30-8058436 9.827025 891 793881 707347971 29-8496231 9.622603 950 902500 857375000 30-8220700 9.830475 892 795664, 709732288 29.8663690 9.626201 951 904401 860085351 30 8382879 9.833923 893 797449 712121957 29-8831056, 9-629797 952 906304 862801 408 30 8544972 9.837369 894 799236 714516984 29·8998328 9-633390 953 908209 865523177 30-8706981 9.840812 895 801025 716917375 29-9165506 9.636981 954 910116 868250664 30.8868904 9.844253| 896 802816 719323136 29·9332591 9 640569 955 912025 870983875 30.9030743 9.847692 -897 804609 721734273 29-9499583 9.644154 956 913936 873722816 30 9192497 9.851128 898 806404 724150792 29.9666481 9.647736 957 915849 876467493 30-9354166 899 808201 726572699 29 9833287 9 651316 958 917764 879217912 30.9515751 900 810000 729000000 30.0 9-654893 959 919684 881974079 30·9677251 9.861421 901 811801 731 432701 30-0166620 9.658468 960 921600 884736000 30.9838668 9.864848) 902 813604 733870808 30 03331 48 9.662040 961 923521 887503681 31 0 9.868272 903 815409 73631 4327 30-0499584 9.665609 962 925444 890277128 31 0161248 9.871694 904 817216 738763264 30-0665928 9-669176 963 927369 893056347 31 0322413 9.875113 905 819025 741217625 30-0832179 9.672740 964 929296 895841344 310483494 9.878530 906 820836 743677416 30-0998339 9.676301 965 931225 898632125 31 0644491 9.881945 907 822649 746142643 30-1164407 9-679860 966 933156 901 428696 31 0805405 9.885357 908 824464 748613312 30-1330383 9.683416 967 935089 904231063 31 0966236 9-888767 909 826281 751089429 30-1496269 9.686970 968 937024 907039232 31·1126984 9.892174 910 828100 753571000 30-1662063 9.690521 969 938961 909853209 311287648 9.895580 911 829921 756058031 30-1827765 9 694069 970 940900 912673000 31·1448230 9.898983 912 831744 758550528 30·1993377 9.697615 971 942841 915498611 31 1608729 9.902383 913 833569 761048497 30-21 58899 9.701158 972 944784 918330048 311769145 9.905781 914 835396 763551944 30-2324329 9.704698 973 946729 921167317 31·1929479 9.909177) 915 837225 766060875 30-2489669 9.708236 974 948676 916 839056 768575296 30-2654919 9.711772 975 950625 917 840889 771095213 30-2820079 9.715305 976 952576 1918 842724 773620632 30-2985148 9.718835 977 954529 919 844561 776151559 30-3150128 9-722363 978 956484 920 846400 778688000 30-331 5018 9.725888 979 958441 921 848241 781229961 30-3479818 9.729410 980 960400 1922 850084 783777448 30.3644529 9-732930 981 962361 923 851929 786330467 30.3809151 9.736448 924 853776 788889024 30-3973683 9 739963 925 855625 791453125 30.4138127 9.743475 926 857476 794022776 30-4302481 9.746985 985 970225 |927 859329'796597983 30-4466747 9.750493 986 972196 928 861184 799178752 30-4630924 9.753998 987 974169 929 863041 801765089 20-4795013 9.757500 988 976144 930 864900 804357000 30.4959014 9.761000 989 978121 1931 866761 806954491 30·5122926 9.764497 990 980100 ¡932 868624 80955756820-5286750 9.767992 991 982081 933 870489 812166237 30-5450487 9-771484 992 984064 934 872356 814780504 30-561 41 36,9-774974 935 874225 817400375 30.5777697 9.778461 936 876096 820025856 30 5941171 9-782946 937 877969 822656953 30 6104557 9 785428 938 879844 825293672 30 6267857 9.788908 939 881721 827936019 30 6431069 9.792386 940 883600 830584000 30 6594194 9.795861 999 998001 997002999 31 6069613 9.996665 941 885481 833237621 30-6757233 9-799333 1000 1000000 1000000000 31 6227767 10.0 982 964324 993 986049 9.929504 9.932883 9.936261 9.939636 9.943009 9.940379 9.949747 924010424 31 2089731 9.912571 9.953113 9.956477 9.983304 9-986648 9.98 9990 9.993328 X SECT. II. GEOMETRY. 874. Geometry is that science which treats of the relations and properties of the boundaries of either body or space. The invention of the science has been referred to a very remote period: by some, to the Babylonians and Chaldeans; by others, to the Egyptians, who are said to have used it for determining the boundaries of their several lands, after the inundations of the Nile. Cassiodorus says that the Egyptians either derived the art from the Babylonians, or invented it after it was known to them. It is supposed that Thales, who died 548 B. C., and Pythagoras of Samos, who flourished about 520 B. c., introduced it from Egypt into Greece. We do not, however, consider it useful here to enter into the history of the science; neither is it necessary to enter into the reasons which have induced us to adopt the system of Rossignol, from whom we extract this section, otherwise than to state that we hope to conduct the student by a simpler and more intelligible method to those results with which he must be acquainted. The limits of body or space are surfaces, and the boundaries of surfaces are lines, and the terminations of lines are points. Bounded spaces are usually called solids, whether occupied by body or not; the subject, therefore, is naturally divided into three parts, lines, surfaces, and solids; and these have two varieties, dependent on their being straight or curved. 875. Geometrical inquiry is conducted in the form of propositions, problems, and demonstrations, being always the result of comparing equal parts or measures. Now, the parts compared may be either lines or angles, or both; hence, the nature of each method should be separately considered, and then the united power of both employed to facilitate the demonstration of propositions. But the reader must first understand the following DEFINITIONS. 1. A solid is that which has length, breadth, and thickness. A slab of marble, for instance, is a solid, since it is long, broad, and thick. 2. A surface is that which has length and breadth, without thickness. A leaf of paper, though not in strictness, inasmuch as it has thickness, may convey the idea of a surface. 3. A line is that which has length, but neither breadth nor thickness. As in the case of a surface, it is difficult to convey the strict notion of a line, yet an infinitely thin line, as a hair, may convey the idea of a line: a thread drawn tight, a straight line. 4. A point is that which has neither length, breadth, nor thickness. sand may give an idea of it. 5. If a line be carried about a point A, so that its other extremity passes from B to C, from C to D, &c. (fig. 223.), the point B, in its revolution, will describe a curve BCDFGLB. This curve line is called the circumference of a circle. The circle is the space enclosed by this circumference. The point A, which, in the formation of the circle is at rest, is called the centre. The right lines AC, AD, AF, &c. drawn from the centre to the circumference, are called radii. A diameter is a right line which passes through the centre, and is terminated both ways by the circumference. The line DAL, for example, is a diameter. An arc is a part of a circumference, as FG. A very fine grain of D B Fig. 223. 6. The circumference of a circle is divided into 360 equal parts, called degrees; each degree is divided into 60 parts, called minutes, and each minute into 60 parts, called seconds. Every circle, without relation to its magnitude, is supposed to be equally divided into degrees, minutes, and seconds. same point, and diverging from each other, form an An angle is commonly 7. Two right lines drawn from the B A Fig. 224. B G R M Fig. 225. 8. The magnitude of an angle does not depend on the lines by which it is formed, but upon their distance from each other. How far soever the lines AB, AC are continued, the angle remains the same. One angle is greater than another when the lines of equal length by which it is formed are more distant. Thus the angle BAL (fig. 223.) is greater than the angle CAB, because the lines AB, AL are more distant from each other or include a greater are than the lines AC, AB. If the legs of a pair of compasses be a little separated, an angle is formed; if they be opened wider, the angle becomes greater; if they be brought nearer, the angle becomes less. 9. If the point of a pair of compasses be applied to the point G (fig. 225.), and a circumference NRB be described, the arc NR contained within the two lines GL, GM will measure the magnitude of the angle LGM. If the arc NR, for example, be an are of 40 degrees, the angle LGM is an angle of 40 degrees. 10. There are three kinds of angles (fig. 226.): a right angle (I), which is an angle of 90 degrees; an obtuse angle (II), which contains more than 90 degrees; and an acute angle (III), which contains less than 90 degrees. II III I 11. One line is perpendicular to another when the two angles it makes with that other line are equal: thus, the line CD (fig. 227.) is perpendicular to the line AB, if Fig. 226. the angles CDA, CDB contain an equal number of degrees. 12. Two lines are parallel when all perpendiculars drawn from one to the other are equal; thus, the lines FG, AB (fig. 228.) are parallel, if all the perpendiculars cd, cd, &c. are equal. C Fdddddddd G 13. A triangle is a surface enclosed by three right lines, called sides (fig. 229.). An equilateral triangle (I) is that which has three sides equal; an isosceles triangle has only two of its sides equal (II); a scalene triangle (III) has its three sides unequal. 14. A quadrilateral figure is a surface enclosed by four right lines, which are called its sides. A B D А сссссссс в Fig. 227. Fig. 228. 15. A parallelogram is a quadrilateral figure, which has its opposite sides parallel; thus, if the side BC (fig. 230.) is parallel to the side AD, and the side AB to the side DC, the quadrilateral figure ABCD is called a parallelogram. 16. A rectangle is a quadrilateral figure all the angles whereof are right angles, as ABCD (fig. 231.). 17. A square is a quadrilateral figure whose sides are all equal and its angles right angles (fig. 232.). 18. A trapezium is any quadrilateral figure not a parallelogram. D A Fig. 231. C B Fig. 232. 19. Those figures are equal which enclose an equal space; thus, a circle and a triangle are equal, if the space included within the circumference of the circle be equal to that contained in the triangle. 20. Those figures are identical which are equal in all their parts; that is, which have all their angles equal and their sides equal, and enclose equal spaces, as BAC, EDG (fig. 233.). It is manifest that two figures are identical which, being placed one upon the other, perfectly coincide, for in that case they must be equal in all their parts. It must be ob- B served, that a line merely so expressed always denotes a right line. AXIOM. Two right lines cannot enclose a space; that category requires at least three lines. RIGHT LINES AND RECTILINEAL FIGURES. 876. PROPOSITION I. The radii of the same circle are all equal. The revolution of the line AD about the point A (fig. 234.) being necessary (Defin. 5.) to form the circle BCDFGLB, when in revolving the point B is upon the point C, the whole line AB must be upon the line AC; otherwise two right lines would enclose a space, which is impossible: wherefore the radius AC is equal to the radius AB. In like manner it may be proved that the radii AB, AF, AG, &c. are all equal to AB, and are therefore equal among themselves. 877. PROP. II. On a given line to describe an equilateral tri. angle. D F Fig. 234. |