together those of each denomination beginning on the right, and divide by 12, to carry to the next higher place, then add these, and so on, as often as there are places in the whole product. Example. Multiply 262 ft. 5 in. by 54 ft. 8 in. Thus, under inches, the products being set down and added, they amount to 2569, which, divided by twelve, gives 197 to carry to the place of feet, and 5 remainder. Then adding the feet together with the quantity carried, it gives the whole number of feet; while the operation is extremely simple and free from the troubles of either side operations or useless stress on the memory. 872. The division of the foot into 12 parts renders the application of the rules of practice very valuable in the computation of duodecimals. The practical rule is to set down the two dimensions one under the other, that is, feet under feet and inches under inches, and multiply each term in the multiplicand by the feet in the multiplier, beginning at the lowest; and, if the numbers be large, put down the inches without carrying 1 for every 12 from inches to feet. Then, instead of multiplying by the inches, take such aliquot parts of the multiplicand as the inches are of a foot; after which add the lines together, carrying 1 for every 12 inches. Example 1.-Multiply 7 ft. 5 in. by 3 ft. 4 in. The same examples have been used to show the relative advantage of the two methods. 873. The abridgment of the labours of practical men is always a matter of importar.ce being identical with the saving of time which is lost in calculation, and which with the architect is of the utmost importance, when it is recollected what multifarious duties he has to discharge. Hence we doubt not that the following table of squares, cubes, and roots of numbers, up to 1000, will be most acceptable to him. An inspection of the table will at once instruct him in the method of using it. The first column shows the number, the second the square of such number, the third exhibits its cube. In the fourth column is found the square root of the number, and in the fifth its cube root. Thus, looking to the number 61 in the first coluinn, we find its square to be 3721, its cube 226981, its square root 7-8102497, and its cube root 3.936497. Again, taking the number 784, we find its square to be 613089, its cube 481890304, its square root 28, and its cube root 9-220872. We presume that we need not further enlarge on instructions on its use. 31 961 32 1024 33 1089 34 1156 35 1225 36 1296 37 1369 38 1444 39 1521 29791 5-5677644 3.141381 94 32768 5.6568542 3.174802| 95 9025 35937 5.7445626 3.207534 96 9216 39304 5.8309519 3·239612 97 9409 42875 5.9160798 3.271066 98 9604 46656 6.0 3.301927 99 9801 50653 60827625 3.332222 100 10000 54872 6.1644140 3.361975 101 10201 59319 6.2449980 3.391211 102 10404| 40 1600 64000 6.3245553 3.419952 103 10609 41 1681 68921 6.4031242 3·448217 104 10816 42 1764 74088 6.4807407 3·476027 105 11025 43 1849 79507 6-5574385 3.503398 106 11236 44 1936 85184 6-6332496 3·530348 107 11449 45 2025 91125 6.7082039 3·556893 108 11664 46 2116 97336 6.7823300 3·583048 109 11881 47 2209 103823 6.8556546 3.608826 110 12100 48 2304 110592 6.9282032 3.634241 111 12321 49 2401 117649 7.0 3.659306 112 12544 50 2500 125000 7-0710678 3.684031 113 12769 51 2601 132651 7.1414284 3.708430 114 12996 52 2704 140608 7.2111026 3.732511 115 13225 53 2809 148877 7-2801099 3.756286 116 13456 54 2916 157464 7.3484692 3.779763 117 13689 55 3025 166375 7-4161985 3.802953 118 13924 56 3136 175616 7.4833148 3.825862 119 14161 57 3249 185193 7.5498344 3.848501 120 14400 58 3364 195112 7.6157731 3.870877 121 14641 59 3481 205379 7-6811457 3 892996 122 14884 60 3600 216000 7.7459667 3.914867 123 15129 61 3721 226981 7.8102497 3.936497 124 15376 238328 7.8740079 3·957892 125 15625 250047 7.9372539 3 979057 126 15876 88588 62 3844 63 3969 1030301 10 0498756 4 657010 1061208 10 0995049 4 672330 1092727 10 14889164 687548 1124864 10.1980390 4.702669 1157625 10.2469508 4.717694 1191016 10.2956301 4.732624 1225043 10 3440804 4.747459 1259712 10.3923048 4.762203 1295029 10-4403065 4.776856 1331000 10 4880885 4.791420 1367631 10.5356538 4.805896 1404928 10.5830052 4.820284 1442897 10 6301458 4.834588 1481544 10.6770783 4.848808 1520875 10 7238053 4.862944 1560896 10 7703296 4.876999 1601613 10.8166538 4.890973 1643032 10.8627805 4.904868, 1685159 10.9087121 4.918685 1728000 10.9544512 4.932424 1771561 11.0 4.946088 1815848 11 0453610 4.959675 1860867 11 0905365 4.9731 90 1906624 11 1355287 4.986631 1953125 11.1803399 5.0 2000376 112249722 5 013298 8836 551368 9.0553851 4.344481 571787 91104336 4·362071 592704 9.1651514 4.379519 614125 9.2195445 4.596830 636056 9-2736185 4 414005 658503 9.3273791 4-431047 681472 9.3808315 4·447960 704969 94339811 4-464745 729000 9.4868330 4.481405 753571 9.5393920 4.497942 778688 9.5916630 4.514357 804357 9.6436508 4.530655 830584 9.6953597 4.546836| 857375 9.7467943 4.562903| 884736 9.7979590 4.578857 912673 9.8488578 4.594701 941192 9.8994949 4·610436 970299 9.9498744 4.626065| 1000000 10.0 4-641589 127 16129 128 16384 129 16641 130 16900 131 17161 132 17424] 133 17689 134 17956 135 18225 136 18496 137 18769 138 19044 139 19321 140 19600 141 19881 142 20164 143 20449 144 20736 145 21025 146 21316 147 21609 148 21904 149 22201 150 22500 151 22801 152 23104 153 23409 154 23716 155 24025 156 24336 157 24649 158 24964 159 25281 160 25600 161 25921 162 26244 163 26569 164 26896 165 27225 166 27556 167 27889 168 28224 169 28561 170 28900 171 29241 172 29584 173 29929 174 30276 175 30625 176 30976 177 31329| 178 31684 179 32041 180 32400 181 32761 182 33124 183 33489 184 33856 185 34225 186 34596 187 34969 188 35344 189 35721 Square Root. Cube Root. No. Square. 2048383 11 2694277 5.026526 190 36100 2097152 11 3137085 5.039684 191 36481 2146689 11.3578167 5·052774 192 36864 2197000 11-4017543 5 065797 | 193 37249 2248091 11·4455231 5 078753 194 37636 2299968 11.4891253 5 091643 195 38025 2352637 11 5325626 5.104469 196 38416 2406104 11 5758369 5·117230 197 38809 2460375 11·6189500 5·129928 198 39204 2515456 11 6619038 5·142563 199 39601 2571353 11·7046999 5·155137 200 40000 2628072 11 7473444 5.167649 201 40401 2685619 11·7898261 5.180101 202 40804 2744000 11-8321596 5·192494 203 41209 2803221 11·8743421 5.204828 204 41616 2863288 11.9163753 5.217103 205 42025 2924207 11 9582607 5.229321 206 42436 2985984 12.0 5.241482 207 42849 3048625 12.0415946 5-253588 208 43264 3112136 12.0830460 5.265637 209 43681 3176523 12 1243557 5.277632 210 44100 3241792 12·1655251 5 289572 211 44521 3307949 12 2065556 5.301459 212 44944 3375000 12.2474487 5·313293 213 45369 3442951 12.2882057 5 325074 214 45796 3511808 12.3288280 5.336803 215 46225 3581577 12-3693169 5348481 216 46656 3652264 12 4096736 5·360108 217 47089 3723875 12-4498996 5 371685 218 47524 3796416 12·4899960 5.383213 219 47961 3869893 12.5299641 5.394690 220 48400 3944312 12.5698051 5.406120 221 48841 4019679 12·6095202 5·417501 222 49284 4096000 12.6491106 5.428835 223 49729 4173281 12.6885775 5.4401 22 224 50176 4251528 12-7279221 5·451362 225 50625 4330747 12-7671453 5.462556 226 51076 4410944 12.8062485 5.473703 227 51529 4492125 12.8452326 5.484806 228 51984 4574296 12.8840987 5.495865 229 52441 4657463 12.9228480 5·506879 230 52900 4741632 12·9614814 5.517848 231 53361 4826809 13.0 5.528775 232 53824 4913000 13 0384048 5.539658 233 54289 5000211 13-0766968 5.550499 234 54756 5088-448 13.1148770 5.561298 235 55225 5177717 13-1529464 5.572054 236 55696 5268024 13·1909060 5 582770 237 56169 5359375 13.2287566 5 593445 238 56644 5451776 13.2664992 5·604079 239 57121 5545233 13.3041347 5 614673 240 57600 5639752 13.3416641 5.625226 241 58081 5735329 13 3790882 5.635741 242 58564 5832000 13 4164079 5646216 243 59049 5929741 13 4536240 5.656652 244 59536 6028568 13.4907376 5.667051 245 60025 6128487 13·5277493 5.677411 246 60516 6229504 13.5646600 5 687734 247 61009 6331625 13 6014705 5.698019 248 61504 6434856 13.6381817 5.708267 249 62001 6539203 13-6747943 5·718479 250 62500 6644672 13.7113092 5·728654 251 63001 6751269 13.7477271 5.738794|252 63504 5.808786 6859000 13.7840488 5.748897 6967871 13·8202750 5 758965 7077888 13.8564065 5.768998 7189057 13 8924440 5.778996 7301384 13 9283883 5 788960 7414875 13 9642400 5·798890 7529536 14.0 7645373 14 0356688 5.818648 7762392 14.0712473 5.828476 7880599 14.1067360 5.838272 8000000 14 1421356 5.848035 8120601 14.1774469 5.857765 8242408 14.2126704 5.867464 8365427 14.2478068 5.877130 8489664 14.2828569 5·886765 8615125 14.3178211 5.896368 874181614-3527001 5·905941 8869743 14.3874946 5.915481 899891214-4222051 5·924991 9123329 14:4568323 5 934473 9261000 14.4913767 5.943911 9393931 14 5258390 5·953341 9528128 14.5602198 5.962731 9663597 14 5945195 5.972091 9800344 14.6287388 5.981426 9938375 14 6628783 5·990727 10077696 14 6969385 6.0 10218313 14-73091996 009244 10360232 14.7648231 6 018363 10503459 14 7986486 6 027650 10648000 14.8323970 6 036811 10793861 14.8660687 6·045943 10941048 14-8996644 6 055048 11089567 14 9331845 6 064126 11239424 14.9666295 6 073177 11390625 15.0 6.082201 11543176 15 0332964 6 091199 11697083 15 0665192 6·100170 11852352 15 0996689 6·109115 12008989 15·1327460 6·118032 12167000 15 1657509 6.126925 12326391 15.1986842 6 135792 12487168 15 2315462 6·114634. 12649337 15.2643375 6 153449 12812904 15.2970585 6 162239 12977875 15:3297097 6·171005 13144256 15 3622915 6.179747 13312053 15.3948043 6·188463 13481272 15.4272486 6.197154 13651919 15 4596248 6·205821 13824000 15 4919334 6 214464 13997521 15.5241747 6 223083 14172488 15.5563492 6 231678 14348907 15.5884573 6.240251 14526784 15.6204994 6.248800 14706125 15 6524758 6.257324 14886936 15-6843871 6 265826 15069223 15-7162336 6 274304 15252992 15.7480157 6.282760 15438249 15.7797338 6 291194 15625000 15.8113883 6.299604 15813251 15.8429795 6 307992 16003008 15.8745079 6.316359) No. Square. Cube. Square Root. Cube Root. No. Square. Cube. Square Root. Cube Root. 31554496 17·7763888 6.811284 31855013 17 8044938 6.818461 32157432 17.8325545 6·825624 32461759 17.8605711 6.832771 32768000 17·8885438 6.839903 33076161 17·9164729 6·847021 33386248 17·9443584 6.854124 33698267 17.9722008 6.861211 34012224 18.0 6.868284 34328125 18 0277564 6.875343 34645976 18 0554701 6·882388 34965783 18-0831 413 6 889419 35287552 18.1107703 6·896435 35611289 18.1383571 6.903436 35937000 18·1659021 6.910423 36264691 18.1934054 6.917396 36594368 18 2208672 6 924355 36926037 18 2482876 6.931300 37259704 18 2756669 6 938232 37595375 18-3030052 6 945149 37933056 18 3305028 6 952053 38272753 18.3575598 6·958943 38614472 18-3847763 6 965819 38958219 18 4119526 6.972682 39304000 18-4390889 6·979532 39651821 18:4661853 6.986369 40001688 18.4932420 6 993191 40353607 18.5202592 70 40707584 18-5472370 7006796 253 64009 16194277 15·9059737 6-324704 316 99856 254 64516 16387064 15.9373775 6 333025 317 100489 255 65025 16581375 15.9687194 6 341325 318 101124 256 65536 16777216 16.0 6.349602 319 101761 257 66049 16974593 16·0312195 6 357859 320 102400 258 66564 17173512 16 0623784 6.366095 321 103041 259 67081 17373979 16 0934769 6.374310 322 103684 260 67600 17576000 16.1245155 6-382504 323 104329 261 68121 17779581 16·1554944 6·390676 324 104976 262 68644 17984728 16 1864141 6.398827 325 105625 263 69169 18191447 16.2172747 6·406958 326 106276 264 69696 18399744 16 2480768 6 415068 327 106929 265 70225 18609625 16.2788206 6·423157 328 107584 266 70756 18821096 16.3095064 6 431226 329 108241 267 71289 19034163 16 3401346 6 439275 330 108900 268 71824 19248832 16 3707055 6.447305|331 109561 269 72361 19465109 16 4012195 6·455314 332 110224 270 72900 19683000 16 4316767 6.463304 333 110889 271 73441 19902511 16:4620776 6-471274 334 111556 272 73984 20123648 16.4924225 6·479224 335 112225 273 74529 20346417 16-5227116 6 487153 336 112896 274 75076 20570824 16 5529454 6.495064 337 113569 275 75625 20796875 16.5831240 6 502956 338 114244 276 76176 21024576 16-6132477 6.510829 339 114921 277 76729 21253933 16.6433170 6.518684 340 115600 278 77284 21484952 16.6733320 6.526519 341 116281 279 77841 21717639 16-7032931 6:534335 342 116964 280 78400 21952000 16 7332005 6.542132 343 117649 281 78961 22188041 16 7630546 6·549911 344 118336 282 79524 22425768 16·7928556 6-557672 345 119025 | 41063625 18:5741756 7·013579 283 80089 22665187 16-8226038 6.565415 346 119716 284 80656 22906304 16·8522995 6.573139 347 120409 285 81225 23149125 16.88194306.580844 348 121104 286 81796 23393656 16.9115345 6.588531 349 121801 287 82369 23639903 16.9410743 6.596202 350 122500 288 82944 23887872 16·9705627 6·603854 351 123201 289 83521 24137569 170 6.611488 352 123904 290 84100 24389000 17-0293864 6.619106 353 124609 291 84681 24642171 17-0587221 6·626705 354 125316 292 85264 24897088 17 0880075 6·634287 355 126025 293 85849 25153757 17.1172428 6·641851 356 126736 294 86436 25412184 17.1464282 6 649399 357 127449 295 87025 25672375 17 1755640 6 656930 358 128164 296 87616 25934336 17·2046505 6·664443 359 128881 297 88209 26198073 17.2336879 6.671940 360 129600 298 88804 26463592 17-2626765 6·679419 361 130321 299 89401 26730899 17-2916165 6 686882 362 131044 300 90000 27000000 17-3205081 6·694328 363 131769 301 90601 27270901 17.3493516 6.701758 364 132496 302 91204 27543608 17.3781472 6.709172 365 133225 303 91809 27818127 17-4068952 6·716569 366 133956 304 92416 28094464 17:4355958 6·723950 367 134689 305 93025 28372625 17·4642492 6·731316 368 135424 306 93636 28652616 17-4928557 6.738665 369 136161 307 94249 28934443 17.5214155 6·745997 370 136900 308 94864 29218112 17-5499288 6·753313 371 137641 309 95481 29503629 17.5783958 6·760614 372 138384 310 96100 29791000 17-6068169 6.767899 373 139129 311 96721 30080231 17 6351921 6·775168 374 139876 312 97344 30371328 17·6635217 6·782422 375 140625 313 97969 30664297 17·6918060 6-789661 376 141375 314 98596 30959144 17 7200451 6.796884 377 142129 315 99225 31255875 17·7482393 6·804091 378 142884 7.120367 41421736 18 6010752 7020349 401|160801 86350888 21 0237960 7.617411 86938307 21 0475652 7·623151 87528384 21 0713075 7·628883 88121125 21 0950231 7·634606| 88716536 21 1187121 7640321 89314623 21·1423745 7·646027 89915392 21·1660105 7.651725 90518849 21·1896201 7·657414 91125000 21.2132034 7·663094 91733851 212367606 7.668766 92345408 21 2602916 7·674430 92959677 21 2837967 7·680085 93576664 21 3072758 7·685732 94196375 21·3307290 7.691371 94818816 21 3541565 7.697002 95443993 21 3775583 7702624 96071912 21·4009346 7.708238 96702579 21 4242853 7713844 97336000 21·4476166 7.719442 97972181 21 4709106 7.725032 98611128 214941853 7 780614 99252847 21.5174348 7·736187 99897344 21 5406592 7.741753 379 143641 54439939 19:4679223 7-236797 442 195364 380 144400 54872000 19 4935887 7-243156 443 196249 381 145161 55306341 19.5192213 7 249504 444 197136 382 145924 55742968 19 5448203 7.255841 445 198025 383 146689 56181887 19.5703858 7.262167 446 198916, 384 147456 56623104 19 5959179 7.268482 447 199809 385148225 5706662519-6214169 7.274786 448 200704 386 148996 57512456 19 6468827 7-281079 449 201601 387 149769 57960603 19 6723156 7·287362 450 202500 388150544 5841107219.6977156 7.293633 451 203401 389 151321 58863869 19-7230829 7.299893 452 204304 390 152100 59319000 19-7484177 7 306143 453 205209 391 152881 59776471 19-7737199 7 312383 454 206116 392 153664 60236288 19.7989899 7.318611 455 207025 393 154449 60698457 19 8242276 7.324829 456 207936 394 155236 6116298419-84943327 331037 457 208849 395 156025 61629875 19.8746069 7.337234 458 209764 396 156816 62099136 19.89974877-343420 459 210681 397 157609 62570773 19·9248588 7349596 460 211600 398 158404 63044792 19′9499373 7355762 461 212521 399 159201 63521199 19.9749844 7.361917462 213444 400 160000 64000000 20.0 7.368063 463 214369 64481201 20-0249844 7.374198 464 215296 402 161604 64964808 20-0499377 7.380322 465 216225 100544625 21 5638587 7.747310 403 162409 65450827 20 0748599 7.386437 466 217156 101194696 215870331 7·752860 404 163216 65939264 20:0997512 7 392542 467 218089 101847563 21-6101828 7.758402 405 164025|| 66430125 20·1246118 7.398636 468 219024|102503232 21·6333077 7 763936 406 164836 66923416 20 1494417 7·404720 469 219961 103161709 21 6564078 7769462 407 165649 67419143 20 1742410 7 410794 470 220900 103823000 21·6794834 7774980 408166464 67917312 20.1990099 7-416859 471 221841 104487111 21 7025344 7·780490 409,167281 68417929 20-2237484 7-422914 472 222784 105154048 21 7255610 7·785992 410 168100 68921000 20 2484567 7-428958 473 223729 105823817 217485632 7.791487 411 168921 69426531 20 2731349 7-434993 474 224676 106496424 21 7715411 7 796974 412169744 69934528 20:2977831 7.441018 475 225625 107171875 21-7944947 7·802453 413 170569 70444997 20 3224014 7·447033 476 226576 107850176 21 8174242 7·807925 414 171396 70951944 20 3469899 7·453039 477 227529 108531333 21 8403297 7·813389 415172225 71473375 20.3715488 7.459036 478 228484 109215352 21 8632111 7 818845 416 173056 71991296 20.3960781 7-465022 479 229441 109902239 21·8860686 7·824294 417 173889 72511713 20.4205779 7-470999 480 230400 110592000 21.9089023 7-829735 418 174724 73034632 20-4450483 7-476966 481 231361 111284641 21.9317122 7-835168 419 175561 73560059 20 46948957-482924 482 232324 111980168 21 9544984 7·840594 420 176400 74088000 20 4939015 7-488872 483 233289 112678587 21 9772610 7·846013 421177241 74618461 20 5182845 7·494810 484 234256 113379904 22.0 7.851424 422 178084 75151448 20.5426386 7.500740 485 235225 114084125 22 0227155 7-856828 423 178929 75686967 20.5669638 7.506660 486 236196 114791256 22 0454077 7·862224 424 179776 76225024 20.5912603 7.512571 487 237169 115501303 22 0680765 7.867613 425 180625 76765625 20 6155281 7.518473 488 238144 116214272 22-0907220 7·872994 426181476 77208776 20-6397674 7.524365 489 239121 116930169 22 1133444 7·878568 427 182329 77854483 20-6639783 7.530248 490 240100 117649000 22-1359436 7·883734 428 183184 78402752 20·6881609 7·536121 491 241081 118370771 22-15851987-889094 429 184041 78953589 20-7123152 7.541986 492 242064 119095488 22:1810730 7·894446 430 184900 79507000 20 7364414 7 547841 492 243049 119823157 22:2036033 7·899791 431 185761 80062991 20-7605395 7.553688 494 244036 120553784 22.2261108 7·905129 432 186624 80621568 20 7846097 7 559525 495 245025 121287375 22.2485955 7·910460 433 187489 81182737 20 8086520 7·565353 496 246016|122023936 22-2710575 7·915784 434 188356 81746504 20.8326667 7.571173 497 247009 122763473 22.2934968 7-921100 435 189225 82312875 20.8566536 7.576984 498 248004 123505992 22.3159136 7·926408 436 190096 82881856 20 8806130 7 582786 499 249001 124251499 22 3383079 7.931710 437 190969 83453453 20.9045450 7 588579 500 250000 125000000 223606798 7·937005 438 191844 84027672 20 9284495 7 594363 501 251001 125751501 22.3830293 7·942293 439 192721 84604519 20·9523268 7·600138 502 252004 126506008 22.4053565 7.947573 440193600 85184000 20 9761770 7 605905 503 253009 127263527 22 4276615 7·952847 441 194481 85766121 21.0 7-611662 504 254016 128024064 22·4499443 7.958114 |