r PROBLEM IX. To meafure heights and difiances by the geometrical square. When the plane is horizontal, the inftrument is to be fupported and placed horizontally at any point A, and it is to be turned till the remote point F, whofe diftance is to be meafured, is feen through the fixed fights; then turn the dex, till, through the fights upon it, you see any acceffible object B; then place the inftrument at the point B, directing the fixed fights to the firft ftation A, and the moveable ones to the point F; and if the index cut the reclined fide of the fquare, as in the point E, then, from fimilar triangles, ES: SB :: as BA : AG; but if the index cut the right fide of the square K, it will be BR: RK:: BA: AF. In either of thefe cafes, the diftance required may be found by the rule of three*. Perpendicular heights, when acceffible, may be obtained by the quadrant only. For example, If you wanted the height of a house, tree, &c. approach towards or retire from the object, till it fubtends an angle of 45°; then fhall the height of the object be equal to its horizontal diftance. Euclid, I. 6. A fimilar obfervation may be made of the other inftruments used for heights and distances; but this, and many more, will daily occur in practice. * The fide DE is called the right fide, E the reclined fide. TABLES. CONTAINING, I. A TABLE OF THE LOGARITHMS OF NUMBERS FROM I TO 10000. II. A TABLE OF LOGARITHMIC SINES, TANGENTS, SECANTS, AND VERSED SINES, TO EVERY Degree and MinutE OF THE QUADRANT. III. A TABLE OF LOGARITHMIC SINES, TANGENTS, AND SECANTS, TO EVERY POINT, HALF POINT, AND QUARTER POINT OF THE COMPASS, A TABLE of the LOGARITHMS of NUMBERS from 1 to 10000, Log. N°. Log. Log N°. Log. (N°. Log. 65 1.81291 851.92942 111.04139 311.49136 511.70757 71 1.85126 911.95904 5_1 6 7 9 100 0000000043 00087 00130 00173 00217 00260 00303 00346 00389 101 00432 00475 00518 00561 00604 00647 00689 00732 00775 00817 102 00860 00903 00945 00988 01030 01072 01115 01157 01199 01242 103 01284 01326 01368 01410 01452 01494 01536 01578 01620 C1662 104 01703 01745 01787 01828 01870 01912 01953 01995 02036 02078 105 02119 02160 02202 02243 C2284 02325 02366 02407 02449 02490 106 02531 02572 02612 02653 02694 02735 02776 02816 02857 02898 107 02938 02979 03019 03060 03100 03141 03181 03222 03262 03302 108 03342 03383 03423 03463 03503 03543 03583 03623 03663 03703 109 03743 03782 03822 03862 03902 03941 03981 04021 04060 04100 110 04139 04179 04218 04258 04297 04336 04376 04415 04454 04493 III 4532 04571 04610 04650 04689 04727 04766 04805 04844 04883 112 04922 04961 04999 05038 05077 05115 05154 05192 05231 05269 113 05308 05346 05385 05423 05461 05500 05538 05576 05614 05652 114 05690 05729 05767 05805 05843 05881 05918 05956 05994 06032 115 06070 06108 06145 06183 06221 06258 06296 06333 06371 06408 116 06446 06483 06521 06558 06595 06633 06670 06707 06744 06781 117 06819 06856 06893 0693006967 07004 07041 07078 07115 07151 118 07180 07225 07262 07298 07335 07372 07408 07445 07482 07518 119 07555 07591 07628 07664 07700 07737 07773 07809 07846 07882 120 07918 07954 0799c|| 08027 08063 08099 08135 08171 08207 08243 121 08279 08314 08350 08386 08422 08458 08493 08529 08565 08600 122 08636 08672 08709 08743 08778 08814 08849 08884 08920 08955 123 08991 09026 09061 09096 09132 09167 09202 09237 09272 09307 124 09342 09377 09412 09447 09482 09517 09552 09587 09621 09656 125 09691 09726 09760 09795 09830 09864 09899 09934 09968|| 10003 126 10037 10072 10106 10140 10175 10209 10243 10278 10312 10346 127 10380 10415 10449 10483 10517 10551 10585 10619 10653 10687 128 10721 10755 10789 10823 10856 10890 10924 10958 10992 11025 129 11059 11093 11126 1116 11193 11227 11260 11294 11327 11361 130 11394 11428 11461 11494 11528 11561 11594 11628 11661 11694 131 11727 11760 11793 11826 1186 11893 11926 11959 11992 12024 132 12057 12090 12123 12156| 12189| 12222 12254 12287| 12320 12352 133 12385 12418 12450 12482 12516 12548 12581 12613 12646 12678 134 12710 12743 12775 12808 12840 12872 12905 12937 12969 13001 13513023 13066 13098 13130 13162 13194 13226 13258 13290 13322 136 13354 13386 13418 13450 13481 13513 13545 13577 13609| 13640 137 13672 13704 13735 13767 13799 13830 13862 13893 13925 13956 138 13988 14019 14051 14082 14114 14145 14176 14208 14239 14270 139 14301 14333 14364 14395 14426 14457 14489 14520 14551 14582 14C 1461314644 14675 14706 14737 14768 14799| 14829| 14850 14891| 141 14922 14053| 14983| 15014 15045 15076| 15106|| 15137 15168| 15198| 142 15220 15259 15290 15320 15351 15381 15412 15442 15473 15503 143 15534 15564 5594 15625 15655 15685 15715 15746 15776 15806 144 15836 15866 15897 15927 15957 15987 16017 16047 16077 16107 145 16137 16167 16197 1622 16256 16286 16316 16346 16376 16406 146 16435 16465 16495 16524 16554 16584 16613 16643 16673 16702 147 16732 16761 16791 16820 1685 16879 1690 16938 16967 16997 14 17026 17056' 17085 17114 17143 17173 17202 17231 17200 17289 149 17319 17348 17377 17406 17435 17464 17493 17522 17551 17580 15 17609 17638 17667 17696 17725 17754 17782 17811 17840 17869 151 17898 17926 17955 17984 18013 18041 18070 18099 18127 18156 152 18184 18213 18241 18270 18298 18327 18355 18384 18412 18441 153 18469 18498 1852 18554 18583 18611 18639 18667 18696 18724 154 18752 18780 18808 18837 18865 18893 18921 18949 18977 19005 155 19033 19061 19089 19117 19145 19173 19201 19229 19257 19285 156 19312 19340 19368 19396 19424 19451 19479 19507 19535 19562 157 19590 19618 9645 19673 19700 19728 19756 19783 19811 19838 152 19866 19893 19921 19948 19976 20003 2003 20058 20085 20112 159 20140 20167 20194 20222 20249 20276 20303 20330 20358 20385 I 2 3 4 5 6 7 8 9 1 N°. 160 20412 20439 20466 20493 20520 20547 20575 20602 20629 20656 161 20683 20710 20736 20763 20790 20817 20844 20871 20898 20925 162 20951 20978 21005 21032 21059 21085 21112 21139 21165 21192 163 21219 21245 21272 21299 21325 21352 21378 21405 21431 21458 164 21484 21511 21537 21564 21590 21617 21643 21669 21696 21722 165 21748 21775 21801 21827 21854 21880 21906 21932 21958 21985 166 22011 22037 22063 22089 22115 22141 22167, 22194 22220 22246 167 22272 22298 22324 22350 22376 22401 22427 22453 22479 22505 168 22531 22557 22583 22608 22634 22660 22686 22712 22737 22763 169 22789 22814 22840 22866 22891 22917 22943 22968 22994 23019 170 23045 23070 23096 23121 23147 23172 23198 23223 23249 23274 171 23300 23325 23350 23376 23401 23426 23452 23477 23502 23528 172 23553 23578 23603 23629 23654 23679 23704 23729 23754 23779 173 23805 23830 23855 23880 23905 23930 23955 23980 24005 24030 174 24055 24080 24105 24130 24155 24180 24204, 24229 24254 24279 175 24304 24329 24353 24378 24403 24428 24452 24477 24502 24527 176 24551 24576 24601 24625 24650 24674 24699 24724 24748 24773 177 24797 24822 24846 24871 24895 24920 24944 24969 24993 25018 178 25042 25066 25091 25115| 25139 25104 25188 25212 25237 25201 179 25285 25310 25334 25358 25382 25406 25431 25455 25479 25503|| 180 25527 25551 25575 25600 25624 25648 25672 25696 25720 25744 181 25768 25792 25816 25840 25864 25888 25912 25935 25959 25983 182 26007 26031 26055 26079 26102 26126 26150 26174 20198 26221 183 26245 26269 26293 26316 26340 26364 26387 26411 26435 26458 184 26482 26505 20529 26553 26576 26600 26623 2664 26670 26694 185 26717 26741 26764 26788 26811 26834 26858 26881 26905 26928 186 26951 26975 26998 27021 27045 27068 27091 27114 27138 27161 187 27184 27207 27231 27254 27277 27300 27323 27346 27370 27393 188 27416 27439 27462 27485 27508 27531 27551 27577 27600 27623 189 27646 27669 27692 27715 27738 27761 27784 27807 27830 27852 19 27875 27898 27921 27944 27967 27989 28012 28035 28058 280811 191 28103 28126 28149 28171 28194 28217 2824 28262 28285 28307 192 28330 28353 28375 28398 28421 28443 28466 28488 28511 28533 193 28556 28578 286c1 28623 28646 28668 28691 28713 28735 28758 194 28780 28803 28825 28847 28870 28892 28914 28937 28959 28981 195 29003 29026 29048 29070 29092 29115 29137 29159 29181 29203 190 29226 29248 29270 29292 29314 29336 29358 29380 29403 29425 197 29447 29460 29491 29513 29535 29557 29579 29601 29623 29645 198 29667 29688 29710 29732 29754 29776 29798 2982c 29842 29863 199 29885 29907 29929 29951 29973 29994 30016 30038 30060 30081 200 30103 30125 30140 30168 30190 30211 30233 30255 30276 30298|| 201 30320 30341 30363 30384 30406 30428 30449 3047 30492 30514 202 30535 30557 30578 30000 30621 30642 30064 30685 30707 307281 203 30750 30771 30792 30814 30835 30856 30878 30899 30920 30942 204 30963 30984 31606 31027 31048 31069 31091 31112 31133 31154 205 31175 31197 31218 31239 31260 31281 31302 31323 31345 31366 206 31387 31408 31429 31450 31471 31492 31513 31534 31555 31576 207 31597 31618 31639 31660 31681 31702 31723 31744 31705 31785 208 31806 31827 31848 31869 31890 31911 31931 31952 31973 31994 209 32015 32035 32056 32077 32098 32118 32139 32160 32181 32201 210 32222 32243 32263 32284 32305 32325 32346 32366 32387 32408 211 32428 32449 32469 32490 32510 32531 32552 32572 32543 32613 212 32634 32654 32675 32695 32715 32736 32756 32777 32797 32818 213 32838 32858 32879 32899 32919 32940 32960 32980 33001 33021 214 33041 33062 33082 33102 33122 33143 33163 33185 33203 33224 215 33244 33264 33284 33304 33325 33345 33365 33385 33405 33425 216 33445 33465 33486 33506 33526 33546 33566 33586 33606 33626 217 33646 33566 33686 337c6 33726 33746 33766 33786 33806 33826 218 33846 33866 33885 33905 33925 33945 33965 33985 34005 34025 219 34044 34064 34084 34104 34124 34143 34163 34183 34203 34223 220 34242 34262 34282|34301 34321 34341 34361 34380, 34400 34420 221 34439 34459 34479 34498 34518 34537 34557 34577 34596 34616 222 34635 34655 34674 34694 34713 34733 34753 34772 34792 34811 223 34830 34850 34869 34889 34908 34928 34947 34967 34986 35005 224 35025 35044 35064 35083 35102 35122 3514 35160 35180 35199 225 35218 35238 35257 35276 35295 35315 35334 35353 35372 35392 226 35411 35430 35449 35468 35488 35507 35526 35545| 35564 35583| 227 35603 35622 35641 35660 35679 35798 35717 35736 35755 35774 228 35703 35813 35832 35851 35870 3589 35908 35927 35946 35965 229 35984 36003 36021 36640 36059 36078 36097 36116 36135 36154 230 36173 36192 36211 36229 36248 36267 36286 36305 36324 36342 231 35361 3638 36399 36418 6436 36455 36474 36493 36511 36530 232 36549 36568 36586 36605 36624 36642 36661 36680 36698 36717 233 36736 36754 36773 36791 35810 36829 36847 36866 36884 36903 234 36922 36940 36959 36977 36996 37014 37033 37051 37070 37088 235 37107 37125 37144 37162 37181 37199 37218 37236 37254 37273 236 37291 3731 37328 37346 37365 37383 37401 37420 37438 37457 237 37475 37493 37511 37530 37548 37566 37585 37603 37621 37639 238 37658 37676 37694 37712 37731 37749 37767 37785 37803 37822 239 37840 37858 37876 37894 37912 37931 37949 37967 37985 38003 240 38021 38039 38057 38075 38093 38112 38130 38148 38166 38184 241 38202 38220 38238 38256 38274 38392 38310 38328 38346 38364| 242 38382 3839 38417 38435 38453 38471 38489 38507 38525 38545 243 38561 38578 38596 38614 38632 38650 38662 38686 38703 38721 244 38729 38757 38775 38792 38810 33828 38846 38863 38881 38899 245 3891738934 3852 3897 38987 39005 39023 39041 39058 39076 246 39094 391 39129 39146 39164 39182 39199 39217 39235 39252 247 39270 39287 39305 39322 39340 39358 39375 39393 39410 39428 248 39445 39463 39480 39198 39515 39533 39550 39568 39585 39602 249 39620 39637 39655 39672 39690 39707 39724 39742 39759 39777 250 39794 39811| 39829 49846 39863|39881| 39898 39915 39933 39950 251 39967 3998540002 40019 40037 40054 40071 40088 40106 40123 252 4014040157 40175 40192 40209 40226-40243 40261 40278 40295 253 40312 40329 40346 40364 40331 40398 40415 40432 40449 40466 1254 40483 40500 40518 40525 40552 40569 40586,40603 40620 40637 255 40654 40671 40688 40705 40722 40739 40756 40773 40790 40807 25640824 40841 40858 40875 40892 40909| 40926| 40943 40959 40976 252 40993 41010 41027 41044 41061 41076 41095 41111 41128 41145 258 41162 41179 41196 41212 41229 41246 41263 41280 41296 41313 259 41330 415 41363 41380 413941414 41430 41447 41464 41481 26c 43497 41544 41531 41547 41564 41581 41597 41614 41631 41647| 261 41664 41681 41607 41714 41731 +1747 41764 41780 41797 41814 262 418 4K47 41863 4183c 418,6 41912 41929 41940 41963 41979 263 41996 42012 42029 42045 42062 42078 42095 42111 42127 42144 1264 42160 42177 42193 42210 42226 42243 42259 42275 42292 42308 265 42325 4234142357 42374 42390 42406 42423 42439 42455 42472 260 42488 42504 42521 42537 42553 42570 42586 42602 42619 42635 26-4265142667 42684 42700 42716 42732 42749 42765 42781 42797 268 4281342830 42846 42862 42878 42894 42911 42927 42943 42959 269 42975 42991 43008 43024 43040 43056 43072 43088 43104 43120 270 43136 43152 43109 43185 43201 43217 43233 43249 43265 43281 271 43297 43313 43329 43345 43361 43377 43393 43409 43425 43441 272 43457 43473 43489 43505 43521 43537 43553 43569 43584 43600 273 43616 43632 43648 43664 43685 43796 43712 43727 43743 43759 274 43775 43791 43807 43823 43838 43854 43870 43886 43902 43917 275 43933, 43949 43965 43981 43996 44012 44028 44044 44059 44075 276 44091 44107 44122 44138 44154 44170 44185 44201 44217 44232 277 44248 44264 44279 44295 44311 44326 44342 44358 44373 4438 278 44404 44420 44436 44451 44467 44583 44498 44514 44529 44545 279 44560 44576 44592 44607 44623 44638 44654 44669 44685 44700 NO 8 9 |