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PROBLEM IX.

To meafure heights and distances by the geometrical square.

When the plane is horizontal, the instrument is to be fupported and placed horizontally at any point A, and it is to be turned till the remote point F, whose distance is to be measured, is seen through the fixed sights; then turn the dex, till, through the fights upon it, you see any accessible object B; then piace the instrument at the point B, directing the fixed fights to the first station A, and the moveable ones to the point 1 ; and if the index cut the reclined side of the square, as in the point E, then, from similar triangles, ES : SB :: as BA : AG; Lut if the index cut the right side of the square K, it will be BR:RK ::BA: AF. In either of these cases, the distance required may be found by the rule of three *.

Perpendicular heights, when accessible, 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 subtends an angle of 45° ; then shall the height of the object be equal to its horizontal dittance. Euclid, I. 6.

A fimilar observation may be made of the other instruments used for heights and distances; but this, and many more, will daily occur in practice.

The side DE is called the right side, E the reclined Gde.

TABLES.

CONTAINING,

1

I. A TABLE OF THE LOGARITHMS OF NUMBERS

FROM I TO 10000.

II. A TABLE OF LOGARITHMIC SINES, TAN.

GENTS, SECANTS, AND VERSED SINES,
TO EVERY DEGREE AND MINUTE OF THE QUA-
DRANT.

III. A TABLE OF LOGARITHMIC SINES, TAN

GENTS, AND SECANTS, TO EVERY POINT,
HALF POINT, AND QUARTER POINT OF THE
COMPASS

A TABLE of the LOGARITHMS of NUMBERS from 1 to 10000,

:

¡N!!! Log. No Log. Log No. Log. No. Log.

I c.00000 211.32222 4111.612781 61 1.78533 811.90848 210.30103 221.34242 42 1.62325 621.79239 82 1.91381 30.47712 23 1.36173) 431.63347 63 1.79934 831.91908

0.60206241.38021 44 1.64345 641.80618 841.92428 5 0.69897 25 1.39794 45 1.65321 651.81291_851.92942 60.77855 261.41497 46 1.06276 66 1.81954 801.93450 70.8451C 271.43136 47 1.67220 67 1.82607 87 1.93952 80.90309 281.44786 481.68124 68 1.83251 881.94448 90.95424 29 1.46240 49 1.69020 69 1.83885 891.94939 101.00poo 301.47712 50 1.69897 701.84510_901.95424) II 1.04139 311.49130 S11.70757 711.85120 911.95904 121.07918 321.50sis 521.71600) 7211.85733 921.96379 131.11394 331.51851 531.72428 7311.86332 93 1.96848 141.14613 341.53148 541.73239 74 1.86923 94.973:3 151.17609 35/1.54407 551.74036_251.87506 2:1.97772 16 1.20412 3617.55636 56 1.74819 761.88081| 901.98227 171.2304,5 37 1.56820 3711.75587| 77 1.88649 91.98677 181.255271 38|1.5797581.76343 786.89204 9811.99123 191.27875 391.5910 591.79085 79 1.89763 991.99563 2011.30103| 40|1.60200 601.778141 811.903071102.00000

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100 0000 00043 00087 00130 00173 00217 00260 00303 00346 00389
101 00432 00475 00518 00561 00604 00647 00689 00732 00775 00819
102 00860 00903 00945 00988 01030 01072 01115 01157 01199) 01242
103 01 284, 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 02284 02325 02366 02407 024491 02490
100 02531 02572 02612 02653 02094 02735 02776 02816 02857 02898
107 02938) 02979 03019 03060 03100 03141 03181 03222 03262 03302
108 03342 03383 03423 03463 0350303543 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
111-453204571 0461004650 04689 04727 04766 04805 04844 04883
112 0492204961 04999 05038 05077 05115 05154 05192 05231 05269
113 05 308 05346 05385 05423 05461 05500 05538 05576 05614 03652
114 05690 05729 057671058057 05843 05881 0591805956 05994 00032
115 0607000108 06145 061831 06221 06258, 0629600333 06371 06408
ist 06446 06483 06521 06558 6595 6633' 06670 06707 06744 06781
117 06819 06850|06893 06930 06.367 07004' 07041 07078 07113 07131
118 07180 07225 072621 07298 07335 07372 074081 07445 07482 07518
119 07555 07591 07628 07664 07700 07737 07773 07809 0784007882
120 07918 07954 07990 08027 08063 08099, 08:35 08171 08207 08243
121 08279 08314 083501 0838608422 458, 08493 08529 08565 08600
122 08636) 08672 08709 08743 08778 08814 08849 08884 08920 08955
123 08991 090261 09061 09096 09132 09167 0,202 09237 09272 09307
124 09342 09377 09412 094471 094821 09317 09552 09587 09627 39656
1125 09691 09726 09760 09795 09830 09864 09899 09934 09968|10003
126 10037 10072 10100 10140 10175 10200 10243 10278 10312 10340
127 10380 10415 10449 10483 10517 10551 10585 10619 10653 10087
128 10721 10755 10789 10823 10856 10890 10924 10958) 10992 11025
124 11059 11093 11126 1116 11193 11227 11260 11294 11327 11361
130 11394 11428 11461 11494 11528 11501 21594 11628 11661 11694
131 11727 1176011793 11826 11860 11893 11926 11959 11992 12024
132 12057 1209012123 12150 12189 12222 12254 12287 12320 12352
133 12385 12418 12450 1248 12516 12548 12581/ 12613 12646 12678)
134 1271012743 1277312808 12840 12872 1290512937 1 2969 13001
135 13023 13060 13098 13130 13162 13194 13220 13258 13240 13322
136 13354 13.386 13418 13450 13481 1351313545 135771 13609 13640
137 13672 13704 13735 13767 13799 13830 1386: 13893 13925 13956
138 13988 1401914051 14082 14114 14145 14176 14208 14239 14270
139 14301 14353 14364 14395 14420 14457 14489 14520 14551 14582
140 14613 14644 14673 14700 14737 14768 1479) 14829 14850 14891
141 14922 14053 149831501455045 15076 15100 15137 15168 15198
142 1522915259 132941 15320 15351 15381 15412 15 442 15473 15503
143 15534 15504 :559415625 15055 15685 15715 15746 15776 15806
144 15830 15866 15897 15927 1595; 15987 10017 16947 16077) 16107
145 10137 16167 161971 1622716256 162:6 16316 16346 16376 16406
146 10435 16405 10495 16.524 16554 16584 10613 16643 16673 16702
147 167.32 16761 16:91 16820 1685 16879 16909 16938 169671 16997
148 17026 17056' 1708 17114 17143 17173 17202 17231 17200 1728.)
(149| 17319 17348 17377117406 17435| 17464 17493 17522 17551 17580
134 17609 17638 17667 17696 17725 17754 17782 1781 1784017869
15017898 17926 17955 1798 : 18013 18041 1807018099 18127 18156
152 18184 18213' 1824118270 18298| 18327 18355 18384 18412 18441
153 18469 18498 1852418554 18583 18611 18634 18667 18696 18724
154 18752 18780 18808 18837 18865 18893) 18921 18949 18977 19005
155 19033 19061 1908919117 19145 19173 19201 1922 19257 19285
150 19312 19340' 19368 19396 19424 19451 194791950719535 19562
1957 19590 19618 9645 19673 19700 147281975619783) 19811 19838
115 19866 19893 19921 1948 1997620003 2003 20058 20085 20112
159 20140 20167 20194 20223 2024920276 20303 20330 20358 20385
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TABLE OF LOGARITHMIC NUMBERS.

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9 160 20412 20439 20466 20493 20520 20547 20575 20602 20629 20656 161 20683 20710 20736 2076320790 20817 20844 20870 20898, 20925 162 20951 20978 21005 21032 21059 21085 21112 21139 21165 21192 163 21219 21245 21272 21299 21325 21352 21378 2140921431 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 231471 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 1173 23805 23830 23855 23880 23905 23930 23955 23980 24005 24030 174 24055 2408024105 24130 241551 24180 24204, 24229 24254 24279 175 24304 24329 24353 24378) 24403| 24428 24452 24477 24502 24527 17024551/ 24576 24601 24625 24650 24674 24099 24724 24748 24773 1771 24797 24822 24840 24871 24895 24920 24944 24969 24993 25018 1781 25042 25066 25091 2511525139) 25164 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 261?4 20198 20221 183 26245 26267 26293 26316 26340 26364 26387 26411 26435 26458 184 26482 2650520529 26553 26570 26600 20623 26647 26070 26694 185) 26717 26741 26764 26788 26811 26834 26858 26881 26905 26928 186 26951 26975) 26998 27021 27045 27068 2709127114 27138127161 187 27184 27207 27231 27254 27277 27300 27323 27346 2737027393 188) 27416 27439 27402 27485 27508 27531 27551 27577 27000 276231 189 27646 25669 276922771527738 27701 27784 27807 27830 27852 19027875 27898 27921| 27944 27967 27989 28012 28035 280581 28081 191 28103) 28126 28149 28171 28194 28217 2824-28262 2¢285 28307 192 28330 28353 28373 28398 28421 23443 28466 28488 28511 28533 193 28556 28578 28601 28623 28646 28668 28691 287131 28735 28758 194 28780 28803 28825 288471 28870 28892 28914 28937 28759 28981 195 29003 29026 29048/ 29070 29092 29115291371 29159 29181 29103 19029226 29248) 292701 29292 29314 2933029358129380 2940329425 1997 294471 29460 24491 29513 29535 29557 29579 29601 29623/ 29645 198 29667 2968829714 29732 29754 29776 29798 29820 29842 29863 199 29885 29907 29929 29951 29973 29994 30016 30038 30000 30081 200 30103 30125 30140 30108 30190 30211 30233/ 30255 30276 30298 201303201 30341 30363 30384 30466 30428 30449 3047 / 30492| 30514 202 30535 30557130578 30000 30621 30042 30064 30685 30707 30728 203 3075030771 30792 30814 30835 30856 3087830899 30920 30942 204 309631 30984 31006 31027) 31048 31069 3109131112 31133 311541 205 31175 31197 35218) 31239 31200 31281 31302 31323 31345 31300 206 31367 31408 31420 31450 31471 31492 31313131534 31555 31576) 207 31597 31618 316,34 31660 31681 31702 31723 31740 31705 31785) 208 31806 31827 31848) 31869 31890' 31911 31931| 319521 31973 31994 209 32015| 32035 320563207732098 32118 32139 32160 32181 32201 210 32222 32243 32263 32284 32365 32325 32 346 32 366 32387 32408 211 32428 32449 32469 32490 32510 32531 32552 32572 32543 32613 212 32634 32654 32675) 32695 32715 32736 3275632777 32797 32818 213 32838 32858 32879 32899 32919 32940 32960! 32980 33001 33021 214 33041 33062 33082 33102 33122 33143) 33103/3310, 33203 33224 215 33244 33264 33284 33304 33325! 33345 33365 33385 33405 33425 216 33445 33465 33486 33506 33526, 33540 33566 33586 33006 33626 217 33646 33666 33686 33706 33726 3374033766 33786 33806 33826 218 33846 33866 33885 33905 33925 33945 3396533985 34005| 34025 219 34044 34064 34084 34104 34124 34143 3416334133 34203 34223 NO

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TABLE OF LOGARITHMIC NUMBERS.

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220 34242 34262 3428234301 34321 34341 34361 34380, 34400, 34420
221 34439, 34459 34479 34498 34518/ 34537| 34557 345771 34596 34616
222 34635 34655| 34674 34694 34713 34733 34753 34772 34792, 34811
223 348 30 3485034869 34889 34908 34928 34947 34967 34986 35005
224 3.5025) 35044 350643508335102 35122 35141 3516035180 35199)
1225 35318 35238| 35257) 35276 35295 3531335,33435353 35372 35302
226 35411 35430 35449 35468) 35488 3550735526 35545 35564 355831
227 35603) 35622 35641 35660 35670 3579235717 35736 35755 35774
228

35703 35813 35832 35851358703548935908 35927 3594635965
22931984 36003| 36021 3604036059 36078 36097 36116 36135 36154
1230 3617336192 36211 36229 36248 3626736286 36305) 36324 36342
231 35361 36.3836399 364181 364 361 36455 36474 36493 36511 36530
232 36549 36568 36586 36605 30024 36642 36661 36680 36698 36717
233 36736 3975436773) 36791 35810 36829 3684736866 36884 36903
234 36922 36940 36959 36977 36996| 37014 37033) 37051 37070 37088
12,35 37107 37125) 37144 37162 37181 3719937218 3723637254 37273

2.36 3729137310 37328 37346 37365 37383 37401 37420 37438 37457
$2371 37475 37492 37511 37530 37548 37566 37585 37603 37621 37639
12.39 37658 37670 376941 37712 377311 37749 377671 37785 37893 37822
239 3401 37858) 37876 37894 37912 37931 37949 37967 37985 38003
240138021 38039 38057 38075 38093 38112 38130 38148 38166 38184
241 382021 38224138238 38256 38274138392 38310 38328 38346 38364
$2421 393821383233417 38435 38453 38471 38189 38507 385251 38545
124338-61 38578 38596 38614 38632/38650 38663 38686 38703 38721
244 3872938757,38775 38792 35.810 33825 38840 3886338881 38899
1245 3891738)3+ 38,52 38970 3898; 39005 390231 390.4739058 39076
246 39094 391139129 39 140 3916.5| 39182 391991 39217 39235 39252
f247 3927039287 39305) 39322 39340 39358) 39.375 39393 39410 39428
248 32445 39463 39480 39.1981 39515| 39533/ 39550 39568 39585 39602
249 39620 39637 39655 39672 39690 39707 39724 39742 39759| 39777
250 39794 39811 34829 49846) 3980s 39881) 39898 39915 3993339950
251 3996;3998|40002 40000 400371 40034 40071/40088 40106 40123
2524014040157 40165 40192 40200 40226 40243 40261 40278 40295
12.53 40312 40329| 40346 403641 4033: 40398|4041540432 40449 404661
254 40483) 405071 40518 40535 40552 40569) 40586,40653 40620| 40637
29.5 40654 400671 30683 40705 40722 40739 40756 40773 40790 40807
z=61 408244-84140858 40875 40892 40409 40926 40943 40959 40976
1257409234101 41027 41044 41061 41076 4109540111 41128| 41145
12584116241174119941212 41224 412.16 41263 41280 41296 41313)
259 41330 41341441363 41380 41397 41414 4143041447 414641 41481
266 45 447 41,514 41531, 41547 4156.5 41501 41397 41614 41637 41647
$261 41664 41681 41607 41714 41731| +1747 41764 41780 41797 41814!
26241839-184747863, 41830 41876 41913 41929 419404196.3 41979
263 41990 42012 42029 42045) 42062 42078 42095 12111 42127 42144
264 42160 42177 42193 42210 42226 4224. 42259 42275 42292 42308
120542325 42341 42357 42374 42390 42406 42423 42439 42455 42472

260 42488 42504 42521 425 37 42553 22570 42586 42602 42619 42635
126-42657142667' 42684 42700 42711 427.32 42749 42765 42781 42792
784281342830 42846) 42862) 42878 42894 42911| 42927 42943 42959
1269 42975 42991 43008 43024 43040 43056 43072 43088 43104 43120
27043136 43152. 43769 43185 43201 432 17 43233| 43249) 43265| 43281
1271| 432971 4 3313 43329 43345 43361 433771 43393 43409, 43425 43441
272, 434571 43.173 43.489 43505 43521 43537 43553 43569 4358443600
27.3 43616 436,32 436 481 43664 4 3680 43796 43712 437271 43743 43759
1274 43775 43791 43807 43823) 43838 438541 43870, 43386 43902 43917
275 43933: 43949. 43365| 43981 43496 44012 44028 44044' 44059 44075
1276 44091 44107 44122 441381 44154 44170 44185 44201 44217 44232
277 44248 44264 44279 14295 44311 44326 44342' 44358 44373/ 44,384
278 44404 44420 44430 44451 44467 4458 1 44498 44514' 44529 44545
279 44560' 44576 44592 +4607 44623 44638) 44654 44659 44685 44700
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