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PROBLEM IX.

To meafure heights and disances 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 place the instrument at the point B, directing the fixed fights to the first station A, and the moveable ones to the point T'; 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 accesible, 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 similar observation may be made of the other instruments used for heights and distances; but this, and many more, will daily occur in practice.

* The fide DE is called the right side, I the reclined lide.

TABLES.

CONTAINING,

1. 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,

Log. No. Log. No. Log No. Log. No. Log. Icocooo 211.32222 471.61278 611.78533 811.90848 20.30103 22 1.34242 42 1.62325 621.79239 82 1.91381 30.47712 231.36173 431.63347 63 1.79934 831.91908 410.60206241.38021 441.64345 641.806181841.92428 5 0.69892 25 1.39794_451.65321 651.81291_851.92942 60.77875 261.41497 46,1.00276661.81954 801.93450 7 0.84510 27 1.43136 471.67220 67 1.82607 871.93952 80.90309 28/1.44786 481.68124 68 1.83251 88 1.94448 90.95424 29 1.46240 49 1.69020 69 1.83885 891.94939 101.00000 301.47712 50 1.69897 70 1.84510_901.95424 II1.04139 31 1.49136 511.707571 71 1.85120 9111.95904 121.07918 32 1.50515 521.71600172 1.85733 921.96379 131.11394 331.5185 1 531.72428 731.86332 931.96848 141.14613 341.53148 541.732391 74 1.86923 941.97313 15 1.17609 35/1.54407 5511.74036) 75 1.8,506 9.51.97772 16 1.20412 3611.55636 561.74819 761.830811 9011.y 2.7 171.23045 37 1.56820 571.75587771.8864) 91.986771 181.25527 381.57978531.76 343 78|1.89209 9811.99123) 191.27875 391.5910 39 1.77085/ 791.89763 90|1.99563 201.3010; 40|1.60201 601.778141 80 1.90109110 12.000

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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 00811
102 00860 00903 00945 00988 01030 01072' 01115 01157 01199) 01242
103 01 284 01326 01368 01410 01452 01494 01536 01578 01620 1662
104 0170301745 01787 01828 01870 01912 01953 01995 02036 02078
105 02119 02100 02202 02243 c2284 02325 02366 02407 02449 02490
100 02531 02572 02612 02653 02044 02735 02776 02816 02857 02898
107 02938) 02979 C3019 03060 0310003141 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
I-4532 04571 0461004650 04689 04727 04766 04805 04844 04883
112 0492204961 04999 05038 05077 OSIIS 05154 05192 05231 05269
113 05 30805346 05385 05423 05461 05500, 05538 05576 05614 05652
114 05690 05 729 05767 058030584305881 0591805956 0599400032
115 C6070 06108 06145 00183 06221 06258, 0629600333 0637106408
11 06446 06483 06521 06558 6595 066;306670 06707 06744 06781
117 06819 06850 06893 06930 06.367 07004' 07041 07078 0711507151
118 07180 07225 07262 07298 07335 07372 07408 07445 07482 07518
119 07555 07591 07628 07664 07700 07737 07773 07809 0784607882
120 07918 07954 07990 08027 08063 08099 08135 08171 08207 08243
121 08279 08314 08350 0838608422 08458, 08493 08529 08565 08600
122 08636 08672 08709 08743 08778 08814 8849 08884 0892008955
129 08991109026 09061 09096 09132 09167 07202 09237 09272 09307
124 09342 09377 09412 09447 094821 09517 09552 095871 09621 09656
125 09691 09726 0976c09795 09830 09864 09899 09934 09968 10003
126 10037 10072 10106 10740 10175 10202 10243 10278 10312 10346
127 10380 10415 10449 10483 10517 10551 10585 10619 10653 10087
128 10721 10755 10789 10823 10856 10890 10924 10958, 10992 11025
1129 11059 11093 11126 1116 11193 11227 11260 11294 11327 11361
130 11394 11428 11461 11494 11528 11501 11594 11628 11661 11694
131 11727 11760 11793 11824 11860 11893 11926 11959 1199212024
132 12057 12090 12123 12150 12189 12222 12254 12287 12320 12352
133 12385 12413 12450 1248 | 12516 12548 12581 12613 26.46 12678
134 12710 12743 12775 12808 12840 12872 12905| 12937 12909) 13001
135 13023 13000 13098 13130 13162 13194 13220 13258 13290 13322
136 13354 13,386 13418 13450 1348313513 13545 135771 13609 13640
137 13672 13704 13735 1376713799 13830 1386 13893 13925 13956
138 13988 14019 14051 14082 14114 14145 14170 14208 14239 14270
139 14301 14353 14364 14395 14426 14457 14484 14520 14551 14582
140 14613 14644 14679 14700 14737| 14768 1479914829 14850 14891
141 14922 14063 14983 15014 15045 15076 15100 151371 15168 15198
142 15229 15259 152901 15320 15351 15381 15412 15442 15473 15503
143 15534 15564:5594 15625 15655 15685 15715 15746 15776 15896
144 1583015866 158971 15927 15957 15987 16017 16047 16077 16107
145 16137 16167 161971 1622; 1625016186 16316 16346 16376 16400
14010435 16465 16495 16524 16554 16584 16613 16643 16673 16702
147 167,32 16761 16791 16820 1685 16879 16900 16938 16967 16997
148 17026 17056' 17085 17114 17143 17173 17202 17231 17200 1728.,
(149 17319 17348 173771 1740617435 17464 17493 17522 17551 17580
1134 17609 17638 17667 17696 17725 17754 17782 1781 1784017869
15017898 17926 17955 1798-1801318041 1807 18099 18127 18156
152 18184 18213 18241 18270 18298 18327 18355 18384 18412 18441
15.318469 18498 1852418554 1858318611 18634 18667 18696 18724
154 18752 18780 18808 18837 18865 10893 18921 18949 18977 19005
155 19033 19061 1908919117 1914 19173 19201 19229 19257 19285
154 19312 19340' 19368 19396 19424 19451 19479 19507 1953 19562
157 19590 19618 9645 19673 19700 19728 1975619783 19811 19838
115&19866 19893. 19921|19948 19976| 20003 2003 20058) 20085 20112
159 20140 20167 20194 20222 2024920276 20303 20330 20358 20385
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162 20951 20978 21005 21032/ 21059 21085 21112 21139 21165 21192 163) 21219 21245) 21272 21299) 21325 21352 21378 2140521431, 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 22683 22712 22737 22763 169 22789 22814 22840 22866 22891 22917 22943 22968 22994 23019 1170 23045 2.3070) 23096 23121 23147 23172 23198 23223 23249 23274 1711 23300 23325 23350 23376 23401 23426 23452 23477 23502 23528 1721 23553 23578 23603 236291 23654 23679 23704 23729 23754 23779 173 23803 23830 23855238801 23905 23930 23955, 23980 24005 24030 174 24055 24000 24105 24130 24155 24180 24204, 24229 24254 24279 175 24304 24329 24353 24378 2440324428 24452 24477 24502 24527 170 24551 24576 24601 24625 24659 24674 24699 24724 24748 24773 177 24797 24822 24846 24871 24895 24920 24944 24969 24993 25018 1781 25042 25066 25091 25115125139 25104 25188 25212 25237, 252013 179 25285 2531025334 25358 25382 25406 25431 2545525479 25503 18025527 25551 25575 25000 256241 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 2620 26293 26316 26340 26364 26387 26411 26435 26458 184 26482 26505 20529 26553 20570 26600 20623 26647 20070 26694 185 26717 26741 26764 26788 26811 26834 2685826881 2690526928 186 26951 26975) 26998 27021 27045 270681 27091| 2711427138/27161 1871 27184 272071 27231 27254 27277 2730027323 27346 2737027393 1881 27476 274391 27462 27485) 27508 27531 27551 27577 27000 27623 189 27646 27059 27692/ 27715 27738 27701 27784 27807 27830 27852 190 27875 27898 27921 27944 27967 27989 28012 28035 20058 28081 191 28103) 28126 28149 28171 28194 28217 28240 28262 28285 28307 1921 28330 28353 28375 283981 28421 28443 28466 28488 28511 28533)

193 28556 28578 28601 28623 28646 28668 28691 28913 2873528758 1941 28780 28803 28825 28847 28870 28892 28914 289371 28959 28981 195 29003 29026 29048 29070 29092! 29115291371 29159 29181) 29203 yo 29226 29248 29270 29292 29314: 29330 29358 29380 2940329425 197 294471 29460 24491 29513 2953; 29557 29579 29601 2962329645 198 29667 29688 29770 29732 29754 29776 29798 29820 29842 29863 199 29885 29907 29929 29951 29973 29994 30016 30038 30000 30081 200 30103 30125 30140 30168 30190 30211 30233 30255 30270 30298 201 30320 30341 30363 30384 30466 30428 30449 30472 30492 30514 202 30535 30557 30578 30000 30621 30042 30064 30685 30707 30728 203 30750 30771 30792 30814, 30835 30856 30878 30899 30920 30942 204 30963 3098431606 310271 31048 31069 3109131112 3113331154 205 31175 3119235218 31239 31260 31281 31302 31323 31345 31300 206 31307/ 31408 31429 3145031471 314923151331534 31555) 31576 2071 31597 31618 316,34 31660 31681 31702 3172331740 31705 31785 208 31806 318271 31040 31869 31890' 31911 31931 319.521 31973 31994 209 32015| 32035 32056 32071 32098 32118 32139 32560 32181 32201 210 32222 32243 32263 32284 32365 32325 32 346 32366323871 32408 211 32428 32449 32469 324901 32510 325311 32552 32572 32593 32613 212 32634 32654 326751 32095 32715 32736 32756 32777 32797 32818 213 32838 32858 32879 32899 32919 32940 32960! 32980 33001 33021 214 33041 33002 33082 33102 33122 33143) 331033310, 3320333224 215| 33244 33 264 33284 33304 33325l 33345 33365 33385 33405 334251 216 33445 3346533486 33506 33526, 33540 33566 33586 33606 33620 217 33646 33666 33686 33706 33726 33740 33766 33786 33806 33826 218 33846 33866 33885 33905 33925 33945 33965 3398534005 34025 219 34044 34064 34084 34104 34124 34143 34163 34133 34203) 34223 NO

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