stationed at each extremity of the base line, it will not be necessary to establish buoys. A boat, provided with an anchor, a sounding line, and a signal flag, has only to throw its anchor, hoist its signal flag, and make the sounding, while the persons at the extremities of the base line measure the angles ;-from these data, the precise place of the boat can be determined.

207. There is also another method of determining the places at which the soundings are made, that admits of great despatch, and which, if the observations be made with care, affords results sufficiently accurate.

Having established, trigonometrically, three points which can be seen from all parts of the harbour, and having provided a sextant, let the sounding be made at any place in the harbour, and at the same time the three angles subtended by lines drawn to the three fixed points, measured with the sextant.

The problem, to find from these data the place of the boat at the time of the sounding, is the same as example 6, page 74.

It is only necessary to measure two of the angles, but it is safest to measure the third also, as it affords a verification of the work.

The great rapidity with which angles can be measured with the sextant, by one skilled in its use, renders this a most expeditious method of sounding and surveying a harbour.

The sextant is not described, nor are its uses explained in these Elements, because its construction combines many philosophical principles, with which the surveyor cannot be supposed conversant.

208. There is yet another method of finding the soundings, which, although not as accurate as those already explained, will, nevertheless, afford results approximating nearly to the truth. It is this:-Let a boat be rowed uniformly across the harbour, from one extremity to the other of any of the lines determined trigonometrically. Let soundings be made continually, and let the precise time of making each be carefully noted. Then, knowing the length of the entire line, the time spent in passing over it, as also the time of making each of the soundings, we can easily find the points of the line at which the several soundings were made; and hence, the depth of water at those points becomes known. Sound

ings may thus be made along any number of known lines, and a comparison of the depths found on different lines, at or near their points of intersection, will show with what degree of accuracy the work has been done.

209. If the soundings are made in tide-waters, the time of high tide must be carefully noted, as also the precise time of making the sounding, so that the exact depth at high or low water may be known. It is considered preferable to reduce the soundings to high-water mark, and the number of feet which the tide rises and falls should be noted on the map.

210. Having plotted the work done with the theodolite, as also the outline of the harbour traced with the compass, it remains to delineate the bottom of the harbour; and this is done by means of horizontal curves (Chap. VI), which have already been used to represent broken or undulating ground.

Let the plane of reference be taken through high-water mark, or to coincide with the surface of the water at high tide. The accuracy with which the bottom of the harbour is to be delineated, will guide us in fixing the distance between the horizontal planes of section.

The first horizontal plane should be passed at a distance below the shallowest point that has been sounded, equal to the number of feet fixed upon for the distance between the planes of section; and the curve, in which it intersects the bottom of the harbour determined as in Chapter VI. And similarly, for the other horizontal planes of section.

Having thus delineated the bottom of the harbour, and noted on the map the distance of each intersecting plane below the plane of reference, let such lines be drawn as will indicate the channels, shoals, sunken rocks, and direction of the current.

In the example given in plate 6, soundings have been made in three directions from the sand-bar in the harbour, and also from the rocky shore across to the light-house.

A TABLE

OF

LOGARITHMS OF NUMBERS

FROM 1 TO 10,000.

N.

8

li

12

13

15

1.176091 40

1.602060 65

Log. N. Log. N. Log. 0.000000 26 1.4:4973 51 1.707570 0.301030 27 1.431364 52 1.716003 0.477121 1.447158 53 1.724276 0.602060 29 1.462398 54 1.732394 79 0.698970 30 1.477121 55 1.740363 80 0.778151 31 1.491362 56 1.748188 81 1.908485 0.845098 32 1.505150 57 1.755875 82 0.903090 33 1.518514 58 1.763428 83 0.954243 3-4 1.531479 59 1.770852 84 1.924279 10 1.000000 35 1.544068 60 1.778151 85 1.929419 1.011393 35 1.556303 61 1.785330 86 1.934498 1.079181 37 1.568202 62 1.792392 87 1.939519 1.113943 38 1.579784 63 1.799341 88 1.944483 1.146128 39 1.591065 1.806180 89 1.812913 90

N. I

76

Log. 1.880814

777 1.886491

78

1.892095

1.897627 1.903090

1.913814

1.919078

1.949390

1.954243

1.819544 91 1.959041 1.826075 92 1.963788 1.832509 93 1.968483 1.838849 94 1.973128 1.845098 95 1.977724

21
1.322219 46 1.662758 71
22 1.342423 47 1.672098 72
23 1.361728
1.681241 73 1.863323
24 1.380211 49 1.690196 74 1.869232
25 1.397940 50 1.698970 75 1.875061

N.B. In the following table, in the last nine columns of each page, where the first or leading figures change from 9's to 0's, points or dots are introduced instead of the O's through the rest of the line, to catch the eye, and to indicate that from thence the annexed first two figures of the Logarithm in the second column stand in the next lower line.

17

18

19

N.

122
123

0

23

819 | D.

56 | 7 | 8 | 9 2100000000, 0434 0868 1301, 1734, 2166| 2598) 3028) 3461| 3891 432 50 101 432147515181 5609 6033 6466 6894 7321 7748 8174 428 102 8600 9026 9451 9876 .300 724 1147 1570 1993 2415 424 10103 012837 3259 3680 4100 4521 4940 5360 5779 6197 6616 419 104 7033 7451 7868 8284 8700 9116 9532 9947.361.775 416 3105 021189 1603 2016 2428 2841 3252 3664 4075 4486 4896 412 4106 5306 5715 6125 6533 6942 7350 7757 8164 8571 8978 408 50107 9384 9789.195 .600 1004 1408 1812 2216| 2619 3021 404 108 033424 3826 4227 4628 5029 5430 5830 6230 6629 7028 400 10 109 7426 7825 8223 8620 9017 9414 9811.207.602.998396 110 041393 1787 2182 2576 2969 3362 3755 4148 4540 4932 393 20 111 5323 5714 6105 6495 6885 7275 7664 8053 8442 8830 389 4 112 9218 9606 9993 .380.766 1153 1538 1924 2309 2694 386 0113 053078 3463 3846 4230 4613 4996 5378 5760 6142 6524 382 114 6905 7286 7666 8046 8426 8805 9185 9563 9942 .320 379 115 060698 1075 1452 1829 2206 2582 2958 3333 3709 4083 376 116 4458 4832 5206 5580 5953 6326 6699 7071 7443 7815 372 117 8186 8557 8928 9298 9668 .38 .407 .776 1145 1514 369 118 071882 2250 2617| 2985 3352 3718 4035 4451 4816 5182 366 119 5547 5912 6276 6640 7004 7368 7731 8094 8457 8819 363 120 079181 9543 9904| .266.626.987 1347 1707 2067 2426| 360 121082785 3144 3503 3801| 4219 4576 4934 5291 5647 6004357 6360 6716 7071 7426| 7781 8136 8490 8845 9193 9552 355 9905 .258 .611 .963 1315| 1667 2018 2370| 2721 3071| 351 124093422| 3772| 4122 4471 48205169 5518 5866 6215| 6562| 349125 6910 7257 7604 7951 8298 8644 8990 9335 9681..26 346 126 100371 0715 1059 1403| 1747 2091 2434 2777 3119 3462 343 127 3904 4146 4487 4823 5169 5510 5851 6191 6531 6871 340 128 7210 7549 7888 8227 8565 8903 9241 9579 9916 .253 333 129 110590 0926 1263 1599 1934 2270 2605 2940 3275| 3609 335 130 113943 4277 4611 4944 5278 5611 5943 6276 6608 6940 333 131 7271 7603 7934 8265 8595 8926 9256 9586 9915 245 330 132 120574 0903 1231 1560 1888 2216 2544 2871 3198 3525328 133 3852 4178 4504 4830 51565481 5806 6131 6456 6781 325 134 7105 74297753 8076 8339 8722 9045 9363 9690 12 323 135 130334 0655 0977 1298 1619| 1939 2260 2530 2900 3219 321 136 3539 3358 4177 4496 4814 5133 5451 5769 6086 6403 318 137 6721 7037 7354 7671 7937 8303 8618 8934 9249 9564 315 138 9879 .194.508.822 1136| 1450 1763 2076 2389 2702 314 139 143015 3327 3639 39514263 4574 4885 5196 5507 5818 311 140 146128 6438 6748 7058 7367 7676 7985 8294 8603 8911 309 141 9219 9527 9835 .142.449.756 1063 1370 1676 1982 307 142 152288 2594 2900 3205 3510 3315 4120 4424 4728| 5032 305 143 5336 56405943 6246 6549 6852 7154 7457 7759 8061 303 144 8362 8664 8965 9266 9567 9868.168 .469 .769 1068 301 145 161368 1667 1967| 2266| 2564| 2863 3161 3460 3758 4055 299 146 4353 4650 4947 5244 5541 5833 6134 6430 6726 7022 297 147 7317 7613 7908 8203 8497 8792 9086 9380 9674 9968 295 148 170262 0555 0848 1141 1434 1726 2019 2311 2603 2895 293 149 3186 3478 3769 4060 4351 4641 4932 5222 5512 5802 291 150 176091 6381 6670 6959 7248 7536 7825 8113 8401 8689 289 151 8977 9264 9552 9839.126 .413 .699 .985 1272 1558 287 152 181844 2129 2415 2700 2985 3270 3555 3839 4123 4407 285 153 4691 4975 5259 5512 5825| 6108 6391 6674 6956 7239 283 154 7521 7803 8084 8365 8647 8928 9209-9490 9771 51 281 155 190332 0612 0892 1171 1451 1730 2010 2289 2567 2846 279 156 3125 3403 3681 3959 4237 4514 4792 5069 5346 5623 278 157 5899 6176 6453 6729 7005 7281 7556 7832 8107 8382 276 158 8657 8932 9206 9481 9755| 29 .303.577 .850 1124 274 159 201397 1670 1943 2216 2488 2761 3033 3305 3577 3848 272

N.

0

171

172 173

2 3 | 4 | 5 | 6 | 7 | 8 | 9 | D. 160| 204120) 43911 4663 4934 52041 5475 5746] 6016; 6286 6556) 271 161 6826 7096 7365 7634 7904 8173 8441 8710 8979 9247 269 162 9515 9783..51 .319 .586 .853 1121 1388 1654 1921 267 163 212188 2454 2720 2986 3252 3518 3783 4049 4314 4579 266 164 4844 5109 5373 5638 5902 6166 6430 6694 6957 7221 234 165 7484 7747 8010 8273 8536 8798 9060 9323 9585 9846 262 166 220108 0370 0631 0892 1153 1414 1675 1936 2196 2456 261 167 2716 2976 3236 3496 3755 4015 4274 4533 4792 5051 259 168 5309 5568 5826 6084 6342 6600 6858 7115 7372 7630| 258 169 7887 8144 8400 8657 8913 9170 9426 9682 9938.193 256 170 230449 0704 0960 1215 1470 1724 1979 2234 2488 2742 254 2996 3250 3504 3757 4011 4264 4517 4770 5023 5276 253 5528 5781 6033 6285 6537 6789 7041 7292 7544 7795 252 8046 8297 8548 8799 9049 9299 9550 9800 ..50 .300 250 174 240549 0799 1048 1297 1546 1795 2044 2293 2541 2790 249 175 3038 3286 3534 3782 4030 4277 4525 4772 5019 5266 248 176 5513 5759 6006 6252 6499 6745 6991 7237 7482 7728 246 177 7973 8219 8464 8709 8954 9198 9443 9687 9932.176 245 178 250420 0664 0908 1151 1395 1638 1881 2125 2368 2610 243) 179 2853 3096 3338 3580| 3822 4064 4306 4548 4790 5031| 242 | 180 255273 5514 5755 5996 6237 6477 6718 6958 7198 7439 241 181 7679 7918 8158 8398 8637 8877 9116 9355 9594 9833 239 182 260071 0310 0548 0787 1025 1263 1501 1739 1976 2214 238 183 2451 2688 2925 3162 3399 3636 3873 4109 4346 4582 237 4818 50545290 5525 5761 5996 6232 6467 6702 6937 235 7172 7406 7641 7875 8110 8344 8578 8812 9046 9279 234 9513 9746 9980.213.446] .679 .912 1144 1377 1609 233 187 271842 2074 2306 2538 2770 3001 3233 3464 3696 3927 232 188 4158 4389 4620 4850 5081 5311 5542 5772 6002 6232 230 189 6462 6692 6921 7151 7380| 7609] 7838 8067 8296 8525 229 190 278754 8982 9211 9439 9667 9895.123 .351 .578.806 228 191 281033 1261 1488 1715 1942 2169 2396 2622 2849 3075 227 192 3301 3527 3753 3979 4205 4431 4656 4882 5107 5332) 226 5557 5782 6007 6232 6456 6681 6905 7130 7354 7578 225 7802 8026 8249 8473 8696 8920 9143 9366 9589 9812 223 195290035 0257 0480 0702 0925 1147 1369 1591 1813 2034 222 2256 2478 2695 2920 3141 3363 3584 3804 4025 4246) 221 4466 4687 4907 5127 5347 5567 5787 6007 6226 6446 220 6665 6884 7104 7323 7542 7761 7979 8198 8416 8635 219 8853 9071 9289 9507 9725 9943.161 .378 .595 .813 218 200 301030 1247 1464 1681 1898 2114 2331 2547 2764 2980 217 201 3196 3412 3628 3844 4059 4275 4491 4706 4921 5136 216 5351 5566 5781 5996 6211 6425 6639 6854 7068 7282 215 7496 7710 7924 8137 8351 8564 8778 8991 9204 9417 213 9630 9843..56.268 .481.693.906 1118 1330 1542 212) 205 311754 1966 2177 2389 2600 2812 3023 3234 3445 3656 211 3867 4078 4289 4499 4710 4920 5130 5340 5551 5760 210 207 5970 6180 6390 6599 6809 7018 7227 7436 7646 7854 209 208 8063 8272 8431 8689 8898 9106 9314 9522 9730 9938 208 209 320146 0354 0562 0769 0977 1184 1391 1598 1805 2012 207 210 322219 2426 2633 2839 3046 3252 3458 3665 3871 4077 206 211

184

185 186

193 194

196

197

198 199

202 203 204

206

212 213

4282 4488 4694 4899 5105 5310 5516 5721 5926 6131 205 6336 6541 6745 6950 7155 7359 7563 7767 7972 8176 204 8380 8583 8787 8991 9191 9398 9601 9805 .8.211 203 214 330414 0617 0819 1022 1225 1427 1630 1832 2034 2236 202 215 2438 2640 2842 3044 3246 3447 3649 3850 4051 4253 202 4454 4655 4856 5057 5257 5458 5658 5859 6059 6260 201 6460 6660 6860 7060 7260 7459 7659 7858 8058 8257 200 8456 8656 8855 9054 9253 9451 9650 9849 ..47.246 199 219 340444 0642 0341 1039 1237 1435 1632 1830 2028′ 2225 198

216 217

218