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2. Given the very acute angles A, B, and the side c to find the remaining parts.

Since A and B are very small, we have very nearly

[blocks in formation]

Hence, sin C = sin (A + B) = sin A cos B + cos A sin B = (A+B)

1.2.3

(A+B)3 ̧

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3. Given the two sides a, b, and the included angle C, which is very obtuse, to find

the other parts of the triangle.

Put C 180° . -a: then, since C is very obtuse, a is very small, and

[blocks in formation]

a2

c2 = a2 — 2ab cos C + b2 = a2 + b2 + 2ab { 1 — 1.3 } = (a + b)2 — ab a2.

Whence c√ (a + b)2. · ab a2 = a + b

α

[blocks in formation]

Again, sin A== sin C = = sin a == {

C

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[ocr errors]
[blocks in formation]

1.2

ab a2 a+b 2

απ

{1+

a3

1.2.3

ab

nearly.

} nearly; and

-1

[ocr errors]
[blocks in formation]

these values, we have

}{a

α

a2 — ab + b2 a2

A3

[ocr errors]

(a + b)

1.2.3

= A

[merged small][ocr errors][merged small]

1.2.3)

sin3A

1.2.3

a2-ab+b2 a2

[blocks in formation]

2

[ocr errors]

ab a2

2(a+b)2

a2 1 1.2.3

nearly; or Asin A +

2(a+b)3 1.2.3} + {a

(a - b) b a2
(a+b)2 1.2.3)

nearly.

[blocks in formation]

(a - b) a a2

nearly.

In the same manner we have

[merged small][ocr errors]

sin A

1.2.3'

(a + b)2 1.2.3)

These angles being understood to be in circular measure, as explained in the preceding note.

For further information on subjects of this nature, the reader is referred to Bonnycastle's Trigonometry, Cagnoli Trigonométrie, several of the authors on Geodesy, especially Puissant, and to the second volume of this work.

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5-19615243-000000

99 9801

[blocks in formation]

9702999-9498744 4-626065 1000000 10-0000009 4641589 1030301 10-0498756 4-657009 1061208 10-0995049 4-672329 1092727 10-14889164-687548 1124864 10-1980390 4-702669 1157625 10-2469508 | 4-717694 1191016 10-2956301 4-732623 1225043 10:3440804 4-747459 1259712 10-3923048 4-762203 1295029 10-4403065 4-776856 1331000 10-4880885 4-791420 1367631 10:5356538 4-805895 1404928 10-58300524-820284 1442897 10-6301458 4·834588 1481544 10-6770783 4848808 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 11-0000000 4-946087 1815848 11:0453610 4-959676 1860867 11:0905365 | 4·973190 1906624 11-1355287 4-986631 1953125 11-1803399 5·000000 2000376 11-2249722 5-013298 2048383 11-2694277 | 5·026526 2097152 11-3137085 5·039684 2146689 11-3578167 5·052774 2197000 11-4017543 5:065797 2248091 11-4455231 5:078753 2299968 11-4891253 | 5-091643 2352637 11-5325626 | 5·104469 2406104 11:5758369 | 5·117230 2460375 11-6189500 | 5·129928 2515456 11-6619038 | 5·142563 2571353 11-7046999 5 155137 2628072 11.7473401 | 5-167649 2685619 11-7898261 5-180101 2744000 11-8321596 5·192494 2803221 11·8743422 | 5·204828 2863288 11-9163753 5 217103 2924207 11-9582607 5-229321 2985984 12 0000000 5-241483

VOL. I.

M m

No. Square.

145 21025
146 21316
147 21609

148 21904

149 22201
150 22500
151 22801
15223104
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

Sq. Root.. Cube Root No. Square.

237 56169

238 56644

Cube.
3048625 12:0415946 5.253588 217 47089
3112136 12-0830460 5.265637 218 47524
3176523 12-1243557 5.277632 219 47961
3241792 12-1655251 5.289572 220 48400
3307949 12-2065556 5.301459 221 48841
3375000 12-2474487 5.313293 222 49284
3442951 12-2882057 5.325074 223 49729
3511808 12-3288280 | 5·336803 224 50176
3581577 12-3693169 5.348481 225 50625
3652264 12-4096736 5.360108 226 51076
3723875 12:4498996 5-371685 227 51529
3796416 12-4899960 5.383213 228 51984
3869893 12.5299641 5.394691 229 52441
3944312 12-5698051 5:406120 230 52900
4019679 12.6095202 | 5·417501 231 53361
4096000 12-6491106 5.428835 232 53824
4173281 12.6885775 5.440122 233 54289
4251528 12-7279221 5.451362 234 54756
4330747 12-7671453 5.462556 235 55225
4410944 12-8062485 5 473704 236 55696
4492125 12-8452326 | 5-484807
4574296 12-8840987 5:495865
4657463 12.9228480 | 5.506878 239 57121
4741632 12.9614814 5.517848 240 57600
4826809 13.0000000 5.528775 241 58081
4913000 13-0384048 | 5·539658 242 58564
5000211 13 0766968 5.550499 243 59049
5088448 13-1148770 5-561298 244 59536
5177717 13-1529464 5.572055 245 60025
5268024 13-1909060 5.582770 246 60516
175 30625 5359375 13-2287566 5.593445 247 61009
176 30976 5451776 13-2664992 5.604079 248 61504
177 31329 5545233 13-3041347 5-614672 249 62001
178 31684 5639752 13.3416641 5.625226 250 62500
179 32041 5735339 13.3790882 5.635741 251 63001
180 32400 5832000 13.4164079 5.646216 252 63504
181 32761 5929741 13 4536240 5.656653 253 64009
182 33124 6028568 13-4907376 5.667051 254 64516
183 33489 6128487 13-5277493 5.677411 255 65025
184 33856 6229504 13.5646600 5.687734 256 65536
185 34225 6331625 13.6014705 5.698019 257 66049
186 34596 6434856 13.6381817 | 5.708267 258 66564
187 34969 6539203 13.6747943 5.718479 259 67081
188 35344 6644672 13.7113092 5.728654 260 67600
189 35721 6751269 13-7477271 5-738794 261 68121
190 36100 6859000 13-7840488 5.748897 262 68644
191 36481 6967871 13-8202750 5.758965 263 69169
192 36864 7077888 13-8564065 5.768998 264 69696
193 37249 7189057 13.8924440 5.778996 265 70225
194 37636 7301384 13.9283883 5·788960 266 70756
195 38025 7414875 13.9642400 5.798890 267 71289
196 38416 7529536 14.0000000 5.808786 268 71824
197 38809 7645373 14-0356688 5.818648 269 72361
198 39204 7762392 14-0712473 5-828477 270 72900
199 39601 7880599 14-1067360 5-838272 271 73441
200 40000 8000000 14-1421356 | 5.848035 272 73984
201 40401 8120601 14-1774469 5.857766 273 74529
202 40804 8242408 14:2126704 5.867464 274 75076
203 41209 8365427 14-2478068 5.877131 275 75625
8489664 14-2828569 5.886765 276 76176
8615125 14-3178211 5.896368 277 76729
874181614-3527001 5.905941 278 77284
8869743 14-3874946 5.915482 279 77841
8998912 14:4222051 | 5.924992 280 78400
9129329 14-4568323 5·934473 281 78961
9261000 14:4913767 5.943922 282 79524
9393931 14:5258390 5.953342 283 80089
9528128 14.5602198 5.962732 284 80656
9663597 14:5945195 5.972093 285 81225
9800344 14-6287388 5.981424 286 81796
9938375 14-6628783 5.990726 287 82369
216 46656 10077696 14-6969385 6.000000 288 82944

204 41616
205 42025

206 42436
207 42849
208 43264
209 43681
210 44100
211 44521
212 44944
213 45369
214 45796
215 46225

[blocks in formation]

10218313 14-7309199 6·009245
10360232 14-7648231 6-018462
10503459 14.7986486 6-027650
10648000 14.8323970 6.036811
10793861 14.8660687 6·045943
10941048 14-8996644 6·055049
11089567 14-9331845 6-064127
11239424 14.9666295 6.073178
11390625 15-0000000 6.082202
11543176 15.0332964 6.091199
11697083 15.0665192 6·100170
11852352 15·0996689 6·109115
12008989 15-1327460 6·118033
12167000 15-1657509 6-126926
12326391 15-1986842 6-135792
12487168 15-2315462 6-144634
12649337 15-2643375 6-153449
12812904 15.2970585 6·162240
12977875 15-3297097 6-171006
13144256 15-3622915 6.179747
13312053 15-3948043 6.188463
13481272 15-4272486 6-197154
13651919 15.4596248 6.205822
13824000 15-4919334 6.214465
13997521 15.5241747 6-223084
14172488 15.5563492 6-231680
14348907 15.5884573 6.240251
14526784 15.6204994 6-248800
14706125 15-6524758 | 6-257325
14886936 15-6843871 | 6-265827
15069223 15-7162336 | 6-274305
15252992 15-7480157 6.282761
15438249 15-7797338 6-291195
15625000 15-8113883 6 299605
15813251 15.8429795 6.307994
16003008 15-8745079 6-316360
16194277 15-9059737 6-324704
16387064 15-9373775 6.333026
16581375 15.9687194 6-341326
16777216 16-0000000 | 6·349604
16974593 16-0312195 | 6·357861
17173512 16.0623784 6-366097
17373979 16.0934769 6-374311
17576000 16-1245155 6.382504
17779581 16-1554944 6.390676
17984728 16-186414] 6.398828
18191447 16-2172747 6-406958
18399744 16.2480768 6-415069
18609625 16-2788206 | 6·423158
18821096 16.3095064 6-431228
19034163 16-3401346 6·439277
19248832 16.3707055 6-447306
19465109 16-4012195 6-455315
19683000 16-43167676-463304
19902511 16-4620776 | 6·471274
20123648 16-4924225 6·479224
20346417 16:5227116 6-487154
2057082416-5529454 6-495065
20796875 16.5831240 6-502957
21024576 16.6132477 6.510830
21253933 16-6433170 6-518684
21484952 16-6733320 | 6-526519
21717639 16-7032931 6·534335
21952000 16·7332005 | 6·542133
22188041 16-7630546 6.549912
22425768 16·7928556 6.557672
22665187 16-8226038 | 6·565414
22906304 16-85229956-573139
23149125 16-8819430 6.580844
23393656 16-9115345 6.588532
23639903 16.9410743 6.596202
23887872 16-9705627 6-603854

No. Square.

Cube. Sq. Root. Cube Root No. Square. Cube.

4943086319-1572441

49836032 19-1833261

5024340919-2093727

7-159599

7·166096

7-172581|

Sq. Root. Cube Root
289 83521 24137569 17·0000000 | 6·611489 | 361|130321| 47045881 19:0000000| 7·120367
290 84100 24389000 17-0293864 6-619106 362|131044 47437928 19-0262976| 7-126936
291 84681 24642171 17·0587221 | 6-626705 363 131769 47832147 19-0525589 7-133492
292 85264 24897088 17-0880075 6-634287 364 132496 48228544 19-0787840 7-140037
293 85849 25153757 17-1172428 6-641852 365 133225 48627125 19-1049732 7·146569|
294 86436 25412184 17-1464282|6-649400 366 133956 49027896 19-1311265 7·153090
295 87025 25672375 17-1755640 6-656930 367 134689
296
87616 25934336 17-2046505 | 6·664444 368 135424
297 88209 26198073 17-2336879 6-671940 369 136161
298 88804 26463592 17-2626765 | 6·679420 370 136900
299 89401 26730899 17-2916165 | 6-686883
300 90000 27000000 17-32050816-694329
301 90601 27270901 17-3493516 6-701759
302 91204 27543608 17-37814726-709173
303 91809 27818127 17:4068952 | 6·716570
304 92416 28094464 17-43559586-723951
93025 28372625 17-4642492 6-731316
93636 28652616 17:4928557 | 6·738664
307 94249 28934443 17-52141556-745997
308 94864 29218112 17-5499288 6 753313
309 95481 29503629 17:5783958 6-760614
310 96100 29791000 17-6068169 | 6-767899

305

306

3338624817-9443584 | 6-854124

50653000 19-2353841
51064811 19-2613603
51478848 19-2873015

7 179054|

7-185516|

7-191966-

7·198405

5189511719-3132079

52313624 19-3390796

7·204832|

371 137641
372 138384
373 139129
374 139876
375 140625 52734375 19-3649167 7-211248|
376 141376 53157376 19-3907194 | 7·217652
377 142129 53582633 19-4164878 7-224045
378 142884

379143641

380 144400

392 153664

393 154449
394 155236

54010152 19-4422221

7-230427

[blocks in formation]

381 145161 55306341 19-5192213 7-249504
382 14592455742968 19-5448203 | 7·255841
311 96721 30080231 17-6351921 6-775169 383 146689 56181887 19-5703858 7:262167
312 97344 30371328 17·6635217 | 6-782423 384 147456
313 97969 30664297 17-6918060 6-789661 385 148225
314 98596 30959144 17-72004516-796884 386 148996
315 99225 31255875 17-7482393 6-804092 || 387 149769
316 99856 31554496 17·77638886·811285 388 150544
317 100489 31855013 17-8044938 6·818462 389 151321
318 101124 32157432 17-8325545 6·825624 390 152100
319 101761 32461759 17-8605711 6·832771 391 152881
320 102400 32768000 17-8885438 6·839904
321 103041 33076161 17-9164729 6 847021
|322 103684
|323 104329 33698267 17·9722008 | 6·861212 395 156025 6162987519-87460697-337234
324 104976 34012224 18-0000000 6·868285 396 156816 62099136 19-89974877-343420|
325 105625 34328125 18:0277564 6·875344 397 157609 62570773 19-9248588 7-349597
326 106276 34645976 18-0554701 | 6·882389 398 158404 63044792 19-9499373 7-355762
327106929 34965783 18:0831413 | 6·889419 399 159201
328 107584 35287552 18-1107703 | 6-896435 400 160000
329 108241 35611289 18:1383571 6·903436 401 160801
330 108900 35937000 18-1659021 6-910423 402 161604
331 109561 36264691 18-1934054 6-917396 403 162409
332110224 36594368 18-2208672 6·924356 404 163216
333 110889 36926037 18-2482876 6.931301 405 164025
334 111556 3725970418-2756669 6938232 406 164836
|335| 112225 37595375 18-3030052 | 6·945150 407 165649
|336112896 | 3793305618-3303028 | 6·952053 408 166464
337 113569 38272753 18-3575598 6-958943 409 167281

339 114921 38958219 18:4119526 6.972683 411 168921
340115600
341 116281
342116964

[blocks in formation]

3930400018·4390889 | 6·979532 412 169744
39651821 18-4661853 6·986368 413 170569
40001688 18:4932420 6·993191 414 171396
343 117649 40353607 18-5202592 7·000000 415 172225
344 118336 40707584 18-5472370 7-006796|| 416 173056
345 119025 41063625 18:5741756 7-013579 417 173889
346 119716 41421736 18-6010752 7·020349 418 174724
347 120409 41781923 18-6279360 7·027106 419 175561
348121104 42144192 18-6547581 7·033850 420 176400
349121801 4250854918-6815417 7·040581 421 177241
|350 122500 42875000 18-7082869 7·047299|| 422 178084
|351123201 43243551|18-7349940 7·054004 423 178929
352 123904 4361420818-7616630 7-060697 424 179776
353 124609 43986977 18-7882942 7-067377 425 180625
354 125316 44361864 18-8148877 7·074044
355 126025 44738875 18-84144377-080699
356 126736 45118016 18-8679623 7087341
357 127449 45499293 18-8944436 7·093971 429 184041
358 128164 45882712 18-9208879 7·100588 430 184900
35912888] 4626827918-9472953 7·107194 431 185761
360 129600 46656000 18-9736660 | 7·113787 432 186624

2M2

No. Square. Cube. Sq. Root. Cube Root No. Square. Cube. Sq. Root. Cube Root
433 187489 81182737 20-8086520 7.565355 505 255025 128787625 22-4722051 7.963374
434 188356 81746504 20-8326667 7-571174 506 256036 129554216 22:4944438 7.968627
435 189225 82312875 20-8566536 7.576985 507 257049130323843 22:5166605 7·973873
+36 190096 82881856 20-8806130 7.582786 508 258064 131096512 22-5388553 7.979112
437190969 83453453 20-9045450 7·588579 509 259081 131872229 22-5610283 7·984344
438 191844 84027672 20-9284495 7.594363 510 260100 132651000 22-5831796 7.989570
439 192721 84604519 20·9523268 7·600138 511 261121 133432831 22-6053091 7.994788
440 193600 85184000 20-9761770 7·605905 512 262144134217728 22-6274170 8.000000
441 194481 85766121 21-0000000 7·611663 513 263169135005697 22·6495033 8 005205
442 195364 86350888 21-0237960 7·617412 514 264196 135796744 22-6715681 8·010403
443 196249 86938307 21.0475652 7·623152 515 265225136590875 22-6936114 8.015595
444 197136 87528384 21-0713075 7.628884 516 266256 137388096 22.7156334 8-020779
445 198025 88121125 21-0950231 7·634607 517 267289 138188413 22-7376340 8·025957
446 198916 88716536 21-1187121 7·640321 518 268324 138991832 22-7596134 8·031129
447 199809 89314623 21·1423745 7·646027 519 269361 139798359 22-7815715 8-036293
448 200704 89915392 21-1660105 7.651725 520 270400 140608000 22-8035085 8·041451
449 201601 90518849 21·1896201 7·657414|| 521 271441 141420761 22-8254244 8.046603
450 202500 91125000 21 2132034 7·663094 || 522 272484 142236648 22-8473193 8-051748
451 203401 91733851 21-2367606 7.668766 523 273529 143055667 22-8691933 8.056886
452 204304 92345408 21-2602916 7·674430 524 274576|14387782422-8910463 8.062018
453 205209 92959677 21-2837967 7·680086 525 275625 144703125 22-9128785 8·067143
454 206116 93576664 21-3072758 7·685733 526 276676 145531576 22-9346899 8:072262
455 207025 94196375 21-3307290 7·691372 527 277729 146363183 22-9564806 8.077374
456 207936 94818816 21-3541565 7.697002 528 278784 147197952 22-9782506 8·082480
457 208849 95443993 21·3775583 7·702625 529 279841148035889 23-0000000 8·087579
458 209764 96071912 21-4009346 7·708239 530 280900148877000 23-0217289 8·092672
459 210681 96702579 21-4242853 7·713845 531 281961 149721291 23-0434372 8·097759
460 211600 97336000 21 4476106 7-719443 532 283024 150568768 23:0651252 8·102839
461 212521 97972181 21-4709106 7-725032 533 284089 151419437 23-0867928 8-107913
462 213444 98611128 21-4941853 7·730614 534 285156152273304 23-1084400 8 112980
463 214369 99252847 21-5174348 7·736188 535 286225 153130375 23-1300670 8-118041
464 215296 99897344 21 5406592 7·741753 536 287296153990656 23-1516738 8-123096
465 216225 100544625 21·5638587 7·747311 537 288369154854153 23·1732605 8·128145
466 217156 101194696 21-5870331 7·752861 538 289444 155720872 23-1948270 8·133187
467 218089 101847563 21-6101828 7.758402 539 290521 156590819 23-2163735 8.138223
468 219024 102503232 21-6333077 7·763936 540 291600 | 157464000 23-2379001 8·143253
469 219961 103161709 21 6564078 7·769462|| 541 292681 158340421 23-2594067 8·148276
470 220900 103823000 21.6794834 7-774980 542 293764159220088 23-2808935 8-153294
471 221841 104487111 21-7025344 7·780490 543 294849 160103007 23-3023604 8·158305
472 222784 105154048 21-7255610 7·785993 544 295936 160989184 23-3238076 8·163310
473 223729 105823817 21-7485632 7-791487 545 297025 161878625 23-3452351 8-168309
474 224676 106496424 21-7715411 7.796974 546 298116 162771336 23-3666429 8-173302
475 225625 107171875 21-7944947 7·802454 547 299209 163667323 23-3880311 8-178289
476 226576 107850176 21-8174242 7·807925 548 300304 164566592 23 4093998 8-183269
477 227529 108531333 21-8403297 7-813389 549 301401 165469149 23-4307490 8-188244
478 228484 109215352 | 21·8632111 7·818846 550 302500 166375000 23-4520788 8-193213
479 229441 109902239 21·8860686 7·824294 551 303601167284151 23-4733892 8.198175
480 230400 110592000 21·9089023 7·829735 552 304704168196608 23-4946802 8-203132
481 231361 111284641 21-9317122 7-835169 553 305809 169112377 23-5159520 8.208082
482 232324 111980168 21-9544984 7·840595 554 306916 170031464 23-5372046 8.213027
483 233289 112678587 21-9772610 7·846013 555 308025 170953875 23:5584380 8.217966
484 234256 113379904 22-0000000 7-851424 556 309136 171879616 23.5796522 8-222898
485 235225 114084125 22-0227155 7·856828
486 236196 114791256 22-0454077 7·862224
487 237169 115501303 22.0680765 7.867613
488 238144 116214272 22-0907220 7.872994
489 239121 116930169 22-1133444 7·878368 561 314721 176558481 23-6854386 8.247474
490 240100 117649000 22-1359436 7-883735 562 315844177504328 23-7065392 8.252371
491 241081 118370771 22:1585198 7-889095 563 316969 178453547 23-7276210 8 257263
492 24206411909548822-1810730 7-894447 564 318096 179406144 23-7486842 8-262149
493 243049 119823157 22-2036033 7.899792 565 319225 180362125 23 7697286 8.267029
494 244036 120553784 22-2261108 7.905129 566 320356 181321496 23 7907545 8-271904
495 245025 121287375 22-2485955 7.910460 567 321489 182284263 23-8117618 8-276773
496 246016 122023936 22-2710575 7915783 568 322624 183250432 23-8327506 8-281635
497 247009 122763473 22-2934968 7.921099 569 323761184220009 23-8537209 8-286493
498 248004 123505992 22-3159136 7-926408 570 324900 185193000 23-8746728 8.291344
499 249001 124251499 22-3383079 7·931710 571 326041 186169411 23-8956063 8-296190
500 250000 125000000 22-3606798 7937005 572 327184 187149248 23-9165215 8-301030
501 251001 125751501 22-3830293 7·942293 573 328329 188132517 23.9374184 8:305865
502 252004 126506008 22-4053565 7.947574 574 329476 189119224 23-9582971 8-310694
503 253009 127263527 22-4276615 7.952848 575 330625 190109375 23-9791576 8.315517
504 254016 128024064 22-4499443 7·958114 576 331776 191102976 24-0000000 8.320335

557 310249 172808693 23-6008474 8-227825
558 311364 173741112 23-6220236 8-232746
559 312481 174676879 23-6431808 8-237661
560 313600 175616000 23-6643191 8-242571

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