by the number of faces, will be the volume of the polyedron.

It only remains to deduce a formula for finding the altitude of the several pyramids, i. e., the distance from the centre to one face of the polyedron.

Conceive a perpendicular OC to be drawn from O, the centre of the polyedron, to one face; the foot of this perpendicular will be the centre of the face. From C, the foot of this perpendicular, draw a perpendicular to one side of the

D

face in which it lies, and connect the point D with the centre of the polyedron. There will thus be formed a right-angled triangle, OCD, whose base, CD, is the apothem of the face, whose angle ODC is half the angle CDL contained between two consecutive faces of the polyedron, and whose altitude OC is the required altitude of the pyramid, or, in other words, the radius of the inscribed sphere. This will be true for any one of the regular polyedrons-the hexaedron is taken here for simplicity of illustration.

Denote the line CD by p, the angle ODC by A, and the perpendicular OC by R. p may be found by the formula, given in Art. 101, for finding the apothem of a regular polygon; A may be found from the formula for sin ¿A, given in Art. 125; then, in the right-angled triangle OCD, we have, formula (3), Art. 37,

R = p tan A.

Compute the area of one of the faces of the given polyedron and multiply it by R, as determined by the formula just given, and multiply the result thus obtained by the number of faces of the polyedron; the final product will be the volume of the given regular polyedron.

The volumes of all the regular polyedrons have been computed on the supposition that their edges are each equal to 1, and the results are given in the following

From the principles demonstrated in Book VII., we may write the following

RULE.-To find the volume of any regular polyedron, multiply the cube of its edge by the corresponding tabular volume; the product will be the volume required.

Examples.

1. What is the volume of a tetraedron, whose edge is 15? Ans. 397.75.

2. What is the volume of a hexaedron, whose edge is 12? Ans. 1728.

3. What is the volume of an octaedron, whose edge is 20? Ans. 3771.236.

4. What is the volume of a dodecaedron, whose edge is 25? Ans. 119736.2328.

5. What is the volume of an icosaedron, whose edge is 20? Ans. 17453.56.

REMARKS. In the following table, in the nine right-hand 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, to catch the eye, and to indicate that from thence the two figures of the Logarithm to be taken from the second column, stand in the next line below.

N.

4

100 000000

101

102

4321
8600

103

012837

104

105

106

5306

107

9384

9789

108

033424

3826

109

7426 7825

1 2 3 5 0434 0868 1301 1734 2166 2598 3029 4751 5181 5609 6038 6466 6894 7321 9026 9451 9876 ◆300 .724 1147 1570 3259 3680 4100 4521 4940 5360 5779 7033 7451 7868 8284 8700 9116 9532 9947 021189 1603 2016 2428 2841 8252 3664 4075 4486 4896 412 5715 6125 6533 6942 7350 7757 8164 8571 8978 408 195 +600 1004 1408 1812 2216 2619 3021 404 4227 4628 5029 5430 5830 6230 6629 7028 400 8223 8620 9017 9414 9811 +207 +602 +998 396 041393 1787 2182 2576 2969 3362 3755 4148 4540 4932 393 5323 5714 6105 6495 6885 7275 7664 8053 8442 8830 389 9218 9606 9993 +380 +766 1153 1538 1924 2309 2694 386 3463 3846 4230 4618 4996 5378 5760 6142 6524 382 7286 7666 8046 8426 8805 9185 9563 9942 +320 379 1075 1452 1829 2206 2582 2958 3333 3709 4083 376 4832 5206 5580 5953 6326 6699 7071 7443 7815 372 8557 8928 9298 9668 ++38 +407 +776 1145 1514 369 2250 2617 2985 3352 3718 4085 4451 4816 5182 366 5547 5912 6276 6640 7004 7368 7731 8094 8457 8819 363 120 079181 9543 9904 +266 +626 .987 1347 1707 2067 2426 360 121 082785 3144 3503 3861 4219 4576 4934 5291 5647 6004 857 122 6360 6716 7071 7426 7781 8136 8490 8845 9198 9552 355 123 9905 +258 .611 +963 1315 1667 2018 2370 2721 3071 351 124 093422 3772 4122 4471 4820 5169 5518 5866 6215 6562 849 125 6910 7257 7604 7951 8298 8644 8990 9335 9681 ♦♦26 346 100371 0715 1059 1403 1747 2091 2434 2777 3119 3462 343 3804 4146 4487 4828 5169 5510 5851 6191 6531 6871 340 7210 7549 7888 8227 8565 8903 9241 9579 9916 +253 338 129 110590 0926 1263 1599 1934 2270 2605 2940 3275 3609 335 6276 6608 6940 333 7271 7603 7934 8265 8595 8926 9256 9586 9915 +245 330 132 120574 0903 1231 1560 1888 2216 2544 2871 3198 3525 328 133 3852 4178 4504 4830 5156 5481 5806 6131 6456 6781 325 7105 7429 7753 8076 8399 8722 9045 9368 9690 +12 323 130334 0655 0977 1298 1619 1939 2260 2580 2900 3219 321 3539 3858 4177 4496 4814 5133 5451 5769 6086 6403 318 6721 7037 7354 7671 7987 8303 8618 8934 9249 9564 315 9879 •194 +508 +822 1136 1450 1763 2076 2389 2702 314 3327 3639 3951 4263 4574 4885 5196 5507 5818 311 146128 6438 6748 7058 7367 7676 7985 8294 8603 8911 309 9219 9527 9835 +142 +449 +756 1063 1370 1676 1982 307 152288 2594 2900 3205 3510 3815 4120 4424 4728 5032 305 5336 5640 5943 6246 6549 6852 7154 7457 7759 8061 303 8362 8664 8965 9266 9567 9868 +168 +469 +769 1068 301 161368 1667 1967 2266 2564 2863 3161 3460 3758 4055 299 4353 4650 4947 5244 5541 5838 6134 6430 6726 7022 297 7317 7613 7908 8203 8497 8792 9086 9380 9674 9968 295 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 5542 5825 6108 6391 6674 6956 7239 283 7521 7803 8084 8366 8647 8928 9209 9490 9771 +51 281 190332 0612 0892 1171 1451 1730 2010 2289 2567 2846 279 3125 3403 3681 3959 4237 4514 4792 5069 5346 5623 278 5899 6176 6453 6729 7005 7281 7556 7832 8107 8382 276 8657 8932 9206 9481 9755 ♦♦29 +303 +577 201397 1670 1943 2216 2488 2761 3033 3305

N.

1 | 2

3 4

161

162

160 204120 6826 9515

163

212188

164

4844

165

7484

166

220108 0370

167

168

169

170

230449

0704

5 4391 4663 4934 5204 5475 6556 271 7096 7365 7634 7904 8178 9247 269 9783 ◆◆51 +319 +586 +853 1121 1388 1654 1921 267 2454 2720 2986 3252 3518 3783 4049 4314 4579 266 5109 5373 5638 5902 6166 6430 6694 6957 7221 264 7747 8010 8273 8536 8798 9060 9323 9585 9846 262 0631 0892 1153 1675 1936 2716 2976 3236 3496 3755 4015 4274 4533 4792 5051 259 5309 5568 5826 6084 6342 6600 6858 7115 7372 7630 258 7887 8144 8400 8657 8913 9170 0960 1215 1470 1724 1979 2234 2488 2742 254 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 240549 0799 1048 1297 1546 1795 2044 2293 2541 2790 249 3038 3286 3534 3782 4030 4277 4525 4772 5019 5266 248 5513 5759 6006 6252 6499 6745 6991 7237 7482 7728 246 7973 8219 8464 8709 8954 9198 9443 9687 9932 +176 245 0664 0908 1151 1395 1638 1881 2125 2368 2610 248 3096 3338 8580 3822 4064 4306 4548 4790 5031 242 5514 5755 5996 6237 6477 6718 6958 7198 7439 241 7918 8158 8398 8637 8877 9116 9355 9594 9833 239 0310 0548 0787 1025 1263 1501 1739 1976 2214 238 2451 2688 2925 3162 3399 3636 8873 4109 4346 4582 237 4818 5054 5290 5525 5761 5996 6232 6467 6702 6937 235 185 7172 7406 7641 7875 8110 8344 8578 8812 9046 9279 234 186 9513 9746 9980 +213 +446 +679 +912 1144 1877 1609 233 187 271842 2074 2306 2538 2770 3001 3233 8464 3696 3927 232 188 4158 4889 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 8527 3753 3979 4205 4431 4656 4882 5107 5332 226 193 5557 5782 6007 6232 6456 1881 6905 7180 7354 7578 225 194 7802 8026 8249 8473 8696 8920 9143 9366 9589 9812 223 195 290035 0257 0480 0702 0925 1147 1369 1591 1813 2034 222 2256 2478 2699 2920 3141 3363 8584 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 +878 +595 813 218 301030 1247 1464 1681 1898 2114 2331 2547 2764 2980 217 3196 3412 3628 3844 4059 4275 4491 4706 4921 5136 216 5351 5566 5781 5996 6211 6425 6639 6854 7068 7282 215 203 7496 7710 7924 8137 8351 8778 8991 9204 9417 213 204 9630 9843 ♦♦56 +268 481 .693 +906 1118 1330 1542 212 205 311754 1966 2177 2389 2600 2812 3023 3234 8445 3656 211 206 3867 4078 4289 4499 4710 4920 5130 5340 5551 5760 210 5970 6180 6390 6599 6809 7018 7227 7436 7646 7854 209 8063 8272 8481 8689 8898 9106 9314 9522 9730 9938 208 209 320146 0354 0562 0769 0977 1184 1391 1598 1805 2012 207

210 822219 2426 2633 2839 3046 3252 3458 3665 3871 4077 206 211 4282 4488 4694 4899 5105 5310 5516 5721 5926 6131 205 212 6336 6541 6745 6950 7155 7359 7563 7767 7972 8176 204 213 8380 8583 8787 8991 9194 9398 9601 9805 +++8 211 203 214 830414 0617 0819 1022 1225 1427 1630 1832 2034 2236 202 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 200 218 8456 8656 8855 9054 9253 9451 9650 9849 ..47 +246 199 219 840444 0642 0841 1039 1237 1435 1632 1830 2028 2225 198