} '^f...^^t2^''p^-t!~^^ ELEMENT AET Mathematical Tables. BY ALEXANDER MACFAELANE, D.Sc, LL.D., Pbopessoe of Physics in the Univebsitt op Texas. Boston, U.S.A., and London: GINN & COMPANY, PUBLISHERS. 1894. Entired at Stationers' Hall. Entered, according to' Act of Congress, in the year 1889, bv ALEXANDER MACFARLANE, in the OflBce of the Librarian of Congress, at Washington, All Rights Reserved. ASTBoiKanr DIPT. Ttpographt by J. 8. Gushing & Co., Boston, U.S.A. Presswobk by Ginn & Co., Boston, U.S.A. ; /\ _L PREFACE. rr^HESE tables are designed to be useful not only in com- -*- puting and in the graphic method, but also in the teaching of arithmetic and in the illustration of the theo- rems of algebra. I have arranged the several tables on a uniform decimal plan, so that the entries for a particular number are generally found in the same position on the page. The arrangement is that of double entry, so that in general the order of reading is the same as for ordinary print. The argument and entry are so expressed that it is easy to find the entry corresponding to any other position of the decimal point in. the argument. In most cases the whole of a table is seen at one opening of the- pages, and the tenth compartment, when not required for the main table, is filled with a short table which is in general auxiliary to the main table. Special acknowledgments are due to Prof. Hastings of Yale University, and Prof. Halsted of this University. In the proof-reading and independent computation, I have re- ceived aid from D. W. Spence and J. C. Nagle, science students of this University. Alexander Macfaklaxe. University of Texas, April, 1889. M298746 CONTENTS. PAGES I. Common logarithms. Proportional parts,* 4-23 . 2-3 The same for 1000-1900. Logarithms for interest . 4-5 II. Aiitilogarithms 6-7 III. Addition logarithms 8-9 IV. Subtraction logarithms. Proportional parts, 1-38 . 10-13 Y. Logarithmic sines and cosines. Proportional parts, 11-30 14-1.5 The same for 0°.00 to 9°. 00. Equivalent of degrees in minutes. Table of S for degrees . . . 16-17 YI. Logarithmic tangents and cotangents. Proportional parts, 31-50 18-19 The same for 0°.00 to 9°.00. Equivalent of minutes in degrees. Table of T for degrees . . . 20-21 YIL Logarithmic sines and cosines for minutes. Propor- tional parts, l~^:Q. Table of S for minutes . 22-23 VIII. Logarithmic tangents and cotangents for minutes. Proportional parts, 50-99. Table of T for min- utes 24-25 IX. Natural sines and cosines. Proportional pai-ts to six, 1-19 26-27 X. Natural tangents and cotangents. Proportional parts to six, 20-59 28-29 XL Natural secants and cosecants. Proportional parts to six, 60-99 30-31 XII. Radians equivalent to degrees. Radians equivalent to m,inutes. Fraction of period, and time, equiva- lent to degrees 32-33 * A caption in Italics indicates a small table appended to the main table. CONTENTS. PAGES Xni. Reciprocals, Multiples of ^ to ^^ . . , . 34-35 XIV. Squares. Square of reciprocal 36-37 XV. Cubes. Cube of reciprocal 38-39 XVI. Square roots. Square root of reciprocal . . . 40-43 XVII. Cube roots. Cube root of reciprocal . . . 44-49 XVIII. Multiples of 100-999 50-67 XIX. Circumference of circle. Constants involving ir . 68-69 XX. Area of circle. Multiples of -^^ and ^^ . . . 70-71 XXI. Content of sphere. Logarithms of factorials and of Powersoft 72-73 XXII. Hyperbolic logarithms. Multiples of the modulus . 74-75 XXIII. Amount at the end of n years .... 76 XXIV. Present value of 1000 77 XXV. Amount of an annuity paid at the end of each year 78 XXVI. Present value of the preceding .... 79 XXVII. Amount of an annuity paid at the beginning of each year 80 XXVIII. Sum to be paid at the end of each of n years to extinguish a debt of 1000 .... 81 XXIX. Least divisors 82-85 XXX. Exponentials 86 Explanation of Tables 87-105 XXXI. Multiples of 1-100 Folding leaf I. COMMON LOGAEITHMS. logn n 1 2 3 4 5 7 8 9 d 10 0000 0043 0086 0128 0170 0212 0253 0294 0334 0374 40 11 12 13 0414 0792 1139 0453 0492 0531 0S2S 0864 0899 1173 1206 1239 0569 0607 0645 0934 0969 1004 1271 1303 1335 0682 0719 0755 1038 1072 1106 1367 1399 1430 37 33 31 U 15 IG 1461 1761 2041 1492 1523 1553 1790 1818 1847 2068 2095 2122 1584 1614 1644 1875 1903 1931 2148 2175 2201 1673 1703 1732 1959 1987 2014 2227 2253 2279 29 27 25 17 IS 19 2304 2553 27SS 2330 2355 2380 2577 260i 2625 2810 2833 2856 2405 2430 2455 2648 2672 2695 2878 2900 2923 2480 2504 2529 2718 2742 2765 2945 2967 2989 24 23 21 20 3010 3032 3054 3075 3096 3118 3139 3160 3181 3201 21 21 22 23 3222 3424 3617 3243 3263 3284 3444 3464 3483 3636 3655 3674 3304 3324 3345 3502 3522 3541 3692 3711 3729 3365 3385 3404 3560 3579 3598 3747 3766 3784 20 19 18 24 25 2G 3802 3979 4150 3820 3838 3856 3997 4014 4031 4166 4183 4200 3874 3892 3909 4048 4065 4082 4216 4232 4249 3927 3945 3962 4099 4116 4133 4265 4281 4298 17 17 16 27 28 29 4314 4472 4624 4330 4346 4362 4487 4502 4518 4639 4654 4669 4378 4393 4409 4533 4548 4564 4683 4698 4713 4425 4440 4456 4579 4594 4609 4728 4742 4757 16 15 14 30 4771 4786 4800 4814 4829 4843 4857 4871 4886 4900 14 31 32 33 4914 5051 5185 4928 4942 4955 5065 5079 5092 5198 5211 5224 4969 4983 4997 5105 5119 5132 5237 5250 5263 5011 5024 5038 5145 5159 5172 5276 5289 5302 13 13 13 34 35 36 5315 5441 5563 5328 5340 5353 5453 5465 5478 5575 5587 5599 5366 5378 5391 5490 5502 5514 5611 5623 5635 5403 5416 5428 5527 5539 5551 5647 5658 5670 13 12 12 37 38 39 5682 5798 5911 5694 5705 5717 5809 5821 5832 5922 5933 5944 5729 5740 5752 5843 5855 5866 5955 5966 5977 5763 5775 5786 5877 5888 5899 5988 5999 6010 12 12 11 40 6021 6031 6042 6053 6064 6075 6085 6096 6107 6117 11 41 42 43 6128 6232 633i 6138 6149 6160 6243 6253 6263 6345 6355 6365 6170 6180 6191 6274 6284 6294 6375 6385 6395 6201 6212 6222 6304 6314 6325 6405 6415 6425 10 10 10 44 45 46 6435 6532 6628 6444 6454 6464 6542 6551 6561 6637 6646 6656 6474 6484 6493 6571 6580 6590 6665 6675 6684 6503 6513 6522 6599 6609 6618 6693 6702 6712 10 10 9 47 48 49 6721 6S12 6902 6730 6739 6749 6821 6830 6839 691] 6920 6928 6758 6767 6776 6848 6857 6866 6937 6946 6955 6785 6794 6803 6875 6884 6893 6964 6972 6981 9 9 9 50 6990 6998 7007 7016 7024 7033 7042 7050 7059 7067 9 61 52 53 7076 7160 7243 7084 7093 7101 7168 7177 7185 7251 7259 7267 7110 7118 7126 7193 7202 7210 7275 7284 7292 7135 7143 7152 7218 7226 7235 7300 7308 7316 8 8 8 54 55 56 7324 7404 7482 7332 7340 7348 7412 7419 7427 7490 7497 7505 7356 7364 7372 7435 7443 7451 7513 7520 7528 7380 7388 7396 7459 7466 7474 7536 7543 7551 8 8 8 57 58 69 7559 7634 77,09 7566 7574 7582 7642 7649 7657 7716 7723 7731 7589 7597 7604 7664 7672 7679 7738 7745 7752 7612 7619 7627 7686 7694 7701 7760 7767 7774 7 8 8 lOgfl I. COMMON LOQAEITHMS. n 12 3 4 5 6 7 8 9 d 60 7782 7789 7796 7803 7810 7818 7825 7832 7839 7846 7 61 7853 7860 7868 7875 7882 7889 7896 7903 7910 7917 7 62 7924 7931 7938 7945 7952 7959 7966 7973 7980 7987 6 63 7993 8000 8007 8014 8021 8028 8035 8041 8048 8055 7 64 8062 8069 8075 8082 8089 8096 8102 8109 8116 8122 7 65 8129 8136 8142 8149 8156 8162 8169 8176 8182 8189 ' 6 66 8195 8202 8209 8215 8222 8228 8235 8241 8248 8254 7 67 8261 8267 8274 8280 8287 8293 8299 8306 8312 8319 6 68 8325 8331 8338 8344 8351 8357 8363 8370 8376 8382 6 69 8388 8395 8401 8407 8414 8420 8426 8432 8439 8445 6 70 8451 8457 8463 8470 8476 8482 8488 8494 8500 8506 7 71 8513 8519 8525 8531 8537 8543 8549 8555 8561 8567 6 72 8573 8579 8585 8591 8597 8603 8609 8615 8621 8627 6 73 8633 8639 8645 8651 8657 8663 8669 8675 8681 8686 6 74 8692 S698 8704 8710 8716 8722 8727 8733 8739 8745 6 75 8751 8756 8762 8768 8774 8779 8785 8791 8797 8802 6 76 8808 8814 8820 8825 SS31 8S37 8842 8848 8854 8859 6 77 8865 8871 8876 8882 8SS7 8893 8899 8904 8910 8915 6 78 8921 8927 8932 8938 8943 8949 8954 8960 8965 8971 5 79 8976 8982 8987 8993 8998 9004 9009 9015 9020 9025 6 80 9031 9036 9042 9047 9053 9058 9063 9069 9074 9079 6 81 9085 9090 9096 9101 9106 9112 9117 9122 9128 9133 82 9138 9143 9149 9154 9159 9165 9170 9175 9180 9186 83 9191 9196 9201 9206 9212 9217 9222 9227 9232 9238 84 9243 9248 9253 9258 9263 9269 9274 9279 9284 9289 85 9294 9299 9304 9309 9315 9320 9325 9330 9335 9340 86 9345 9350 9355 9360 9365 9370 9375 9380 9385 9390 87 9395 9400 9405 9410 9415 9420 9425 9430 9435 9440 88 9445 9450 9455 9460 9465 9469 9474 9479 9484 9489 89 9494 9499 9504 9509 9513 9518 9523 9528 9533 9538 90 9542 9547 9552 9557 9562 9566 9571 9576 9581 9586 91 9590 9595 9600 9605 9609 9614 9619 9624 9628 9633 92 9638 9643 9647 9652 9657 9661 9666 9671 9675 9680 93 9685 9689 9694 9699 9703 9708 9713 9717 9722 9727 94 9731 9736 9741 9745 9750 9754 9759 9763 9768 9773 95 9777 9782 9786 9791 9795 9800 9805 9809 9814 9818 96 9823 9827 9832 9836 9841 9845 9850 9854 9859 9S63 97 9868 9872 9877 9881 9886 9890 9894 9899 9903 9908 98 9912 9917 9921 9926 9930 9934 9939 9943 9948 9952 99 9956 9961 9965 9969 9974 9978 9983 9987 9991 9996 pp. 4 5 6789 10 11 12 13 , 14 15 16 17 18 19 20 21 22 23 1 1 1 1 1 1 T 1 1 1 2 2 ~2 2 2 ~2 2 2 2 2 1 1 1 1 2 2 2 2 2 3 3 3 3 4 4 4 4 4 5 3 1 2 2 2 2 3 3 3 4 4 5 5 5 5 6 6 6 7 7 4 2 2 2 3 3 4 4 4 5 6 6 6 1 7 8 8 8 9 9 5 2 3 3 4 4 5 S 6 6 7 8 8 9 9 10 10 11 11 12 6 2 3 4 4 5 5 6 7 7 8 8 9 10 IC 11 11 12 13 13 14 7 3 4 4 5 6 6 7 8 8 9 10 11 11 12 13 13 14 15 15 16 8 3 4 5 6 6 7 8 9 10 10 11 12 13 14 14 15 16 17 18 18 9 4 5 5 6 7 8 9 10 11 12 13 14 14 IS 16 17il8 19 20 21 I. COMMON LOGAEITHMS logn n 12 3 4 5 6 7 8 9 d pp. 100 0000 0004 0009 0013 0017 0022 0026 0030 0035 0039 4 5 101 102 103 0043 00S6 012S 0048 0052 0056 0090 0095 0099 0133 0137 0141 0060 0065 0069 0103 0107 0111 0145 0149 0154 0073 0077 0082 0116 0120 0124 0158 0162 qi66 4 4 4 1 1 2 104 105 100 0170 0212 0253 0175 0179 0183 0216 0220 0224 0257 0261 0265 0187 0191 0195 0228 0233 0237 0269 0273 0278 0199 0204 0208 0241 0245 0249 0282 0286 0290 4 4 4 2 3 3 107 lOS 109 0294 0334 0374 029S 0302 0306 0338 0342 0346 0378 0382 0386 0310 0314 0318 0350 0354 0358 0390 0394 0398 0322 0326 0330 0362 0366 0370 0402 0406 0410 4 4 4 4 4 5 110 0414 041S 0422 0426 0430 0434 0438 0441 0445 0449 4 111 112 113 0453 0492 0531 0457 0461 0465 0496 0500 0504 0535 0538 0542 0469 0473 0477 0508 0512 0515 0546 0550 0554 0481 0484 0488 0519 0523 0527 0558 0561 0565 4 4 4 lU 115 IIG 0569 0607 0645 0573 0577 05S0 0611 0615 0618 0648 0652 0656 0584 0588 0592 0622 0626 0630 0660 0663 0667 0596 0599 0603 0633 0637 0641 0671 0674 0678 4 4 4 117 118 119 06S2 0719 0755 0686 0689 0693 0722 0726 0730 0759 0763 0766 0697 0700 0704 0734 0737 0741 0770 0774 0777 0708 0711 0715 0745 0748 0752 0781 0785 0788 4 3 4 120 0792 0795 0799 0803 0806 0810 0813 0817 0821 0824 4 4 121 122 123 0S28 0S64 0899 083 1 0835 0839 0867 0S71 0874 0903 0906 0910 0842 0846 0849 0878 0881 0885 0913 0917 0920 0853 0856 0860 0888 0892 0896 0924 0927 0931 4 3 3 1 1 124 125 126 0934 0969 1004 0938 0941 0945 0973 0976 0980 1007 1011 1014 0948 0952 0955 0983 0986 0990 1017 1021 1024 0959 0962 0966 0993 0997 1000 1028 1031 1035 3 4 3 2 2 2 127 128 129 1038 1072 1106 1041 1045 1048 1075 1079 1082 1109 1113 1116 1052 1055 1059 1086 1089 1092 1119 1123 1126 1062 1065 1069 1096 1099 1103 1129 1133 1136 3 3 3 3 3 4 130 1139 1143 1146 1149 1153 1156 1159 1163 1166 1169 4 131 132 133 1173 1206 1239 1176 1179 1183 1209 1212 1216 1242 1245 1248 1186 1189 1193 1219 1222 1225 1252 1255 1258 1196 1199 1202 1229 1232 1235 1261 1265 1268 4 4 3 134 135 13G 1271 1303 1335 1274 1278 1281 1307 1310 1313 1339 1342 1345 1284 1287 1290 1316 1319 1323 1348 1351 1355 1294 1297 1300 1326 1329 1332 1358 1361 1364 3 3 3 137 138 139 1367 1399 1430 1370 1374 1377 1402 1405 1408 1433 1436 1440 1380 1383 1386 1411 1414 1418 1443 1446 1449 1389 1392 1396 1421 1424 1427 1452 1455 1458 3 3 3 140 1461 1464 1467 1471 1474 1477 1480 1483 1486 1489 3 3 141 142 143 1492 1523 1553 1495 1498 1501 1526 1529 1532 1556 1559 1562 1504 1508 1511 1535 1538 1541 1565 1569 1572 1514 1517 1520 1544 1547 1550 1575 1578 1581 3 3 3 1 1 144 145 140 1584 1614 1644 1587 1590 1593 1617 1620 1623 1647 1649 1652 1596 1599 1602 1626 1629 1632 1655 1658 1661 1605 1608 1611 1635 1638 1641 1664 1667 1670 3 3 3 1 2 2 147 148 149 1673 1703 1732 1676 1679 1682 1706 1708 1711 1735 1738 1741 1685 1688 1691 1714 1717 1720 1744 1746 1749 1694 1697 1700 1723 1726 1729 1752 1755 1758 3 3 3 2 2 3 logn CONTINUED. n 12 3 4 5 6 7 8 9 d pp. 150 1761 1764 1767 1770 1772 1775 1778 1781 1784 1787 3 3 151 152 153 1790 1818 1847 1793 1796 1798 1821 1824 1827 1850 1853 1855 ISOl 1804 1807 1830 1833 1836 1858 1861 1864 1810 1813 1816 1838 1841 1844 1867 1870 1872 2 3 3 1 1 154 155 156 1875 1903 1931 1878 1881 1SS4 1906 1909 1912 1934 1937 1940 1886 1889 1892 1915 1917 1920 1942 1945 1948 1895 1898 1901 1923 1926 1928 1951 1953 1956 2 3 3 1 2 2 157 158 159 1959 1987 2014 1962 1965 1967 1989 1992 1995 . 2017 2019 2022 1970 1973 1976 1998 2000 2003 2025 2028 2030 1978 1981 1984 2006 2009 2011 2033 2036 2038 3 3 3 2 2 3 160 2041 2044 2047 2049 ! 2052 2055 2057 2060 2063 2066 2 161 162 163 2068 2095 2122 2071 2074 2076 2098 2101 2103 2125 2127 2130 2079 2082 2084 2106 2109 2111 2133 2135 2138 2087 2090 2092 2114 2117 2119 2140 2143 2146 3 3 2 164 165 166 2148 2175 2201 2151 2154 2156 2177 2180 2183 2204 2206 2209 2159 2162 2164 2185 2188 2191 2212 2214 2217 2167 2170 2172 2193 2196 2198 2219 2222 2225 3 3 2 167 168 169 2227 2253 2279 2230 2232 2235 2256 2258 2261 2281 2284 22S7 2238 2240 2243 2263 2266 2269 2289 2292 2294 2245 2248 2251 2271 2274 2276 2297 2299 2302 2 3 2 170 2304 2307 2310 2312 2315 2317 2320 2322 2325 2327 3 2 171 172 173 2330 2355 2380 2333 2335 2338 2358 2360 2363 2383 2385 2388 2340 2343 2345 2365 2368 2370 2390 2393 2395 2348 2350 2353 2373 2375 2378 2398 2400 2403 2 2 2 1 174 175 176 2405 2430 2455 2408 2410 2413 2433 2435 2438 2458 2460 2463 2415 2418 2420 2440 2443 2445 2465 2467 2470 2423 2425 2428 2448 2450 2453 2472 2475 2477 2 2 3 1 1 1 177 178 179 180 2480 2504 2529 2482 2485 2487 2507 2509 2512 2531 2533 2536 2490 2492 2494 2514 2516 2519 2538 2541 2543 2497 2499 2502 2521 2524 2526 2545 2548 2550 2 3 3 1 2 2 2553 2555 2558 2560 2562 2565 2567 2570 2572 2574 3 181 182 183 2577 2601 2625 2579 2582 2584 2603 2605 2608 2627 2629 2632 2586 2589 2591 2610 2613 2615 2634 2636 2639 2594 2596 2598 2617 2620 2622 2641 2643 2646 3 3 2 184 185 186 2648 2672 2695 2651 2653 2655 2674 2676 2679 2697 2700 2702 2658 2660 2662 2681 2683 2686 2704 2707 2709 2665 2667 2669 2688 2690 2693 2711 2714 2716 3 2 2 187 188 189 2718 2742 2765 2721 2723 2725 2744 2746 2749 2767 2769 2772 2728 2730 2732 2751 2753 2755 2774 2776 2778 2735 2737 2739 2758 2760 2762 2781 2783 2785 3 3 3 n 1 2 3 4 5 6 7 8 9] d 430 1.00 .00 000 0434 0S6S 1301 1734 2166 2598 3029 3461 38 ^1 1.01 1.02 1.03 .00 432 .00 860 .01 283 1 7 4751 5181 5609 9026 9451 9S76 3259 3680 4100 6038 6466 6894 *0300*0724*1147 4521 4940 5360 7321 7748 81 *1570*1993*24 5779 6197 66 74 15 16 426 422 417 1.04 1.05 1.06 .01 703 .02 lis .02 53C 3 9 6 7451 7868 8284 1603 2016 2428 5715 6125 6533 8700 9116 9532 2841 3252 3664 6942 7350 7757 9947*0361*07 4075 4486 48 8164 8571 89 75 96 78 414 410 406 1.07 |.02 935 1.081.03 342 1.091.03 742 4 4 6 9789*0195*0600 3826 4227 4628 7825 8223 8620 *1004*140S*1812 5029 5430 5830 9017 9414 9811 *22] 6*2619*30 6230 6629 70 *0207*0602*09 21 28 98 403 398 395 11. ANTILOGAEITHMS. 10 " In 12 3 4 5 (J 7 8 9 d pp. 00 1000 1002 1005 1007 1009 1012 1014 1016 1019 1021 2 234 01 02 03 1023 1047 1072 1026 1028 1030 1050 1052 1054 1074 1076 1079 1033 1035 1057 1059 1081 1084 1038 1062 1086 10-10 1042 1045 1064 1067 1069 1089 1091 1094 2 3 2 000 01 1 1 1 1 04 05 00 1096 1122 IHS 1099 1102 1104 1125 1127 1130 115l 1153 1156 1107 1109 1132 1135 1159 1161 1112 1138 1164 1114 1140 1167 1117 1119 1143 1146 1169 1172 3 2 3 1 1 2 1 22 122 07 08 09 1175 1202 1230 1178 1180 11S3 1205 1208 1211 1233 1236 1239 1186 1189 1213 1216 1242 1245 1191 1219 1247 1194 1222 1250 1197 1199 1225 1227 1253 1256 3 3 3 123 223 234 10 1259 1262 1265 1268 1271 1274 1276 1279 1282 1285 3 56 7 11 12 13 12SS 131S 1349 1291 1294 1297 1321 1324 1327 1352 1355 1358 1300 1303 1330 1334 •1361 1365 1306 1337 1368 1309 1340 1371 1312 1315 1343 1346 1374 1377 3 3 3 1 1 1 1 11 222 14 15 IG 13S0 1413 1445 1384 1387 1390 1416 1419 1422 1449 1452 1455 1393 1396 1400 1426 1429 1432 1459 1462 1466 1403 1435 1469 1406 1409 1439 1442 1472 1476 4 3 3 223 334 344 17 18 19 1479 1514 1549 1483 1486 1489 1517 1521 1524 1552 1556 1560 1493 1496 1528 1531 1563 1567 1500 1535 1570 1503 1538 1574 1507 1510 1542 1545 1578 1581 4 4 4 445 456 556 20 15S5 1589 1592 1596 1600 1603 1607 1611 1614 1618 4 8 9 21 23 1622 1660 1698 1626 1629 1633 1663 1667 1671 1702 1706 1710 1637 1641 1644 1675 1679 1683 1714 1718 1722 1648 1652 1656 1687 1690 1694 1726 1730 1734 4 4 4 1 1 2 2 2 3 24 25 26 1738 1778 1820 1742 1746 1750 1782 1786 1791 1824 1828 1832 1754 1758 1795 1799 1837 1841 1762 1803 1845 1766 1770 1774 1807 1811 1816 1849 1854 1858 4 4 4 3 4 4 5 5 5 27 28 29 1862 1905 1950 1866 1871 1875 1910 1914 1919 1954 1959 1963 1879 1884 1923 1928 1968 1972 1888 1932 1977 1892 1936 1982 1897 1901 1941 1945 1986 1991 4 5 4 6 6 6 7 7 8 30 1995 2000 2004 2009 2014 2018 2023 2028 2032 2037 5 1011 31 32 33 2042 2089 2138 2046 2051 2056 2094 2099 2104 2143 2148 2153 2061 2065 2109 2113 2158 2163 2070 2118 2168 2075 2123 2173 2080 2084 2128 2133 2178 2183 5 5 5 1 1 2 2 3 3 34 35 30 2188 2239 2291 2193 2198 2203 2244 2249 2254 2296 2301 2307 2208 2213 2259 2265 2312 2317 2218 2270 2323 2223 2275 2328 2228 2234 2280 2286 2333 2339 5 5 5 4 4 5 6 6 7 37 38 39 2344 2399 2455 2350 2355 2360 2404 2410 2415 2460 2466 2472 2366 2371 2421 2427 2477 2-183 2377 2432 2^89 2382 2438 2495 2388 2393 2443 2449 2500 2506 6 6 6 7 8 8 9 9 10 40 2512 2518 2523 2529 2535 2541 2547 2553 2559 2564 6 1213 41 42 43 2570 2630 2692 2576 2582 2588 2636 2642 2649 2698 2704 2710 2594 2600 2606 2655 2661 2667 2716 2723 2729 2612 2673 2735 2618 2624 2679 2685 2742 2748 6 7 6 1 1 2 3 4 4 44 45 40 2754 2818 2884 2761 2767 2773 2825 2831 2838 2891 2897 2904 2780 2786 2793 28 H 2851 2858 2911 2917 2924 2799 2S05 2812 2864 2871 2877 2931 2938 2944 6 5 5 6 7 7 8 47 2951 48 3020 49 , 3090 2958 2965 2972 3027 3034 3041 3097 3105 3112 2979 2985 2992 3048 3055 3062 3119 3126 3133 2999 3006 3013 3069 3076 3083 3141 3148 3155 8 9 10 10 11 12 10 11. ANTILOGAEITHMS. n 12 3 4 5 6 7 8 9 d pp. 50 3162 3170 3177 3184 3192 3199 3206 3214 3221 3228 8 1415 51 52 53 3236 3311 3388 3243 3251 3258 3319 3327 3334 3396 3404 3412 3266 3273 3281 3342 3350 3357 3420 3428 3436 3289 3296 3304 3365 3373 3381 3443 3451 3459 7 7 8 1 2 3 3 4 5 54 55 56 3467 3548 3631 3475 3483 3491 3556 3565 3573 3639 3648 3656 3499 3508 3516 3581 3589 3597 3664 3673 3681 3524 3532 3540 3606 3614 3622 3690 3698 3707 8 9 8 6 6 7 8 8 9 57 58 59 60 3715 3802 3S90 3724 3733 3741 3811 3819 3828 3899 3908 3917 3750 3758 3767 3837 3846 3855 3926 3936 3945 3776 3784 3793 3864 3873 3882 3954 3963 3972 9 8 9 1011 1112 13 14 3981 3990 3999 4009 4018 4027 4036 4046 4055 4064 lo!l617| 61 62 63 4074 4169 4266 4083 4093 4102 4178 41SS 4198 4276 4285 4295 4111 4121 4130 4207 4217 4227 4305 4315 4325 4140 4150 4159 4236 4246 4256 4335 4345 4355 10 10 10 11 5 5 64 65 66 4365 4467 4571 4375 4385 4395 4477 4487 4498 4581 4592 4603 4406 4416 4426 4508 4519 4529 4613 4624 4634 4436 4446 4457 4539 4550 4560 4645 4656 4667 10 11 10 6 7 8 9 1010 67 68 69 4677 4786 4898 4688 4699 4710 4797 4808 4819 4909 4920 4932 4721 4732 4742 4831 4842 4853 4943 4955 4966 4753 4764 4775 4864 4875 4887 4977 4989 5000 11 11 12 1112 13 14 1415 70 5012 5023 5035 5047 5058 5070 5082 5093 5105 5117 12|l819| 71 72 73 5129 5248 5370 5140 5152 5164 5260 5272 5284 5383 5395 5408 5176 5188 5200 5297 5309 5321 5420 5433 5445 5212 5224 5236 5333 5346 5358 5458 5470 5483 12 12 12 2 2 4 4 5 6 74 75 76 5495 S623 5754 5508 5521 5534 5636 5649 5662 5768 5781 5794 5546 5559 5572 5675 5689 5702 5808 5821 5834 5585 5598 5610 5715 5728 5741 5848 5861 5875 13 13 13 7 8 9 10 nil 77 78 79 5888 6026 6166 5902 5916 5929 6039 6053 6067 6180 6194 6209 5943 5957 5970 6081 6095 6109 6223 6237 6252 5984 5998 6012 6124 6138 6152 6266 6281 6295 14 14 15 13 13 1415 1617 80 6310 6324 6339 6353 ! 6368 6383 6397 6412 6427 6442 15 202l| 81 82 83 6457 6607 6761 6471 6486 6501 6622 6637 6653 6776 6792 6S0S 6516 6531 6546 6668 6683 6699 6823 6839 6855 6561 6577 6592 6714 6730 6745 6871 6887 6902 15 16 16 2 2 4 4 6 6 84 85 86 6918 7079 7244 6934 6950 6966 7096 7112 7129 7261 7278 7295 6982 6998 7015 7145 7161 7178 7311 7328 7345 7031 7047 7063 7194 7211 7228 7362 7379 7396 16 16 17 8 8 1011 12 13 87 88 89 7413 7586 7762 7430 7447 7464 7603 7621 7638 7780 7798 7816 7482 7499 7516 7656 7674 7691 7834 7852 7870 7534 7551 7568 7709 7727 7745 7889 7907 7925 IS 17 IS 14 15 1617 1819 90 7943 7962 7980 7998 8017 8035 8054 8072 8091 8110 18 2223 91 92 93 8128 8318 8511 8147 8166 8185 8337 8356 8375 8531 8551 8570 8204 8222 8241 8395 8414 8433 8590 8610 8630 8260 8279 8299 8453 8472 8492 8650 8670 8690 19 19 20 2 2 4 5 7 7 94 95 96 8710 8913 9120 S730 8750 8770 8933 8954 8974 9141 9162 9183 8790 8810 8831 8995 9016 9036 9204 9226 9247 8851 8872 SS92 9057 9078 9099 9268 9290 9311 21 21 22 9 9 1112 13 14 97 98 99 9333 9550 9772 9354 9376 9397 9572 9594 9616 9795 9817 9840 94-19 9441 9462 9638 9661 9683 9863 9886 9908 9484 9506 9528 9705 9727 9750 9931 9954 9977 22 22 23 15 16 1818 20 21 III. ADDITION LOGAKITHMS. log(l + n) li^ 1 2 3 1 4 5 « 7 8 9 d pp. «. o.uuuo 0001 0001 0001 0001 0001 0002 0002 0003 0003 1 7.0 1 2 3 U.0004 OOCH OOOi 0005 0005' 0006 0006 0006 0007 0007 0007 0007 0009 0009 0009 0009 0005 0005 0005 0006 0006 0006 0008 0008 0008 0009 0010 0010 0005 0005 0005 0006 0007 0007 0008 0008 0008 0010 0010 0011 1 12 00 00 01 4 5 « 0011 0011 0011 0012 0014 0014 0014 0015 0017| 0018 0018 0018 0012 0012 0013 0015 0015 0016 0019 0019 0020 0013 0013 0016 0016 0020 0021 0013 0017 0021 1 1 01 1 1 1 1 7 8 9 0022 0027 0034 0022 0023 0023 0028 0029 0029 0035 0036 0037 0024 0024 0025 0030 0031 0031 0038 0039 0039 0025 0026 0032 0033 0040 0041 0027 C034 0042 1 1 1 12 1 2 34 8.0 0.0043 0044 0045 0046 0047 0048 0050 0051 0052 0053 1 1 0054] 0056 0057 0058 2 ' 006S, 0070 0071 0073 3 0086 0088 0090 0092 0060 0061 0062 0075 0077 0078 0094 0096 0098 0064 0065 0080 0082 0101 0103 0067 0084 0105 1 2 3 00 1 1 11 4 i 0108 5 0135 (J 0170 0110 0113 0115 0138 0141 0145 0173 0177 0181 0118 0121 0123 0148 0151 0155 0186 0190 0194 0126 0129 0158 0162 0199 0203 0132 0166 0208 3 4 4 1 2 22 22 7 0212 0266 0332 0217 0222 0227 0272 0278 0284 0339 0347 0355 0232 0238 0243 0291 0297 0304 0363 0371 0379 0248 0254 0260 0311 0318 0325 0387 0396 0405 6 7 9 23 23 34 9.0 1 0414 0515 0423 0432 0442 0526 0538 0550 0452 0462 0472 0562 0574 0586 0482 0493 0599 0612 0504 0625 11 14 9.20 0.0639 0640 0642 0643 0644 0646 0647 0649 0650 0651 2 5(J 21 06531 0654 0656 0657 22 0667; 0668 0670 0671 23 0681 0683 0684 0686 0658 0660 0661 0673 0674 0675 06S7 0689 0690 0663 0664 0677 0678 0691 0693 0665 0680 0694 2 1 2 1 1 1 1 22 24 0696 0697 0699 0700 25 0711 0712 0714 0715 20] 0726i 0728 0729 0731 0702 0703 0705 0717 07-18 0720 0732 0^34 0735 0706 0708 0709 0721 0723 0725 0737 0738 0740 2 1 2 22 33 34 27 28 29 0742 0757 0774 0743 0745 0746 0759 0761 0762 0775 0777 0779 0748 0749 0751 0764 0766 0767 0780 0782 0783 0753 0754 0756 0769 0770 0772 0785 0787 0788 1 2 2 44 45 5 5 9.30 0.0790 0792 0793 0795 0797 0798 0800 0802 0804 0805 2 78 31 ' 0807 0809 0810 0812 32: 08241 0826 0827 0829 33 0841 0843 OS45 0847 0814 0815 0817 0831 0833 0834 0849 0850 0852 0819 0821 0822 0836 0838 0840 0854 0856 0857 2 1 2 1 1 12 22 34' 0859 0861 0863 0865 35 1 0877: 0879 0881 0883 3G' 0S96 0898 0900 0901 0866 0868 0870 0885 0887 0888 0903 0905 0907 0872 0874 0890 0892 0909 0911 0876 0894 0913 1 2 2 33 44 45 37 1 0915 38 0934 39: 0953 0917 0918 0920 0936 0938 0940 0955 0957 0959 0922 0924 0926 0942 0944 0945 0961 0963 0965 0928 0930 0932 0947 0949 0951 0967 0969 0971 2 2 2 56 66 67 9.40 0.U973 0975 0977 0979 0981 0983 0985 0987 0989 0991 2 9 41 42 43 0993 0996 0998 1000 1014 1016 leiS 1020 1035 1 1037 1039 1042 1002 1004 1006 1022 1025 1027 1044 1046 1048 lOOS 1010 1029 1031 1050 1052 1012 1033 1054 2 2 3 1 2 3 44 45 4G 10571 1059 1061 1063 1078 1081 1083 1085 1101 1103 1105 1107 1065 1067 1070 1087 1089 1092 1110 1112 1114 1072 1074 1094 1096 1116 1119 1076 1098 1121 2 3 2 4 5 5 47| 1123] 1125 1128 1130 48 11461 1148 1151 1153 49 1169! 1172 1174 1177 1132 1135 1137 1155 1158 1160 1179 1181 1184 1139 1141 1162 1165 1186 1189 1144 1167 1191 2 2 2 6 7 8 log(l + n) III. ADDITION LOGAEITHMS. log/? 1 2 3 4 5 6 7 8 9 d 3 pp. 9.50 0.1193 1196 1198 1201 1203 1205 1208 1210 1213 1215 10 51 52 53 1218 1242 1267 1220 1222 1225 1245 1247 1250 1270 1272 1275 1227 1230 1232 1252 1255 1257 1277 1280 1283 1235 1237 1240 1260 1262 1265 1285 1288 1290 2 2 3 1 2 3 54 55 56 1293 1319 1345 1295 1321 1348 1298 1301 1324 1327 1351 1353 1303 1306 1308 1329 1332 1335 1356 1359 1361 1311 1337 1364 1314 1316 1340 1343 1367 1369 3 2 3 4 5 6 57 58 59 1372 1399 1427 1375 1377 1380 1402 1405 1408 1430 1433 1436 1383 1386 1388 1410 1413 1416 1438 1441 1444 1391 1419 1447 1394 1397 1422 1424 1450 1453 2 3 2 3 7 8 9 9.60 0.1455 1458 1461 1464 1467 1470 1473 1475 1478 1481 11 61 62 63 14S4 1513 1543 1487 1516 1546 1490 1493 1519 1522 1549 1552 1496 1499 1502 1525 1528 1531 1555 1558 1561 1504 1534 1564 1507 1510 1537 1540 1567 1570 3 3 3 1 2 3 M 65 66 1573 1604 1635 1576 1607 1638 1579 1582 1610 1613 1641 1644 1585 1588 1591 1616 1619 1622 1647 1651 1654 1594 1625 1657 1598 1601 1629 1632 1660 1663 3 3 3 4 6 7 67 68 69 9.70 1666 1699 1731 0.1764 1670 1702 1735 1673 1676 1705 1708 1738 1741 1679 1682 1686 1712 1715 1718 1744 1748 1751 1689 1721 1754 1692 1695 1725 1728 1758 1761 4 3 3 7 8 9 10 1768 1771 1774 1778 1781 178f 1788 1791 1795 12 71 72 73 1798 1832 1867 1801 1836 1870 1805 1808 1839 1842 1874 1877 1812 1815 1818 1846 1849 1853 1881 1884 1888 1822 1856 1891 1825 1829 1860 1863 1895 1898 3 4 4 1 2 4 74 75 76 1902 1938 1974 1906 1941 1978 1909 1913 1945 1949 1981 1985 1916 1920 1923 1952 1956 1959 1989 1992 1996 1927 1931 1934 1963 1967 1970 2000 2003 2007 4 4 4 5 6 7 77 78 79 2011 2048 2086 2015 2052 2090 2018 2022 2056 2059 2094 2097 2026 2029 2033 2063 2067 2071 2101 2105 2109 2037 2075 2113 2041 2044 2078 2082 2117 2121 4 4 3 4 8 10 11 9.80 0.2124 2128 2132 2136 2140 2144 2148 2152 2156 2159 13 81 82 83 2163 2203 2243 2167 2207 2247 2171 2175 2211 2215 2251 2255 2179 2183 2187 2219 2223 2227 2259 2263 2267 2191 2231 2271 2195 2199 2235 2239 2275 2279 4 4 5 1 3 4 84 85 86 2284 2325 2366 2288 2329 2371 2292 2296 2333 2337 2375 2379 2300 2304 2308 2341 2346 2350 2383 2388 2392 2312 2316 2321 2354 2358 2362 2396 2400 2405 4 4 4 5 7 8 87 88 89 9.90 2409 2452 2495 2413 2417 2422 2456 2460 2465 2499 2504 2508 2426 2430 2434 2469 2473 2478 2513 2517 2521 2439 2443 2447 2482 2486 2491 2526 2530 2535 5 4 4 5 9 10 12 0.2539 2543 2548 2552 2557 2561 2566 2570 2575 2579 14 91 92 93 2584 2629 2674 2588 2593 2597 2633 2638 2642 2679 2684 2688 2602 2606 2611 2647 2651 2656 2693 2697 2702 2615 2661 2707 2620 2624 2665 2670 2711 2716 5 4 5 1 3 4 94 95 96 2721 2767 2815 2725 2730 2735 2772 2777 2782 2820 2824 2829 2739 2744 2749 2786 2791 2796 2834 2839 2844 2753 2758 2763 2801 2805 2810 2848 2853 2S5S 4 5 5 6 7 8 97 98 99 2863 2911 2961 2868 2873 2877 2916 2921 2926 2966 2970 2975 2882 2887 2892 2931 2936 2941 2980 2985 2990 2897 2902 2907 2946 2951 2956 2995 3000 3005 4 5 5 10 11 13 10 IV. SUBTEAOTION LOGAKITHMS. log logr/i 1 2 3 0001 0001 0001 4 5 6 0001 0001 0002 7 8 9 d 1 pp. 6. 0.0000 0002 0003 0003 7.0 0.0004 OOCH OOOi OOOi 0005 0005 0005 0005 0005 0005 1 2 1 2 3 0005 0007 ooo<; 0006 0006 0006 0007 0007 0007 0009 0009 0009 0006 0006 0006 0008 0008 0008 0010 0010 0010 0006 0007 0007 0008 0008 0008 0010 0010 0011 1 1 4 5 6 0011 0014 0017 0011 0011 0012 0014 0014 0015 0018 OOIS 0019 0012 0012 0013 0015 0015 0016 0019 0019 0020 0013 0013 0013 0016 0017 0017 0020 0021 0021 1 1 1 1 1 1 1 7 8 9 0022 0027 0035 0022 0023 0023 0028 0029 0029 0035 0036 0037 0024 0024 0025 0030 0031 0032 0038 0039 0040 0026 0026 0027 0032 0033 0034 0041 0042 0043 1 1 1 1 1 2 1 2 8.0 0.0044 0045 0046 0047 0048 0049 0050 0051 0053 0054 1 3 4 1 2 3 0055 0069 0088 0056 0058 0059 0071 0073 0074 0090 0092 0094 0060 0062 0063 0076 0078 0080 0096 0098 0101 0065 0066 0068 0082 0084 0086 0103 0105 0108 1 2 2 1 1 1 1 4 5 6 0110 0140 0176 0113 0116 0118 0143 0146 0150 0181 0185 0189 0121 0124 0127 0153 0157 0161 0194 0198 0203 0130 0133 0136 0164 0168 0172 0208 0213 0218 4 4 5 1 2 2 2 2 2 7 8 9 0223 0283 0359 0229 0234 0240 0290 0297 0304 0368 0377 0386 0245 0251 0257 0311 0319 0327 0396 0405 0415 0264 027Q 0276 0335 0343 0351 0425 0436 0447 7 8 11 14 18 2 3 2 3 3 4 9.0 1 0458 0584 0469 0480 0492 0599 0614 0629 0504 0517 0530 0645 0661 0678 0543 0556 0570 0695 0713 0731 9.20 0.0749 0751 0753 0755 0757 0759 0761 0763 0765 0767 1 56 21 0768 22; 078S 23 0808 0770 0772 0774 0790 0792 0794 0810 0812 0814 0776 0778 0780 0796 0798 0800 0817 0819 0821 0782 0784 0786 0802 0804 0806 0823 0825 0827 2 2 2 1 1 1 1 2 2 24 0829 25 1 OS.SO 26 0872 0831 0833 0835 0853 0855 0857 0875 0877 0879 0838 0840 0842 0859 0861 0863 0881 0884 0886 0844 0846 0848 0866 0868 0870 0888 0890 0893 2 2 2 2 2 3 3 3 4 27 0895 28 0918 |29 0942 0897 0899 0902 0920 0923 0925 0944 0947 0949 0904 0906 0909 0928 0930 0932 0952 0954 0957 0911 0913 0916 0935 0937 0940 0959 0962 0964 2 2 3 3 4 4 4 5 5 5 9.30 0.0967 0969 0972 0974 0977 0979 0982 0984 0987 0989 7 8 31 0992 32 1018 33j 1045 0994 0997 1000 1020 1023 1026 1047 1050 1053 1002 1005 1007 1028 1031 1034 1056 1058 1061 1010 1013 1015 1037 1039 1042 1064 1067 1069 3 3 3 1 1 1 2 2 2 34 1072 35 1101 36 1130 1075 1078 1081 1104 1106 1109 1133 1136 1139 1084 1086 1089 1112 1115 1118 1142 1145 1148 1092 1095 1098 1121 1124 1127 1151 1154 1157 3 3 3 3 3 4 4 4 5 37 38 39 1160 1191 1223 1163 1166 1169 1194 1198 1201 1226 1230 1233 1172 1176 1179 1204 1207 1210 1236 1240 1243 1182 1185 1188 1214 1217 1220 1246 1250 1253 3 3 3 3 5 6 6 6 6 7 9.40 0.1256 1260 1263 1266 1270 1273 1277 1280 1283 1287 9 11 41 1290 42 1325 43 1362 1294 1297 1301 1329 1333 1336 1365 1369 1373 1304 1308 1311 1340 1343 1347 1377 1380 1384 1315 1318 1322 1351 1354 1358 1388 1392 1395 3 4 4 1 1 2 2 3 3 44 1399 45 1438 46 1478 1403 1407 1411 1442 1446 1450 1482 1486 1490 1414 1418 1422 1454 1458 1462 1494 1498 1502 1426 1430 1434 1466 1470 1474 1506 1511 1515 4 4 4 4 4 5 6 5 7 47 1519 48| 1561 49 1605 1523 1527 1531 1566 1570 1574 1610 1614 1619 1536 1540 1544 1579 15S3 1588 1623 1628 1633 1549 15.53 1.557 1592 1596 1601 1637 1642 1646 1 5 6 8 7 9 8 10 log I IV. SUBTKAOTION LOGAEITHMS. 11 log /I 12 3 4 5 6 7 8 9 a pp. 9.50 0.1651 1656 1660 1665 1670 1674 1679 1684 1688 1693 5 1213 51 52 53 1698 1747 1797 1703 1708 1712 1752 1757 1762 1802 1807 1812 1717 1722 1727 1767 1772 1777 1818 1823 1828 1732 1737 1742 1782 1787 1792 1833 1839 1844 5 1 1 5 2 3 5 4 4 54 55 1849 1903 1959 1854 1860 1865 1909 1914 1920 1965 1971 1976 1871 1876 1881 1925 1931 1936 1982 1988 1994 1887 1892 1898 1942 1948 1953 ( 2000 2005 2011 ( 5 5 5 3 6 7 S 7 8 57 58 59 2017 2077 2140 2023 2029 2035 2084 2090 2096 2146 2153 2159 2041 2047 2053 2102 2108 2115 2166 2172 2179 2059 2065 2071 ( 2121 2127 2134 ( 2185 2192 2198 ' ■-> 8 9 S 1010 7 1112 9.60 0.2205 2211 2218 2225 2231 2238 2245 2252 2259 2265 ' J 1415 61 62 63 2272 2342 2415 2279 2286 2293 2350 2357 2364 2423 2430 2438 2300 2307 2314 2371 2378 2386 2445 2453 2460 2321 2328 2335 ' 2393 2400 2408 ' 2468 2476 2483 i J 1 2 J 3 3 5 4 5 64 65 66 2491 2570 2653 2499 2507 2515 2578 2586 2595 2661 2670 2678 2522 2530 2538 2603 2611 2619 2687 2695 2704 2546 2554 2562 { 2628 2636 2644 ' 2713 2721 2730 « I 6 6 } 7 8 } 8 9 67 68 69 2739 2829 2922 2748 2756 2765 2838 2847 2856 2932 2942 2951 2774 2783 2792 2866 2S75 2884 2961 297i 2981 2801 2810 2819 1 2894 2903 2913 < 2991 3001 3011 1 1011 } 1112 1314 9.70 0.3021 3031 3041 3051 3061 3071 3082 3092 3103 3113 ] 1617 71 72 73 3123 3231 3344 3134 3145 3155 3242 3253 3265 3356 3368 3380 3166 3177 3188 3276 3287 3299 3391 3403 3415 3 198 3209 3220 1 3310 3321 3333 1 3427 3439 3451 1 1 2 2 1 3 3 2 5 5 74 75 76 3463 3589 3721 3476 3488 3500 3602 3614 3627 3734 3748 3762 3513 3525 3538 3641 3654 3667 3775 3789 3803 3550 3563 3576 1 3680 3694 3707 1 3817 3831 3846 1 3 6 7 4 8 9 4 1010 77 78 79 9.80 3860 4007 4163 3874 3889 3903 4022 4038 4053 4180 4196 4212 3918 3933 3947 4069 4084 4100 4229 4245 4262 3962 3977 3992 1 4116 4131 4147 1 4278 4295 4312 1 5 1112 6 1314 7 1415 0.4329 4346 4364 4381 4398 4416 4434 4452 4470 4488 1 8 181t 81 82 83 4506 4694 4896 4524 4542 4561 4714 4733 4753 4917 4938 4959 45S0 4598 4617 4773 4793 4813 4980 5002 5024 4636 4655 4675 1 4834 4854 4875 2 5046 5068 5090 2 9 2 2. 1 4 4 2 5 6 84 85 86 5112 5345 5598 5135 5157 5180 5370 5394 5419 5624 5651 5678 5203 5227 5250 5444 5469 5494 5705 5732 5760 5273 5297 5321 2 5520 5546 5572 2 5787 5815 5844 2 4 7 8 6 9 10 8 1111 87 88 89 5872 6172 6502 5901 5930 5959 6204 6236 6268 6537 6572 6608 5989 6019 6049 6300 6333 6366 6644 6680 6717 6079 6110 6141 3 6400 6434 6468 3 6754 6792 6830 3 1 1313 4 1415 5 1617 pp. 1 2 3 20 21 22 23 24 25 26 1 27 28 29 ■ 30 3132 33 34 35 36, 37 38 2 4 6 2 2 2 4 4 5 6 7 7 2 3 3 5 5 5 7 8 8 3 3 3 5 6 6 8 8 9 3 6 9 3 3 3 6 6 7 9 10 10 3 4 4 7 7 7 10 11 11 4 4 7 8 11 11 4 5 6 8 10 12 8 9 9 11 11 12 13 13 14 10 10 10 12 13 13 14 15 16 11 11 12 14 14 15 16 17 17 12 15 18 12 13 13 16 16 17 19 19 20 14 14 14 17 18 18 20 21 22 15 15 19 19 22 23 7 8 9 14 16 18 15 15 16 17 18 18 19 20 21 17 18 18 19 20 21 22 23 23 19 20 20 22 22 23 24 25 26 21 24 27 22 22 23 25 26 26 28 29 30 24 25 25 27 28 29 31 32 32 26 27 30 30 ii 34 12 IV. SUBTEAOTION LOGARITHMS '«^(r^) logn 12 3 4 5 6 7 8 9 d pp. 9.9(K) 0.6S6S 6872 6876 6880 6884 6888 6892 6895 6899 6903 4 5 901 6907 9(»2 6946 903 1 6986 6911 6915 6919 6950 6954 6958 6990 6994 6998 6923 6927 6931 6962 6966 6970 7002 7006 7010 6935 6938 6942 6974 6978 6982 7014 7018 7022 1 1 1 1 2 904 905 906 7026 7030 7034 7038 7067 7071 7075 7079 7108 7112 7116 7121 7042 7047 7051 7083 7087 7092 7125 7129 7133 7055 7059 7063 7096 7100 7104 7137 7141 7146 2 2 2 3 2 3 907 908 909 7150 7154 7158 7162 7192 7196 7200 7205 7234 7239 7243 7247 7167 7171 7175 7209 7213 7217 7252 7256 7260 7179 7183 7188 7222 7226 7230 7265 7269 7273 — 3 4 3 4 4 5 9.910 0.7278 17282 7286 7291 7295 7299 7304 7308 7313 7317 6 7 911 912 913 7321 7366 7410 7326 7330 7335 7370 7375 7379 7415 7419 7424 7339 7343 7348 7383 7388 7392 7428 7433 7438 7352 7357 7361 7397 7401 7406 7442 7447 7451 1 1 1 1 2 2 914 915 916 7456 7502 7548 7460 7465 7469 7506 7511 7516 7553 7558 7562 7474 7479 7483 7520 7525 7530 7567 7572 7577 7488 7492 7497 7534 7539 7544 7581 7586 7591 2 3 3 4 4 4 917 918 919 7595 7643 7692 7600 7605 7610 7648 7653 7658 7697 7701 7706 7614 7619 7624 7663 7667 7672 7711 7716 7721 7629 7634 7638 7677 7682 7687 7726 7731 7736 — 4 5 5 6 5 6 9.920 0.7741 7746 7751 7756 7761 7766 7771 7776 7781 7786 8 9 921 7791 922 7841 923 7892 7796 7801 7806 7846 7851 7856 7897 7903 7908 7811 7816 7821 7861 7867 7872 7913 7918 7923 7826 7831 7836 7877 7882 7887 7929 7934 7939 5 1 1 2 2 2 3 924 925 926 7944 7997 8050 7949 7955 7960 8002 8007 8013 8056 8061 8066 7965 7970 7976 8018 8023 8029 8072 8077 8083 7981 7986 7992 8034 8040 8045 8088 8094 8099 6 3 4 4 5 5 5 927 928 929 8105 8160 8215 8110 8115 8121 8165 8171 8176 8221 8227 8232 8126 8132 8137 I 8143 8148 8154 8182 8187 8193 8199 8204 8210 8238 8244 8249 , 8255 8261 8266 6 5 6 6 6 6 6 7 7 8 9.93010.8272 8278 8284 8289 8295 8301 8307 8312 8318 8324 1011 931 8330 932 83S8 933 8448 8336 8341 8347 8394 8400 8406 8154 8460 8466 8353 8359 8365 8412 8418 8424 8472 8478 8484 8371 8377 8382 8430 8436 8442 8490 8496 8502 6 6 6 1 1 2 2 3 3 934 8508 935' 8570 936; 8632 8514 8520 8527 8576 8582 8588 8638 8645 8651 8533 8539 8545 8595 8601 8607 8657 8664 8670 8551 8557 8563 8613 8620 8626 8676 8683 8689 7 6 7 4 4 5 6 6 7 937 1 8696 18702 8708 8715 938 1 8760 ! 8767 8773 8780 939 8826 8833 8839 SS46 8721 8728 8734 8786 8793 8800 8853 8859 8866 8741 8747 8754 8806 8813 8819 8873 8879 8886 6 7 7 7 7 8 8 9 9 10 9.9400.8893 8900 8906 8913 8920 8927 8934 8940 8947 8954 1213 941 8961 942 9030 943j 9101 8968 8975 8982 ! 8989 8996 9002 9037 9044 9051 9058 9065 9073 9108 9115 9122 ; 9130 9137 9144 9009 9016 9023 9080 9087 9094 9151 9158 9166 7 7 7 1 1 2 3 4 4 944| 9173 945 9246 946, 9321 9180 9188 9195 9254 9261 9269 9329 9336 9344 9202 9209 9217 9276 9284 9291 9351 9359 9367 9224 9232 9239 9299 9306 9314 9374 9382 9390 7 7 7 5 5 6 7 7 8 947 9397 948 1 9475 949 9555 9405 9413 9421 9483 9491 9499 9563 9571 9579 9428 9436 9444 9507 9515 9523 9587 9595 9603 9452 9460 9467 9531 9539 9547 96119619 9628 8 8 8 8 9 10 10 11 12 log( 1 1-n ) CONTINUED. 13 logn 1 2 3 1 4 5 6 7 8 9 (1 pp. 1415 9.9o00.963f ) 9644 9652 9660 9669 9677 9685 9694 9702 9710 9 951 952 953 97]9: 9727 9735 9744 9752 9761 9769! 9778 9786 9795' 8 9803. 9812 9820 9S29J 9838 9846 9S55{ 9864 9872 9SS!! 9 9890 9899 9907 9916' 9925 9934 9943 9951 9960 9969 9 1 2 3 3 4 5 954 955 956 9978' 9987 9996*0005*0014*0023*0032*0041*0050*00601 9 1.0069' 0078 0087 0096i 0106 0115 01241 0133 0143 01521 9 0161 01710180 0190: 0199 0209 0218 0228 0237 0247i 9 6 6 7 8 8 9 957 958 959 0256 0266 0276 0285 0295 0305 0314 0354 0364 0373 0383 0393 0403 0413 0453| 0464 0474 0484 0494 0504 0514 0324 0334 0344 10 0423 0433 0443 10 0525 0535 0545 11 10 11 1112 13 14 1617 9.9601.05561 0566 0576 05S7I 0597 060S 0618 0629 0640 0650 11 961 962 963 0661: 06710682 0693! 0704 0714 0725 0769' 0780 0791 0S02| 0813 0824 0835 0S80| 08910902 0913 1 0925 0936 0948 0736 0747 075s'll 0846 0857 0868 12 0959 0971 0982 12 11 5 5 964 965 966 0994 1111 1232 1005 1017 1028 1040 1052 1064 1075 1087 109912 1123 1135 1147, 1159 1171 1183! 1195 1207 1220 12 1244 1257 1269 1281 1294 1306! 1319 1331 1344 13 6 7 8 9 1010 967 968 969 1357 1485 1618 13691382 1395 1408 1420 1433 14981512 1525! 1538 1551 1565 1632 1645 1659 1673 1686 1700 1446 1459 1472 13 1578 1591 1605 13 1714 1728 1742 14 1112 13 14 14 15 1819 9.970 1.1756 17701784 1798 1S12 1826 1841 1855 1869 1884 14 971 972 973 1898; 1913 1927 1942: 1956 1971 1986 2001 2016 203115 2046: 20612076 2091, 2106 2121 2137' 2152 2167 2183 16 2199 2214 2230 22461 2261 2277 2293| 2309 2325 2341 16 u 974 975 976 2357J 2374 2390 2406 2423 2439 24561 2473 2489 250617 2523 2540 2557 2574 2591 2608 2625! 2643 2660 2678 17 2695| 2713 2731 2748 2766 2784 28021 2820 2838 2857 18 7 8 910 1111 977 978 979 2875 2893 2912 2931 2949 2968 2987 3063 3082 3102 3121 3141 3161 3180 3260^ 3280 3301 3321 3342 3362 3383 3006 3025 3044 19 3200 3220 3240 20 3404 3425 3446 21 13 13 14 15 1617 2021 9.980 1.3467 3488 3510 3531 3553 3575 3596 3618 3640 3663 22 981 982 983 3685 3707 3730 3753 3915 3939 3962 3986 4158: 4183 4208 4234 3775 3798 3821 4010 4035 4059 4259 4285 4311 3845 3868 3891 '24 4084 4108 4133 25 4337 4363 4390 26 2 2 4 4 6 6 984 985 986 4416' 4443 4470 4497 4692 4720 4749 4778 49S6| 5017 5048 5079 4524 4552 4579 4807 4836 4866 5110 5142 5174 4607 4635 4663 29 4896 4926 4956 30 5206 5238 5270 33 8 8 1011 12 13 987 988 989 5303 5646 6019 5336 5370 5403 5682 5718 5754 6058 6097 6137 5437 5471 5505 5791 5828 5866 6178 6218 6259 5540 5575 5610 36 5903 5942 5980 39 6301 6343 6385,43 1415 1617 1819 2223 9.990 1.6428 6471 6514 6559 6603 6648 6693 6739 6786 6833^47 991 992 993 6880' 6928 6977 7026 7387 1 7441 7496 7551 7962 8024 8087 8151 7076 7126 7177 7608 7665 7722 8215 8281 8348 7228 7281 7333 54 7781 7840 7901 !61 8416 8485 8555 71 2 2 4 5 7 7 994 995 996 8626:' 9413 2.0377 8699 8773 8848 9500 9589 9680 0487 0599 0714 8924 9002 9081 9773 9868 99651 0833 0955 1080 9162 9244 932S!85 *0065*0166*0270i 1209 1342 1480 9 9 1112 13 14 997 998 999 1622 3378 6383 1768 1920 2078 3600 3834 4082 6840 7351 7930 2241 2411 2588 4345 4624 4924 8599 9391*0359| 2772 2965 3166 5245 5592 5969 *1608*3369*6378 15 16 18 18 20 21 u V. LOGARITHMIC SINES AND COSINES loir sin H logrcosn^ n^! .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 d 7. 2419 5429 7190 8439 9408*0200 *0870*1450*1961 *2419 89 1 2 S.2419 542S 71SS 2832 3210 35.58 5640 5842 6035 7330 7468 7602 3880 4179 4459 6220 6397 6567 ■7731 7857 7979 4723 4971 5206 6731 6889 7041 8098 8213 8326 5428 7188 8436 88 87 86 4 6 <» 8436 9403 9.0192 8543 8647 8749 9489 9573 9655 0264 0334 0403 8849 89-16 9042 9736 9816 9894 0472 0539 060i 9135 9226 9315 9970*0046*0120 0670 0734 0797 9403 *0192 0859 S5 84 83 88 72 62 7 8 9 0S59 1436 1943 0920 0981 1040 1489 1542 1594 1991 2038 2085 1099 1157 1214 1646 1697 1747 2131 2176 2221 1271 1326-1381 1797 1847 1895 2266 2310 2353 1436 1943 2397 82 81 80 ^9 55 48 44 39 To 9.2397 2439 2482 2524 2565 2606 2647 2687 2727 2767 2806 11 12 la 2806 3179 3521 2845 2883 2921 3214 3250 3284 3554 3586 3618 2959 2997 3034 3319 3353 3387 3650 3682 3713 3070 3107 3143 3421 3455 3488 374i 3775 3806 3179 3521 3837 78 77 76 36 33 31 14 15 1<> 3837 4130 4403 3867 3897 3927 4158 4186 4214 4430 4456 4482 3957 3986 4015 4242 4269 4296 4508 4533 4559 4044 4073 4102 4323 4350 4377 4584 4609 4634 4130 4403 4659 75 74 73 28 26 25 17 18 19 4659 4900 5126 4684 4709 4733 4923 4946 4969 5148 5170 5192 4757 4781 4805 4992 5015 5037 5213 5235 5256 4829 4853 4876 5060 5082 5104 5278 5299 5320 4900 5126 5341 72 71 70 24 22 21 20 20 9.5341 5361 5382 5402 5423 5443 5463 5484 5504 5523 5543 69 21 22 23 5543 5736 5919 5563 5583 5602 57.54 5773 5792 5937 5954 5972 5621 5641 5660 5810 5828 5847 5990 6007 6024 5679 5698 5717 5865 5883 5901 6042 6059 6076 5736 5919 6093 68 67 66 19 18 17 24 25 26 6093 6259 6418 6110 6127 6144 6276 6292 6308 6434 6449 6465 6161 6177 6194 6324 6340 6356 6480 6495 6510 6210 6227 6243 6371 6387 6403 6526 6541 6556 6259 6418 6570 65 64 63 16 15 14 27 28 29 30 6570 6716 6856 6585 6600 6615 6730 6744 6759 6869 6883 6896 6629 6644 6659 6773 6787 6801 6910 6923 6937 6673 6687 6702 6814 6828 6842 6950 6963 6977 6716 6856 6990 62 61 60 14 14 13 12 9.6990 7003 7016 7029 7042 7055 7068 7080 7093 7106 7118 59 31 32 33 7118 7242 7361 7131 7144 7156 7254 7266 7278 7373 7384 7396 7168 7181 7193 7290 7302 7314 7407 7419 7430 7205 7218 7230 7326 7338 7349 7442 7453 7464 7242 7361 7476 58 57 56 12 12 12 34 35 36 7476 7586 7692 7487 7-198 7509 7597 7607 7618 7703 7713 7723 7520 7531 7542 7629 7640 7650 7734 7744 7754 7553 7564 7575 7661 7671 76S2 7764 7774 7785 7586 7692 7795 55 54 53 11 10 10 37 38 39 40 7795 7893 7989 7805 7815 7825 7903 7913 7922 7998 8007 8017 7835 7844 7854 7932 7941 7951 8026 8035 8044 7864 7874 7884 7960 7970 7979 8053 8063 8072 7893 7989 8081 52 51 50 9 10 9 ~8 9.8081 8090 8099 8108 8117 8125 8134 8143 8152 8161 8169 49 41 42 43 8169 8255 8338 8178 8187 8195 8264 8272 8280 8346 8354 8362 8204 8213 8221 8289 8297 8305 8370 8378 8386 8230 8238 8247 8313 8322 8330 8394 8402 8410 8255 8338 8418 48 47 46 8 8 8 44 45 46 8418 8495 8569 8426 8433 8441 8502 8510 8517 8577 8584 8591 8449 8457 8464 8525 8532 8540 8598 8606 8613 8472 8480 8487 8.547 8555 8562 8620 8627 8634 8495 8569 8641 45 44 43 8 7 7 47 48 49 8641 8711 8778 8648 8655 8662 8718 8724 8731 8784 8791 8797 8669 8676 8683 8738 8745 8751 8804 8810 8817 8690. 8697 8704 8758 8765 8771 8823 8830 8836 8711 8778 8843 42 41 40 7 7 7 1.0 1 .9 .S .7 .6 .5 .4 .3 .2 .1 .0 n" d V. LOaAEITHMIO SINES AND COSINES. 15 log sin n° log COS»l,° n° .0 .1 .2 .3 .4 .5 .6 j .7 .8 .9 1.0 d 6 50 9.8843 8849 8855 8862 8868 8874 8880 8887 8893 8899 8905 39 51 52 53 8905 8965 9023 8911 8917 8923 8971 8977 8983 9029 9035 9041 8929 8935 8941 8989 8995 9000 9046 9052 9057 8947 8953 8959 9006 9012 9018 9063 9069 9074 8965 9023 9080 38 37 36 6 5 6 54 55 56 9080 9134 9186 9085 9091 9096 9139 9144 9149 9191 9196 9201 9101 9107 9112 9155 9160 9165 9206 9211 9216 9118 9123 9128 9170 9175 9181 9221 9226 9231 9134 9186 9236 35 34 33 6 5 5 57 58 59 9236 9284 9331 9241 9246 9251 9289 9294 9298 9335 9340 9344 9255 9260 9265 9303 9308 9312 9349 9353 9358 9270 9275 9279 9317 9322 9326 9362 9367 9371 9284 9331 9375 32 31 30 29 5 5 4 T 60 9.9375 9380 93S4 9388 9393 9397 9401 9406 9410 9414 9418 61 62 63 9418 9459 9499 9422 9427 9431 9463 9467 9471 9503 9507 9510 9435 9439 9443 9475 9479 9483 9514 9518 9522 9447 9451 9455 9487 9491 9495 9525 9529 9533 9459 9499 9537 28 27 26 4 4 4 64 65 66 9537 9573 9607 9540 9544 9548 9576 9580 9583 9611 9614 9617 9551 9555 9558 9587 9590 9594 9621 9624 9627 9562 9566 9569 9597 9601 9604 9631 9634 9637 9573 9607 9640 25 24 23 4 3 3 67 68 69 9640 9672 9702 9643 9647 9650 9675 9678 9681 9704 9707 9710 9653 9656 9659 9684 9687 9690 9713 9716 9719 9662 9666 9669 9693 9696 9699 9722 9724 9727 9672 9702 9730 22 21 20 19 3 3 3 3 70 9.9730 9733 9735 9738 9741 9743 9746 9749 9751 9754 9757 71 72 73 9757 9782 9806 9759 9762 9764 9785 9787 9789 9808 98119813 9767 9770 9772 9792 9794 9797 9815 9817 9820 9775 9777 9780 9799 9801 9804 9822 9824 9826 9782 9806 9828 18 17 16 2 2 2 74 75 76 9828 9849 9869 9831 9833 9835 9851 9853 9855 9871 9873 9875 9837 9839 9841 9857 9859 9861 9876 9878 9880 9843 9845 9847 9863 9865 9867 9882 9884 9885 9849 9869 9887 15 14 13 2 2 2 77 78 79 9887 9904 9919 9889 9891 9892 9906 9907 9909 9921 9922 9924 9894 9896 9897 9910 9912 9913 9925 9927 9928 9899 9901 9902 9915 9916 9918 9929 9931 9932 9904 9919 9934 12 11 10 2 1 2 80 9.9934 9935 9936 9937 9939 9940 9941 9943 9944 9945 9946 9 1 81 82 83 9946 9958 9968 9947 9949 9950 9959 9960 9961 9968 9969 9970 9951 9952 9953 9962 9963 9964 9971 9972 9973 9954 9955 9956 9965 9966 9967 9974 997i 9975 9958 9968 9976 8 7 6 2 1 1 84 85 86 9976 9983 9989 9977 9978 9978 9984 9985 9985 9990 9990 9991 9979 9980 9981 9986 9987 9987 9991 9992 9992 9981 9982 9983 9988 9988 9989 9993 9993 9994 9983 9989 9994 5 4 3 87 88 89 9994 9997 9^99 9994 9995 9995 9998 9998 9998 9999 0000 0000 9996 9996 9996 9998 9999 9999 0000 0000 0000 9996 9997 9997 9999 9999 9999 0000 0000 0000 9997 9999 0000 2 1 d 1.0 1 .9 .8 .7 .6 .5 .4 .3 .2 .1 .0 u° pp. 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 2627 28 29 36] 1 2 3 1 1 1 2 2 3 3 4 4 1 2 2 3 3 3 4 5 5 2 2 2 3 4 4 5 5 6 2 4 6 2 2 2 4 4 5 6 7 7 2 3 3 5 5 5 7 8 8 3 3 3 5 6 6 8 8 9 3 6 9 4 5 6 4 5 5 6 6 7 7 7 8 6 6 6 7 8 8 8 9 10 7 7 8 9 9 10 10 11 11 8 10 12 8 9 9 11 11 12 13 13 14 10 10 10 12 13 13 14 15 16 11 1112 14 14 15 16 17 17 12 15 18 7 8 9 8 8 9 9 10 10 10 11 12 10 11 11 11 12 13 13 14 14 12 13 13 14 14 15 15 16 17 14 16 18 15 15 16 17 18 18 19 20 21 17 18 18 19 20 21 22 23 23 19 20 20 22 22 23 24 25 26 21 24 27 LOGAKITHMIO SINES AND COSINES lofr sin )( log- COS n° II () 1 2 3 4 5 6 7 8 9 10 d 0.0 6. 2-41'; 5429 7190 8439 9408*0200 *0870*1450*1961 *2419 89.9 0.l'7.2419 2833 3211 355S 0.21 5429 5641 5S43 6036 0.3l 7190 7332 7470 7604 3880 4180 4460 6221 6398 6568 7734 7859 7982 4723 4971 5206 6732 6890 7043 8101 8217 8329 5429 7190 8439 89.8 89.7 89.6 0.4 0.5 O.G 8439 8547 S651 8753 SS53 8951 9046 9408 9494 9579 9661 9743 9822 9901 8.0200 0272 034: 0412 0480 0548 0614 9140 9231 9321 9977*0053*0127 0679 0744 0807 9408:89.5 *0200| 89.4 0870 89.3 87 73 63 0.7 0.8 0.9 08700931 0992 1052^1111 1169 1227 1450 1503 1557 1609 1661 1713 1764 1961 12009 2056 2103|2150 2196 2241 1284 1340 1395 1814 1863 1912 2286 2331 2375 1450'89.2 1961! 89.1 2419j89.0 S5 49 44 39 1.0 S.2419'2462 2505 2547 2589 2630 2672 2712 2753 2793 2832 88.9 1.1 1.2 lis 2832 3210 3558 2872 29112949 3246 3282 3317 35913624 3657 2988 3025 3063 3353 3388 3422 3689 3722 3754 3100 3137 3174 3456 3491 3524 3786 3817 3848 3210 88.8 3558:88.7 3880|88.6 36 34 32 1.4 1.5 1.6 3880 4179 4459 391139413972 4208 4237 4265 4486 4513 4540 4002 4032 4062 4293 4322 4349 4567 4593 4619 4091 4121 4150 4377 4405 4432 4645 4671 4697 4179 4459 4723 88.5 88.4 88.3 29 27 26 1.7 1.8 1.9 4723 4971 5206 4748 4773 479914824 4848 4873 4995 5019 5043^5066 5090 5113 5228 5251 5274|5296 5318 5340 4898 4922 4947 5136 5160 5183 5363 5385 5406 4971 5206 5428 88.2 88.1 88.0 24 23 22 2.0;S.542S 5450 54715493^5514 5535 5557 5578 5598 5619 5640 87.9 21 2.1 2.2 2.3 5640 5842 6035 5661 5681 5702: 5 722 5742 5762 5862 5881 5901 5920 5939 5959 6054 6072 60916110 6128 6147 5782 5802 5822 5978 5997 6016 6165 6183 6201 5842 6035 6220 87.8 87.7 87.6 20 19 19 2.4 2.5 2.G 6220 6397 6567 6238 6256 6274 6291 6309 6327 6414 6431 6449 6466 6483 6500 6584 6600 6617 6633 6650 6666 6344 6362 6379 6517 6534 6550 6682 6699 6715 6397 6567 6731 87.5 87.4 87.3 18 17 16 2.7 2.8 2.9 6731 6889 7041 6747 6763 6779 6795 6810 6826 6904 6920 693516950 6965 6981 7056 70717086 7100 7115 7130 6842 6858 6873 6996 7011 7026 7144 7159 7174 6889 7041 7188 87.2 87.1 87.0 16 15 14 14 3.0 8.7188 7202 7217 7231 7245 7260 7274 7288 7302 7316 7330 86.9 3.1 3.2 3.3 7330 7344 7358 7372 746817482 7495 7508 7602 7615 7628 7641 7386 7400 7413 7522 7535 7549 7654 7667 7680 7427 7441 7454 7562 7575 7588 7693 7705 7718 7468 7602 7731 86.8 86.7 86.6 14 14 13 3.4 3.5 3.6 7731 7744 7756 7769 77S2 7794 7807 7857:7869 7881 7894,7906 7918 7930 7979|7991 8003 8015 8027 8039 8051 7819 7832 7844 7943 7955 7967 8062 8074 8086 7857 7979 8098 86.5 86.4 86.3 13 12 12 3.7 3.8 3.9 4^ 8098:8109 8121 81338144 8156 8168 8213 8225 8236 8248 8259 8270 8281 8326|8337 8348 8359 S370 8381 8392 8179 8191 8202 8293 8304 8315 8403 8414 8425 8213 8326 8436 86.2 86.1 86.0 11 11 11 11 8.8436 8447 8457 8468 8479 8490 8500 8511 8522 8532 8543 85.9 4.1 4.2 4.3 8543 8553 8564 85 75 8585 8595 8606 8647 8658 8668 8678 8688 8699 8709 8749 8759 8769 8780 8790 8799 8809 8616 8627 8637 8719 8729 8739 8819 8829 8839 8647 8749 8849 85.8 85.7 85.6 10 10 10 4.4 4.5 4.6 8849 8859 8869 8878 8888 8898 8908 8946 8956 8966 8975 8985 8994 9004 9042 9051 9060 9070 9079 90S9 9098 8917 8927 8937 9013 9023 9032 9107 9116 9126 8946 9042 9135 85.5 85.4 85.3 9 10 9 4.7 4.8 4.9 9135 9226 9315 9144 9153 9162 9172 9181 9190 ^235 9244 9253 9262 9271 9280 ;324 9333 9342 9351 9359 9368 9199 9208 9217 9289 9298 9307 9377 9386 9394 9226 9315 9403 85.2 85.1 85.0 9 8 9 d lj 10 1 987|664 3 2 1 n° CONTINUED. 17 log Sinn o log- COS n° n° 12 3 4 5 6 7 8 9 10 d 5.0 8.9403 9412 9420 9429 9437 9446 9455 9463 9472 9480 9489 84.9 9 5.1 5.2 5.3 9489 9497 9506 9514 9573 9581 9589 959S 9655:9664 9672 9680 9523 9531 9539 9606 9614 9623 9688 9696 9704 9548 9556 9565 9631 9639 9647 9712 9720 9728 9573 9655 9736 84.8'8 84.7:8 84.6:8 5.4 5.5 5.6 9736 9744 9752 9760 9816 9824 9831 9839 9894 9901 9909 9917 9768 9776 9784 9S47 9855 9863 9925 9932 9940 9792 9800 9808 9870 9878 9886 9948 9955 9963 9816 84.5'8 9894 84.4 S 9970 84.3,7 5.7 5.8 5.9 670 9970 9.0046 0120 9.0192 9978 9986 9993 0053 0061 0068 0127 0134 0142 *0001*0008*0016 0075 0083 0090 0149 0156 0163 *0023*0031*003S 0098 0105 0112 0171 0178 0185 *004684.28 012084.118 0192 84.0 7 0200 0207 0214 0221 0228 0235 0243 0250 0257 0264 83.9 7 6.1 6.2 6.3 0264 0271 0278 0285 0334:0341 0348 0355 0403 0410 0417 0424 0292 0299 0306 0362 0369 0376 0431 0438 0444 0313 0320 0327 0383 0390 0397 0451 0458 0465 0334 83.87 0403|S3.7 6 0472;83.6 7 6.4 6.5 6.6 0472 0478 0485 0492 0539 0545 0552 0558 0605 0611 0618 0624 0498 0505 0512 0565 0572 0578 0631 0637 0644 0519 0525 0532 0585 0591 0598 0650 0657 0663 0539J83.5 7 060583.4:7 0670 83.3 7 6.7 6.8 6.9 7.0 0670 0676 0683 0689 07340740 0746 0753 0797 0803 0S09 0816 0695 0702 0708 0759 0765 0772 0822 0828 0834 0715 0721 0727 0778 0784 0790 0840 0847 0853 0734 83.2 7 0797 83.1 7 0859183.0 6 9.0859|0S65 0871 0877 0884 0890 0896 0902 0908 0914 0920 82.9 6 7.1 7.2 7.3 0920 0926 0932 0938 0981 0987 0993 0999 1040 1046 1052 1058 0945 0951 0957 1005 1011 1017 1064 1070 1076 0963 0969 0975 1022 1028 1034 1081 1087 1093 098182.8 6 1040 82.7 6 1099 82.6 6 7.4 7.5 7.6 1099 1105 1111 1116 1157 1163 1168 1174 1214 1220 1226 1231 1122 1128 1134 1180 1186 1191 1237 1242 1248 1140 1145 1151 1197 1203 1208 1254 1259 1265 1157 82.5 6 1214S2.4|6 127182.36 7.7 7.8 7.9 8.0 1271 1326 1381 9.1436 1276 1282 1287 1332 1337 1343 1387 1392 1398 1293 1299 1304 1348 1354 1359 1403 1409 1414 1310 1315 1321 1365 1370 1376 1419 1425 1430 132682.2'5 138182.115 1436 82.0 6 1441 1446 1452 1457 1462 1468 1473 1478 1484 1489 81.9 5 8.1 8.2 8.3 1489 1494 1500 1505 1542 1547 1553 1558 1594 1600 1605 1610 1510 1516 1521 1563 1568 1574 1615 1620 1625 1526 1532 1537 1579 1584 1589 1631 1636 1641 1542 81.8 5 1594:81.7 5 164681.65 8.4 8.5 8.6 1646' 1651 1656 1661 169711702 1707 1712 1747 1752 1757 1762 1666 1672 1677 1717 1722 1727 1767 1772 1777 1682 1687 1692 1732 1737 1742 1782 1787 1792 1697 '81. 5 5 1747 81.4 5 1797 81.3 5 8.7 8.8 8^ 1797 1847 1895 10 1802.1807 1812 1851 1856 1861 1900 1905 1910 1817 1822 1827 1866 1871 1876 1915 1919 1924 1832 1837 1842 1881 1886 1890 1929 1934 1939 1847 81.2 5 1895! 81. 15 1943 81.0 4 9 8 7 6 5 4 3 2 1 n° o 12 3 4 5 6 7 8 9 S lo^ sin .1 .2 .3 6 12 IS 6.6 7.2 7.8 12.6 13.2 13.8 18.6 19.2 19.8 8.4 9 9.6 14.4 15 15.6 20.4 21 21.6 10.2 16.2 22.2 lO.S 11.4 16.8 17.4 22.8. 23.4 0.00 1.11 2.40 .7581 .7582 8.2872 L.7583 8.6220 .4 .5 .6 24 30 36 24.6 25.2 25.8 30.6 31.2 31.8 36.6 37.2 37.8 26.4 27 27.6 28.2 32.4 33 33.6 34.2 38.4 39 39.6 40.2 28.8 29.4 34.8 35.4 40.8 41.4 3.21 3.85 4.40 1.7584 8.7482 L.7585 8.8270 8.8849 .7 .8 .9 42 48 54 42.6 43.2 43.8 48.6 49.2 49.8 54.6 55.2 55.8 14.4 45 45.6^46.2 50.4 51 51.6,52.2 S6.4 57 57.6I58.2 46.8 47.4 52.8 53.4 58.8 59.4 IS VI. LOGARITHMIO TANGENTS AND COTANGENTS. loy tan H lofiT cotan n° n^ .0 1 .1 .2 .3 .4 .6 .6 .7 .8 .9 1.0 d 7.' 2419 5429 7190 8439 9409*0200 *0870*145Q*1962 *2419 89 1 >) 3 S.2419 2S33 3211 3559 5431 5643 5S45 603S 7194 7337 74 7i 7609 3881 4181 4461 6223 6401 6571 7739 7865 7988 472i 4973 5208 6736 6894 7046 8107 8223 8336 5431 7194 8446 88 87 86 4 5 (t S446 8554 8659 8762 9420 9506 9591 9674 9.0216 0289 0360 0430 8862 8960 9056 9756 9836 9915 0499 0567 0633 9150 9241 9331 9992*0068*0143 0699 0764 0828 9420 *0216 0891 85 84 83 89 73 63 7 8 9 0891 0954 1015 1076 1478 1533 1587 1640 1997, 2046 2094 2142 1135 1194 1252 1693 1745 1797 2189 2236 2282 1310 1367 1423 1848 1898 1948 2328 2374 2419 1478 1997 2463 82 81 80 55 49 44 10 9.2463 2507 2551 2594 2637 2680 2722 2764 2805 2846 2887 79 41 11 12 13 2887 2927 2967 3006 3275 3312 3349 3385 3634 3668 3702 3736 3046 3085 3123 3422 3458 3493 3770 3804 3837 3162 3200 3237 3529 3564 3599 3870 3903 3935 3275 3634 3968 78 77 76 38 35 33 U 15 16 3968 4000 4032 4064 428r4311 4341 4371 4575 4603 4632 4660 4095 4127 415S 4400 4430 4459 4688 4716 4744 4189 4220 4250 4488 4517 4546 4771 4799 4826 4281 4575 4853 75 74 31 29 27 17 18 19 20 4853 4880 4907 4934 511815143 5169 5195 5370 5394 5419 5443 4961 4987 5014 5220 5245 5270 5467 5491 5516 5040 5066 5092 5295 5320 5345 5539 5563 5587 5118 5370 5611 72 71 70 26 25 24 23 9.5611 5634 5658 5681 5704 5727 5750 5773 5796 5819 5842 69 22 23 5842 6064 6279 5864 5887 5909 6086 6108 6129 6300 6321 6341 5932 5954 5976 6151 6172 6194 6362 6383 6404 5998 6020 6042 6215 6236 6257 6424 6445 6465 6064 6279 6486 68 67 66 22 22 21 24 25 26 6486 6687 6882 6506 6527 6547 6706 6726 6746 6901 6920 6939 6567 6587 6607 6765 6785 6804 6958 6977 6996 6627 6647 6667 6824 6843 6863 7015 7034 7053 6687 6882 7072 65 64 63 20 19 19 27 28 29 7072 7257 7438 7090 7109 7128 7275 7293 7311 7455 7473 7491 7146 7165 7183 7330 7348 7366 7509 7526 7544 7202 7220 7238 7384 7402 7420 7562 7579 7597 7257 7438 7614 62 61 60 19 18 17 17 30 9.7614 7632 7649 7667 7684 7701 7719 7736 7753 7771 7788 59 31 ti 7788 7958 8125 7805 7822 7839 7975 7992 8008 8142 8158 8175 7856 7873 7890 8025 8042 8059 8191 8208 8224 7907 7924 7941 8075 8092 8109 8241 8257 8274 7958 8125 8290 58 57 56 17 16 16 It 36 8290 8452 8613 8306 8323 8339 8468 8484 8501 8629 8644 8660 8355 8371 8388 8517 8533 8549 8676 8692 8708 8404 8420 8436 8565 8581 8597 8724 8740 8755 8452 8613 8771 55 54 53 16 16 16 37 38 39 40 8771 8787 SS03 8818 89281 8944 8959 S97i 9084 9099 9115 9130 8S34 SS50 8865 8990 9006 9022 9146 9161 9176 8881 8897 8912 9037 9053 9068 9192 9207 9223 8928 9084 9238 52 51 50 16 16 15 16 9.9238 9254 9269 9284 9300 9315 9330 9346 9361 9376 9392 49 41 42 43 9392' 9407 9422 9438 9544 9560 9575 9590 9697 9712 9727 9742 9453 9468 9483 9605 9621 9636 9757 9772 9788 9499 9514 9529 9651 9666 9681 9803 9818 9833 9544 9697 9848 48 47 46 15 16 15 44 45 46 9848 9864 9879 9S94 0.0000 0015 0030 0045 0152,0167 01S2 0197 9909 9924 9939 0061 0076 0091 0212 0228 0243 9955 9970 9985 0106 0121 0136 0258 0273 0288 *0000 0152 0303 45 44 43 15 16 15 47 48 49 0303 0319 0334 03491 0364 0379 0395 0456 0471 0486 05011 0517 0532 0547 0608 0624 0639 0654 0670 0685 0700 0410 0425 0440 0562 0578 0593 0716 0731 0746 0456 0608 0762 42 41 40 16 15 16 1.0 .9 .8 .7 .6 .5 .4 .3 .2 .1 .0 In^ VI. LOGARITHMIC TANGENTS AND COTANGENTS. 19 logtan?t° log cotan n° .0 0.0762 0916 10 1229 1387 1548 1710 1875 2042 2212 0.2386 2562 2743 2928 3118 3313 3514 3721 3936 4158 0.4389 4630 4882 5147 5425 5719 6032 6366 6725 7113 .2 .3 0777 0932 1088 1245 1403 1564 1726 1891 2059 2229 0793 0808 0947 0963 1103 1119 1260 1276 1419 1435 1580 1596 1743 1759 1908 1925 2076 2093 2247 2264 2403 2580 2762 2947 3137 3333 3535 3743 3958 4181 2421 2438 2598 2616 2780 2798 2966 2985 3157 3176 33S3 3313 3555 3576 3764 3785 3980 4002 4204 4227 0.7537 8003 8522 9109 9784 1.0580 1554 2806 4569 7581 1.0 4413 4655 4908 5174 5454 5750 6065 6401 6763 7154 4437 4461 4680 4705 4934 4960 5201 5229 5483 5512 5780 5811 6097 6130 6436 6471 6800 6838 7195 7236 .5 0824 0978 1135 1292 1451 1612 1776 0839 0854 0994 1010 1150 1166 1308 1324 1467 1483 1629 1645 1792 1809 1941 1958 1975 2110 2127 2144 2281 2299 2316 2456 2474 2491 2634 2817 3004 3196 3393 3596 3806 4024 4250 2652 2670 2835 2854 3023 3042 3215 3235 3413 3433 3617 3638 3828 3849 4046 4068 4273 4296 7581 8052 8577 9172 9857 0669 1664 2954 4792 8038 7626 7672 8102 8152 8633 8690 9236 9301 9932*0008 0759 0850 1777 1893 3106 3264 5027 5275 8550 9130 4484 4730 4986 5256 5541 5842 6163 4509 4533 4755 4780; 5013 5039 5284 5312 5570 5600 5873 5905 6196 6230 .9 0870 0885 0901 1025 1041 1056 1182 1197 1213 1340 1356 1371 1499 1516 1532 1661 1677 1694 1825 1842 1858 1992 2008 2161 2178 2333 2351 2025 2195 2368 2509 2527 2689 2707 2872 2891 3061 3080 325 t 3274 3453 3473 3659 3679 3871 3892 4091 4113 4319 4342 2545 2725 2910 3099 3294 3494 3700 39^4 4136 4366 6507 6542 6578 6877 6915 6954 7278 7320 7363 7718 7764 7811 8203 8748 9367 ^0085^ 0944 2012 3429 5539 9800^ 8255 8307 8806 8865 9433 9501 0164*0244 1040 1138 2135 2261 3599 3777 5819 6119 0591*1561 4557 4581 4805 4831 5066 5093 5340 5368 5629 5659 5936 5968 6264 6298 4606 4857 5120 5397 5689 6000 6332 6615 6651 6688 6994 7033 7073 7406 7449 7493 .6 A 7858 7906 7954 8360 8413 8467 8924 8985 9046 9570 9640 9711 "0326*0-109*0494 1238 1341 1446 2391 2525 2663 3962 4155 4357 6441 6789 7167 ■2810*4571*7581 1.0 091639 107238 122937 138730 1548 35 171034 187533 2042 32 2212!31 2386 '30 2562 29 274328 2928 27 3118 26 3313 25 3514 24 3721 23 3936 '22 415821 4389 20 4630 19 4882 5147 5425 5719 6032^14 63o6[13 6725 12 711311 7537 10:44 8003 8522 9109 9784 ►05 SO 1554 2806 4569 7581 .3 .1 I .0 9j49 855 7 63 pp. 31 32 33 34 35 36 17 38 39 40 41 42 43 44 45 46 47 48 49 4 8 12 J 4 4 '% 8 9 12 13 13 4 5 5 9 9 9 13 14 14 5 5 5 9 10 10 14 14 15 16 20 24 16 17 17 21 21 22 25 25 26 18 18 18 22 23 23 26 27 28 19 19 20 24 24 25 28 29 29 28 32 36 29 29 30 33 34 34 37 38 39 3132 32 35 36 37 40 41 41 33 34 34 38 38 39 42 43 44 50 3 3 3 6 6 7 9 10 10 12 13 13 16 16 17 19 19 20 22 22 23 25 26 26 28 29 30 3 4 4 7 7 7 10 11 11 14 14 14 17 18 18 20 21 22 24 25 25 27 28 29 31 32 32 4 4 4 7 8 8 11 11 12 15 15 16 19 19 20 22 23 23 26 27 27 30 30 31 33 34 35 20 VI. LOGAEITHMIO TANGENTS AND COTANGENTS loj7 tan « ' 1<>^ votannP CONTINUED. 21 log tan n° loff cotan n° n° 1 2 3 4 5 6 7 8 9 10 84.9 d 5.0 8.942( ) 9428 9437 9446 9454 9463 9472 9480 9489 9497 9506 5.1 5.2 5.3 950( 959] 967^ ) 9515 9523 9532 9599 9608 9616 9682 9690 9699 9540 9549 9557 9624 9633 9641 9707 9715 9723 9565 9574 9582 9649 9657 9666 9731 9739 9747 9591 9674 9756 84.8 84.7 84.6 9 8 9 5.4 5.5 5.6 975^ 9S36 991i 9764 9772 9780 9844 9852 9860 9922 9930 9938 9788 9796 9S04 9867 9875 9883 9946 9953 9961 9812 9820 9828 9891 9899 9907 9969 9977 9984 9836 9915 9992 84.5 84.4 84.3 8 8 8 5.7 5.8 5.9 9992 9.006e 0142 *0000*0007*0015 0075 0083 0090 0150 0157 0165 *0022*0030*003S 0098 0105 0113 0172 0180 0187 *0045*0053*0060 0120 0128 0135 0194 0202 0209 ^0068 0143 0216 0289 84.2 84.1 84.0 83.9 8 8 7 8 6.0 9.021^ 0223 0231 0238 0245 0253 0260 0267 0274 0281 6.1 6.2 6.3 02SS 036C 043C 0296 0303 0310 0367 0374 0381 0437 0444 0451 0317 0324 0331 0388 0395 0402 0457 0464 0471 0338 0346 0353 0409 0416 0423 0478 0485 0492 0360 -0430 0499 83.8 83.7 83.6 7 7 7 6.4 6.5 6.6 049? 0567 0633 0506 0512 0519 0573 0580 0587 0640 0647 0653 0526 0533 0540 0593 0600 0607 0660 0667 0673 0546 0553 0560 0614 0620 0627 0680 0686 0693 0567 0633 0699 83.5 83.4 83.3 7 6 6 6.7 6.8 6.9 069C 07(A 082S 0/06 0712 0719 0771 0777 0784 0835 0841 0847 0725 0732 0738 0790 0796 0803 0854 0860 0866 0745 0751 0758 0809 0816 0822 0873 0879 0885 0764 0828 0891 83.2 83.1 83.0 82.9 6 6 6 7 7.0 9.0891 0898 0904 0910 0916 0923 0929 0935 0941 0947 0954 7.1 7.2 7.3 095^ 101 = 107^ 0960 0966 0972 1021 1027 1033 1082 1088 1094 0978 0984 0991 1039 1045 1051 1100 1106 1112 0997 1003 1009! 1015 1058 1064 1070 1076 1117 1123 1129 1135 82.8 82.7 82.6 6 6 6 7.4 7.5 7.6 113f 119^ 1252 1141 1147 1153 1200 1206 1212 1258 1264 1270 1159 1165 1171 1218 1223 1229 1276 1281 1287 1177 1183 1188 1235 1241 1247 1293 1299 1304 1194 1252 1310 82.5 82.4 82.3 6 5 6 7.7 7.8 7.9 131C 136/ 142^ ) 1316 1321 1327 1372 1378 1384 1428 1434 1439 1333 1338 1344 1389 1395 1400 1445 1450 1456 1350 1355 1361 1406 1412 1417 1461 1467 1473 1 1367 1423 1478 82.2 82.1 82.0 6 6 5 8.0 9.1475 1484 1489 1494 1500 1503 151] 1516 1522 1527 1533 81.9 6 8.1 8.2 8.3 1533 158/ 164C 1538 1544 1549 1592 1597 1603 ) 16^5 1651 1656 1554 1560 1565 1608 1613 1619 1661 1667 1672 1571 1576 1581 1624 1629 1635 1677 1682 1688 1587 1640 1693 81.8 81.7 81.6 6 5 5 8.4 8.5 8.6 16931 1698 1703 1709 17i5 1750 1755 1761 1797 1802 1807 1812 1714 1719 1724 1766 1771 1776 1817 1822 1827 1729 1735 1740 1781 1786 1791 1832 1837 1842 1745 1797 1848 81.5 81.4 81.3 5 6 6 8.7 8.8 8.9 1848 1853 1858 1863 1898 1903 1908 1913 1948' 1953 1958 1963 1868 1873 1878; 1883 1888 1893 1918 1923 1928 1933 1938 1943 1968 1973 1977 1982 1987 1992 1898 19-18 1997 81.2 81.1 81.0 11° 5 5 5 10 ! 9 8 7 6 5 4 3 2 1 1 f 1 .16 1 2 3 4 5 6 7 8 9 o ,T log tan .016 .03 .05 .183 .2 .216 .06 .083 .1 .23 .25 .26 .116 .13 .15 .283 .3 .316 0.00 1.28 1.98 1.7581 1.7580 8.3492 1.7579 8.5387 2 8 .3 .5 .35 .36 .383 .516 .53 .55 .^ »416 .43 .56 .583 .6 .45 .46 .483 .616 .63 .65 2.49 2.91 3.27 1.7578 8.6384 1.7577 8.7061 1.7576 8.7569 4 5 .6 .83 .683 .7 .716 .85 .86 .883 .73 .75 .76 .9 .916 .93 .783 .8 .816 .95 .96 .983 3.60 3.90 4.18 1.7575 8.7988 1.7574 8.8336 8.8638 22 VII. LOGAKITHMIO SINES AND COSINES u"^^ 0' 10' 20' 30' 40' 50' j 60' d PP 7.4637 7648 9408 *065S*1627*2419i89 45 46 47 48 49 1 2 3 S.2419 542b 7ISS 3088 5776 7423 3668 6097 7645 4179 6397 7857 4637 5050 6677 6940 8059 8251 5428 88 7188 87 8436 86 5 5 5 5 5 9 9 9 10 10 14 14 14 14 15 4 o 8436 9403 9.0192 8613 9545 0311 8783 9682 0426 8946 9816 0539 9104 9256 9945*0070 0648 0755 9403 *0192 0859 85 84 83 18 18 19 19 20 23 23 24 24 25 27 28 28 29 29 7 8 9 0S.S9 1436 1943 0961 1525 2022 1060 1612 2100 1157 1697 2176 1252 1345 1781 1863 2251 2324 1436 1943 2397 82 81 80 91 80 73 32 32 33 34 34 36 37 38 38 39 41 41 42 43 44 9.2397 2468 2538 2606 2674 2740 2806 79 66 40 41 42 43 44 11 12 13 2806 3179 3521 2870 3238 3575 2934 3296 3629 2997 3353 3682 3058 3119 3410 3466 3734 3786 3179 3521 3837 78 77 76 60 55 51 4 4 4 4 4 8 8 8 9 9 12 12 13 13 13 U 15 10 3837 4130 4403 3887 4177 4447 3937 4223 4491 3986 4269 4533 4035 4083 4314 4359 4576 4618 4130 4403 4659 75 74 73 47 44 41 16 16 17 17 18 20 21 21 22 22 24 25 25 26 26 17 18 19 20 4659 4900 5126 4700 4939 5163 4741 4977 5199 4781 .5015 5235 4821 4861 5052 5090 5270 5306 4900 5126 5341 72 71 70 39 36 35 28 29 29 30 31 32 33 34 34 35 36 37 38 39 40 9.5341 5375 5409 5443 5477 5510 5543 69 33 35 36 37 38 39 21 ^2 23 5543 5736 5919 5576 5767 5948 5609 5798 5978 5641 5828 6007 5673 5704 5859 5889 6036 6065 5736 5919 6093 68 67 66 32 30 28 4 4 4 4 4 7 7 7 8 8 11 11 11 11 12 24 25 26 6093 6259 6418 6121 6286 6444 6149 6313 6470 6177 6340 6495 6205 6232 6366 6392 6521 6546 6259 6418 6570 65 64 63 27 26 24 14 14 15 15 16 15 18 19 19 20 21 22 22 23 23 27 28 29 6570 6716 6856 6595 6740 6878 6620 6763 6901 6644 6787 6923 6668 6692 6810 6833 6946 6968 6716 6856 6990 62 61 60 24 23 22 25 25 26 27 27 28 29 30 30 31 32 32 33 34 35 30 9.6990 7012 7033 7055 7076' 7097 7118 59 21 30 31 32 33 34 31 32 33 7118 7242 7361 7139 7262 7380 7160 7282 7400 7181 7302 7419 7201 7222 7322 7342 7438 7457 7242 7361 7476 58 57 56 20 19 19 3 3 3 3 3 6 6 6 7 7 9 9 10 10 10 34 35 36 7476 7586 7692 7494 7604 771$ 7513 7622 7727 7531 7640 7744 7550 7568 7657 7675 7761 7778 7586 7692 7795 55 54 53 18 17 17 12 12 13 13 14 15 16 16 17 17 18 19 19 20 20 37 38 39 40 7795 7893 7989 7811 7910 8004 7828 7926 8020 7844 7941 8035 7861 7877 7957 7973 8050 8066 7893 7989 8081 52 51 50 16 16 15 21 22 22 23 24 24 25 26 26 27 27 28 29 30 31 9.8081 8096 8111 8125 8140 8155 8169 49 14 25 26 27 28 29 41 42 43 8169 8255 8338 8184 8269 8351 8198 8283 8365 8213 8297 8378 8227 8241 8311 8324 8391 8405 8255*48 8338 47 8418 46 14 14 13 3 3 3 3 3 5 5 5 6 6 8 8 8 8 9 44 45 46 8418 8495 8569 8431 8507 8582 8444 8520 8594 8457 8532 8606 8469 8482 8545 8557 8618 8629 8495 8569 8641 45 44 43 13 12 12 10 10 11 11 12 13 13 14 14 15 15 16 16 17 17 47 48 49 8641 8711 8778 8653 8722 8789 50' 8665 8733 8800 8676 8745 8810 86SS 8699 8756 8767 SS21 8832 8711 8778 8843 42 41 40 12 11 11 18 IS 19 20 20 20 21 22 22 23 23 23 24 25 26 60' 40' 30' 20' 10' 1 0' 11° d rOK MINUTES. 23 log: sin logr cos n° 0' 10' 20' 30' 40' 50' 60' (1 PP 1 50 9.8S43 8853 8864 8874 8884 8895 8905 39 10 20 21 22 23 24 51 52 53 8905 8965 9023 8915 8975 9033 8925 8985 9042 8935 8995 9052 8945 8955 9004 90i4 9061 9070 8965 9023 9080 38 37 36 10 9 10 2 4 6 2 4 6 2 4 7 2 2 5 5 7 7 54 55 56 9080 9134 9186 9089 9142 9194 9098 9151 9203 9107 9160 9211 9116 9125 9169 9177 9219 9228 9134 9186 9236 35 34 33 9 9 8 8 10 12 8 11 13 9 11 13 9 10 12 12 14 14 57 58 59 9236 9284 9331 9244 9292 9338 9252 9300 9346 9260 9308 9353 9268 9276 9315 9323 9361 9368 9284 9331 9375 32 31 30 8 8 7 14 15 15 16 17 16 17 18 18 19 15 19 20 21 22 60 9.9375 9383 9390 9397 9404 9411 9418 29 7 15 16 17 18 19| 61 62 63 9418 9459 9499 9425 9466 9505 9432 9473 9512 9439 9479 9518 9446 9453 9486 9492 9524 9530 9459 9499 9537 28 27 26 6 7 7 2 3 5 2 3 5 2 3 5 2 2 4 4 5 6 64 65 66 9537 9573 9607 9543 9579 9613 9549 9584 9618 9555 9590 9624 9561 9567 9596 9602 9629 9635 9573 9607 9640 25 24 23 6 5 5 6 8 9 6 8 10 7 9 10 7 8 9 10 11 11 67 68 69 9640 9672 9702 9646 9677 9706 9651 9682 9711 9656 9687 9716 9661 9667 9692 9697 9721 9725 9672 9702 9730 22 21 20 5 5 5 11 12 11 13 14 12 14 15 13 13 14 15 16 17 70 9.9730 9734 9739 9743 9748 9752 9757 19 5 10 11 12 13 14 71 72 73 9757 9782 9806 9761 9786 9810 9765 9790 9814 9770 9794 9817 9774 9778 979S 9802 9821 9825 9782 9806 9828 18 17 16 4 4 3 1 2 3 1 2 3 1 2 4 1 1 3 3 4 4 74 75 76 9828 9849 9869 9832 9853 9872 9836 9856 9875 9839 9859 9878 9843 9846 9863 9866 9881 9884 9849 9869 9887 15 14 13 I 3 4 5 6 4 6 7 5 6 7 5 6 7 7 8 8 77 78 79 9887 9904 9919 9890 9907 9922 9893 9909 9924 9896 9912 9927 9899 9901 9914 9917 9929 9931 9904 9919 9934 12 11 10 3 2 3 7 8 9 8 8 9 10 10 11 9 10 10 11 12 13 80 9.9934 9936 9938 9940 9942 9944 9946 9 2 5 6 7 8 9 81 82 83 9946 9958 9968 9948 9959 9969 9950 9961 9971 9952 9963 9972 9954 9956 9964 9966 9973 9975 9958 9968 9976 8 7 6 2 2 1 1 1 2 1 1 2 1 1 2 1 1 2 2 2 3 84 85 86 9976 9983 9989 9977 9985 9990 9979 9986 9991 9980 9987 9992 9981 9982 9988 9989 9993 9993 9983 9989 9994 5 4 3 1 1 2 3 3 2 3 4 3 4 4 3 4 4 5 5 5 87 88 89 9994 9997 9999 9995 9998 *0000 9995 9998 *0000 9996 9999 *0000 9996 9997 9999 9999 *0000 *0000 9997 9999 *0000 2 1 4 4 5 4 5 5 5 6 6 6 6 6 7 7 8 60' 50' 40' 30' 20' 10' 0' n° d 1 1 1 2 3 4 1 1 1 1 1 2 2 2 2 2 o t ' S log sin o t / S 1 og sin 00 1 51 2 50 1 1 3.5363 1 3.5364 8.5090 ■0 3.5365 8.6940 5 07 5 32 5 56 30 33 35 7 3.5369 I 3.5370 5 3.5371 8.95 8.95 9.01 03 42 44 8 32 4 08 4 39 2 2- 2 2 3.5366 8.7898 \S 3.5367 8.8578 ■9 3.5368 8.9089 6 18 6 39 6 59 37 39 41 S 3.5372 9 3.5373 9 9.0-1 9.0^ 9.05 03 37 49 1 1 1 2 2 2 3 2 3 3 4 2J VIII. lot; tllll LOGARITHMIC TANGENTS AND COTANGENTS Utii cotan n^" 0' 10' 20' 30' 40' 50' j 60' d PP 7.4637 7648 940"; *0658 *1627 *2419 89 95 96 97 98 99 8.2419 5431 7194 3089 5779 7429 3669 6101 7652 4181 6401 786i 4638 6682 8067 5053 6945 8261 5431 7194 8446 88 87 86 10 10 10 10 10 19 19 J9 20 20 29 29 29 29 30 4 5 « 8446 9420 9.0216 8624 9563 0336 8795 9701 0453 8960 9836 0567 9118 9966 0678 9272 *0093 0786 9420| 85 *0216l84 0891,83 105 38 38 39 39 40 48 48 49 49 50 57 58 58 59 59 7 i 0891 8 1478 9 1997 099i 1569 2078 1096 1658 2158 1194 1745 2236 1291 1831 2313 1385 1915 2389 1478 1997 2463 80 93 82 74 67 67 68 69 69 76 77 78 78 79 86 86 87 88 89 10 9.2463 2536 2609 2680 2750 2819 2887 79 68 90 91 92 93 94 11 2887 12 3275 13 3634 2953 3336 3691 3020 3085 3397 3458 3748 3804 3149 3517 3859 3212 3576 3914 3275 3634 3968 78 77 76 63 58 54 9 9 9 9 9 18 IS 18 19 19 27 27 28 28 28 14 3968 15 42SI 1<> 457i 4021 4331 4622 4074 4381 4669 4127 4430 4716 4178 4479 4762 4230 4527 4808 4281 4575 4853 75 74 73 51 48 45 36 36 37 37 38 45 46 46 47 47 54 55 55 56 56 17 IS ly 4853 5118 5370 4898 5161 5411 4943 5203 5451 4987 5245 5491 5031 5287 5531 5075 5329 5571 5118 5370 5611 72 71 70 43 41 40 63 64 64 65 66 72 73 74 74 75 81 82 83 84 85 20 9.5611 5650 5689 5727 ■5766 5804 5842 69 38 85 86 87 88 89 21 22 23 5842 6064 6279 5879 6100 6314 5917 6136 6348 5954 6172 6383 5991 6208 6417 6028 6243 6452 6064 6279 6486 68 67 66 36 36 34 9 9 9 9 9 17 17 17 18 18 26 26 26 26 27 24 25 26 6486 6687 6882 6520 6720 6914 6553 6752 6946 6587 6785 6977 6620 6817 7009 6654 6850 7040 6687 6882 7072 65 64 63 33 32 32 34 34 35 35 36 43 43 44 44 45 51 52 52 53 53 27 28 29 7072 7257 7438 7103 7287 7467 7134 7317 7497 7165 7348 7526 7196 7378 7556 7226 7408 7585 7257 7438 7614 62 61 60 31 30 29 60 60 61 62 62 68 69 70 70 71 77 77 78 79 SO 30 ^9.7614 7644 7673 7701 7730 7759 7788 59 29 80 81 82 83 84 31 32 33 7788 7958 8125 7816 7986 8153 7845 8014 8180 7873 8042 8208 7902 8070 8235 7930 8097 8263 7958 8125 8290 58 57 56 28 28 27 S 8 S 8 8 16 16 16 17 17 24 24 25 25 25 34 35 36 8290 8452 8613 8317 8479 8639 8344 8506 8666 8371 8533 8692 8398 8559 8718 8425 8586 8745 8452 8613 8771 55 54 53 27 27 26 32 32 33 33 34 40 41 41 42 42 48 49 49 50 50 37 38 39 8771 8928 9084 8797 8954 9110 SS24 8980 9135 8850 9006 9161 8876 9032 9187 8902 9058 9212 S92S 9084 9238 52 51 50 26 26 26 56 57 57 58 59 64 65 66 66 67 72 73 74 75 76 40 9.9238 9264 9289 9315 9341 9366 9392 49 26 75 76 77 78 79 41 9392 42 9544 43 9697 9417 9570 9722 9443 9595 9747 9468 9621 9772 9494 9646 9798 9519 9671 9823 9544 9697 9848 48 47 46 25 26 25 8 8 8 8 8 15 15 15 16 16 23 23 23 23 24 44 9848 45 0.0000 46 0)52 9874 0025 0177 9899 0051 0202 9924 0076 0228 9949 0101 0253 9975 0126 0278 *0000 0152 0303 45 44 43 25 26 25 30 30 31 31 32 38 38 39 39 40 45 46 46 47 47 47 0303 48 04 .56 49 ' 0608 0329 0481 0634 0354 0506 0659 0379 0532 0685 0405 0557 0711 0430 0583 0736 0456 0608 0762 42 41 40 26 25 26 53 53 54 55 55 60 61 62 62 63 68 68 69 70 71 60' 50' 40' 30' 20' 10' j 0' n^ d FOE MINUTES. 25 log- tan log- cotan n° 0' 10' 20' 30' 40' 50' 60' d PP 50 0.0762 0788 0813 0839 0865 0890 0916 39 26 70 71 72 73 74 51 52 53 0916 1072 1229 0942 0968 0994 1020 1046 1098 1124 1150 1176 1203 1255 1282 1308 1334 1361 1072 1229 1387 38 37 36 26 26 26 7 7 7 7 7 14 14 14 IS 15 21 21 22 22 22 54 55 56 1387 1548 1710 1414 1441 1467 1494 1521 1575 1602 1629 1656 1683 1737 1765 1792 1820 1847 1548 1710 1875 35 34 33 27 27 28 28 28 29 29 30 35 36 36 37 37 42 43 43 44 44 57 58 59 1875 2042 2212 1903 1930 1958 1986 2014 2070 2098 2127 2155 2184 2241 2270 2299 2327 2356 2042 2212 2386 32 31 30 28 28 30 49 50 50 51 52 56 57 58 58 59 63 64 65 66 67 60 0.2386 2415 2444 2474 2503 2533 2562 29 29 65 66 67 68 69 61 62 63 2562 2743 2928 2592 2622 2652 2683 2713 2774 2804 2835 2866 2897 2960 2991 3023 3054 30S6 2743 2928 3118 28 27 26 30 31 32 7 7 7 7 7 13 13 13 14 14 20 20 20 20 21 64 65 66 3118 3313 3514 3150 3183 3215 3248 3280 3346 3380 3413 3447 3480 3548 3583 3617 3652 3686 3313 3514 3721 25 24 23 33 34 35 26 26 27 27 28 33 33 34 34 35 39 40 40 41 41 67 68 69 3721 3936 4158 3757 3792 3828 3864 3900 3972 4009 4046 4083 4121 4196 4234 4273 4311 4350 3936 4158 4389 21 20 36 37 39 46 46 47 48 48 52 53 54 54 55 59 59 60 61 62 70 0.4389 4429 4469 4509 4549 4589 4630 19 41 60 61 62 63 64 71 72 73 4630 4882 5147 4671 4713 4755 4797 4839 4925 4969 5013 5057 5102 5192 5238 5284 5331 5378 4882 5147 5425 18 17 16 43 45 47 6 6 6 6 6 12 12 12 13 13 18 18 19 19 19 74 75 76 5425 5719 6032 5473 5521 5570 5619 5669 5770 5822 5873 5926 5979 6086 6141 6196 6252 6309 5719 6032 6366 15 14 13 50 53 57 24 24 25 25 26 30 31 31 32 32 36 37 37 38 38 77 78 79 6366 6725 7113 6424 6483 6542 6603 6664 6788 6851 6915 6980 7047 7181 7250 7320 7391 7464 6725 7113 7537 12 11 10 61 66 73 42 43 43 44 45 48 49 50 SO 51 54 55 56 57 58 80 0.7537 7611 7687 7764 7842 7922 8003 9 81 55 56 57 58 59 81 82 83 8003 8522 9109 8085 8169 8255 8342 8431 8615 8709 8806 8904 9005 9214 9322 9433 9547 9664 8522 9109 9784 8 7 6 91 104 120 6 6 6 6 6 11 11 11 12 12 17 17 17 17 18 84 85 86 9784 1.0580 1554 9907*0034*0164*0299*0437 0728 0882 1040 1205 1376 1739 1933 2135 2348 2571 *0580 1554 2806 5 4 3 22 22 23 23 24 28 28 29 29 30 33 34 34 35 35 87 88 89 2806 4569 7581 3055 3318 3599 3899 4221 4947 5362 5819 6331 6911 8373 9342*0591*2352*5363 4569 7581 2 1 39 39 40 41 41 44 45 46 46 47 50 50 51 52 53 60' 50' 40' 30' 20' 10' 0' n° d 50 51 52 53 54 5 5 5 5 5 10 10 10 11 11 15 15 16 16 16 r ' T log tan o r ' T log tan 00 44 1 41 3.5363 44 3.5362 8.1072 101 3.5361 8.4682 4 19 4 34 4 49 259 3.5354 8.8778 274 3.5353 8.9024 289 3.5352 8.9256 2 15 2 43 3 06 135 3.5360 8.5943 163 3.5359 8.6762 186 3.5358 8.7337 5 03 5 16 5 28 303 3.5351 S.9463 316 3.5350 8.9646 328 3.5349 8.9809 20 20 21 21 22 25 26 26 27 27 30 31 31 32 32 3 27 3 46 4 03 207 3.5357 8.7802 226 3..5356 8.8185 243 3.5355 8.8501 5 41 5 52 6 04 341 3.5348 8.9979 352 3.5347 9.0118 364 9.0265 35 36 36 37 38 40 41 42 42 43 45 46 47 48 49 IX. NATURAL SIlfES AND COSINES. sin n cos n° vt" .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 (1 0' 0' 13' 18' 24' 30' 30' 42' 48' 54' CO' ~0 o.ouuu 0017 003^0052 0070 0087 OlOi 0122 0140 0157 "0l75 "89 78' 1 0175 0192 020<; 0227 0244 0262 0279 0297 0314 0332 0341; 8S 17 2 0349 0366 0384 0401 0419 0436 0454 0471 0488 0506 0523 87 17 3 0523 0541 0558 0576 0593 0610 0628 0645 0663 0680 069S 86 18 4 0(^.98' 0715 0732 0750 0767 0785 0802 0819 0837 0854 0872 85 18 5 0872 0SS9 0906 0924 0941 0958 0976 0993 1011 1028 1045 84 17 6 1045, 1063 lOSO 1097 IIU 1132 1149 1167 1184 1201 1219 83 IS 7 1219 1236 1253 1271 1288 1305 1323 1.340 1357 1374 1392 82 18 8 1392 1409 1426 1444 1461 1478 1495 1513 1530 LS47 1564 81 17 9 1564 1 582 1.599 1616 1633 1650 1668 1685 1702 1719 1736 80 17 1010.1736 1754 1771 1788 1805 1822 1840 1857 1874 1891 1908 79 17 11 190Si 1925 1942 1959 1977 1994 2011 2028 2045 2062 2079 78 17 12 2079| 2096 2113 2130 2147 2164 2181 21982215 22.33 2250 77 17 13 2250, 2267 22S4 2300J 2317 2334 2351 2368 2385 2402 2419 76 17 14 2419 2436 2453 2470 2487 2504 2521 2.538 25.54 2571 2588 75 17 15 2588 2605 2622 2639 2656 2672 2689 2706 2723 2740 2756 74 16 10 2756, 2773 2790 2807 2823 2840 2857 2874 2890 2907 2924 73 17 17 29241 2940 2957 2974 2990 3007 3024 3040 3057 3074 3090 72 16 18 3090 3107 3123 3140 3156 3173 3190 3206 3223 3239 3256 71 17 19 3256 3272 3289 3305 3322 3.338 ?,ZSS\ 3371 3.387 3404 3420 70 16 20 ;0.3420; 3437 3453 3469| 3^86 3502 3518 3535 35513567 3584 69 17 21 35S4 3600 3616 3633 3649 3665 3681 3697 3714 3730 3746 C8 16 oo 3746, 3762 3778 3795 .38113827 3843 3859 3875 3891 3907 67 16 23 3907 3923 3939 3955 3971 3987 4003 4019 4035 4051 4067 66 16 24 4067, 4083 4099 4115 4131 4147 4163 4179 4195 4210 4226! 65 16 . 25 4226, 4242 4258 4274 4289 4305 4321 4337 4352 4368 4384 64 16 26 4384 4399 4415 4431 4446 4462 4478 4493 4509 4524 4540 63 16 27 . 4540 4555 4571 4586 4602 4617 4633 4648 4664 4679 4695 62 16 28 4695 4710 4726 4741 4756 4772 4787 4802 4818 4833 4848 61 15 2^ 4848 4863 4879 4894 4909 4924 4939 4955 4970 4985 5000 60 15 30 0.5000 5015 5030 5045 5060 5075 5090 5105 5120 5135 5150 59 15 31 5150! 5 165 5 ISO 5 195 5210 5225 5240 5255 5270 5284 5299 58 15 32 5299| 5314 5329 5344 5358 5373 5388 5402 5417 5432 5446 57 14 33 5446 5461 5476 5490 5.505 5519 5534 5548 5563 5577 5592 56 15 34 5592 5606 5621 5635 5650 5664 5678 5693 5707 5721 5736 55 15 35 5736 5750 5764 57791 5793 5807 5821 5835 5850 5864 5878 54 14 36 5878 5892 5906 5920 5934 5948 5962 5976 5990 6004 6018 53 14 37 6018 6032 6046 6O60I 6074 6088 6101 6115 6129 6143 6157 52 14 38 6157 6170 6184 6198 6211 6225 6239 6252 6266 6280 6293 lo 13 39 6293, 6307 6320 6334; 6347 6361 6374 6388 6401 6414 6428 ii 40 0.6428 6441 6455 6468 6481 6494 6508 6521 6534 6547 6561 49 14 41 6561 6574 6587 66O0! 6613 6626 6639 6652 6665 6678 6691 48 13 42 6691. 6704 6717 6730 6743 6756 6769 6782 6794 6807 6820 47 13 43 6820 6S33 6845 6858 6871 6884 6896 6909 6921 6934 6947 46 13 44 6947 6959 6972 6984 6997 7009 7022 7034 7046 7059i 7071 45 12 45 7071 7083 7096 7108 7120 7133 7145 7157 7169 7181 7193 44 12 46 7193 7206 7218 72301 7242 7254 7266 7278 7290 7302 7314 43 12 47 7314! 7325 7337 7349 7361 7373 7385 7396 7408 7420 7431 42 11 48 7431 7443 7455 7466 7478 7490 7501 7513 7524 7.536 7547 41 11 49 7547 7559 7570 7581 7593 7604 7615 7627 763S 7649 7660 40 11 ec 54' 48' 42' 3(5' 30' 24' 18' 12' 6' 0' 1.0 .9 .8 .7 .6 .5 .4 .3 .2 .1 .0 n" d IX. NATUEAL SIlfES AND COSINES. 27 sinn° COS n° n° .0 a .2 .3 .4 .5 .6 .7 .8 .9 1.0 d 0' 6' 13' 18' 24' 30' 36' 42' 48' 54' 60' 50 0.7660 7672 7683 7694 7705 7716 7727 7738 7749 776C 7771 "39 IT 51 7771 7782 7793 7804! 7815 7826 7837 7848 7859 7869 7880 38 11 52 7SS0 7891 7902 7912' 7923 7934 7944 7955 '7965 7976 7986^ 37 10 53 79S6 7997 8007 8018 8028 8039 8049 8059 8070 8080 8090' 36 10 54 8090 8100 81118121 8131 8141 8151 8161 8171 8181 8192 35 11 55 8192 8202 8211 8221 8231 8241 8251 8261 8271 8281 8290! 34 9 56 8290 8300 8310 8320 8329 8339 8348 8358 8368 8377 8387 33 10 57 8387 8396 8406 8415,8425 8434 8443 8453 8^62 8471 8480 32 9 58 8480 8490 8499 8508,8517 8526 8536 8545 8554 8563 8572 31 9 59 8572 8581 8590 8599 8607 8616 8625 8634 8643 8652 8660 30 8 60 0.8669 8669 8678 8686 8695 8704 8712 8721 8729 8738 8746 ~2d T 61 8746 8755 8763 8771 '8780 8788 8796 8805 8813 8821 8829 28 8 62 8829 8838 8846 8854! 8862 SS70 8878 8886 8894 8902 8910| 27 8 63 8910 8918 8926 8934 8942 8949 8957 896i 8973 8980 8988 26 S 64 8988 8996 9003 9011 9018 9026 9033 9041 9048 9056 9063! 25 7 65 9063 9070 9078 9085 9092 9100 9107 9114 9121 9128 9135 24 7 66 9135 9143 9150 9157 9164 9171 9178 9184 9191 9198 9205 23 7 67 9205 9212 9219 9225 9232 9239 9245 9252 9259 9265 9272 22 7 68 9272 9278 9285 9291 9298 9304 9311 9317 9323 9330 9336 21 6 69 9336 9342 9348 9354 9361 9367 9373 9379 9385 9391 9397 20 6 To 0.9397 9403 9409 9415 9421 9426 9432 9438 9444 9449 9455 Id 6 71 9455 9461 9466 9472 9478 9483 9489 9494 9500 9505 9511 18 6 72 9511 9516 95219527 9532 9537 9542 9548 9553 9558 9563 17 5 73 9563 9568 9573 9578 9583 9588 9593 9598 9603 9608 9613 16 5 74 9613 9617 9622 9627 9632 9636 9641 9646 9650 9655 9659 15 4 75 9659 9664 9663 9673 9677 9681 9686 9690 9694 9699 9703 14 4 76 9703 9707 97119715 9720 9724 9728 9732 9736 9740 9744 13 4 77 9744 9748 97519755 9759 9763 9767 9770 9774 9778 9781 12 3 78 9781 9785 9789 9792 9796 9799 9803 9806 9810 9813 9816 11 3 79 9816 9820 9823 9826 9829 9833 9836 9839 9842 9845 9848 10 3 80 0.9848 9851 9854 9857 9860 9S63 9866 9869 9871 9874 9877 ~~9 T 81 9S77 9880 9882 9885 9888 9890 9893 9895 9898 9900 9903 8 3 82 9903 9905 9907 9910 9912 9914 9917 9919 9921 9923 9925 7 2 83 9925 9928 9930 9932 9934 9936 9938 9940 9942 9943 9945 6 2 84 9945 9947 9949 9951 9952 9954 9956 9957 9959 9960 9962 5 2 85 9962 9963 9965 9966 9968 9969 9971 9972 9973 9974 9976 4 2 86 9976 9977 9978 9979 9980 9981 9982 9983 9984 9985 9986 3 1 87 9986 9987 9988 9989 9990 9990 9991 9992 9993 9993 9994 2 1 88 9994: 9995 9995 9996 9996 9997 9997! 9';97 9998 9998 9998 1 89 9998 9999 9999 9999 9999 1.000 1.000 J 1.000 1.000 1.000 1.000 60' 54' 48' 42' 36' 30' 24' 18' 12' 6' 0' 1.0 .9 .8 .7 .6 .5 .4 .3 .2 .1 .0 11° d 12 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 2 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 3 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 4 1 1 2 3 3 4 5 5 6 7 7 8 9 9 10 11 11 12 13 5 1 2 3 3 4 5 6 7 8 8 9 10 11 1 I U U 14 15 16 28 X. NATUEAL TANGENTS AND COTANGENTS. tan »» cot nP if .0 0' .1 .2 .3 0' 12' 18' .4 .5 .G 24' 30' 36' .7 .8 .9 42' 48' 54' 1.0 60' d 2 :1 4 5 7 8 9 0.0000 0175 0349 0524 0699 0S75 1051 122S 1405 15S4 0017 003i 0052 0192 0209 0227 0367 0384 a J 02 0.542 05.59 0577 0717 07.34 0752 0892 0910 092S 1069 1086 1104 1246 1263 1281 1423 1441 14.59 1602 1620 1638 0070 0087 0105 0244 0262 0279 0419 0437 0454 1 0594 0612 0629 0769 0787 0805 0945 0963 0981 1122 1139 1157 1299 1317 1334 1477 1495 1512 1655 167.3 1691 0122 0140 0157 0297 0314 0.332 04 72 0489 0.507 0647 0664 0682 0822 0840 0857 0998 1016 1033 1175 1192 1210 1352 1370 1388 15.30 1548 1566 1709 1727 1745 0175 0349 0524 0699 0875 1051 1228 1405 1584 1763 :89 'HS S7 86 85 84 83 82 81 80 18 17 !5 18 IS IS 17 18 18 10 11 12 13 u 15 1« 17 IS 19 0.1763 1944 2126 2309 2493 2679 2S67 3057 3249 3443 1781 1799 1817 1962 1980 1998 2144 2162 2180 2327 2345 2364 2512 25.30 2549 2698 2717 2736 2886 2905 2924 3076 3096 3115 3269 3288 3307 3463 34S2 3502 1835 1853 1871 2016 2035 2053 2199 2217 2235 2382 2401 2419 2568 2586 2605 2754 2773 2792 2943 2962 2981 3134 3153 3172 3327 3346 3365 3522 3541 3561 1890 1908 1926 2071 2089 2107 2254 2272 2290 2438 2456 2475 2623 2642 2661 2811 2830 2849 3000 3019 3038 3191 3211 3230 3385 3404 3424 3581 3600 3620 1944 2126 2309 2493 2679 2867 3057 3249 3443 3640 79 78 77 76 75 74 73 71 70 18 19 19 18 18 18 19 19 19 20 20 21 2'* 23 24 25 2(i 27 2H 29 0.3640 3659 3679 3699 3S39; 3S59 3879 3S99 4040 4061 40S1 4101 4245 4265 42S6 4307 4452 4473 4494 4515 4663' 46S4 4706 4727 4S77 4S99 4921 4942 5095 5117 5139 5161 5317 5340 5362 53S4 5543] 5566 5589 5612 3719 3739 3759 3919 3939 3959 4122 4142 4163 4327 4348 4369 4536 4557 4578 4748 4770 4791 4964 4986 5008 5184 5206 5228 5407 5430 5452 5635 5658 5681 3779 3799 3819 3979 4000 4020 4183 4204 4224 4390 4411 4431 4599 4621 4642 4813 4834 4856 5029 5051 5073 5250 5272 5295 5475 5498 5520 5704 5727 5750 3839 60 4040 6H 4245 67 4452 66 4663165 4877' 64 5095 63 5317 62 5543 61 5774 60 20 20 21 21 21 21 22 22 23 24 30 31 32 33 34 35 3G 37 38 39 0.5774 5797 5820 5844 6009 6032 6056 6080 62491 6273 6297 6322 6494! 6519 6544 6569 6745; 67716796 6822 7002 7028 7054 70S0 7265 7292 7319 7346 7536 7563 7590 7618 7813 7841 7869 7S9S S09S 8127 8156 8185 5867 5890 5914 6104 6128 6152 6346 6371 6395 6594 6619 6644 6847 6873 6899 7107 7133 7159 7373 7400 7427 7646 7673 7701 7926 7954 7983 8214 8243 8273 5938 5961 5985 6176 6200 6224 6420 6445 6469 6669 6694 6720 6924 6950 6976 7186 7212 72.39 7454 7481 7508 7729 7757 7785 8012 8040 8069 8302 8332 8361 6009 6249 6494 6745 7002 7265 7536 7813 8098 8391 59 58 57 56 55 54 53 52 51 50 24 25 25 25 26 26 28 28 29 30 40 41 42 43 44 45 4G 47 48 49 0.8391 ' 8421 8451 84S1 8693 8724 8754 8785 9004 90.36 9067 9099 9325 9358 9391 9424 9657| 9691 9725 9759 1.0000 0035 0070 0105 0355 0392 0428 0464 0724' 0761 0799 083 7 11061 1145 1184 1224 1504 1544 1585 1626 S51] 8541 8571 8816 8847 8878 9131 9163 9195 9457 9490 9523 9793 9827 9861 01410176 0212 0501 0538 05 7i 0^5 0913 0951 1263 1303 1343 1667 1708 1750 8601 8632 8662 8910 8941 8972 9228 9260 9293 9556 9590 9623 9896 9930 9965 0247 0283 0319 0612 0649 0686 0990 1028 1067 1383 1423 1463 1792 1833 1875 8693 9004 9325 9657 *0000 0355 0724 1106 1504 1918 49 48 47 46 45 44 43 42 41 40 31 32 32 34 35 36 38 39 41 43 60' 54' 48' 42' 1.0 .9 .8 .7 36' 30' 24' .6 .5 .4 18' 13' 6' .3 .2 .1 0' .0 n° d X. taii»*° NATURAL TANGENTS AND COTANGENTS. 29 cot»i-° n° .0 .1 .2 .3 1 .4 .5 .6 .7 .8 .9 1.0 d 50 0' 6' 13' 18' 1 34' 30' 36' 42' 48' 54' 60' 39 ^ 1.1918 1960 2002 2045 2088 2131 2174 2218 2261 2305 2349 51 2349 2393 2437 2482: 2527 2572 2617 2662 2708 2753 2799 38 46 52 2799 2846 2892 2938 2985 3032 3079 3127 3175 3222 3270 37 48 53 3270 3319 3367 3416 3465 3514 3564 3613 3663 3713 3764 36 51 54 3764 3814 3865 3916 3968 4019 4071 4124 4176 4229 4281 35 52 55 4281 4335 4388 4442 4496 4550 4605 4659 4715 4770 4826 34 56 56 4826 4882 4938 4994, 5051 5108 5166 5224 5282 5340 5399 33 59 57 5399 5458 5517 5577 5637 5697 5757 5818 5880 5941 6003 32; 62| 58 6003 6066 6128 6191 6255 6319 6383 6447 6512 6577 6643 31 66 59 60 6643 1.7321 6709 6775 6842 6909 6977 7045 7113 7182 7251 7321 30 29 70 74 7391 7461 7532 7603 7675 7747 7S20 7893 7966 8040 61 8040 8115 8190 8265 8341 8418 8495 8572 8650 8728 8807 28 79 62 8807 8887 8967 9047 9128 9210 9292^ 9375 9458 9542' 9626 27 84 63 9626 9711 9797 9883 9970*0057*0145 *0233*0323*0413 *0503 26 90 64:2.0503 0594 0686 0778 0872 0965 1060 1155 1251 1348 1445 25 1 97 65 1445 1543 1642 1742 1842 1943 2045! 2148 2251 2355 2460 24105 66 2460 2566 2673 2781 2889 2998*3109 32^0 3332 3445 3559 23J114 67 3559 3673 3789 3906 4023 4142 4262 4383 4504 4627 4751 22! 124 68 4751 4876 5002 5129' 5257 5386 5517 5649 5782 5916 6051 21 135 69 70 6051 6187 6325 6464 6605 6746 6889 7034 7179 7326i 7475 20 149 19164 2.7475 7625 7776 7929 8083 8239 8397 8556 8716 8878 9042 71 9042 9208 9375 9544 9714 9887*0061 *0237*04 15*0595 1*07 7 7 is: 182 72 3.0777 0961 1146 1334 1524 1716 1910 2106 2305 2506 2709 17;203 73 2709 2914 3122 3332: 3544 3759 3977 4197 4420 4646 4874 16J228 74 4874 5105 5339 5576, 5816 6059 6305 6554 6806 7062 7321 15 75 7321 7583 7848 8118! 8391 8667 8947 9232 9520 9812*0108 14 76 4.0108 0408 0713 1022, 1335 1653 1976 2303 2635 2972 3315 13 77 3315 3662 4015 4373' 4737 5107 5483 5864 6252 6646! 7046 12 78 7046 7453 7867 82SS, 8716 9152 9594 *0045*0504*0970:*1446 11 79 80 5.1446 1929 2422 2924 3435 3955 4486 .5026 55 -8 6140| 6713 f2*2432:*3138 10 9 — 5.6713 7297 7894 8502 9124 9758*0405 *1066*]7- 81 6.3138 3859 4596 5350, 6122 6912 7720 S548 9395*0264 i*l 1.54 8 827.1154 2066 3002 3962 4947 5958 6996 8062 91 58*0285!* 1443 7 83 8.1443 2636 3863 5126 6427 7769 9152 ♦0579*2052*3572 *5144 6 84 9.514 9.677 9.845 10.02 10.20 10.3910.58 10.7810.9911.20 11.43 6 85 11.43 11.6611.9112.16 12.43 12.7113.00 13.30 13.62 13.95 14.30 4 86 14.30 14.6715.0615.46 15.89 16.35 16.83 17.34 17.8918.46 19.0S 3 87 19.08 19.74 20.45 21.20 22.02 22.90 23.86 24.90 26.03 27.27 28.64 2 88 28.64 30.1431.82 33.69 35.80 38.19 40.92 44.07 47.74 52.08 57.29 1 89 57.29 63.66 71.62 81.85 95.49 114.6 143.2 191.0 286.5 573.0 T 1.0 .9 .8 .7 1 .6 .5 .4 .3 .2 .1 .0 20 21 22 23 24 25 26 27 28 29 30 ^1 32 33 34 35 3637 38 39| T 3 4 4 4 4 4 4 5 5 5 5 5 5 6 6 6 6 6 6 7 2 7 7 7 8 8 8 9 9 9 10 10 10 11 11 11 12 12 12 13 13 3 10 11 11 12 12 13 13 14 14 15 15 16 16 17 U''IS 18 19 19 20 4 13 14 15 15 16 17 17 IS 19 19 20 21 21 22 23 23 24 25 25 26 5 17 18 IS 19 20 21 22 23 23 24 25 26 27 28 28 29 30 31 32 33 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 1 7 7 7 7 7 8 8 8 8 8 8 9 9 9 9 9 9 [10 10 10 2 13 14 14 14 15 15 15 16 16 16 17 17 17 18 18 18 19 [ 19 19 20 3 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 i 29 29 30 4 27 27 28 29 29 30 31 31 32 33 33 34 35 35 36 37 37 38 39 39 5 133 34 35 36 37 38 38 39 40 41 42 43 43 44 45 46 47 ' 48 48 i^ 30 XI. NATUEAL SECANTS AND COSECANTS. sec n° cosecn*^ u° .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 d O' C 12' 18' 24' :ui' 3<>' 42' 48' 54' oooiTTooi oooT CO' "0002 89 T 1.0000 0000 0000 0000 0000 0000 0001 1 2 8 0002 0002 0002 0003 1 0003 0003 0004 0006 0007 0007 UU08 0009 0010 0010 0014 OOli U016 0U17 0018 0019 0020 0004 0005 0006 0011 0012 0013 0021 0022 0023 0006 0014 002^ 88 87 80 1 1 4 5 (i 0024 0026 0027 0028 0030 0031 0032 0034 0035 0037 0038 i 0040 004 1 0W3 0045 0U46 0048 0050 0051 0053 0055 1 0057 0059 0061 | 0063 0065 0067 ] 0069 0071 0073 0038 0055 0075 85 84 83 1 2 2 7 8 9 0075 0093 0125 0077 0079 0082 0084 00S6 00S9 0101 0103 0106! 0108 0111 0114 0127 01.^>0 0133i 0136 0139 0142 0091 0093 0096 0116 0119 0122 0145 0148 0151 0098 0125 0154 82 81 80 2 3 3 10:i.0154 0157 01610164 0167 0170 0174 0177 0180 0184 0187 79 3 11 01S7 12 0223 13' 02C3 01910194 0198 0227 02310235 0267 0271 0276 0201 0205 0209 0239 0243 0247 0280 0284 0288 0212 0216 0220 0251 0255 0259 0293 0297 0302 0223 0263 0306 78 77 76 3 4 4 14 0306 15! ^-'^-^ l«i 0403 03110315 0320 0358 0363 0367 0408 0413 (H 19 0324 0329 0334 0372 0377 0382 0424 0429 0435 0338 0343 03^8 0388 0393 0398 0440 04^6 0451 0353 0403 0457 75 74 73 5 5 6 17 18 19 0457 0515 0576 0^63 0468 0474 05210527 0533 0583 0589 0595 0480 0485 0491 0539 0545 0551 0602 0608 0615 0497 0503 0509 0557 0564 0570 0622 0628 0635 0515 0576 0642 72 71 70 6 6 7 20 1.0642 0649 0655 0662 0669 0676 0683 0690 0697 0704 0711 69 7 21 22 23 0711 0785 0864 0719 0726 0733 0793 0801 0808 0872 0880 0888 0740 0748 0755 0816 0824 0832 0896 0904 0913 0763 0770 0778 0840 0848 0856 0921 0929 0938 0785 0864 0946 68 67 66 7 8 8 241 0946 25; 1034 2«j 1126 0955 0963 0972 1043 1052 1061 11361145 1155 0981 0989 0998 1070 1079 1089 1164 1174 1184 1007 1016 1025 1098 1107 1117 1194 1203 1213 1034 1126 1223 65 64 63 9 9 10 27 1223 28 i 1326 29 1434 1233 1243 1253 1336 1347 1357 1445 1456 1467 1264 1274 1284 1368 1379 1390 1478 1490 1501 1294 1305 1315 HOI 1412 1423 1512 1524 1535 1326 1434 1547 62 61 60 11 11 12 12 30 1.1547 1559 1570 1582 1594 1606 1618 1630 1642 1654 1666 59 31 1666 167916911703 32 1792 11805.18181831 33 ' 1924 1937 1951 1964 1716 1728 1741 1844 1857 1870 1978 1992 2006 1753 1766 1779 1883 1897 1910 2020 2034 2048 1792 1924 2062 58 57 56 13 14 14 34 35 3G 2062 2208 2361 2076 2091 2105 2223 2238 2253 2376 2392 2408 2120 2134 2149 2268 2283 2299 2424 2440 2456 2163 2178 2193 2314 2329 2345 2472 24S9 2505 2208 2361 2521 55 54 53 15 16 16 37 38 39 2521 2538 2554 2571 2690 ' 2708 2725 2742 2868 ' 2886 2904 2923 2588 2605 2622 2760 2778 2796 2941 2960 2978 2639 2656 2673 2813 2831 2849 2997 3016 .3035 2690 2868 3054 52 51 50 17 19 19 40 1.3054,3073 3093 3112 3131 31513171 3190 3210 3230 3250 49 20 41 42 43 3250 I 3270 3291 3311 3456 3478 3499 3520 3673 3696 3718 3741 3331 3352 3373 3542 3563 3585 3763 3786 3809 3393 .3414 3435 3607 3629 3651 3832 3855 3878 3456 3673 3902 48 47 46 21 22 24 44 45 46 3902 1 3925 3949 3972 4142,4167 4192 4217 4396 4422 4448 4474 3996 4020 4044 4242 4267 4293 4501 4527 4554 4069 4093 4118 4318 4344 4370 4581 4608 4635 4142 4396 4663 45 44 43 24 26 28 47 48 49 4663 4945 5243 4690 4718 4746 4974 5003 5032 5273 5304 5335 4774 4802 4830 5062 5092 5121 5366 5398 5429 4859 4887 4916 5151 5182 5212 5461 5493 5525 4945 5243 5557 42 41 40 29 31 32 CO' 1.0 .'54' 48' 42' .9 .8 .7 3<>' ;jO' 24' .6 .5 .4 18' 12' G' .3 .2 .1 0' .0 n° d XI. NATUEAL SECANTS AND COSECANTS. 31 sec n° cosec n° 1.5557 5890 6243 6616 7013 7434 78S3 8361 SS7I 9416 2.0000 0627 1301 2027 2812 3662 4586 5593 6695 7904 2.9238 3.0716 2361 4203 6280 8637 4.1336 4454 8097 5.2408 801 5.7588 816.3925 82 7.1853 83'8.20S5 9.567 11.47 14.34 19.11 28.65 57.30 .1 .2 .3 6' 12' 18' 5590 5925 6279 6655 7054 7478 7929 8410 8924 9473 5622 5655 5959 5994 6316 6353 6694 6733 7095 7137 7522 7566 7976 8023 8460 8510 8977 9031 9530 9587 0061 0692 1371 2103 2894 3751 4683 5699 681 1 8032 0122 0183 0757 0824 1441 1513 2179 2256 2976 3060 3841 3931 4780 4879 5805 5913 6927 7046 8161 8291 9379 9521 0872 1030 2535 2712 4399 4598 6502 6727 8890 9147 1627 1923 4793 5137 8496 8901 2883 3367 9665 1190 2891 4799 6955 9408 2223 5486 9313 3860 8164 8751 9351 4637 5366 6111 2757 3684 4635 3238 4457 5711 9.7289.89510.07 11.7111.95 12.20 14.7015.0915.50 19.77 20.47 21.23 30.1631.84 33.71 63.66 71.62 81.85 1.0 .9 .8 .7 34' 30' 3G' 5688 5721 6029 6064 6390 6427 6772 6812 7179 7221 7610 7655 8070 8118 8561 8612 9084 9139 9645 9703 5755 6099 6464 6852 7263 7700 8166 8663 9194 9762 0245 0890 1584 2333 3144 4022 4978 6022 7165 8422 0308 0371 0957 1025 1657 1730 2412 2490 3228 3314 4114 4207 5078 5180 6131 6242 7285 7407 8555 8688 9811 9957*0106 1352 1515 1681 3072 3255 3440 5003 5209 5418 7186 7420 7657 9672 9939*0211 2527 2837 3150 5841 6202 6569 9732*0159*0593 4362 4874 5396 .7 .8 .9 1.0 43' 48' 54' 60' 5788 5S22 6135 6171 6502 6540 6892 6932 7305 7348 7745 7791 8214 8263 8714 8766 9249 9304 9821 9880 5856 6207 6578 6972 7391 7837 8312 8818 9360 9940 0434 1093 1803 2570 3400 4300 5282 6354 7529 8824 0498 0562 1162 1231 1877 1952 2650 2730 3486 3574 4395 4490 5384 5488 6466 6580 7653 7778 8960 9099 *0256*0407*056] 1848 2017 2188 3628 3817 4009 5629 5843 6060 7897 8140 8387 *04S6*0765*104S 3469 3792 4121 6942 7321 7706 ♦1034*1484*1942 5928 6470 7023 9963*0589 1227 6874 7655 8454 5611 6613 7642 7004 8337 9711 10.25 10.43 10.63 12.47 12.75 13.03 15.93 16.38 16.86 22.04 22.93 23.88 35.8138.20 40.93 95.49114.6143.2 .5 A 1880 2546 3228 9273*0112*0972 8700 9787*0905 n 129*2593*4105 10.83 11.03 11.25 13.34 13.65 13.99 17.37 17.91 18.49 24.92 26.05 27.29 44.08 47.75 52.09 191.0 286.5 573.0 5890 39 6243 38 6616137 7013j36 7434 35 7883 34 8361 133 8871 9416 =0000 34 36 38 41 43 46 49 321 53 31 1 .56 30 i 60 0627 1301 2027 2812 3662 4586 5593 6695 7904 ^238 20 ^0716 19 2361 4203 6280 8637 4336 4454 8097 *2408 _7588 3925 •■1853 "2055 ^5668 11.47 14.34 19.11 28.65 57.30 .1 .0 n^ d 60 61 62 63 64 65 66 67 68 69l 70 71 72 73 10 11 11 11 11 11 11 12 )2 12 21 21 21 22 22 22 23 23 23 24 31 32 32 33 33 34 34 35 35 36 41 42 43 43 44 45 45 46 47 47 52 53 53 54 55 : 56 57 58 58 59 82 83J84 85 86,87 88 89 90 91 14 14 14 14 14 15 15 15 15 15 27 28 28 28 29 29 29 30 30 130 41 42 42 43 43 44 44 45 45 46 55 55 56 57 57 58 59 59 60 61 68 69 70 71 72 73 73 74 75 76 12 12 24 24 36 37 48 49 60 61 92 93 15 16 31 31 46 47 61 62 77 78 74 75 76 12 13 13 25 25 25 37 38 38 49 50 51 62 63 63 94 95 96 16 16 16 31 32 32 47 48 48 63 63 64 78 79 80 77 78 79 13 13 13 26 26 26 39 39 40 51 52 53 64 65 66 97 98 99 16 16 17 32 33 33 49 49 50 65 65 66 81 82 S3 32 XII. RADIANS. nir 180 n^ .0 .1 .2 .3 .4 .5 .0 .7 .8 .9 I> h in 10 0.1745 1763 1780 1798 1815 1833 1850 1868 1885 1902 .0278 40 11 13 1920 2094 2269 1937 1955 1972 2112 2129 2147 2286 2304 2321 1990 2007 2025 2164 2182 2199 2339 2356 2374 2042 2059 2077 2217 2234 2251 2391 2409 2426 .0306 .0333 .0361 44 48 52 U 15 1« 2443 261S 2793 2461 2478 2496 2635 2653 2670 2810 2827 2845 2513 2531 2548 2688 2705 2723 2862 2880 2897 2566 2583 2601 2740 2758 2775 2915 2932 2950 .0389 .0417 .0444 56 1 1 4 17 IS 19 20 2967 3142 3316 0.3491 2985 3002 3019 3159 3176 3194 3334 3351 3368 3037 3054 3072 3211 3229 3246 3386 3403 3421 3089 3107 3124 3264 3281 3299 3438 3456 3473 .0472 .0500 .0528 1 8 1 12 1 16 3.i08 3526 3.543 3560 3578 3595 3613 3630 3648 .0556 1 20 21 2>> 2:1 3665 3S40 4014 3683 3700 3718 3857 3875 3892 4032 4049 4067 3735 3752 3770 3910 3927 3944 4084 4102 4119 3787 3805 3822 3962 3979 3997 4136 4154 4171 .0583 .0611 .0639 124 1 28 1 32 24 25 20 41S9 4363 4.S3S 4206 4224 4241 4381 4398 4416 4555 4573 4590 4259 4276 4294 4433 4451 4468 4608 4625 4643 4311 4328 4346.0667 4485 4503 4520 .0694 4660 4677 4695 .0722 1 36 1 40 1 44 27 2S 29 30 4712 4SS7 5061 4730 4747 4765 4904 4922 4939 5079 5096 5114 4782 4800 4817 4957 4974 4992 5131 5149 5166 4835 4852 4869 5009 5027 5044 5184 5201 5219 .0750 .0778 .0806 1 48 1 52 1 56 0.5236 5253 5271 5288 5306 5323 5341 5358 5376 5393 .0833 2 31 32 33 5411 55S5 5760 5428 5445 5463 5603 5620 5637 5777 5794 5812 5480 5498 5515 5655 5672 5690 5829 5847 5864 5533 55.50 5568 5707 5725 5742 5882 5899 5917 .0861 .0889 .0917 2 4 2 8 2 12 34 35 30 5934 6109 6283 5952 5969 5986 6126 6144 6161 6301 6318 6336 6004 6021 6039 6178 6196 6213 6353 6370 6388 6056 6074 6091 6231 6248 6266 6405 6423 6440 .0944 .0972 .1000 2 16 2 20 2 24 37 38 39 40 6458 6632 6807 6475 6493 6510 6650 6667 6685 6824 6842 6859 6528 6545 6562 6702 6720 6737 6877 6894 6912 6580 6597 6615 6754 6772 6789 6929 6946 6964 .1028 .1056 .1083 2 28 2 32 2 36 0.6981 6999 7016 7034 7051 7069 7086 7103 7121 7138 .1111 2 40 41 42 43 7156 7330 7505 7173 7191 7208 7348 7365 7383 7522 7540 7557 7226 7243 7261 7400 7418 7435 7575 7592 7610 7278 7295 7313 7453 7470 7487 7627 7645 7662 .1139 .1167 .1194 2 44 2 48 2 52 44 45 40 7679 7854 8029 7697 7714 7732 7871 7889 7906 8046 8063 8081 7749 7767 7784 7924 7941 7959 8098 8116 8133 7802 7819 7837 7976 7994 8011 8151 8168 8186 .1222 .1250 .1278 2 56 3 3 4 47 4.S 49 5^ S203 8378 8552 8221 8238 8255 8395 8412 8430 8570 8587 8604 8273 8290 8308 S447 8465 8482 8622 8639 8657 8325 8343 8360 8500 8517 8535 8674 8692 8709 .1306 .1333 .1361 3 8 3 12 3 16 0.8727 8744 8762 8779 8796 8814 8831 8849 8866 8884 .1389 3 20 51 52 53 8901 9076 9250 8919 8936 8954 9093 9111 9128 9268 9285 9303 8971 8988 9006 9146 9163 9180 9320 9338 9355 9023 9041 9058 9198 9215 9233 9372 9390 9407 .1417 .1444 .1472 3 24 3 28 3 32 54 55 50 9425 9599 9774 9442 9460 9477 9617 9634 9652 9791 9809 9826 9495 9512 9529 9669 9687 9704 9844 9861 9879 9547 9564 9582 .1500 9721 9739 9756 .1528 9896 9913 9931 '.1556 3 36 3 40 3 44 57 5S 59 9948 1.0123 0297 9966 9983*0001 0140 0158 0175 0315 0332 0350 *0018*0036*0053|*O071*008S*O]05'.15S3 0193 0210 0228 0245 0263 0280.1611 0367 0385 0402 0420 0437 0455 .1639 3 48 3 52 3 56 180 XII. EADIANS. 60 .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 p h 111 1.0472 0489 0507 0524 0542 0559 0577 0594 0612 0629 .1667 4 61 62 63 0647 0821 0996 0664 0681 0699 0S38 0856 0873 1013 1030 1048 0716 0734 0751 0891 0908 0926 1065 1083 1100 0769 0786 0804 0943 0961 0978 1118 1135 1153 .1694 .1722 .1750 4 4 4 8 4 12 64 65 66 1170 1345 1519 1188 1205 1222 1362 1380 1397 1537 1554 1572 1240 1257 1275 1414 1432 1449 1589 1606 1624 1292 1310 1327 1467 1484 1502 1641 1659 1676 .1778 .1806 .1833 4 16 4 20 4 24 67 68 69 70 1694 1868 2043 1711 1729 1746 1886 1903 1921 2060 2078 2095 1764 1781 1798 1938 1956 1973 2113 2130 2147 1816 1833 1851 1990 2008 2025 2165 2182 2200 .1861 .1889 .1917 4 28 4 32 4 36 1.2217 2235 2252 2270 2287 2305 2322 2339 2357 2374 .1944 4 40 71 72 73 2392 2566 2741 2409 2427 2444 2584 2601 2619 2758 2776 2793 2462 2479 2497 2636 2654 2671 2811 2828 2846 2514 2531 2549 2689 2706 2723 2863 2881 2898 .1972 .2000 .2028 4 44 4 48 4 52 74 75 76 2915 3090 3265 2933 2950 2968 3107 3125 3142 3282 3299 3317 2985 3003 3020 3160 3177 3195 3334 3352 3369 3038 3055 3073 3212 3230 3247 3387 3404 3422 .2056 .2083 .2111 4 56 5 5 4 77 78 79 3439 3614 3788 3456 3474 3491 3631 3648 3666 3806 3823 3840 3509 3526 3544 3683 3701 3718 3858 3875 3893 3561 3579 3596 3736 3753 3771 3910 3928 3945 .2139 .2167 .2194 5 8 5 12 5 16 80 1.3963 3980 3998 4015 4032 4050 4067 4085 4102 4120 .2222 5 20 81 82 83 4137 4312 4486 4155 4172 4190 4329 4347 4364 4504 4521 4539 4207 4224 4242 4382 4399 4416 4556 4573 4591 4259 4277 4294 4434 4451 4469 4608 4626 4643 .2250 .2278 .2306 5 24 5 28 5 32 84 85 86 4661 4835 5010 4678 4696 4713 4853 4870 4888 5027 5045 5062 4731 4748 4765 4905 4923 4940 5080 5097 5115 4783 4800 4818 4957 4975 4992 5132 5149 5167 .2333 .2361 .2389 5 36 5 40 5 44 87 88 89 5184 5359 5533 5202 5219 5237 5376 5394 5411 5551 5568 5586 5254 5272 5289 5429 5446 5464 5603 5621 5638 5307 5324 5341 5481 5499 5516 5656 5673 5691 .2417 .2444 .2472 5 48 5 52 5 56 90 1.5708 5725 5743 5760 5778 5795 5813 5830 5848 5865 .2500 6 91 92 98 5S82 6057 6232 5900 5917 5935 6074 6092 6109 6249 6266 6284 5952 5970 5987 6127 6144 6162 6301 6319 6336 6005 6022 6040 6179 6197 6214 6354 6371 6389 .2528 .2556 .2583 6 4 6 8 6 12 94 95 96 6406 6581 6755 6424 6441 6458 6598 6616 6633 6773 6790 6808 6476 6493 6511 6650 6668 6685 6825 6842 6860 6528 6546 6563 6703 6720 6738 6877 6895 6912 .2611 .2639 .2667 6 16 6 20 6 24 97 98 99 6930 7104 7279 6947 6965 6982 7122 7139 7157 7296 7314 7331 7000 7017 7034 7174 7191 7209 7349 7366 7383 7052 7069 7087 7226 7244 7261 7401 7418 7436 .2694 .2722 .2750 6 28 6 32 6 36 T 2 3 12 3 4 5 6 7 8 9 P s 4 S 12 .00 2909! 3200 3491 3782 .00 5818 6109 6400 6690 .00 8727 9018 9308 9599 4072 4363 4654 6981 7272 7563 9890*0181*0472 4945 5236 5527 7854 8145 8436 *0763*1054*1345 .000 0463 0926 1389 4 5 6 .01 163 .01 454 .01 745 6 1926 2217 2508 4 4835 5126 5417 3 7744 8035 8326 2799 3090 3381 5708 5999 6290 8617 8908 9199 3672 3963 4254 6581 6872 7162 9490 9780*0071 1852 2315 2778 16 20 24 7 8 9 .02 036 .02 327 .02 618 2 0653 0944 1235 1 3562 3853 4144 d 6471 6762 7053 1526 1817 2108 4435 4725 5016 7343 7634 7925 2398 2689 2980 5307 5598 5SS9 8216 8507 8798 3241 3704 4167 28 32 36 34 XIII. EEOIPROOALS. n 1 2 3 4 5 6 7 8 9 d 1.0 1.0000 9901 9804 9709 9615 9524 9434 9346 9259 9174 83 1.1 1.- l!ii 0.9091 S333 7692 9009 8929 8850 8264 8197 8130 7634 7576 7519 8772 8696 8621 8065 8000 7937 7463 7407 7353 8547 8475 8403 7874 7813 7752 7299 7246 7194 70 60 51 1.4 1.5 l.G 7143 6667 6250 7092 7042 6993 6623 6579 6536 6211 6173 6135 6944 6897 6849 6494 6452 6410 6098 6061 6024 6803 6757 6711 6369 6329 6289 5988 5952 5917 44 39 35 1.7 1.8 1.9 5SS2 5556 5263 5848 5814 5780 5525 5495 5464 5236 5208 5181 5747 5714 5682 5435 5405 5376 5155 5128 5102 5650 5618 5587 5348 5319 5291 5076 5051 5025 31 28 25 2.0 0.5000 4975 4950 4926 4902 4878 4854 4831 4808 4785 23 2.1 2.2 2.3 4762 4545 434S 4739 4717 4695 4525 4505 4484 4329 4310 4292 4673 4651 4630 4464 4444 4425 4274 4255 4237 4608 4587 4566 4405 4386 4367 4219 4202 4184 21 19 17 2.4 2.5 2.0 4167 4000 3S46 4149 4132 4115 3984 3968 3953 3831 3817 3S02 4098 4082 4065 3937 3922 3906 3788 3774 3759 4049 4032 4016 3891 3876 3861 3745 3731 3717 16 15 13 2.7 2.8 2.9 3704 3571 3448 3690 3676 3663 3559 3546 3534 3436 3425 3413 3650 3636 3623 3521 3509 3497 3401 3390 3378 3610 3597 3584 3484 3472 3460 3367 3356 3344 13 12 11 3.0 03333 3322 3311 3300 3289 3279 3268 3257 3247 3236 10 3.1 3.2 3.3 3226 3125 3030 3215 3205 3195 3115 3106 3096 3021 3012 3003 3185 3175 3165 3086 3077 3067 2994 2985 2976 3155 3145 3135 3058 3049 3040 2967 2959 2950 10 10 9 3.4 3.5 3.6 2941 2857 2778 2933 2924 2915 2849 2841 2833 2770 2762 2755 2907 2899 2890 2825 2817 2809 2747 2740 2732 2882 2874 2865 2801 2793 2786 2725 2717 2710 8 8 7 3.7 3.8 3.9 2703 2632 2564 2695 2688 2681 2625 2618 2611 2558 2551 2545 2674 2667 2660 2604 2597 2591 2538 2532 2525 2653 2646 2639 2584 2577 2571 2519 2513 2506 7 7 6 4.0 0.2500 2494 2488 2481 2475 2469 2463 2457 2451 2445 6 4.1 4.2 4.3 2439 2381 2326 2433 2427 2421 2375 2370 2364 2320 2315 2309 2415 2410 2404 2358 2353 2347 2304 2299 2294 2398 2392 2387 2342 2336 2331 2288 2283 2278 6 5 5 4.4 4.5 4.6 2273 2222 2174 226S 2262 2257 2217 2212 2208 2169 2165 2160 2252 2247 2242 2203 2198 2193 2155 2151 2146 2237 2232 2227 2188 2183 2179 2141 2137 2132 5 5 4 4.7 4.8 4.9 2128 2083 2041 2123 2119 2114 2079 2075 2070 2037 2033 2028 2110 2105 2101 2066 2062 2058 2024 2020 2016 2096 2092 20SS 2053 2049 2045 2012 2008 2004 5 4 4 5.0 0.2000 1996 1992 1988 1984 1980 1976 1972 1969 1965 4 5.1 5.2 5.3 1961 1923 18S7 1957 1953 1949 1919 1916 1912 1883 1880 1876 1946 1942 1938 1908 1905 1901 1873 1869 1866 1934 1931 1927 1898 1894 1890 1862 1859 1855 4 3 3 5.4 5.5 5.6 1852 1818 1786 1848 1845 1842 1815 1812 1808 1783 1779 1776 1838 1835 1832 1805 1802 1799 1773 1770 1767 1828 1825 1821 1795 1792 1789 1764 1761 1757 3 3 3 5.7 5.8 5.9 1754 1724 1695 1751 1748 1745 1721 1718 1715 1692 1689 1686 1742 1739 1736 1712 1709 1706 1684 1681 1678 1733 1730 1727 1704 1701 1698 1675 1672 1669 3 3 2 1 n XIII. EEOIPROOALS. 35 n 12 3 4 5 6 7 8 9 d 6.0 0.1667 1664 1661 1658 1656 1653 1650 1647 1645 1642 3 6.1 6.2 6.3 1639 1613 1587 1637 1634 1631 1610 1608 1605 1585 1582 1580 1629 1626 1623 1603 1600 1597 1577 1575 1572 1621 159i 157C 1618 1616 1592 1590 1567 1565 3 3 2 6.4 6.5 6.6 1563 1538 1515 1560 1558 1555 1536 1534 1531 1513 1511 1508 1553 1550 1548 1529 1527 1524 1506 1504 1502 154C 1522 1495 1543 1541 1520 1517 1497 1495 3 2 2 6.7 6.8 6.9 1493 1471 1449 1490 1488 1486 1468 1466 1464 1447 1445 1443 1484 1481 1479 1462 1460 1458 1441 1439 1437 1477 1456 1435 1475 1473 1453 1451 1433 1431 2 2 2 7.0 0.1429 1427 1425 1422 1420 1418 1416 1414 1412 1410 2 7.1 7.2 7.3 1408 13Sv 1370 1406 1404 1403 1387 1385 1383 1368 1366 1364 1401 1399 1397 1381 1379 1377 1362 1361 1359 1395 1376 1357 1393 1391 1374 1372 1355 1353 2 2 2 7.4 7.5 7.6 1351 1333 1316 1350 134S 1346 1332 1330 1328 1314 1312 1311 1344 1342 1340 1326 1325 1323 1309 1307 1305 1339 1321 1304 1337 1335 1319 1318 1302 1300 2 2 7.7 7.8 7.9 1299 1282 1266 1297 1295 1294 1280 1279 1277 1264 1263 1261 1292 1290 1289 1276 1274 1272 1259 1258 1256 1287 1285 1284 1271 1269 1267 1255 1253 1252 2 1 2 8.0 0.1250 1248 1247 1245 1244 1242 1241 1239 1238 1236 1 8.1 8.2 8.3 1235 1220 1205 1233 1232 1230 1218 1217 1215 1203 1202 1200 1229 1227 1225 1214 1212 1211 1199 1198 1196 1224 1209 1195 1222 1221 1208 1206 1193 1192 1 1 2 8.4 8.5 8.6 1190 1176 1163 1189 1188 11S6 1175 1174 1172 1161 1160 1159 1185 1183 1182 1171 1170 1168 1157 1156 1155 1181 1167 1153 1179 1178 1166 1164 1152 1151 2 1 2 8.7 8.8 8.9 1149 1136 1124 1148 1147 1145 1135 1134 1133 1122 1121 1120 1144 1143 1142 1131 1130 1129 1119 1117 1116 1140 1127 1115 1139 1138 1126 1125 1114 1112 2 1 1 9.0 0.1111 1110 1109 1107 1106 1105 1104 1103 1101 1100 1 9.1 9.2 9.3 1099 1087 1075 1098 1096 1095 1086 1085 1083 1074 1073 1072 1094 1093 1092 1091 1082 1081 1080 i 1079 1071 1070 1068 ! 1067 1089 1088 1078 1076 1066 1065 1 1 1 9.4 9.5 9.6 1064 1053 1042 1063 1062 1060 1052 1050 1049 1041 1040 1038 1059 1058 1057 I 1056 1048 1047 1046 ; 1045 1037 1036 1035 , 1034 1055 1054 1044 1043 1033 1032 1 1 1 9.7 9.8 9.9 1031 1020 1010 1030 1029 1028 1019 1018 1017 1009 1008 1007 1027 1026 1025 1024 1016 1015 1014 ! 1013 1006 1005 1004 ! 1003 1022 1021 1012 1011 1002 1001 1 1 1 2 3 4 5 6 7 8 9 11 12 13 14 15 16 1 2 3 .5 . (3) (6) .25 .2.1(6) .5 .4.(3) .75 .6 .5 .1429 .125 .2857 .25 .4286 .375 .(2) .(3) .(09) .(IS) .(27) 08(3) .0769 1(6) .1538 25 .2308 .0714 .0(6) .0625 .1429.1(3) .125 .2143 .2 .1875 4 5 6 .8 .(6) .8(3) .5714 .5 .7143 .625 .8571 .75 •(4) .(5) .(6) .(36) •(45) .(54) (3) .3077 41(6) .3846 5 .4615 .2857 .2(6) .25 .3571.(3) .3125 .4286 .4 .375 7 8 9 .875 :S .(63) .(72) ■(SI) 58(3) .5385 (6) .6154 75 .6923 .5 .4(6) .4375 .5714.5(3) .5 .6429 .6 .5625 3G XIV. SQUARES. u^ n 1 2 3 4 6 6 7 8 9 d 1.0 1.000 1.020 1.040 1.061 1.082 1.103 1.124 1.145 1.166 1.188 22 1.1 1.2 1.3 1.210 1.440 1.690 1.232 1.254 1.277 1.464 1.488 1.513 1.716 1.742 1.769 1.300 1.323 1.346 1.538 1..S63 1.588 1.796 1.823 1.850 1.369 1.392 1.416 1.613 1.638 1.664 1.877 1.904 1.932 24 26 28 1.4 1.5 1.6 l.%0 2.250 2..S60 1.9SS 2.016 2.045 2.280 2.310 2.341 2.592 2.624 2.657 2.074 2.103 2.132 2.372 2.403 2.434 2.690 2.723 2.756 2.161 2.190 2.220 2.465 2.496 2.528 2.789 2.822 2.856 30 32 34 1.7 1.8 1.9 2.S90 3.240 3.610 2.924 2.958 2.993 3.276 3.312 3.349 3.648 3.686 3.725 3.028 3.063 3.098 3.386 3.423 3.460 3.764 3.803 3.842 3.133 3.16S 3.204 3.497 3.534 3.572 3.881 3.920 3.960 36 38 40 2.0 4.000 4.040 4.080 4.121 4.162 4.203 4.244 4.285 4.326 4.368 42 2.1 2.2 2!3 4.410 4.S40 5.290 4.452 4.494 4.537 4.884 4.928 4.973 5.336 5.382 5.429 4.580 4.623 4.666 5.018 5.063 5.108 5.476 5.523 5.570 4.709 4.752 4.796 5.153 5.198 5.244 5.617 5.664 5.712 44 46 48 2.4 2.5 2.G 5.760 6.250 6.760 5.808 5.856 5.905 6.300 6.350 6.401 6.812 6.864 6.917 5.954 6.003 6.052 6.452 6.503 6.554 6.970 7.023 7.076 6.101 6.150 6.200 6.605 6.656 6.708 7.129 7.182 7.236 50 52 54 2.7 2.8 2.9 7.290 7.S40 8.410 7.344 7.398 7.453 7.896 7.952 8.009 8.468 8.526 8.585 7.508 7.563 7.618 8.066 8.123 S.ISO 8.644 8.703 8.762 7.673 7.728 7.784 8.237 8.294 8.352 8.821 8.880 8.940 56 58 60 3.0 9.000 9.060 9.120 9.181 9.242 9.303 9.364 9.425 9.486 9.548 62 3.1 3.2 3.3 9.610 10.24 10.89 9.672 9.734 9.797 10.30 10.37 10.43 10.96 11.02 11.09 9.S60 9.923 9.986 10.50 10.56 10.63 11.16 11.22 11.29 10.05 10.11 10.18 10.69 10.76 10.82 11.36 11.42 11.49 6 7 7 3.4 3.5 3.6 11.56 12.25 12.96 11.63 11.70 11.76 12.32 12.39 12.46 13.03 13.10 13.18 11.83 11.90 11.97 12.53 12.60 12.67 13.25 13.32 13.40 12.04 12.11 A2.18 12.74 12.82 12.89 13.47 13.54 13.62 7 7 7 3.7 3.8 3.9 13.69 14.44 15.21 13.76 13.84 13.91 14.52 14.59 14.67 15.29 15.37 15.44 13.99 14.06 14.14 14.75 14.82 14.90 15.52 15.60 15.68 14.21 14.29 14.36 14.98 15.05 15.13 15.76 15.84 15.92 8 8 8 4.0 16.00 16.08 16.16 16.24 16.32 16.40 16.48 16.56 16.65 1673 8 4.1 4.2 4.3 16.81 17.64 18.49 16.89 16.97 17.06 17.72 17.81 17.89 18.58 18.66 18.75 17.14 17.22 17.31 17.98 18.06 18.15 18.84 18.92 19.01 17.39 17.47 17.56 18.23 18.32 18.40 19.10 19.18 19.27 8 9 9 4.4 4.5 4.6 19.36 20.25 21.16 19.45 19.54 19.62 20.34 20.43 20.52 21.25 21.34 21.44 19.71 19.80 19.89 20.61 20.70 20.79 21.53 21.62 21.72 19.98 20.07 20.16 20.88 20.98 21.07 21.81 21.90 22.00 9 9 9 4.7 4.8 4.9 22.09 1 22.18 22.28 22.37 23.04 23.14 23.23 23.33 24.01 j 24.11 24.21 24.30 22.47 22.56 22.66 23.43 23.52 23.62 24.40 24.50 24.60 22.75 22.85 22.94 23.72 23.81 23.91 24.70 24.80 24.90 10 10 10 5.0 25.00 , 25.10 25.20 25.30 25.40 25.50 25.60 25.70 25.81 25.91 10 6.1 5.2 6.3 26.01 26.11 26.21 26.32 27.04 i 27.14 27.25 27.35 28.09 1 28.20 28.30 28.41 26.42 26.52 26.63 27.46 27.56 27.67 28.52 28.62 28.73 26.73 26.83 26.94 27.77 27.88 27.98 28.84 28.94 29.05 10 11 11 6.4 6.5 5.6 29.16 29.27 29.38 29.48 30.25 ! 30.36 30.47 30.58 31.36 i 31.47 31.58 31.70 29.59 29.70 29.81 30.69 30.80 30.91 31.81 31.92 32.04 29.92 30.03 30.14 31.02 31.14 31.25 32.15 32.26 32.38 11 11 11 5.7 5.8 5.9 32.49 1 32.60 32.72 32.83 33.64 1 33.76 33.87 33.99 .34.81 1 34.93 35.05 35.16 32.95 33.06 33.18 34.11 34.22 34.34 35.28 35.40 35.52 33.29 33.41 33.52 34.46 34.57 34.69 35.64 35.76 35.88 12 12 12 n^ XIV. SQUAEES. 37 n 12 3 4 5 6 7 8 9 d 6.0 36.00 36.12 36.24 36.36 36.48 36.60 36.72 36.84 36.97 37.09 12 6.1 6.2 6.3 37.21 38.44 39.69 37.33 37.45 37.58 38.56 38.69 38.81 39.82 39.94 40.07 37.70 37.82 37.95 38.94 39.06 39.19 40.20 40.32 40.45 38.07 38.19 38.32 39.31 39.44 39.56 40.58 40.70 40.83 12 13 13 6.4 6.5 6.6 40.96 42.25 43.56 41.09 41.22 41.34 42.38 42.51 42.64 43.69 43.82 43.96 41.47 41.60 41.73 42.77 42.90 43.03 44.09 44.22 44.36 41.86 41.99 42.12 43.16 43.30 43.43 44.49 44.62 44.76 13 13 13 6.7 6.8 6.9 44.89 46.24 47.61 45.02 45.16 45.29 46.38 46.51 46.65 47.75 47.89 48.02 45.43 45.56 45.70 46.79 46.92 47.06 48.16 48.30 48.44 45.83 45.97 46.10 47.20 47.33 47.47 48.58 48.72 48.86 14 14 14 7.0 49.00 49.14 49.28 49.42 49.56 49.70 49.84 49.98 50.13 50.27 14 7.1 7.2 7.3 50.41 51.84 53.29 50.55 50.69 50.84 51.98 52.13 52.27 53.44 53.58 ,53.73 50.98 51.12 51.27 52.42 52..S6 52.71 53.88 54.02 54.17 51.41 51.55 51.70 52.85 53.00 53.14 54.32 54.46 54.61 14 15 15 7.4 7.5 7.6 54.76 56.25 57.76 54.91 55.06 55.20 56.40 56.55 56.70 57.91 58.06 58.22 55.35 55.50 55.65 56.85 57.00 57.15 58.37 58.52 58.68 55.80 55.95 56.10 57.30 57.46 57.61 58.83 58.98 59.14 15 15 15 7.7 7.8 7.9 59.29 60.84 62.41 59.44 59.60 59.75 61.00 61.15 61.31 62.57 62.73 62.88 59.91 60.06 60.22 61.47 61.62 61.78 63.04 63.20 63.36 60.37 60.53 60.68 61.94 62.09 62.25 63.52 63.68 63.84 16 16 16 8.0 64.00 64.16 64.32 64.48 64.64 64.80 64.96 65.12 65.29 65.45 16 8.1 8.2 8.3 65.61 67.24 68.89 65.77 65.93 66.10 67.40 67.57 67.73 69.06 69.22 69.39 66.26 66.42 66.59 67.90 68.06 68.23 69.56 69.72 69.89 66.75 66.91 67.08 68.39 68.56 68.72 70.06 70.22 70.39 16 17 17 8.4 8.5 8.6 70.56 72.25 73.96 70.73 70.90 71.06 72.42 72.59 72.76 74.13 74.30 74.48 71.23 71.40 71.57 72.93 73.10 73.27 74.65 74.82 75.00 71.74 71.91 72.08 73.44 73.62 73.79 75.17 75.34 75.52 17 17 17 8.7 8.8 8.9 75.69 77.44 79.21 75.86 76.04 76.21 77.62 77.79 77.97 79.39 79.57 79.74 76.39 76.56 76.74 78.15 78.32 78.50 79.92 80.10 80.28 76.91 77.09 77.26 78.68 78.85 79.03 80.46 80.64 80.82 18 18 18 9.0 81.00 81.18 81.36 81.54 81.72 81.90 82.08 82.26 82.45 82.63 18 9.1 9.2 9.3 82.81 84.64 86.49 82.99 83.17 83.36 84.82 85.01 85.19 86.68 86.86 87.05 83.54 83.72 83.91 85.38 85.56 85.75 87.24 87.42 87.61 84.09 84.27 84.46 85.93 86.12 86.30 87.80 87.98 88.17 18 19 19 9.4 9.5 9.6 88.36 90.25 92.16 88.55 88.74 88.92 90.44 90.63 90.82 92.35 92.54 92.74 89.11 89.30 89.49 91.01 91.20 91.39 92.93 93.12 93.32 89.68 89.87 90.06 91.58 91.78 91.97 93.51 93.70 93.90 19 19 19 9.7 9.8 9.9 94.09 96.04 98.01 94.28 94.48 94.67 96.24 96.43 96.63 98.21 98.41 98.60 94.87 95.06 95.26 96.83 97.02 97.22 98.80 99.00 99.20 95.45 95.65 95.84 97.42 97.61 97.81 99.40 99.60 99.80 20 20 20 n .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 d 1 2 3 1.000 0.250 0.111 *826 *694 *592 227 207 189 104 *977 *91S *510 *444 *391 174 160 148 *S65 *816 *772 *346 *309 *277 137 128 119 *730 *693 *657 27 8 32 4 5 6 0.0 625 0.0 400 0.0 278 595 567 541 384 370 356 269 260 252 517 494 473 343 331 319 244 237 230 453 434 416 308 297 287 223 216 210 16 9 6 7 8 9 0.0 204 0.0 156 0.0 123 198 193 188 152 149 145 121 118 116 183 178 173 142 138 135 113 111 109 169 164 160 132 129 126 106 104 102 4 3 2 38 XV. CUBES. 11-^ n 1 2 8 4 5 6 7 8 9 d 1.0 1.000 1.030 1.061 1.093 1.125 1.158 1.191 1.225 1.260 1.295 36 1.1 1 '' l!3 1.331 1.728 2.197 1.368 1.405 1.443 1.772 1.816 1.861 2.248 2.300 2.353 1.482 1.521 1.561 1.907 1.953 2.000 2.406 2.460 2.515 1.602 1.643 1.685 2.048 2.097 2.147 2.571 2.628 2.686 43 50 58 1.4 1.5 1.6 2.744 2.803 2.863 2.924 3.375 3.443 3.512 3.582 4.096 4.173 4.252 4.331 2.986 3.049 3.112 3.652 3.724 3.796 4.411 4.492 4.574 3.177 3.242 3.308 3.870 3.944 4.020 4.657 4.742 4.827 67 76 86 1.7 l.H 1.9 2^ 4.913 5.000 5.0S8 5.178 5.832: 5.930 6.029 6.128 6.859^ 6.968 7.078 7.189 5.268 5.359 5.452 6.230 6.332 6.435 7.301 7.415 7.530 5.545 5,640 5.735 6.539 6.645 6.751 7.645 7.762 7.881 97 108 119 8.000 8.121 8.242 8.365 8.490 8.615 S.742 8.870 8.999 9.129 132 2.1 2.2 2!3 9.261:9.394 9.528 9.664 10.648 10.79 10.94 11.09 12.167 12.33 12.49 12.65 9.800 9-.938 10.08 11.24 11.39 11.54 12.81 12.98 13.14 10.22 10.36 10.50 11.70 11.85 12.01 13.31 13.48 13.65 15 16 17 2.4' 13.824] 14.00 14.17 14.35 2.5 15.625 15.81 16.00 16.19 2.6 17.576 17.78 17.98 18.19 14.53 14.71 14.89 16..39 16,58 16.78 18.40 18.61 18.82 15.07 15.25 15.44 16.97 17.17 17.37 19.03 19.25 19.47 19 21 21 2.7 19.683 19.90 20.12 20.35 2.S! 21.952| 22.19 22.43 22.67 2.9; 24.3S9| 24.64 24.90 25.15 20.57 20.80 21.02 22.91 23.15 23.39 25.41 25.67 25.93 21.25 21.48 21.72 23.64 23.89 24.14 26.20 26.46 26.73 23 25 27 3.0 27.000 27.27 27.54 27.82 28.09 28.37 28.65 28.93 29.22 29.50 29 3.1! 29.791 3.2 32.768 3.3j 35.937 30.08 30.37 30.66 33.08 33.39 33.70 36.26 36.59 36.93 30.96 31.26 31.55 34.01 34.33 34.65 37.26 37.60 37.93 31.86 32.16 32.46 34.97 35.29 35.61 38.27 38.61 38.96 31 33 34 3.4! 39.304 3.5, 42.875 3.6 46.656 39.65 40.00 40.35 43.24 43.61 43.99 47.05 47.44 47.83 40.71 41.06 41.42 44.36 44.74 45.12 48.23 48.63 49.03 41.78 42.14 42.51 45.50 45.88 46.27 49.43 49.84 50.24 37 39 41 3.7 50.653' 51.06 51.48 51.90 3.8 54.872 55.31 55.74 56.18 3.9 59.319 59.78 60.24 60.70 52.31 52.73 53.16 56.62 57.07 57.51 61.16 61.63 62.10 53.58 54.01 54.44 57.96 58.41 58.86 62.57 63.04 63.52 43 46 48 4.0 64.000 64.48 64.96 65.45 65.94 66.43 66.92 67.42 67.92 68.42 50 4.1 68.921 69.43 69.93 70.44 4.2, 74.088 74,62 75.15 75.69 4.3 79.507 80.06 80.62 81.18 70.96 71.47 71.99 76.23 76.77 77.31 81.75 82.31 82.88 72.51 73.03 73.56 77.85 78.40 78.95 83.45 84.03 84.60 53 56 58 4.4 85.1841 85.77 86.35 86.94 4..> 91.125' 91.73 92.35 92.96 4.6 97.336, 97.97 98.61 99.25 87.53 88.12 88.72 93.58 94.20 94.82 99.90 100.5 101.2 89.31 89.92 90.52 95.44 96.07 96.70 101.8 102.5 103.2 61 64 6 4.7 4.S 4.9 103.823, 104.5 105.2 105.8 110.592! 111.3 112.0 112.7 117.6^9 118.4 119.1 119.8 106.5 107.2 107.9 113.4 114,1 114.8 120.6 121.3 122.0 108.5 109.2 109.9 115.5 116.2 116.9 122.8 123.5 124.3 7 7 7 5.0 125.000; 125.8 126.5 127.3 128.0 128.8 129.6 130.3 131.1 131.9 8 5.1 132.651S 133.4 134.2 135.0 5.2 140.608; 141.4 142.2 143.1 5.3 148.877 149.7 150.6 151.4 135.8 136.6 137.4 143.9 144.7 145.5 152.3 153.1 154.0 138.2 139.0 139.8 146.4 147.2 148.0 154.9 155.7 156.6 8 9 9 5.4 157.464 158.3 159.2 160.1 5.5 I66.375: 167.3 168.2 169.1 5.6 175.616 176.6 177.5 178.5 161.0 161.9 162.8 170.0 171.0 171.9 179.4 180.4 181.3 163.7 164.6 165.5 172.8 173.7 174.7 182.3 183.3 184.2 9 9 10 5.7 185.193 1S6.2 187.1 188.1 5.S 195.n2: 196.1 197.1 198.2 5.9 205.379 206.4 207.5 208.5 189.1 190.1 191.1 199.2 200.2 201.2 209.6 210.6 211.7 192.1 193.1 194.1 202.3 203.3 204.3 212.8 213.8 214.9 10 11 11 XV. CUBES. 39 n 1 2 3 4 5 6 7 8 9 d 6.( ) 216.000 217.1 218.2 219.3 220.3 221.4 222.5 223.6 224.8 225.9 11 6.] 6.1 6.9 L 226.981 5 238.328 250.047 228.1 229.2 230.3 239.5 240.6 241.8 251.2 252.4 253.6 231.5 232.6 233.7 243.0 244.1 245.3 254.8 256.0 257.3 234.9 236.0 237.2 246.5 247.7 248.9 258.5 259.7 260.9 11 11 12 6.4 6.a 6.1 [ 262.144 274.625 » 287.496 263.4 264.6 265.8 275.9 277.2 278.4 285.5 290.1 291.4 267.1 268.3 269.6 279.7 281.0 282.3 292.8 294.1 295.4 270.8 272.1 273.4 283.6 284.9 286.2 296.7 298.1 299.4 12 13 14 6.: 6.S 300.763 314.432 328.509 302.1 303.5 304.8 315.8 317.2 318.6 329.9 331.4 332.8 306.2 307.5 30S.9 320.0 321.4 322.8 334.3 335.7 337.2 310.3 311.7 313.0 324.2 325.7 327.1 338.6 340.1 341.5 14 14 15 7.C 343.000 344.5 345.9 347.4 348.9 350.4 3il.9 353.4 354.9 356.4 15 7.1 7.2 7.3 357.911 373.248 389.017 359.4 360.9 362.5 374.8 376.4 377.9 390.6 392.2 393.8 364.0 365.5 367.1 379.5 381.1 382.7 395.4 397.1 398.7 368.6 370.1 371.7 384.2 385.8 387.4 400.3 401.9 403.6 IS 16 16 7.4 7.5 7.G 405.224 421.875 438.976 406.9 408.5 410.2 423.6 425.3 427.0 440.7 442.5 444.2 411.8 413.5 415.2 428.7 430.4 432.1 445.9 447.7 449.5 416,8 418.5 420.2 433.8 435.5 437.2 451.2 453.0 454.8 17 18 17 7.7 7.8 7.9 456.533 ■474.552 493.039 458.3 460.1 461.9 476.4 478.2 480.0 494.9 496.8 498.7 463.7 465.5 467.3 481.9 483.7 485.6 500.6 502.5 504.4 469.1 470.9 472.7 487.4 489.3 491.2 506.3 508.2 510.1 19 IS 19 8.C 512.000 513.9 515.8 517.8 519.7 521.7 523.6 525.6 527.5 529.5 19 8.1 8.2 8.3 531.441 551.368 571.787 533.4 535.4 537.4 553.4 555.4 557.4 573.9 575.9 578.0 539.4 541.3 543.3 559.5 561.5 563.6 580.1 582.2 584.3 545.3 547.3 .549.4 565.6 567.7 569.7 586.4 588.5 590.6 20 21 21 8.4 8.5 8.fc 592.704 614.125 636.056 594.8 596.9 599.1 616.3 618.5 620.7 638.3 6H0.5 642.7 601.2 603.4 605.5 622.8 625.0 627.2 645.0 647.2 649.5 607.6 609.8 612.0 629.4 631.6 633.8 651.7 654.0 656.2 21 23 23 8.7 8.8 8.9 658.503 681.472 704.969 660.8 663.1 665.3 683.8 686.1 6885 707.3 709.7 712.1 667.6 669.9 672.2 690.8 693.2 695.5 714.5 716.9 719.3 674.5 676.8 679.2 697.9 700.2 702.6 721.7 724.2 726.6 23 24 24 9.0 729.000 731.4 733.9 736.3 738.8 741.2 743.7 746.1 748.6 751.1 25 9.1 9.2 9.3 753.571 778.688 804.357 756.1 758.6 761.0 781.2 783.8 786.3 807.0 809.6 812.2 763.6 766.1 768.6 771.1 773.6 776.2 788.9 791.5 794.0 796.6 799.2 801.8 814.8 817.4 820.0 ! 822.7 825.3 827.9 25 26 27 9.4 830.584 9.5 857.375 9.6 884.736 833.2 835.9 838.6 860.1 862.8 865.5 887.5 890.3 893.1 841.2 843.9 846.6 849.3 852.0 854.7 868.3 871.0 873.7 ; 876.5 879.2 882.0 895.8 898 6 901.4 : 904.2 907.0 909.9 27 27 28 9.7 912.673 9.8 941.192 9.9,970.299 915.5 918.3 921.2 944.1 947.0 949.9 973.2 976.2 979.1 924.0 926.9 929.7 952.8 955.7 958.6 982.1 985.1 988.0 932.6 935.4 938.3 961.5 964.4 967.4 991.0 994.0 997.0 29 29 30 n T 9 3 .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 d 1000 0.125 0.0 370 *757 *579 *455 108 *939 *822 336 305 278 *364 *296 *244 *723 *640 *569 254 233 214 *204 *171 *146 *.508 *456 *410 197 182 169 21 40 13 4 5 6 0.0 156 0.00 800 0.00 463 145 135 126 754 711 672 441 420 400 117 110 103 635 601 569 381 364 348 *963 *904 *850 540 513 487 332 318 304 50 24- 12 7 8 9 0.00 292 0.00 195 0.00 137 279 268 257 188 181 175 133 129 124 247 237 228 169 163 157 120 117 113 219 211 203 152 147 142 110 106 103 8 5 3 40 XVI. SQUAEE KOOTS V/i n 12 3 4 5 6 7 8 9 d 1.0 1.0000 0050 0100 0149 0198 0247 0296 0344 0392 0440 48 1.1 1.2 1.3 04SS 0954 1402 0536 0583 0630 1000 1045 1091 1446 1489 1533 0677 0724 0770 1136 1180 1225 1576 1619 1662 0817 0863 0909 1269 1314 1358 1705 1747 1790 45 44 42 1.4 1.5 1.0 1S32 2247 2649 1874 1916 1958 2288 2329 2369 2689 2728 2767 2000 2042 2083 2410 2450 2490 2806 2845 2884 2124 2166 2207 2530 2570 2610 2923 2961 3000 40 39 38 1.7 1.8 1.9 303S 3416 37S4 3077 3 Hi 3153 3454 3491 3528 3820 3856 3892 3191 3229 3266 3565 3601 3638 3928 3964 4000 3304 3342 3379 3675 3711 3748 4036 4071 4107 37 36 35 2.0 1.4142 4177 4213 4248 4283 4318 4353 4387 4422 4457 34 2.1 2.2 2.3 4491 4832 5166 4526 4560 4595 4866 4900 4933 5199 5232 5264 4629 4663 4697 4967 5000 5033 5297 5330 5362 4731 4765 4799 5067 5100 5133 5395 5427 5460 33 33 32 2.4 2.5 2.6 5492 5811 6125 5524 5556 5588 5843 5875 5906 6155 6186 6217 5620 5652 5684 5937 5969 6000 6248 6279 6310 5716 5748 5780 6031 6062 6093 6340 6371 6401 31 32 31 2.7 2.8 2.9 6432 6733 7029 6462 6492 6523 6763 6793 6823 7059 7088 7117 6553 6583 6613 6852 6882 6912 7146 7176 7205 6643 6673 6703 6941 6971 7000 7234 7263 7292 30 29 29 3.0 1.7321 7349 7378 7407 7436 7464 7493 7521 7550 7578 29 3.1 3.2 3.3 7607 7889 8166 7635 7664 7692 7916 7944 7972 8193 8221 8248 7720 7748 7776 8000 8028 8055 8276 8303 8330 7804 7833 7861 8083 8111 8138 8358 8385 8412 28 28 27 3.4 3.5 3.6 8439 8708 8974 S466 8493 8520 8735 8762 8788 9000 9026 9053 8547 8574 8601 8815 8841 8868 9079 9105 9131 8628 8655 8682 8894 8921 8947 9157 9183 9209 26 27 26 3.7 3.8 3.9 9235 9494 9748 9261 9287 9313 9519 9545 9570 9774 9799 9824 9339 9365 9391 9596 9621 9647 9849 9875 9900 9416 9442 9468 9672 9698 9723 9925 9950 9975 26 25 25 4.0 2.0000 0025 0050 0075 0100 0125 0149 0174 0199 0224 24 4.1 4.2 4.3 0248 0494 0736 0273 0298 0322 0518 0543 0567 0761 0785 0809 0347 0372 0396 0591 0616 0640 0833 0857 0881 0421 0445 0469 0664 0688 0712 0905 0928 0952 25 24 24 4.4 4.5 4.6 0976 1213 1448 1000 1024 1048 1237 1260 1284 1471 1494 1517 1071 1095 1119 1307 1331 1354 1541 1564 1587 1142 1166 1190 1378 1401 1424 1610 1633 1656 23 24 23 4.7 4.8 4.9 1679 1909 2136 1703 1726 1749 1932 1954 1977 2159 2181 2204 1772 1794 1817 2000 2023 2045 2226 2249 2271 1840 1863 1886 2068 2091 2113 2293 2316 2338 23 23 23 5.0 2.2361 2383 2405 2428 2450 2472 2494 2517 2539 2561 22 5.1 5.2 5.3 2583 2804 3022 2605 2627 2650 2S25 2847 2869 3043 3065 3087 2672 2694 2716 2891 2913 2935 3108 3130 3152 2738 2760 2782 2956 2978 3000 3173 3195 3216 22 22 22 5.4 5.5 5.6 3238 3452 3664 3259 3281 3302 3473 3495 3516 3685 3707 3728 3324 3345 3367 3537 3558 3580 3749 3770 3791 3388 3409 3431 3601 3622 3643 3812 3833 3854 21 21 21 5.7 5.8 5.9 3875 4083 4290 3896 3917 3937 4104 4125 4145 4310 4331 4352 3958 3979 4000 4166 4187 4207 4372 4393 4413 4021 4042 4062 4228 4249 4269 4434 4454 4474 21 21 21 V** riEST PAET. 41 n 12 3 4 5 6 7 8 9 d 6.0 2.4495 4515 4536 4556 4576 4597 4617 4637 4658 4678 20 6.1 6.2 6.3 4698 4900 5100 4718 4739 4759 4920 4940 4960 5120 5140 5159 4779 4799 4819 4980 5000 5020 5179 5199 5219 4839 4860 4SS0 5040 5060 5080 5239 5259 5278 20 20 20 6.4 6.5 6.6 5298 5495 5690 5318 5338 5357 5515 5534 5554 5710 5729 5749 5377 5397 5417 5573 5593 5612 5768 5788 5807 5436 5456 5475 5632 5652 5671 5826 5846 5865 20 19 19 6.7 6.8 6.9 5884 6077 6268 5904 5923 5942 6096 6115 6134 6287 6306 6325 5962 5981 6000 6153 6173 6192 6344 6363 6382 6019 6038 6058 6211 6230 6249 6401 6420 6439 19 19 19 7.0 2.6458 6476 6495 6514 6533 6552 6571 6589 6608 6627 19 7.1 7.2 7.3 6646 6833 7019 6665 6683 6702 6851 6870 6SS9 7037 7055 7074 6721 6739 6758 6907 6926 6944 7092 7111 7129 6777 6796 6814 6963 6981 7000 7148 7166 7185 19 19 18 7.4 7.5 7.6 7203 7386 7568 7221 7240 7258 7404 7423 7441 7586 7604 7622 7276 7295 7313 7459 7477 7495 7641 7659 7677 7331 7350 7368 7514 7532 7550 7695 7713 7731 18 18 18 7.7 7.8 7.9 7749 7928 8107 7767 7785 7803 7946 7964 7982 8125 8142 8160 7821 7839 7857 8000 8018 8036 8178 8196 8213 7875 7893 7911 8054 8071 8089 8231 8249 8267 17 18 17 8.0 2.8284 8302 8320 8337 8355 8373 8390 8408 8425 8443 17 8.1 8.2 8.3 8460 8636 8810 8478 8496 8513 8653 8671 8688 8827 8844 8862 8531 8548 8566 8705 8723 8740 8879 8896 8914 8583 8601 8618 8758 8775 8792 8931 8948 8965 18 18 IS 8.4 8.5 8.6 8983 9155 9326 9000 9017 9034 9172 9189 9206 9343 9360 9377 9052 9069 9086 9223 9240 9257 9394 9411 9428 9103 9120 9138 9275 9292 9309 9445 9462 9479 17 17 17 8.7 8.8 8.9 9496 9665 9833 9513 9530 9547 9682 9698 9715 9850 9866 9883 9563 9580 9597 9732 9749 9766 9900 9917 9933 9614 9631 9648 9783 9799 9816 9950 9967 9983 17 17 17 "l6" 9.0 3.0000 0017 0033 0050 0067 0083 0100 0116 0133 0150 9.1 9.2 9.3 0166 0332 0496 0183 0199 0216 0348 0364 0381 0512 0529 0545 0232 0249 0265 0282 0299 0315 0397 0414 0430 ' 0447 0463 0480 0561 0578 0594 0610 0627 0643 17 16 16 9.4 9.5 9.6 0659 0822 0984 0676 0692 0708 0838 0854 0871 1000 1016 1032 0725 0741 0757 0773 0790 0806 0887 0903 0919 0935 0952 0968 1048 1064 1081 1097 1113 1129 16 16 16 9.7 9.8 9.9 1145 1305 1464 1161 1177 1193 1321 1337 1353 1480 1496 1512 1209 1225 1241 1369 13S5 1401 1528 1544 1559 1257 1273 1289 1417 1432 1448 1575 1591 1607 16 16 16 n .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 d 1 2 3 1.000 0.707 0.577 *953 *913 *877 690 674 659 568 559 550 *845 *816 *791 645 632 620 542 535 527 *767 *745 *725 609 598 587 520 513 506 18 10 6 4 5 6 0.500 0.447 0.408 494 488 482 443 439 434 405 402 398 477 471 466 430 426 423 395 392 389 461 456 452 419 415 412 386 383 381 5 4 3 7 8 9 0.378 0.354 0.333 375 373 370 351 349 347 331 330 328 368 365 363 345 343 341 326 324 323 360 358 356 339 337 335 321 319 318 2 2 2 42 XVI. SQUARE EOOTS: Vn n 10 .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 d 3.1623 1780 1937 2094 2249 2404 2558 2711 2863 3015 151 11 J.J 13 3166 4641 6056 3317 3466 3615 4785 4928 5071 6194 6332 6469 3764 3912 4059 5214 5355 5496 6606 6742 6S7S 4205 4351 4496 5637 5777 5917 7014 7148 7283 145 139 134 14 15 1« 7417 8730 4.0000 7550 7683 7815 88.59 8987 9115 0125 0249 0373 7947 8079 8210 9243 9370 9497 0497 0620 0743 8341 8471 8601 9623 9749 9875 0866 0988 1110 129 125 121 17 18 19 1231 2426 3589 1352 1473 1593 2544 2661 2778 3704 3818 3932 1713 1833 1952 2895 3012 3128 4045 4159 4272 2071 2190 2308 3243 3359 3474 4385 4497 4609 118 115 112 20 4.4721 4833 4944 5056 5166 5277 5387 5497 5607 5717 109 21 23 5826 6904 7958 5935 6043 6152 70li 7117 7223 8062 8166 8270 6260 6368 6476 7329 7434 7539 8374 8477 8580 6583 6690 6797 7645 7749 7854 86S3 8785 S8SS 107 104 102 24 25 26 8990 5.0000 0990 9092 9193 9295 0100 0200 0299 1088 1186 1284 9396 9497 9598 0398 0498 0596 1381 1478 1575 9699 9800 9900 0695 0794 0892 1672 1769 1865 100 98 97 27 28 29 1962 2915 3852 2058 2154 2249 3009 3104 3198 3944 4037 4129 2345 2440 2536 3292 3385 3479 4222 4314 4406 2631 2726 2820 3572 3666 3759 4498 4589 4681 95 93 91 30 5.4772 4863 4955 5045 5136 5227 5317 5408 5498 55SS 90 31 32 33 5678 6569 7446 5767 5857 5946 6657 6745 6833 7533 7619 7706 6036 6125 6214 6921 7009 7096 7793 7879 7966 6303 6391 64 SO 71S4 7271 7359 8052 8138 8224 89 87 86 34 35 36 8310 9161 6.0000 8395 8481 8566 9245 9330 9414 0083 0166 0249 8652 8737 8822 9498 9582 9666 0332 0415 0498 8907 8992 9076 9749 9833 9917 0581 0663 0745 85 83 S3 37 38 39 0828 1644 2450 0910 0992 1074 1725 1806 1887 2530 2610 2690 1156 1237 1319 1968 2048 2129 2769 2849 2929 1400 1482 1563 2209 2290 2370 3008 3087 3166 81 80 80 40 6.3246 3325 3403 3482 3561 3640 3718 3797 3875 3953 78 41 42 43 4031 4807 5574 4109 4187 4265 4885 4962 5038 5651 5727 5803 4343 4420 4498 5115 5192 5269 5879 5955 6030 4576 4653 4730 5345 5422 5498 6106 6182 6257 77 76 75 44 45 46 6332 7082 7823 6408 6483 6558 7157 7231 7305 7897 7971 8044 6633 6708 6783 73SO 7454 7528 8118 8191 8264 6858 6933 7007 7602 7676 7750 8337 8411 8484 75 73 73 47 48 49 8557 9282 7.0000 8629 8702 8775 9354 9426 9498 0071 0143 0214 8848 8920 8993 9570 9642 9714 0285 0356 0427 9065 9138 9210 9785 9857 9929 0498 0569 0640 72 11 70 50 7.0711 0781 0852 0922 0993 1063 1134 1204 1274 1344 51 52 53 1414 2111 2801 1484 1554 1624 2180 2250 2319 2870 2938 3007 1694 1764 1833 2388 2457 2526 3075 3144 3212 1903 1972 2042 2595 2664 2732 3280 3348 3417 69 69 68 64 55 56 3485 4162 4833 3553 3621 3689 4229 4297 4364 4900 4967 5033 3756 3824 3892 4431 4498 4565 5100 5166 5233 3959 4027 4095 4632 4699 4766 5299 5366 5432 67 67 66 57 58 59 5498 6158 6811 5565 5631 5697 6223 6289 6354 6877 6942 7006 5763 5829 5895 6420 6485 6551 7071 7136 7201 5961 6026 6092 6616 6681 6746 7266 7330 7395 66 65 65 Vn SECOND PAET. 43 n .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 (1 60 7.7460 7524 7589 7653 7717 7782 7846 7910 7974 8038 64 61 62 63 8102 8740 9373 8166 8230 8294 SS04 8867 8930 9436 9498 9561 8358 8422 8486 8994 9057 9120 9624 9687 9750 8549 8613 8677 91S3 9246 9310 9812 9875 9937 63 63 63 64 65 66 8.0000 0623 1240 0062 0125 0187 0685 0747 0S08 1302 1363 1425 0250 0312 0374 0870 0932 0994 1486 1548 1609 0436 0498 0561 1056 1117 1179 1670 1731 1792 62 61 62 67 68 69 1854 2462 3066 1915 1976 2037 2523 2583 2644 3126 3187 3247 2098 2158 2219 2704 2765 2825 3307 3367 3427 2280 2341 2401 2885 2946 3006 3487 3546 3606 61 60 60 70 8.3666 3726 3785 3845 3905 3964 4024 4083 4143 4202 59 71 72 73 4261 4853 5440 4321 4380 4439 4912 4971 5029 5499 5557 5615 4499 4558 4617 5088 5147 5206 5674 5732 5790 4676 4735 4794 5264 5323 5381 5849 5907 5965 59 59 58 74 75 76 6023 6603 7178 6081 6139 6197 6660 6718 6776 7235 7293 7350 6255 6313 6371 6833 6891 6948 7407 7464 7521 6429 6487 6545 7006 7063 7121 7579 7636 7693 58 57 57 77 78 79 7750 8318 8882 7807 7864 7920 8374 8431 8487 8938 8994 9051 7977 8034 8091 8544 8600 8657 9107 9163 9219 8148 8204 8261 8713 8769 8826 9275 9331 9387 57 56 56 80 8.9443 9499 9554 9610 9666 9722 9778 9833 9889 9944 56 81 82 83 9.0000 0554 1104 0056 0111 0167 0609 0664 0719 1159 1214 1269 0222 0277 0333 0774 0830 0885 1324 1378 1433 0388 0443 0499 0940 0995 1049 148S 1542 1597 55 55 55 84 85 86 1652 2195 2736 1706 1761 1815 2250 2304 2358 2790 2844 2S98 1869 1924 1978 2412 2466 2520 2952 3005 3059 2033 2087 2141 2574 2628 26S2 3113 3167 3220 54 54 54 87 88 89 3274 3808 4340 3327 33S1 3434 3862 3915 3968 4393 4446 4499 34SS 3541 3595 4021 4074 4128 4552 4604 4657 3648 3702 3755 4181 4234 4287 4710 4763 4816 53 53 52 90 9.4868 4921 4974 5026 5079 5131 5184 5237 5289 5341 53 91 92 93 5394 5917 6437 5446 5499 5551 5969 6021 6073 6488 6540 6592 5603 5656 5708 6125 6177 6229 6644 6695 6747 5760 5812 5864 6281 6333 6385 6799 6850 6902 53 52 52 94 95 96 6954 7468 7980 7005 7057 7108 7519 7570 7622 8031 S0S2 8133 7160 7211 7263 7673 7724 7775 8184 8234 S2S5 7314 7365 7417 7826 7877 7929 S336 8387 S438 51 51 51 97 98 99 8489 8995 9499 8539 8590 8641 9045 9096 9146 9549 9599 9649 S691 8742 8793 9197 9247 9298 9700 9750 9800 8843 8894 8944 9348 9398 9448 9850 9900 9950 51 51 50 n 12 3 4 5 6 7 8 9 d 1 2 3 0.316 0.224 0.183 302 289 277 219 213 209 ISO 177 174 267 258 250 204 200 196 171 169 167 243 236 229 192 189 186 164 162 160 5 3 2 4 5 6 0.158 0.141 0.129 156 154 152 140 139 137 128 127 126 151 149 147 136 135 134 125 124 123 146 144 143 132 131 130 122 121 120 2 1 7 8 9 0.120 0.112 0.105 119 lis 117 111 110 110 105 104 104 116 115 115 109 108 108 103 103 102 114 113 113 107 107 106 102 101 101 1 1 1 XVII. CUBE ROOTS: 600 606 612 618 624 630 636 642 648 654 7 700 707 714 721 728 735 742 749 756 763 s 8U0 SOS 816 824 832 840 848 856 864 872 9 900 909 918 927 936 945 954 963 972 981 110 111 112 113 114 115 110 117 118 119 1 110 111 112 113 114 115 116 117 118 119 2 220 222 224 226 228 230 232 234 236 238 3 330 333 336 339 342 345 348 351 354 357 4 440 444 448 452 456 460 464 468 472 476 5 550 555 560 565 570 575 580 585 590 595 6 660 666 672 678 684 690 696 702 708 714 7 770 777 784 791 798 805 812 819 826 833 8 8S0 SS8 896 904 912 920 928 936 944 952 9 990 999 1008 1017 1026 1035 1044 1053 1062 1071 120 121 122 123 124 125 12G 127 128 129 1 120 121 122 123 124 "T25 126 ~127 128 129 2 240 242 244 246 248 250 252 254 256 258 3 360 363 366 369 372 375 378 381 3S4 387 4 4S0 484 488 492 496 500 504 508 512 516 5 600 605 610 615 620 625 630 635 640 645 6 720 726 732 738 744 750 756 762 768 774 7 840 847 854 861 868 875 882 889 896 903 8 960 968 976 984 992 1000 1008 1016 1024 1032 9 lOSO 1089 1098 1107 1116 1125 1134 1143 1152 1161 130 131 132 133 134 135 136 137 138 139 1 130 131 132 133 134 135 136 137 138 139 2 260 262 264 266 268 270 272 274 276 278 3 390 393 396 399 402 405 408 411 414 417 4 520 524 528 532 536 540 544 548 552 556 5 650 655 660 665 670 675 680 685 690 695 6 780 786 792 798 804 810 816 822 828 834 7 910 917 924 931 938 945 952 959 966 973 8 1040 1048 1056 1064 1072 lOSO 10S8 1096 1104 1112 9 1170 1179 IISS 1197 1206 1215 1224 1233 1242 1251 140 141 142 143 144 145 146 147 148 149 1 140 141 142 143 144 ~145~ 146 147 148 149 2 280 282 284 2S6 288 290 292 294 296 298 3 420 423 426 429 432 435 438 441 444 447 4 560 564 568 572 576 580 584 588 592 596 5 700 705 710 715 720 725 730 735 740 745 6 840 846 852 858 864 870 876 882 888 894 7 980 987 994 1001 lOOS 1015 1022 1029 1036 1043 8 1120 1128 1136 1144 1152 1160 1168 1176 1184 1192 9 1260 1269 1278 1287 1296 1305 1314 1323 1332 1341 XVIII. MULTIPLES. 51 150 151 152 153 154 155 156 157 158 159 1 150 151 152 153 154 155 156 157 158 159 2 300 302 304 306 308 310 312 314 316 318 3 450 453 456 459 462 465 468 471 474 477 4: 600 604 60S 612 616 620 624 628 632 636 5 750 755 760 765 770 775 780 785 790 795 6 900 906 912 918 924 930 936 942 948 954 7 1050 1057 1064 1071 1078 1085 1092 1099 1106 1113 8 1200 1208 1216 1224 1232 1240 1248 1256 1264 1272 9 1350 1359 1368 1377 1386 1395 1404 1413 1422 1431 160 161 162 163 164 165 166 167 168 169 1 160 161 162 163 164 165 166 167 168 169 2 320 322 324 326 328 330 332 334 336 338 3 480 483 486 489 492 495 498 501 504 507 4 640 644 648 652 656 660 664 668 672 676 6 800 805 810 815 820 825 830 835 840 845 6 960 966 972 978 984 990 996 1002 1008 1014 7 1120 1127 1134 1141 1148 1155 1162 1169 1176 1183 8 1280 1288 1296 1304 1312 1320 1328 1336 1344 1352 9 1440 1449 1458 1467 1476 1485 1494 1503 1512 1521 170 171 172 173 174 175 176 177 178 179 1 170 171 172 173 174 175 176 177 178 179 2 340 342 344 346 348 350 352 354 356 358 3 510 513 516 519 522 525 528 531 534 537 4 680 684 688 692 696 700 704 708 712 716 5 850 855 860 865 870 875 880 SS5 890 895 6 1020 1026 1032 1038 1044 1050 1056 1062 1068 1074 7 1190 1197 1204 1211 1218 1225 1232 1239 1246 1253 8 1360 1368 1376 1384 1392 1400 1408 1416 1424 1432 9 1530 1539 1548 1557 1566 1575 1584 1593 1602 1611 180 181 182 183 184 185 186 187 188 189 1 180 181 182 183 184 185 186 187 188 189 2 360 362 364 366 368 370 372 374 376 378 3 540 543 546 549 552 555 558 561 564 567 4 720 724 728 732 736 740 744 748 752 756 5 900 905 910 915 920 925 930 935 940 945 6 1080 1086 1092 1098 1104 1110 1116 1122 1128 1134 7 1260 1267 1274 1281 1288 1295 1302 1309 1316 1323 8 1440 1448 1456 1464 1472 14S0 1488 1496 1504 1512 9 1620 1629 1638 1647 1656 1665 1674 1683 1692 1701 190 191 192 193 194 195 196 197 198 199 1 190 191 192 193 194 195 196 197 198 199 2 380 382 384 386 388 390 392 394 396 398 3 570 573 576 579 582 585 588 591 594 597 4 760 764 768 772 776 780 784 788 792 796 5 950 955 960 965 970 975 980 985 990 995 6 1140 1146 1152 1158 1164 1170 1176 1182 1188 1194 7 1330 1337 1344 1351 1358 1365 1372 1379 1386 1393 8 1520 1528 1536 1544 1552 1560 1568 1576 1584 1592 9 1710 1719 1728 1737 1746 1755 1764 1773 1782 1791 52 XVIII. MULTIPLES. 200 201 202 203 204 205 206 207 208 209 1 2 3 200 400 600 201 202 203 402 404 406 603 606 609 204 408 612 205 410 615 206 412 618 207 414 621 208 209 416 418 624 627 4 5 G SOO 1000 1200 804 808 812 1005 1010 1015 1206 1212 1218 816 1020 1224 820 824 1025 1030 1230 1236 828 1035 1242 832 836 1040 1045 1248 1254 7 8 9 1400 1600 ISOO 1407 1414 1421 1608 1616 1624 1809 1818 1827 1428 1632 1836 1435 1640 1845 1442 1648 1854 1449 1656 1863 1456 1463 1664 1672 1872 1881 210 211 212 213 214 215 216 217 218 219 1 2 3 210 420 630 211 212 213 422 424 426 633 636 639 214 428 642 215 430 645 216 432 648 217 434 651 218 219 436 438 654 657 4 5 84 1050 1260 844 848 852 1055 1060 1065 1266 1272 1278 856 1070 1284 860 1075 1290 864 1080 1296 868 1085 1302 872 876 1090 1095 1308 1314 7 8 9 1470 1680 1890 1477 1484 1491 1688 1696 1704 1899 1908 1917 1498 1712 1926 1505 1720 1935 1512 1728 1944 1519 1526 1533 1736 1744 1752 1953 1962 1971 220 221 222 223 224 225 226 227 228 229 1 2 3 220 440 660 221 222 223 442 444 446 663 666 669 224 448 672 225 450 675 226 452 678 227 454 681 228 229 456 458 684 687 4 5 6 SSO 1100 1320 884 888 892 1105 1110 1115 1326 1332 1338 896 900 904 1120 1125 1130 1344 1350 1356 908 1135 1362 912 916 1140 1145 1368 1374 7 8 9 1540 1760 1980 1547 1554 1561 1768 1776 17S4 1989 1998 2007 1568 1575 1792 1800 2016 2025 1582 1808 2034 1589 1596 1603 1816 1824 1832 2043 2052 2061 230 231 232 233 234 235 236 237 238 239 1 2 3 230 460 690 231 232 233 462 464 466 693 696 699 234 468 702 235 470 705 236 472 708 237 474 711 238 239 476 478 714 717 4 5 6 920 1150 1380 924 928 932 1155 1160 1165 1386 1392 1398 936 1170 1404 940 1175 1410 944 1180 1416 948 1185 1422 952 956 1190 1195 1428 1434 7 8 9 1610 1840 2070 1617 1624 1631 1848 1856 1864 2079 2088 2097 1638 1645 1872 1880 2106 2115 1652 1888 2124 1659 1896 2133 1666 1673 1904 1912 2142 2151 240 241 242 243 244 245 246 247 248 249 1 2 3 240 480 720 241 242 243 482 484 486 723 726 729 244 488 732 245 490 735 246 492 738 247 494 741 248 249 496 498 744 747 4 5 6 960 1200 1440 964 968 972 1205 1210 1215 1446 1452 1458 976 1220 1464 980 1225 1470 984 1230 1476 988 1235 1482 992 996 1240 1245 1488 1494 7 8 9 1680 1920 2160 1687 1694 1701 1928 1936 1944 2169 2178 2187 1708 1952 2196 1715 1960 2205 1722 1968 2214 1729 1736 1743 1976 1984 1992 2223 2232 2241 XVIII. MULTIPLES. 53 250 251 252 253 254 255 256 257 258 259 1 250 251 252 253 254 255 256 257 258 259 2 500 502 504 506 508 510 512 514 516 518 3 750 753 756 759 762 765 768 771 774 777 4 1000 1004 1008 1012 1016 1020 1024 1028 1032 1036 5 1250 1255 1260 1265 1270 1275 1280 1285 1290 1295 6 1500 1506 1512 1518 1524 1530 1536 1542 1548 1554 7 1750 1757 1764 1771 1778 1785 1792 1799 1806 1813 8 2000 2008 2016 2024 2032 2040 2048 2056 2064 2072 9 2250 2259 2268 2277 2286 2295 2304 2313 2322 2331 260 261 262 263 264 265 266 267 268 269 1 260 261 262 263 264 265 266 267 268 269 2 520 522 524 526 528 530 532 534 536 538 3 780 783 786 789 792 795 798 801 804 807 4 1040 1044 1048 1052 1056 1060 1064 1068 1072 1076 5 1300 1305 1310 1315 1320 1325 1330 1335 1340 1345 6 1560 1566 1572 1578 1584 1590 1596 1602 1608 1614 7 1820 1827 1834 1841 1848 1855 1862 1869 1876 1883 8 2080 2088 2096 2104 2112 2120 2128 2136 2144 2152 9 2340 2349 2358 2367 2376 2385 2394 2403 2412 2421 270 271 272 273 274 275 276 277 278 279 1 270 271 272 273 274 275 276 277 278 279 2 540 542 544 546 548 550 552 554 556 558 3 810 813 816 819 822 825 828 831 834 837 4 1080 1084 1088 1092 1096 1100 1104 1108 1112 1116 5 1350 1355 1360 1365 1370 1375 13S0 1385 1390 1395 6 1620 1626 1632 1638 1644 1650 1656 1662 1668 1674 7 1890 1897 1904 1911 1918 1925 1932 1939 1946 1953 8 2160 2168 2176 2184 2192 2200 2208 2216 2224 2232 9 2430 2439 2448 2457 2466 2475 2484 2493 2502 2511 280 281 282 283 284 285 286 287 288 289 1 280 281 282 283 284 285 286 287 288 289 2 560 562 564 566 568 570 572 574 576 578 3 840 843 846 849 852 855 858 861 864 867 4 1120 1124 1128 1132 1136 1140 1144 1148 1152 1156 5 1400 1405 1410 1415 1420 1425 1430 1435 1440 1445 6 1680 1686 1692 1698 1704 1710 1716 1722 1728 1734 7 1960 1967 1974 1981 19SS 1995 2002 2009 2016 2023 8 2240 2248 2256 2264 2272 2280 2288 2296 2304 2312 9 2520 2529 2538 2547 2556 2565 2574 2583 2592 2601 290 291 292 293 294 295 296 297 298 299 1 290 291 292 293 294 295 296 297 298 299 2 580 582 584 586 588 590 592 594 596 598 3 870 873 876 879 882 885 888 891 894 897 4 1160 1164 1168 1172 1176 1180 1184 1188 1192 1196 5 1450 1455 1460 1465 1470 1475 1480 1485 1490 1495 6 1740 1746 1752 1758 1764 1770 1776 1782 1788 1794 7 2030 2037 2044 2051 2058 2065 2072 2079 2086 2093 8 2320 2328 2336 2344 2352 2360 2368 2376 2384 2392 9 2610 2619 2628 2637 2646 2655 2664 2673 26S2 2691 XVIII. MULTIPLES. 800 801 802 303 304 305 306 307 308 309 1 2 8 300 600 900 301 602 903 302 604 906 303 606 909 304 608 912 305 610 915 306 612 918 307 614 921 308 309 616 618 924 927 4 5 6 1200 1500 ISOO 1204 1505 1806 1208 1510 1812 1212 1515 1818 1216 1520 1824 1220 1525 1830 1224 1530 1836 1228 1535 1842 1232 1236 1540 1545 1848 1854 7 H 9 2100 2400 2700 2107 2408 2709 2114 2416 2718 2121 2424 2727 2128 2432 2736 2135 2440 2745 2142 2448 2754 2149 2456 2763 2156 2163 2464 2472 2772 2781 310 311 312 313 314 315 316 317 318 319 1 2 3 310 620 930 311 622 933 312 624 936 313 626 939 314 628 942 315 630 945 316 632 948 317 634 951 318 319 636 638 954 957 4 5 6 1240 1550 1860 1244 1555 1866 1248 1560 1872 1252 1565 1878 1256 1570 1884 1260 1575 1890 1264 1580 1896 1268 1585 1902 1272 1276 1590 1595 1908 1914 7 8 9 2170 2480 2790 2177 2488 2799 2184 2496 2808 2191 2504 2817 2198 2512 2826 2205 2520 2835 2212 2528 2844 2219 2536 2853 2226 2233 2544 2552 2862 2871 320 321 322 323 324 325 326 327 328 329 1 2 3 320 640 960 321 642 963 322 644 966 323 646 969 324 648 972 325 650 975 326 652 978 327 654 981 328 329 656 658 984 987 4 5 6 1280 1600 1920 1284 1605 1926 1288 1610 1932 1292 1615 1938 1296 1620 1944 1300 1625 1950 1304 1630 1956 1308 1635 1962 1312 1316 1640 1645 1968 1974 7 8 9 2240 2560 2880 2247 2568 2889 2254 2576 2898 2261 2584 2907 2268 2592 2916 2275 2600 2925 2282 2608 2934 2289 2616 2943 2296 2303 2624 2632 2952 2961 330 331 332 333 334 335 336 337 338 339 1 2 3 330 660 990 331 662 993 332 664 996 333 666 999 334 668 1002 335 670 1005 336 672 1008 337 674 1011 338 339 676 678 1014 1017 4 5 6 1320 1650 1980 1324 1655 1986 1328 1660 1992 1332 1665 1998 1336 1670 2004 1340 1675 2010 1344 1680 2016 1348 1685 2022 1352 1356 1690 1695 2028 2034 7 8 9 2310 2640 2970 2317 2648 2979 2324 2656 2988 2331 2664 2997 2338 2672 3006 2345 2680 3015 2352 2688 3024 2359 2696 3033 2366 2373 2704 2712 3042 3051 340 341 342 343 344 345 346 347 348 349 1 2 3 340 680 1020 341 682 1023 342 684 1026 343 686 1029 344 688 1032 345 690 1035 346 692 1038 347 694 1041 348 349 696 698 1044 1047 4 5 6 1360 1700 2040 1364 1705 2046 1368 1710 2052 1372 1715 2058 1376 1720 2064 1380 1725 2070 1384 1730 2076 1388 1735 2082 1392 1396 1740 1745 2088 2094 7 8 9 2380 2720 3060 2387 2728 3069 2394 2736 3078 2401 2744 3087 2408 2752 3096 2415 2760 3105 2422 2768 3114 2429 2776 3123 2436 2443 2784 2792 3132 3141 XVIII. MULTIPLES. 350 351 352 353 354 355 356 357 358 359 1 2 3 350 700 1050 351 702 1053 352 704 1056 353 706 1059 354 355 70S 710 1062 1065 356 712 1068 357 714 1071 358 716 1074 359 718 1077 4 5 6 1400 1750 2100 1404 1755 2106 1408 1760 2112 1412 1765 2118 1416 1420 1770 1775 2124 2130 1424 1780 2136 1428 1785 2142 1432 1790 2148 1436 1795 2154 7 8 9 2450 2S00 3150 2457 2808 3159 2464 2816 3168 2471 2824 3177 2478 2485 2832 2840 3186 3195 2492 2848 3204 2499 2856 3213 2506 2864 3222 2513 2872 3231 360 361 362 363 364 365 366 367 368 369 1 2 3 360 720 1080 361 722 1083 362 724 1086 363 726 1089 364 365 728 730 1092 1095 366 732 1098 367 734 1101 368 736 1104 369 738 1107 4 5 6 1440 1800 2160 1444 1805 2166 1448 1810 2172 1452 1815 2178 1456 1460 1820 1825 2184 2190 1464 1830 2196 1468 1835 2202 1472 1840 2208 1476 1845 2214 7 8 9 2520 2880 3240 2527 2888 3249 2534 2896 3258 2541 2904 3267 2548 2555 2912 2920 3276 3285 2562 2928 3294 2569 2936 3303 2576 2944 3312 2583 2952 3321 370 371 372 373 374 375 376 377 378 379 1 2 3 370 740 1110 371 742 1113 372 744 1116 373 746 1119 374 375 748 750 1122 1125 376 752 1128 377 754 1131 378 756 1134 379 758 1137 4 5 6 1480 1850 2220 1484 1855 2226 1488 1860 2232 1492 1865 2238 1496 1500 1870 1875 2244 2250 1504 1880 2256 1508 1885 2262 1512 1890 2268 1516 1895 2274 7 8 9 2590 2960 3330 2597 2968 3339 2604 2976 3348 2611 2984 3357 2618 2625 2992 3000 3366 3375 2632 3008 3384 2639 3016 3393 2646 3024 3402 2653 3032 3411 380 381 382 383 384 385 386 387 388 389 1 2 380 760 1140 381 762 1143 382 764 1146 383 766 1149 384 385 768 770 1152 1155 386 772 1158 387 774 1161 388 776 1164 389 778 1167 4 5 6 1520 1900 2280 1524 1905 2286 1528 1910 2292 1532 1915 2298 1536 1540 1920 1925 2304 2310 1544 1930 2316 1548 1935 2322 1552 1940 2328 1556 1945 2334 7 8 9 2660 3040 3420 2667 3048 3429 2674 3056 3438 2681 3064 3447 2688 2695 3072 3080 3456 3465 2702 3088 3474 2709 3096 3483 2716 3104 3492 2723 3112 3501 390 391 392 393 394 395 396 397 398 399 1 2 3 390 780 1170 391 782 1173 392 784 1176 393 786 1179 394 395 788 790 1182 1185 396 792 1188 397 794 1191 398 796 1194 399 798 1197 4 5 6 1560 1950 2340 1564 1955 2346 1568 1960 2352 1572 1965 2358 1576 1580 1970 1975 2364 2370 1584 1980 2376 1588 1985 2382 1592 1990 2388 1596 1995 2394 7 8 9 2730 3120 3510 2737 3128 3519 2744 3136 3528 2751 3144 3537 2758 2765 3152 3160 3546 3555 2772 3168 3564 2779 3176 3573 2786 3184 3582 2793 3192 3591 56 XVIII. MULTIPLES. 400 401 402 403 404 405 406 407 408 409 1 400 401 402 403 404 405 406 407 408 409 2 SOO 802 804 806 SOS 810 812 814 816 818 3 1200 1203 1206 1209 1212 1215 121S 1221 1224 1227 4 1600 1604 1608 1612 1616 1620 1624 1628 1632 1636 5 2000 2005 2010 2015 2020 2025 2030 2035 2040 2045 2400 2406 2412 2418 2424 2430 2436 2442 2448 2454 7 2S00 2807 2814 2821 2828 2835 2842 2849 2856 2863 8 3200 3208 3216 3224 3232 3240 324B 3256 3264 3272 9 3600 3609 3618 3627 3636 3645 3654 3663 3672 3681 410 411 412 413 414 415 416 417 418 419 1 410 411 412 413 414 415 416 417 418 419 2 S20 822 824 826 828 830 832 834 836 838 3 1230 1233 1236 1239 1242 1245 1248 1251 1254 1257 4 1640 1644 1648 1652 1656 1660 1664 1668 1672 1676 5 2050 2055 2060 2065 2070 2075 2080 2085 2090 2095 6 2460 2466 2472 2478 2484 2490 2496 2502 2508 2514 7 2870 2877 2884 2891 2898 2905 2912 2919 2926 2933 8 3280 3288 3296 3304 3312 3320 3328 3336 3344 3352 9 3690 3699 3708 3717 3726 3735 3744 3753 3762 3771 420 421 422 423 424 425 426 427 428 429 1 420 421 422 423 424 425 426 427 428 429 2 840 842 844 846 848 850 852 854 856 858 3 1260 1263 1266 1269 1272 1275 1278 1281 1284 1287 4 1680 1684 1688 1692 1696 1700 1704 1708 1712 1716 5 2100 2105 2110 2115 2120 2125 2130 2135 2140 2145 6 2520 2526 2532 2538 2544 2550 2556 2562 2568 2574 7 2940 2947 2954 2961 2968 2975 29S2 2989 2996 3003 8 3360 3368 3376 3384 3392 3400 3408 3416 3424 3432 9 3780 3789 3798 3807 3816 3825 3834 3843 3852 3861 430 431 432 433 434 435 436 437 438 439 1 430 431 432 433 434 435 436 437 438 439 2 860 862 864 866 868 870 872 874 876 878 3 1290 1293 1296 1299 1302 1305 1308 1311 1314 1317 4 1720 1724 1728 1732 1736 1740 1744 1748 1752 1756 5 2150 2155 2160 2165 2170 2175 2180 2185 2190 2195 6 2580 2586 2592 2598 2604 2610 2616 2622 2628 2634 7 3010 3017 3024 3031 3038 3045 3052 3059 3066 3073 8 3440 3448 3456 3464 3472 3480 3488 3496 3504 3512 9 3870 3879 3SSS 3897 3906 3915 3924 3933 3942 3951 440 441 442 443 444 445 446 447 448 449 1 440 441 442 443 444 445 446 447 44S 449 2 880 882 884 886 888 890 892 894 896 898 3 1320 1323 1326 1329 1332 1335 1338 1341 1344 1347 4 1760 1764 1768 1772 1776 1780 1784 1788 1792 1796 5 2200 2205 2210 2215 2220 2225 2230 2235 2240 2245 6 2640 2646 2652 2658 2664 2670 2676 2682 2688 2694 7 3080 3087 3094 3101 3108 3115 3122 3129 3136 3143 8 3520 3528 3536 3544 3552 3560 3568 3576 3584 3592 9 3960 3969 3978 3987 3996 4005 4014 4023 4032 4041 XVIII. MULTIPLES. 57 450 451 452 453 454 455 456 457 458 459 1 2 3 450 900 1350 451 902 1353 452 904 1356 453 906 1359 454 908 1362 455 910 1365 456 912 1368 457 914 1371 458 916 1374 459 918 1377 4 5 6 ISOO 2250 2700 1804 2255 2706 1808 2260 2712 1812 2265 2718 1816 2270 2724 1820 2275 2730 1824 2280 2736 1828 2285 2742 1832 2290 2748 1836 2295 2754 7 8 9 3150 3600 4050 3157 3608 4059 3164 3616 4068 3171 3624 4077 3178 3632 4086 3185 3640 4095 3192 3648 4104 3199 3656 4113 3206 3664 4122 3213 3672 4131 460 461 462 463 464 465 466 467 468 469 1 2 3 460 920 1380 461 922 1383 462 924 1386 463 926 1389 464 928 1392 465 930 1395 466 932 1398 467 934 1401 468 936 1404 469 938 1407 4 5 6 1840 2300 2760 1844 2305 2766 1848 2310 2772 1852 2315 2778 1856 2320 2784 1S60 2325 2790 1864 2330 2796 1868 2335 2802 1872 2340 2808 1876 2345 2814 7 1 3220 3680 4140 3227 3688 4149 3234 3696 4158 3241 3704 4167 3248 3712 4176 3255 3720 4185 3262 3728 4194 3269 3736 4203 3276 3744 4212 3283 3752 4221 470 471 472 473 iU 475 476 477 478 479 1 2 3 470 940 1410 471 942 1413 472 944 1416 473 946 1419 474 948 1422 475 950 1425 476 952 1428 477 954 1431 478 956 1434 479 958 1437 4 5 6 1880 235a 2820 1884 2355 2826 1888 2360 2832 1892 2365 2838 1896 2370 2844 1900 2375 2850 1904 2380 2856 1908 2385 2862 1912 2390 2868 1916 2395 2874 7 8 9 3290 3760 4230 3297 3768 4239 3304 3776 4248 3311 3784 4257 3318 3792 4266 3325 3800 4275 3i32 3808 4284 3339 3816 4293 3346 3824 4302 3353 3832 4311 480 481 482 483 484 485 486 487 488 489 1 2 3 480 960 1440 481 962 1443 482 964 1446 483 966 1449 484 968 1452 485 970 1455 486 972 1458 487 974 1461 488 976 1464 489 978 1467 4 5 6 1920 2400 2S80 1924 2405 2886 1928 2410 2892 1932 2415 2898 1936 2420 2904 1940 2425 2910 1944 2430 2916 1948 2435 2922 1952 2440 2928 1956 2445 2934 7 8 9 3360 3840 4320 3367 3848 4329 3374 3856 4338 3381 3864 4347 3388 3872 4356 3395 3880 4365 3402 3888 4374 3409 3896 4383 3416 3904 4392 3423 3912 4401 490 491 492 493 494 495 496 497 498 499 1 2 3 490 980 1470 491 982 1473 492 984 1476 493 986 1479 494 988 1482 495 990 1485 496 992 1488 497 994 1491 498 996 1494 499 998 1497 4 5 6 1960 2450 2940 1964 2455 2946 1968 2460 2952 1972 2465 2958 1976 2470 2964 1980 2475 2970 1984 2480 2976 1988 2485 2982 1992 2490 2988 1996 2495 2994 7 8 9 3430 3920 4410 3437 3928 4419 3444 3936 4428 3451 3944 4437 3458 3952 4446 3465 3960 4455 3472 3968 4464 3479 3976 4473 3486 3984 4482 3493 3992 4491 58 XVIII. MULTIPLES. 6(M> 501 502 503 504 505 506 507 608 509 1 2 8 500 1000 1500 501 1002 1503 502 1004 1506 503 1006 1509 504 1008 1512 505 1010 1515 506 1012 1518 507 1014 1521 508 1016 1524 509 1018 1527 4 5 2000 2500 3000 2004 2505 3006 2008 2510 3012 2012 2515 3018 2016 2520 3024 2020 2525 3030 2024 2530 3036 2028 2535 3042 2032 2540 3048 2036 2545 3054 s 9 3500 4000 4500 3507 4008 4509 3514 4016 4518 3521 4024 4527 3528 4032 4536 3535 4040 4545 3542 4048 4554 3549 4056 4563 3556 4064 4572 3563 4072 4581 510 511 612 513 514 515 510 517 518 519 1 2 3 510 1020 1530 511 1022 1533 512 1024 1536 513 1026 1539 514 1028 1542 515 1030 1545 516 1032 1548 517 1034 1551 518 1036 1554 519 1038 1557 4 5 6 2(H0 2550 3060 2044 2555 3066 2048 2560 3072 2052 2565 3078 2056 2570 3084 2060 2575 3090 2064 2580 3096 2068 2585 3102 2072 2590 3108 2076 2595 3114 7 8 9 3570 4080 4590 3577 4088 4599 3584 4096 4608 3591 4104 4617 3598 4112 4626 3605 4120 4635 3612 4128 4644 3619 4136 4653 3626 4144 4662 3633 4152 4671 520 521 522 523 524 525 526 527 528 529 1 2 3 520 1040 1560 521 1042 1563 522 1044 1566 523 1046 1569 524 1043 1572 525 1050 1575 526 1052 1578 527 1054 1581 528 1056 1584 529 1058 1587 4 5 6 20S0 2600 3120 2084 2605 3126 2088 2610 3132 2092 2615 3138 2096 2620 3144 2100 2625 3150 2104 2630 3156 2108 2635. 3162 2112 2640 3168 2116 2645 3174 7 S 9 3640 4160 4680 3647 4168 4689 3654 4176 4698 3661 4184 4707 3668 4192 4716 3675 4200 4725 3682 4208 4734 3689 4216 4743 3696 4224 4752 3703 4232 4761 530 531 532 533 534 535 536 537 538 539 1 2 3 530 1060 1590 531 1062 1593 532 1064 1596 533 1066 1599 534 1068 1602 535 1070 1605 536 1072 1608 537 1074 1611 538 1076 1614 539 1078 1617 4 5 6 2120 2650 3180 2124 2655 3186 2128 2660 3192 2132 2665 3198 2136 2670 3204 2140 2675 3210 2144 26S0 3216 2148 2685 3222 2152 2690 3228 2156 2695 3234 7 8 9 3710 4240 4770 3717 4248 4779 3724 4256 4788 3731 4264 4797 3738 4272 4806 3745 4280 4815 3752 42SS 4824 3759 4296 4833 3766 4304 4842 3773 4312 4851 540 541 542 543 544 545 546 547 548 549 1 2 3 540 1080 1620 541 1082 1623 542 1084 1626 543 1086 1629 544 1088 1632 545 1090 1635 546 1092 1638 547 1094 1641 548 1096 1644 549 1098 1647 4 a 6 2160 2700 3240 2164 2705 3246 216S 2710 3252 2172 2715 3258 2176 2720 3264 2180 2725 3270 2184 2730 3276 2188 2735 32S2 2192 2740 3288 2196 2745 3294 7 8 9 3780 4320 4860 3787 4328 4869 3794 4336 4878 3801 4344 48S7 3808 4352 4896 3815 4360 4905 3822 4368 4914 3829 4376 4923 3836 4384 4932 3843 4392 4941 XVIII. MULTIPLES. 59 550 551 552 553 554 555 556 557 558 559 1 2 3 550 1100 1650 551 1102 1653 552 1104 1656 553 1106 1659 554 1108 1662 555 1110 1665 556 1112 1668 557 1114 1671 558 1116 1674 559 1118 1677 4 5 6 2200 2750 3300 2204 2755 3306 2208 2760 3312 2212 2765 3318 2216 2770 3324 2220 2775 3330 2224 2780 3336 2228 2785 3342 2232 2790 3348 2236 2795 3354 7 8 9 3S50 4400 4950 3857 4408 4959 3864 4416 4968 3871 4424 4977 3878 4432 4986 3885 4440 4995 3892 4448 5004 3899 4456 5013 3906 4^64 5022 3913 4472 5031 560 561 562 563 564 565 566 567 568 569 1 2 3 560 1120 16S0 561 1122 1683 562 1124 1686 563 1126 1689 564 1128 1692 565 1130 1695 566 1132 1698 567 1134 1701 568 1136 1704 569 1138 1707 4 5 2240 2800 3360 2244 2805 3366 2248 2810 3372 2252 2815 3378 2256 2820 3384 2260 2825 3390 2264 2830 3396 2268 2835 3402 2272 2840 3408 2276 2845 3414 7 8 9 3920 4480 5040 3927 4488 5049 3934 4496 5058 3941 4504 5067 3948 4512 5076 3955 4520 5085 3962 4528 5094 3969 4536 5103 3976 4544 5112 3983 4552 5121 570 571 572 573 574 575 576 577 578 579 1 2 3 570 1140 1710 571 1142 1713 572 1144 1716 573 1146 1719 574 1148 1722 575 1150 1725 576 1152 1728 577 1154 1731 578 1156 1734 579 1158 1737 4 5 G 2280 2850 3420 2284 2855 3426 2288 2860 3432 2292 2865 3438 2296 2870 3444 2300 2875 3450 2304 2880 3456 2308 2885 3462 2312 2890 3468 2316 2895 3474 7 8 9 3990 4560 5130 3997 4568 5139 4004 4576 5148 4011 4584 5157 4018 4592 5166 4025 4600 5175 4032 4608 5184 4039 4616 5193 4046 4624 5202 4053 4632 5211 580 581 582 583 584 585 586 587 588 589 1 2 3 580 1160 1740 581 1162 1743 582 1164 1746 583 1166 1749 584 1168 1752 585 1170 1755 586 1172 1758 587 1174 1761 588 1176 1764 589 1178 1767 4 5 6 2320 2900 3480 2324 2905 3486 2328 2910 3492 2332 2915 3498 2336 2920 3504 2340 2925 3510 2344 2930 3516 2348 2935 3522 2352 2940 3528 2356 2945 3534 7 8 9 4060 4640 5220 4067 4648 5229 4074 4656 5238 4081 4664 52-17 4088 4672 5256 4095 4680 5265 4102 4688 5274 4109 4696 5283 4116 4704 5292 4123 4712 5301 590 591 592 593 594 595 596 597 598 599 1 2 3 590 1180 1770 591 1182 1773 592 1184 1776 593 1186 1779 594 1188 1782 595 1190 1785 596 1192 1788 597 1194 1791 598 1196 1794 599 1198 1797 4 5 6 2360 2950 3540 2364 2955 3546 2368 2960 3552 2372 2965 3558 2376 2970 3564 2380 2975 3570 2384 2980 3576 2388 2985 3582 2392 2990 3588 2396 2995 3594 7 8 9 4130 4720 5310 4137 4728 5319 4144 4736 5328 4151 4744 5337 4158 4752 5346 4165 4760 5355 4172 4768 5364 4179 4776 5373 4186 4784 5382 4193 4792 5391 60 XVIII. MULTIPLES. 600 601 602 603 604 605 606 607 608 609 1 2 8 600 1200 1800 601 1202 1803 602 1204 1806 603 1206 1809 604 1208 1812 605 1210 1815 606 1212 1818 607 1214 1821 608 609 1216 1218 1824 1827 4 5 6 2400 3000 3600 2404 3005 3606 2408 3010 3612 2412 3015 3618 2416 3020 3624 2420 3025 3630 2424 3030 3636 2428 3035 3642 2432 2436 3040 3045 3648 3654 7 1 4200 4800 5400 4207 4808 5409 4214 4816 5418 4221 4824 5427 4228 4832 5436 4235 4840 5445 4242 4848 5454 4249 4856 5463 4256 4263 4864 4872 5472 5481 610 611 612 613 614 615 616 617 618 619 1 2 3 610 1220 1830 611 1222 1833 612 1224 1836 613 1226 1839 614 1228 1842 615 1230 1845 616 1232 1848 617 1234 1851 618 619 1236 1238 1854 1857 4 5 2440 3050 3660 2444 3055 3666 2448 3060 3672 2452 3065 3678 2456 3070 3684 2460 3075 3690 2464 3080 3696 2468 3085 3702 2472 2476 3090 3095 3708 3714 7 8 9 4270 4SS0 5490 4277 4888 5499 4284 4896 5508 4291 4904 5517 4298 4912 5526 4305 4920 5535 4312 4928 5544 4319 4936 5553 4326 4333 4944 4952 5562 5571 6-'0 621 622 623 624 625 626 627 628 629 1 2 3 620 1240 I860 621 1242 1863 622 1244 1866 623 1246 1869 624 1248 1872 625 1250 1875 626 1252 1878 627 1254 1881 628 629 1256 1258 1884 1887 4 5 6 2480 3100 3720 2484 3105 3726 2488 3110 3732 2492 3115 3738 2496 3120 3744 2500 3125 3750 2504 3130 3756 2508 3135 3762 2512 2516 3140 3145 3768 3774 7 8 9 4340 4960 5580 4347 4968 5589 4354 4976 5598 4361 4984 5607 4368 4992 5616 4375 5000 5625 4382 5008 5634 4389 5016 5643 4396 4403 5024 5032 5652 5661 G30 631 632 633 634 635 636 637 638 639 1 2 3 630 1260 1890 631 1262 1893 632 1264 1896 633 1266 1899 634 1268 1902 635 1270 1905 636 1272 1908 637 1274 1911 638 639 1276 1278 1914 1917 4 5 G 2520 3150 3780 2524 3155 3786 2528 3160 3792 2532 3165 3798 2536 3170 3804 2540 3175 3810 2544 3180 3816 2548 3185 3822 2552 2556 3190 3195 3828 3834 7 8 9 4410 5040 5670 4417 5048 5679 4424 5056 5688 4431 5064 5697 4438 5072 5706 4445 5080 5715 4452 5088 5724 4459 5096 5733 4466 4473 5104 5112 5742 5751 640 641 642 643 644 645 646 647 648 649 1 2 3 640 1280 1920 641 1282 1923 642 1284 1926 643 1286 1929 644 1288 1932 645 1290 1935 646 1292 1938 647 1294 1941 648 649 1296 1298 1944 1947 4 5 6 2560 3200 3840 2564 3205 3846 2568 3210 3852 2572 3215 3858 2576 3220 3864 2580 3225 3870 2584 3230 3876 2588 3235 3882 2592 2596 3240 3245 3888 3894 7 8 9 4480 5120 5760 4487 5128 5769 4494 5136 5778 4501 5144 5787 4508 5152 5796 4515 5160 5805 4522 5168 5814 4529 5176 5823 4536 4543 5184 5192 5832 5841 XVIII. MULTIPLES. 61 650 651 652 653 654 655 656 657 658 659 1 2 3 650 1300 1950 651 1302 1953 652 1304 1956 653 1306 1959 654 1308 1962 655 1310 1965 656 1312 1968 657 1314 1971 658 659 1316 1318 1974 1977 4 5 6 2600 3250 3900 2604 3255 3906 2608 3260 3912 2612 3265 3918 2616 3270 3924 2620 3275 3930 2624 3280 3936 2628 3285 3942 2632 2636 3290 3295 3948 3954 7 8 9 4550 5200 5850 4557 5208 5859 4564 5216 5868 4571 5224 5877 4578 5232 5886 4585 5240 5895 4592 5248 5904 4599 5256 5913 4606 4613 5264 5272 5922 5931 660 661 662 663 664 665 666 667 668 669 1 2 3 660 1320 1980 661 1322 1983 662 1324 19S6 663 1326 1989 664 1328 1992 665 1330 1995 666 1332 1998 667 1334 2001 668 669 1336 1338 2004 2007 4 5 6 2640 3300 3960 2644 3305 3966 2648 3310 3972 2652 3315 3978 2656 3320 3984 2660 3325 3990 2664 3330 3996 2668 3335 4002 2672 2676 3340 3345 4008 4014 7 8 9 4620 5280 5940 4627 5288 5949 4634 5296 5958 4641 5304 5967 4648 5312 5976 4655 5320 59S5 4662 5328 5994 4669 5336 6003 4676 4683 5344 5352 6012 6021 670 671 672 673 674 675 676 677 678 679 1 2 3 670 1340 2010 671 1342 2013 672 1344 2016 673 1346 2019 674 1348 2022 675 1350 2025 676 1352 2028 677 1354 2031 678 679 1356 1358 2034 2037 4 5 6 2680 3350 4020 2684 3355 4026 2688 3360 4032 2692 3365 4038 2696 3370 4044 2700 3375 4050 2704 3380 4056 2708 3385 4062 2712 2716 3390 3395 4068 4074 7 8 9 4690 5360 6030 4697 5368 6039 4704 5376 6048 4711 5384 6057 4718 5392 6066 4725 5400 6075 4732 5408 6084 4739 5416 6093 4746 4753 5424 5432 6102 6111 680 681 682 683 684 685 686 687 688 689 1 3 680 1360 2040 681 1362 2043 682 1364 2046 683 1366 2049 684 1368 2052 685 1370 2055 686 1372 2058 687 1374 2061 688 689 1376 1378 2064 2067 4 5 6 2720 3400 4080 2724 3405 40S6 2728 3410 4092 2732 3415 4098 2736 3420 4104 2740 3425 4110 2744 3430 4116 2748 3435 4122 2752 2756 3440 3445 4128 4134 7 8 9 4760 5440 6120 4767 5448 6129 4774 5456 6138 4781 5464 6147 4788 5472 6156 4795 5480 6165 4802 5488 6174 4809 5496 6183 4816 4823 5504 5512 6192 6201 690 691 692 693 694 695 696 697 698 699 1 3 690 1380 2070 691 1382 2073 692 1384 2076 693 1386 2079 694 1388 2082 695 1390 2085 696 1392 2088 697 1394 2091 698 699 1396 1398 2094 2097 4 5 6 2760 3450 4140 2764 3455 4146 2768 3460 4152 2772 3465 4158 2776 3470 4164 2780 3475 4170 2784 3480 4176 2788 3485 4182 2792 2796 3490 3495 4188 4194 7 8 9 4830 5520 6210 4837 5528 6219 4844 5536 6228 4851 5544 6237 4858 5552 6246 4865 5560 6255 4872 5568 6264 4879 5576 6273 4886 4893 5584 5592 6282 6291 62 XVIII. MULTIPLES. 1 700 701 702 703 704 706 706 707 708 709 1 700 701 702 703 704 705 706 707 708 709 2 1400 1402 1404 1406 1408 1410 1412 1414 1416 1418 3 2100 2103 2106 2109 2112 2115 2118 2121 2124 2127 4 2S00 2804 2808 2812 2816 2820 2824 2828 2832 2836 5 3500 3505 3510 3515 3520 3525 3530 3535 3540 3545 6 4200 4206 4212 4218 4224 4230 4236 4242 4248 4254 7 4900 4907 4914 4921 4928 4935 4942 4949 4956 4963 8 5600 5608 5616 5624 5632 5640 5648 5656 5664 5672 {> 6300 6309 6318 6327 6336 6345 6354 6363 6372 6381 710 711 712 713 714 715 71G 717 718 719 1 710 711 712 713 714 715 716 717 718 719 2 1420 1422 1424 1426 1428 1430 1432 1434 1436 1438 3 2130 2133 2136 2139 2142 2145 2148 2151 2154 2157 4 2S40 2844 2848 2852 2856 2860 2864 2868 2872 2876 5 3550 3555 3560 3565 3570 3575 3580 3585 3590 3595 G 4260 4266 4272 4278 4284 4290 4296 4302 4308 4314 7 4970 4977 4984 4991 4998 5005 5012 5019 5026 5033 H 56S0 5688 5696 5704 5712 5720 5728 5736 5744 5752 \) 6390 6399 6408 6417 6426 6435 6444 6453 6462 6471 720 721 722 723 724 725 726 727 728 729 1 720 721 722 723 724 ~^7 726 727 728 729 2 1440 1442 1444 1446 1448 1450 1452 1454 1456 1458 3 2160 2163 2166 2169 2172 2175 2178 2181 2184 2187 4 28S0 2884 2888 2892 2896 2900 2904 2908 2912 2916 5 3600 3605 3610 3615 3620 3625 3630 3635 3640 3645 6 4320 4326 4332 4338 4344 4350 4356 4362 4368 4374 7 5040 5047 5054 5061 5068 5075 5082 5089 5096 5103 H 5760 5768 5776 5784 5792 5S00 5808 5816 5824 5832 9 6480 6489 6498 6507 6516 6525 6534 6543 6552 6561 730 731 732 733 734 735 736 737 738 739 1 730 731 732 733 734 735 736 737 738 739 2 1460 1462 1464 1466 1468 1470 1472 1474 1476 1478 3 2190 2193 2196 2199 2202 2205 2208 2211 2214 2217 4 2920 2924 2928 2932 2936 2940 2944 2948 2952 2956 5 3650 3655 3660 3665 3670 3675 3680 3685 3690 3695 G 4380 4386 4392 4398 4404 4410 4416 4422 4428 4434 7 5110 5117 5124 5131 5138 5145 5152 5159 5166 5173 8 5840 5848 5856 5864 5872 5880 5888 5896 5904 5912 9 6570 6579 6588 6597 6606 6615 6624 6633 6642 6651 740 741 742 743 744 745 746 747 748 749 1 740 741 742 743 744 745 746 747 748 749 2 1480 1482 1484 1486 1488 1490 1492 1494 1496 1498 3 2220 2223 2226 2229 2232 2235 2238 2241 2244 2247 4 2960 2964 2968 2972 2976 2980 2984 2988 2992 2996 5 3700 3705 3710 3715 3720 3725 3730 3735 3740 3745 G 4440 4446 4452 4458 4464 4470 4476 4482 4488 4494 7 5180 5187 5194 5201 5208 5215 5222 5229 5236 5243 8 5920 5928 5936 5944 5952 5960 5968 5976 5984 5992 9 6660 6669 6678 6687 6696 6705 6714 6723 6732 6741 XVIII. MULTIPLES. G3 750 751 752 753 754 755 756 757 758 759 1 2 3 750 1500 2250 751 1502 2253 752 753 1504 1506 2256 2259 754 1508 2262 755 1510 2265 756 1512 2268 757 1514 2271 758 759 1516 1518 2274 2277 4 5 6 3000 3750 4500 3004 3755 4506 3008 3012 3760 3765 4512 4518 3016 3770 4524 3020 3775 4530 3024 3780 4536 3028 3785 4542 3032 3036 3790 3795 4548 4554 7 8 9 5250 6000 6750 5257 6008 6759 5264 5271 6016 6024 6768 6777 5278 6032 6786 5285 6040 6795 5292 6048 6804 5299 6056 6813 5306 5313 6064 6072 6822 6831 760 761 762 763 764 765 766 767 768 769 1 2 3 760 1520 2280 761 1522 2283 762 763 1524 1526 2286 2289 764 1528 2292 765 1530 2295 766 1532 2298 767 1534 2301 768 769 1536 1538 2304 2307 4 5 6 3040 3800 4560 3044 3805 4566 3048 3052 3810 3815 4572 4578 3056 3820 4584 3060 3825 4590 3064 3830 4596 3068 3835 4602 3072 3076 3840 3845 4608 4614 7 8 9 5320 6080 6840 5327 6088 6849 5334 5341 6096 6104 6858 6867 5348 6112 6876 5355 6120 6885 5362 6128 6894 5369 6136 6903 5376 5383 6144 6152 6912 6921 770 771 772 773 774 775 776 777 778 779 1 2 3 770 1540 2310 771 1542 2313 772 773 1544 1546 2316 2319 774 1548 2322 775 1550 2325 776 1552 2328 777 1554 2331 778 779 1556 1558 2334 2337 4 5 6 3080 3850 4620 3084 3855 4626 3088 3092 3860 3865 4632 4638 3096 3870 4644 3100 3875 4650 3104 3880 4656 3108 3885 4662 3112 3116 3890 3895 4668 4674 7 8 9 5390 6160 6930 5397 6168 6939 5404 5411 6176 6184 6948 6957 5418 6192 6966 5425 6200 6975 5432 6208 6984 5439 6216 6993 5446 5453 6224 6232 7002 7011 780 781 782 783 784 785 786 787 788 789 1 2 3 780 1560 2340 781 1562 2343 782 783 1564 1566 2346 2349 784 1568 2352 785 1570 2355 786 1572 2358 787 1574 2361 788 789 1576 1578 2364 2367 4 5 6 3120 3900 4680 3124 3905 4686 3128 3132 3910 3915 4692 4698 3136 3920 4704 3140 3925 4710 3144 3930 4716 3148 3935 4722 3152 3156 3940 3945 4728 4734 7 8 9 5460 6240 7020 5467 6248 7029 5474 5481 6256 6264 7038 7047 5488 6272 7056 5495 6280 7065 5502 6288 7074 5509 6296 7083 5516 5523 6304 6312 7092 7101 790 791 792 793 794 795 796 797 798 799 1 2 3 790 1580 2370 791 1582 2373 792 793 1584 1586 2376 2379 794 1588 2382 795 1590 2385 796 1592 2388 797 1594 2391 798 799 1596 1598 2394 2397 4 5 6 3160 3950 4740 3164 3955 4746 3168 3172 3960 3965 4752 4758 3176 3970 4764 3180 3975 4770 3184 3980 4776 3188 3985 4782 3192 3196 3990 3995 4788 4794 7 8 9 5530 6320 7110 5537 6328 7119 5544 5551 6336 6344 7128 7137 5558 6352 7146 5565 6360 7155 5572 6368 7164 5579 6376 7173 5586 5593 6384 6392 7182 7191 64 XVIII. MULTIPLES. 800 801 802 803 804 805 806 807 808 809 1 800 801 802 803 804 805 806 807 808 809 2 1600 1602 1604 1606 1608 1610 1612 1614 1616 1618 3 2400 2403 2406 2409 2412 2415 2418 2421 2424 2427 4 3200 3204 3208 3212 3216 3220 3224 3228 3232 3236 5 4000 4005 4010 4015 4020 4025 4030 4035 4040 4045 6 4S00 4806 4812 4818 4824 4830 4836 4842 4848 4854 7 5600 5607 5614 5621 5628 5635 5642 5649 5656 5663 8 6400 6408 6416 6424 6432 6440 6448 6456 6464 6472 9 7200 7209 7218 7227 7236 7245 7254 7263 7272 72S1 810 811 812 813 814 815 816 817 818 819 1 810 811 812 813 814 815 816 817 ~81S" 819 2 1620 1622 1624 1626 1628 1630 1632 1634 1636 1638 3 2430 2433 2436 2^39 2442 2445 2448 2451 2454 2457 4 3240 3244 3248 3252 3256 3260 3264 3268 3272 3276 5 4050 4055 4060 4065 4070 4075 40S0 4085 4090 4095 6 4S60 4866 4872 4878 4884 4890 4896 4902 4908 4914 7 5670 5677 5684 5691 5698 5705 5712 5719 5726 5733 8 6480 6488 6496 6504 6512 6520 6528 6536 6544 6552 9 7290 7299 7308 7317 7326 7335 7344 7353 7362 7371 820 821 822 823 824 825 826 827 828 829 1 820 821 822 823 824 825 826 827 828 829 2 1640 1642 1644 1646 1648 1650 1652 1654 1656 1658 3 2460 2463 2466 2469 2472 2475 2478 2481 2484 2487 4 3280 3284 3288 3292 3296 3300 3304 3308 3312 3316 5 4100 4105 4110 4115 4120 4125 4130 4135 4140 4145 6 4920 4926 4932 4938 4944 4950 4956 4962 4968 4974 7 5740 5747 5754 5761 5768 5775 5782 5789 5796 5803 8 6560 6568 6576 6584 6592 6600 6608 6616 6624 6632 9 73S0 7389 739S 7407 7416 7425 7434 7443 7452 7461 830 831 832 833 834 835 836 837 838 839 1 830 831 832 833 834 "835^ 836 837 838 839 2 1660 1662 1664 1666 1668 1670 1672 1674 1676 1678 3 2490 2493 2496 2499 2502 2505 2508 2511 2514 2517 4 3320 3324 3328 3332 3336 3340 3344 3348 3352 3356 5 4150 4155 4160 4165 4170 4175 4180 4185 4190 4195 6 4980 4986 4992 4998 5004 5010 5016 5022 5028 5034 7 5810 5817 5824 5831 5838 5845 5852 5859 5866 5873 8 6640 6648 6656 6664 6672 6680 6688 6696 6704 4712 9 7470 7479 7488 7^97 7506 7515 7524 7533 7542 7551 840 841 842 843 844 845 846 847 848 849 1 840 841 842 843 844 845 846 847 848 849 2 1680 1682 1684 1686 1688 1690 1692 1694 1696 1698 3 2520 2523 2526 2529 2532 2535 2538 2541 2544 2547 4 3360 3364 3368 3372 3376 3380 3384 3388 3392 3396 5 4200 4205 4210 4215 4220 4225 4230 4235 4240 4245 6 5040 5046 5052 5058 5064 5070 5076 5082 5088 5094 7 5880 5887 5894 5901 5908 5915 5922 5929 5936 5943 8 6720 6728 6736 6744 6752 6760 6768 6776 6784 6792 9 7560 7569 7578 7587 7596 7605 7614 7623 7632 7641 XVIII. MULTIPLES. 65 850 851 852 853 854 855 856 857 858 859 1 2 3 850 1700 2550 851 1702 2553 852 1704 2556 853 1706 2559 854 1708 2562 855 1710 2565 856 1712 2568 857 1714 2571 858 1716 2574 859 1718 2577 4 5 6 3400 4250 5100 3404 4255 5106 3408 4260 5112 3412 4265 5118 3416 4270 5124 3420 4275 5130 3424 4280 5136 3428 4285 5142 3432 4290 5148 3436 4295 5154 7 8 9 5950 6800 7650 5957 6808 7659 5964 6816 7668 5971 6824 7677 5978 6832 7686 5985 6840 7695 5992 6848 7704 5999 6856 7713 6006 6864 7722 6013 6872 7731- 860 861 862 863 864 865 866 867 868 869 1 2 3 860 1720 2580 861 1722 2583 862 1724 2586 863 1726 2589 864 1728 2592 865 1730 2595 866 1732 2598 867 1734 2601 868 1736 2604 869 1738 2607 4 5 6 3440 4300 5160 3444 4305 5166 3448 4310 5172 3452 4315 5178 3456 4320 5184 3460 4325 5190 3464 4330 5196 3468 4335 5202 3472 4340 5208 3476 4345 5214 7 8 9 6020 6880 7740 6027 6888 7749 6034 6896 7758 6041 6904 7767 6048 6912 7776 6055 6920 7785 6062 6928 7794 6069 6936 7803 6076 6944 7812 6083 6952 7821 870 871 872 873 874 875 876 877 878 879 1 2 3 870 1740 2610 871 1742 2613 872 1744 2616 873 1746 2619 874 1748 2622 875 1750 2625 876 1752 2628 877 1754 2631 878 1756 2634 879 1758 2637 4 5 6 3480 4350 5220 3484 4355 5226 3488 4360 5232 3492 4365 5238 3496 4370 5244 3500 4375 5250 3504 4380 5256 3508 4385 5262 3512 4390 5268 3516 4395 5274 7 8 9 6090 6960 7830 6097 6968 7839 6104 6976 7848 6111 6984 7857 6118 6992 7866 6125 7000 7875 6132 7008 7884 6139 7016 7893 6146 7024 7902 6153 7032 7911 880 881 882 883 884 885 886 887 888 889 1 2 3 880 1760 2640 881 1762 2643 882 1764 2646 883 1766 2649 884 1768 2652 885 1770 2655 886 1772 2658 887 1774 2661 888 1776 2664 889 1778 2667 4 5 6 3520 4400 5280 3524 4405 5286 3528 4410 5292 3532 4415 5298 3536 4420 5304 3540 4425 5310 3544 4430 5316 3548 4435 5322 3552 4440 5328 3556 4445 5334 7 8 9 6160 7040 7920 6167 7048 7929 6174 7056 7938 6181 7064 7947 6188 7072 7956 6195 7080 7965 6202 7088 7974 6209 7096 7983 6216 7104 7992 6223 7112 8001 890 891 892 893 894 895 896 897 898 899 1 2 3 890 1780 2670 891 1782 2673 892 1784 2676 893 1786 2679 894 1788 2682 895 1790 2685 896 1792 2688 897 1794 2691 898 1796 2694 899 1798 2697 4 5 6 3560 4450 5340 3564 4455 5346 3568 4460 5352 3572 4465 5358 3576 4470 5364 3580 4475 5370 3584 4480 5376 3588 4485 5382 3592 4490 5388 3596 4495 5394 7 8 9 6230 7120 8010 6237 7128 8019 6244 7136 8028 6251 7144 8037 6258 7152 8046 6265 7160 8055 6272 7168 8064 6279 7176 8073 6286 7184 8082 6293 7192 8091 Gfi XVIII. MULTIPLES. 900 901 902 903 904 905 906 907 908 909 1 900 901 902 903 904 905 906 907 908 909 2 ISOO 1802 1804 1806 1808 1810 1812 1814 1816 1818 3 2700 2703 2706 2709 2712 2715 2718 2721 2724 2727 4 3600 3604 3608 3612 3616 3620 3624 3628 3632 3636 5 4500 4505 4510 4515 •1520 4525 4530 4535 4540 4545 6 5400 5406 5412 5418 5424 5430 5436 5442 54-18 5454 7 6300 6307 6314 6321 6328 6335 6342 6349 6356 6363 8 7200 7208 7216 7224 7232 7240 7248 7256 7264 7272 9 SlOO 8109 SllS 8127 8136 8145 8154 8163 8172 8181 910 911 912 913 914 915 916 917 918 919 1 910 911 912 913 914 915 916 917 ~918" 919 2 1S20 1822 1824 1826 1828 1830 1832 1834 1836 1838 3 2730 2733 2736 2739 2742 2745 2748 2751 2754 2757 4 3640 3644 3648 3652 3656 3660 3664 3668 3672 3676 5 4550 4555 4560 4565 4570 4575 4580 4585 4590 4595 6 5460 5466 5472 5478 5484 5490 5496 5502 5508 5514 7 6370 6377 6384 6391 6398 6405 6412 6419 6426 6433 8 72S0 7288 7296 7304 7312 7320 7328 7336 7344 7352 9 8190 8199 8208 8217 8226 8235 8244 8253 8262 8271 920 921 922 923 924 925 926 927 928 929 1 920 921 922 923 924 925 926 927 928 929 2 1840 1842 1844 1846 1848 1850 1852 1854 1856 1858 3 2760 2763 2766 2769 2772 2775 2778 2781 2784 2787 4 3680 3684 3688 3692 3696 3700 3704 3708 3712 3716 5 4600 4605 4610 4615 4620 4625 4630 4635 4640 4645 6 5520 5526 5532 5538 5544 5550 5556 5562 5568 5574 7 6440 6447 6454 6461 6468 6475 6482 6489 6496 6503 8 7360 7368 7376 7384 7392 7400 7-1 OS 7416 7424 7432 9 8280 8289 8298 8307 8316 8325 8334 8343 8352 8361 930 931 932 933 934 935 936 937 938 939 1 930 931 932 933 934 935 936 937 938 939 2 1860 1862 1864 1866 1868 1870 1872 1874 1876 1878 3 2790 2793 2796 2799 2802 2805 2808 2811 2814 2817 4 3720 3724 3728 3732 3736 3740 37-14 3748 3752 3756 5 4650 4655 4660 4665 4670 4675 4680 4685 4690 4695 6 5580 5586 5592 5598 5604 5610 5616 5622 5628 5634 7 6510 6517 6524 6531 6538 6545 6552 6559 6566 6573 8 7440 7448 7456 7464 7472 7480 7488 7496 7504 7512 9 8370 8379 8388 8397 8406 8415 8424 8433 8442 8451 940 941 942 943 944 945 946 947 948 949 1 940 941 942 943 944 945 946 947 948 949 2 1880 1882 1884 1886 ISSS 1890 1892 1894 1896 1898 3 2820 2823 2826 2829 2832 2835 2838 2841 2844 2847 4 3760 3764 3768 3772 3776 3 780 3784 3788 3792 3796 5 4700 4705 4710 4715 4720 4725 4730 4735 4740 4745 6 5640 5646 5652 5658 5664 5670 5676 5682 5688 5694 7 6580 6587 6594 6601 6608 6615 6622 6629 6636 6643 8 7520 7528 7536 7544 7552 7560 7568 7576 7584 7592 9 8460 8469 8478 8487 84(1^ 8505 8514 8523 8532 8541 XVIII. MULTIPLES. 67 950 951 952 953 954 955 956 957 958 959 1 950 951 952 953 954 955 956 957 958 959 2 1900 1902 1904 1906 1908 1910 1912 1914 1916 1918 3 2850 2853 2856 2859 2862 2865 2868 2871 2874 2877 4 3800 3804 3808 3812 3816 3820 3824 3828 3832 3836 5 4750 4755 4760 4765 4770 4775 4780 4785 4790 4795 6 5700 5706 5712 571S 5724 5730 5736 5742 5748 5754 7 6650 6657 6664 6671 6678 6685 6692 6699 6706 6713 8 7600 7608 7616 7624 7632 7640 7648 7656 7664 7672 9 8550 8559 8568 8577 8586 8595 8604 8613 8622 8631 960 961 962 963 964 965 966 967 968 969 1 960 961 962 963 964 965 966 967 968 969 2 1920 1922 1924 1926 1928 1930 1932 1934 1936 1938 3 2880 2883 2886 2889 2892 2895 2898 2901 2904 2907 4 3840 3844 3848 3852 3856 3860 3864 3868 3872 3876 5 4800 4805 4810 4815 4820 4825 4830 483 5 4840 4845 6 5760 5766 5772 5778 5784 5790 5796 5802 5808 5814 7 6720 6727 6734 6741 6748 6755 6762 6769 6776 6783 8 7680 7688 7696 7704 7712 7720 7728 7736 7744 7752 9 86+0 8649 8658 8667 8676 8685 8694 8703 8712 8721 970 971 972 973 974 975 976 977 978 979 1 970 971 972 973 974 975 976 977 978 979 2 1940 1942 1944 1946 1948 1950 1952 1954 1956 1958 3 2910 2913 2916 2919 2922 2925 2928 2931 2934 2937 4 3880 3884 3888 3892 3896 3900 3904 3908 3912 3916 5 4850 4855 4860 4865 4870 4875 4SS0 4885 4890 4895 6 5820 5826 5832 5838 5844 5850 5856 5862 5868 5874 7 6790 6797 6804 6811 6818 6825 6832 6839 6846 6853 8 7760 7768 7776 7784 7792 7800 7808 7816 7824 7832 9 8730 8739 8748 8757 8766 8775 8784 8793 8802 8811 980 981 982 983 984 985 986 987 988 989 1 980 981 982 983 984 985 986 987 ~98S" 989 2 1960 1962 1964 1966 1968 1970 1972 1974 1976 1978 3 2940 2943 2946 2949 2952 2955 2958 2961 2964 2967 4 3920 3924 3928 3932 3936 3940 3944 3948 3952 3956 5 4900 4905 4910 4915 4920 4925 4930 4935 4940 4945 6 5880 5886 5892 5898 5904 5910 5916 5922 5928 5934 7 6860 6867 6874 6881 6888 6895 6902 6909 6916 6923 8 7840 7848 7856 7864 7872 7880 7888 7896 7904 7912 9 8820 8829 8S3S 8847 8856 8865 8874 8883 8892 8901 990 991 992 993 994 995 996 997 998 999 1 990 991 992 993 994 995 996 997 998 999 2 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 3 2970 2973 2976 2979 2982 2985 2988 2991 2994 2997 4 3960 3964 3968 3972 3976 3980 3984 3988 3992 3996 5 4950 4955 4960 4965 4970 4975 4980 4985 4990 4995 6 5940 5946 5952 5958 5964 5970 5976 5982 5988 5994 7 6930 6937 6944 6951 6958 6965 6972 6979 6986 6993 8 7920 7928 7936 7944 7952 7960 7968 7976 7984 7992 9 8910 8919 8928 8937 8946 8955 8964 8973 8982 8991 68 XIX. CIROUMFERENOE OF OIROLE nn n 12 3 4 5 6 7 8 9 PP 1.0 3.142 3.173 3.204 3.236 3.267 3.299 3.330 3.362 3.393 3.424 32 1.1 1.2 1.3 3.456 3.770 4.0S4 3.4S7 3.519 3.550 3.801 3.833 3.864 4.115 4.147 4.178 3.581 3.613 3.644 3.896 3.927 3.958 4.210 4.241 4.273 3.676 3.707 3.738 3.990 4.021 4.053 4.304 4.335 4.367 3 6 10 1.4 4.39S 1.5 4.712 1.6 5.027 4.430 4.461 4.492 4.744 4.775 4.807 5.058 5.089 5.121 4.524 4.555 4.587 4.838 4.869 4.901 5.152 5.184 5.215 4.618 4.650 4.681 4.932 4.964 4.995 5.246 5.278 5.309 13 16 19 1.7 5.341 l.S 5.65i 1.9 1 5.969 5.372 5.404 5.435 5.686 5.718 5.749 6.000 6.032 6.063 5.466 5.498 5.529 5.781 5.812 5.843 6.095 6.126 6.158 5.561 5.592 5.623 5.875 5.906 5.938 6.189 6.220 6.252 22 26 29 2.0 i 6.2S3 6.315 6.346 6.377 6.409 6.440 6.472 6.503 6.535 6.566 31 2.1 2.2 2.3 6.597 6.912 7.226 6.629 6.660 6.692 6.943 6.974 7.006 7.257 7.2SS 7.320 6.723 6.754 6.786 7.037 7.069 7.100 7.351 7.383 7.414 6.817 6.849 6.880 7.131 7.163 7.194 7.446 7.477 7.508 3 6 9 2.4 2.5 2.6 7.540 7.S54 8.168 7.571 7.603 7.634 7.885 7.917 7.948 8.200 8.231 8.262 7.665 7.697 7.728 7.980 8.011 8.042 8.294 8.325 8.357 7.760 7.791 7.823 8.074 8.105 8.137 8.388 8.419 8.451 12 16 19 2.7 2.8 2.9 8.482 8.796 9.111 8.514 8.545 8.577 8.828 8.859 8.891 9.142 9.173 9.205 8.608 8.639 8.671 8.922 8.954 8.985 9.236 9.268 9.299 8.702 8.734 8.765 9.016 9.048 9.079 9.331 9.362 9.393 22 25 28 3.0 9.425 9.456 9.488 9.519 9.550 9.582 9.613 9.645 9.676 9.708 4 3.1 3.2 3.3 9.739 10.05 10.37 9.770 9.802 9.833 10.08 10.12 10.15 10.40 10.43 10.46 9.865 9.896 9.927 10.18 10.21 10.24 10.49 10.52 10.56 9.959 9.990 10.02 10.27 10.30 10.34 10.59 10.62 10.65 1 1 3.4 3.5 3.6 10.68 11.00 11.31 10.71 10.74 10.78 11.03 11.06 11.09 11.34 11.37 11.40 10.81 10.84 10.87 11.12 11.15 11.18 11.44 11.47 11.50 10.90 10.93 10.96 11.22 11.25 11.28 11.53 11.56 11.59 2 2 2 3.7 3.8 3.9 11.62 11.94 12.25 11.66 11.69 11.72 11.97 12.00 12.03 12.28 12.32 12.35 11.75 11.78 11.81 12.06 12.10 12.13 12.38 12.41 12.44 11.84 11.88 11.91 12.16 12.19 12.22 12.47 12.50 12.53 3 3 4 4.0 12.57 12.60 12.63 12.66 12.69 12.72 12.75 12.79 12.82 12.85 3 4.1 4.2 4.3 12.88 13.19 13.51 12.91 12.94 12.97 13.23 13.26 13.29 13.54 13.57 13.60 13.01 13.04 13.07 13.32 13.35 13.38 13.63 13.67 13.70 13.10 13.13 13.16 13.41 13.45 13.48 13.73 13.76 13.79 1 1 4.4 4.5 4.6 13.82 14.14 14.45 13.85 13.89 13.92 14.17 14.20 14.23 14.48 14.51 14.55 13.95 13.98 14.01 14.26 14.29 14'33 14.58 14.61 14.64 14.04 14.07 14.11 14.36 14.39 14.42 14.67 14.70 14.73 1 2 2 4.7 4.8 4.9 5.0 14.77 15.08 15.39 14.80 14.83 14.86 15.11 15.14 15.17 15.43 15.46 15.49 14.89 14.92 14.95 15.21 15.24 15.27 15.52 15.55 15.58 14.99 15.02 15.05 15.30 15.33 15.36 15.61 15.65 15.68 2 2 3 15.71 15.74 15.77 15.80 15.83 15.87 15.90 15.93 15.96 15.99 5.1 5.2 5.3 16.02 16.34 16.65 16.05 16.08 16.12 16.37 16.40 16,43 16.68 16.71 16.74 16.15 16.18 16.21 16.46 16.49 16.52 16.78 16.81 16.84 16.24 16.27 16.30 16.56 16.59 16.62 16.87 16.90 16.93 5.4 5.0 5.6 16.96 17.28 17.59 17.00 17.03 17.06 17.31 17.34 17.37 17.62 17.66 17.69 17.09 17.12 17.15 17.40 17.44 17.47 17.72 17.75 17.78 17.18 17.22 17.25 17.50 17.53 17.56 17.81 17.84 17.88 5.7 5.8 5.9 17.91 18.22 18.54 17.94 17.97 18.00 18.25 18.28 18.32 18.57 18.60 18.63 18.03 18.06 18.10 18.35 18.38 18.41 18.66 18.69 18.72 18.13 18.16 18.19 18.44 18.47 18.50 18.76 18.79 18.82 im 71 BEING DIAMETER. 69 n 6.0 12 3 4 5 6 7 8 9 PP 18.85 18.88 18.91 18.94 18.98 19.01 19.04 19.07 19.10 19.13 4 6.1 19.16 19.20 19.23 19.26 19.29 19.32 19.35 19.38 19.42 19.45 6.2 19.48 19.51 19.54 19.57 19.60 19.63 19.67 19.70 19.73 19.76 1 6.3 19.79 19.82 19.85 19.89 19.92 19.95 19.98 20.01 20.04 20.07 1 6.4 20.11 20.14 20.17 20.20 20.23 20.26 20.29 20.33 20.36 20.39 2 6.5 20.42 20.45 20.48 20.51 20.55 20.58 20.61 20.64 20.67 20.70 2 6.6 20.73 20.77 20.80 20.83 20.86 20.89 20.92 20.95 20.99 21.02 2 6.7 21.05 21.08 21.11 21.14 21.17 21.21 21.24 21.27 21.30 21.33 3 6.8 21.36 21.39 21.43 21.46 21.49 21.52 21.55 21.58 21.61 21.65 3 6.9 21.68 21.71 21.74 21.77 21.80 21.83 21.87 21.90 21.93 21.96 4 7.0 21.99 22.02 22.05 22.09 22.12 22.15 22.18 22.21 22.24 22.27 7.1 22.31 22.34 22.37 22.40 22.43 22.46 22.49 22.53 22.56 22.59 7.2 22.62 22.65 22.68 22.71 22.75 22.78 22.81 22.84 22.87 22.90 7.3 22.93 22.97 23.00 23.03 23.06 23.09 23.12 23.15 23.18 23.22 7.4 23.25 23.28 23.31 23.34 23.37 23.40 23.44 23.47 23.50 23.53 7.5 23.56 23.59 23.62 23.66 23.69 23.72 23.75 23.78 23.81 23.84 7.6 23.88 23.91 23.94 23.97 24.00 24.03 24.06 24.10 24.13 24.16 7.7 24.19 24.22 24.25 24.28 24.32 24.35 24.38 24.41 24.44 24.47 7.8 24.50 24.54 24.57 24.60 24.63 24.66 24.69 24.72 24.76 24.79 7.9 24.82 24.85 24.88 24.91 24.94 24.98 25.01 25.04 25.07 25.10 8.0 25.13 25.16 25.20 25.23 25.26 25.29 25.32 25.35 25.38 25.42 T 8.1 25.45 25.48 25.51 25.54 25.57 25.60 25.64 25.67 25.70 25.73 8.2 25.76 25.79 25.82 25.86 25.89 25.92 25.95 25.98 26.01 26.04 1 8.3 26.08 26.11 26.14 26.17 26.20 26.23 26.26 26.30 26.33 26.36 1 8.4 26.39 26.42 26.45 26.48 26.52 26.55 26.58 26.61 26.64 26.67 1 8.5 26.70 26.73 26.77 26.80 26.83 26.86 26.89 26.92 26.95 26.99 2 8.6 27.02 27.05 27.08 27.11 27.14 27.17 27.21 27.24 27.27 27.30 2 8.7 27.33 27.36 27.39 27.43 27.46 27.49 27.52 27.55 27.58 27.61 2 8;8 27.65 27.68 27.71 27.74 27.77 27.80 27.83 27.87 27.90 27.93 2 8.9 27.96 27.99 28.02 28.05 28.09 28.12 28.15 28.18 28.21 28.24 3 9.0 28.27 28.31 28.34 28.37 28.40 28.43 28.46 28.49 28.53 28.56 9.1 28.59 28.62 28.65 28.68 28.71 28.75 28.78 28.81 28.84 28.87 9.2 28.90 28.93 28.97 29.00 29.03 29.06 29.09 29.12 29.15 29.19 9.3 29.22 29.25 29.28 29.31 29.34 29.37 29.41 29.44 29.47 29.50 9.4 29.53 29.56 29.59 29.63 29.66 29.69 29.72 29.75 29.78 29.81 9.5 29.85 29.88 29.91 29.94 29.97 30.00 30.03 30.07 30.10 30.13 9.6 30.16 30.19 30.22 30.25 30.28 30.32 30.35 30.38 30.41 30.44 9.7 30.47 30.50 30.54 30.57 30.60 30.63 30.66 30.69 30.72 30.76 9.8 30.79 30.82 30.85 30.88 30,91 30.94 30.98 31.01 31.04 31.07 9.9 31.10 31.13 31.16 31.20 31.23 31.26 31.29 31.32 31.35 31.38 value log recip. yfo 0.0175 2.2419 57.2958 TT 3.1416 0.4971 0.3183 ■giro' 0.0087 3.9408 114.5916 27r 6.2832 0.7982 0.1592 TT- 9.8696 0.9943 0.1013 47r 12.5664 1.0992 0.0796 Vn 1.7725 0.2486 0.5642 7r 2" 1.5708 0.1961 0.6366 ir^ 31.006 1.4914 0.0323 TT = 3.1415926536 f 0.7854 1.8951 1.2732 r. 1.4646 0.1657 0.6828 f 0.5236 1.7190 1.9099 Vi 0.8060 1.9063 1.2407 47r 3 4.1888 0.6221 0.2387 W 1.6120 0.2074 0.6204 XX. AEEA or OIKGLE n^ n 1 2 3 4 5 G 7 8 9 d 1.0 0.785 1 0.801 0.817 0.833 0.849 0.866 0.882 0.899 0.916 0.933 17 1.1 1.2 1.3 0.950 1.131 1.327 0.968 0.985 1.003 1.1.50 1.169 1.18S 1.348 1.368 1.389 1.021 1.039 1.057 1.208 1.227 1.247 1.410 1.431 1.453 1.075 1.094 1.112 1.267 1.287 1.307 1.474 1.496 1.517 19 20 22 1.4 1.5 l.U 1.539 1.767 2.011 1.561 1..5S4 1.606 1.791 1.815 1-839 2.036 2.061 2.087 1.629 1.651 1.674 1.863 1.887 1.911 2.112 2.138 2.164 1.697 1.720 1.744 1.936 1.961 1.986 2.190-2.217 2.243 23 25 27 1.7 1.8 1.9 2.270 2.545 2.835 2.297 2.324 2.351 2.573 2.602 2.6.30 2.865 2.895 2.926 2.378 2.405 2.433 2.6.59 2.6SS 2.717 2.956 2.986 3.017 2.461 2.488 2.516 2.746 2.776 2.806 3.048 3.079 3.110 29 29 32 2.0 '3.142 3.173 3.205 3.237 3.269 3.301 3.333 3.365 3.398 3.431 33 2.1 13.464 2.2 '3.801 2.3 !4.15i 3.497 3.530 3.563 3.8.36 3.871 3.906 4.191 4.227 4.264 3.597 3.631 3.664 3.941 3.976 4.011 4.301 4.337 4.374 3.698 3.733 3.767 4.047 4.083 4.119 4.412 4.449 4.486 34 36 38 2.4 2.5 2.G 4.524 4.909 5.309 4.562 4.600 4.638 4.948 4.9SS 5.027 5.350 5.391 5.433 4.676 4.714 4.753 5.067 5.107 5.147 5.474 5.515 5.557 4.792 4.831 4.870 5.187 5.228 5.269 5.599 5.641 5.683 39 40 43 2.7 2.8 2.9 5.726 6.158 6.605 5.76S 5.811 5.S53 6.202 6.246 6.290 6.651 6.697 6.743 5.896 5.940 5.983 6.335 6.379 6.424 6.789 6.835 6.881 6.026 6.070 6.114 6.469 6.514 6.560 6.928 6.975 7.022 44 45 47 3.0 7.069 7.116 7.163 7.211 7.258 7.306 7.354 7.402 7.451 7.499 49 3.1 3.2 3.3 7.548 8.042 8.553 7.596 7.645 7.694 8.093 8.143 8.194 8.605 8.657 8.709 7744 7.793 7.843 8.245 8.296 8.347 8.762 8.814 8.867 7.892 7.942 7.992 8.398 8.450 8..S01 8.920 8.973 9.026 50 52 53 3.4 3.5 3.6 9.079 9.621 10.18 9.133 9.186 9.240 9.676 9.731 9.787 10.24 10.29 10.35 9.294 9.348 9.402 9.842 9.898 9.954 10.41 10.46 10.52 9.457 9.511 9.566 10.01 10.07 10.12 10.58 10.64 10.69 55 6 6 3.7 ! 10.75 3.8 11.34 3.9 11.95 10.81 10.87 10.93 11.40 11.46 11.52 12.01 12.07 12.13 10.99 11.04 11.10 11.58 11.64 11.70 12.19 12.25 12.32 11.16 11.22 11.28 11.76 11.82 11.88 12.38 12.44 12.50 6 7 7 4.0 1 12.57 12.63 12.69 12.76 12.82 12.88 12.95 13.01 13.07 13.14 6 4.1 4.2 4.3 13.20 13.85 14.52 13.27 13.33 13.40 13.92 13.99 14.05 14.59 14.66 14.73 13.46 13.53 13.,59 14.12 14.19 14.25 14.79 14.86 14.93 13.66 13.72 13.79 14.32 14.39 14.45 15.00 15.07 15.14 6 7 7 4.4 4.5 4.G 15.21 15.90 16.62 15.27 15.34 15.41 15.98 16.05 16.12 16.69 16.76 16.84 15.48 15.55 15.62 16.19 16.26 16.33 16.91 16.98 17.06 15.69 15.76 15.83 16.40 16.47 16.55 17.13 17.20 17.28 7 7 7 4.7 j 17.35 4.8 ' 18.10 4.9 i 18.86 17.42 17..S0 17.57 18.17 18.25 18.32 18.93 19.01 19.09 17.65 17.72 17.80 18.40 18.47 18.55 19.17 19.24 19.32 17.87 17.95 18.02 18.63 18.70 18.78 19.40 19.48 19.56 8 8 7 5.0 1 19.63 19.71 19.79 19.87 19.95 20.03 20.11 20.19 20.27 20.35 8 5.1 5.2 5.3 20.43 21.24 22.06 20.51 20.59 20.67 21.32 21.40 21.^8 22.15 22.23 22.31 20.75 20.83 20.91 21.57 21.65 21.73 22.40 22.48 22.56 20.99 21.07 21.16 21.81 21.90 21.98 22.65 22.73 22.82 8 8 8 5.4 5.5 5.G 22.90 23.76 24.63 22.99 23.07 23.16 23.84 23.93 24.02 24.72 24.81 24.89 23.24 23.33 23.41 24.11 24.19 24.2S 24.98 25.07 25.16 23.50 23.59 23.67 24.37 24.45 24.54 25.25 25.34 25.43 9 9 9 5.7 =25.52 5.S 26.42 5.9 27.34 25.61 25.70 25.79 26.51 26.60 26.69 27.43 27.53 27.62 25.88 25.97 26.06 26.79 26. S8 26.97 27.71 27.81 27.90 26.15 26.24 26.33 27.06 27.15 27.25 27.99 28.09 28.18 9 9 9 n BEING DIAMETE3. 71 n 3 4 5 8 6.0 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.y 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 8.0 8.1 8.2 8.3 8.4 8.5 8.6 8.7 28.27 29.22 30.19 31.17 32.17 33.18 34.21 35.26 36.32 37.39 28.37 29.32 30.29 31.27 32.27 33.29 34.32 3.S.36 36.42 37.50 28.46 28.56 29.42 29.51 30.39 30.48 31.37 31.47 32.37 32.47 33.39 33.49 34.42 34.52 35.47 35.57 36.53 36.64 37.61 37.72 28.65 29.61 30.58 31.57 32.57 33.59 34.63 35.68 36.75 37.83 28.75 28.84 29.71 29.80 30.68 30.78 31.67 31.77 32.67 32.78 33.70 33.80 34.73 34.84 35.78 35.89 36.85 36.96 37.94 38.05 28.94 29.03 29.90 30.00 30.88 30.97 31.87 31.97 32.88 32.98 33.90 34.00 34.94 35.05 36.00 36.10 37.07 37.18 38.16 38.26 29.13 9 30.09 , 10 31.07 32.07 33.08 ! 10 34.11 35.15 36.21 37.28 38.37 38.48 39.59 40.72 41.85 43.01 44.18 45.36 46.57 47.78 49.02 38.59 38.70 39.70 39.82 40.83 40.94 41.97 42.08 43.12 43.24 44.30 44.41 45.48 45.60 46.69 46.81 47.91 48.03 49.14 49.27 38.82 39.93 41.06 42.20 43.36 44.53 45.72 46.93 48.15 49.39 50.27 51.53 52.81 54.11 55.42 56.75 58.09 59.45 60.82 62.21 50.39 51.66 52.94 54.24 55.55 56.88 58.22 59.58 60.96 62.35 50.52 50.64 51.78 51.91 53.07 53.20 54.37 54.50 55.68 55.81 57.01 57.15 58.36 58.49 59.72 59.86 61.10 61.24 62.49 62.63 38.93 40.04 41.17 42.31 43.47 44.65 45.84 47.05 48.27 49.51 39.04 39.15 40.15 40.26 41.28 41.40 42.43 42.54 43.59 43.71 44.77 44.89 45.96 46.08 47.17 47.29 48.40 48.52 49.64 49.76 50.77 52.04 53.33 54.63 55.95 57.28 58.63 59.99 61.38 62.77 50.90 51.02 52.17 52.30 53.46 53.59 54.76 54.89 56.08 56.21 57.41 57.55 58.77 58.90 60.13 60.27 61.51 61.65 62.91 63.05 39.26 40.38 41.51 42.66 43.83 45.01 46.20 47.42 48.65 49.89 39.37 40.49 41.62 42.78 43.94 45.13 46.32 47.54 48.77 50.01 39.48 11 40.60 ' 41.74 42.89 44.06 45.25 46.45 47.66 48.89 ; 13 50.14 ; 13 51.15 52.42 53.72 55.02 56.35 57.68 59.04 60.41 61.79 63.19 51.28 52.55 53.85 55.15 56.48 57.82 59.17 60.55 61.93 63.33 51.40! 13 52.68 13 53.98 13 55.29 i 13 56.61 14 57.95 I 14 59.31 I 14 60.68 14 62.07 14 63.48 14 9.0 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 63.62 65.04 66.48 67.93 69.40 70.88 72.38 73.90 75.43 76.98 63.76 65.18 66.62 68.08 69.55 71.03 72.53 74.05 75.58 77.13 63.90 64.04 65.33 65.47 66.77 66.91 68.22 68.37 69.69 69.84 71.18 71.33 72.68 72.84 74.20 74.36 75.74 75.89 77.29 77.44 64.18 65.61 67.06 68.51 69.99 71.48 72.99 74.51 76.05 77.60 64.33 64.47 65.76 65.90 67.20 67.35 68.66 68.81 70.14 70.29 71.63 71.78 73.14 73.29 74.66 74.82 76.20 76.36 77.76 77.91 64.61 66.04 67.49 68.96 70.44 71.93 73.44 74.97 76.51 7S.07 64.75 64.90 66.19 66.33 67.64 67.78 69.10 69.25 70.58 70.73 72.08 72.23 73.59 73.75 75.12 75.28 76.67 76.82 78.23 78.38 32nds .03125 .0625 .09375 .125 .15625 .1875 .21875 .25 .28125 36ths 10 .02(7) .0(5) .08(3) 11 12 13 •(1) .13(8) .1(6) 14 15 16 .19(4) .(2) .25 17 18 19 3125 ,34375 ,375 ,40625 .4375 .46875 .5 .53125 .5625 .59375 .2(7) 20 .30(5) .(3) .36(1) 21 22 23 .3(8) 24 25 26 .47(2) .5 27 28 .52(7) 29 .625 .(5) 30 .65625 ..58(3) 31 .6875 .6(1) 32 .71875 .63(8) 33 .75 .(6) 34 .78125 .69(4) 35 .8125 •7(2) 36 .84375 .75 .875 .(7) .90625 .80(5) .9375 .8(3) .96875 .86(1) .(8) .91(6) .9(4) .97(2) XXI. CONTENT OF SPHEKE il3 n 1 2 3 j 4 5 6 7 8 9 d 1.0 .5236 .5395 .5556 .5722 .5890 .6061 .6236 .6414 .6596 .6781 188 1.1 1.2 l.*3 .6969 .9048 1.1.50 .7161 .7356 .7555 .9276 .9508 .974.? 1.177 1.204 1.232 .7757 .7963 .8173 .9983 1.023 1.047 1.260 1.288 1.317 .8386 .8603 .8823 1.073 1.098 1.124 1.346 1.376 1.406 225 26 31 1.4 '1.437 1.5 1.767 l.G 2.145 1.468 1.499 1.531 1.803 1.8.39 1.875 2.185 2.226 2.268 1.563 1.596 1.630 1.912 1.9.50 1.988 2.310 2.352 2.395 1.663 1.697 1.732 2.026 2.065 2.105 2.439 2.483 2.527 35 40 45 1.7 2.572 1.8 13.0.54 1.9 3.591 2.618 2.664 2.711 3.105 3.157 3.209 3.648 3.706 3.764 2.758 2.806 2.855 3.262 3.315 3.369 3.823 3.882 3.942 2.903 2.953 3.003 3.424 3.479 3..535 4.003 4.064 4.126 51 56 63 2.0 4.189 4.252 4.316 4.380 4.445 4.511 4.577 4.644 4.712 4.780 69 2.1 4.849 2.2 , 5.575 2.3 6.371 4.919 4.989 5.060 5.652 5.729 5.806 6.454 6.538 6.623 5.131 5.204 5.277 5.885 5.964 6.044 6.709 6.795 6.882 5.350 5.425 5.500 6.125 6.206 6.288 6.970 7.059 7.148 75 83 90 2.4 7.238 2.5 8.1 SI 2.0 ! 9.203 7.329 7.421 7.513 8.280 8.379 8.479 9.309 9.417 9.525 7.606 7.700 7.795 8.580 8.682 8.785 9.634 9.744 9.855 7.890 7.986 8.083 8.888 8.992 9.097 9.966 10.08 10.19 98 106 12 2.7 10.31 2..S' 11.49 2.9 12.77 10.42 10.54 10.65 11.62 11.74 11.87 12.90 13.04 13.17 10.77 10.89 11.01 11.99 12.12 12.25 13.31 13.44 13.58 11.13 11.25 11.37 12.38 12.51 12.64 13.72 13.86 14.00 12 13 14 3.0 1 14.14 14.28 14.42 14.57 14.71 14.86 15.00 15.15 15.30 15.45 15 3.1 ' 15.60 3.2 17.16 3.3 j 18.82 15.75 15.90 16.06 17.32 17.48 17.64 18.99 19.16 19.33 16.21 16.37 16.52 17.81 17.97 18.14 19.51 19.68 19.86 16.68 16.84 17.00 18.31 18.48 18.65 20.04 20.22 20.40 16 17 18 3.4 20.58 3.5 1 22.45 3.6 1 24.43 20.76 20.94 21.13 22.64 22.84 23.03 24.63 24.84 25.04 21.31 21.50 21.69 23.23 23.43 23.62 25.25 25.46 25.67 21.88 22.07 22.26 23.82 24.02 24.23 25.88 26.09 26.31 19 20 21 3.7 3.8 3.9 26.52 28.73 31.06 26.74 26.95 27.17 28.96 29.19 29.42 31.30 31.54 31.78 27.39 27.61 27.83 29.65 29.88 30.11 32.02 32.27 32.52 28.06 28.28 28.50 30.35 30.58 30.82 32.76 33.01 33.26 23 24 25 4.0 33.51 33.76 34.02 34.27 34.53 34.78 35.04 35.30 35.56 35.82 27 4.1 4.2 4.3 36.09 38.79 41.63 36.35 36.62 36.88 39.07 39.35 39.63 41.92 42.21 42.51 37.15 37.42 37.69 39.91 40.19 40.48 42.80 43.10 43.40 37.97 38.24 38.52 40.76 41.05 41.34 43.70 44.00 44.30 27 29 30 4.4 4.5 4.G 44.60 47.71 50.97 44.91 45.21 45.52 48.03 48.35 48.67 51.30 51.63 51.97 45.83 46.14 46.45 49.00 49.32 49.65 52.31 52.65 52.99 46.77 47.08 47.40 49.97 50.30 50.63 53.33 53.67 54.02 31 34 34 4.7 4.8 4.9 54.36 57.91 61.60 54.71 55.06 55.41 58.27 58.63 59.00 61.98 62.36 62.74 55.76 56.12 56.47 59.37 59.73 60.10 63.12 63.51 63.89 56.83 57.19 57.54 60.48 60.85 61.22 64.28 64.67 65.06 37 38 39 5.0 65.45 65.84 66.24 66.64 67.03 67.43 67.83 68.24 68.64 69.05 41 5.1 5.2 5.3 69.46 73.62 77.95 69.87 70.28 70.69 74.05 74.47 74.90 78.39 78.84 79.28 71.10 71.52 71.94 75.33 75.77 76.20 79.73 80.18 80.63 72.36 72.78 73.20 76.64 77.07 77.51 81.08 81.54 81.99 42 44 46 5.4 5.5 5.6 82.45 87.11 91.95 82.91 83.37 83.83 87.59 88.07 88.55 92.45 92.94 93.44 84.29 84.76 85.23 89.03 89.51 90.00 93.94 94.44 94.94 85.70 86.17 86.64 90.48 90.97 91.46 95.44 95.95 96.46 47 49 51 5.7 ' 96.97 5.8 102.2 5.9 107.5 97.48 97.99 98.51 102.7 103.2 103.8 108.1 108.6 109.2 99.02 99.54 100.1 104.3 104.8 105.4 109.7 110.3 110.9 100.6 101.1 101.6 105.9 106.4 107.0 111.4 112.0 112.5 6 5 6 ^^3 n BEING DIAMETER. 73 6.0 .1 6.2 6.3 J.4 6.5 6.6 6.7 6.8 6.9 113.1 11S.8 124.8 130.9 137.3 143.8 150.5 157.5 164.6 172.0 113.7 119.4 125.4 131.5 137.9 144.5 151.2 158.2 165.4 172.8 114.2 114.8 120.0 120.6 126.0 126.6 132.2 132.8 138.5 139.2 145.1 145.8 151.9 152.6 158.9 159.6 166.1 166.8 173.5 174.3 115.4 121.2 127.2 133.4 139.8 146.5 153.3 160.3 167.6 175.0 115.9 116.5 121.8 122.4 127.8 128.4 134.1 134.7 140.5 141.2 147.1 147.8 154.0 154.7 161.0 161.7 168.3 169.0 175.8 176.5 117.1 123.0 129.1 135.3 141.8 14S.5 155.4 162.5 169.8 177.3 117.7 118.3 123.6 124.2 129.7 130.3 136.0 136.6 142.5 143.1 149.2 149.8 156.1 156.8 163.2 163.9 170.5 171.3 178.1 178.8 7.0 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 179.6 187.4 195.4 203.7 212.2 220.9 229.8 239.0 248.5 258.2 180.4 1SS.2 196.2 204.5 213.0 221.8 230.8 240.0 249.4 259.1 181.1 181.9 189.0 189.8 197.1 197.9 205.4 206.2 213.9 214.8 222.7 223.6 231.7 232.6 240.9 241.8 250.4 251.4 260.1 261.1 182.7 190.6 198.7 207.1 215.6 224.4 233.5 242.8 252.3 262.1 183.5 184.3 191.4 192.2 199.5 200.4 207.9 208.8 216.5 217.4 225.3 226.2 234.4 235.3 243.7 244.7 253.3 254.3 263.1 264.1 185.0 193.0 201.2 209.6 218.3 227.1 236.3 245.6 255.2 265.1 185.8 186.6 193.8 194.6 202.0 202.9 210.5 211.3 219.1 220.0 228.0 228.9 237.2 238.1 246.6 247.5 256.2 257.2 266.1 267.1 8.0 8.1 8.2 .3 A 8.7 8.8 268.1 278.3 288.7 299.4 310.3 321.6 333.0 344.8 356.8 369.1 269.1 279.3 289.8 300.5 311.4 322.7 334.2 346.0 358.0 370.4 270.1 271.1 280.3 281.4 290.8 291.9 301.6 302.6 312.6 313.7 323.8 325.0 335.4 336.5 347.2 348.4 359.3 360.5 371.6 372.9 272.1 282.4 292.9 303.7 314.8 326.1 337.7 349.6 361.7 374.1 273.1 274.2 283.4 284.5 294.0 295.1 304.8 305.9 315.9 317.0 327.3 328.4 338.9 340.1 3.50.8 352.0 362.9 364.2 375.4 376.6 275.2 285.5 296.2 307.0 318.2 329.6 341.2 353.2 365.4 377.9 276.2 286.6 297.2 308.1 319.3 330.7 342.4 354.4 366.6 379.2 277.2 287.6 298.3 309.2 320.4 331.9 343.6 355.6 367.9 380.4 9.0 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 381.7 394.6 407.7 421.2 434.9 448.9 463.2 477.9 492.8 508.0 383.0 395.9 409.1 422.5 436.3 450.3 464.7 479.4 494.3 509.6 384.3 385.5 397.2 398.5 410.4 411.7 423.9 425.2 437.7 439.1 451.8 453.2 466.1 467.6 480.8 482.3 495.8 497.3 511.1 512.7 386.8 388.1 399.8 401.1 413.1 414.4 426.6 428.0 440.5 441.9 454.6 456.0 469.1 470.5 483.8 485.3 •198.9 500.4 514.2 515.8 389.4 402.4 415.7 429.4 443.3 457.5 472.0 486.8 501.9 517.3 390.7 403.7 417.1 430.7 444.7 458.9 473.5 488.3 503.4 518.9 392.0 405.1 418.4 432.1 446.1 460.4 474.9 489.8 505.0 520.5 393.3 406.4 419.8 433.5 447.5 461.8 476.4 491.3 506.5 522.0 log n ! 0.0000 0.3010 0.7782 1.3802 2.0792 2.8573 3.7024 4.6055 5.5598 log 2» 0.3010 0.6021 0.9031 1.2041 1.5051 1.8062 2.1072 2.4082 2.7093 10 11 12 13 14 15 16 17 18 19 6.5598 7.6012 8.6803 9.7943 10.9404 12.1165 13.3206 14.5511 15.8063 17.0851 3.0103 20 3.3113 3.6124 3.9134 21 22 23 4.2144 4.5154 4.8165 24 25 26 5.1175 5.4185 5.7196 27 28 29 18.3861 19.7083 21.0508 22.4125 23.7927 25.1906 26.6056 28.0370 29.4841 30.9465 6.0206 6.3216 6.6227 6.9237 7.2247 7.5257 7.8268 8.1278 8.4288 8.7299 74 XXII. HYPERBOLIC LOGAEITHMS. logen n\ 1 2 3 4 5 6 7 8 9 d i.o'o.oooo 0100 0198 0296 0392 0488 0583 0677 0770 0862 91 1.1 1 0953 1.2 1S23 1.3 2624 1044 1133 1222 1906 1989 2070 2700 2776 2852 1310 1398 1484 2151 2231 2311 2927 3001 3075 1570 1655 1740 2390 2469 2546 3148 3221 3293 83 78 72 1.4 336i 1.5 i 4055 1.61 4700 3436 3507 3577 4121 4187 4253 4762 4824 4886 3646 3716 3784 4318 4383 4447 4947 5008 5068 3853 3920 3988 4511 4574 4637 5128 5188 5247 67 63 59 1.7 1.8 1.9 5306 5S78 6419 5365 5423 5481 5933 5988 6043 6471 6523 6575 5539 5596 5653 6098 6152 6206 6627 6678 6729 5710 5766 5822 6259 6313 6366 6780 6831 6881 56 53 50 2.0 0.6931 6981 7031 7080 7129 7178 7227 7275 7324 7372 47 2.1 2.2 2.3 7419 7SS5 8329 7467 7514 7561 7930 7975 8020 8372 8416 8459 7608 7655 7701 8065 8109 8154 8502 8544 8587 7747 7793 7839 8198 8242 8286 8629 8671 8713 46 43 42 2.4 2.5 2.6 8755 9163 9555 8796 8838 8879 9203 9243 9282 9594 9632 9670 8920 8961 9002 9322 9361 9400 9708 9746 9783 9042 9083 9123 9439 9478 9517 9821 9858 9895 40 38 38 2.7 2.8 2.9 9933 1.0296 0647 9969*0006*0043 0332 0367 0403 0682 0716 0750 *0080*01 16*0152 0438 0473 0508 0784 0818 0852 *0188*0225*0260 0543 0578 0613 0886 0919 0953 36 34 33 3.0 1.09S6 1019 1053 1086 1119 1151 1184 1217 1249 1282 32 3.1 3.2 3.3 1314 1632 1939 1346 1378 1410 1663 1694 1725 1969 2000 2030 1442 1474 1506 1756 1787 1817 2060 2090 2119 1537 1569 1600 1848 1878 1909 2149 2179 2208 32 30 30 3.4 3.5 3.6 2238 2528 2S09 2267 2296 2326 2556 2585 2613 2837 2865 2892 2355 2384 2413 2641 2669 2698 2920 2947 2975 2442 2470 2499 2726 2754 2782 3002 3029 3056 29 27 27 3.7 3.8 3.9 3083 3350 3610 3110 3137 3164 3376 3403 3429 3635 3661 3686 3191 3218 3244 3455 3481 3507 3712 3737 3762 3271 3297 3324 3533 3558 3584 3788 3813 3838 26 26 25 4.0 1.3863 3888 3913 3938 3962 3987 4012 4036 4061 4085 25 4.1 4.2 4.3 4110 4351 4586 4134 4159 4183 4375 4398 4422 4609 4633 4656 4207 4231 4255 4446 4469 4493 4679 4702 4725 4279 4303 4327 4516 4540 4563 4748 4770 4793 24 23 23 4.4 4.5 4.6 4S16 5041 5261 4839 4861 4884 5063 5085 5107 5282 5304 5326 4907 4929 4951 5129 5151 5173 5347 5369 5390 4974 4996 5019 5195 5217 5239 5412 5433 5454 22 22 22 4.7 4.8 4.9 5476 5686 5892 5497 5518 5539 5707 5728 5748 5913 5933 5953 5560 5581 5602 5769 5790 5810 5974 5994 6014 5623 5644 5665 5831 5851 5872 6034 6054 6074 21 20 20 5.0 1.6094 6114 6134 6154 6174 6194 6214 6233 6253 6273 19 5.1 5.2 5.3 6292 6487 6677 6312 6332 6351 6506 6525 6544 6696 6715 6734 6371 6390 6409 6563 6582 6601 6752 6771 6790 6429 6448 6467 6620 6639 6658 6808 6827 6845 20 19 19 5.4 5.5 5.6 6864 7047 7228 6882 6901 6919 7066 7084 7102 7246 7263 7281 6938 6956 6974 7120 7138 71.56 7299 7317 7334 6993 7011 7029 7174 7192 7210 7352 7370 7387 18 18 18 5.7 5.8 5.9 7405 7579 7750 7422 7440 7457 7596 7613 7630 7766 7783 7800 7475 7492 7509 7647 7664 7681 7817 7834 7851 7527 7544 7561 7699 7716 7733 7867 7884 7901 18 17 17 lOge^i XXII. HYPEEBOLIO LOGAEITHMS. n 12 3 4 5 6 7 8 9 d 6.0 1.7918 7934 7951 7967 7984 8001 8017 8034 8050 8066 17 6.1 6.2 6.3 8083 8245 8405 8099 8116 8132 8262 8278 8294 8421 8437 8453 8148 8165 8181 8310 8326 8342 8469 8485 8500 8197 8213 8229 8358 8374 8390 8516 8532 8547 16 15 16 6.4 6.5 6.6 8563 8718 8871 8579 8594 8610 8733 8749 8764 8886 8901 8916 8625 8641 8656 8779 8795 8810 8931 8946 8961 8672 8687 8703 8825 8840 8856 8976 8991 9006 15 15 15 6.7 6.8 6.9 7.0 9021 9169 9315 9036 9051 9066 9184 9199 9213 9330 9344 9359 9081 9095 9110 9228 9242 9257 9373 9387 9402 9125 9140 9155 9272 9286 9301 9416 9430 9445 14 14 14 1.9459 9473 9488 9502 9516 9530 9544 9559 9573 9587 14 7.1 7.2 7.3 9601 9741 9879 9615 9629 9643 9755 9769 9782 9892 9906 9920 9657 9671 9685 9796 9810 9824 9933 9947 9961 9699 9713 9727 9838 9851 9865 9974 9988*0001 14 14 14 7.4 7.5 7.6 2.0015 0149 0281 0028 0042 0055 0162 0176 0189 0295 0308 0321 0069 0082 0096 0202 0215 0229 0334 0347 0360 0109 0122 0136 0242 0255 0268 0373 0386 0399 13 13 13 7.7 7.8 7.9 0412 0541 0669 0425 0438 0451 0554 0567 0580 0681 0694 0707 0464 0477 0490 0592 0605 0618 0719 0732 0744 0503 0516 0528 0631 0643 0656 0757 0769 0782 13 13 12 8.0 2.0794 0807 0819 0832 0844 0857 0869 0882 0894 0906 13 8.1 8.2 8.3 0919 1041 1163 0931 0943 0956 1054 1066 1078 1175 1187 1199 0968 0980 0992 1090 1102 1114 1211 1223 1235 1005 1017 1029 1126 1138 1150 1247 1258 1270 12 13 12 8.4 8.5 8.6 1282 1401 1518 1294 1306 1318 1412 1424 1436 1529 1541 1552 1330 1342 ]353 1448 1459 1471 1564 1576 1587 1365 1377 1389 1483 1494 1506 1599 1610 1622 12 12 11 8.7 8.8 8.9 1633 1748 1861 1645 1656 1668 1759 1770 1782 1872 1883 1894 1679 1691 1702 1793 1804 1815 1905 1917 1928 1713 1725 1736 1827 1838 1849 1939 1950 1961 12 12 11 9.0 2.1972 1983 1994 2006 2017 2028 2039 2050 2061 2072 11 9.1 9.2 9.3 2083 2192 2300 2094 2105 2116 2203 2214 2225 2311 2322 2332 2127 2138 2148 2235 2246 2257 2343 2354 2364 2159 2170 2181 2268 2279 2289 2375 2386 2396 11 11 11 9.4 9.5 9.6 2407 2513 2618 2418 2428 2439 2523 2534 2544 2628 2638 2649 2450 2460 2471 2555 2565 2576 2659 2670 2680 2481 2492 2502 2586 2597 2607 2690 2701 2711 11 11 10 9.7 9.8 9.9 2721 2824 2925 2732 2742 2752 2834 2844 2854 2935 2946 2956 2762 2773 2783 2865 2875 2885 2966 2976 2986 2793 2803 2814 2895 2905 2915 2996 3006 3016 10 10 10 n 1 in 1st 2d S A 4th in logelO'V logelO-« 1 2.3026 0.2303 0.0230 O.C 023 0.0002 0.4343 2.3026 3.6974 2 4.6052 0.4605 0.0461 0.0 046 0.0005 0.8686 4.6052 5.3948 3 6.9078 0.6908 0.0691 0.0 069 0.0007 1.3029 6.9078 7.0922 4 9.2103 0.9210 0.0921 0.0 092 0.0009 1.7372 9.2103 10.7897 5 11.5129 1.1513 0.1151 O.C 115 0.0012 2.1715 11.5129 12.4871 6 13.8155 1.3816 0.1382 O.C 138 0.0014 2.6058 13.8155 14.1845 7 16.1181 1.6118 0.1612 O.C 161 0.0016 3.0401 16.1181 17.8819 8 18.4207 1.8421 0.1842 O.C 184 0.0018 3.4744 18.4207 19.5793 9 20.7233 2.0723 0.2072 O.C 207 0.0021 3.9087 20.7233 21.2767 JC, XXIII. AMOUNT AT THE END OF u YEAES. . WO n 3 3A 4 4} 6 6 7 8 9 10 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1 1.030 1.03.^ 1.040 1.045 1.050 1.060 1.070 1.080 1.090 1.100 o 1.061 1.071 1.0S2 1.092 1.103 1.124 1.145 1.166 1.188 1.210 8 1.093 1.109 1.125 1.141 1.158 1.191 1.225 1.260 1.295 1.331 4 1.126 1.H8 1.170 1.193 1.216 1.262 1.311 1.360 1.412 1.464 5 1.1-S9 1.1S8 1.217 1.246 1.276 1.338 1.403 1.469 1.539 1.611 6 1.194 1.229 1.265 1.302 1.340 1.419 1.501 1.587 1.677 1.772 7 1.230 1.272 1.316 1.361 1.407 1..504 1.606 1.714 1.828 1.949 8 1.267 1.317 1.369 1.422 1.477 1.594 1.718 1.851 1.993 2.144 . 9 1.30i 1.363 1.423 1.486 1.551 1.689 1.838 1.999 2.172 2.358 10 1.344 1.411 1.480 1.5.53 1.629 1.791 1.967 2.159 2.367 2.594 11 1.384 1.460 1..S39 1.623 1.710 1.S9S 2.105 2.332 2.580 2.853 12 1.426 1..S11 1.601 1.696 1.796 2.012 2.252 2.518 2.813 3.138 13 1.469 1.564 1.665 1.772 1.886 2.133 2.410 2.720 3.066 3.452 U 1.513 1.619 1.732 1.852 1.980 2.261 2.579 2.937 3.342 3.797 15 1.558 1.675 1.801 1.935 2.079 2.397 2.759 3.172 3.642 4.177 IG 1.605 1.734 1.873 2.022 2.183 2..540 2.952 3.426 3.970 4.595 17 1.653 1.795 1.948 2.113 2.292 2.693 3.159 3.700 4.328 5.054 18 1.702 1.857 2.026 2.208 2.407 2.854 3.380 3.996 4.717 5.560 19 1.754 1.923 2.107 2.308 2.527 3.026 3.617 4.316 5.142 6.116 20" 1.806 1.990 2.191 2.412 2.653 3.207 3.870 4.661 5.604 6.727 21 1.860 2.059 2.279 2.520 2.786 3.400 4.141 5.034 6.109 7.400 .).> 1.916 2.132 2.370 2.634 2.925 3.604 4.430 5.437 6.659 8.140 23 1.974 2.206 2.465 2.752 3.072 3.820 4.741 5.871 7.258 8.954 24 2.033 2.283 2.563 2.876 3.225 4.049 5.072 6.341 7.911 9.850 25 2.094 2.363 2.666 3.005 3.386 4.292 5.427 6.848 8.623 10.83 20 2.157 2.446 2.772 3.141 3.556 4.549 5.807 7.396 9.399 11.92 27 2.221 2.532 2.883 3.282 3.733 4.822 6.214 7.988 10.25 . 13.11 28 2.288 2.620 2.999 3.430 3.920 5.112 6.649 8.627 11.17 14.42 29 2.357 2.712 3.119 3.584 4.116 5.418 7.114 9.317 12.17 15.86 30 2.427 2.807 3.243 3.745 4.322 5.743 7.612 10.06 13.27 17.45 31 2.500 2.905 3.373 3.914 4.538 6.088 8.145 10.87 14.46 19.19 32 2.575 3.007 3.508 4.090 4.765 6.453 8.715 11.74 15.76 21.11 33 2.652 3.112 3.648 4.274 5.003 6.841 9.325 12.68 17.18 23.23 34 2.732 3.221 3.794 4.466 5.253 7.251 9.978 13.69 18.73 25.55 35 2.814 3.3.34 3.946 4.667 5.516 7.686 10.68 14.79 20.41 28.10 36 2.898 3.450 4.104 4.877 5.792 8.147 11.42 15.97 22.25 30.91 37 2.985 3.571 4.268 5.097 6.081 8.636 12.22 17.25 24.25 34.00 38 3.075 3.696 4.439 5.326 6.385 9.154 13.08 18.63 26.44 37.40 39 3.167 3.825 4.616 5.566 6.705 9.704 13.99 20.12 28.82 41.14 40 3.262 3.959 4.801 5.816 7.040 10.29 14.97 21.72 31.41 45.26 41 3.360 4.098 4.993 6.078 7.392 10.90 16.02 23.46 34.24 49.79 42 3.461 4.241 5.193 6.352 7.762 11.56 17.14 25.34 37.32 54.76 43 3.565 4.390 5.400 6.637 8.150 12.25 18.34 27.37 40.68 60.24 44 3.671 4.543 5.617 6.936 8.557 12.99 19.63 29.56 44.34 66.26 45 3.782 4.702 5.841 7.248 8.985 13.76 21.00 31.92 48.33 72.89 46 3.895 4.867 6.07i 7.574 9.434 14.59 22.47 34.47 52.68 80.18 47 4.012 5.037 6.318 7.915 9.906 15.47 24.05 37.23 57.42 88.20 48 4.132 5.214 6.571 8.271 10.40 16.39 25.73 40.21 62.59 97.02 49 4.256 5.396 6.833 8.644 10.92 17.38 27.53 43.43 68.22 106.7 1000 1 + 100 J XXIV. PRESENT VALUE OF 1000 DUE u YEARS HENOE. 77 n 3 3^ 4 4} 5 6 7 8 9 10 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1 2 3 970.9 966.2 961.5 942.6 933.5 924.6 915.1 901.9 SS9.0 956.9 952.4 943.4 915.7 907.0 890.0 876.3 863.8 839.6 934.6 925.9 917.4 873.4 857.3 841.7 816.3 793.8 772.2 909.1 826.4 751.3 4 5 6 855.5 871.4 854.8 862.6 842.0 821.9 837.5 813.5 790.3 838.6 822.7 792.1 802.5 783.5 747.3 767.9 746.2 705.0 762.9 735.0 708.4 713.0 680.6 649.9 666.3 630.2 596.3 683.0 620.9 564.5 7 8 9 813.1 786.0 759.9 789.4 759.4 730.7 766.4 733.7 702.6 734.8 710.7 665.1 703.2 676.8 627.4 672.9 644.6 591.9 622.7 5S3.5 547.0 582.0 540.3 501.9 543.9 500.2 460.4 513.2 466.5 424.1 10 744.1 708.9 675.6 643.9 613.9 558.4 508.3 463.2 422.4 385.5 11 12 13 722.4 684.9 649.6 701.4 661.8 624.6 681.0 639.4 600.6 616.2 584.7 526.8 589.7 556.8 497.0 564.3 530.3 468.8 475.1 428.9 387.5 444.0 397.1 355.5 415.0 367.7 326.2 350.5 318.6 289.7 14 15 IG 661.1 617.8 577.5 641.9 596.9 555.3 623.2 576.7 533.9 540.0 505.1 442.3 516.7 481.0 417.3 494.5 458.1 393.6 387.8 340.5 299.2 362.4 315.2 274.5 338.7 291.9 251.9 263.3 239.4 217.6 17 18 19 605.0 557.2 513.4 587.4 538.4 493.6 570.3 520.2 474.6 473.2 436.3 371.4 452.8 415.5 350.3 433.3 395.7 330.5 316.6 270.3 231.1 295.9 250.2 212.0 276.5 231.7 194.5 197.8 179.9 163.5 20 553.7 502.6 456.4 414.6 376.9 311.8 258.4 214.5 178.4 148.6 21 22 23 537.5 485.6 438.8 521.9 469.2 422.0 506.7 453.3 405.7 396.8 358.9 294.2 379.7 341.8 277.5 363.4 325.6 261.8 241.5 198.7 163.7 225.7 183.9 150.2 210.9 170.3 137.8 135.1 122.8 111.7 24 25 26 491.9 438.0 390.1 477.6 423.1 375.1 463.7 408.8 360.7 347.7 310.1 247.0 332.7 295.3 233.0 318.4 281.2 219.8 197.1 157.7 126.4 184.2 146.0 116.0 172.2 135.2 106.4 101.5 92.30 ■ 83.91 27 28 29 450.2 395.0 346.8 437.1 381. 7 333.5 424.3 368.7 320.7 304.7 267.8 207.4 291.6 255.1 195.6 279.0 242.9 184.6 160.9 125.2 97.61 150.4 115.9 89.55 140.6 107.3 82.15 76.28 69.34 63.04 30 412.0 356.3 308.3 267.0 231.4 174.1 131.4 99.38 75.37 57.31 31 32 33 400.0 344.2 296.5 388.3 332.6 285.1 377.0 321.3 274.1 255.5 220.4 164.3 244.5 209.9 155.0 234.0 199.9 146.2 122.8 92.02 69.15 114.7 85.20 63.44 107.2 78.89 58.20 52.10 47.36 43.06 34 35 36 366.0 310.5 263.6 355.4 300.0 253.4 345.0 2S9.8 243.7 223.9 190.4 137.9 214.3 181.3 130.1 205.0 172.7 122.7 100.2 73.05 53.39 93.66 67.63 48.99 87.54 62.62 44.94 39.14 35.58 32.35 37 38 39 40 335.0 2S0.0 234.3 325.2 270.6 225.3 315.8 261.4 216.6 196.2 164.4 115.8 1S7.8 156.6 109.2 179.7 149.1 103.1 81.81 57.99 41.23 76.46 53.69 37.83 71.46 49.71 34.70 29.41 26.73 24.30 306.6 252.6 20S.3 171.9 142.0 97.22 66.78 46.03 31.84 22.09 41 42 43 297.6 244.0 200.3 289.0 235.8 192.6 280.5 227.8 185.2 164.5 135.3 91.72 157.4 128.8 86.53 150.7 122.7 81.63 62.41 42.62 29.21 58.33 39.46 26.80 54.51 36.54 24.58 20.09 18.26 16.60 44 45 46 272.4 220.1 178.0 264.4 212.7 171.2 256.7 205.5 164.6 144.2 116.9 77.01 138.0 111.3 72.65 132.0 106.0 68.54 50.95 33.83 22.55 47.61 31.33 20.69 44.50 29.01 18.98 15.09 13.72 12.47 47 48 49 249.3 198.5 158.3 242.0 191.8 152.2 235.0 185.3 146.3 126.3 100.9 64.66 120.9 96.14 61.00 115.7 91.56 57.55 41.59 26.86 17.42 38.87 24.87 15.98 36.32 23.03 14.66 11.34 10.31 9.370 78 XXV. AMOUNT OF AN ANNUITY PAID AT THE END OP EAOH YEAR. i'^mj 100 n 3 3i 4 4!, 6 6 7 8 9 10 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1 2 3 1.000 1.000 1.000 2.030 2.035 2.040 3.091 3.106 3.122 1.000 1.000 1.000 2.045 2.050 2.060 3.137 3.153 3.184 1.000 1.000 2.070 2.080 3.215 3.246 1.000 2.090 3.278 1.000 2.100 3.310 4 5 G 4.1S4 4.215 4.246 ,S.309 5.362 5.416 6.468 6.550 6.633 4.278 4.310 4.37i 5.471 5.526 5.637 6.717 6.802 6.975 4.440 4.506 4.573 5.751 5.867 5.985 7.153 7.336 7.523 4.641 6.105 7.716 7 8 9 7.662 7.779 7.S9S S.S92 9.052 9.214 10.16 10.37 10.58 8019 8.142 8.394 9.380 9.549 9.897 10.80 11.03 11.49 8.654 8.923 9.200 10.26 10.64 11.03 11.98 12.49 13.02 9.487 11.44 13.58 10 11.46 11.73 12.01 12.29 12.58 13.18 13.82 14.49 15.19 15.94 11 12 13 12.81 13.14 13.49 14.19 14.60 15.03 15.62 16.11 16.63 13.84 14.21 14.97 15.46 15.92 16.87 17.16 17.71 18.88 15.78 16.65 17.89 18.98 20.14 21.50 17.56 20.14 22.95 18.53 21.38 24.52 U 15 16 17.09 17.68 18.29 18.60 19.30 20.02 20.16 20.97 21.82 18.93 19.60 21.02 20.78 21.58 23.28 22.72 23.66 25.67 22.55 24.21 25.13 27.15 27.89 30.32 26.02 29.36 33.00 27.97 31.77 35.95 17 18 19 21.76 22.71 23.70 23.41 24.50 25.65 25.12 26.36 27.67 24.74 25.84 28.21 26.86 28.13 30.91 29.06 30.54 33.76 30.84 33.75 34.00 37.45 37.38 41.45 36.97 41.30 46.02 40.54 45.60 51.16 57.27 20 26.87 28.28 29.78 31.37 33.07 36.79 41.00 45.76 51.16 21 22 23 28.68 30.27 31.97 30.54 32.33 34.25 32.45 34.46 36.62 33.78 35.72 39.99 36.30 38.51 43.39 38.94 41.43 47.00 44.87 50.42 56.76 49.01 55.46 62.87 53.44 60.89 69.53 64.00 71.40 79.54 24 25 26 34.43 36.67 39.08 36.46 38.95 41.65 38.55 41.31 44.31 41.69 44.50 50.82 44.57 47.73 54.86 47.57 51.11 59.16 58.18 66.76 63.25 73.11 68.68 79.95 76.79 84.70 93.32 88.50 98.35 109.2 27 28 29 40.71 43.76 47.08 42.93 46.29 49.97 45.22 48.91 52.97 50.71 54.67 63.71 53.99 58.40 68.53 57.42 62.32 73.64 74.48 87.35 80.70 95.34 87.35 104.0 102.7 113.0 124.1 121.1 134.2 148.6 30 47.58 51.62 56.08 61.01 66.44 79.06 94.46 113.3 136.3 164.5 31 32 33 50.00 54.43 59.33 52.50 57.33 62.70 55.08 60.34 66.21 64.75 70.76 84.80 68.67 75.30 90.89 72.76 80.06 97.34 102.1 123.3 110.2 134.2 118.9 146.0 149.6 164.0 179.8 181.9 201.1 222.3 34 35 36 57.73 63.45 69.86 60.46 66.67 73.65 63.28 70.01 77.60 77.03 85.07 104.2 81..S0 90.32 111.4 86.16 95.84 119.1 128.3 158.6 138.2 172.3 148.9 187.1 197.0 215.7 236.1 245.5 271.0 299.1 37 38 39 66.17 73.46 81.70 69.16 77.03 85.97 72.23 80.72 90.41 91.04 101.6 127.3 96.14 107.7 135.9 101.5 114.1 145.1 160.3 203.1 172.6 220.3 185.6 238.9 258.4 282.6 309.1 330.0 364.0 401.4 40 75.40 84.55 95.03 107.0 120.8 154.8 199.6 259.1 337.9 442.6 41 42 43 78.66 88.51 99.83 82.02 92.61 104.8 85.48 96.85 110.0 112.8 127.8 165.0 118.9 135.2 176.0 125.3 143.0 187.5 214.6 280.8 369.3 230.6 304.2 403.5 247.8 329.6 440.8 487.9 537.6 592.4 44 45 46 89.05 101.2 115.4 92.72 105.8 121.0 96..S0 110.5 126.9 131.9 151.1 199.8 138.8 1.59.7 212.7 146.1 168.7 226.5 266.1 356.9 285.7 386.5 306.8 418.4 481.5 525.9 574.2 652.6 718.9 791.8 47 4^ 4!) 100.4 115.4 132.9 104.4 120.4 139.3 108 5 125.6 145.8 1.53.7 178.1 241.1 161.6 188.0 256.6 169.9 1984 273.0 329.2 452.9 626.9 353.3 490.1 684.3 379.0 530.3 746.9 872.0 960.2 1057 (i + ^Y V loo; 100 ' /+4)' XXVL PRESENT VALUE OF AN ANNU- ITY PAID AT THE END OP EACH YEAR. n 3 3i 4 4,V 5 6 7 8 9 10 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1 2 3 .9709 .9662 .9615 1.913 1.900 1.8S6 2.829 2.802 2.775 .9569 .9524 .9434 1.873 1.859 1.833 2.749 2.723 2.673 .9346 .9259 .9174 1.808 1.783 1.759 2.624 2.577 2.531 .9091 1.736 2.487 4 5 6 3.717 3.673 3.630 4.580 4.515 4.452 5.417 5.329 5.242 3.5SS 3.546 3.465 4.390 4.329 4.212 5.158 5.076 4.917 3.387 3.312 3.240 4.100 3.993 3.890 4.767 4.623 4.486 3.170 3.791 4.355 7 8 9 6.230 6.115 6.002 7.020 6.874 6.733 7.786 7.608 7.435 5.893 5.7S6 5.582 6.596 6.463 6.210 7.269 7.108 6.802 5.389 5.206 5.033 5.971 5.747 5.535 6.515 6.247 5.995 4.868 5.335 5.759 10 8.530 8.317 8.111 7.913 7.722 7.360 7.024 6.710 6.418 6.145 11 12 13 9.253 9.002 8.760 9.954 9.663 9.385 10.63 10.30 9.986 8.529 8.306 7.887 9.119 8.863 8.384 9.683 9.394 8.853 7.499 7.139 6.805 7.943 7.536 7.161 8.358 7.904 7.487 6.495 6.814 7.103 14 15 16 11.30 10.92 10.56 11.94 11.52 11.12 12.56 12.09 11.65 10.22 9.899 9.295 10.74 10.38 9.712 11.23 10.84 10.11 8.745 8.244 7.786 9.108 8.559 8.061 9.447 8.851 8.313 7.367 7.606 7.824 17 18 19 13.17 12.65 12.17 13.75 13.19 12.66 14.32 13.71 13.13 11.71 11.27 10.48 12.16 11.69 10.83 12.59 12.09 11.16 9.763 9.122 8.544 10.06 9.372 8.756 10.34 9.6(H 8.950 8.022 8.201 8.365 20 14.88 14.21 13.59 13.01 12.46 11.47 10.59 9.818 9.129 8.514 21 22 23 15.42 14.70 14.03 15.94 15.17 14.45 16.44 15.62 14.86 13.40 12.82 11.76 13.78 13.16 12.04 14.15 13.49 12.30 10.84 10.02 9.292 11.06 10.20 9.442 11.27 10.37 9.580 8.649 8.772 8.883 24 25 26 16.94 16.06 15.25 17.41 16.48 15.62 17.88 16.89 15.98 14..50 13.80 12.55 14.83 14.09 12.78 15.15 14.38 13.00 11.47 10.53 9.707 11.65 10.67 9.823 11.83 10.81 9.929 8.985 9.077 9.161 27 28 29 30 18.33 17.29 16.33 18.76 17.67 16.66 19.19 18.04 16.98 15.45 14.64 13.21 15.74 14.90 13.41 16.02 15.14 13.59 11.99 10.94 10.03 12.14 11.05 10.12 12.28 11.16 10.20 9.237 9.307 9.370 19.60 18.39 17.29 16.29 15.37 13.76 12.41 11.26 10.27 9.427 31 32 33 20.00 18.74 17.59 20.39 19.07 17.87 20.77 19.39 18.15 16.54 15.59 13.93 16.79 15.80 14.08 17.02 16.00 14.23 12..S3 11.35 10.34 12.65 11.43 10.41 12.75 11.51 10.46 9.479 9.526 9.569 34 35 36 21.13 19.70 18.41 21.49 20.00 18.66 21.83 20.29 18.91 17.25 16.19 14.37 17.46 16.37 14.50 17.67 16.55 14.62 12.85 11.59 10.52 12.95 11.65 10.57 13.04 11.72 10.61 9.609 9.644 9.677 37 38 39 22.17 20.57 19.14 22.49 20.84 19.37 22.81 21.10 19.58 17.86 16.71 14.74 18.05 16.87 14.85 18.23 17.02 14.95 13.12 11.78 10.65 13.19 11.83 10.69 13.26 11.88 10.73 9.706 9.733 9.757 40 23.11 21.36 19.79 18.40 17.16 15.05 13.33 11.92 10.76 : 9.779 | 41 42 43 23.41 21.60 19.99 23.70 21.83 20.19 23.98 22.06 20.37 18.57 17.29 15.14 18.72 17.42 15.22 18.87 17.55 15.31 13.39 11.97 10.79 13.45 12.01 10.81 13.51 12.04 10.84 9.799 9.817 9.834 44 45 46 24.25 22.28 20.55 24.52 22.50 20.72 24.78 22.70 20.88 19.02 17.66 15.38 19.16 17.77 15.46 19.29 17.88 15.52 13.56 12.08 10.86 13.61 12.11 10.88 13.65 12.14 10.90 9.849 9.863 9.875 47 48 49 25.02 22.90 21.04 25.27 23.09 21.20 25.50 23.28 21.34 19.41 17.98 15.59 19..S4 18.08 15.65 19.65 18.17 15.71 13.69 12.16 10.92 13.73 12.19 10.93 13.77 12.21 10.95 9.887 9.897 9.906 80 XXVII. AMOUNT OF AN AN- NUITY WHEN PAID AT THE BEGINNING OF EACH YEAR. 1 + 1 + 100 y 100 n 8 3^ 4 4,V 5 6 7 8 9 10 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1 2 3 1.030 1.035 1.040 2.0*;! 2.106 2.122 3.184 3.215 3.246 1.045 1.050 1.060 2.137 2.153 2.184 3.278 3.310 3.375 1.070 1.080 1.090 2.215 2.246 2.278 3.440 3.506 3.573 1.100 2.310 3.641 4 6 6 4.309 4.362 4.416 5.468 5.5.50 5.633 6.662 6.779 6.898 4.471 4.526 4.637 5.717 5.802 5.975 7 019 7.142 7.394 4.751 4.867 4.985 6.153 6.3.36 6.523 7.654 7.923 8.200 5.105 6.716 8.487 7 8 9 7.S92 S0S2 8.214 9.159 9..368 9.583 10.46 10.73 11.01 8.380 8.549 8.897 9.802 10.03 10.49 11.29 11 58 12.18 9.260 9.637 10.03 10.98 11.49 12.02 12.82 13.49 14.19 10.44 12.58 14.94 10 11.81 12.14 12.49 12.84 13.21 13.97 14.78 15.65 16.56 17.53 11 12 13 13.19 13.60 14.03 14.62 15.11 15.63 16.09 16.68 17 29 14.46 14.92 15.87 16.16 16.71 17.88 17.93 18.60 2O02 16.89 17.98 19.14 19.14 20.50 21.95 21.55 23.21 25.02 20.38 23.52 26.97 14 15 16 17.60 18.30 19.02 19.16 19.97 2082 20.76 21.71 22.70 19.78 20.58 22.28 21.72 22.66 24.67 23.74 24.84 27 21 24.13 26.15 28.36 26.89 29.32 32.00 29.84 32.75 35.97 30.77 34.95 39.54 17 18 19 22.41 23..SO 24 65 24.12 25.36 26.67 25.87 27.28 28.78 25.86 27.13 29.91 28.06 29.54 32.76 30.37 32.07 35.79 33.00 36.45 40.30 36.38 40.45 45.02 40.00 44.76 5016 44.60 50.16 56.27 20 27.68 29.27 30.97 32.78 34.72 38.99 43.87 49.42 55.76 63.00 21 22 23 29.54 31.33 33.25 31.45 33.46 35.62 33.43 35.67 38.08 35.30 37.51 42,39 37.94 40.43 46.00 40.69 43.50 49.82 48.01 54.46 61.87 52.44 59.89 68.53 57.18 65.76 75.79 70.40 78.54 87.50 24 25 26 35.46 37.95 40.65 37.55 40.31 43.31 39.71 42.76 46.08 43.57 46.73 53.86 46.57 50.11 58.16 49.71 53.67 62.71 62.25 72.11 83.70 67.68 78.95 92.32 73.48 86.35 101.7 97.35 108.2 120.1 27 28 29 41.93 45.29 48.97 44.22 47.91 51.97 46.58 50.62 55.08 52.99 57.40 67.53 56.42 61.32 72.64 60.01 65.44 78.06 79.70 94.34 112.0 86.35 103.0 123.1 93.46 112.3 135.3 133.2 147.6 163.5 30 49.00 53.43 58.33 63.75 69.76 83.30 101. 1 122.3 148.6 180.9 31 32 33 51.50 56.33 61.70 54.08 59.34 65.21 56.73 62.45 6S.S6 67.67 74.30 89.89 71.76 79.06 96.34 76.03 84.07 103.2 109.2 133.2 163.0 117.9 145.0 17S.S 127.3 157.6 196.0 200.1 221.3 244.5 34 35 36 .5'9.46 65.67 72.65 62.28 69.01 76.60 65.17 72.46 80.70 80.50 89.32 110.4 85.16 94.84 118.1 90.04 100.6 126.3 137.2 171.3 214.7 147.9 186.1 235.1 159.3 202.1 257.4 270.0 298.1 329.0 37 38 39 40 68.16 76.03 84.97 71.23 79.72 89.41 74.40 83.55 94.03 95.14 106.7 134.9 100.5 113.1 144.1 106.0 119.8 153.8 171.6 219.3 281.6 184.6 237.9 308.1 198.6 258.1 336.9 363.0 400.4 441.6 77.66 87.51 98.83 111.8 126.8 164.0 213.6 279.8 368.3 486.9 41 42 43 81.02 91.61 103.8 84.48 95.85 109.0 88.05 100.2 114.4 117.9 134.2 175.0 124.3 142.0 186.5 130.9 150.1 198.8 229.6 303.2 402.5 246.8 328.6 439.8 265.1 355.9 480.5 536.6 591.4 651.6 44 45 46 91.72 104.8 1200 95 50 109.5 125.9 99.40 114.4 131.9 137.8 158.7 211.7 145.1 167.7 225.5 152.7 177.1 240.1 284.7 385.5 524.9 305.8 417.4 573.2 328.2 451.9 625.9 717.9 790.8 871.0 47 4S 103.^ 119.4 138.3 107.5 124.6 144.8 Ills 1,300 151.7 160.6 187.0 255.6 168.9 197.4 272.0 177.5 208.3 289.3 352.3 489.1 683.3 378.0 529.3 745.9 405.5 572.8 814.1 959.2 1056 1163 iooo(i + -^Yj:~ [ 100 J 100 1 + looy XXVIII. SUM TO BE PAID AT THE END or EACH OF u YEAES TO EXTINGUISH A DEBT OP 1000. 81 n 3 3^ 4 4i 5 6 7 8 9 10 1 2 3 1030 1035 1040 522.6 526.4 530.2 353.5 356.9 360.3 1045 1050 1060 534.0 537.8 545.4 363.8 367.2 374.1 1070 1080 1090 553.1 560.8 568.5 381.1 388.0 395.1 1100 576.2 402.1 4 5 6 269.0 272.3 275.5 218.4 221.5 224.6 184.6 187.7 190.8 278.7 2S2.0 288.6 227.8 231.0 237.4 193.9 197.0 203.4 295.2 301.9 308.7 243.9 250.5 257.1 209.8 216.3 222.9 315.5 263.8 229.6 7 8 9 160.5 163.5 166.6 142.5 145.5 148.5 128.4 131.4 134.5 169.7 172.8 179.1 151.6 154.7 161.0 137.6 140.7 147.0 185.6 192.1 198.7 167.5 174.0 180.7 153.5 160.1 166.8 205.4 187.4 173.6 10 117.2 120.2 123.3 126.4 129.5 135.9 142.4 149.0 155.8 162.7 11 12 13 108.1 111.1 114.1 100.5 103.5 106.6 94.03 97.06 100.1 117.2 120.4 126.8 109.7 112.8 119.3 103.3 106.5 113.0 133.4 140.1 146.9 125.9 132.7 139.7 119.7 126.5 133.6 154.0 146.8 140.8 14 15 16 88.53 91.57 94.67 83.77 86.83 89.94 79.61 82.68 85.82 97.82 101.0 107.6 93.11 96.34 103.0 89.02 92.27 98.95 114.3 121.3 128.4 109.8 116.8 124.1 105.9 113.0 120.3 135.7 131.5 127.8 17 18 19 75.95 79.04 82.20 72.71 75.82 78.99 69.81 72.94 76.14 85.42 88.70 95.44 82.24 85.55 92.36 79.41 82.75 89.62 102.4 109.6 117.0 99.41 106.7 114.2 96.75 104.1 111.7 124.7 121.9 119.5 20 67.22 70.36 73.58 76.88 80.24 87.18 94.39 101.9 109.5 117.5 21 22 23 64.87 68.04 71.28 62.75 65.93 69.20 60.81 64.02 67.31 74.60 78.00 85.00 72.55 75.97 83.05 70.68 74.14 81.28 92.29 99.83 107.6 90.41 98.03 105.9 88.71 96.42 104.4 115.6 114.0 112.6 24 25 26 59.05 62.27 65.59 57.43 60.67 64.01 55.94 59.21 62.57 68.99 72.47 79.68 67.44 70.95 78.23 66.02 69.56 76.90 87.19 94.98 103.0 85.81 93.68 101.8 84.56 92.51 100.7 111.3 110.2 109.2 27 28 29 30 54.56 57.85 61.24 53.29 56.60 60.01 52.11 55.45 58.88 64.72 68.29 75.70 63.52 67.12 74.59 62.41 66.05 73.58 83.43 91.45 99.73 82.39 90.49 98.85 81.45 89.62 98.06 108.3 107.5 106.7 51.02 54.37 57.83 61.39 65.05 72.65 80.59 88.83 97.34 106.1 31 32 33 50.00 53.37 56.86 49.05 52.44 55.95 48.16 51.57 55.10 60.44 64.13 71.79 59.56 63.28 71.00 58.74 62.49 70.27 79.80 88.11 96.69 79.07 87.45 96.10 78.41 86.85 95.56 105.5 105.0 104.5 34 35 36 47.32 50.76 54.31 46.54 50.00 53.58 45.80 49.28 52.89 57.98 61.76 69.60 57.27 61.07 68.97 56.61 60.43 68.39 77.80 86.30 95.08 77.23 85.80 94.64 76.72 85.34 94.24 104.1 103.7 103.3 37 38 39 45.11 48.61 52.24 44.46 47.98 51.63 43.84 47.39 51.06 55.98 59.84 67.86 55.40 59.28 67.36 54.86 58.76 66.89 76.24 84.92 93.87 75.80 84.54 93.54 75.39 84.19 93.24 103.0 102.7 102.5 40 43.26 46.83 50.52 54.34 58.28 66.46 75.01 83.86 92.96 102.3 41 42 43 42.71 46.30 50.02 42.19 45.80 49.54 41.70 45.33 49.09 53.86 57.82 66.06 53.41 57.39 65.68 52.98 56.99 65.33 74.66 83.56 92.71 74.34 83.29 92.48 74.04 83.03 92.27 102.0 101.9 101.7 44 45 46 41.23 44.88 48.66 40.79 44.45 48.26 40.36 44.05 47.88 52.58 56.62 65.01 52.20 56.26 64.70 51.84 55.93 64.41 73.76 8280 92.08 73.50 82.59 91.90 73.26 82.39 91.74 101.5 101.4 101.3 47 48 49 39.96 43.67 47.52 39.58 43.31 47.18 39.21 42.96 46.86 51.51 55.61 64.15 73.04 82.21 91.60 51.19 55.32 63.90 72.83 82.04 91.46 50.89 55.04 63.66 7264 81.89 91.34 101.1 101.0 100.9 XXIX. LEAST DIVISOES. 1 V 3 4 n 1 3 7 9 1 8 7 9 I 3 7 9 1 3 7 9 13 7 9 - 3 3 3 3 — 3 7 1 2 3 3 7 7 3 3 11 — 3 3 3 3 7 7 3 11 11 13 3 3 17 3 3 7 3 7 — - 3 3 - 3 3 11 3 - - 3 13 3 11 7 4 5 6 3 13 3 11 — 3 - — 3 3 7 13 7 3 3 11 3 1 3 7 7 3 17 23 3 17 3 — 19 13 3 3 7 7 3 3- .3- - 3 - 3 11 7 8 9 ~3 17 19 11 3 7 — 3 - — 3 3 23 — 3 — 11 3 19 7 3 7 3 3 13 3 3 17 3 7 — 11 7 3 3 — 3 3—37 29 3 7 3 — 23 — 13 10 7 17 19 - 3 — 3 — — 3 13 3 — - 17 — 3 7 3 — 11 12 13 3 3 3 — 17 3 — 7 11 7 3 3 13 3 3 23 19 3 3 7 3 - 3 3 11 11 3 3 — 31 7 17 3 13 7 3 31 3 17 11 29 - 3 17 3 19 14 15 16 3 19 23 3 7 3 - 11 3 17 3 3 13 17 37 — 3 3 7 7 3 3 3 ", 3—3 — 3 29 7 23 — 3 11 11 3 — 3 23 — 7 — 3 31 3 17 17 18 19 3 13 3 11 3 — 13 3 - 23 29 "3 3 17 7 23 - 3 3 17 19 3 17 — 11 7 - 3 31 3 41 3 3 — 3 37 3 11 3 — 13 7 — 3-3 7 19 - 43 3 29 3 — 20 3 — 3 7 — 3 — 3 43 7 — — 3 19 3 — 13 3 23 3 21 23 11 31 3 3 7 7 3 — 47 3 — J 3 29 3 7 13 7 3 3 11 11 3 23 3 17 13 3 17 23 3 3 — 7 — — 3 3 19 7 3 — 3 13 -3—3 24 25 26 7 41 3 3 19 29 3 23 13 3 — 3 7 19 7 3 3 41 11 3 3 — 3 3 7 43 37 7 3 11 11 3 3 - 17 43 — 3 3 7 — 7 — 31 3 - 3 - 19 3 — 3 27 28 29 37 3 3 — 3 7 53 3 - 3 29 41 3 11 3 3 3 7 7 3 23 37 3 11 3 29 19 3 3 7 7 3 3 17 — 13 41 - 3-37 17 3 7 3 30 — 3 31 3 — 23 7 — 3 — 3 13 7 3 — 3 - 17 11 — 31 32 33 7 29 13 — 3 — 3 — — 3 — 3 3 13 7 11 3 3 31 3 3 3 11 53 7 3 3 31 3 13 — 43 53 3 41 3 47 3 3 7 3 47 7 3 17 3 13 17 34 35 36 19 41 3 31 13 3 — 7 3 11 — 3 3 23 3 3 13 3 7 11 7 3 3 13 23 3 3 19 i - 3 3 - 19 3 3 11 3 - -3-3 11 — 7 41 37 38 39 40^ 3 47 7 3 11 - 3 13 — 3 3 47 37 3 - 7 3 11 3 61 3 3 43 3 3 7 7 3 — 37 — 3 3 31 11 3 3 19 3 23 23 3 — 3 7 11 — — - 19 3 _ 3 — — 3 — 3 29 37 11 7 3 13 3 - 41 42 43 3 11 11 3 13 3 7 7 3 59 31 3 3 11 19 23 3 3 7 13 3 29 7 41 3 3 3 3 61 — 3 3 19 7 — 3 41 3 11 3 31 7 3 43 3 — 44 45 46 3 7 43 7 3 3 — — 3 17 11 11 13 3 3 7 7 3 3 31 3 3 19 43 3 7 7 3 3 23 11 11 3 3 13 41 - 23 3 — 3-3 19 7 3 - 3 - 47 4S 49 3 13 3 3 17 11 3 7 — 7 17 3 3 17 53 3 3 61 3 7 7 3 29 3 13 11 3 3 — 3 3 7 3 11 11 3 47 3 47 29 37 13 3-37 XXIX. LEAST DIVISOES. 5 « 7 8 9 u 1 3 7 9 1 3 7 9 1 3 7 9 1 3 7 9 1 3 7 9 3 — 3 — — 3 - 3 7 — 3 3 7 3-3 1 2 3 3 3 11 3 3 7 7 3 19 3 - 13 3 — - 3 3 — 3 — 3 — 7 — 13 3 3 3 11 7 3 3 17 3 17 — 3 13 3—3 4 5 6 11 19 3 3 7 3 3 13 3 3 23 3 3 11 3 11 ~3 7 13 7 3 3 11 3 3 19 13 3 17 7 - - 3 - 3 17 3 7 8 9 23 3 3 3 3 7 3 31 7 3 13 - 3 11 — 3 3—3 13 3 - — 7 — 19 3 11 11 3 3 - 3 — 7 3 23 7 3 13 — 17 19 3 29 3-3 10 — 3 7 3 — _ 11 — 3 29 3 13 23 3 — 3 — 7 11 12 13 1 7 7 3 13 3 23 19 3 3 13 3 29 3 7 7 3 — 37 — 3 11 31 19 — 3—3 3 7 1 7 3 3 19 29 1 3 13 - 3 11 3 — 3 7 11 — 14 15 16 3 13 3 31 3 3 3 7 11 7 3 3 13 - 3 - 3 7 — 11 19 3 7 3 3 23 3 41 3 3 7 7 3 3 37 19 3—3 17 18 19 17 3 17 3 7 3 19 11 3 3 41 — 3 37 13 3 29 — 3 7 11 7 3 — 3-3 3 _ 13 3 7 7 3 3 3 3 31 11 11 3 7 3 7 3 20 7 11 29 3 3 — 19 3 31 3 — — — — 3 7 3 - 21 22 23 3 — 3 3 37 13 — 17 3 7 3 17 11 3 3 23 13 41 7 - 3 - 3 43 - 3 — 3 3 37 — 3 3 7 11 3 7 29 3 3 13 3 11 — 3 — 24 25 26 3 11 11 3 7 3 3 23 13 3 3 11 — 3 17 7 3 17 3 31 3 37 3 3 29 7 13 3 3 13 19 3 47 3 3 11 3 - 7 23 — 3 - 27 28 29 3 13 3 3 31 3 11 11 3 3—3 7 47 19 17 47 — 3 13 3 — 3 13 7 3 3 43 11 11 3 3 — 19 29 3 7 — 3 — 3 7 11 — 13 3 41 3 — 30 3 43 3 7 3 — 3 37 7 17 — 3 — 3 11 3 19 3 31 32 33 23 3 3 7 7 3 3 29 3 13 3 3 7 7 3 3 19 3 - 3 29 11 11 3 31 17 3 3 7 17 19 3 3 11 3 31 23 7 37 3 — 3 43 3 34 35 36 7 53 3 3 11 13 3 3 3 7 7 3 3 43 19 3 3 23 3 — 3 7 7 3 13 59 3 3 29 11 3 17 37 3 7 3 7 13 — — 3 59 3-3 37 38 39 11 3 3 59 13 3 7 17 3 37 3 17 S3 3 3 53 - 3 3 7 3 7 3 — 11 29 41 3 23 19 3 3 11 7 7 13 3 3 17 3 13 29 17 3 7 3 7 3 40 — 3 — 3 31 17 7 13 3—3 — 7 3 61 3 — - 17 - 41 42 43 7 3 19 3 3 3 3 7 23 3 3 11 17 3 11 17 43 3 — 3—3 3 11 29 37 47 53 3—3 13 3 41 59 "3 3 7 7 3 13 3-3 23 - 53 44 45 46 3 61 29 3 — 7 3 47 — 3 3 59 3 3 41 — 3 13 7 17 3 11 7 17 23 3—3 3 19 3 31 3 7 67 3 13 43 3 3 — 3 11 3—3 13 7 37 47 48 49 - 7 67 — 3 23 3 43 — 3—3 3 11 11 3 7 3 19 31 3 13 3 17 — 11 — 3-3 3 7 13 7 3 17 19 3 3 3 3 67 7 - 3 - 3 59 3 — 19 — 84 XXIX. LEAST DIVISORS. 1 2 3 4 u 1 3 7 9 1 1 3 7 9 13 7 9 13 7 9 13 7 9 60 3 — 3 — — 3 29 3 11 47 3 7 3- 71 3 7 3 51 52 58 — 3 - 3 7 11 41 - 3 - 3 — 19 — 7 - 3 13 3 17 47 3 13 3 3 47 3 23 23 3 — 3 17 — 7 73 7 3 11 3 13 3 - 3 19 53 37 — 19 3 7 3 29 7 3—3 54 55 56 11 3 — 3 ~3 13 1 71 3 37 1 - 31 3 41 3 3 11 3 61 - 3 - 3 7 — 17 13 - 3 - 3 — 11 7 29 3 43 3 — 13 — 3 23 3 31 - 3 - 3 57 58 59 - 3 13 3 - 7 - 37 3 — 3 19 - 29 - 7 3 — 3 11 23 3 61 3 3 59 3 17 — 3—3 31 7 11 3 - 3 7 19 13 - 3 17 3 - 3 — 3 — 13 3 19 3 60 17 3 — 3 — 7 11 13 3 19 3 — 37 3 - 3 7 23 61 62 63 — 17 31 41 3—37 -373 3 — 3 29 — 3-3 - 59 — 71 - 3 11 3 - 7 13 - 3 - 3 - 17 7 3 23 3 17 13 3 - 3 3 — 3 11 79 3 - 3 17—11 7 64 65 66 37 19 43 13 3 7. 3 23 7 3—3 3 11 3 7 17 3 7 3 11 17 13 — — 3 — 3 — 11 61 — 3 37 3 7 59 7 41 47 3 47 3 13 19 3 - 3 3 17 3 — 31 3 - 3 29 7 17 61 67 68 69 19 - 3 - 3 11 67 3 - 3 3 7 3- 7 3 17 3 - 31 - 11 11 3 7 3 19 3 7 3 13 53 23 3—37 29 3 7 3 3 11 3 17 — 3 41 3 11 53 70 — 47 7 43 3 - 3 - 7 3-3 79 13 31 — 3-37 71 72 73 3 — 3 — 19 3 - 3 7 67 13 3 11 3 3 71 3 13 - 17 3 31 3 - - 3 17 3 3 7 3 11 7 3—3 11 41 37 3 7 3 13 11 3 7 3- 74 75 76 3 11 3 31 13 3 - 3 11 7 -3-3 7 11 - 73 3 23 3 19 41 13 7 17 3 — 3 — — 3 29 3 3 - 3 43 17 3 — 3 13 17 7 — 7 3 11 3 -19 3 - 3 - 77 78 79 3 - 3 13 29 3 37 3 — 7 — 11 11 3 - 3 73 13 - 7 3 41 3 - 7 59 3 - 3 - 89 3 - 3 3 11 3 71 41 3 17 3 7 17 — 3 61 3 — 11 7 47 3 13 3 - 80 3 53 3 - -3-3 13 71 23 7 3 29 3 - 11 3 13 3 81 82 83 - 3 11 3 59 13 29 — 3 19 3 7 - 7 — 23 3 43 3 - — 3—3 3 - 3 11 — 3 19 3 53 7 11 — 47 3 79 3 3 13 3 31 7 17 — 29 3 — 3 73 19 3 17 3 84 85 86 31 3 7 3 - 11 47 67 3 7 3- 13 47 19 - 3-37 79 3 7 3 3 - 3 - -3-3 37 — 3 11 3 19 7 3 89 3 53 23 7 3 - 3 83 — 3-3 87 88 89 7 3-3 13 23 3 29 3 59 31 — 23 — 3 7 3 — 7 3 37 3 3 11 3 7 — 373 11 - 79 - — 3—3 — 11 3—37 — 7 — 13 3 37 3 - — 3 23 3 90 — 3-3 71 29 3 7 3 — 11 3 7 3 83 - 91 92 93 19— 7 - 3 — 3 — 71 3 41 3 3 13 3 11 61 3 13 3 - 67 7 - 7 3-3 - 23 - 11 3 - 3 19 23 13 3 7 3- 7 3-3 3 41 3 7 — 373 13 — 94 95 96 7 - 23 97 3 13 3 37 - 3 13 3 3 - 3 - - 3 31 3 7 — 59 — — 3 11 3 — 89 7 13 3 — 3 — 3 - 3 - - 3 23 3 3 7 3 11 7 3-3 31 - 11 - 97 98 99 89 31 17 7 3 — 3 17 - 3 - 3 3 11 3 — -3-3 11 23 47 7 - 3 71 3 7 11 31 - 3 — 3 — 37 - 7 - 3 - 3 - — 3 19 3 3 - 3 - 13 3 43 3 - 61 7 - XXIX. LEAST DIVISORS. 85 5 6 7 8 9 n 1 3 7 9 13 7 9 1 3 7 9 1 3 7 9 1 3 7 9 50 — 31 13 — 3 61 3 37 TT 3 — 3 - 13 _ 7 3 11 3 — 51 3 — 3 7 13 3 — 3 7 31 - 3 71 3 29 3 — 3 52 59 3 7 3 — 19 23 11 3 — 3 17 3 11 67 — 7 53 — 53 11 23 3 31 3 7 41 3 19 3 — 7 — 17 3 54 3 7 3 53 43 3 7 3 13 3 - 3 11 17 3 23 3 55 7 3 — 3 67 — 19 — 3 - 3 7 - 3 37 3 7 29 11 56 — — 3 7 3 — 53 3 7 3 13 — 11 3 - 3 41 57 3 11 3 13 7 3 73 3 29 23 53 — 3 — 3 7 — 3 11 3 58 — 3 — 3 - 11 3 7 3 — — 3 7 3 43 71 - 17 59 11 — 7 59 3 67 3 47 7 3 43 3 — 31 — 53 3 13 3 7 60 ~3 — 3 73 11 3- 3 13 — 59 — 3 7 3 3 7 3 61 3 47 3 61 - 7 31 3 - 3 37 7 3 23 3 41 11 62 7 13 — 11 3 — 3 — 3—3 11 61 19 3 7 3 — 63 3 - 3 — — 3 — 3 23 - 7 — 3 13 3 7 3-3 64 — 3 11 3 7 23 29 — 3 — 3 11 — 3 13 3 43 73 67 65 — — 79 7 3 - 3 - — 3—3 — 29 7 11 3 19 3 — 66 3 — 3 — - 3 59 3 7 — 11 — 3 41 3 — — 3 37 3 67 43 3 29 3 67 7 3 13 3 - - 3 11 3 - 7 13 68 13 7 — 19 3 — 3 — •s_ 3 13 3 7 — 71 83 3 61 3—1 69 3 17 3 — — 3—3 — 19—7 3 — 3 29 — 3- 3 70 TT 3 — 3 23 7 37 — ~T 11 3 - 73 3 l9 T 7 41 47 31 1 71 23 17 — 3 13 3 67 71 3 — 3 43 11 7 3 — 3 23 72 3 — 3 7 53 3 13 3 11 7 19 29 3 — 3 37 23 3 - 3 73 — 3 7 3 17 37 53 - 3 73 3 47 11 3 83 3 19 - 13 7 74 — 29 — — 3 17 3 7 31 3 - 3 - 7 _ 3 59 3 - 75 3 7 3 — — 373 67 11 3 — 3 3 71 3 76 7 3 13 3 47 79 11 — 3 - 3 7 - 3 3 — 7 43 — 77 23 — 3 7 3 17 19 3 7 3 31 43 13 — 3 - 3 11 78 3 — 3 29 7 3—3 17 3 — 3 7 13 3 53 3 79 — 3 73 3 19 - 31 13 3 7 3 79 23 3 7 3 61 - 11 19 80 S3 — 7 — 3 11 3 - 7 3 41 3 — 59 T — 37 81 3 31 3 41 -3-3 11 13 - 3 7 3 19 3 7 3 82 37 3 23 3 11 — 7 — 3 - 3 17 7 3 3 43 83 7 — 61 13 3 — 3 — 11 3—3 17 83 — 3 7 3 37 84 3 79 3 11 -3-3 43 37 7 61 3 17 3 13 7 3 29 3 85 17 3 43 3 7 - 13 11 3 — 3 23 — 3 31 3 11 13 86 41 17 11 7 3 — 3 - 13 3 — 3 — 19 7 — 3 — 3 — 87 3 — 3 19 — 3 11 3 7 31 67 - 3 - 3 11 59 3 19 3 88 53 3 17 3 7 3 19 3 13 83 3 3 17 — 7 11 89 — 7 13 17 3 — 3 — 3 47 3 7 13 11 89 3 17 3 — 90 3 11 3 — 13 3 - 3 47 43 29 7 3 31 ~3 61 — 3 11 3 91 — 3 - 3 — 7 89 53 3 — 3 67 — 3 — 3 7 29 17 - 1 92 11 19 - 47 3 59 3 13 73 3—3 37 7 3 - 3 17 93 3 47 3 7 11 3 17 3 — 7 — 83 3 11 3 41 — 3-3 94 13 3 7 3 17 3 — 3 — 19 3 53 3 11—7 95 41 19 11 3 73 3 7 17 3 61 3 11 7 — 43 3 53 3 29 96 3 7 3 13 — 373 19 17 3 23 3 — 11 3—3 97 7 3 11 3 43 13 3 29 3 7 — 3 — 3 — 7 97 41 98 — 59 — — 3 7 3 71 — 3 7 3 41 — — 11 3 13 3 19 99 3 37 3 23 7 3—3 13 - 11 17 3 67 3 7 97 3 13 3 86 XXX. EXPONENTIALS. n 1 2 8 * 5 « 7 8 9 .0 1.000 1.010 1.020 1.030 1.041 1.051 1.062 1.073 1.083 1.094 .1 .2 .3 1.105 1.221 1.350 1.116 1.127 1.139 1.234 1.246 1.2.59 1.363 1.377 1.391 1.1.50 1.271 1.40^ 1.162 1.284 1.419 1.174 1.297 1.433 1.185 1.310 1.448 1.197 1.209 1.323 1..336 1.462 1.477 .4 .6 .6 1.492 1.649 1.S22 1.507 1.522 1.537 1.665 1.682 1.699 1.840 1.859 1.878 1.5.53 1.716 1.896 1.568 1.733 1.916 1.584 1.751 1.935 1.600 1.768 1.954 1.616 1.6.32 1.786 1.804 1.974 1.994 .7 .8 .9 2.014 2.226 2.460 2.0.34 2.0.54 2075 2.248 2.270 2.293 2.484 2.509 2.53i 2.096 2.117 2.138 2.316 2.340 2.363 2.560 2.586 2.612 2.160 2.181 2.203 2.387 2.411 2.435 2.638 2.664 2.691 1. 2.71S 7.389 20.09 3.004 3.320 3.669 8.166 9.025 9.974 22.20 24.53 27.11 4.055 4.482 4.9.S3 11.02 12.18 13.46 29.96 33.12 36.60 5.474 6.050 6.686 14.88 16.44 18.17 40.45 44.70 49.40 4. 5. 6. 54.60 148.4 403.4 60.34 66.69 73.70 164.0 181.3 200.3 445.9 492.7 544.6 81.45 90.02 99.48 221.4 244.7 270.4 601.8 665.1 735.1 109.9 121.5 134.3 298.9 330.3 365.0 812.4 897.8 992.3 7. 8. 9. 1097 2981 8103 1212 1,339 1480 3294 3641 4024 8955 9897 10938 1636 1808 1998 4447 4915 5432 12088 1336014765 2208 2441 2697 6003 6634 7332 16318 18034 19930 1 n 1 2 3 4 5 6 7 8 9 .0 1.000 *990 *980 *970 *961 *951 *942 *932 *923 *914 .1 9 .*3 0.905 0.819 0.741 896 887 878 811 803 795 733 726 719 869 787 712 861 779 70S 852 771 698 844 763 691 835 827 756 748 684 677 .4 .5 .6 0.670 0.607 0.549 664 657 651 600 595 589 543 538 533 644 583 527 638 577 522 631 571 517 625 566 512 619 613 560 554 507 502 .7 .8 .9 0.497 0.449 0.407 492 487 482 445 440 436 403 399 395 477 432 391 472 427 387 468 423 383 463 419 379 458 454 415 411 375 372 k 3. 0.368 0.135 0.0498 333 301 273 122 111 100 450 408 369 247 *907 334 223 *821 302 202 ♦743 273 183 *672 247 165 150 *608 *550 224 202 4. 5. 6. 0.0183 0.00674 0.00248 166 150 136 610 552 499 224 203 184 123 452 166 111 409 150 101 370 136 *910 335 123 *823 *745 303 274 111 101 7. 8. 9. 0.000912 0.000335 0.000123 825 747 676 304 275 249 112 101 091 611 225 083 5.53 203 075 501 184 068 453 167 061 410 371 151 136 055 050 n nc "J 1 e" 1 e " e (IT 2 HIT e 3 1 2 3 2.718 5.436 8.154 282 .367879 564 .735759 845 1.103638 E.718 1.649 1.396 .368 .607 .717 4.81 23.1 111 1 4 3 .208 .0432 .00898 4 6 6 10.873 13.591 16.309 127 1.471518 409 1.839397 691 2.207277 1.284 1.221 1.181 .779 .819 .847 535 257 123 5 6 92 .00187 .000388 .0000807 7 8 9 19.027 21.746 24.464 973 2.575156 255 2.943036 536 3.310915 1.154 1.133 l.llS .867 .883 .895 596 286 137 10 752 9406 .0000168 .00000349 .000000725 EXPLANATION OF TABLES. Throughout these tables the figures of the argument are printed in thick type, the initial figures being printed in the left-hand column, and the terminal figures in the top row. The entry is found in the intersection of the row of the initial figures with the colvimn of the terminal figure. A bar below a terminal 5 or shows that the final 5 has been increased ; hence, when the entry is further contracted, the terminal fig- ure ought not to be increased by one. I. Common Logarithms, pp. 2-5. Pages 2 and 3 give the logarithm to fou'r places of any sequence of three significant figures. The column headed d gives the difference between the last logarithm of the row and the first logarithm of the next row ; it facilitates the finding of the difference between any two successive logarithms in the row. The small table at the bottom of page 3 gives the pro- portional parts of the tabular differences from 4 to 23. The tabular difference is printed in the top row, and the tenths in the left-hand column. Pages 4 and 5 give the logarithms of any sequence of 4 figures from 1000 to 1900. The proportional parts of the tabular differences are given in the right column ; only the difference is printed, the tenth being understood from the loca- tion of the proportional part. The small table at the bottom of page 5 gives the logarithm to six places of the numbers 1.000 to 1.100 which occur in cal- culations of interest. The initial pair of figures are given only for the entry and are understood for the remaining entries of the row ; unless an asterisk is printed in front, which indi- cates that the initial figures are those printed in front of the next row. 88 EXPLANATION OF TABLES. Given a number to find its logarithm, or, to use the table directly. The characteristic or integral part of the required logarithm is obtained by counting the number of places by which the first figure of the number is removed to the left or to the right of the unit's i)lace ; if to the left, the characteristic is posi- tive ; if to the right, negative. Thus the characteristic of 1234 is 3, while that of .01234 is 2. The mantissa, or fractional part of the required logarithm, is obtained from the table thus. If the number has not more than two significant figures, then the mantissa is found in the column headed 0. If there are three significant figures, the mantissa is found in the intersection of the row for the first two figures with the column headed by the third figure. For example, the mantissa of 23 is .3617, that of 234 is .3G92. When the number contains four significant figures, tlie man- tissa can be obtained directly from pages 4 and 5 provided the number is not greater than 1900. AVhen the number is greater, the required mantissa is found from pages 2 and 3 by interpola- tion. Find the mantissa for the first three significant figures ; find the difference between it and the next higher mantissa in the table (it will in general be nearly equal to that printed under d) ; find from the table at the bottom of page 3 the proportional part of this tabular difference for the fourth significant figure and add it to the lower mantissa. Thus log 2345 is .3692 plus five-tenths of 19, that is 10 ; hence .3702. For log 23456 we add besides six-hundredths of 19, that is 1 ; hence .3703. Or we multiply 19 by .56 aud take the nearest integer to the value, namely, 11. Given a logarithm to Jiyul the corresponding number; or, to use the table inversely. When the number is wanted to not more than three significant figures, find the mantissa in the table which is nearest to the given mantissa ; the corresponding argument gives three figures of the number, and the position of the decimal point is deter- mined by the characteristic. Thus the number corresponding to 2.5015 is 317, that to 2.5020 is .0318. Suppose four significant figures are wanted. If the mantissa does not exceed .2785, the nearest mantissa on pages 4-5 will point out the number to four figures. If it exceeds the above EXPLANATION OF TABLES. 89 number, the fourth figure is obtained by interpolation. Find the next lower mantissa ; find the difference between the said mantissa and the next higher, also the difference between said mantissa and the given mantissa, and find from the table of page 3 what proportional part the latter difference is of the former. For example, the next lower to .7370 is .7364, hence the first part of the number is 545, and the fourth figure is that part of ten which 6 is of 8 : namely 7 ; hence 5.457. To find the logarithm of a product. Take the sum of the logarithms of the factors. Thus, log (123x4567)= 2.0899 3.6590 6 6.7495 To find the logarithm of a quotient. Subtract the logarithm of the denominator from the loga- rithm of the numerator. Thus, log 1^^ = 2.0899 3.6596 2.4303 To find the logarithn of a poiver. Multiply the logarithm of the base by the index of the power. Thus, log 9872 ^ 2 (2.9943) = 5.9886 log (.987)- = 2(1.9943) =1.9886 To find the logarithm of a root. Divide the logarithm of the base by the index of the root. Thus, log V987 = i (2.9943) = 1.4972 log V:987 = i (1.9943) = 1.9972 II. Antilogarithms, pp. 6-7. This is a table of the fractional powers of 10 from 10 -^^ to 10-^. The first two figures of the fraction or mantissa are given in the left column, and the third in the top row. 90 EXPLANATION OF TABLES. To find the number corresponding to a logarithm. This is given by a direct use of the table. Find the entry for the first three figures of the mantissa, take the difference between that entry and the next higher, and from the column of proportional parts find the part which requires to be added to the entry on account of the fourth figure. Insert the deci- mal point in the place indicated by the characteristic of the logarithm. For example, to find the antilogarithm of 2.9876. For 987 we have 9705, difference is 22 ; the proportional part of 22 for 6 is 13, therefore 9705 + 13 = 9718, and inserting the decimal point, 971.8. The direct use of a table of antilogarithms serves the same purpose as the inverse use of a table of logarithms. III. Addition Logarithms, pp. 8-9. The argument is log n, where n is a fraction less than unity. Thus the characteristic of log n is negative ; it is not printed so, but is indicated by its complement to 10. Thus the argu- ment 9.713 means 9.713 - 10, or 1.713. Given the logarithm of each of two numbers, to find the loga- rithm of their sum. Let a and b denote the two numbers, of which b is the less. Then log (a + 6) = log a + log^'l + ^\ (1) and log n = log - = log 6 — log a. (2) a Having found log n by means of (2), we get log (1 + n) from the table, and by adding it to log a obtain log (a +6). Example, log a = 3.8060, log & = 2.1618. Therefore, log n = log- = 2.1618 " 3.8060 2.3558 therefore, from the table, log (1 + n) = 0.0097 therefore, log (a + 6) = 3.8060 0.0097 3.8157 EXPLANATION OF TABLES. 91 To solve the same question by means of Tables I. and II. : a = 6397, b = 1449 + 2 = 145.1, a + 6 = 6542, log (a + 6) = 8156 + 1 = 3.8157. IV. Subtraction Logarithms, pp. 10-13. This table is arranged similarly to the preceding, excepting that the tenth decade is expanded on pages 12-13. Given the logarithm of each of two numbers, to find the loga- rithm of their difference. Let a and 6 denote the two numbers, of which b is the less. Then, log (a - 6) = log a(l _ ^^ = log a + log (l -^^ = loga— log 1-^ a Let log n = log -, then from the table we get log , which a 1 — n subtracted from log a, gives log (a — 6). Example. Given log a = 1.9876 and log6 = L5432. Then log n = log- = 1.5432 " 1.9876 1.5556 i.e., 9.5556 - 10; Now, from the table, 9.555 gives 0.1931 and 6 gives 3, ... log^— =0.1934, 1 — n .-. log(a-&)=L9876 0.1934 1.7942 Suppose that log (1 + n) is known (n being less than unity) and its position in Table III., then log can be found in 1 — n the corresponding position in Table IV., and log can be found by adding these two logarithms together. Logarithms of addition and subtraction are sometimes called Gaussian logarithms. 92 EXPLANATION OF TABLES. V. Logarithmic Sines and Cosines, pp. 14-17.. Pages 14-15 give the logarithm of the sine to every tenth of a degree, that is, every six minutes. The logarithm of the cosine is obtained by taking the riglit-hand argument and read- ing backwards. The d column gives the difference between the two last entries of a row, the last entry of one row being identical with the first of the succeeding. As the sines and cosines are all less than unity, the characteristics of the loga- rithms are all negative ; they are indicated by their complement to 10. Pages 16-17 give the sines and cosines for the first nine degrees to every hundredth of a degree. Given an arc, to find its log sin. If the arc is less than 90°, its log sin is found by the direct use of the table. Por example, to find log sin 17°.66. By the table log sin 17°.6 is 9.4805-10, the difference is 24, and the proportional part for 6 is 14 ; hence, 9.4819 — 10. If the arc is > 90° but < 180°, find the log sin of the difference be- tween 180° and the arc ; if > 180° but < 270° find that of the difference between the arc and 180°, and if > 270° but < 360°, find that of the difference between 360° and the arc. Given the log sin, to find the arc. Por example, to find the arc in degrees the log sin of which is 9.6669. The next lower log sin in the table is 9.6659, which corresponds to 27°. 6 ; the tabular difference is 14, and the given difference is 10 ; hence, the arc is 27°. 67. At the bottom of page 17 there are two auxiliary tables. The one gives the equivalent in minutes of the fractions of a de- gree ; thus, 0°.63 is equivalent to 37 '.8. The other is called a Delambre's table ; it is used to find the log sin of a small arc. On account of the table, pages 16-17, this auxiliary table is not required, except Avhen the arc is less than 0°.4. By S is here meant the logarithm of the ratio of the number expressing the degree to the number expressing the corresponding sine. To find the sine of a small arc. Let n denote the number of degrees ; then, log sin n° = log n — S. EXPLANATION OF TABLES.. 93 Example : to find log sin 0°.123. Log .123 is 9.0899 — 10, and S is 1.7581 ; hence, logsin0°.123 is 7.3318 - 10. To find a small arc, given its log sin. We have log n = log sin 91° + S. For example, given log sin to be 7.1234 — 10. As the log sin is less than 8.2872-10, the value of S to add is 1.7581. Hence, 8.8815 — 10, the number corresponding to which is .0761, hence 0°.0761. VI. Logarithmic Tangents and Cotangents, pp. 18-21. This table is similar to the preceding. After 45° the tan- gent is greater than unity, the characteristic is no longer negative ; hence, the true characteristic is printed. Of the two auxiliary tables at the bottom of page 21, the one gives the equivalent in degrees of so many minutes ; the other is a Delambre's table of T for the first four degrees. By T is here meant the logarithm of the ratio of the number of degrees to the number expressing the tangent. It is used in finding the log tan of a very small arc. We have, log tan n° = log n— T and log n = log tan n° + T. VII. Logarithmic Sines and Cosines for Minutes, pp. 22-23. Here the log sin is given directly to every ten minutes, and by interpolation to every minute. The same table gives log cos when read backwards. Pages 22-25 give the proportional parts for all the differences from 1 to 100. Example: to find log cos 19° 28'. Log cos 19° 20' is 9.9748 — 10, the tabular difference is —5, the proportional part of —5 for 8 is -4; hence, 9.9744-10. At the end we have a table of S for the range between 0° and 7°, where the change in the value of the tabular difference is too rapid to allow of interpolation by proportional parts. Here S is the logarithm of the ratio of the number of minutes expressing the arc to the sine of the arc. Thus, 3.5372 is the log of the ratio of 378 to .1097. 94 EXPLANATION OF TABLES. VIII. Logarithmic Tangents and Cotangents for Minutes, pp. 24-25. Tills table is similar to the preceding. By T is meant the log of the ratio of the number of minutes expressing the arc to the tangent of the arc. IX. Natural Sines and Cosines, pp. 26-27. The natural sine is given to each tenth of a degree ; that is, to every six minutes. The equivalent minutes are printed alongside of the tenths of a degree. At the bottom of page 27 there is a table of proportional parts, the whole interval being six, to facilitate the interpolation to a minute. What is the sine of 34° 46' ? The sine of 34° 42' is .5693, the tabular difference is 14, and the pp. of 14 for 4 is 9 ; hence, .5702. What is the arc whose cosine is .4326 ? The arc of .4321 is 64° 24', the tabular difference is 16, the difference of given cosine is 5, corresponding to a pp. of 5 for a tabular difference of 16 we have 2' ; hence, the arc is 64° 22'. X. Natural Tangents and Cotangents, pp. 28-29. XI. Natural Secants and Cosecants, pp. 30-31. These tables are similar to the preceding. At the end of each we have a continuation of the table of proportional parts, the interval being six. XII. Radians, pp. 32-33. By a radian is meant the unit of circular measure of an angle. The table gives directly the number of radians equiv- alent to any number of degrees expressed by not more than three significant figures. The integer figure of the entry is printed only in the column. Thus, the equivalent of 67°.8 is 1.1833 radians, and the equivalent of 1 radian is 57°.3. The tabular difference is either 17 or 18 ; hence, to find the equiva- lent for 4 significant figures, we add the proper pp. of either 18 or 17, as the case may be. The column headed h m gives the equivalent in hours and minutes of the corresponding number of degrees in the left EXPLANATION OF TABLES. 95 column ; and the adjacent column headed p gives that fraction of a whole period or perigon which is equivalent to the ratio of the corresponding number of degrees to 360°. The small table at bottom of page 33 gives the number of radians equivalent to the given number of minutes, while the column headed p gives that fraction of a period or perigon which the corresponding number of minutes bears to 360°. When the decimal point is changed by any number of places in the argument, the decimal point is changed by an equal number of places in the entry. Thus, 3°.6=. 06283 and 360°= 6.283. XIII. Reciprocals, pp. 34-35. The reciprocal is given directly for any sequence of three figures, the decimal point being after the first. When the decimal point in the argument is shifted any number of places, the decimal point in the entry is shifted an equal number of places in the opposite direction. Thus, J_ = . 1267, ^ = .01267, 7.89 ' 78.9 -^ = .001267, — = 1.267. 789 ' .789 At the bottom of page 35 we have the first nine multiples of the fractions \, \, etc., up to ^. A bracket indicates that the figures included repeat themselves. XIV. Squares, pp. 36-37. This table gives directly to four significant figures the square of any sequence of three figures, the decimal point being after the first. When the decimal point changes in the number, the decimal point in the square changes by double the number of places in the same direction. Thus the square of 3.76 is 14.14, that of 37.6 is 1414, and that of .376 is .1414. When the number consists of more than three figures, the square may be found by means of the table of proportional parts. For example, to find the square of 1889 to four signi- ficant figures. The square of 188 is 35,340, the pp. of 38 for 9 is 34 ; therefore the square of 1889 is 3,568,000. Here the zeros are not significant, but only indicate the position of the decimal point. 96 EXPLANATION OF TABLES. To find the complete square for any sequence of three figures. The complete square of any two figures is given in the zero eohimn. If the number of three figures is less than 317, we have to find the square of the third figure, and a})pend the terminal figui'e to the entry of the table, diminishing the ter- minal figure of the entry by one if the number appended is equal to or greater than 5. For instance, take 234. The square of 4 is 16, hence 6 is to be appended to 5476, but the fourth figure reduces to 5 because it has been increased by one when the 6 was cut off. Hence the complete square is 54,756. When the number exceeds 316, find the square of the two terminal figures in the zero column, take the last two figures of it and append them to the entry, diminishing the terminal figure of the entry by one if the addendum equals or exceeds 50. For example, the last two figures of the square of 96 is 16, the entry for 896 is 8028, hence the complete square is 802,816. The table at the bottom of page 37 gives the square of the reciprocal of any number of two digits. Thus the square of — is .0865. When the decimal point is shifted in the 3.4 ^ argument, the decimal point of the entry requires to be shifted by twice the number of places in the opposite direction. Thus the square of -— is 1.32. .87 XV. Cubes, pp. 38-39. This table gives to four figures the cube of any number of three figures, and in the column the complete cube of any number of two figures. When the decimal point is shifted in the number, the decimal point of the cube requires to be shifted thrice the number of places in the same direction. Thus the cube of 1.23 is 1.861, that of 12.3 is 1801, that of .123 is .001861. The small table of page 39 gives the cube of the reciprocal of the number. Thus the cube of — is .00142. When the 8.9 decimal point is shifted in the number, the decimal point in the reciprocal of the cube is shifted thrice the number of places in the opposite direction. I EXPLANATION OF TABLES. 97 XVI. Square Roots, pp. 40-43. Tlie first part of the table, pages 40-41, gives the square root of any number of three significant figures, when the decimal point is after the first figure, or is any even number of places to the right or left of that position ; while the second part of the table, pages 42-43, gives the square root, when the decimal point is after the second figure or any even number of places to the right or left of that position. The square root is given to five figures, the initial figure being printed only in the column. When the decimal point of the number is shifted any even number of j)laces from its position after either the first digit or after the second digit, the decimal point in the correspond- ing entry shifts by half the number of places in the same direction. Thus, Va87= 3.1417, V98J = 9.9348, V9S7 = 31.417, V9870 = 99.348, Vi987 = .99348, V.0987 = .31417. The small table of page 41 gives the square root of the recipro- cal of any number of two figures, the decimal point being after the first figure ; while the small table of page 43 gives the same when the decimal point is after the second figure. Thus, _4:_ = .319 Ji_ = .101, J-- = 1.01, —4= = .0319. V9.8 V98 V.98 V980 XVII. Cube Roots, pp. 44-49. The first part of the table gives the cube root, when the decimal point is after the first significant figure, or when dis- placed any multiple of three places to the right or left of that position ; the second part similarly when the decimal point is after the second significant figure ; and the third part when it is after the third. A displacement of three places in the num- ber causes a displacement of one place in the same direction in the cube root. Thus, a/i:23 = 1.0714, ■v/12:3 = 2.3084, a/123 = 4.9732, ■v/i230 = 10.714, -v/:0l23 = .23084, -v^m = .49732. Similarly the three small tables give the cube root of the reciprocal of any two figures for the three distinct positions of the decimal point. 98 EXPLANATION OF TABLES. XVIII. Multiples, pp. 50-67. This table gives tlie first nine multiples of any number of three figures, and the folding table at the end gives the same for any number of two figures. By means of this table and our knowledge of the ordinary multiplication table we can write down any of the nine multiples of a number of four figures, and with the help of the folding table we can do the same for any number of five figures. By a double reference to the table we obtain a multiple of six figures, and so on. Thus, 8 times 789 = 6312 8 times 6789 = 6312 48 54312 8 times 56789= 6312 448 454332 8 times 456789 = 6312 3648 3654312 To multiply any tivo numbers together. Consider, for example, the product of 123,456,789 and 6987. Turn up the multiples of 789, and write down the 7, 8, 9, and 6 multiples under one another in the usual manner, only space is to be left between each pair of multiples for another row of figures ; then turn up the multiples of 456, write down the 7 multiple with its initial figure below the fourth figure of the 7 multiple of 789, and similarly for the other multiples ; then turn up the multiples of 123, write down the 7 multiple with its first figure above the fourth figure of the 7 multiple of 456, and so on, as follows .• 861 5523 3192 984 6312 3648 1107 7101 4104 738 4734 2736 86259258474 3 EXPLANATION OF TABLES. 99 To divide one number by another. For example, to divide 4,567,890 by 567. Turn up the mul- tiples of 567 ; find the next lower to 4567, deduct it ; take down another figure, find the next lower multiple to the number so formed, and so on, as follows : 567)4567890(8056 4536 If the divisor consist of four figures, as 5678, turn up the multiples of 567 and correct them mentally for the additional figure 8. If there are five figures, as 56,789, correct the multi- ples of 567 by adding the multiples of 89 from the folding table. XIX. Circumference of Circle, pp. 68-69. When the decimal point is changed in the diameter, the decimal point in the circumference changes by an equal num- ber of places in the same direction. When n represents the radius, the circumference is obtained by doubling the entry. The small table, page 69, gives the value, the logarithm, and the reciprocal of frequently occurring constants, which involve TT. The mantissa of the logarithm of the reciprocal is the com- plement to 1 of the mantissa of the logarithm of the constant. Thus, log - is 1.5029. XX. Area of Circle, pp. 70-71. When n denotes the radius, the area is obtained by multi- plying the entry by 4. When the decimal point is changed in the diameter, the decimal point of the area changes by double the number of places in the same direction. Thus, when the diameter is 3.96 the area is 12.32, when 39.6 then 1232, when .396 then .1232. The diameter of a circle of given area is obtained by the inverse use of the table. When n denotes the diameter of a sphere, the surface is 100 EXPLANATION OF TABLES. ttw*. Hence the surface of a sphere of given diameter is ob- tained by multiplying the entry of the table by 4. The auxiliary table, page 71, gives the decimal equivalents of the binary divisions of the inch, and also the decimal equiv- alents of a number of inches as part of the foot or of the yard. Thus, the area of a circle of 3f inch diameter, is that of 3.375 inch; hence, 8.920 + 26, that is, 8.946 square inches. XXI. Content of Sphere, pp. 72-73. This table gives the content of a sphere of which n is the diameter. When the radius is given, the spherical content is obtained by multiplying the tabular entry by 8. When the decimal point is changed in the diameter, the decimal point of the content is changed thrice the number of places in the same direction. The small table at the bottom of page 73 gives the logarithm of the product of successive integers from 1 up to n, and the logarithm of the powers of 2 up to the 29th. For example, logl.2.3.4.5.6 is 2.8573. XXII. Hyperbolic Logarithms, pp. 74-75. This table gives directly the hyperbolic or natural logarithm of any sequence of three signiticant figures, the decimal point being after the first. When the decimal point of the sequence is shifted n places to the right from the above position, find the logarithm of 10" in the auxiliary table and add it to the entry ; and when the decimal point is shifted n places to the left, add the logarithm of 10"". Thus, log 56.7 = 1.7352 log 567 = 1.7352 2.3026 4.6052 4.0378 6.3404 log .567 = 1.7352 log. .0507 = 1.7352 3.6974 5.3948 1.4326 3.1300 By m in the auxiliary table is meant the modulus or multi- plier for converting the natural or hyperbolic logarithm of a number into the common logarithm of the number, and by — is m meant the reciprocal modulus or multiplier for converting the common log of a number into the natural. EXPLANATION OF TABLES. 101 Example : To fiud the hyperbolic log of 1889, given the com- mon log to be 3.2762. The equivalent of 3 =6.9078 The equivalent of .2 = 0.4605 The equivalent of .07 = 0.1612 The equivalent of .006 = 0.0138 The equivalent of .0002 = 0.0005 Therefore hyp. log. of 1889 = 7.5438 XXIII. Amount of One Unit of Money at the End of a Given Number of Years, p. 76. The argument in the left-hand column is the number of years during which one unit of money (whether dollar, pound, franc, or mark) has been allowed to accumulate at compound interest, while the argument in the top row is the rate of interest expressed as so much per cent per year. The general expression for the amount of one unit in n years at r per cent per year is fl + ^^ • Given the prindixd, the number of years, and the rate, to find the amount. Find from the table the amount of one unit of money for the given number of years and rate, and multiply that number by the principal. For example, to find the amount of $123 at the end of 25 years at 6 per cent per year. The entry for 25 years and 6 per cent is 4.292 ; to find the i)roduct of this number and 123, turn up the multiples of 123. 4.292 X 123 = 246 1107 24 6 492 527.916 As the fourth figure of 4.292 is inexact, the figures 1 and 6 of the product are not significant; hence the result is $527.9. Given the amount, the number of years, and the rate, to find the principal. Find from the table the amount of one unit for the given number of years and rate, and divide the total amount by it ; the quotient is the principal. 102 EXPLANATION OF TABLES. Given the principal^ the rate, and the amount, to find the num- ber of years. Divide the amount by the principal and compare the quo- tient with the entries in the column under the given rate. For example, to find the number of years in which $456 becomes 1^742.82 at 5 per cent per year. Dividing 742.82 by 456, we get 1.629, which is the entry in the 5 per cent column for 10 years. Given the principal, the amount, and the member of years, to find the rate. Divide the amount by the principal and compare the quotient with the entries in the row of the given number of years. To find the amount of a unit of money for a number of years and a fraction of a year. Find the difference between the entry for the number of years and the next higher entry, multiply it by the fraction of the year, and add the resvdt to the lower entry. For ex- ample, to find it for 7 years and 3 months, the rate of interest being 8 per cent. The entry for 7 years is 1.714, and that for 8 is 1.851 ; the difference is 137, the fourth part of which is 34, which added to 1.714 gives 1.748. To find the amount of a unit of money for an intermediate rate of interest. The value may be found approximately by applying the principle of proportional parts as above. For example, the amount for 9 years at 5^ per cent is 1.551, plus one-half of 138; hence, 1.620. To find the amount of one unit of money for a number of years greater than 49. Break the number of years into parts each not greater than 49, multiply together the entries for the several parts ; the result is the amount for the given number of years. For example, to find the amount of one unit of money for 70 years at 10 per cent. The entry for 40 years is 45.26, and that for 30 years is 17.45. To find the product of 45.26 by 17.45, turn up the multiples of 745 and correct them for the 1 ; the result is 789.7870, but the last three figures are not significant; hence, 789.8. The true value is 789.747. EXPLANATION OF TABLES. 103 This problem may also be solved by means of the small table of logarithms, page 5, where the logs of the coefficients from 1.000 up to 1.100 are given to six places in order that their multiples may be obtained exact to four places. The log of 1.10 is .041393, which multiplied by 70 gives 2.8975, the antilogarithm of which is 789.8. XXIV. Present Value of 1000 Units of Money, p. 77. This table gives the present value of 1000 units of money due n years hence, the rate of interest having any one of the values in the top row. The entry is given, not for 1, but for 1000, in order to simplify the specification of the decimal point. When an entry is taken out, the decimal point ought to be shifted three places to the left. The method of using this table is the same as that fop Table XXIII. XXV. Amount of an Annuity when paid at the End of Bach Year, p. 78. This table gives the amount of an annuity of one unit of money per year, when the annuity is allowed to accumulate for n years, the first payment being made at the end of one year from the time of reckoning. The method of using Table XXIII, applies to this table, " one unit of money per year" being substituted for "one unit of money," excepting the rule at the end for extending the table. In order to extend the table, the value of ( 1 -| — — ] must be found by that rule, and the result substituted in the formula at the top of the table. XXVI. Present Value of the Preceding, p. 79. This table gives the value at the beginning of the time of reckoning of an annuity of one unit of money per year allowed to accumulate for a given number of years, the first payment being made at the end of one year from the beginning of the time of reckoning. The method of using the table is the same as for Table XXV. 104 EXPLANATION OF TABLES. XXVII. Amount of an Annuity when paid at the Be- ginning of Each Year, p. 80. Tliis table gives the timount of au annuity of one unit of money per year, when the several payments are allowed to grow at any one of the rates of interest specified, the first pay- ment being made at the beginning of the time of reckoning. The method of using the table is the same as for Table XXV. XXVIII. Annuity required to extinguish a Debt of 1000, p. 81. This table gives the annual sum to be paid for a given num- ber of years, the first payment being made one year from the present time, in order to extinguish a present debt of 1000 units of money. Here the 1000 is introduced for the same reason as in the case of Table XXIV. To extend the table, the extended value of (1 + r)" must be found and substituted in the formula printed at the top. XXIX. Least Divisors, pp. 82-85. This table gives the least divisor of any number up to 10,000. The first two figures of the number are given in the left-hand column, the third figure in the top row and the terminal figure in the row beneath. The only terminal figures entered are 1, 3, 7, 9, because any number which terminates otherwise is evidently divisible by 2 or 5. To find the factors of any number less than 10,000. If it is an even number, divide out 2 until the remainder is odd ; if it then ends in 5, divide out the power of 5 ; then enter the table with the remaining quotient to find its least divisor ; divide out that divisor, and with the then remaining quotient enter the table again ; and so on until the remaining quotient is a prime, which is indicated in the table by a bar. Example : 1889 is a prime. 9876 = 2 X 4938 = 2^ x 2469. Xow 2469 has least divisor 3, and quotient is 823, and 823 is a prime. Hence, 9876 = 22 X 3 X 823. EXPLANATION OF TABLES. 105 XXX. Exponentials, p. 86. The upper part of the table contains the ascending powers of e from .00 to .99 and from 1.0 to 9.9; and the lower part the corresponding descending powers. The upper part forms a small table of hyperbolic antilogarithms. At the bottom of the page we have the first nine multiples of e and of the reciprocal of e, the first nine fractional powers of e, both positive and negative, and the poAvers of e given by the first nine multiples of ^, both positive and negative. XXXI. Multiples (Folding Leaf). This table contains the first nine multiples of the numbers from 1 to 99. It may be used as a table of proportional parts for tenths by inserting a decimal point before the last figure, and for hundredths by inserting the point before the second last figure. XXXI. MULTIPLES. 1 1 2 8 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 2 3 4 5 6 7 8 9 •10 11 12 13 14" 15 16 n IS 19 2 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 3 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 1 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 58 72 76 5 5 10 15 20 25 30 35 40 45 SO 55 .» 65 70 75 80 85 90 95 6 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 102 108 114 7 7 14 21 28 35 42 49 56 63 70 77 34 91 98 105 112 119 126 133 8 16 24 32 40 48 56 64 72 80 88 % 1(M 112 120 128 136 144 152 9 9 18 27 36 45 54 63 72 81 90 99 108 117 126 135 144 153 162 171 20 21 22 23 24 25 26 27 28 29 80 81 a 33 34 35 36 37 38 39 1 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 2 40 42 44 46 48 SO 52 54 56 58 60 62 .i4 66 68 70 72 74 76 78 3 60 63 66 69 72 75 78 81 84 87 90 93 % 99 102 105 108 111 114 117 4 80 84 88 92 96 100 104 108 112 116 120 124 128 132 136 140 144 148 152 156 5 100 105 110 115 120 125 130 135 140 145 ISO 155 160 165 170 175 180 185 190 195 6 120 126 132 138 144 1.50 156 162 168 174 180 186 192 198 204 210 216 222 228 234 7 140 147 154 161 163 175 182 189 196 203 210 217 224 231 238 245 252 259 256 273 8 160 168 176 184 192 200 208 216 224 232 240 248 256 264 272 280 288 296 304 312 9 ISO 189 198 207 216 225 234 243 252 261 270 1279 2SS 297' 306 315 324 333 342 351 40 41 42 43 44 45 46 47 48 49 50 51 5i 53 54 55 56 57 58 59 ^ 40 41 42 43 44 45 46 47 48 49 ~5'(r 51 52 53 54 55 55 57 58 59 2 82 84 86 88 90 92 94 96 98 100 102 lot 106 108 110 112 114 116 118 3 120 123 126 129 132 135 138 141 144 147 150 153 156 159 162 165 158 171 174 177 4 160 164 168 172 176 180 184 188 192 196 200 204 208 212 216 220 224 228 232 236 5 200 205 210 215 220 225 230 235 240 245 2.50 255 260 265 270 275 280 285 290 295 6 240 246 252 258 264 270 276 282 288 294 300 305 312 318 324 330 336 342 348 354 7 280 287 294 301 308 315 322 329 336 343 3.50 357 364 371 378 385 392 399 406 413 H 320 32S 336 344 352 iM 3r,8 376 384 392 400 408 416 424 432 440 448 456 464 472 9 360 369 378 387 396 405 414 423 432 441 450 4.59 468 477 486 495 S(M 513 522 531 (10 61 62 63 64 65 66 67 68 69 70 71 73 78 74 75 76 77 78 79 ^ 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 2 120 122 124 126 128 1.30 132 134 136 138 140 142 144 146 148 150 152 154 1.56 158 183 186 189 192 195 198 201 204 207 210 213 216 219 222 225 228 231 234 237 4 240 2U 248 252 256 260 2f>4 268 272 276 280 284 2SS 292 296 300 304 308 312 316 9 300 .305 310 315 320 325 3.TO 335 340 345 350 355 iM 365 370 375 380 385 390 395 6 360 366 372 378 384 390 396 402 408 414 420 426 432 438 444 4.50 456 462 468 474 7 420 427 434 441 448 455 462 469 476 483 490 497 .504 511 518 525 .532 539 545 553 488 496 504 512 520 528 536 .544 552 560 .568 576 584 592 500 608 616 624 632 9 540 .549 558 567 576 585 594 603 612 621 630 639> 648 657 565 675 584 693 702 711 80 81 82 83 84 85 86 87 88 8!) 90 91 92 98 94 95 96 97 98 99 1 80 81 82 83 84 85 86 87 88 89 90 , 91 92 93 94 95 96 97 98 99 2 160 162 164 166 168 170 172 174 176 178 ISO 1S2 1S4 186 188 190 192 194 196 198 3 240 243 246 249 252 255 258 261 264 267 270 273 276 279 282 2S5 288 291 2'H 297 4 320 324 328 332 336 MO 344 348 352 3.S6 360 364 368 372 375 380 384 388 392 396 5 400 405 410 415 420 425 430 . 435 -l-IO 445 4.50 455 460 465 470 475 480 485 4'X) 495 » 480 486 492 498 504 510 516 522 528 534 540 546 552 5.58 564 570 576 582 5SS 594 7 560 .567 574 581 5SS 595 602 609 616 623 630 637 644 651 658 665 672 679 686 593 S 640 648 656 664 672 680 6S8 696 704 712 720 728 73f. 744 752 760 768 776 784 792 9 720 729 738 747 756 765 774 783 792 801 810 819 82S 837 846 855 864 873 SS2 891 14 DAY USE RETURN TO DESK FROM ^JOiWCH .BftaftOWED STATISTICS LIBRARY This book is due on the last date stamped below, or on the date to which renewed. Renewed books are subject to immediate recall. MTFT^hi wM^si-^ oow-^-wo H A B/lX i LD 21-40m.-10,'65 (F77638l0)476 General Library ' University of California Berkeley k MJe98?46 4-^o.^n Cv^Tcl I U C. BERKELEY L)BBAR1ES lllllll