TABLE DS1.1: Some important constants Name Symbol Value Speed of light in vacuum Charge of electron Gravitational constant Planck constant Boltzmann constant Avogadro number Universal gas constant Mass of electron Mass of neutron Mass of proton Electron-charge to mass ratio Faraday constant Rydberg constant Bohr radius Stefan-Boltzmann constant Wien’s constant Permittivity of free space Permeability of free space c e G h k NA R m e m n m p e/m e F R a0 σ b ε0 1/4πε0 μ0 2.9979 108 m s–1 1.602 10–19 C 6.673 10–11 N m2 kg –2 6.626 10–34 J s 1.381 10–23 J K–1 6.022 1023 mol–1 8.314 J mol–1 K–1 9.110 10–31 kg 1.675 10–27 kg 1.673 10–27 kg 1.759 1011 C/kg 9.648 104 C/mol 1.097 107 m–1 5.292 10–11 m 5.670 10–8 W m–2 K–4 2.898 10–3 m K 8.854 10–12 C2 N–1 m–2 8.987 109 N m2 C–2 4π 10–7 T m A–1 ≅ 1.257 10–6 Wb A–1m–1 TABLE DS 1.2: Other useful contstants TABLE DS 2: E. M.F. of cells Name Symbol Value Mechanical equivalent of heat Standard atmospheric pressure Absolute zero Electron volt Unified Atomic mass unit Electron rest energy Energy equivalent of 1 u Volume of ideal gas (0 °C and 1 atm) Acceleration due to gravity (sea level, at equator) J 1 atm 0 K 1 eV 1 u mc2 1 uc2 V g 4.186 J cal–1 1.013 105 Pa – 273.15 °C 1.602 10–19 J 1.661 10–27 kg 0.511 MeV 931.5 MeV 22.4 L mol–1 9.78049 m s–2 Cell E.M.F. (V) Cell E.M.F. (V) Daniell 1.08 – 1.09 Cadmium at 20°C 1.018 54 Grove 1.8 – 1.9 Lead accumulator 1.9 – 2.2 Lechlanche 1.45 Edison cell 1.45 Voltaic 1.01 Clarke 1.43 Bunsen 1.95 Ni-Fe 1.20 TABLE DS 3: Refractive index of substances : For sodium light λ= 5896 10–10 m Solid Refractive index Liquid Refractive Index Diamond 2.417 Canada balsam 1.53 Glass (crown) 1.48 – 1.61 Water 1.333 Glass (flint) 1.53 – 1.96 Alcohol (ethyl) 1.362 Glass(soda) 1.50 Aniline 1.595 Benzene 1.501 Ice 1.31 Cedar oil 1.516 Mica 1.56 – 1.60 Chloroform 1.450 Rock-salt 1.54 Ether 1.350 Glycerine 1.47 Quartz (O - Ray) 1.5443 Olive oil 1.46 Quartz (E - Ray) 1.5534 Paraffin oil 1.44 Quartz (fused) 1.458 Kerosene oil 1.39 Turpentine oil 1.44 TABLE DS 4: Wavelength of spectral lines : (in Å, 1Å = 10–10 m) [The visible spectrum colours are indicated – r, o, y, g, b, i, v ] Hydrogen Helium Mercury Neon Sodium 3970v 3889v 4047v 5765 y (D2) 5890 o 4102v 4026v 4078v 5853 y (D1) 5896 o 4340b 4471b 4358v 5882 o 4861 gb 5876y 4916 b, g 6597 r 6563 r 6678 r 4960 g 7245 r 7065 r 5461 g 5770y 5791y 6152o 6322o 288 TABLE DS 5: Electromagnetic spectrum (wavelength) Range Wavelength Wireless waves 5 m and above Infra-red 3.0 10–4 m to 7.5 10–7 m Visible Red 7.5 10–7 m to 6.5 10–7 m Visible Orange 6.5 10–7 m to 5.9 10–7 m Visible Yellow 5.9 10–7 m to 5.3 10–7 m Visible Green 5.3 10–7 m to 4.9 10–7 m Visible Blue 4.9 10–7 m to 4.2 10–7 m Visible Indigo 4.2 10–7 m to 3.9 10–7 m Ultra Violet 3.9 10–7 m to 1.8 10–7 m Soft X-Rays 2.0 10–7 m to 1.0 10–7 m Hard X-Rays 1.0 10–10 m to 1.0 10–11 m Gamma Rays 5.0 10–11 m to 5.0 10–12 m TABLE DS 6: Standard Wire Gauges (S.W.G.) and resistances for a metre length of constantan and manganin S.W.G. Diameter Resistance (Ω) No. (mm) Copper Constantan Manganin (60% Cu, 40% Ni) (84% Cu, 4% Ni, 12% Mn) 10 3.25 0.0021 0.057 0.051 12 2.64 0.0032 0.086 0.077 14 2.03 0.0054 0.146 0.131 16 1.63 0.0083 0.228 0.204 18 1.22 0.0148 0.405 0.361 20 0.914 0.0260 0.722 0.645 22 0.711 0.0435 1.20 1.07 24 0.559 0.070 1.93 1.73 26 0.457 0.105 2.89 2.58 28 0.374 0.155 4.27 3.82 30 0.315 0.222 6.08 5.45 32 0.274 0.293 8.02 7.18 34 0.234 0.404 11.1 9.9 36 0.193 0.590 16.2 14.5 38 0.152 0.950 26.0 23.2 40 0.122 1.48 40.6 36.3 42 0.102 2.10 58.5 53.4 44 0.081 3.30 91.4 81.7 46 0.061 5.90 162.5 145.5 289 TABLE DS 7: Dielectric Constants of Common Materials Material Temperature Frequency Dielectric (°C) (Hz) Constant Amber 20 106 2.8 Amber 20 3 109 2.6 Soda glass 20 106 7.5 Fused quartz 20 103 to 108 3.8 Liquid paraffin (Medical Grade) 20 103 2.2 Transformer oil (Class B) 20 103 2.2 Marble 20 106 8 Sand (dry) 20 106 3 Sandstone 20 106 10 Paper (Oil impregnated condenser tissue) 20 103 2.3 Mica 20 103 to 108 5.4 to 7 Epoxy resin (e.g. Araldite) 20 106 3.3 Cellulose Acetate 20 106 3.5 Vinyl Acetate (Plasticised) 20 106 4 Vinyl Chloride (P.V.C.) 20 106 4 Ebonite (Pure) 20 106 3 Rubber (Vulcanised soft) 20 106 3.2 Rubber, Synthetic 20 106 2.5 Paraffin wax 20 106 2.2 Sulphur 20 3 109 3.4 Walnut wood (dry) 20 107 2.0 Walnut (17% moisture) 20 107 5 Vacuum NA any 1.00000 Air 20 Upto 3 109 1.00054 Porcelain 20 106 5.5 Barium titanate 20 106 1200 Rutile group 20 106 to 109 40 to 80 Water 20 109 80 Water 20 1010 64 290 TABLE DS 8: Typical objects with electrostatic charge Object C V Q Energy E = 1 2 C V2 1. Balloon of 20cm diameter rubbed all round by nylon cloth 11 pF 200 V 2.2 nC 0.22 μJ* 2. Metal sphere on insulated stand rubbed by silk (dia 9 cm) 5 pF 500 V 2.5 nC 0.62 μJ* 3. Metal sphere charged by a school type Van-de-Graaff generator 5 pF 0.25 MV 1.25 μC 0.16 J 4. Boy on an insulated stool repeatedly charged by electrophorus 50 pF 3000 V 150 nC 225 μJ* 5. 30 cm 30 cm improvised capacitor with wax soaked tissue paper dielectric (K = 2.7, A = 700 cm and d = 0.4 mm) and charged by a 9 V battery. 4 nF 9 V 36 nC 162 nJ* 6. -do- plates separated to 2 cm apart. 80 pF 450 V 36 nC 8.1 μJ* 7. Flash gun capacitors (professional type) 500 pF 400 V 0.2 nC 40 J 8. Average lighting between earth and cloud (h = 1 to 5 km) 108 V to 109 V 20 C 109 to 1010 J 9. (a) Earth and ionised air of high conductivity at top of stratosphere (h = 50 km) (b) Ionisation current between earth and ionosphere in fair weather. -- 0.4 MV 0.4 MV 5.7 105 C 1800 C/s 1011 J 7 108 J/s 10. Charged plate of good electrophorous (diameter 20 to 30 cm) in dry weather, after lifting it up. 10 pF 3000 V 30 nC 45 μJ* *These bodies in your school laboratory are not dangerous to touch. 291 TABLE DS 9: Electrical resistivities of typical metals and alloys Resistivity (10–8 ohm metre) Material 0°C 100°C 300°C 700°C Temperature coefficient at 0°C (over range 0°C-100°C (10–4 ) Aluminium 2.45 3.55 5.9 24.7 45 Chromium 12.7 16.1 25.2 47.2 Copper 1.56 2.24 3.6 6.7 43 Iron 8.9 14.7 31.5 85.5 65 Lead 19.0 27.0 50 107.6 42 Gold 2.04 2.84 40 Mercury 94.0766* 103.5 128 10 Nickel 6.14 10.33 22.5 68 Platinum 9.81 13.65 21.0 34.3 39.2 Platinum- Rhodium (87/13) 19.0 22.0 15.6 Platinum- Rhodium (90/10) 18.7 21.8 16.6 Platinum- Iridium (90/10) 24.8 28.0 13 Silver 1.51 2.13 3.42 6.5 41 Tin 11.5(20°C) 15.8 50 60 46 Tungsten 4.9 7.3 12.4 24 48 Zinc 5.5 7.8 13.0 37(500°C) 42 Mercury at 0°C is used as a secondary standard to realise the standard unit of resistance, ohm. 292 TABLE DS 10: Electrical resistivities of common insulators and semiconductors Substance Resistivity Substance (Resistivity (ohm metre) (ohm metre) Diamond 1010 to 1011 Carbon 0°C 3.5 10–5 Ebonite 1014 Carbon 500°C 2.7 10–5 Glass (Soda lime) 5 109 Carbon 1000°C 2.1 10–5 Glass (Pyrex) 1012 Carbon 2000°C 1.1 10–5 Glass (conducting) 5 106 Carbon 2500°C 0.9 10–5 Mica 1011 1015 Germanium 0°C 0.46 Paper (dry) 1010 Silicon 0°C 2300 Paraffin wax 1014 Porcelain 1010 to 1013 Sulphur (rhombic) 2 1021 TABLE DS : 11 Data for intrinsic and extrinsic semiconductors Material Energy Gap (eV) No. density per m3* electron-hole pairs at 300 K Mobility Electrons (m2 V–1 s–1) Mobility holes (m2 V–1 s–1) Conductivity (S m–1) Density (kg m–3) Semiconductors Germanium 0.76 6 1019 0.39 0.19 2.18 5320 Silicon 1.12 7 1015 0.135 0.048 4.4 10–4 2300 Doped Si with P with Insulators 0.045 2.5 1021 0.135 ... 2.5 10 2300 Diamond 6 to 12 ≈107 *Values are approximate, they give an idea about order of magnitude only. 293 TABLE DS 12: Alloys of high resistance Alloy Resistivity 20 °C Temperature coefficient Max. Operating temp(10–8 ohm metre) in range 0-100°C (10–4) erature (°C) Constantan (58.8% Cu, 40% Ni,1.2% Mn) 44 to 52 – 0.4 to + 0.1 500 German Silver (65% Cu, 20% Zn, 15% Ni) 28 to 35 + 0.4 150 to 200 Manganin (85% Cu, 12% Mn, 3% Ni) 42 to 48 0.3 100 Nickeline (54% Cu, 20% Zn, 26% Ni) 39 to 45 0.2 150 - 200 Nichrome (67.5% Ni, 15% Cr, 16% Fe, 1.5% Mn) 100 to 110 2.0 1000 TABLE DS 13: Transition temperature to the superconducting state Substance Transition Substance Transition temperature (K) temperature (K) Metals: Compounds: Cadmium 0.6 NiBi 4.2 Zinc 0.8 PbSe 5.0 Aluminium 1.2 NbB 6.0 Uranium 1.3 Nb2C 9.2 Tin 3.7 nBC 10.1 to 10.5 Mercury 4.7 nBN 15 to 16 Lead 7.3 Nb3Sn 18 Niobium 9.2 YBa2Cu3O7 90 294 LOGARITHMS TABLE I N 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 10 0000 0043 0086 0128 0170 0212 0253 0294 0334 0374 5 4 9 8 13 12 17 16 21 26 2O 24 30 28 34 38 32 36 11 0414 0453 0492 0531 0569 0607 0645 0682 0719 0755 44 8 7 12 11 16 15 20 23 18 22 27 26 31 35 29 33 12 0792 0828 0864 0899 0934 0969 1004 1038 1072 1106 3 3 7 7 11 10 14 14 18 21 17 20 25 24 28 32 27 31 13 1139 1173 1206 1239 1271 1303 1335 1367 1399 1430 3 3 6 7 10 10 13 13 16 19 16 19 23 22 26 29 25 29 14 1461 1492 1523 1553 1584 1614 1644 1673 1703 1732 3 3 6 6 9 9 12 12 15 19 14 17 22 20 25 28 23 26 15 1761 1790 1818 1847 1875 1903 1931 1959 1987 2014 3 3 6 6 9 8 11 11 14 17 14 17 20 19 23 26 22 25 16 2041 2068 2095 2122 2148 2175 2201 2227 2253 2279 3 3 6 5 8 8 11 10 14 16 13 16 19 18 22 24 21 23 17 2304 2330 2355 2380 2405 2430 2455 2480 2504 2529 3 3 5 5 8 8 10 10 13 15 12 15 18 17 20 23 20 22 18 2553 2577 2601 2625 2648 2672 2695 2718 2742 2765 2 2 5 4 7 7 9 9 12 14 11 14 17 16 19 21 18 21 19 2788 2810 2833 2856 2878 2900 2923 2945 2967 2989 2 2 4 4 7 6 9 8 11 13 11 13 16 15 18 20 17 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 3010 3222 3424 3617 3802 3979 4150 4314 4472 4624 4771 4914 5051 5185 5315 5441 5563 5682 5798 5911 6021 6128 6232 6335 6435 6532 6628 6721 6812 6902 3032 3243 3444 3636 3820 3997 4166 4330 4487 4639 4786 4928 5065 5198 5328 5453 5575 5694 5809 5922 6031 6138 6243 6345 6444 6542 6637 6730 6821 6911 3054 3263 3464 3655 3838 4014 4183 4346 4502 4654 4800 4942 5079 5211 5340 5465 5587 5705 5821 5933 6042 6149 6253 6355 6454 6551 6646 6739 6830 6920 3075 3284 3483 3674 3856 4031 4200 4362 4518 4669 4814 4955 5092 5224 5353 5478 5599 5717 5832 5944 6053 6160 6263 6365 6464 6561 6656 6749 6839 6928 3096 3304 3502 3692 3874 4048 4216 4378 4533 4683 4829 4969 5105 5237 5366 5490 5611 5729 5843 5955 6064 6170 6274 6375 6474 6471 6665 6758 6848 6937 3118 3324 3522 3711 3892 4065 4232 4393 4548 4698 4843 4983 5119 5250 5378 5502 5623 5740 5855 5966 6075 6180 6284 6385 6484 6580 6675 6767 6857 6946 3139 3345 3541 3729 3909 4082 4249 4409 4564 4713 4857 4997 5132 5263 5391 5514 5635 5752 5866 5977 6085 6191 6294 6395 6493 6590 6684 6776 6866 6955 3160 3365 3560 3747 3927 4099 4265 4425 4579 4728 4871 5011 5145 5276 5403 5527 5647 5763 5877 5988 6096 6201 6304 6405 6503 6599 6693 6785 6875 6964 3181 3385 3579 3766 3945 4116 4281 4440 4594 4742 4886 5024 5159 5289 5416 5539 5658 5775 5888 5999 6107 6212 6314 6415 6513 6609 6702 6794 6884 6972 3201 3404 3598 3784 3962 4133 4298 4456 4609 4757 4900 5038 5172 5302 5428 5551 5670 5786 5899 6010 6117 6222 6325 6425 6522 6618 6712 6803 6893 6981 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4 4 4 4 4 3 3 3 3 3 3 33 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 6 6 6 6 5 5 5 5 5 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 8 8 8 7 7 7 7 6 6 6 6 6 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 4 4 4 11 13 10 12 10 12 9 11 9 11 9 10 8 10 8 9 8 9 7 9 7 9 7 8 7 8 6 8 6 8 6 7 6 7 6 7 6 7 5 7 5 6 5 6 5 6 5 6 5 6 5 6 5 6 5 5 4 5 4 5 15 14 14 13 12 12 11 11 11 10 10 10 9 9 9 9 8 8 8 8 8 7 7 7 7 7 7 6 6 6 17 19 16 18 15 17 15 17 14 16 14 15 13 15 13 14 12 14 12 13 11 13 11 12 11 12 10 12 10 11 10 11 10 11 9 10 9 10 9 10 9 10 8 9 8 9 8 9 8 9 8 9 7 8 7 8 7 8 7 8 295 LOGARITHMS TABLE 1 (Continued) N 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 50 6990 6998 7007 7016 7024 7033 7042 7050 7059 7067 1 2 3 3 4 5 6 7 8 51 7076 7084 7093 7101 7110 7118 7126 7135 7143 7152 1 2 3 3 4 5 6 7 8 52 7160 7168 7177 7185 7193 7202 7210 7218 7226 7235 1 2 2 3 4 5 6 7 7 53 7243 7251 7259 7267 7275 7284 7292 7300 7308 7316 1 2 2 3 4 5 6 6 7 54 7324 7332 7340 7348 7356 7364 7372 7380 7388 7396 1 2 2 3 4 5 6 6 7 55 7404 7412 7419 7427 7435 7443 7451 7459 7466 7474 1 2 2 3 4 5 5 6 7 56 7482 7490 7497 7505 7513 7520 7528 7536 7543 7551 1 2 2 3 4 5 5 6 7 57 7559 7566 7574 7582 7589 7597 7604 7612 7619 7627 1 2 2 3 4 5 5 6 7 58 7634 7642 7649 7657 7664 7672 7679 7686 7694 7701 1 1 2 3 4 4 5 6 7 59 7709 7716 7723 7731 7738 7745 7752 7760 7767 7774 1 1 2 3 4 4 5 6 7 60 7782 7789 7796 7803 7810 7818 7825 7832 7839 7846 1 1 2 3 4 4 5 6 6 61 7853 7860 7768 7875 7882 7889 7896 7903 7910 7917 1 1 2 3 4 4 5 6 6 62 7924 7931 7938 7945 7952 7959 7966 7973 7980 7987 1 1 2 3 3 4 5 6 6 63 7993 8000 8007 8014 8021 8028 8035 8041 8048 8055 1 1 2 3 3 4 5 5 6 64 8062 8069 8075 8082 8089 8096 8102 8109 8116 8122 1 1 2 3 3 4 5 5 6 65 8129 8136 8142 8149 8156 8162 8169 8176 8182 8189 1 1 2 3 3 4 5 5 6 66 8195 8202 8209 8215 8222 8228 8235 8241 8248 8254 1 1 2 3 3 4 5 5 6 67 8261 8267 8274 8280 8287 8293 8299 8306 8312 8319 1 1 2 3 3 4 5 5 6 68 8325 8331 8338 8344 8351 8357 8363 8370 8376 8382 1 1 2 3 3 4 4 5 6 69 8388 8395 8401 8407 8414 8420 8426 8432 8439 8445 1 1 2 2 3 4 4 5 6 70 8451 8457 8463 8470 8476 8482 8488 8494 8500 8506 1 1 2 2 3 4 4 5 6 71 8513 8519 8525 8531 8537 8543 8549 8555 8561 8567 1 1 2 2 3 4 4 5 5 72 8573 8579 8585 8591 8597 8603 8609 8615 8621 8627 1 1 2 2 3 4 4 5 5 73 8633 8639 8645 8651 8657 8663 8669 8675 8681 8686 1 1 2 2 3 4 4 5 5 74 8692 8698 8704 8710 8716 8722 8727 8733 8739 8745 1 1 2 2 3 4 4 5 5 75 8751 8756 8762 8768 8774 8779 8785 8791 8797 8802 1 1 2 2 3 3 4 5 5 76 8808 8814 8820 8825 8831 8837 8842 8848 8854 8859 1 1 2 2 3 3 4 5 5 77 8865 8871 8876 8882 8887 8893 8899 8904 8910 8915 1 1 2 2 3 3 4 4 5 78 8921 8927 8932 8938 8943 8949 8954 8960 8965 8971 1 1 2 2 3 3 4 4 5 79 8976 8982 8987 8993 8998 9004 9009 9015 9020 9025 1 1 2 2 3 3 4 4 5 80 9031 9036 9042 9047 9053 9058 9063 9069 9074 9079 1 1 2 2 3 3 4 4 5 81 9085 9090 9096 9101 9106 9112 9117 9122 9128 9133 1 1 2 2 3 3 4 4 5 82 9138 9143 9149 9154 9159 9165 9170 9175 9180 9186 1 1 2 2 3 3 4 4 5 83 9191 9196 9201 9206 9212 9217 9222 9227 9232 9238 1 1 2 2 3 3 4 4 5 84 9243 9248 9253 9258 9263 9269 9274 9279 9284 9289 1 1 2 2 3 3 4 4 5 85 9294 9299 9304 9309 9315 9320 9325 9330 9335 9340 1 1 2 2 3 3 4 4 5 86 9345 9350 9355 9360 9365 9370 9375 9380 9385 9390 1 1 2 2 3 3 4 4 5 87 9395 9400 9405 9410 9415 9420 9425 9430 9435 9440 0 1 1 2 2 3 3 4 4 88 9445 9450 9455 9460 9465 9469 9474 9479 9484 9489 0 1 1 2 2 3 3 4 4 89 9494 9499 9504 9509 9513 9518 9523 9528 9533 9538 0 1 1 2 2 3 3 4 4 90 9542 9547 9552 9557 9562 9566 9571 9576 9581 9586 0 1 1 2 2 3 3 4 4 91 9590 9595 9600 9605 9609 9614 9619 9624 9628 9633 0 1 1 2 2 3 3 4 4 92 9638 9643 9647 9652 9657 9661 9666 9671 9675 9680 0 1 1 2 2 3 3 4 4 93 9685 9689 9694 9699 9703 9708 9713 9717 9722 9727 0 1 1 2 2 3 3 4 4 94 9731 9736 9741 9745 9750 9754 9759 9763 9768 9773 0 1 1 2 2 3 3 4 4 95 9777 9782 9786 9791 9795 9800 9805 9809 9814 9818 0 1 1 2 2 3 3 4 4 96 9823 9827 9832 9836 9841 9845 9850 9854 9859 9863 0 1 1 2 2 3 3 4 4 97 9868 9872 9877 9881 9886 9890 9894 9899 9903 9908 0 1 1 2 2 3 3 4 4 98 9912 9917 9921 9926 9930 9934 9939 9943 9948 9952 0 1 1 2 2 3 3 4 4 99 9956 9961 9965 9969 9974 9978 9983 9987 9997 9996 0 1 1 2 2 3 3 3 4 296 ANTILOGARITHMS TABLE II N 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 00 1000 1002 1005 1007 1009 1012 1014 1016 1019 1021 0 0 1 1 1 1 2 2 2 .01 1023 1026 1028 1030 1033 1035 1038 1040 1042 1045 0 0 1 1 1 1 2 2 2 .02 1047 1050 1052 1054 1057 1059 1062 1064 1067 1069 0 0 1 1 1 1 2 2 2 .03 1072 1074 1076 1079 1081 1084 1086 1089 1091 1094 0 0 1 1 1 1 2 2 2 .04 1096 1099 1102 1104 1107 1109 1112 1114 1117 1119 0 1 1 1 1 2 2 2 2 .05 1122 1125 1127 1130 1132 1135 1138 1140 1143 1146 0 1 1 1 1 2 2 2 2 .06 1148 1151 1153 1156 1159 1161 1164 1167 1169 1172 0 1 1 1 1 2 2 2 2 .07 1175 1178 1180 1183 1186 1189 1191 1194 1197 1199 0 1 1 1 1 2 2 2 2 .08 1202 1205 1208 1211 1213 1216 1219 1222 1225 1227 0 1 1 1 1 2 2 2 3 .09 1230 1233 1236 1239 1242 1245 1247 1250 1253 1256 0 1 1 1 1 2 2 2 3 .10 1259 1262 1265 1268 1271 1274 1276 1279 1282 1285 0 1 1 1 1 2 2 2 3 .11 1288 1291 1294 1297 1300 1303 1306 1309 1312 1315 0 1 1 1 2 2 2 2 3 .12 1318 1321 1324 1327 1330 1334 1337 1340 1343 1346 0 1 1 1 2 2 2 2 3 .13 1349 1352 1355 1358 1361 1365 1368 1371 1374 1377 0 1 1 1 2 2 2 3 3 .14 1380 1384 1387 1390 1393 1396 1400 1403 1406 1409 0 1 1 1 2 2 2 3 3 .15 1413 1416 1419 1422 1426 1429 1432 1435 1439 1442 0 1 1 1 2 2 2 3 3 .16 1445 1449 1452 1455 1459 1462 1466 1469 1472 1476 0 1 1 1 2 2 2 3 3 .17 1479 1483 1486 1489 1493 1496 1500 1503 1507 1510 0 1 1 1 2 2 2 3 3 .18 1514 1517 1521 1524 1528 1531 1535 1538 1542 1545 0 1 1 1 2 2 2 3 3 .19 1549 1552 1556 1560 1563 1567 1570 1574 1578 1581 0 1 1 1 2 2 3 3 3 .20 1585 1589 1592 1596 1600 1603 1607 1611 1614 1618 0 1 1 1 2 2 3 3 3 .21 1622 1626 1629 1633 1637 1641 1644 1648 1652 1656 0 1 1 2 2 2 3 3 3 .22 1660 1663 1667 1671 1675 1679 1683 1687 1690 1694 0 1 1 2 2 2 3 3 3 .23 1698 1702 1706 1710 1714 1718 1722 1726 1730 1734 0 1 1 2 2 2 3 3 4 .24 1738 1742 1746 1750 1754 1758 1762 1766 1770 1774 0 1 1 2 2 2 3 3 4 .25 1778 1782 1786 1791 1795 1799 1803 1807 1811 1816 0 1 1 2 2 2 3 3 4 .26 1820 1824 1828 1832 1837 1841 1845 1849 1854 1858 0 1 1 2 2 3 3 3 4 .27 1862 1866 1871 1875 1879 1884 1888 1892 1897 1901 0 1 1 2 2 3 3 3 4 .28 1905 1910 1914 1919 1923 1928 1932 1936 1941 1945 0 1 1 2 2 3 3 4 4 .29 1950 1954 1959 1963 1968 1972 1977 1982 1986 1991 0 1 1 2 2 3 3 4 4 .30 1995 2000 2004 2009 2014 2018 2023 2028 2032 2037 0 1 1 2 2 3 3 4 4 .31 2042 2046 2051 2056 2061 2065 2070 2075 2080 2084 0 1 1 2 2 3 3 4 4 .32 2089 2094 2099 2104 2109 2113 2118 2123 2128 2133 0 1 1 2 2 3 3 4 4 .33 2138 2143 2148 2153 2158 2163 2168 2173 2178 2183 0 1 1 2 2 3 3 4 4 .34 2188 2193 2198 2203 2208 2213 2218 2223 2228 2234 1 1 2 2 3 3 4 4 5 .35 2239 2244 2249 2254 2259 2265 2270 2275 2280 2286 1 1 2 2 3 3 4 4 5 .36 2291 2296 2301 2307 2312 2317 2323 2328 2333 2339 1 1 2 2 3 3 4 4 5 .37 2344 2350 2355 2360 2366 2371 2377 2382 2388 2393 1 1 2 2 3 3 4 4 5 .38 2399 2404 2410 2415 2421 2427 2432 2438 2443 2449 1 1 2 2 3 3 4 4 5 .39 2455 2460 2466 2472 2477 2483 2489 2495 2500 2506 1 1 2 2 3 3 4 5 5 .40 2512 2518 2523 2529 2535 2541 2547 2553 2559 2564 1 1 2 2 3 4 4 5 5 .41 2570 2576 2582 2588 2594 2600 2606 2612 2618 2624 1 1 2 2 3 4 4 5 5 .42 2630 2636 2642 2649 2655 2661 2667 2673 2679 2685 1 1 2 2 3 4 4 5 6 .43 2692 2698 2704 2710 2716 2723 2729 2735 2742 2748 1 1 2 3 3 4 4 5 6 .44 2754 2761 2767 2773 2780 2786 2793 2799 2805 2812 1 1 2 3 3 4 4 5 6 .45 2818 2825 2831 2838 2844 2851 2858 2864 2871 2877 1 1 2 3 3 4 5 5 6 .46 2884 2891 2897 2904 2911 2917 2924 2931 2938 2944 1 1 2 3 3 4 5 5 6 .47 2951 2958 2965 2972 2979 2985 2992 2999 3006 3013 1 1 2 3 3 4 5 5 6 .48 3020 3027 3034 3041 3048 3055 3062 3069 3076 3083 1 1 2 3 3 4 5 6 6 .49 3090 3097 3105 3112 3119 3126 3133 3141 3148 3155 1 1 2 3 3 4 5 6 6 297 ANTILOGARITHMS TABLE II (Continued) N 0 1 .50 3162 3170 .51 3236 3243 .52 3311 3319 .53 3388 3396 .54 3467 3475 .55 3548 3556 .56 3631 3639 .57 3715 3724 .58 3802 3811 .59 3890 3899 .60 3981 3990 .61 4074 4083 .62 4169 4178 .63 4266 4276 .64 4365 4375 .65 4467 4477 .66 4571 4581 .67 4677 4688 .68 4786 4797 .69 4898 4909 .70 5012 5023 .71 5129 5140 .72 5248 5260 .73 5370 5383 .74 5495 5508 .75 5623 5636 .76 5754 5768 .77 5888 5902 .78 6026 6039 .79 6166 6180 .80 6310 6324 .81 6457 6471 .82 6607 6622 .83 6761 6776 .84 6918 6934 .85 7079 7096 .86 7244 7261 .87 7413 7430 .88 7586 7603 .89 7762 7780 .90 7943 7962 .91 8128 8147 .92 8318 8337 .93 8511 8531 .94 8710 8730 .95 8913 8933 .96 9120 9141 .97 9333 9354 .98 9550 9572 .99 9772 9795 2 3 3177 3184 3251 3258 3327 3334 3404 3412 3483 3491 3565 3573 3648 3656 3733 3741 3819 3828 3908 3917 3999 4009 4093 4102 4188 4198 4285 4295 4385 4395 4487 4498 4592 4603 4699 4710 4808 4819 4920 4932 5035 5047 5152 5164 5272 5284 5395 5408 5521 5534 5649 5662 5781 5794 5916 5929 6053 6067 6194 6209 6339 6353 6486 6501 6637 6653 6792 6808 6950 6966 7112 7129 7278 7295 7447 7464 7621 7638 7798 7816 7980 7998 8166 8185 8356 8375 8551 8570 8750 8770 8954 8974 9162 9183 9376 9397 9594 9616 9817 9840 4 5 3192 3199 3266 3273 3342 3350 3420 3428 3499 3508 3581 3589 3664 3673 3750 3758 3837 3846 3926 3936 4018 4027 4111 4121 4207 4217 4305 4315 4406 4416 4508 4519 4613 4624 4721 4732 4831 4842 4943 4955 5058 5070 5176 5188 5297 5309 5420 5433 5546 5559 5675 5689 5808 5821 5943 5957 6081 6095 6223 6237 6368 6383 6516 6531 6668 6683 6823 6839 6982 6998 7145 7161 7311 7328 7482 7499 7656 7674 7834 7852 8017 8035 8204 8222 8395 8414 8590 8610 8790 8810 8995 9016 9204 9226 9419 9441 9638 9661 9863 9886 6 7 8 9 3206 3214 3221 3228 3281 3289 3296 3304 3357 3365 3373 3381 3436 3443 3451 3459 3516 3524 3532 3540 3597 3606 3614 3622 3681 3690 3698 3707 3767 3776 3784 3793 3855 3864 3873 3882 3945 3954 3963 3972 4036 4046 4055 4064 4130 4140 4150 4159 4227 4236 4246 42S6 4325 4335 4345 4355 4426 4436 4446 4457 4529 4539 4550 4560 4634 4645 4656 4667 4742 4753 4764 4775 4853 4864 4875 4887 4966 4977 4989 5000 5082 5093 5105 5117 5200 5212 5224 5236 5321 5333 5346 5358 5445 5458 5470 5483 5572 5585 5598 5610 5702 5715 5728 5741 5834 5848 5861 5875 5970 5984 5998 6012 6109 6124 6138 6152 6252 6266 6281 6295 6397 6412 6427 6442 6546 6561 6577 6592 6699 6714 6730 6745 6855 6871 6887 6902 7015 7031 7047 7063 7178 7194 7211 7228 7345 7362 7379 7396 7516 7534 7551 7568 7691 7709 7727 7745 7870 7889 7907 7925 8054 8072 8091 8110 8241 8260 8279 8299 8433 8453 8472 8492 8630 8650 8670 8690 8831 8851 8872 8892 9036 9057 9078 9099 9247 9268 9290 9311 9462 9484 9506 9528 9683 9705 9727 9750 9908 9931 9954 9977 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 5 3 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 7 7 7 4 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 7 7 7 7 7 78 8 8 8 8 8 9 9 9 5 6 4 4 4 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 5 5 5 6 5 6 5 6 5 6 5 6 5 6 5 6 5 7 6 7 6 7 6 7 6 7 6 7 6 8 6 8 7 8 7 8 7 8 7 8 7 9 7 9 8 9 8 9 8 9 8 10 8 10 8 10 9 10 9 11 9 11 9 11 9 11 10 12 10 12 10 12 10 12 11 13 11 13 11 13 11 14 7 5 5 5 6 6 6 6 6 6 6 6 7 7 7 7 7 7 8 8 8 8 8 9 9 9 9 9 10 10 10 10 11 11 1111 12 12 12 12 13 13 13 14 14 14 15 15 15 16 16 8 9 6 7 6 7 6 7 6 7 6 7 7 7 7 8 7 8 7 8 7 8 7 8 8 9 8 9 8 9 8 9 8 9 9 10 9 10 9 10 9 10 9 11 10 11 10 11 10 11 10 12 10 12 11 12 11 12 11 13 11 13 12 13 12 14 12 14 1314 13 15 13 15 13 15 14 16 14 16 14 16 15 17 15 17 15 17 16 18 16 18 17 19 17 19 17 20 18 20 18 20 298 NATURAL SINES TABLE I 0' 6' 12' 18' 24' 30' 36' 42' 48' 54' Mean 0°.0 0°.1 0°.2 0°.3 0°.4 0°.5 0°.6 0°.7 0°.8 0°.9 Differences 1' 2' 3' 4' 5' 0 .0000 0017 0035 0052 0070 0087 0105 0122 0140 0157 3 6 9 12 15 1 .0175 0192 0209 0227 0244 0262 0279 0297 0314 0332 3 6 9 12 15 2 .0349 0366 0384 0401 0419 0436 0454 0471 0488 0506 3 6 9 12 15 3 .0523 0541 0558 0576 0593 0610 0628 0645 0663 0680 3 6 9 12 15 4 .0698 0715 0732 0750 0767 0785 0802 0819 0837 0854 3 6 9 12 15 5 .0872 0889 0906 0924 0941 0958 0976 0993 1011 1028 3 6 9 12 14 6 .1045 1063 1080 1097 1115 1132 1149 1167 1184 1201 3 6 9 12 14 7 .1219 1236 1253 1271 1288 1305 1323 1340 1357 1374 3 6 9 12 14 8 .1392 1409 1426 1444 1461 1478 1495 1513 1530 1547 3 6 9 12 14 9 .1564 1582 1599 1616 1633 1650 1668 1685 1702 1719 3 6 9 12 14 10 .1736 1754 1771 1788 1805 1822 1840 1857 1874 1891 3 6 9 12 14 11 .1908 1925 1942 1959 1977 1994 2011 2028 2045 2062 3 6 9 11 14 12 .2079 2096 2113 2130 2147 2164 2181 2198 2215 2232 3 6 9 11 14 13 .2250 2267 2284 2300 2317 2334 2351 2368 2385 2402 3 6 8 11 14 14 .2419 2436 2453 2470 2487 2504 2521 2538 2554 2571 3 6 8 11 14 15 .2588 2605 2622 2639 2656 2672 2689 2706 2723 2740 3 6 8 11 14 16 .2756 2773 2790 2807 2823 2840 2857 2874 2890 2907 3 6 8 11 14 17 .2924 2940 2957 2974 2990 3007 3024 3040 3057 3074 3 6 8 11 14 18 .3090 3107 3123 3140 3156 3173 3190 3206 3223 3239 3 6 8 11 14 19 .3256 3272 3289 3305 3322 3338 3355 3371 3387 3404 3 5 8 11 14 20 .3420 3437 3453 3469 3486 3502 3518 3535 3551 3567 3 5 8 11 14 21 .3584 3600 3616 3633 3649 3665 3681 3697 3714 3730 3 5 8 11 14 22 .3746 3762 3778 3795 3811 3827 3843 3859 3875 3891 3 5 8 11 14 23 .3907 3923 3939 3955 3971 3987 4003 4019 4035 4051 3 5 8 11 14 24 .4067 4083 4099 4115 4131 4147 4163 4179 4195 4210 3 5 8 11 13 25 .4226 4242 4258 4274 4289 4305 4321 4337 4352 4368 3 5 8 11 13 26 .4384 4399 4415 4431 4446 4462 4478 4493 4509 4524 3 5 8 10 13 27 .4540 4555 4571 4586 4602 4617 4633 4648 4664 4679 3 5 8 10 13 28 .4695 4710 4726 4741 4756 4772 4787 4802 4818 4833 3 5 8 10 13 29 .4848 4863 4879 4894 4909 4924 4939 4955 4970 4985 3 5 8 10 13 30 .5000 5015 5030 5045 5060 5075 5090 5105 5120 5135 3 5 8 10 13 31 .5150 5165 5180 5195 5210 5225 5240 5255 5270 5284 2 5 7 10 12 32 .5299 5314 5329 5344 5358 5373 5388 5402 5417 5432 2 5 7 10 12 33 .5446 5461 5476 5490 5505 5519 5534 5548 5563 5577 2 5 7 10 12 34 .5592 5606 5621 5635 5650 5664 5678 5693 5707 5721 2 5 7 10 12 35 .5736 5750 5764 5779 5793 5807 5821 5835 5850 5864 2 5 7 10 12 36 .5878 5892 5906 5920 5934 5948 5962 5976 5990 6004 2 5 7 9 12 37 .6018 6032 6046 6060 6074 6088 6101 6115 6129 6143 2 5 7 9 12 38 .6157 6170 6184 6198 6211 6225 6239 6252 6266 6280 2 5 7 9 11 39 .6293 6307 6320 6334 6347 6361 6374 6388 6401 6414 2 4 7 9 11 40 .6428 6441 6455 6468 6481 6494 6508 6521 6534 6547 2 4 7 9 11 41 .6561 6574 6587 6600 6613 6626 6639 6652 6665 6678 2 4 7 9 11 42 .6691 6704 6717 6730 6743 6756 6769 6782 6794 6807 2 4 6 9 11 43 .6820 6833 6845 6858 6871 6884 6896 6909 6921 6934 2 4 6 8 11 44 .6947 6959 6972 6984 6997 7009 7022 7034 7046 7059 2 4 6 8 10 299 NATURAL SINES TABLE I (Continued) 0' 6' 12' 18' 24' 30' 36' 42' 48' 54' Mean 0°.0 0°.1 0°.2 0°.3 0°.4 0°.5 0°.6 0°.7 0°.8 0°.9 Differences 1' 2' 3' 4' 5' 45 .7071 7083 7096 7108 7120 7133 7145 7157 7169 7181 2 4 6 8 10 46 .7193 7206 7218 7230 7242 7254 7266 7278 7290 7302 2 4 6 8 10 47 .7314 7325 7337 7349 7361 7373 7385 7396 7408 7420 2 4 6 8 10 48 .7431 7443 7455 7466 7478 7490 7501 7513 7524 7536 2 4 6 8 10 49 .7547 7558 7570 7581 7593 7604 7615 7627 7638 7649 2 4 6 8 9 50 .7660 7672 7683 7694 7705 7716 7727 7738 7749 7760 2 4 6 7 9 51 .7771 7782 7793 7804 7815 7826 7837 7848 7859 7869 2 4 5 7 9 52 .7880 7891 7902 7912 7923 7934 7944 7955 7965 7976 2 4 5 7 9 53 .7986 7997 8007 8018 8028 8039 8049 8059 8070 8080 2 3 5 7 9 54 .8090 8100 8111 8121 8131 8141 8151 8161 8171 8181 2 3 5 7 8 55 .8192 8202 8211 8221 8231 8241 8251 8261 8271 8281 2 3 5 7 8 56 .8290 8300 8310 8320 8329 8339 8348 8358 8368 8377 2 3 5 6 8 57 .8387 8396 8406 8415 8425 8434 8443 8453 8462 8471 2 3 5 6 8 58 .8480 8490 8499 8508 8517 8526 8536 8545 8554 8563 2 3 5 6 8 59 .8572 8581 8590 8599 8607 8616 8625 8634 8643 8652 1 3 4 6 7 60 .8660 8669 8678 8686 8695 8704 8712 8721 8729 8738 1 3 4 6 7 61 .8746 8755 8763 8771 8780 8788 8796 8805 8813 8821 1 3 4 6 7 62 .8829 8838 8846 8854 8862 8870 8878 8886 8894 8902 1 3 4 5 7 63 .8910 8918 8926 8934 8942 8949 8957 8965 8973 8980 1 3 4 5 6 64 .8988 8996 9003 9011 9018 9026 9033 9041 9048 9056 1 3 4 5 6 65 .9063 9070 9078 9085 9092 9100 9107 9114 9121 9128 1 2 4 5 6 66 .9135 9143 9150 9157 9164 9171 9178 9184 9191 9198 1 2 3 5 6 67 .9205 9212 9219 9225 9232 9239 9245 9252 9259 9265 1 2 3 4 6 68 .9272 9278 9285 9291 9298 9304 9311 9317 9323 9330 1 2 3 4 5 69 .9336 9342 9348 9354 9361 9367 9373 9379 9385 9391 1 2 3 4 5 70 .9397 9403 9409 9415 9421 9426 9432 9438 9444 9449 1 2 3 4 5 71 .9455 9461 9466 9472 9478 9483 9489 9494 9500 9505 1 2 3 4 5 72 .9511 9516 9521 9527 9532 9537 9542 9548 9553 9558 1 2 3 3 4 73 .9563 9568 9573 9578 9583 9588 9593 9598 9603 9608 1 2 3 3 4 74 .9613 9617 9622 9627 9632 9636 9641 9646 9650 9655 1 2 2 3 4 75 .9659 9664 9668 9673 9677 9681 9686 9690 9694 9699 1 1 2 3 4 76 .9703 9707 9711 9715 9720 9724 9728 9732 9736 9740 1 1 2 3 3 77 .9744 9748 9751 9755 9759 9763 9767 9770 9774 9778 1 1 2 3 3 78 .9781 9785 9789 9792 9796 9799 9803 9806 9810 9813 1 1 2 2 3 79 .9816 9820 9823 9826 9829 9833 9836 9839 9842 9845 1 1 2 2 3 80 .9848 9851 9854 9857 9860 9863 9866 9869 9871 9874 0 1 1 2 2 81 .9877 9880 9882 9885 9888 9890 9893 9895 9898 9900 0 1 1 2 2 82 .9903 9905 9907 9910 9912 9914 9917 9919 9921 9923 0 1 1 2 2 83 .9925 9928 9930 9932 9934 9936 9938 9940 9942 9943 0 1 1 1 2 84 :9945 9947 9949 9951 9952 9954 9956 9957 9959 9960 0 1 1 1 2 85 .9962 9963 9965 9966 9968 9969 9971 9972 9973 9974 0 0 1 1 1 86 .9976 9977 9978 9979 9980 9981 9982 9983 9984 9985 0 0 1 1 1 87 .9986 9987 9988 9989 9990 9990 9991 9992 9993 9993 0 0 0 1 1 88 .9994 9995 9995 9996 9996 9997 9997 9997 9998 9998 0 0 0 0 0 89 .9998 9999 9999 9999 9999 1.000 1.000 1.000 1.000 1.000 0 0 0 0 0 90 1.000 300 NATURAL COSINES TABLE II 0' 6' 12' 18' 24' 30' 36' 42' 48' 54' Mean 0°.0 0°.1 0°.2 0°.3 0°.4 0°.5 0°.6 0°.7 0°.8 0°.9 Differences 1' 2' 3' 4' 5' 0 1.000 1.000 1.000 1.000 1.000 1.000 .9999 9999 9999 9999 0 0 0 0 0 1 .9998 9998 9998 9997 9997 9997 9996 9996 9995 9995 0 0 0 0 0 2 .9994 9993 9993 9992 9991 9990 9990 9989 9988 9987 0 0 0 1 1 3 .9986 9985 9984 9983 9982 9981 9980 9979 9978 9977 0 0 1 1 1 4 .9976 9974 9973 9972 9971 9969 9968 9966 9965 9963 0 0 1 1 1 5 .9962 9960 9959 9957 9956 9954 9952 9951 9949 9947 0 1 1 1 2 6 .9945 9943 9942 9940 9938 9936 9934 9932 9930 9928 0 1 1 1 2 7 .9925 9923 9921 9919 9917 9914 9912 9910 9907 9905 0 1 1 2 2 8 .9903 9900 9898 9895 9893 9890 9888 9885 9882 9880 0 1 1 2 2 9 .9877 9874 9871 9869 9866 9863 9860 9857 9854 9851 0 1 1 2 2 10 .9848 9845 9842 9839 9836 9833 9829 9826 9823 9820 1 1 2 2 3 11 .9816 9813 9810 9806 9803 9799 9796 9792 9789 9785 1 1 2 2 3 12 .9781 9778 9774 9770 9767 9763 9759 9755 9751 9748 1 1 2 3 3 13 .9744 9740 9736 9732 9728 9724 9720 9715 9711 9707 1 1 2 3 3 14 .9703 9699 9694 9690 9686 9681 9677 9673 9668 9664 1 1 2 3 4 15 .9659 9655 9650 9646 9641 9636 9632 9627 9622 9617 1 2 2 3 4 16 .9613 9608 9603 9598 9593 9588 9583 9578 9573 9568 1 2 2 3 4 17 .9563 9558 9553 9548 9542 9537 9532 9527 9521 9516 1 2 3 3 4 18 .9511 9505 9500 9494 9489 9483 9478 9472 9466 9461 1 2 3 4 5 19 .9455 9449 9444 9438 9432 9426 9421 9415 9409 9403 1 2 3 4 5 20 .9397 9391 9385 9379 9573 9367 9361 9354 9348 9342 1 2 3 4 5 21 .9336 9330 9323 9317 9311 9304 9298 9291 9285 9278 1 2 3 4 5 22 .9272 9265 9259 9252 9245 9239 9232 9225 9219 9212 1 2 3 4 6 23 .9205 9198 9191 9184 9178 9171 9164 9157 9150 9143 1 2 3 5 6 24 .9135 9128 9121 9114 9107 9100 9092 9085 9078 9070 1 2 4 5 6 25 .9063 9056 9048 9041 9033 9026 9018 9011 9003 8996 1 3 4 5 6 26 .8988 8980 8973 8965 8957 8949 8942 8934 8926 8918 1 3 4 5 6 27 .8910 8902 8894 8886 8878 8870 8862 8854 8838 1 3 4 5 7 28 .8829 8821 8813 8805 8796 8788 8780 8771 8763 8755 1 3 4 6 7 29 .8746 8738 8729 8721 8712 8704 8695 8686 8678 8669 1 3 4 6 7 30 .8660 8652 8643 8634 8625 8616 8607 8599 8590 8581 1 3 4 6 7 31 .8572 8563 8554 8545 8536 8526 8517 8508 8499 8490 2 3 5 6 8 32 .8480 8471 8462 8453 8443 8434 8425 8415 8406 8396 2 3 5 6 8 33 .8387 8377 8368 8358 8348 8339 8329 8320 8310 8300 2 3 5 6 8 34 .8290 8281 8271 8261 8251 8241 8231 8221 8211 8202 2 3 5 7 8 3S .8192 8181 8171 8161 8151 8141 8131 8121 8111 8100 2 3 5 7 8 36 .8090 8080 8070 8059 8049 8039 8028 8018 8007 7997 2 3 5 7 8 37 .7986 7976 7965 7955 7944 7934 7923 7912 7902 7891 2 4 5 7 9 38 .7880 7869 7859 7848 7837 7826 7815 7804 7793 7782 2 4 5 7 9 39 .7771 7760 7749 7738 7727 7716 7705 7694 7683 7672 2 4 6 7 9 40 .7660 7649 7638 7627 7615 7604 7593 7581 7570 7559 2 4 6 8 9 41 .7547 7536 7524 7513 7501 7490 7478 7466 7455 7443 2 4 6 8 10 42 .7431 7420 7408 7396 7385 7373 7361 7349 7337 7325 2 4 6 8 10 43 .7314 7302 7290 7278 7266 7254 7242 7230 7218 7206 2 4 6 8 10 44 .7193 7181 7169 7157 7145 7133 7120 7108 7096 7083 2 4 6 8 10 301 NATURAL COSINES TABLE II (Continued) 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 0' 0°.0 .7071 .6947 .6820 .6691 .6561 .6428 .6293 .6157 .6018 .5878 .5736 .5592 .5446 .5299 .5150 .5000 .4848 .4695 .4540 .4384 .4226 .4067 .3907 .3746 .3584 .3420 .3256 .3090 .2924 .2756 .2588 .2419 .2250 .2079 .1908 .1736 .1564 .1392 .1219 .1045 .0872 .0698 .0523 .0349 .0175 .0000 6' 0°.1 7059 6934 6807 6678 6547 6414 6280 6143 6004 5864 5721 5577 5432 5284 5135 4985 4833 4679 4524 4368 4210 4051 3891 3730 3567 3404 3239 3074 2907 2740 2571 2402 2233 2062 1891 1719 1547 1374 1201 1028 0854 0680 0506 0332 0157 12' 0°.2 7046 6921 6794 6665 6534 6401 6266 6129 5990 5850 5707 5563 5417 5270 5120 4970 4818 4664 4509 4352 4195 4035 3875 3714 3551 3387 3223 3057 2890 2723 2554 2385 2215 2045 1874 1702 1530 1357 1184 1011 0837 0663 0488 0314 0140 18' 0°.3 7034 6909 6782 6652 6521 6388 6252 6115 5976 5835 5693 5548 5402 5255 5105 4955 4802 4648 4493 4337 4179 4019 3859 3697 3535 3371 3206 3040 2874 2706 2538 2368 2198 2028 1857 1685 1513 1340 1167 0993 0819 0645 0471 0297 0122 24' 0°.4 7022 6896 6769 6639 6508 6374 6239 6]01 5962 5821 5678 5534 5388 5240 5090 4939 4787 4633 4478 4321 4163 4003 3843 3681 3518 3355 3190 3024 2857 2689 2521 2351 2181 2011 1840 1668 1495 1323 1149 0976 0802 0628 0454 0279 0105 30' 0°.5 7009 6884 6756 6626 6494 6361 6225 6088 5948 5807 5664 55]9 5373 5225 5075 4924 4772 4617 4462 4305 4147 3987 3827 3665 3502 3338 3173 3007 2840 2672 2504 2334 2164 1994 1822 1650 1478 1305 1132 0958 0785 0610 0436 0262 0087 36' 0°.6 6997 6871 6743 6613 6481 6347 6211 6074 5934 5793 5650 5505 5358 5210 5060 4909 4756 4602 4446 4289 4131 3971 3811 3649 3486 3322 3156 2990 2823 2656 2487 2317 2147 1977 1805 1633 1461 1288 1115 0941 0767 0593 0419 0244 0070 42' 0°.7 6984 6858 6730 6600 6468 6334 6198 6060 5920 5779 5635 5490 5344 5195 5045 4894 4741 4586 4431 4274 4115 3955 3795 3633 3469 3305 3140 2974 2807 2639 2470 2300 2130 1959 1788 1616 1444 1271 1097 0924 0750 0576 0401 0227 0052 48' 0°.8 6972 6845 6717 6587 6455 6320 6184 6046 5906 5764 5621 5476 5329 5180 5030 4879 4726 4571 4415 4258 4099 3939 3778 3616 3453 3289 3123 2957 2790 2622 2453 2284 2113 1942 1771 1599 1426 1253 1080 0906 0732 0558 0384 0209 0035 54'0°.9 6959 6833 6704 6574 6441 6307 6170 6032 5892 5750 5606 5461 5314 5165 5015 4863 4710 4555 4399 4242 4083 3923 3762 3600 3437 3272 3107 2940 2773 2605 2436 2267 2096 1925 1754 1582 1409 1236 1063 0889 0715 0541 0366 0192 0017 1' 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 Mean Differences 2' 3' 4'4 6 8 4 6 8 4 6 9 4 7 9 4 7 9 4 7 9 5 7 9 5 7 9 5 7 9 5 7 9 5 7 10 5 7 10 5 7 10 5 7 10 5 8 10 5 8 10 5 8 10 5 8 10 5 8 10 5 8 11 5 8 11 5 8 11 5 8 11 5 8 11 5 8 11 5 8 11 6 8 11 6 8 11 6 8 11 6 8 11 6 8 11 6 8 11 6 9 11 6 9 11 6 9 11 6 9 12 6 9 12 6 9 12 6 9 12 6 9 12 6 9 12 6 9 12 6 9 12 6 9 12 6 9 12 5'101111111111111112121212121213131313131313141414141414141414141414141414141414141515151515 302 NATURAL TANGENTS TABLE III 0' 6' 12' 18' 24' 30' 36' 42' 48' 54' Mean 0°.0 0°.1 0°.2 0°.3 0°.4 0°.5 0°.6 0°.7 0°.8 0°.9 Differences 1' 2' 3' 4' 5' 0 .0000 0017 0035 0052 0070 0087 0105 0122 0140 0157 3 6 9 12 15 1 .0175 0192 0209 0227 0244 0262 0279 0297 0314 0332 3 6 9 12 15 2 .0349 0367 0384 0402 0419 0437 0454 0472 0489 0507 3 6 9 12 15 3 .0524 0542 0559 0577 0594 0612 0629 0647 0664 0682 3 6 9 12 15 4 .0699 0717 0734 0752 0769 0787 0805 0822 0840 0857 3 6 9 12 15 5 .0875 0892 0910 0928 0945 0963 0981 0998 1016 1033 3 6 9 12 15 6 .1051 1069 1086 1104 1122 1139 1157 1175 1192 1210 3 6 9 12 15 7 .1228 1246 1263 1281 1299 1317 1334 1352 1370 1388 3 6 9 12 15 8 .1405 1423 1441 1459 1477 1495 1512 1530 1548 1566 3 6 9 12 15 9 .1584 1602 1620 1638 1655 1673 1691 1709 1727 1745 3 6 9 12 15 10 .1763 1781 1799 1817 1835 1853 1871 1890 1908 1926 3 6 9 12 15 11 .1944 1962 1980 1998 2016 2035 2053 2071 2089 2107 3 6 9 12 15 12 .2126 2144 2162 2180 2199 2217 2235 2254 2272 2290 3 6 9 12 15 13 .2309 2327 2345 2364 2382 2401 2419 2438 2456 2475 3 6 9 12 15 14 .2493 2512 2530 2549 2568 2586 2605 2623 2642 2661 3 6 9 12 16 15 .2679 2698 2717 2736 2754 2773 2792 2811 2830 2849 3 6 9 13 16 16 .2867 2886 2905 2924 2943 2962 2981 ‘3000 3019 3038 3 6 9 13 16 17 .3057 3076 3096 3115 3134 3153 3172 3191 3211 3230 3 6 10 13 16 18 .3249 3269 3288 3307 3327 3346 3365 3385 3404 3424 3 6 10 13 16 19 .3443 3463 3482 3502 3522 3541 3561 3581 3600 3620 3 7 10 13 16 20 .3640 3659 3679 3699 3719 3739 3759 3779 3799 3819 3 7 10 13 17 21 .3839 3859 3879 3899 3919 3939 3959 3979 4000 4020 3 7 10 13 17 22 .4040 4061 4081 4101 4122 4142 4163 4183 4204 4224 3 7 10 14 17 23 .4245 4265 4286 4307 4327 4348 4369 4390 4411 4431 3 7 10 14 17 24 .4452 4473 4494 4515 4536 4557 4578 4599 4621 4642 4 7 11 14 18 25 .4663 4684 4706 4727 4748 4770 4791 4813 4834 4856 4 7 11 14 18 26 .4877 4899 4921 4942 4964 4986 5008 5029 5051 5073 4 7 11 15 18 27 .5095 5117 5139 5161 5184 5206 5228 5250 5272 5295 4 7 11 15 18 28 .5317 5340 5362 5384 5407 5430 5452 5475 5498 5520 4 8 11 15 19 29 .5543 5566 5589 5612 5635 5658 5681 5704 5727 5750 4 8 12 15 19 30 .5774 5797 5820 5844 5867 5890 5914 5938 5961 5985 4 8 12 16 20 31 .6009 6032 6056 6080 6104 6128 6152 6176 6200 6224 4 8 12 16 20 32 .6249 6273 6297 6322 6346 6371 6395 6420 6445 6469 4 8 12 16 20 33 .6494 6519 6544 6569 6594 6619 6644 6669 6694 6720 4 8 13 17 21 34 .6745 6771 6796 6822 6847 6873 699 6924 6950 6976 4 9 13 17 21 35 .7002 7028 7054 7080 7107 7133 7159 7186 7212 7239 4 9 13 18 22 36 .7265 7292 7319 7346 7373 7400 7427 7454 7481 7508 5 9 14 18 23 37 .7536 7563 7590 7618 7646 7673 7701 7729 7757 7785 5 9 14 18 23 38 .7813 7841 7869 7898 7926 7954 7983 8012 8040 8069 5 9 14 19 24 39 .8008 8127 8156 8185 8214 8243 8273 8302 8332 8361 5 10 15 20 24 40 .8391 8421 8451 8481 8511 8541 8571 8601 8632 8662 5 10 15 20 25 41 .8693 8724 8754 8785 8816 8847 8878 8910 8941 8972 5 10 16 21 26 42 .9004 9036 9067 9099 9131 9163 9195 9228 9260 9293 5 11 16 21 27 43 .9325 9358 9391 9424 9457 9490 9523 9556 9590 9623 6 11 17 22 28 44 .9657 9691 ‘ 9725 9759 9793 9827 9861 9896 9930 9965 6 11 17 23 29 303 NATURAL TANGENTS TABLE III (Continued) 0' 6' 12' 18' 24' 30' 36' 42' 48' 54' Mean 0°.0 0°.1 0°.2 0°.3 0°.4 0°.5 0°.6 0°.7 0°.8 0°.9 Differences 1' 2' 3' 4' 5' 45 1.0000 0035 0070 0105 0141 0176 0212 0247 0283 0319 6 12 18 24 30 46 1.0355 0392 0428 0464 0501 0538 0575 0612 0649 0686 6 12 18 25 31 47 1-0724 0761 0799 0837 0875 0913 0951 0990 1028 1067 6 13 19 25 32 48 1-1106 1145 1184 1224 1263 1303 1343 1383 1423 1463 7 13 20 27 33 49 1.1504 1544 1585 1626 1667 1708 1750 1792 1833 1875 7 14 21 28 34 50 1-1918 1960 2002 2045 2088 2131 2174 2218 2261 2305 7 14 22 29 35 51 1.2349 2393 2437 2482 2527 2572 2617 2662 2708 2753 8 15 23 30 38 52 1.2799 2846 2892 2938 2985 3032 3079 3127 3175 3222 8 16 24 31 39 53 1.3270 3319 3367 3416 3465 3514 3564 3613 3663 3713 8 16 25 33 41 54 1.3764 3814 3865 3916 3968 4019 4071 4124 4176 4229 9 17 26 34 43 55 1-4281 4335 4388 4442 4496 4550 4605 4659 4715 4770 9 18 27 36 45 56 1-4826 4882 4938 4994 5051 5108 5166 5224 5282 5340 10 19 29 38 48 57 1.5399 5458 5517 5577 5637 5697 5757 5818 5880 5941 10 20 30 40 50 58 1.6003 6066 6128 6191 6255 6319 6383 6447 6512 6577 11 21 32 43 53 59 1.6643 6709 6775 6842 6909 6977 7045 7113 7182 7251 11 23 34 45 56 60 1-7321 7391 7461 7532 7603 7.675 7747 7820 7893 7966 12 24 36 48 60 61 1.8040 8115 8190 8265 8341 8418 8495 8572 8650 8728 13 26 38 51 64 62 1.8807 8887 8967 9047 9128 9210 9292 9375 9458 9542 14 27 41 55 68 63 1.9626 9711 9797 9883 9970 2.0057 2.0145 2.0233 2.0323 2.0413 15 29 44 58 73 64 2.0503 0594 0686 0778 0872 0965 1060 1155 1251 1348 16 31 47 63 78 65 2.1445 1543 1642 1742 1842 1943 2045 2148 2251 2355 17 34 51 68 85 66 2.2460 2566 2673 2781 2889 2998 3109 3220 3332 3445 18 37 55 73 92 67 2.3559 3673 3789 3906 4023 4142 4262 4383 4504 4627 20 40 60 79 99 68 2.4751 4876 5002 5129 5257 5386 5517 5649 5782 5916 22 43 65 87 108 69 2.6051 6187 6325 6464 6605 6746 6889 7034 7179 7326 24 47 71 95 119 70 2.7475 7625 7776 7929 8083 8239 8397 8556 8716 8878 26 52 78 104 131 71 2.9042 9208 9375 9544 9714 9887 3.0061 3.0237 3.0415 3.0595 29 58 87 116 145 72 3.0777 0961 1146 1334 1524 1716 1910 2106 2305 2500 32 64 96 129 161 73 3.2709 2914 3122 3332 3544 3759 3977 4197 4420 4646 36 72 108 144 180 74 3.4874 5105 5339 5576 5816 6059 6305 6554 6806 7062 41 811 22 163 204 75 3.7321 7583 7848 8118 8391 8667 8947 9232 9520 9812 46 93 139 186 232 76 4.0108 0408 0713 1022 1335 i653 1976 2303 2635 2972 53 107 160 213 267 77 4.3315 3662 4015 4374 4737 5107 5483 5864 6252 6646 78 4.7046 7453 7867 8288 8716 9152 9594 5.0045 5.0504 5.0970 Mean differences cease 79 5.1446 1929 2422 2924 3435 3955 4486 5026 5578 6140 to be sufficiently accurate. 80 5.6713 7297 7894 8502 9124 9758 6.0405 6.1066 6.1742 6.2432 81 6.3138 3859 4596 5350 6122 6912 7720 8548 9395 7.0264 82 7.1154 2066 3002 3%2 4947 5958 6996 8062 9158 8.0285 83 8.1443 2636 3863 5126 6427 7769 9152 9.0579 9.2052 9.3572 84 9.5144 9.677 9.845 10.02 10.20 10.39 10.58 10.78 10.99 11-20 85 1143 11.66 11.91 12.16 12.43 12.71 13.00 13.30 13.62 13.95 86 14.30 14.67 15.06 15.46 15.89 16.35 16.83 17.34 17.89 18.46 87 19.08 19.74 20.45 21.20 22.02 22.90 23.86 24.90 26.03 27.27 88 28.64 30.14 31.82 33.69 35.80 38.19 40.92 44.07 47.74 52.08 89 57.29 63.66 71.62 81.85 95.49 114.6 143.2 191.0 286.5 573.0 90 not defined 304 Notes 305 Notes 306

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