GIFT OF TABLES FOR ENGINEERING CALCULATIONS BY RICHARD C. POWELL ASSOC. MEM. A. I. E. E. SECOND EDITION REVISED AND ENLARGED BERKELEY, CAL. C. POWELL, PUBLISHER 1907 COPYRIGHT, 1907, BY RICHARD C. PCTWELL PRICE $2,00 PRESS OF BERKELEY REPORTER INC. BOOKBINDERS AND PRINTERS BERKELEY, CAL. PREFACE THE present voltfme is the seco.nd edition of the "Handbook of Tables for Engineering Calculations," compiled by Mr. William E. Hawley and the author. The first edition has been almost entirely rewritten and much new matter added, being nearly doubled in size. Although descriptions and illustrations of appa- ratus, and discussions of best practice are invalu- able to the engineer, it is believed that there is use for a handbook not burdened with these things, but one that furnishes to the student and engineer those formulae and tables which enable him to perform with the least amount of time and effort the calculations incident to engineering. The author has attempted to collect the most reliable material and arrange it for greatest con- venience of use; those tables which are more fre- quently in simultaneous use have been put together and in nearly every case two-page tables are on pages which face. Very little has been inserted unrevised with the exception of the tables for rolled steel shapes. These the author was enabled to reproduce from the "Cambria Steel" Handbook through the gen- erosity of the Cambria Steel Company. The author has had the advantage of many sug- gestions, and to all those who made them, and also to others who assisted in numerous ways, he is very grateful. His best thanks are due to Pro- fessor LeConte of the University of California for his assistance in the preparation of the section on hydraulics, which was for the most part compiled by him. THE AUTHOR. Berkeley, August^ IJKfo ^ o * > CONTENTS Greek Letters 2 Logarithms Explanation 3 Table Natural Logarithms 1.00 to 9.99 6 Table Natural Logarithms 1 to 999 8 Table Common Logarithms 1 to 9999 10 Trigonometric Functions Table Natural and Logarithmic Sine, Cosine, Tangent and Cotangent 31 Table Minutes in Decimal Parts of Degrees 40 Table Functions of Certain Angles 49 Table Angles Greater than 90 to Angles Less than 90... 40 Formulae 41 Any Plane Triangle 42 Plane Oblique Triangles 43 Right-Angled Spherical Triangles 44 Any Spherical Triangle 44 Circles Formulae Arcs, Cords and Segments 45 1*able Lengths of Circular Arcs for Unit Radius 46 Table Length of Cord, Rise and Area of Segments 47 Table Circumferences and Areas for Diameters in Inches.. 49 Table Areas in Square Feet for Diameters in Inches 51 Functions of the Natural Numbers Table Square, Cube, Square and Cube Roots, Reciprocal, Area and Circumference of Circles 52 Table Fifth Roots and Fifth Powers 72 Table Square and Cube Roots of Fractions 72 CONTENTS v Conversion of Units Table Decimal Equivalents of Fractions of One Inch 73 Table Inches in Decimal Parts of a Foot 73 Table U. S. and British Units 74 Table Metric Units 75 Table U. S. to Metric and Metric to U. S. Lengths, Areas, Volumes, Weights, Pressures and Miscellaneous 76 Table Conversion Factors. Angles, Lengths, Areas, Vol- umes, Water Discharges, Weights, Weights per Unit Length, Densities, Water Weights and Volumes, Forces, Work and Heat, Power, Angular and Linear Velocities, Temperatures, Electrostatic to Electromag- netic ' 78 Wire and Sheet Metal Gauges - 85 Powers and Multiples of ir e and g 86 Hyperbolic Functions Formulae 87 Table Tangent 87 Table Sine 88 Table Cosine 89 Solution of Equations Quadratic Equation Algebraic and Slide Rule Solutions... 90 Cubic Equation Slide Rule Solution 91 Descartes' Rule of Signs 92 Homer's Method 92 Newton's Method 94 By Plotting or by the Use of Tables 94 Series 95 Integrals Table Integrals 96 Definite Integrals Approximate Evaluation 98 Plane Curves Cycloid, Logarithmic Spiral and Catenary 99 vi CONTENTS Probable Error Formulae 100 Table Constants 101 Sections and Solids Guldin's Rules 101 Prismoidal Formula 101 Moments of Inertia 102 Table Area, Center of Mass, Moments of Inertia, Radius of Gyration and Section Modulus of Sections 102 Table Surface, Volume, Center of Mass and Moments of Inertia of Solids 107 Coefficients of Strength and Deflection for Steel .Shapes 109 Properties of Simple and Compound Rolled Shapes 109 Table Properties of Standard I-Beams 110 Table Properties of Bulb Beams 112 Table Properties of Standard Channels 114 Table Properties of Standard Angles 116 Table Properties of T-bars 124 Table Properties of Z-bars 126 Strength of Materials Table Tension, Compression, Shear and Coefficient of Elas- ticity 128 Columns Formulae 128 Beams Mechanics of Beams 129 Table Reactions, Bending Moment and Deflection 131 Valency and Atomic Weights of the Elements Table 134 Densities of Substances Table 134 Heat Table Comparison of Centigrade and Fahrenheit 136 Table Coefficients of Expansion and Specific Heats 138 CONTENTS vii Heat Con 't. Table Density and Volume of Water 139 Table Melting and Boiling Points 139 Table Boiling Points of Water for Various Pressures 139 Table Latent Heats 140 Gases Formulae 140 Table Density and Constant of Gases 141 Table Specific Heat of Gases 141 Factor of Evaporation Formulae 141 Table Factors of Evaporation 142 Table Properties of Saturated Steam ?r 143 Table Properties of Saturated Vapor of Ammonia 150 Table Properties of Saturated Vapor of Sulphur Dioxide 150 Magnetic Constants Table Magnetic Properties of Iron and Steel 151 Table Loss Due to Hysteresis 151 Electrical Constants Table Specific Resistances and Temperature Coefficients 152 Table Specific Inductive Capacities 153 Table Rate of Electrolytic Deposition 153 Wires and Cables Table Fusing of Wires 153 Table Properties of Copper Wire 154 Table Properties of Aluminum Wire 155 Table Properties of Copper Cables 156 Table Properties of Aluminum Cables 156 Table Diameters of Cotton-Covered Wire 157 B. & S. Gauge Law and Slide Rule Formulae 157 Transmission Line Constants Formulae 158 Table Capacity, Charging Current and Reactance for Ratios of Distance between Wires to Diameter 158 Table Capacity, Charging Current, Reactance and Imped- ance of Copper and Aluminum for Distances between Wires. Three Phase 60 cycles 159 viii CONTENTS Transmission Line Constants Con't. Table Capacity, Charging Current, Reactance and Imped- ance of Copper and Aluminum for Distances between Wires. Single Phase 25 cycles 160 Hydraulics Immersed Rectangle Formulae 161 Impact Due to a Jet Formula , 161 Orifices Formulae 162 Flow Over Weirs Formulae 163 Flow Through Pipes Formulae 164 Flow in Open Channels Formulae 165 Table I Coefficients of Efflux for Circular Vertical Orifices. .166 Table II Coefficients of Efflux for Square Vertical Orifices. .166 Table III Imperfect Contraction 167 Table IV Coefficient for Conical Orifices 167 Table V Coefficients for Conical Tubes 167 Table VI Discharge Coefficients for Weirs, Francis' For- mula 168 Tables VII, VIII, IX and X Discharge Coefficients for Weirs, Smith's Formula 168 3 / Table XI Values of h /2 . 169 Table XII Weir Measurement 170 Table XIII Coefficients of Flow for Iron Pipes 171 Table XIV Loss of Head by Friction in Iron Pipes 172 Table XV Coefficients of Flow for Wood Stave Pipes 173 Table XVI Coefficients of Friction for Elbows 174 Table XVII Coefficients of Friction for Curves 174 Table XVIII Kutter's Coefficients 174 Index . ..177 GREEK LETTERS. Greek Letters A a Alpha B /3 Beta 7 Gamma 5 Delta e Epsilon f Zeta T; Eta Theta 1 Iota K Kappa X Lambda M Mu N v Nu Xi o Omicron n TT Pi P p Rho 2 Phi X x Chi * V Psi 1} o> Omega LOGARI1WMS LOGARITHMS Explanation The logarithm of a number to a given base is the index of the power to which the base must be raised to produce the given number. If a* = n, x is the logarithm of n to the base a, and is written x = log a n. The only systems of logarithms in practical use are the so- called natural (Hyperbolic) and the common (Briggs.) In the former the base is the incommensurable number 2.718282 usually denoted by e. In the common system the base is 10. The relations between logarithms in these two systems are: Iog 10 n = 0.434294 log, n, log, n = 2.302585 log, n. The natural logarithm is often designated by In; the com- mon by log. Thus, In x is read natural logarithm of x while log x is read common logarithm of x. The logarithm of a product is the sum of the logarithms of its factors, thus log (a b c n) = log a -f log b -f log c + + log n. The logarithm of a quotient is the logarithm of the divi- dend minus the logarithm of the divisor, thus c log = log a log b 4 LOG vRITHMS" The logarithm of a number raised to any power, integral or fractional, is the logarithm.- of the number multiplied by the index of the power, thus log n x = x log n and log * / n = log n These rules hold for any base. Explanation of the Tables. Natural Logarithms. The natural logarithms of the num- bers from 1 to 9.99 are given on pages 6 and 7, and from 1 to 999 on pages 8 and 9. The In of numbers less than 1 are negative Common Logarithms. In the common system, the logarithm is considered as consisting of a positive decimal part called the mantissa and an integral part called the characteristic, which may be either positive or negative. The mantissa is the same for all numbers having the same sequence of figures, and hence is independent of the decimal point or of ciphers which preceed or follow this given sequence of figures. The characteristic depends upon the decimal point and is determined by the following rules: The characteristic of the logarithm of a number greater than unity is positive and is less by one than the number of digits in its integral part. The characteristic of the logarithm of a decimal fraction is nega- tive and is greater by one than the number of ciphers immediately after the decimal point . By reversing these rules, the decimal point in the sequence of figures corresponding to any mantissa is readily found. The logarithms of the numbers from 1 to 100, and the mantissas of the numbers from 100 to 10,000 are given on pages 10 to 30. The column D gives the tabular differences, and where LOGARITHMS 5 these are large, the proportional parts are given at the foot of the page. Examples. The mantissa for 757.6 is 0.87944 and the characteristic is 2. Hence, log 757.6 = 2.87944. To find the number whose log is 1.70131, we first find the sequence of figures corresponding to the mantissa 0.70131, This is (page 21) 5027, and from the rule for the characteristic we find the number to be 50.27. The mantissa for 0.0438 is 0.64147 and the characteristic -2, hence, log 0.0438 ="2". 64147, a bar being placed over the characteristic to indicate that it alone is negative. To avoid confusion, it is better to add and subtract 10 from the log. Thus the log 0.0438 is written log 0.0438 =8.64147 10. To extract the root of a decimal fraction we add and subtract a multiple of 10; as for example to find the cube root of 0.0438 we proceed by adding and subtracting 20 from the log as just written, and then dividing by 3, thus 3^28.641470 30 9.54715610 The figures corresponding to the mantissa 0.547156 (p. 17) are easily found to be 3525, and the characteristic determines the number to be 0.3525. Trigonometric Functions. The natural sines, tangents cotangents and cosines with their logarithms are given on pages 31 to 39. The values are for angles between and 90 degrees, varying by tenths of degrees, i. e., by 6 minutes. For angles between and 45 degrees read down the left-hand side of the page; for angles between 45 and 90 degrees read up the right- hand side. The characteristics of the logarithmic functions are printed in full-faced type at or near the top of the columns headed Log. A glance at the natural function will determine whether or not 10 is to be subtracted from the characteristic given. For reducing minutes to decimal parts of a degree, and vice versa, use the table on page 40. NATURAL LOGARITHMS Natural Logarithms of Numbers 1.00 9.99 and 1 999 N 1 2 3 4 5 6 7 8 9 1.0 1.1 1.2 1.3 0.0000 0.09531 0.1823 0.2624 0.00995 0.1044 0.1906 0.2700 0.01980 0.1133 0.1988 0.2776 0.02956 0.1222 0.2070 0.2852 0.03922 0.1310 0.2151 0.2927 0.04879 0.1398 0.2231 0.3001 0.05827 0.1484 0.2311 0.3075 0.06766 0.1570 0.2390 0.3148 0.07696 0.1655 0.2469 0.3221 0.08618 0.1740 0.2546 0.3293 1.4 1.5 1.6 0.3365 0.4055 0.4700 0.3436 0.4121 0.4762 0.3507 0.4187 0.4824 0.3577 0.4253 0.4886 0.3646 0.4318 0.4947 0.3716 0.4383 0.500*, 0.3784 0.4447 0.5068 0.3853 0.4511 0.5128 0.3920 0.4574 0.5188 0.3988 0.4637 0.5247 1.7 l.S 1.9 0.5306 0.5878 0.6419 0.5365 0.5933 0.6471 0.5423 0.5988 0.6523 0.5481 0.6043 0.6575 0.5539 0.6098 0.6627 0.5596 0.6152 0.6678 0.5653 0.6206 0.6729 0.5710 0.6259 0.6780 0.5766 0.6313 0.6831 0.5822 0.6366 0.6881 2.0 2.1 2.2 2.3 0.6931 0.7419 0.7885 0.8329 0.6981 0.7467 0.7930 0.8372 0.7031 0.7514 0.7975 0.8416 0.7080 0.7561 0.8020 0.8459 0.7129 0.7608 0.8065 0.8502 0.7178 0.7655 0.8109 0.8544 0.7227 0.7701 0.8154 0.8587 0.7275 0.7747 0.8198 0.8629 0.7324 0.7793 0.8242 0.8671 0.7372 0.7839 0.828G 0.8713 2.4 2.5 2.6 0.8755 0.9163 0.9555 0.8796 0.9203 0.9594 0.8838 0.9243 0.9632 0.8879 0.9282 0.9670 0.8920 0.9322 0.9708 0.8961 0.9361 0.9746 0.9002 0.9400 0.9783 0.9042 0.9439 0.9821 0.9083 0.9478 0.9858 0.9123 0.9517 0.9895 2.7 2.8 2.9 0.9933 1.0296 1.0647 0.9969 1.0332 1.0682 1.0006 1.0367 1.0716 1.0043 1.0403 1.0750 1.0080 1.0438 1.0784 1.0116 1.0473 1.0818 1.0152 1.0508 1.0852 1.0188 1.0543 1.0886 1.0225 1.0578 1.0919 1. 1; 1. 3.0 3.1 3.2 3.3 1.0986 1.1314 1.1632 1.1939 1.1019 1.1346 1.1663 1.1969 1.1053 1.1378 1.1694 1.2000 1.1086 1.1410 1.1725 1.2030 1.1119 1.1442 1.1756 1.2060 1.1151 1.1474 1.1787 1.2090 1.1184 1.1506 1.1817 1.2119 1.1217 1.1537 1.1848 1.2149 1.1249 1.1569 1.1878 1.2179 1. 1. 1. 1.1 3.4 3.5 3.6 1.2238 1.2528 1.2809 1.2267 1.2556 1.2837 1.2296 1.2585 1.2865 1.2326 1.2613 1.2892 1.2355 1.2641 1.2920 1.2384 1.2669 1.2947 1.2413 1.2698 1.2975 1.2442 i:2726 1.3002 1.2470 1.2754 1.3029 i. 1. 1. 3.7 3.8 3.9 1.3083 1.3350 1.3610 1.3110 1.3376 1.3635 1.3137 1.3403 1.3661 1.3164 1.3429 1.3686 1.3191 1.3455 1.3712 1.3218 1.3481 1.3737 1.3244 1.3507 1.3762 1.3271 1.3533 1.3788 1.3297 1.3558 1.3813 1, 1. i.- 4.0 4.1 4. "s" 1.3863 1.4110 1.4351 1.4586 1.3888 1.4134 1.4375 1.4609 1.3913 1.4159 1.4398 1.4633 1.3938 1.4183 1.4422 1.4656 1.3962 1.4207 1.4446 1.4679 1.3987 1.4231 1.4469 1.4702 1.4012 1.4255 1.4493 1.4725 1.4036 1.4279 1.4516 1.4748 1.4061 1.4303 1.4540 1.4770 i. i. i.' i. 4.4 4.5 4.6 1.4816 1.5041 1.5261 1.4839 1.5063 1.5282 1.4861 1.5085 1.5304 1.4884 1.5107 1.5326 1.4907 1.5129 1.5347 1.4929 1.5151 1.5369 1.4951 1.5173 1.5390 1.4974 1.5195 1.5412 1.4996 1.5217 1.5433 1.1 i. i. 4.7 4.8 4.9 1.5476 1.5686 1.5892 1.5497 1.5707 1.5913 1.5518 1.5728 1.5933 1.5539 1.5748 1.5953 1.5560 1.5769 1.5974 1.5581 1.5790 1.5994 1.5602 1.5810 1.6014 1.5623 1.5831 1.6034 1.5644 1.5851 1.6054 i. 1.1 i. NATURAL LOGARITHMS N 1 2 3 4 5 6 7 8 9 5.0 5.1 5.2 5.3 1.6094 1.6292 1.6487 1.6677 1.6114 1.6312 1.6506 1.6696 1.6134 1.6332 1.6525 1.6715 1.6154 1.6351 1.6544 1.6734 1.6174 1.6371 1.6563 1.6752 1.6194 1.6390 1.6582 1.6771 1.6214 1.6409 1.6601 1.6790 1.6233 1.6429 1.6620 1.6808 1.6253 1.6448 1.6639 1.6827 1.6273 1.6467 1.6658 1.6845 5.4 5.5 5.6 1.6864 1.7047 1.7228 1.6882 1.7066 1.7246 1.6901 1.7084 1.7263 1.6919 1.7102 1.7281 1.6938 1.7120 1.7299 1.6956 1.7138 1.7317 1.6974 1.7156 1.7334 1.6993 1.7174 1.7352 1.7011 1.7192 1.7370 1.7029 1.7210 1.7387 5.7 5.8 5.9 1.7405 1.7579 1.7750 1.7422 1.7596 1.7766 1.7440 1.7613 1.7783 1.7457 1.7630 1.7800 1.7475 1.7647 1.7817 1.7492 1.7664 1.7834 1.7509 1.7681 1.7851 1.7527 1.7699 1.7867 1.7544 1.7716 1.7884 1.7561 1.7733 1.7901 6.0 6.1 6.2 6.3 1.7918 1.8083 1.8245 1.8405 1.7934 1.8099 1.8262 1.8421 1.7951 1.8116 1.8278 1.8437 1.7967 1.8132 1.8294 1.8453 1.7984 1.8148 1.8310 1.8469 1.8001 1.8165 1.8326 1.8485 1.8017 1.8181 1.8342 1.8500 1.8034 1.8197 1.8358 1.8516 1.8050 1.8213 1.8374 1.8532 18.066 1.8229 1.8390 1.8547 6.4 6.5 6.6 1.8563 1.8718 1.8871 1.8579 1.8733 1.8886 1.8594 1.8749 1.8901 1.8610 1.8764 1.8916 1.8625 1.8779 1.8931 1.8641 1.8795 1.8946 1.8656 1.8810 1.8961 1.8672 1.8825 1.8976 1.8687 1.8840 1.8991 1.8703 1.8856 1.9006 S.7 6.8 6.9 1.9021 1.9169 1.9315 1.9036 1.9184 1.9330 1.9051 1.9199 1.9344 1.9066 1.9213 1.9359 1.9081 1.9228 1.9373 1.9095 1.9242 1.9387 1.9110 1.9257 1.9402 1.9125 1.9272 1.9416 1.9140 1.9286 1.9430 1.9155 1.9301 1.9445 7.0 7.1 7.2 7.3 1.9459 1.9601 1.9741 1.9879 1.9473 1.9615 1.9755 1.9892 1.9488 1.9629 1.9769 1.9906 1.9502 1.9643 1.9782 1.9920 1.9516 1.9657 1.9796 1.9933 1.9530 1.9671 1.9810 1.9947 1.9544 1.9685 1.9824 1.9961 1.9559 1.9699 1.9838 1.9974 1.9573 1.9713 1.9851 1.9988 1.9587 1.9727 1.9865 2.0001 M 7.5 7.6 2.0015 2.0149 2.0281 2.0028 2.0162 2.0295 2.0042 2.0176 2.0308 2.0055 2.0189 2.0321 2.0069 2.0202 2.0334 2.0082 2.0215 2.0347 2.0096 2.0229 2.0360 2.0109 2.0242 2.0373 2.0122 2.0255 2.0386 2.0136 2.0268 2.0399 r.7 r.s 7.9 2.0412 2.0541 2.0669 2.0425 2.0554 2.0681 2.0438 2.0567 2.0694 2.0451 2.0580 2.0707 2.0464 2.0592 2.0719 2.0477 2.0605 2.0732 2.0490 2.0618 2.0744 2.0503 2.0631 2.0757 2.0516 2.0643 2.0769 2.0528 2.0656 2.0782 8.0 5.1 5.2 5.3 2.0794 2.0919 2.1041 2.1163 2.0807 2.0931 2.1054 2.1175 2.0819 2.0943 2.1066 2.1187 2.0832 2.0956 2.1078 2.1199 2.0844 2.0968 2.1090 2.1211 2.0857 2.0980 2.1102 2.1223 2.0869 2.0992 2.1114 2.1235 2.0882 2.1005 2.1126 2.1247 2.0894 2.1017 2.1138 2.1258 2.0906 2.1029 2.1150 2.1270 5.4 5.5 5.6 2.1282 2.1401 2.1518 2.1294 2.1412 2.1529 2.1306 2.1424 2.1541 2.1318 2.1436 2.1552 2.1330 2.1448 2.1564 2.1342 2.1459 2.1576 2.1353 2.1471 2.1587 2.1365 2.1483 2.1599 2,1377 2.1494 2.1610 2.1389 2.1506 2.1622 5.7 1.8 5.9 2.1633 2.1748 2.1861 2.1645 2.1759 2.1872 2.1656 2.1770 2.1883 2.1668 2.1782. 2.1894 2.1679 2.1793 2.1905 2.1691 2.1804 2.1917 2.1702 2.1815 2.1928 2.1713 2.1827 2.1939 2.1725 2.1838 2.1950 2.1736 2.1849 2.1961 >.o ).l ).2 >.3 2.1972 2.2083 2.2192 2.2300 2.1983 2.2094 2.2203 2.2311 2.1994 2.2105 2.2214 2.2322 2.2006 2.2116 2.2225 2.2332 2.2017 2.2127 2.2235 2.2343 2.2028 2.2138 2.2246 2.2354 2.2039 2.2148 2.2257 2.2364 2.2050 2.2159 2.2268 2.2375 2.2061 2.2170 2.2279 2.2386 2.2072 2.2181 2.2289 2.2396 ).4 ).5 ).6 2.2407 2.2513 2.2618 2.2418 2.2523 2.2628 2.2428 2.2534 2.2638 2.2439 2.2544 2.2649 2.2450 2.2555 2.2659 2.2460 2.2565 2.2670 2.2471 2.2576 2.2680 2.2481 2.2586 2.2690 2.2492 2.2597 2.2701 2.2502 2.2607 2.2711 ).7 ).8 1.9 2.2721 2.2824 2.2925 2.2732 2.2834 2.2935 2.2742 2.2844 2.2946 2.2752 2.2854 2.2956 2.2762 2.2865 2.2966 2.2773 2.2875 2.2976 2.2783 2.2885 2.2986 2.2793 2.2895 2.2996 2.2803 2.2905 2.3006 2.2814 2.2915 2.3016 NATURAL LOGARITHMS N 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 38 39 40 41 42 43 44 45 46 47 48 49 0.00000 0.69315 1.09861 1.38629 2.30259 2.39790 2.48491 2.56495 2.63906 2.99573 3.04452 3.09104 3.13549 3.17805 3.40120 3.43399 3.46574 3.49651 3.52636 3.68888 3.71357 3.73767 3.76120 3.78419 3.91202 3.93183 3.95124 3.97029 3.98898 4.09434 4.11087 4.12713 4.14313 4.15888 ,24850 4.26268 ,38203 4.39445 49981 4.51086 .27667 4.29046 4.30407 .40672 4.41884 4.43082 .52179 4.53260 4.54329 .60517 4.61512 .62497 4.63473 4.64439 .70048 4.70953 .71850 4.72739 4.73620 .78749 4.79579 4.80402 4.81218 4.82028 4.86753 4.87520 4.88280 4.89035 4.89784 4.94164 4.94876 4.95583 4.96284 4.96981 5.01064 5.01728 5.02388 5.03044 5.03695 5.07517 5.08140 5.08760 5.09375 5.09987 5.13580 5.14166 5.14749 5.15329 5.15906 5.19296 5.19850 5.20401 5.20949 5.21494 5.24702 5.25227 5.25750 5.26269 5.26786 5.29832 5.30330 5.30827 5.31321 5.31812 5.34711 5.35186 5.35659 5.36129 5.36598 5.39363 5.39816 5.40268 5.40717 5.41165 5.43808 5.44242 5.44674 5.45104 5.45532 5.48064 5.48480 5.48894 5.49306 5.49717 5.52146 5.52545 5.52943 5.53339 5.53733 5.56068 5.56452 5.56834 5.57215 5.57595 5.59842 5.60212 5.60580 5.60947 5.61313 5.63479 5.63835 5.64191 5.64545 5.64897 5.66988 5.67332 5.67675 5.68017 5.68358 5.70378 5.70711 5.71043 5.71373 5.71703 5.73657 5.73979 5.74300 5.74620 5.74939 5.76832 5.77144 5.77455 5.77765 5.78074 5.79909 5.80212 5.80513 5.80814 5.81114 5.82895 5.83188 5.83481 5.83773 5.84064 5.85793 5.86079 5.86363 5.86647 5.86930 5.88610 5.88888 5.89164 5.89440 5.89715 5.91350 5.91620 5.91889 5.92158 5.92426 5.94017 5.94280 5.94542 5.94803 5.95064 5.96615 5.96871 5.97126 5.97381 5.97635 5.99146 5.99396 5.99645 5.99894 6.00141 6.01616 6.01859 6.02102 6.02345 6.02587 6.04025 6.04263 6.04501 6.04737 6.04973 6.06379 6.06611 6.06843 6.07074 6.07304 6.08677 6.08904 6.09131 6.09357 6.09582 6.10925 6.11147 6.11368 6.11589 6.11810 6.13123 6.13340 6.13556 6.13773 6.13988 6.15273 6.15486 6.15698 6.15910 6.16121 6.17379 6.17587 6.17794 6.18002 6.18208 6.19441 6.19644 6.19848 6.20051 6.20254 1.60944 1.79176 1.94591 2.07944 2.19722 2.70805 2.77259 2.83321 2.89037 2.94444 3.21888 3.25810 3.29584 3.33220 3.36730 3.55535 3.58352 3.61092 3.63759 3.66356 3.80666 3.82864 3.85015 3.87120 3.89182 4.00733 4.02535 4.04305 4.06044 4.07754 4.17439 4.18965 4.20469 4.21951 4.23411 4.31749 4.33073 4.34381 4.35671 4.36945 4.44265 4.45435 4.46591 4.47734 4.48864 4.55388 4.56435 4.57471 4.58497 4.59512 4.65396 4.66344 4.67283 4.68213 4.69135 4.74493 4.75359 4.76217 4.77068 4.77912 4.82831 4.83628 4.84419 4.85203 4.85981 4.90527 4.91265 4.91998 4.92725 4.93447 4.97673 4.98361 4.99043 4.99721 5.00395 5.04343 5.04986 5.05625 5.06260 5.06890 5.10595 5.11199 5.11799 5.12396 5.12990 5.16479 5.17048 5.17615 5.18178 5.19739 5.22036 5.22575 5.23111 5.23644 5.24175 5.27300 5.27811 5.28320 5.28827 5.29330 5.32301 5.32788 5.33272 5.33754 5.34233 5.37064 5.37528 5.37990 5.38450 5.38907 5.41610 5.42053 5.42495 5.42935 5.43372 5.45959 5.46383 5.46806 5.4.7227 5.47646 5.50126 5.50533 5.50939 5.51343 5.51745 5.54126 5.54518 5.54908 5.55296 5.55683 5.57973 5.58350 5.58725 5.59099 5.59471 5.61677 5.62040 5.62402 5.62762 5.63121 5.65249 5.65599 5.65948 5.66296 5.66643 5.68698 5.69036 5.69373 5.69709 5.70044 5.72031 5.72359 5.72685 5.73010 5.73334 5.75257 5.75574 5.75890 5.76205 5.76519 5.78383 5.78690 5.78996 5.79301 5.79606 5.81413 5.81711 5.82008 5.82305 5.82600 5.84354 5.84644 5.84932 5.85220 5.85507 5.87212 5.87493 5.87774 5.88053 5.88332 5.89990 5.90263 5.90536 5.90808 5.91080 5.92693 5.92959 5.93225 5.93489 5.93754 5.95324 5.95584 5.95842 5.96101 5.96358 5.97889 5.98141 5.98394 5.98645 5.98896 6.00389 6.00635 6.00881 6.01127 6.01372 6.02828 6.03069 6.03309 6.03548 6.03787 6.05209 6.05444 6.05678 6.05912 6.06146 6.07535 6.07764 6.07993 6.08222 6.08450 6.09807 6.10032 6.10256 6.10479 6.10702 6.12030 6.12249 6.12468 6.12687 6.12905 6.14204 6.14419 6.14633 6.14847 6.15060 6.16331 6.16542 6.16752 6.16961 6.17170 6.18415 6.18621 6.18826 6.19032 6.19236 6.20456 6.20658 6.20859 6.21060 6.21261 6 9 NATURAL LOGARITHMS 3 50 6.21461 6.21661 6.21860 6.22059 6.22258 51 6.23441 6.23637 6.23832 6.24028 6.24222 52 6.25383 6.25575 6.25767 6.25958 6.26149 53 6.27288 6.27476 6.27664 6.27852 6.28040 54 6.29157 6.29342 6.29527 6.29711 6.29895 55 6.30992 6.31173 6.31355 6.31536 6.31716 56 6.32794 6.32972 6.33150 6.33328 6.33505 57 6.34564 6.34739 6.34914 6.35089 6.35263 58 6.36303 6.36475 6.36647 6.36819 6.36990 59 6.38012 6.38182 6.38351 6.38519 6.38688 60 6.39693 6.39859 6.40026 6.40192 6.40357 61 6.41346 6.41510 6.41673 6.41836 6.41999 62 6.42972 6.43133 6.43294 6.43455 6.43615 63 6.44572 6.44731 6.44889 6.45047 6.45205 64 6.46147 6.46303 6.46459 6.46614 6.46770 65 6.47697 6.47851 6.48004 6.48158 6.48311 66 6.49224 6.49375 6.49527 6.49677 6.49828 67 6.50728 6.50877 6.51026 6.51175 6.51323 68 6.52209 6.52356 6.52503 6.52649 6.52796 69 6.53669 6.53814 6.53959 6.54103 6.54247 70 6.55108 6.55251 6.55393 6.55536 6.55678 71 6.56526 6.56667 6.56808 6.56948 6.57088 72 6.57925 6.58064 6.58203 6.58341 6.58479 73 6.59304 6.59441 6.59578 6.59715 6.59851 74 6.60665 6.60800 6.60935 6.61070 6.61204 75 6.62007 6.62141 6.62274 6.62407 6.62539 76 6.63332 6.63463 6.63595 6.63726 6.63857 77 6.64639 6.64769 6.64898 6.65028 6.65157 78 6.65929 6.66058 6.66185 6.66313 6.66441 79 6.67203 6.67330 6.67456 6.67582 6.67708 80 6.68461 6.68586 6.68711 6.6S835 6.68960 81 6.69703 6.69827 6.69950 6.70073 6.70196 82 6.70930 6.71052 6.71174 6.71296 6.71417 83 6.72143 6.72263 6.72383 6.72503 6.72623 84 6.73340 6.73459 6.73578 "6.73697 6.73815 85 6.74524 6.74641 6.74759 6.74876 6.74993 86 6.75693 6.75809 6.75926 6.76041 6.76157 87 6.76849 6.76964 6.77079 6.77194 6.77308 88 6.77992 6.78106 6.78219 6.78333 6.78446 89 6.79122 6.79234 6.79347 6.79459 6.79571 90 6.80239 6.80351 6.80461 6.80572 6.80683 91 6.81344 6.81454 6.81564 6.81674 6.81783 92 6.82437 6.82546 6.82655 6.82763 6.82871 93 6.83518 6.83626 6.83733 6.83841 6.83948 94 6.84588 6.84694 6.84801 6.84907 6.85013 95 6.85646 6.85751 6.85857 6.85961 6.86066 96 6.86693 6.86797 6.86901 6.87005 6.87109 97 6.87730 6.87833 6.87936 6.88038 6.88141 98 6.88755 6.88857 6.88959 6.89061 6.89163 99 6.89770 6.89871 6.89972 6.90073 6.90174 6.22456 6.22654 6.22851 6.23048 6.23245 6.24417 6.24611 6.24804 6.24998 6.25190 6.26340 6.26530 6.26720 6.26910 6.27099 6.28227 6.28413 6.28600 6.28786 6.28972 6.30079 6.30262 6.30445 6.30628 6.30810 6.31897 6.32077 6.32257 6.32436 6.32615 6.33683 6.33859 6.34036 6.34212 6.34388 6.35437 6.35611 6.35784 6.35957 6.36130 6.37161 6.37332 6.37502 6.37673 6.37843 6.38856 6.39024 6.39192 6.39359 6.39526 6.40523 6.40688 6.40853 6.41017 6.41182 6.42162 6.42325 6.42487 6.42649 6.42811 6.43775 6.43935 6.44095 6.44254 6.44413 6.45362 6.45520 6.45677 6.45834 6.45990 6.46925 6.47080 6.47235 6.47389 6.47543 6.48464 6.48616 6.48768 6.48920 6.49072 6.49979 6.50129 6.50279 6.50429 6.50578 6.51471 6.51619 6.51767 6.51915 6.52062 6.52942 6.53088 6.53233 6.53379 6.53524 6.54391 6.54535 6.54679 6.54822 6.54965 6.55820 6.55962 6.56103 6.56244 6.56386 6.57228 6.57368 6.57508 6.57647 6.57786 6.58617 6.58755 6.58893 6.59030 6.59167 6.59987 6.60123 6.60259 6.60394 6.60530 6.61338 6.61473 6.61607 6.61740 6.61874 6.62672 6.62804 6.62936 6.63068 6.63200 6.63988 6.64118 6.64249 6.64379 6.64509 6.65286 6.65415 6.65544 6.65673 6.65801 6.66568 6.66696 6.66823 6.66950 6.67077 6.67834 6.67960 6.68085 6.68211 6.68336 6.69084 6.69208 6.69332 6.69456 6.69580 6.70319 6.70441 6.70564 6.70686 6.70808 6.71538 6.71659 6.71780 6.71901 6.72022 6.72743 6.72863 6.72982 6.73102 6.73221 6.73934 6.74052 6.74170 6.74288 6.74406 6.75110 6.75227 6.75344 6.75460 6.75577 6.76273 6.76388 6.76504 6.76619 6.76734 6.77422 6.77537 6.77651 6.77765 6.77878 6.78559 6.78672 6.78784 6.78897 6.79010 6.79682 6.79794 6.79906 6.80017 6.80128 6.80793 6.80904 6.81014 6.81124 6.81235 6.81892 6.82002 6.82111 6.82220 6.82329 6.82979 6.83087 6.83195 6.83303 6.83411 6.84055 6.84162 6.84268 6.84375 6.84482 6.85118 6.85224 6.85330 6.85435 6.85541 6.86171 6.86276 6.86380 6.86485 6.86589 6.87213 6.87316 6.87420 6.87523 6.87626 6.88244 6.88346 6.88449 6.88551 6.88653 6.89264 6.89366 6.89467 6.89568 6.89669 6.90274 6.90375 6.90475 6.90575 6.90675 6 8 9 10 COMMON LOGARITHMS Logarithms of Numbers 1 100 N- Log. N Log. N Log. N Log. N Log. 1 0.00000 21 1 .32222 41 1 .61278 61 1.78533 81 1.90849 2 0.30103 22 1 .34242 42 1 .62325 62 1.79239 82 1.91381 3 0.47712 23 1 .36173 43 1 .63347 63 1.79934 83 1.91908 4 0.60206 24 1.38021 44 1 64345 64 1.80618 84 1. 92428 5 0.69897 25 1 .39794 45 1 .65321 65 1.81291 85 1.92942 6 0.77815 26 1 .41497 46 1 .66276 66 1.81954 86 1.93450 7 0.84510 27 1 .43136 47 1 .67210 67 .82607 87 1.93952 8 0.90309 28 1 .44716 48 1 .68124 68 .83251 88 1.94448 9 0.95424 29 1 .46240 49 1 ,69020 69 .83885 89 1.94939 10 1.00000 30 1 ,47712 50 1 .69897 70 .84510 90 1.95424 11 1.04139 31 1 .49136 51 1 .70757 71 .85126 91 1.95904 12 1.07918 32 1 .50515 52 1 .71600 72 1.85733 92 1.96379 13 1.11394 33 1 .51851 53 1 .72428 73 1.86332 93 1.96848 14 1.14613 34 1 .53148 54 1 .73239 74 1.86923 94 1.97313 15 1.17609 35 1 .54407 55 1 .74036 75 1.87506 95 1.97772 16 1.20412 36 1 .55630 56 1 .74819 76 1.88081 96 1.98227 17 1.23045 37 1 .56820 57 1 .75587 77 1.88649 97 1.98677 18 1.25527 38 1 .57978 58 1 .76343 78 1.89209 98. 1.99123 19 1.27875 39 1 .59106 59 1 .77085 79 1.89763 99 1.99564 20 1.30103 40 1 .60206 60 1 .77815 80 1.90309 100 2.000UO Mantissas of Numbers 10010000 N 1 2 3 4 5 6 7 8 9 D 100 101 00000 00432 00043 00475 00087 00518 00130 00561 00173 00604 00217 00647 00260 00689 00303 00732 00346 00775 00389 00817 43 1Q2 103 104 00860 01284 01703 00903 01326 01745 00945 01368 01787 00988 01410 01828 01030 01452 01870 01072 01494 01912 01115 01536 01953 01157 01578 01995 01199 01620 02036 01242 01662 02078 42 105 106 02119 02531 02160 02572 02202 02612 02243 02653 02284 02694 02325 02735 02366 02776 02407 02816 02449 02857 02490 02898 41 107 02938 02979 03019 03060 03100 108 03342 03383 03423 03463 03503 109 03743 03782 03822 03862 03902 03141 03181 03222 03262 03302 03543 03583 03623 03663 03703 40 03941 03981 04021 04060 04100 Proportional Parts D 1 2 3 4 5 6 7 8 9 44 4.4 8.8 13.2 17.6 22.0 26.4 30.8 35.2 39.6 43 4.3 8.6 12.9 17.2 21.5 25.8 30.1 34.4 38.7 42 4.2 8.4 12.6 16.8 21.0 25.2 29.4 33.6 37.8 41 4.1 8.2 12.3 16.4 20.5 24.6 28.7 32.8 36.9 40 4.0 8.0 12.0 16.0 20.0 24.0 28.0 32.0 36.0 COMMON LOGARITHMS 11 N 1 2 3 4 5 6 7 8 9 D 110 111 112 04139 04532 04922 04179 04571 04961 04218 04610 04999 04258 04650 05038 04297 04689 05077 04336 04727 05115 04376 04766 05154 04415 04805 05192 04454 04844 05231 04493 04883 05269 39 113 114 115 05308 05690 06070 05346 05729 06108 05385 05767 06145 05423 05805 06183 05461 05843 06221 05500 05881 06258 05538 05918 06296 05576 05956 06333 05614 05994 06371 05652 06032 06408 38 116 117 118 06446 06819 07188 06483 06,856 07225 06521 06893 07262 06558 06930 07298 06595 06967 07335 06633 07004 07372 06670 07041 07408 06707 07078 07445 06744 07115 07482 06781 07151 07518 37 119 120 121 07555 07918 08279 07591 07954 08314 07628 07990 08350 07664 08027 08386 07700 08063 08422 07737 08099 08458 07773 08135 08493 07809 08171 08529 07846 08207 08565 07882 08243 08600 36 122 123 124 125 08636 08991 09342 09691 08672 09026 09377 09726 08707 09061 09412 09760 08743 09096 09447 09795 08778 09132 09482 09830 08814 09167 09517 09864 08849 09202 09552 09899 08884 09237 09587 09934 08920 09272 09621 09968 08955 09307 09656 10003 35 126 127 128 129 10037 10380 10721 11059 10072 10415 10755 11093 10106 10449 10789 11126 10140 10483 10823 11160 10175 10517 10857 11193 10209 10551 10890 11227 10243 10585 10924 11261 10278 10619 10958 11294 10312 10653 10992 11327 10346 10687 11025 11361 34 130 131 132 133 11394 11727 12057 12385 11428 11760 12090 12418 11461 11793 12123 12450 11494 11826 12156 12483 11528 11860 12189 12516 11561 11893 12222 12548 11594 11926 12254 12581 11628 11959 12287 12613 11661 11992 12320 12646 11694 12024 12352 12678 33 134 135 136 137 12710 13033 13354 13672 12743 13066 13386 13704 12775 13098 13418 13735 12808 13130 13450 13767 12840 13162 13481 13799 12872 13194 13513 13830 12905 13226 13545 13862 12937 13258 13577 13893 12969 13290 13609 13925 13001 13322 13640 13956 32 138 139 13988 14301 14019 14333 14051 14364 14082 14395 14114 14426 14145 14457 14176 14489 14208 14520 14239 14551 14270 14582 31 N 1 2 3 4 5 6 7 8 9 D Proportional Parts D 1 2 3 4 5 6 7 8 9 39 3.9 7.8 11.7 15.6 19.5 23.4 27.3 31.2 35.1 38 3.8 7.6 11.4 15.2 19.0 22.8 26.6 30.4 34.2 37 3.7 7.4 11.1 14.8 18.5 22.2 25.9 29.6 33.3 36 3.6 7.2 10.8 14.4 18.0 21.6 25.2 28.8 32.4 35 3.5 7.0 10.5 14.0 17.5 21.0 24.5 28.0 31.5 34 3.4 6.8 10.2 13.6 17.0 20.4 23.8 27.2 30.6 33 3.3 6.6 9.9 13.2 16.5 19.8 23.1 26.4 29.7 32 3.2 6.4 9.6 12.8 16.0 19.2 22.4 25.6 28.8 31 3.1 6.2 9.3 12.4 15.5 18.6 21.7 24.8 27.9 12 COMMON LOGARITHMS N 1 2 3 4 5 6 7 8 9 D 140 141 142 14613 14922 15229 14644 14953 15259 14675 14983 15290 14706 15014 15320 14737 15045 15351 14768 15076 15381 14799 15106 15412 14829 15137 15442 14860 15168 15473 14891 15198 15503 31 143 144 145 146 15534 15836 16137 16435 15564 15866 16167 16465 15594 15897 16197 16495 15625 15927 16227 16524 15655 15957 16256 16554 15685 15987 16286 16584 15715 16017 16316 16613 15746 16047 16346 16643 15776 16077 16376 16673 15806 16107 16406 16702 30 147 148 149 16732 17026 17319 16761 17056 17348 16791 17085 17377 16820 17114 17406 16850 17143 17435 16879 17173 17464 16909 17202 17493 16938 17231 17522 16967 17260 17551 16997 17289 17580 nq 150 151 152 17609 17898 18184 17638 17926 18213 17667 17955 18241 17696 17984 18270 17725 18013 18298 17754 18041 18327 17782 18070 18355 17811 18099 18384 17840 18127 18412 17869 18156 18441 153 154 18469 18752 18498 18780 18526 18808 18554 18837 18583 18865 18611 18893 18639 18921 18667 18949 18696 18977 18724 19005 155 156 157 19033 19312 19590 19061 19340 19618 19089 19368 19645 19117 19396 19673 19145 19424 19700 19173 19451 19728 19201 19479 19756 19229 19507 19783 19257 19535 19811 19285 19562 19838 28 158 159 19866 20140 19893 20167 19921 20194 19948 20222 19976 20249 20003 20276 20030 20303 20058 20330 20085 20358 20112 20385 160 161 162 163 20412 20683 20952 21219 20439 20710 20978 21245 20466 20737 21005 21272 20493 20763 21032 21299 20520 20790 21059 21325 20548 20817 21085 21352 20575 20844 21112 21378 20602 20871 21139 21405 20629 20898 21165 21431 20656 20925 21192 21458 27 164 165 166 167 168 169 21484 21748 22011 22272 22531 22789 21511 21775 22037 22298 22557 22814 21537 21801 22063 22324 22583 22840 21564 21827 22089 22350 22608 22866 21590 21854 22115 22376 22634 22891 21617 21880 22141 22401 22660 22917 21643 21906 22167 22427 22686 22943 21669 21932 22194 22453 22712 22968 21696 21958 ^2220 22479 22737 22994 21722 21985 22246 22505 22763 23019 26 N 1 2 3 4 5 6 7 8 9 D Proportional Parts D 1 2 3 4 5 6 7 8 9 31 3.1 6.2 9.3 12.4 15.5 18.6 21.7 24.8 27.9 30 3.0 6.0 9.0 12.0 15.0 18.0 21.0 24.0 27.0 29 2,9 5.8 8.7 11.6 14.5 17,4 20.3 23.2 26.1 28 2.8 5.6 8.4 11.2 14.0 16.8 19.6 22.4 25.2 27 2.7 5.4 8.1 10.8 13.5 16.2 18.9 21.6 24.3 26 2.6 5.2 7.8 10.4 13.0 15.6 18.2 20.8 23.4 COMMON LOGARITHMS 13 N01234 56789D 170 23045 23070 23096 23121 23147 23172 23198 23223 23249 23274 171 23300 23325 23350 23376 23401 23426 23452 23477 23502 23528 172 23553 23578 23603 23629 23654 23679 23704 23729 23754 23779 173 23805 23830 23855 23880 23905 23930 23955 23980 24005 24030 174 24055 24080 24105 24130 24155 24180 24204 24229 24254 24279 175 24304 24329 24353 24378 24403 24428 24452 24477 24502 24527 176 24551 24576 24601 24625 24650 24674 246J9 24724 24748 24773 177 24797 24822 24846 24871 24895 24920 24944 24969 24993 25018 . 178 25042 25066 25091 25115 25139 25164 25188 25212 25237 25261 179 25285 25310 25334 25358 25382 25406 25431 25455 25479 25503 180 25527 25551 25575 25600 25624 25648 25672 25696 25720 25744 181 25768 25792 25816 25840 25864 25888 25912 25935 25959 25983 24 182 26007 26031 26055 26079 26102 26126 26150 26174 26198 26221 183 26245 26269 26293 26316 26340 26364 26387 26411 26435 26458 184 26482 26505 26529 26553 26576 26600 26623 26647 26670 26694 185 26717 26741 26764 26788 26811 26834 26858 26881 26905 26928 186 26951 26975 26998 27021 27045 27068 27091 27114 27138 27161 187 27184 27207 27231 27254 27277 27300 27323 27346 27370 27393 188 27416 27439 27462 27485 27508 27531 27554 27577 27600 27623 189 27646 27669 27692 27715 27738 27761 27784 27807 27830 27852 23 190 27875 27898 27921 27944 2796-7 27989 28012 28035 28058 28081 191 28103 8126 28149 28171 28194 28217 28240 28262 28285 28307 192 28330 28353 28375 28398 28421 28443 28466 28488 28511 28533 193 28556 28578 28601 28623 28646 28668 28691 28713 28735 28758 194 28780 28803 28825 28847 28870 28892 28914 28937 28959 28981 195 29003 29026 29048 29070 29092 29115 29137 29159 29181 29203 196 29226 29248 29270 29292 29314 29336 29358 29380 29403 29425 197 29447 29469 29491 29513 29535 29557 29579 29601 29623 29645 22 198 29667 29688 29710 29732 29754 29776 29798 29820 29842 29863 199 29885 29907 29929 29951 29973 29994 30016 30038 30060 30081 200 30103 30125 30146 30168 30190 30211 30233 30255 30276 30298 201 30320 30341 30363 30384 30406 30428 30449 30471 30492 30514 202 30535 30557 30578 30600 30621 30643 30664 30685 30707 30728 203 30750 30771 30792 30814 30835 30856 30878 30899 30920 30942 21 204 30963 30984 31006 31027 31048 31069 31091 31112 31133 31154 N01234 5678 Proportional Parts D 1 2 3 4 5 6 7 8 9 25 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 22.5 24 2*4 4.8 7.2 9.6 12.0 14.4 16.8 19.2 21.6 23 2.3 4.6 6.9 9.2 11.5 13.8 16.1 18.4 20.7 22 2.2 4.4 6.6 8.8 11.0 13.2 15.4 17.6 19.8 21 2.1 4.2 6.3 8.4 10.5 12.6 14.7 16.8 18.9 14 COMMON LOGARITHMS N 1 2 3 4 5 6 7 8 9 D 205 31175 31197 31218 31239 31260 31281 31302 31323 31345 31366 206 31387 31408 31429 31450 31471 31492 31513 31534 31555 31576 207 31597 31618 31639 31660 31681 31702 31723 31744 31765 31785 21 208 31806 31827 31848 31869 31890 31911 31931 31952 31973 31994 209 32015 32035 32056 32077 32098 32118 32139 32160 32181 32201 210 32222 32243 32263 32284 32305 32325 32346 32366 32387 32408 211 32428 32449 32469 32490 32510 32531 32552 32572 32593 32613 212 32634 32654 32675 32695 32715 32736 32756 32777 32797 32818 213 32838 32858 32879 32899 32919 32940 32960 32980 33001 33021 214 33041 33062 33082 33102 33122 33143 33163 33183 33203 33224 215 33244 33264 33284 33304 33325 33345 33365 33385 33405 33425 216 33445 33465 33486 33506 33526 33546 33566 33586 33606 33626 20 217 33646 33666 33686 33706 33726 33746 33766 33786 33806 33826 218 33846 33866 33885 33905 33925 33945 33965 33985 34005 34025 219 34044 34064 34084 34104 34124 34143 34163 34183 34203 34223 220 34242 34262 34282 34301 34321 34341 34361 34380 34400 34420 221 34439 34459 34479 34498 34518 34537 34557 34577 34596 34616 222 34635 34655 34674 34694 34713 34733 34753 34772 34792 34811 223 34830 34850 34869 34889 34908 34928 34947 34967 34986 35005 224 35025 35044 35064 35083 35102 35122 35141 35160 35180 35199 225 35218 35238 35257 35276 35295 35315 35334 35353 35372 35392 226 35411 35430 35449 35468 35488 35507 35526 35545 35564 35583 227 35603 35622 35641 35660 35679 35698 35717 35736 35755 35774 1 9 228 35793 35813 35832 35851 35870 35889 35S08 35927 35946 35965 Xj 229 35984 36003 36021 36040 36059 36078 36097 36116 36135 36154 230 36173 36192 36211 36229 36248 36267 36286 36305 36324 36342 231 36361 36380 36399 36418 36436 36455 36474 36493 36511 36530 232 36549 36568 36586 36605 36624 36642 36661 36680 36698 36717 233 36736 36754 36773 36791 36810 36829 36847 36866 36884 36903 234 36922 36940 36959 36977 36996 37014 37033 37051 37070 37088 235 37107 37125 37144 37162 37181 37199 37218 37236 37254 37273 236 37291 37310 37328 37346 37365 37383 37401 37420 37438 37457 18 237 37475 37493 37511 37530 37548 37566 37585 37603 37621 37639 238 37658 37676 37694 37712 37731 37749 37767 37785 37803 37822 239 37840 37858 37876 37894 37912 37931 37949 37967 37985 38003 N 1 2 3 4 5 6 7 8 9 D Proportional Parts D 1 2 3 4 5 6 7 8 9 21 2.1 4.2 6.3 8.4 10.5 12.6 14.7 16.8 18.9 20 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 19 1.9 3.8 5.7 7.6 9.5 11.4 13.3 15.2 17.1 18 1.8 3.6 5.4 7.2 9.0 10.8 12.6 14.4 16.2 COMMON LOGARITHMS 15 N 1 2 3 4 5 6 7 8 9 D 240 241 242 38021 38202 38382 38039 38220 38399 38057 38238 38417 38075 38256 38435 38093 38274 38453 38112 38292 38471 38130 38310 38489 38148 38328 38507 38166 38346 38525 38184 38364 38543 243 244 245 38561 38739 38917 38578 38757 38934 38596 38775 38952 38614 38792 38970 38632 38810 38987 38650 38828 39005 38668 38846 39023 38686 38863 39041 38703 38881 39058 38721 38899 39076 18 246 247 248 39094 39270 39445 39111 39287 39463 39129 39305 39480 39146 39322 39498 39164 39340 39515 39182 39358 39533 39199 39375 39550 39217 39393 39568 39235 39410 39585 39252 39428 39602 249 250 251 252 39620 39794 39967 40140 39637 39811 39985 40157 39655 39829 40002 40175 39672 39846 40019 40192 39690 39863 40037 40209 39707 39881 40054 40226 39724 39898 40071 40243 39742 39915 40088 40261 39759 39933 40106 40278 39777 39950 40123 40295 253 54 255 40312 40483 40654 40329 40500 40671 40346 40518 40688 40364 40535 40705 40381 40552 40722 40398 40569 40739 40415 40586 40756 40432 40603 40773 40449 40620 40790 40466 40637 40807 17 256 257 258 40824 40993 41162 40841 41010 41179 40858 41027 41196 40875 41044 41212 40892 41061 41229 40909 41078 41246 40926 41095 41263 40943 41111 41280 40960 41128 41296 40976 41145 41313 259 260 261 41330 41497 41664 41347 41514 41681 41363 41531 41697 41380 41547 41714 41397 41564 41731 41414 41581 41747 41430 41597 41764 41447 41614 41780 41464 41631 41797 41481 41641 41814 262 263 264 265 41830 41996 42160 42325 41847 42012 42177 42341 41863 42029 42193 42357 41880 42045 42210 42374 41896 42062 42226 42390 41913 42078 42243 42406 41929 42095 4-J259 42423 41946 42111 42275 42439 41963 42127 42292 42455 41979 42144 42308 42472 266 267 269 42488 42651 42813 42975 42504 42667 42830 42991 42521 42684 42846 43008 42537 42700 42862 43024 42553' 42716 42878 43040 42570 42732 42894 43056 42586 42749 42911 43072 42602 42765 42927 43088 42619 42781 42943 43104 42635 42797 42959 43120 270 271 272 273 43136 43297 43457 43616 43152 43313 43473 43632 43169 43329 43489 43648 43185 43345 43505 43664 43201 43361 43521 43680 43217 43377 43537 43696 43233 43393 43553 43712 43249 43409 43569 43727 43265 43425 43584 43743 43281 43441 43600 43759 16 274 275 276 43775 43933 44091 43791 43949 4*107 43807 43965 44122 43823 43981 44138 43838 43996 44154 43854 44012 44170 43870 44028 44185 43886 44044 44201 43902 44059 44217 43917 44075 44232 277 279 44248 44404 44560 44264 44420 44576 44279 44436 44592 44295 44451 44607 44311 44467 44623 44326 44483 44638 44342 44498 44654 44358 44514 44669 44373 44529 44685 44389 44545 44700 N 1 2 3 4 5 6 7 8 9 D Proportional Parts D 1 2 3 4 5 6 7 8 9 18 17 16 1.8 1.7 1.6 3.6 3.4 3.2 5 5 4. .4 7.2 .1 6.8 8 6.4 9.0 10.8 8.5 10.2 8.0 9.6 12.6 11.9 11.2 14.4 13.6 12.8 16.2 15.3 14.4 16 COMMON LOGARITHMS N 1 2 3 4 5 6 7 8 9 D 80 44716 44731 44747 44762 44778 44793 44809 44824 44840 44855 281 44871 44886 44902 44917 44932 44948 44963 44979 44994 45010 282 45025 45040 45056 45071 45086 45102 45117 45133 45148 45163 283 45179 45194 45209 45225 45240 45255 45271 45286 45301 45317 284 45332 45347 45362 45378 45393 45408 45423 45439 45454 45469 285 45484 45500 45515 45530 45545 45561 45576 45591 45606 45621 286 45637 45652 45667 45682 45697 45712 45728 45743 45758 45773 287 45788 45803 45818 45834 45849 45864 45879 45894 45909 45924 288 45939 45954 45969 45984 46000 46015 46030 46045 46060 46075 289 46090 46105 46120 46135 46150 46165 461*80 46195 46210 46225 15 290 46240 46255 46270 46285 46300 46315 46330 46345 46359 46374 291 46389 46404 46419 46434 46449 46464 46479 46494 46509 46523 292 46538 46553 46568 46583 46598 46613 46627 46642 46657 46672 293 46687 46702 46716 46731 46746 46761 46776 46790 46805 46820 294 46835 46850 46864 46879 46894 46909 46923 46938 46953 46967 295 46982 46997 47012 47026 47041 47056 47070 47085 47100 47114 296 47129 47144 47159 47173 47188 47202 47217 47232 47246 47261 297 47276 47290 47305 47319 47334 47349 47363 47378 47392 47407 298 47422 47436 47451 47465 47480 47494 47509 47524 47538 4755? 299 47567 47582 47596 47611 47625 47640 47654 47669 47683 47698 300 47712 47727 47741 47756 47770 47784 47799 47813 4782S 47842 301 47857 47871 47885 47900 47914 47929 47943 47958 47972 47986 302 48001 48015 48029 48044 48058 48073 48087 48101 48116 48130 303 48144 48159 48173 48187 48202 48216 48230 48244 48259 48273 304 48287 48302 48316 48330 48344 48359 48373 48387 48401 48416 305 48430 48444 48458 48473 48487 48501 48515 48530 48544 48558 306 48572 48586 48601 48615 48629 48643 48657 48671 48686 48700 307 48714 48728 48742 48756 48770 48785 48799 48813 48827 48841 308 48855 48869 48883 48897 48911 48926 48940 48954 48968 48982 309 48996 49010 49024 49038 49052 49066 49080 49094 49108 49122 14 310 49136 49150 49164 49178 49192 49206 49220 49234 49248 49262 311 49276 49290 49304 49318 49332 49346 49360 49374 49388 49402 312 49415 49429 49443 49457 49471 49485 49499 49513 49527 49541 313 49554 49568 49582 49596 49610 49624 49638 49651 49665 49679 314 49693 49707 49721 49734 49748 49762 49776 49790 49803 49817 315 49831 49845 49859 49872 49886 49900 49914 49927 49941 49955 316 49969 49982 49996 50010 50024 50037 50051 50065 50079 50092 317 50106 50120 50133 50147 50161 50174 50188 50202 50215 50229 318 50243 50256 50270 50284 50297 50311 50325 50338 50352 50365 319 50379 50393 50406 50420 50433 50447 50461 50474 50488 50501 N 1 2 3 4 5 6 7 8 9 D Proportional Parts 1.5 1.4 3.0 4.5 4.2 6.0 5.6 7.5 7.0 9.0 8.4 10.5 9.8 12.0 11.2 13.5 12.6 COMMON LOGARITHMS N 1 2 3 4 5 6 7 8 9 D 320 321 50515 50651 50529 50664 50542 50678 50556 50691 50569 50705 50583 50718 50596 50732 50610 50745 50623 50759 50637 50772 14 322 323 324 325 50786 50920 51055 51188 50799 50934 51068 51202 50813 50947 51081 51215 50826 50961 51095 51228 50840 50974 51108 51242 50853 50987 51121 51255 50866 51001 51135 51268 50880 51014 51148 51282 50893 51028 51162 51295 50907 51041 51175 51308 326 327 328 51322 51455 51587 51335 51468 51601 51348 51481 51614 51362 51495 51627 51375 51508 51640 51388 51521 51654 51402 51534 51667 51415 51548 51680 51428 51561 51693 51441 51574 51706 329 330 S31 51720 51851 51983 51733 51865 51996 51746 51878 52009 51759 51891 52022 '51772 51904 52035 51786 51917 52048 51799 51930 52061 51812 51943 52075 51825 51957 52088 51838 51970 52101 332 333 334 335 336 337 338 52114 52244 52375 52504 52634 52763 52892 52127 52257 52388 52517 52647 52776 52905 52140 52270 52401 52530 52660 52789 52917 52153 52284 52414 52543 52673 52802 52930 52166 52297 52427 52556 52686 52815 52943 52179 52310 52440 52569 52699 52827 52956 52192 52323 52453 52582 52711 52840 52969 52205 52336 52466 52595 52724 52853 52982 52218 52349 52479 52608 52737 52866 52994 52231 52362 52492 52621 52750 52879 53007 13 339 340 341 53020 53148 53275 53033 53161 53288 53046 53173 53301 53058 53186 53314 53071 53199 53326 53084 53212 53339 53097 53224 53352 53110 53237 53364 53122 53250 53377 53135 53263 53390 342 343 344 53403 53529 53656 53415 53542 53668 '53428 53555 53681 53441 53567 53694 53453 53580 53706 53466 53593 53719 53479 53605 53732 53491 53618 53744 53504 53631 53757 53517 53648 53769 345 346 S47 53782 53908 54033 53794 53920 54045 53807 53933 54058 53820 53945 54070 53832 53958 54083 53845 53970 54095 53857 53983 54108 53870 53995 54120 53882 54008 54133 53895 54020 54145 348 349 54158 54283 54170 54295 54183 54307 54195 54320 54208 54332 54220 54345 54233 54357 54245 54370 54258 54382 54270 54394 350 351 352 353 54407 54531 54654 54777 54419 54543 54667 54790 54432 54555 54679 54802 54444 54568 54691 54814 54456 54580 54704 54827 54469 54593 54716 54839 54481 54605 54728 54851 54494 54617 54741 54864 54506 54630 54753 54876 54518 54642 54765 54888 354 355 356 54900 55023 55145 54913 55035 55157 54925 55047 55169 54937 55060 55182 54949 55072 55194 54962 55084 55206 54974 55096 55218 54986 55108 55230 54998 55121 55242 55011 55133 55255 12 357 358 359 55267 55388 55509 55279 55400 55522 55291 55413 55534 55303 55425 55546 55315 55437 55558 55328 55449 55570 55340 55461 55582 55352 55473 55594 55364 55485 55606 55376 55497 55618 N 1 2 3 4 5 6 7 8 9 Proportional Parts D 1 2 3 4 5 6 7 8 9 14 13 12 1.4 1.3 1.2 2.8 2.6 2.4 4.2 3.9 3.6 5.6 5.2 4.8 7.0 6.5 6.0 8.4 7.8 7.2 9.8 9.1 8.4 11.2 10.4 9.6 12.6 11.7 10.8 CPMMON LOGARITHMS N 1 2 3 4 5 6 7 8 9 D 360 361 362 363 55630 55751 55871 55991 55642 55763 55883 56003 55654 55775 55895 56015 55666 55787 55907 56027 55678 55799 55919 56038 55691 55811 55931 56050 55703 55823 55943 56062 55715 55835 55955 56074 55727 55847 55967 56086 55739 55859 55979 56098 364 365 366 56110 56229 56348 56122 56241 56360 56134 56253 56372 56146 56265 56384 56158 56277 56396 56170 56289 56407 56182 56301 56419 56194 56312 56431 56205 56324 56443 56217 56336 56455 367 368 369 56467 56585 56703 56478 56597 56714 56490 56608 56726 56502 56620 56738 56514 56632 56750 56526 56644 56761 56538 56656 56773 56549 56667 56785 56561 56679 56797 56573 56691 56808 12 370 371 372 373 56820 56937 57054 57171 56832 56949 57066 57183 56844 56961 57078 57194 56855 56972 57089 57206 56867 56984 57101 57217 56879 56996 57113 57229 56891 57008 57124 57241 56902 57019 57136 57252 56914 57031 57148 57264 56926 57043 57159 57276 374 375 376 . 57287 57403 57519 57299 57415 57530 57310 57426 57542 57322 57438 57553 57334 57449 57565 57345 57461 57576 57357 57473 57588 57368 57484 57600 57380 57496 57611 57392 57507 57623 377 378 379 57634 57749 57864 57646 57761 57875 57657 57772 57887 57669 57784 57898 57680 57795 57910 57692 57807 57921 57703 57818 57933 57715 57830 57944 57726 57841 57955 57738 57852 57967 380 381 382 383 57978 58092 58206 58320 57990 58104 58218 58331 58001 58115 58229 58343 58013 58127 58240 58354 58024 58138 58252 58365 58035 58149 58263 58377 58047 58161 58274 58388 58058 58172 58286 58399 58070 58184 58297 58410 58081 58195 58309 58422 S84 385 386 58433 58546 58659 58444 58557 58670 58456 58569 58681 58467 58580 58692 58478 58591 58704 58490 58602 58715 58501 58614 58726 58512 58625 58737 57524 58636 58749 58535 58647 58760 S87 388 389 58771 58883 58995 58782 58894 59006 58794 58906 59017 58805 58917 59028 58816 58928 59040 58827 58939 59051 58838 58950 59062 58850 58961 59073 58861 58973 59084 58872 58984 59095 11 390 391 392 393 59106 59218 59329 59439 59118 59229 59340 59450 59129 59240 59351 59461 59140 59251 59362 59472 59151 59262 59373 59483 59162 59273 59384 59494 59173 59284 59395 59506 59184 59295 59406 59517 59195 59306 59417 59528 59207 59318 59428 59539 394 395 396 59550 59660 59770 59561 59671 59780 59572 59682 59791 59583 59693 59802 59594 59704 59813 59605 59715 59824 59616 59726 59835 59627 59737 59846 59638 59748 59857 59649 59759 59868 397 398 399 59879 59988 60097 59890 59999 60108 59901 60010 60119 59912 60021 60130 59923 60032 60141 59934 60043 60152 59945 60054 60163 59956 60065 60173 59966 60076 60184 59977 60086 60195 N 1 2 3 4 5 6 7 8 9 D Proportional Parts 1.2 1.1 2.4 2.2 3.6 3.3 4.8 4.4 6.0 5.5 7.2 6.6 8.4 7.7 9.6 8.8 10.8 9.9 COMMON LOGARITHMS 19 N0123456789D 400 60206 60217 60228 60239 60249 60260 60271 60282 60293 60304 401 60314 60325 60336 60347 60358 60369 60379 60390 60401 60412 402 60423 60433 60444 60455 60466 60477 60487 60498 60509 60520 403 60531 60541 60552 60563 60574 60584 60595 60606 60617 60627 404 60638 60649 60660 60670 60681 60692 60703 60713 60724 60735 405 60746 60756 60767 60778 60788 60799 60810 60821 60831 60842 406 60853 60863 60874 60885 60895 60906 60917 60927 60938 60949 ^ 407 60959 60970 60981 60991 61002 61013 61023 61034 61045 61055 408 61066 61077 61087 61098 61109 61119 61130 61140 61151 61162 409 61172 61183 61194 61204 61215 61225 61236 61247 61257 61268 410 61278 61289 61300 61310 61321 61331 61342 61352 61363 61374 411 61384 61395 61405 6141G 61426 61437 61448 61458 61469 61479 412 61490 61500 61511 61521 61532 61542 61553 61563 61574 61584 413 61595 61606 61616 61627 61637 61648 61658 61669 61679 61690 414 61700 61711 61721 61731 61742 61752 61763 61773 61784 61794 415 61805 61815 61826 61836 61847 61857 61868 61878 61888 61899 416 61909 61920 61930 61941 61951 61962 61972 61982 61993 62003 417 62014 62024 62034 62045 62055 H2066 62076 62086 62097 62107 418 62118 62128 62138 62149 62159 62170 62180 62190 62201 62211 419 62221 62232 62242 62252 62263 62273 62284 62294 62304 62315 420 62325 62335 62346 62356 62366 62377 62387 62397 62408 62418 421 62428 62439 62449 62459 62469 62480 62490 62500 62511 62521 422 62531 62542 62552 62562 62572 62-583 62593 62603 62613 62624 423 62634 62644 62655 62665 62675 62685 62696 62706 62716 62726 424 62737 62747 62757 62767 62778 62788 62798 62808 62818 62829 425 62839 62849 62859 62870 62880 62890 62900 62910 62921 62931 426 62941 62951 62961 62972 62982 62992 63002 63012 63022 63033 427 63043 63053 63063 63073 63083 63094 63104 63114 63124 63134 428 63144 63155 63165 63175 63185 63195 63205 63215 63225 63236 429 63246 63256 63266 63276 63286 63296 63306 63317 63327 63337 430 63347 63357 63367 63377 63387 63397 63407 63417 63428 63438 431 63448 63458 63468 63478 63488 63498 63508 63518 63528 63538 10 432 63548 63558 63568 63579 63589 63599 63609 63619 63629 63639 433 63649 63659 63669 63679 63689 63699 63709 63719 63729 63739 434 63749 63759 63769 63779 63789 63799 63809 63819 63829 63839 435 63849 63859 63869 63879 63889 63899 "63909 63919 63929 63939 436 63949 63959 63969 63979 63988 63998 64008 64018 64028 64038 437 64048 64058 64068 64078 64088 64098 64108 64118 64128 64137 438 64147 64157 64167 64177 64187 64197 64207 64217 64227 64237 439 64246 64256 64266 4276 64286 64296 64306 64316 64326 64335 440 64345 64355 64365 64375 64385 64395 64404 64414 64424 64434 441 64444 64454 64464 64473 64483 64493 64503 64513 64523 64532 442 64542 64552 64562 64572 64582 64591 64601 64611 64621 64631 443 64640 64650 64660 64670 64680 64689 64699 64709 64719 64729 444 64738 64748 64758 64768 64777 64787 64797 64807 64816 64826 445 64836 64846 64856 64865 64875 64885 64895 64904 64914 64924 446 64933 64943 64953 64963 64972 64982 64992 65002 65011 65021 447 65031 65040 65050 65060 65070 65079 65089 65099 65108 65118 448 65128 65137 65147 65157 65167 65176 65186 65196 65205 65215 449 65225 65234 65244 65254 65263 65273 65283 65292 65302 65312 N 01 23 4 56 7 8 9 D COMMON LOGARITHMS N 1 2 3 4 5 6 7 8 9 D 450 451 452 453 65321 65418 65514 65610 65331 65427 65523 65619 65341 65437 65533 65629 65350 65447 65543 65639 65360 65456 65552 6564S 65369 65466 65562 65658 65379 65475 65571 65667 65389 65485 65581 65677 65398 65495 65591 65686 65408 65504 65600 65696 10 454 455 456 65706 65801 65896 65715 65811 65906 65725 65820 65916 65734 65830 65925 65744 65839 65935 65753 65849 65944 65763 65858 65954 65772 65868 65963 65782 65877 65973 65792 65887 65982 457 458 459 65992 66087 66181 66001 66096 66191 66011 66106 66200 66020 66115 66210 66030 66124 66219 66039 66134 66229 66049 66143 66238 66058 66153 66247 66068 66162 66257 66077 66172 66266 460 461 462 463 66276 66370 66464 66558 66285 66380 66474 66567 66295 66389 66483 66577 66304 66398 66492 66586 66314 66408 66502 66596 66323 66417 66511 66605 66332 66427 66521 66614 66342 66436 66530 66624 66351 66445 66539 66633 66361 66455 66549 66642 464 465 466 66652 66745 66839 66661 66755 66848 66671 66764 66857 66680 66773 66867 66689 66783 66876 66699 66792 66885 66708 66801 66894 66717 66811 66904 66727 66820 66913 66736 66829 66922 467 468 469 66932 67025 67117 66941 67034 67127 66950 67043 67136 66960 67052 67145 66969 67062 67154 66978 67071 67164 66987 67080 67173 66997 67089 67182 67006 67099 67191 67015 67108 67201 470 471 472 473 67210 67302 67394 67486 67219 67311 67403 67495 67228 67321 67413 67504 67237 67330 67422 67514 67247 67339 67431 67523 67256 67348 67440 67532 67265 67357 67449 67541 67274 67367 67459 67550 67284 67376 67468 67560 67293 67385 67477 67569 474 475 476 477 478 479 67578 67669 67761 67852 67943 68034 67587 67679 67770 67861 67952 68043 67596 67688 67779 67870 67961 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69223 69311 69055 69144 69232 69320 69064 69152 69241 69329 69073 69161 69249 69338 69082 69170 69258 69346 69090 69179 69267 69355 69099 69188 69276 69364 494 495 496 69373 69461 69548. 69381 69469 69557 69390 69478 69566 69399 69487 69574 69408 69496 69583 69417 69504 69592 69425 69513 69601 69434 69522 69609 69443 69531 69618 69452 69539 69627 497 498 499 69636 69723 69810 69644 69732 69819 69653 69740 69827 69662 69749 69836 69671 69758 69845 69679 69767 69854 69688 69775 69862 69697 69784 69871 69705 69793 69880 69714 69801 69888 COMMON LOGARITHMS 21 N 1 2 3 4 5 6 7 8 9 D 500 501 502 503 69897 69984 70070 70157 69906 69992 70079 70165 69914 70001 70088 70174 69923 70010 70096 70183 69932 70018 70105 70191 69940 70027 70114 70200 69949 70036 70122 70209 69958 70044 70131 70217 69966 70053 70140 70226 69975 70062 70148 70234 504 505 506 70243 70329 70415 70252 70338 70424 70260 70346 70432 70269 70355 70441 70278 70364 70449 70286 70372 70458 70295 70381 70467 70303 70389 70475 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88547 88604 88497 88553 88610 88502 88559 88615 88508 88564 88621 88513 88570 88627 88519 88576 88632 88525 88581 88638 88530 88587 88643 770 771 772 773 88649 88705 88762 88818 88655 88711 88767 88824 88660 88717 88773 88829 88666 88722 88779 88835 88672 88728 88784 88840 88677 88734 88790 88846 88683 88739 88795 88852 88689 88745 88801 88857 88694 88750 88807 88863 88700 6 88756 88812 88868 774 775 776 88874 88930 88986 88880 88936 88992 88885 88941 88997 88891 88947 89003 88897 88953 89009 88902 88958 89014 88908 88964 89020 88913 88969 89025 88919 88975 89031 88925 88981 89037 777 778 779 89042 89098 89154 89048 89104 89159 89053 89109 89165 89059 89115 89170 89064 89120 89176 89070 89126 89182 89076 89131 89187 89081 89137 89193 89087 89143 89198 89092 89148 89204 780 781 782 783 89209 89265 89321 89376 89215 89271 89326 89382 89221 89276 89332 8938J 89226 89282 89337 89393 89232 89287 89343 89398 89237 89293 89348 89404 89243 89298 89354 89409 89248 89304 89360 89415 89254 89310 89365 89421 89260 89315 89371 89426 784 785 786 89432 .89487 89542 89437 89492 89548 89443 89498 89553 89448 89504 89559 89454 89509 89564 89459 89515 89570 89465 89520 89575 89470 89526 89581 89476 89531 89586 89481 89537 89592 787 788 789 89597 89653 89708 89603 89658 89713 89609 89664 89719 89614 89669 89724 89620 89675 89730 89625 89680 89735 89631 89686 89741 89636 89691 89746 89642 89697 89752 89647 89702 89757 790 791 792 793 89763 89818 89873 89927 89768 89823 89878 89933 89774 89829 89883 89938 89779 89834 89889 89944 89785 89840 89894 89949 89790 89845 89900 89955 89796 89851 89905 89960 89801 89856 89911 89966 89807 89862 89916 89971 89812 89867 89922 89977 794 795 796 9982 90037 90091 89988 90042 90097 89993 90048 90102 89998 90053 90108 90004 90059 90113 90009 90064 90119 90015 90069 90124 90020 90075 90129 90026 90080 90135 90031 90086 90140 5 797 798 799 90146 90200 90255 90151 90206 90260 90157 90211 90266 90162 90217 90271 90168 90222 90276 90173 90227 90282 90179 90233 90287 90184 90238 90293 90189 90244 90298 90195 90249 90304 COMMON LOGARITHMS 27 N0123456789D 800 90309 90314 90320 90325 90331 90336 90342 90347 90352 90358 801 90363 90369 90374 90380 90385 90390 90396 90401 90407 90412 802 90417 90423 90428 90434 90439 90445 90450 90455 90461 90466 803 90472 90477 90482 90488 90493 90499 90504 90509 90515 90520 804 90526 90531 90536 90542 90547 90553 90558 90563 90569 90574 805 90580 90585 90590 90596 90601 90607 90612 90617 90623 90628 806 90634 90639 90644 90650 90655 90660 90666 90671 90677 90682 807 90687 90693 90698 90703 90709 90714 90720 90725 90730 90736 808 90741 90747 90752 90757 90763 90768 90773 90779 90784 90789 809 90795 90800 90806 90811 90816 90822 90827 90832 90838 90843 810 90849 90854 90859 90865 90870 90875 90881 90886 90891 90897 811 90902 90907 90913 90918 90924 90929 90934 90940 90945 90950 812 90956 90961 90966 90972 90977 90982 90988 90993 90998 91004 813 91009 91014 91020 91025 91030 91036 91041 91046 91052 91057 814 91062 91068 91073 9107S 91084 91089 91094 91100 91105 91110 815 91116 91121 91126 91132 91137 91142 91148 91153 91158 91164 816 91169 91174 91180 91185 91190 91196 91201 91206 91212 91217 817 91222 91228 91233 91238 91243 91249 91254 91259 91265 91270 818 91275 91281 91286 91291 91297 91302 91307 91312 91318 91323 819 91328 91334 91339 91344 91350 91355 91360 91365 91371 91376 820 91381 91387 91392 91397 91403 91408 91413 91418 91424 91429 821 91434 91440 91445 91450 91455 91461 91466 91471 91477 91482 822 91487 91492 91498 91503 91508 91514 91519 91524 91529 91535 823 91540 91545 91551 91556 91561 91566 91572 91577 91582 91587 824 91593 91598 91603 91609 91614 91619 91624 91630 91635 91640 825 91645 91651 91656 91661 91666 91672 91677 91682 91687 91693 5 826 91698 91703 91709 91714 91719 91724 91730 91735 91740 91745 827 91751 91756 91761 91766 91772 91777 91782 91787 91793 91798 828 91803 91808 91814 91819 91824 91829 91834 91840 91845 91850 829 91855 91861 91866 91871 91876 91882 91887 91892 91897 91903 830 91908 91913 91918 91924 91929 91934 91939 91944 91950 91955 831 91960 91965 91971 91976 91981 91986 91991 91997 92002 92007 832 92012 92018 92023 92028 92033 92038 92044 92049 92054 92059 833 92065 92070 92075 92080 92085 92091 92096 92101 92106 92111 834 92117 92122 92127 92132 92137 92143 92148 92153 92158 92163 835 92169 92174 92179 92184 92189 92195 92200 92205 92210 92215 836 92221 92226 92231 92236 92241 92247 92252 92257 92262 92267 837 92273 92278 92283 92288 92293 92298 92304 92309 92314 92319 838 92324 92330 92335 92340 92345 92350 92355 92361 92366 92371 839 92376 92381 92387 92392 92397 92402 92407 92412 92418 92423 840 92428 92433 92438 92443 92449 92454 92459 92464 92469 92474 841 92480 92485 92490 92495 92500 92505 92511 92516 92521 92526 842 92531 92536 92542 92547 92552 92557 92562 92567 92572 92578 843 92583 92588 92593 92598 92603 92609 92614 92619 92624 92629 844 92634 92639 92645 92650 92655 92660 92665 92670 92675 92681 845 92686 92691 92696 92701 92706 92711 92716 92722 92727 92732 846 92737 92742 92747 92752 92758 92763 92768 92773 92778 92783 847 92788 92793 92799 92804 92809 92814 92819 92824 92829 92834 848 92840 92845 92850 92855 92860 92865 92870 92875 92881 92886 849 92891 92896 92901 92906 92911 92916 92921 92927 92932 92937 N012345678 9 D 28 COMMON LOGARITHMS N 1 2 3 4 5 6 7 8 9 D 850 851 852 853 92942 92993 93044 93095 92947 92998 93049 93100 92952 93003 93054 93105 92957 93008 93059 93110 92962 93013 93064 93115 92967 93018 93069 93120 92973 93024 93075 93125 92978 93029 93080 93131 92983 93034 93085 9313G 92988 93039 930SO 93141 854 855 856 93146 93197 93247 93151 93202 93252 93156 93207 93258 93161 93212 93263 93166 93217 93268 93171 93222 93273 93176 93227 93278 93181 93232 93283 93186 93237 93288 93192 93242 93293 857 858 859 93298 93349 93399 93303 93354 93404 93308 93359 93409 93313 93364 93414 93318 93369 93420 93323 93374 93425 93328 93379 93430 93334 93384 93435 93339 93389 93440 93344 93394 93445 860 861 862 863 93450 93500 93551 93601 93455 93505 93556 93606 93460 93510 93561 93611 93465 93515 93566 93616 93470 93520 93571 93621 93475 93526 93576 93626 93480 93531 93581 93631 93485 93536 93586 93636 93490 93541 93591 93641 93495 93546 93596 93646 864 865 866 93651 93702 93752 93656 93707 93757 93661 93712 93762 93666 93717 93767 93671 93722 93772 93676 93727 93777 93682 93732 93782 93687 93737 93787 93692 93742 93792 93697 93747 93797 867 868 869 93802 93852 93902 93807 93857 93907 93812 93862 93912 93817 93867 93917 93822 93872 93922 93827 93877 93927 93832 93882 93932 93837 93887 93937 93842 93892 93942 93847 93897 93947 870 871 872 873 93952 94002 94052 94101 93957 94007 94057 94106 93962 94012 94062 94111 93967 94017 94067 94116 93972 94022 94072 94121 93977 94027 94077 94126 93982 94032 94082 94131 93987 94037 94086 94136 93992 94042 94091 94141 93997 94047 94096 94146 874 875 876 94151 94201 94250 94156 94206 94255 94161 94211 94260 94166 94216 94265 94171 94221 94270 94176 94226 94275 94181 94231 94280 96186 94236 94285 94191 94240 94290 94196 94245 5 94295 877 878 879 94300 94349 94399 94305 94354 94404 94310 94359 94409 94315 94364 94414 94320 94369 94419 94325 94374 94424 94330 94379 94429 94335 94384 94433 94340 94389 94438 943*45 94394 94443 880 881 882 883 94448 94498 94547 94596 94453 94503 94552 94601 94458 94507 94557 94606 94463 94512 94562 94611 94468 94517 94567 94616 94473 94522 94571 94621 94478 94527 94576 94626 94483 94532 94581 94630 94488 94537 94586 94635 94493 94542 94591 94640 884 885 886 94645 94694 94743 94650 94699 94748 94655 94704 94753 94660 94709 94758 94665 94714 94763 94670 94719 94768 94675 94724 94773 94680 94729 94778 94685 94734 94783 94689 94738 94787 887 888 889 94792 94841 94890 94797 94846 94895 94802 94851 94900 94807 94856 94905 94812 94861 94910 94817 94866 94915 94822 94871 94919 94827 94876 94924 94822 94880 94929 94836 94885 94934 890 891 892 893 94939 94988 95036 95085 94944 94993 95041 95090 94949 94998 95046 95095 94954 95002 95051 95100 94959 95007 95056 95105 94963 95012 95061 95109 94968 95017 95066 95114 94973 95022 95071 95119 94978 95027 95075 95124 94983 95032 95080 95129 894 895 896 95134 95182 95231 95139 95187 95236 95143 95192 95240 95148 95197 95245 95153 95202 95250 95158 95207 95255 95163 95211 95260 95168 95216 95265 95173 95221 95270 95177 95226 95274 897 898 899 95279 95328 95376 95284 95332 95381 95289 95337 95386 95294 95342 95390 95299 95347 95395 95303 95352 95400 95308 95357 95405 95313 95361 95410 95318 95366 95415 95323 95371 95419 COMMON LOGARITHMS 29 N 1 2 3 4 5 6 7 8 9 D 900 901 902 903 95424 95472 95521 95569 95429 95477 95525 95574 95434 95482 95530 95578 95439 95487 95535 95583 95444 95492 95540 95588 95448 95497 95545 95593 95453 95501 95550 95598 95458 95506 95554 95602 95463 95511 95559 95607 95468 95516 95564 95612 904 905 906 95617 95665 95713 95622 95670 95718 95626 95674 95722 95631 95679 95727 95636 95684 95732 95641 95689 95737 95646 95694 95742 95650 95698 95746 95655 95703 95751 95660 95708 95756 907 908 909 95761 95809 95856 95766 95813 95861 95770 95818 95866 95775 95823 95871 95780 95828 95875 95785 95832 95880 95789 95837 95885 95794 95842 95890 95799 95847 95895 95804 95852 95899 910 911 912 913 95904 95952 95999 96047 95909 95957 96004 96052 95914 95961 96009 96057 95918 95966 96014 96061 95923 95971 96019 96066 95928 95976 96023 96071 95933 95980 96028 96076 95938 95985 96033 96080 95942 95990 96038 96085 95947 95995 96042 96090 914 915 916 96095 96142 96190 96099 96147 96194 96104 96152 96199 96109 96156 96204 96114 96161 96209 96118 96166 96213 96123 96171 96218 96128 96175 96223 96133 96180 96227 96137 96185 96232 917 918 919 96237 96284 96332 96242 96289 96336 96246 96294 96341 96251 96298 96346 96256 96303 96350 96261 96308 96355 96265 96313 96360 96270 96317 96365 96275 96322 96359 96280 96327 96374 920 921 922 923 96379 96426 96473 96520 96384 96431 96478 96525 96388 96435 96483 96530 96393 96440 96487 96534 96398 96445 96492 96539 96402 96450 96497 96544 96407 96454 96501 96548 96412 96459 96506 96553 96417 96464 96511 96558 96421 96468 96515 96562 924 925 926 96567 96614 96661 96572 96619 96666 96577 96624 96670 96581 96628 96675 96586 96633 96680 96591 96638 96685 96595 96642 96689 96600 96647 96694 96605 96652 96699 96609 5 96656 96703 927 928 929 96708 96755 96802 96713 96759 96806 96717 96764 96811 96722 96769 96816 96727 96774 96820 96731 96778 96825 96736 96783 96830 96741 96788 96834 96745 96792 96839 96750 96797 96844 930 931 932 933 96848 96895 96942 96988 96853 96900 96946 96993 96858 96904 96951 96997 96862 96909 96956 97002 96867 96914 96960 97007 96872 96918 96965 97011 96876 96923 96970 97016 96881 96928 96974 97021 96886 96932 96979 97025 96890 96937 96984 97030 934 935 936 97035 97081 97128 97039 97086 97132 97044 97090 97137 97049 97095 97142 97053 97100 97146 97058 97104 97151 97063 97109 97155 97067 97114 97160 97072 97118 97165 97077 97123 97169 937 938 939 97174 97220 97267 9717'9 97225 97271 97183 97230 97276 97188 97234 972.80 97192 97239 97285 97197 97243 97290 97202 97248 97294 97206 97253 97299 97211 97257 97304 97216 97262 97308 940 941 942 943 97313 97359 97405 97451 97317 97364 97410 97456 97322 97368 97414 97460 97327 97373 97419 97465 97331 97377 97424 97470 97336 97382 97428 97474 97340 97387 97433 97479 97345 97391 97437 97483 97350 97396 97442 97488 97354 97400 97447 97493 944 945 946 97497 97543 97589 97502 97548 97594 97506 97552 97598 97511 97557 97603 97516 97562 97607 97520 97566 97612 97525 97571 97617 97529 97575 97621 97534 97580 97626 97539 97585 97630 947 948 949 97635 97681 97727 97640 97685 97731 97644 97690 97736 97649 97695 97740 97653 97699 97745 97658 97704 97749 97663 97708 97754 97667 97713 97759 97672 97717 97763 97676 97722 97768 30 COMMON LOGARITHMS N 1 2 3 4 5 6 7 8 9 D 950 951 952 953 97772 97818 97864 97909 97777 97823 97868 97914 97782 97827 97873 97918 97786 97832 97877 97923 97791 97836 97882 97928 97795 97841 97886 97932 .97800 97845 97891 97937 97804 97850 97896 97941 97809 97855 97900 97946 97813 97859 97905 97950 954 955. 956 97955 98000 98046 97959 98005 98050 97964 98009 98055 97968 98014 98059 97973 98019 98064 97978 98023 98068 97982 98028 98073 97987 98032 98078 97991 98037 98082 97996 98041 98087 957 958 959 98091 98137 98182 98096 98141 98186 98100 98146 98191 98105 98150 98195 98109 98155 98200 98114 98159 98204 98118 98164 98209 98123 98168 98214 98127 98173 98218 98132 98177 98223 5 960 961 962 963 98227 98272 98318 98363 98232 98277 98322 98367 98236 98281 98327 98372 98241 98286 98331 98376 98245 98290 98336 98381 98250 98295 98340 98385 98254 98299 98345 98390 98259 98304 98349 98394 98263 98308 98354 98399 98268 98313 98358 98403 964 98408 98412 98417 98421 98426 98430 98435 98439 98444 98448 965 966 98453 98498 98457 98502 98462 98507 98466 98511 98471 98516 98475 98520 98480 98525 98484 98529 98489 98534 98493 98538 967 968 969 98543 98588 98632 98547 98592 98637 98552 98597 98641 98556 98601 98646 98561 98605 98650 98565 98610 98655 98570 98614 98659 98574 98619 98664 98579 98623 98668 98583 98628 98673 970 971 972 973 98677 98722 98767 98811 98682 98726 98771 98816 98686 98731 98776 98820 98691 98735 98780 98825 98695 98740 98784 98829 98700 98744 98789 98834 98704 98749 98793 98838 98709 98753 98798 98843 98713 98758 98802 98847 98717 98762 98807 98851 974 975 976 98856 98900 98945 98860 98905 98949 98865 98909 98954 98869 98914 98958 98874 98918 98963 98878 98923 98967 98883 98927 98972 98887 98932 98976 98892 98936 98981 98896 98941 98985 977 978 979 98989 99034 99078 98994 99038 99083 98998 99043 99087 99003 99047 99092 99007 99052 99096 99012 99056 99100 99016 99061 99105 99021 99065 99109 99025 99069 99114 99029 99074 99118 980 981 982 983 99123 99167 99211 99255 99127 99171 99216 99260 99131 99176 99220 99264 99136 99180 99224 99269 99140 99185 99229 99273 99145 99189 99233 99277 99149 99193 99238 99282 99154 99198 99242 99286 99158 99202 99247 99291 99162 99207 99251 99295 4 984 985 986 99300 99344 99388 99304 99348 99392 99308 99352 99396 99313 99357 99401 99317 99361 99405 99322 99366 99410 99326 99370 99414 99330 99374 99419 99335 99379 99423 99339 99383 99427 987 988 989 99432 99476 99520 99436 99480 99524 99441 99484 99528 99445 99489 99533 99449 99493 99537 99454 99498 99542 99458 99502 99546 99463 99506 99550 99467 99511 99555 99471 99515 99559 990 991 992 993 99564 99607 99651 99695 99568 99612 99656 99699 99572 99616 99660 99704 99577 99621 99664 99708 99581 99625 99669 99712 99585 99629 99673 99717 99590 99634 99677 99721 99594 99638 99682 99726 99599 99642 99686 99730 99603 99647 99691 99734 994 995 996 99739 99782 99826 99743 99787 99830 99747 99791 99835 99752 99795 9983D 99756 99800 99843 99760 99804 99848 99765 99808 99852 99769 99813 99856 99774 99817 99861 99778 99822 99865 997 998 999 99870 99913 99957 99874 99917 99961 99878 99922 99965 99883 99926 99970 99887 99930 99974 99891 99935 99978 99896 99939 99983 99900 99944 99987 99904 99948 99991 99909 99952 99996 TRIGONOMETRIC FUNCTIONS Sine Tangent Cotangent Cosine Ang, . Nat. Log. Nat. Log. Nat. Log. Nat. Log. 7 7 2 9 1.00000 90 .1 .00175 .24188 .00175 .24188 572.96 .75812 1.00000 .99999 .9 .2 .00349 .54291 .00349 .54291 286.48 .45709 .99999 .99999 .8 .3 .00524 .71900 .00524 .71900 190.98 .28100 .99999 .99999 .7 .4 .00698 .84393 .00698 .84394 143.24 .15606 .99998 .99999 .6 .5 .00873 .94084 .00873 .94086 114.59 .05914 .99996 .99998 .5 8 8 1 9 .6 .01047 .02002 .01047 .02004 95.489 .97996 .99995 .99998 .4 .7 .01222 .08697 .01222 .08700 81.847 .91300 .99993 .99997 .3 .8 .01396 -.14495 .01396 .14500 71.615 .85500 .99990 .99996 .2 .9 .01571 .19610 .01571 .19616 63.657 .80384 .99988 .99995 .1 1.0 .01745 .24186 .01746 . 24192 57.290 .75808 .99985 .99993 89 .1 .01920 .28324 .01920 .28332 52.081 .71668 .99982 .99992 9 .2 .02094 .32103 .02095 .32112 47.740 .67888 .99978 .99990 .8 .3 .02269 .35578 .02269 .35590 44.066 .64410 .99974 .99989 .7 .4 .02443 .38796 .02444 .38809 40.917 .61191 .99970 .99987 .6 .5 .02618 .41792 .02619 .41807 38.188 .58193 .99966 .99985 5 .6 .02792 .44594 .02793 .44611 35.801 .55389 .99961 .99983 .4 .7 .02967 .47226 .02968 .47245 33.694 .52755 .99956 .99981 .3 .8 .03141 .49708 .03143 .49729 31.821 .50271 .99951 .99979 2 .9 .03316 .52055 .03317 .52079 30.145 .47921 .99945 .99976 !i 2.0 .03490 .54282 .03492 .54308 28.636 .45692 .99939 .99974 88 .1 .03664 .56400 .03667 .56429 27.271 .43571 .99933 .99971 .9 2 .03839 .58419 .03842 .58451 26.031 .41549 .99926 .99968 .8 !3 .04013 .60349 .01016 .60384 24.898 .39616 .99919 .99965 .7 .4 .04188 .62196 .04191 .62234 23.859 .37766 .99912 .99962 .6 .5 .04362 .63968 .04366 .64009 22.904 .35991 .99905 .99959 .5 .6 .04536 .65670 .04541 .65715 22.022 .34285 .99897 .99955 .4 .7 .04711 .67308 .04716 .67356 21.205 .32644 .99889 .99952 .3 .8 .04885 .68886 .04891 .68938 20.446 .31062 .99881 .99948 .2 .9 .05059 .70409 .05066 .70465 19.740 .29535 .99872 .99944 .1 3.0 .05234 .71880 .05241 .71940 19.081 .28060 .99863 .99940 87 .1 .05408 .73303 .05416 .73366 18.464 .26634 .99854 .99936 .9 2 .05582 .74680 .05591 .74748 17.886 .25252 .99844 .99932 .8 !s .05756 .76015 .05766 .76087 17.343 .23913 .98334 .99928 .7 .4 .05931 .77310 .05941 .77387 16.832 .22613 .99824 .99923 .6 .5 .06105 .78568 .06116 .78649 16.350 .21351 .99813 .99919 .5 .6 .06279 .79789 .06291 .79875 15.895 .20125 .99803 .99914 .4 .7 .06453 .80978 .06467 .81068 15.464 .18932 .99792 .99909 .3 .8 .06627 .82134 .06642 .82230 15.056 .17770 .99780 .99904 .2 .9 .06802 .83261 .06817 .83361 14.669 .16639 .99768 .99899 .1 4.0 .06976 .84358 .06993 .84464 14.301 .15536 .99756 .99894 86 .1 .07150 .85429 .07168 .85540 13.951 .14460 .99744 .99889 .9 2 .07324 .86474 .07344 .86591 13.617 .13409 .99731 .99883 .8 !s .07498 .87494 .07519 .87616 13.300 .12384 .99719 .99878 .7 .4 .07672 .88490 .07695 .88619 12.996 .11382 .99705 .99872 .6 .5 .07846 .89464 .07870 .89598 12.706 .10402 .99692 .99866 .5 .6 .08020 .90417 .08046 .90557 12.429 .09443 .99678 .99860 .4 .7 .08194 .91349 .08221 .91495 12.163 .08505 .99664 .99854 .3 .8 .08368 .92261 .08397 .92414 11.909 .07586 .99649 .99847 .2 .9 .08542 .93154 .08573 .93313 11.664 .06687 .99635 .99841 .1 5.0 .08716 .94030 .08749 .94195 11.430 .05805 .99619 .99834 85 Nat. Log. Nat. Log. Nat. Log. Nat. Log.Ang. Cosine Cotangent Tangent Sine TRIGONOMETRIC FUNCTIONS Sine Tangent Cotangent Cosine Ang . Nat. Log. Nat. Log. Nat. Log. Nat. Log. 8 8 1 9 5.0 .08716 .94030 .08749 .04195 11.430 .05805 .99619 .99834 85 .1 .08889 .94887 .08925 .95060 11.205 .04940 .99604 .99828 .9 .2 .09063 .95728 .09101 .95908 10.988 .04092 .99588 .99821 .8 .3 .09237 .96553 .09277 .96739 10.780 .03261 .99572 .99814 .7 .4 .09411 .97363 .09453 .97556 10.579 .02444 .99556 .99807 .6 .5 .09585 .98157 .09629 .98358 10.385 .01642 .99540 .99800 .5 .6 .09758 .98937 .09805 .99145 10.199 .00855 .99523 .99792 .4 .7 .09932 .99704 .09981 .99919 10.019 .00081 .99506 .99785 .3 9 9 10 9 .8 .10106 .00456 .10158 .00679 9.4480 .99321 .99488 .99777 .2 .9 .10279 .01196 .10334 .01427 9.6768 .98573 .99470 .99769 .1 6.0 .10453 .01923 .10510 .02162 9.5144 .97838 .99452 .99761 84 .1 .10626 .02639 .10687 .02885 9.3572 .97115 .99434 .99753 .9 .2 .10800 .03342 .10863 .03597 9.2052 .96403 .99415 .99745 .8 .3 .10973 .04034 .11040 .04297 9.0579 .95703 .99396 .99737 .7 .4 .11147 .04715 .11217 .04987 8.9152 .95013 .99377 .99729 .6 .5 .11320 .05386 .11394 .05666 8.7769 .94334 .99357 .99720 .5 .6 .11494 .06046 .11570 .06335 8.6427 .93665 .99337 .99711 .4 .7 .11667 .06696 .11747 .06994 8.5126 .93006 .99317 .99702 .3 .8 .11840 .07337 .11924 .07643 8.3863 .92357 .99297 .99693 .2 .9 .12014 .07968 .12101 .08283 8.2636 .91717 .99276 .99684 .1 7.0 .12187 .08589 .12278 .08914 8.1443 .91086 .99255 .99675 83 .1 .12360 .09202 .12456 .09537 8.0285 .90463 .99233 .99666 .3 .2 .12533 .09807 .12633 .10150 7.9158 .89850 .99211 .99656 .8 .3 .12706 .10403 .12810 .10756 7.8062 .89244 .99189 .99647 .7 .4 .12880 .10990 .12988 .11353 7.6996 .88647 .99167 .99637 .6 .5 .13053 .11570 .13165 .11943 7.5958 .88057 .99144 .99627 .5 .6 .13226 .12142 .13343 .12525 7.4947 .87475 .99122 .99617 .4 .7 .13399 .12706 .13521 .13099 7.3962 .86901 .99098 .99607 .3 .8 .13572 .13263 .13698 .13667 7.3002 .86333 .99075 .99596 .2 .9 .13744 .13813 .13876 .14227 7.2066 .85773 .99051 .99586 .1 8.0 .13917 .14356 .14054 .14780 7.1154 .85220 .99027 .99575 82 .1 .14090 .14891 .14232 .15327 7.0264 .84673 .99002 .99565 .9 .2 .14263 .15421 .14410 .15867 6.9395 .84133 .98978 .99554 .8 .3 .14436 .15944 .14588 .16401 6.8548 .83599 .98953 .99543 .7 .4 .14608 .16460 .14767 .16928 6.7720 .83072 .98927 .99532 .6 .5 .14781 .16970 .14945 .17450 6.6912 .82550 .98902 .99520 .5 .6 .14954 .17474 .15124 .17965 6.6122 .82035 .98876 .99509 .4 .7 .15126 .17973 .15302 .18475 6.5350 .81525 .98849 .99497 .3 .8 .15299 .18465 .15481 .18979 6.4596 .81021 .98823 .99486 .2 .9 .15471 .18952 .15660 .19478 6.3859 .80522 .98796 .99474. .1 9.0 .15643 .19433 .15838 .19971 6.3138 .80029 .98769 .99462 81 .1 .15816 .19909 .16017 .20459 6.2432 .79541 .98741 .99450 .9 .2 .15988 .20380 .16196 .20942 6.1742 .79058 .98714 .99438 .8 .3 .16160 .20845 .16376 .21420 6.1066 .78580 .98686 .99425 .7 .4 .16333 .21306 .16555 .21893 6.0405 .78107 .98657 .99413 .6 .5 .16505 .21761 .16734 .22361 5.9758 .77639 .98629 .99400 .5 .6 .16677 .22211 .16914 . 22824 5.9124 .77176 .98600 .99388 .4 .7 .16849 .22657 .17093 . 23283 5.8502 .76717 .98570 .99375 .3 .8 .17021 .23098 .17273 .23737 5.7894 .76263 .98541 .99362 2 .9 .17193 .23535 .17453 .24187 5.7297 .75814 .98511 .99348 !i 10 .17365 .23967 .17633 .24632 5.6713 .75368 .98481 .99335 80 Nat. Log. Nat. Log. Nat. Log. Nat. Log.Ang. Cosine Cotangent Tangent Sine TRIGONOMETRIC FUNCTIONS 33 Sine Tangent Cotangent Cosine Ang Nat. Log. Nat. Log. Nat. Log. Nat. Log. 9 9 10 9 10 .17365 .23967 .17633 .24632 5.6713 .75368 .98481 .99335 80 .1 .17537 .24395 .17813 .25073 5.6140 .74927 .98450 .99322 .9 2 .17708 .24818 .17993 .25510 5.5578 .74490 .98420 .99308 .8 .3 .17880 .25237 .18173 .25943 5.5026 .74057 .98389 .99294 .7 .4 .18052 .25652 .18353 .26372 5.4486 .73628 .98357 .99281 .6 .5 .18224 .26063 .18534 .26797 5.3955 .73203 .98325 .99267 .5 .6 .18395 .26470 .18714 .27218 5.3435 .72782 .98294 .99253 .4 .7 .18567 .26873 .18895 .27635 5.2924 .72365 .98261 .99238 .3 .8 .18738 .27273 .19076 .28049 5.2422 .71951 .98229 .99224 .2 .9 .18910 .27668 .19257 .28459 5.1929 .71541 .98196 .99209 .1 11 .19081 .28060 .19438 .28865 5.1446 .71135 .98163 .99195 79 .1 .19252 .28448 .19619 .29268 5.0970 .70732 .98129 .99180 .9 2 .19423 .28833 .19801 .29668 5.0504 .70332 .98096 .99165 .8 .'3 .19595 .29214 .19982 .30064 5.0045 .69936 .98061 .99150 .7 .4 .19766 .29591 .20164 .30457 4.9594 .69543 .98027 .99135 .6 .5 .19937 .29966 .20345 .30846 4.9152 .69154 .97992 .99119 .5 .6 .20108 .30336 .20527 .31233 4.8716 .68767 .97958 .99104 .4 .7 .20279 .30704 .20709 .31616 4.8288 .68384 .97922 .99088 .3 .8 .20450 .31068 .20891 .31996 4.7867 .68004 .97887 .99072 .2 .9 .20620 .31430 .21073 .32373 4.7453 .67627 .97851 .99056 .1 12 .20791 .31788 .21256 .32747 .7046 .67253 .97815 .99040 . 78 .1 .20962 .32143 .21438 .33119 .6646 .66881 .97778 .99024 .9 .2 .21132 .32495 .21621 .33487 .6252 .66513 .97742 .99008 .8 .3 .21303 .32844 .21804 .33853 .5864 .66147 .97705 .98991 .7 .4 .21474 .33190 .21986 .34215 .5483 .65785 .97667 .98975 .6 .5 .21644 .33534 .21169 .34576 .5107 .65424 .97630 .98958 .5 .6 .21814 .33874 .22353 .34933 .4737 .65067 .97592 .98941 .4 .7 .21985 .34212 .22536 .35288 4.4373 .64712 .97553 .98924 .3 .8 .22155 .34547 .22719 .35640 4.4015 .64360 .97515 .98907 .2 .9 .22325 .34879 .22903 .35989 4.3662 .64011 .97476 .98890 .1 13 .22495 .35209 .23087 .36336 4.3315 .63664 .97437 .98872 77 .1 .22665 .35536 .23271 .36681 4.2972 .63319 .97398 .98855 .9 .2 .22835 .35860 .23455 .37023 4.2635 .62977 .97358 .98837 .8 .3 .23005 .36182 .23639 .37363 .2303 .62637 .97318 .98819 .7 .4 .23175 .36502 .23823 .37700 .1976 .62300 .97278 .98801 .6 .5 .23345 .36819 .24008 .38035 .1653 .61965 .97237 .98783 .5 .6 .23514 .3/133 .24193 .38368 .1335 .61632 .97196 .98765 .4 .7 .23684 .37445 .24377 .38699 .1022 .61301 .97155 .98746 .3 .8 .23853 .37755 .24562 .39027 .0713 .60973 .97113 .98728 .2 .9 .24023 .38062 .24747 .39353 .0408 .60647 .97072 .98T09 .1 14 .24192 .38368 .24933 .39677 4.0108 .60323 .97030 .98690 76 .1 .24362 .38670 .25118 .39999 3.9812 .60001 .96987 .98671 .9 2 .24531 .38971 .25304 .40319 3.9520 .59681 .96945 .98652 .8 '.3 .24700 .39270 .25490 .40636 3 9232 .59364 .96902 .98633 .7 .4 .24869 .39566 .25676 .40952 s!8947 .59048 .96858 .98614 -.6 .5 .25038 .39860 .25862 .41266 3.8667 .58734 .96815 .98594 .5 .6 .25207 .40152 .26048 .41578 3.8391 .58422 .96771 .98574 .4 .7 .25376 .40442 .26235 .41887 3.8118 .58113 .96727 .98555 .3 .8 .25545 .40730 .26421 .42195 3.7848 .57805 .96682 .98535 2 .9 .25713 .41016 .26608 .42501 3.7583 .57499 .96638 .98515 .'l 15 .25882 .41300 .26795 .42805 3.7321 .57195 .96593 .98494 75 Nat. LOST. Nat. Log. Nat. Log. Nat. Log.Ang. Cosine Cotangent Tangent Sine 34 TRIGONOMETRIC FUNCTIONS Sine Tangent Cotangent Cosine Ang. , Nat. Log. Nat. Log. Nat. Log. Nat. Log. 9 9 10 9 15 .25882 .41300 .26795 .42805 3.7321 .57195 .96593 .98494 75 .1 .26050 .41582 .26982 .43108 3.7062 .56892 .96547 .98474 .9 .2 .26219 .41861 .27169 .43408 3.6806 .56592 .96502 .98453 .8 .3 .26387 .42140 .27357 .43707 3.6554 .56293 .96456 .98433 .7 .4 .26556 .42416 .27545 .44004 3.6305 .55996 .96410 .98412 .6 .5 .26724 .42690 .27732 .44299 3.6059 .55701 .96363 .98391 .5 .6 .26892 .42962 .27921 .44592 3.5816 .55408 .96316 .98370 .4 .7 .27060 .43233 .28109 .44884 3.5576 .55116 .96269 .98349 .3 .8 .27228 .43502 .28297 .45174 3.5339 .54826 .96222 .98327 .2 .9 .27396 .43769 .28486 .45463 3.5105 .54537 .96174 .98306 .1 16 .27564 .44034 .28675 .45750 3.4874 .54250 .96126 .98284 74 .1 .27731 .44297 .28864 .46035 3.4646 .53965 .96078 .98262 .9 .2 .27899 .44559 .29053 .46319 3.4420 .53681 .96029 .98240 .8 .3 .28067 .44819 .29242 .46601 3.4197 .53399 .95981 .98218 .7 A .28234 .45077 .29432 .46881 3.3977 .53119 .95931 .98196 .6 .5 .28402 .45334 .29621 .47161 3.3759 .52840 .95882 .98174 .5 .6 .28569 .45589 .29811 .47438 3.3544 .52562 .95832 .98151 .4 .7 .28736 .45843 .30001 .47714 3.3332 .52286 .95782 .98129 .3 .8 . 2890? .46095 .30192 .47989 3.3122 .52011 .95732 .98106 .2 .9 .29070 .46345 .30382 .48262 3.2914 .51738 .95681 .98083 .1 17. .29237 .46594 .30573 .48534 3.2709 .51466 .95630 .98060 73 .1 .29404 .46841 .3076* .48804 3.2506 .51196 .95579 .98036 .9 .2 .29571 .47086 .30955 .49073 3.2305 .50927 .95528 .98013 .8 .3 .29737 .47330 .31147 .49341 3.2106 .50659 .95476 .97989 .7 .4 .29904 .47573 .31338 .49607 3.1910 .50393 .95424 .97966 .6 .5 .30071 .47814 .31530 .49872 3.1716 .50128 .95372 .97942 .5 .6 .30237 .48054 .31722 .50136 3.1524 .49864 .95319 .97918 .4 .7 .30403 .48292 .31914 .50398 3.1334 .49602 .95266 .97894 .3 .8 .30570 .48529 .32106 .50659 3.1146 .49341 .95*213 .97870 .2 .y .30736 .48764 .32299 .50919 3.0961 .49081 .95159 .97845 .1 18 .30902 .48998 .32492 .51178 3.0777 .48822 .95106 .97821 72 .1 .31068 .49231 .32685 .51435 3.0595 .48565 .95052 .97796 .9 .2 .31233 .49462 .32878 .51691 3.0415 .48309 .94997 .97771 .8 .3 .31399 .49692 .33072 .51946 3.0237 .48054 .94943 .97746 .7 .4 .31565 .49920 .33266 .52200 3.0061 .47801 .94888 .97721 .6 .5 .31730 .50148 .33460 .52452 2.9887 .47548 .94832 .97696 .5 .6 .31896 .50374 .33654 .52703 2.9714 .47297 .94777 .97670 .4 .7 .32061 .50598 .33348 .52953 2.9544 .47047 .94721 .97645 .3 .8 .32227 .50821 .34043 .53203 2.9375 .46798 .94665 .97619 .2 .9 .32392 .51043 .34238 .53450 2.9208 .46550 .94609 .97593 .1 19 .32557 .51264 .34433 .53697 2.9042 .46303 .94552 .97567 71 .1 .32722 .51484 .34628 .53943 2.8878 .46057 .94495 .97541 .9 .2 .32887 .51702' .34824 .54187 2.8716 .45813 .94438 .97515 .8 .3 .33051 .51919 .35020 .54431 2.8556 .45569 .94380 .97488 .7 .4^ .33216 .52135 .35216 .54673 2.8397 .45327 .94322 .97461 .6 .5 .33381 .52350 .35412 .54915 2.8239 .45085 .94264 .97435 .5 .6 .33545 .52563 .35608 .55155 2.8083 .44845 .94206 .97408 .4 .7 .33710 .52775 .35805 .55395 2.7929 .44605 .94147 .97381 .3 .8 .33874 .52986 .36002 .55633 2.7776 .44367 .94088 .97353 .2 .9 .34038 .53196 .36199 .55870 2.7625 .44130 .94029 .97326 .1 20 .34202 .53405 .36397 .56107 2.7475 .43893 .93969 .97299 70 Nat. Log. Nat. Log. Nat. Log. Nat. Log.Ang. Cosine Cotangent Tangent Sine TRIGONOMETRIC FUNCTIONS 35 Sine Tangent Cotangent Cosine Ang, . Nat. Log. Nat. Log. Nat. Log. Nat. Log. 9 9 10 9 20 .34202 .53405 .36397 .56107 2.7475 .43893 .93969 .97299 70 .1 .34366 .53613 .36595 .56342 2.7326 .43658 .93909 .97271 .9 .2 .34530 .53819 .36793 .56576 2.7179 .43424 .93849 .97243 .8 .3 .34694 .54025 .36991 .56810 2.7034 .43190 .93789 .97215 .7 .4 .34857 .54229 .37190 .57042 2.6889 .42958 .93728 .97187 .6 .5 .35021 .54433 .37388 .57274 2.6746 .42726 .93667 .97159 .5 .6 .35184 .54635 .37588 .57504 2.6605 .42496 .93606 .97130 .4 .7 .35347 .54836 .37787 .57734 2.6464 .42266 .93544 .97102 .3 .8 .35511 .55036 .37986 .57963 2.6325 .42037 .93483 .97073 .2 .9 .35674 .55235 .38186 .58191 2.6187 .41809 .93420 .97044 .1 21 .35837 .55433 .38386 .58418 2.6051 .41582 .93358 .97015 69 .1 .36000 .55630 .38587 .58644 2.5916 .41356 .93295 .96986 .9 .2 .36162 .55826 .38787 .58869 2.5782 .41131 .93232 .96957 .8 .3 .36325 .56021 .38988 .59094 2.5649 .40906 .93169 .96927 .7 .4 .36488 .56215 .39190 .59317 2.5517 .40683 .93106 .96898 .6 .5 .36650 .56408 .39391 .59540 2.5386 .40460 .93042 .96868 .5 .6 .36812 .56600 .39593 .59762 2.5257 .40238 .92978 .96838 .4 .7 .36975 .56790 .39795 .59983 2.5129 .40017 .92913 .96808 .3 8 .37137 .56980 .39997 .60203 2.5002 .39797 .92849 .96778 .2 .9 .37299 .57169 .40200 .60422 2.4876 .39578 .92784 .96747 .1 22 .37461 .57358 .40403 .60641 2.4751 .39359 .92718 .96717 68 .1 .37622 .57545 .40606 .60859 2.4627 .39141 .92653 .96686 .9 .2 .37784 .57731 .40809 .61076 2.4504 .38924 .92587 .96655 .8 .3 .37946 .57916 .41013 .61292 2.4383 .38708 .92521 .96624 .7 .4 .38107 .58101 .41217 .61508 2.4262 .38492 .92455 .96593 .6 .5 .38268 .58284 .41421 .61722 2.4142 .38278 .92388 .96562 .5 .6 .38430 .58467 .41626 .61936 2.4023 .38064 .92321 .96530 .4 .7 .38591 .58648 .41831 .62150 2.3906 .37850 .92254 .96498 .$ .8 .38752 .58829 .42036 .62362 2.3789 .37638 .92186 .96467 .2 .9 .38912 .59009 .42242 .62574 2.3673 .37426 .92119 .96435 .1 23 .39073 .59188 .42447 .62785 2.3559 .37215 .92050 .96403 67 .1 .39234 .59366 .42654 .62996 2.3445 .37004 .91982 .96370 .9- .2 .39394 .59543 .42860 .63205 2.3332 .36795 .91914 .96338 .8 .3 .39555 .59720 .43067 .63414 2.3220 .36586 .91845 .96305 .7 .4 .39715 .59895 .43274 .63623 2.3109 .36377 .91775 .96273 .& .5 .39875 .60070 .43481 .63830 2.2998 .36170 .91706 .96240 .5 .6 .40035 .60244 .43689 .64037 2.2889 .35963 .91636 .96207 .4 .7 .40195 .60417 .43897 .64243 2.2781 .35757 .91566 .96174 .3 .8 .40355 .60589 .44105 .64449 2.2673 .35551 .91496 .96140 .2 .9 .40514 .60761 .44314 .64654 2.2566 .35346 .91425 .96107 .1 24 .40674 .60931 .44523 .64858 2.2460 .35142 .91355 .96073 66 .1 .40833 .61101 .44732 .65062 2.2355 .34938 .91283 .96039 .9 .2 .40992 .61270 .44942 .65265 2.2251 .34735 .91212 .96005 .8 .3 .41151 .61439 .45152 .65467 2.2148 .34533 .91140 .95971 .r .4 .41310 .61606 .45362 .65669 2.2045 .34331 .91068 .95937 .6 .5 .41469 .61773 .45573 .65870 2.1943 .34130 .90996 .95902 .5 .6 .41628 .61939 .45784 .66071 2.1842 .33929 .90924 .95868 .4 .7 .41787 .62104 .45995 .66271 2.1742 .33729 .90851 .95833 .3 .8 .41945 .62268 .46206 .66470 2.1642 .33530 .90778 .95798 .2 .9 .42104 .62432 .46418 .66669 2.1543 .33331 .90704 .95763 .1 25 .42262 .62595 .46631 .66867 2.1445 .33133 .90631 .95728 65 Nat. Log. Nat. Log. Nat. Log. Nat. Loer.Ane:. Cosine Cotangent Tangent Sine TRIGONOMETRIC FUNCTIONS Sine Tangent Cotangent Cosine Ang. Nat. Log. Nat. Log. Nat. Log. Nat. Log. 9 9 10 9 25 .42262 .62595 .46631 .66867 2.1445 .33133 .90631 .95728 65 .1 .42420 .62757 .46843 .67065 .1348 .32935 .90557 .95692 .9 .2 .42578 .62918 .47056 .67262 .1251 .32738 .90483 .95657 .8 3 .42736 .63079 .47270 .67458 .1155 .32542 .90408 .95621 .7 .4 .42894 .63239 .47483 .67654 .1060 .32346 .90334 .95585 .6 .5 .43051 .63398 .47698 .67850 .0965 .32150 .90259 .95549 .5 .6 .43209 .63557 .47912 .68044 .0872 .31956 .90183 .95513 .4 .7 .43366 .63715 .48127 .68239 .0778 .31761 .90108 .95476 .3 .8 .43523 .63872 .48342 .68432 .0686 .31568 .90032 .95440 .2 .9 .43680 .64028 .48557 .68626 .0594 .31374 .89956 .95403 .1 26 .43837 .64184 .48773 .68818 .0503 .31182 .89879 .95366 64 .1 .43994 .64339 .48989 .69010 .0413 .30990 .89803 .95329 .9 2 .44151 .64494 .49206 .69202 .0323 .30798 .89726 .95292 .8 '.3 .44307 .64647 .49423 .69393 .0233 .30607 .89649 .95254 .7 .4 .44464 .64800 .49640 .69584 .0145 .30416 .89571 .95217 .6 .5 .44620 .64953 .49858 .69774 .0057 .30226 .89493 .95179 .5 .6 .44776 .65104 .50076 .69963 1.9970 .30037 .89415 .95141 .4 .7 .44932 .65255 .50295 .70152 L.9883 .29848 .89337 .95103 .3 .8 .45088 .65406 .50514 .70341 1.9797 .29659 .89259 .95065 .2 .9 .45243 .65556 .50733 .70529 1.9711 .29471 .89180 .95027 .1 27 .45399 .65705 .50953 .70717 1.9626 .29283 .89101 .94988 63 .1 .45554 .65853 .51173 .70904 1.9542 .29096 .89021 .94949 .9 .45710 .66001 .51393 .71090 1.9458 .28910 .88942 .94911 .8 !s .45865 .66148 .51614 .71277 1.9375 .28723 .88862 .94871 .7 .4 .46020 .66295 .51835 .71462 1.9292 .28538 .88782 .94832 .6 .5 .46175 .66441 .52057 .71648 1.9210 .28352 .88701 .94793 .5 .6 .46330 .66586 .52279 .71833 1.9128 .28167 .88620 .94753 .4 .7 .46484 .66731 .52501 .72017 1.9047 .27983 .88539 .94714 .3 .8 .46639 .66875 .52724 .72201 1.8967 .27799 .88458 .94674 .2 .9 .46793 .67018 .52947 .72384 1.8887 .27616 .88377 .94634 .1 28 .46947 .67161 .53171 .72567 1.8807 .27433 .88295 .94593 62 .1 .47101 .67303 .53395 .72750 1.8728 .27250 .88213 .94553 .9 .2 .47255 .67445 .53620 .72932 ' 1.8650 .27068 .88130 .94513 .8 .3 .47409 .67586 .53844 .73114 1.8572 .26886 .88048 .94472 .7 .4 .47562 .67726 .54070 .73295 1.8495 .26704 .87965 .94431 .6 .5 .47716 .67863 .54296 .73476 1.8418 .26524 .87882 .94390 .5 .6 .47869 .68006 .54522 .73657 1.8341 .26343 .87798 .94349 .4 .7 .48022 .68144 .54748 .73837 1.8265 .26163 .87715 .94307 .3 .8 .48175 .68283 .54975 .74017 1.8190 .25983 .87631 .94266 .2 .9 .48328 .68420 .55203 .74196 1.8115 .25804 .87546 .94224 .1 29 .48481 .68557 .55431 .74375 1.8040 .25625 .87462 .94182 61 .1 .48634 .68694 .55659 .74554 1.7966 .25446 .87377 .94140 .9 .2 .48786 .68829 .55888 .74732 1.7893 .25268 .87292 .94098 .8 .3 .48938 .68965 .56117 .74910 1.7820 .25090 '.87207 .94055 .7 .4 .49090 .69100 .56347 .75087 1.7747 .24913 .87121 .94012 .6 .5 .49242 .69234 .56577 .75264 1.7675 .24736 .87036 .93970 .5 .6 .49394 .69368 .56808 .75441 1.7603 .24559 .86949 .93927 .4 .7 .49546 .69501 .57039 .75617 1.7532 .24383 .86863 .93884 .3 .8 .49697 .69633 .57271 .75793 1.7461 .24207 .86777 .93840 o .9 ,49849 .69765 .57503 .75969 1.7391 .24031 .86690 .93797 '.1 30 .50000 .69897 .57735 .76144 1.7321 .23856 .86603 .93753 60 Nat. Log. Nat. Log. Nat. Log. Nat. Log.Ang. Cosine Cotangent Tangent Sine TRIGONOMETRIC FUNCTIONS Sine Tangent Cotangent Cosine Ang. , Nat. Log. Nat. Log. Nat. Log. Nat. Log. 9 9 10 9 30 .50000 .69897 .57735 .76144 1.7321 .23856 .86603 .93753 60 .1 .50151 .70028 .57968 .76319 1.7251 .23681 .86515 .93709 .9 2 .50302 .70159 .58201 .76493 1.7182 .23507 .86427 .93665 .8 .'S .50453 .70288 .58435 .76668 1.7113 .23332 .86340 .93621 .7 .4 .50603 .70418 .58670 .76841 1.7045 .23159 .86251 .93577 .6 .5 .50754 .70547 .58905 .77015 1.6977 .22985 .86163 .93532 .5 .6 .50904 .70675 .59140 .77188 1.6909 .22812 .86074 .93487 .4 .7 .51054 .70803 .59376 .77361 1.6842 .22639 .85985 .93442 .3 .8 .51204 .70931 .59612 .77533 1.6775 .22467 .85896 .93397 .2 .9 .51354 .71058 .59849 .77706 1.6709 .22294 .85806 .93352 .1 31 .51504 .71184 .60086 .77877 1.6643 .22123 .85717 .93307 59 .1 .51653 .71310 .60324 .78049 1.6577 .21951 .85627 .93261 .9 .2 .51803 .71435 .60562 .78220 1.6512 .21780 .85536 .93215 .8 .3 .51952 .71560 .60801 .78391 1.6447 .21609 .85446 .93169 .7 .4 .52101 .71685 .61040 .78562 1.6383 .21438 .85355 .93123 .6 .5 .52250 .71809 .61280 .78732 1.6319 .21268 .85264 .93077 .5 .6 .52399 .71932 .61520 .78902 1.6255 .21098 .85173 .93030 .4 .7 .52547 .72055 .61761 .79072 1.6191 .20928 .85081 .92983 .3 .8 .52696 .72177 .62003 .79241 1.6128 .20759 .84989 .92936 .2 .9 .52844 .72299 .62245 .79410 1.6066 .20590 .84897 .92889 .1 32 .52992 .72421 .62487 .79579 1.6003 .20421 .84805 .92842 58 .1 .53140 .72542 .62730 .79747 1.5941 .20253 .84712 .92795 .9 .2 .53288 .72663 .62973 .79916 1.5880 .20084 .84619 .92747 .8 .3 .53435 .72783 .63217 .80084 1.5818 .19916 .84526 .92699 .7 .4 .53583 .72902 .63462 .80251 1.5757 .19749 .84433 .92651 .6 .5 .53730 .73022 .63707 .80419 1.5697 .19581 .84339 .92603 .5 .6 .53877 .73140 .63953 .80586 1.5637 .19414 .84245 .92555 .4 .7 .54024 .73259 .64199 .80753 1.5577 .19247 .84151 .92506 .3 .8 .54171 .73377 .64446 .80919 1.5517 .19081 .84057 .92457 .2 .9 .54317 .73494 .64693 .81086 1.5458 .18914 .83962 .92408 .1 33 .54464 .73611 .64941 .81252 1.5399 .18748 .83867 .92359 57 .1 .54610 .73727 .65189 .81418 1.5340 .18582 .83772 .92310 .9 .2 .54756 .73843 .65438 .81583 1 . 5 -Sil .18417 .83676 .92260 .8 .3 .54902 .73959 .65688 .81748 1.5224 .18252 .83581 .92211 .7 .4 .55048 .74074 .65938 .81913 1.5166 .18087 .83485 .92161 .6 .5 .55194 .74189 .66189 .82078 1.5108 .17922 .83389 .92111 .5 .6 .55339 .74303 .66440 .82243 1.5051 .17757 .83292 .92060 .4 .7 .55484 .74417 .66692 .82407 1.4994 .17593 .83195 .92010 .3 .8 .55630 .74531 .66944 .82571 1.4938 .17429 .83098 .91959 2 .9 .55775 .74644 .67197 .82735 1.43*2 .17265 .83001 .91909 !i 34 .55919 .74756 .67451 .82899 1.4826 .17101 .82904 .91857 56 .1 .56064 .74868 .67705 .83062 1.4770 .16938 .82806 .91806 .9 2 .56208 .74980 .67960 .83225 1.4715 .16775 .82708 .91755 .8 !s .56353 .75091 .68215 .83388 1.4659 .16612 .82610 .91703 .4 .56497 .75202 .68471 .83551 1.4605 .16449 .82511 .91651 !e .5 .56641 .75313 .68728 .83713 1.4550 .16287 .82413 .91599 .5 .6 .56784 .75423 .68985 .83876 1.4496 .16124 .82314 .91547 .4 .7 .56928 .75533 .69243 .84038 1.4442 .15962 .82214 .91495 .3 .8 .57071 .75642 .69502 .84200 1.4388 .15800 .82115 .91442 .2 .9 .57215 .75751 .69761 .84361 1.4335 .15639 .82015 .91389 .1 35 .57358 .75859 .70021 .84523 1.4281 .15477 .81915 .91336 55 Nat. Log. Nat. Log. Nat. Log. Nat. Losr.Ane:. Cosine Cotangent Tangent Sine 38 TRIGONOMETRIC FUNCTIONS Sine Ang. Nat. Log. Tangent Nat. Log. Cotangent Nat. Log. Cosine Nat. Log. 9 9 10 9 35 .57358 .75859 .70021 .84523 1 .4281 .15477 .81915 .91336 55 .1 .57501 .75967 .70281 .84684 1 .4229 .15316 .81815 .91283 .9 .2 .57643 .76075 .70542 .84845 1 .4176 .15155 .81714 .91230 .8 .3 .57786 .76182 .70804 .85006 1 .4124 .14994 .81614 .91176 .7 .4 .57928 .76289 .71066 .85166 1 .4071 .14834 .81513 .91123 .6 .5 .58070 .76395 .71329 .85327 1 .4019 .14673 .81412 .91069 .5 .6 .58212 .76502 .71593 .85487 1 .3968 .14513 .81310 .91014 .4 .7 .58354 .76607 .71857 .85647 1 .3916 .14353 .81208 .90960 .3 .8 .58496 .76712 .72122 .85807 1 .3865 .14193 .81106 .90906 .2 .9 .58637 .76817 .72388 .85967 1 .3814 .14033 .81004 .90851 .1 36 .58779 .76922 .72654 .86126 1 .3764 .13874 .80902 .90796 54 .1 .58920 .77026 .72921 .86285 1 .3713 .13715 .80799 .90741 .9 2 .59061 .77130 .73189 .86445 1 .3663 .13555 .80696 .90685 .8 '.3 .59201 .77233 .73457 .86604 1 .3613 .13397 .80593 .90630 .7 .4 .59342 .77336 .73726 .86762 1 .3564 .13238 .80489 .90574 .6 .5 .59482 .77439 .73996 .86921 1 .3514 .13079 .80386 .90518 .5 .6 .59622 .77541 .74267 .87079 1 .3465 .12921 .80282 .90462 .4 .7 .59763 .77643 .74538 .87238 1 .3416 .12762 .80178 .90405 .3 .8 .59902 .77744 .74810 .87396 1 .3367 .12604 .80073 .90349 2 .9 .60042 .77846 .75082 .87554 1 .3319 .12446 .79968 .90292 !i 37 .60182 .77946 .75355 .87711 1 .3270 .12289 .79864 .90235 53 .1 .60321 .78047 .75629 .87869 1 .3222 .12131 .79758 .90178 .9 .2 .60460 .78147 .75904 .88027 1 .3175 .11073 .79653 .90120 .8 .3 .60599 .78246 .76180 .88184 1 .3127 .11816 .79547 .90063 .7 .4 .60738 .78346 .76456 .88341 1.3079 .11659 .79441 .90005 .6 .5 .60876 .78445 .76733 .88498 1 .3032 .11502 .79335 .89947 .5 .6 .61015 .78543 .77010 .88655 1 .2985 .11345 .79229 .89888 .4 .7 .61153 .78642 .77289. .88812 1 .2938 .11188 .79122 .89830 .3 .8 .61291 .78739 .77568 .88968 1 .2892 .11032 .79016 .89771 .2 .9 .61429 .78837 .77848 .89125 1 .2846 .10875 .78908 .89712 .1 38 .61566 .78934 .78129 .89281 1 .2799 .10719 .78801 .89653 52 .1 .61704 .79031 .78410 .89437 1 .2753 .10563 .78694 .89594 .9 2 .61841 .79128 .78692 .89593 1 .2708 .10407 .78586 .89534 .8 '.3 .61978 .79224 .78975 .89749 1 .2662 .10251 .78478 .89475 .7 .4 .62115 .79319 .79259 .89905 1 .2617 .10095 .78369 .89415 .6 .5 .62251 .79415 .79544 .90061 1 .2572 .09939 .78261 .89354 .5 .6 .62388 .79510 .79829 .90216 1 .2527 .09784 .78152 .89294 .4 .7 .62524 .79605 .80115 .90371 1 .2482 .09629 .78043 .89233 .3 .8 .62660 .79699 .80402 .90527 1 .2437 .09473 .77934 .89173 .2 .9 .62796 .79793 .80690 .90682 1 .2393 .09318 .77824 .89112 .1 39 .62932 .79887 .80978 .90837 1 .2349 .09163 .77715 .89050 51 .1 .63068 .79981 .81268 .90992 1 .2305 .09008 .77605 .88989 .9 .2 .63203 .80074 .81558 .91147 1 .2261 .08853 .77494 .88927 .8 .3 .63338 .80166 .81849 .91301 1 .2218 .08699 .77384 .88865 .7 .4 .63473 .80259 .82141 .91456 1 .2174 .08544 .77273 .88803 .6 .5 .63608 .80351 .82434 .91610 1 .2131 .08390 .77162 .88741 .5 .6 .63742 .80443 .82727 .91765 1 .2088 .08235 .77051 .88678 .4 .7 .63877 .80534 .83022 .91919 1 .2045 .08081 .76940 .88615 .3 .8 .64011 .80625 .83317 .92073 1 .2002 .07927 .76828 .88552 .2 .9 .64145 .80716 .83613 .92227 1 .1960 .07773 .76717 .88489 .1 40 .64279 .80807 .83910 .92381 1 .1918 .07619 .76604 .88425 51 Nat. LOST. Nat. Log. Nat. Log. Nat. Log.Ane. Cosine ~ Cotangent Tangent Sine TRIGONOMETRIC FUNCTIONS 39 Sine Ang. Nat. Log. Tangent Nat. Log. Cotangent Nat. Log. Cosine Nat. Log. 9 9 10 9 40 .64279 .80807 .83910 .92381 1 .1918 .07619 .76604 .88425 50 .1 .64412 .80897 .84208 .92535 1.1875 .07465 .76492 .88362 .9 2 .64546 .80987 .84507 .92689 1 .1833 .07311 .76380 .88298 .8 is .64679 .81076 .84806 .92843 1.1792 .07157 .76267 .88234 .7 .4 .64812 .81166 .85107 .92996 1 .1750 .07004 .76154 .88169 .6 .5 .64945 .81254 .85408 .93150 1 .1708 .06850 .76041 .88105 .5 .6 .65077 .81343 .85710 .93303 1.1667 .06697 .75927 .88040 .4 .7 .65210 .81431 .86014 .93457 1 .1626 .06543 .75813 .87975 .3 .8 .65342 .81519 .86318 .93610 1 .1585 .06390 .75700 .87909 2 .9 .65474 .81607 .86623 .93763 1 .1544 .06237 .75585 .87844 ii 41 .65606 .81694 .86929 .93916 1 .1504 .06084 .75471 .87778 49 .1 .65738 .81781 .87236 .94069 1 .1463 .05931 .75356 .87712 .9 2 .65869 .81868 .87543 .94222 1 .1423 .05778 .75241 .87646 .8 .3 .66000 .81955 .87852 .94375 1 .1383 .05625 .75126 .87579 .7 .4 .66131 .82041 .88162 .94528 1 .1343 .05472 .75011 .87513 .6 .5 .66262 .82126 .88473 .94681 1 .1303 .05319 .74896 .87446 .5 .6 .66393 .82212 .88784 .94834 1 .1263 .05166 .74780 .87378 .4 .7 .66523 .82297 .89097 .94986 1 .1224 .05014 .74664 .87311 .3 .8 .66653 .82382 .89410 .95139 1 .1184 .04861 .74548 .87243 2 .9 .66783 .82467 .89725 .95291 1 .1145 .04709 .74431 .87175 '.1 42 .66913 .82551 .90040 .95444 1 .1106 .04556 .74314 .87107 48 .1 .67043 .82635 .90357 .95596 1 .1067 .04404 .74198 .87039 .9 .2 .67172 .82719 .90674 .95749 1 .1028 .04252 .74080 .86970 .8 .3 .67301 .82802 .90993 .95901 1 .0990 .04099 .73963 .86902 .7 .4 .67430 .82885 .91313 .96053 1 .0951 .03947 .73846 .86832 .6 .5 .67559 .82968 .91633 .96205 1 .0913 .03795 .73728 .86763 .5 .6 .67688 .83051 .91955 .96357 1 .0875 .03643 .73610 .86694 .4 .7 .67816 .83133 .92277 .96510 1 .0837 .03490 .73491 .86624 .3 .8 .67944 .83215 .92601 .96662 1 .0799 .03338 .73373 .86554 .2 .9 .68072 .83297 .92926" .96814 I .0761 .03186 .73254 .86483 .1 43 .68200 .83378 .93252 .96966 1 .0724 .03034 .73135 .86413 47 .1 .68327 .83460 .93578 .97118 1 .0686 .02882 .73016 .86342 .9 2 .68455 .83540 .93906 .97269 1 .0649 .02731 .72897 .86271 .8 '.3 .68582 .83621 .94235 .97421 1 .0612 .02579 .72777 .86200 .7 .4 .68709 .83701 .94565 .97573 1 .0575 .02427 .72657 .86128 .6 .5 .68835 .83781 .94896 .97725 i .0538 .02275 .72537 .8605& .5 .6 .68962 .83861 .95229 .97877 i .0501 .02123 .72417 .85984 .4 .7 .69088 .83940 .95562 .98029 i .0464 .01971 .72297 .85912 .3 .8 .69214 .84020 .95897 .98180 i .0428 .01820 .72176 .85839 .2 .9 . 69340 .84099 .96232 .98332 i .0392 .01668 .72055 .85766 .1 44 .69466 .84177 .96569 .98484 i .0355 .01516 .71934 .85693 46 .1 .69591 .84255 .96907 .98635 i .0319 .01365 .71813 .85620 .9 .2 .69717 .84334 .97246 .98787 i .0283 .01213 .71691 .85547 .8 .3 .69842 .84411 .97586 .98939 i .0247 .01061 .71569 .85473 .7 .4 .69966 .84489 .97927 .99090 i .0212 .00910 .71447 .85399 .6 .5 .70091 .84566 .98270 .99242 i .0176 .00758 .71325 .85324 .5 .6 .70215 .84643 .98613 .99394 i .0141 .00606 .71203 .85250 .4 .7 .70339 .84720 .98958 .99545 i .0105 .00455 .71080 .85175 .3 .8 .70463 .84796 .99304 .99697 i .0070 .00303 .70957 .85100 .2 .9 .70587 .84873 .99652 .99848 i .0035 .00152 .70834 .85024 .1 45 .70711 .84949 1.00000 .00000 1.0000 .00000 .70711 .84949 45 Nat. Log. Nat. Log Nat. Log. Nat. Log.Ansr. Cosine Cotangent Tangent Sine TRIGONOMETRIC FUNCTIONS Minutes in Decimal Parts of Degrees. Min. Deg. 1 0.01667 2 0.03333 3 0.05000 4 0.06667 0.08333 0.10000 0.11667 0.13333 0.15000 0.16667 0.18333 0.20000 0.21667 0.23333 Min. 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 Deg. 0.25000 0.26667 0.28333 0.30000 0.31667 0.33333 0.35000 0.36667 0.38333 0.40000 0.41667 0.43333 0.45000 0.46667 0.48333 Min. Deg. 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 0.50000 0.51667 0.53333 0.55000 0.56667 0.58333 0.60000 0.61667 0.63333 0.65000 0.66667 0.68333 0.70000 0.71667 0.73333 Min. 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 Deg. 0.75000 0.76667 0.78333 0.80000 0.81667 0.83333 0.85000 0.86667 0.88333 0.90000 0.91667 0.93337 0.95000 0.96667 0.98333 Functions of Certain Angles. degrees 30 45 60 90 18 27 36 sin cos= tan -1 oo ctn ./ 3 1 oo Angles greater than 90 to Angles less than 90. The following scheme gives, first, the sign of the trigono- metric functions for angles in the various quadrants, and, second, the reduction of the functions for angles greater than 90 degrees to the functions for angles less than 90 degrees. TRIGONOMETRIC FUNCTIONS Angles Greater than 90 to Angles less than 90 C 41 Angle u between Angle u in degrees 90 180 270 and and and and 90 180 270 360 =fc u 90 u 180 u 270 u sin u -f- -j- =fc sin u -f cos u -+- sin u cos u cos u -{- ~h + cos u =F sin u cos u + sin u tan u = -}- + + tan u =f ctn u + tan u HF ctn u ctn u = + -f- | + ctn u + tan u + ctn u + tan u sin (45 u) = cos (45 =F u) tan (45 u) = ctn (45 =F u) Formulae. tan a 2. cosa = csc a i/l-fctn 2 a Vl-ftan a a 1 __ 1 ctn_a sec a V 7 l+tan 2 a l/l+ctn 2 a o 1 sin a 3. tana = ctn a cos a 4. sin 2 a -f cos 2 a = 1. 5. csc 2 a = 1 -f ctn 2 a. ' 9. cos y z a = V] 10. sin 2 a = 2 sin a cos a. 6. sec* a = 1 tan 2 a. 7. sin K a--=V)(l cos a). -r cos a). 8. tan % a = \U ~ cos a cos a 11. cos 2 a == cos 2 a sin a = 1 ? sin 2 a 2 cos 2 a 1. 12. sin 3 a = 3 sin a 4 sin 3 a. 14. cos 3 a = 4 cos 3 a 3 cos a. 13. tan2a= -_ _ 1 tan 2 a 15. ctn 2 a = ctn a 42 TRIGONOMETRIC FUNCTIONS Formulae 16. sin (a + ft) = sin a. cos 6 + cos a. sin b. 17. cos (a. + 6) = cos a. cos 6 =F sin a. sin ft. 18. tan (a ft) = tan tan ft . 1 + tan a. tan ft 19. ctn 6 =b ctn a 20. sin a sin ft = 2 sin % (a + ft) .cos > (a + ft). 21. cos a + cos ft = 2 cos ) (a + 6) cos V (a ft). 22. cos a - cos 6 = 2 sin ^ (a -|~ &) -sin )'g ( a 23. cos a. cos ft 24. sin a . sin ft 25. sin 2 a sin 2 ft = cos 2 ft cos 2 a = sin (a + 6) sin (a ft). 26. cos 2 a sin 2 ft = cos 2 ft sin 2 a cos (a ft) cos (a - ft). sin A sin B sin C 3 a sin ^ + sin B = tan ^ (^ + ^) -^ ctn - a 6 sin A sin B tan > (.4 5) tan }/ (A B} 4. sm^A=- b(8 ^' 5. cos>^^- 6. tanM^-= ) 7. Area = K be sin .4 = T/ S ( s _ a ) ( s __ &) ( 8 _ c ). NOTE: In both plane and spherical triangles a, b and c are the sides opposite the angles A, B and C respectively. TRIGONOMETRIC FUNCTIONS 43 Plane Oblique Triangles Given To Find Formula 7,2 i r s a 2 Or, formula 4 for cos ^ = 9 7~ - small angles and 5 for angles near 90. a,b,A B smB-=- a C C = 180 (A + B). a sin C sin A b cos A + va 2 b 2 sin If a > b, then 5 < 96? and B < A. If /> >> a >> & sin J, then for one of the tri- angles with the given elements, A < B < 90 and for the other one B > 90. a, A, B b, c b = CT sin # r -?in_C r = asin(J + B) sin yf sin A sin ^4. a, b, C A, B tan A = t a sm C B^=180(A + C) o a cos C Or, determine A and 5 from tan ( A ~ B \ = ^. ctn ^ 2 a + 6 2 c ^V 7 ^ +"/> 2 ^2^aZ) cos (7 = asinC = a -^A sin ^4 cos u ^ F a in which tan -M = 44 TRIGONOMETRIC FUNCTIONS Right-Angled Spherical Triangle NAPIER'S RULES The five circular parts a, b, (T/2-J), (ir/2-c), (v/2-B), are supposed to be ranged around a circle in the order in which they naturally occur with respect to the triangle. If any one of the parts is taken for the middle part, then the two parts next to it are the adjacent parts and the re- maining two parts are the opposite parts. Sine of middle part = product of tangents of adjacent parts. Sine of middle part = product of cosines of opposite parts. Any Spherical Triangle smj? = sjnC. cQg fl = cQg 6 cos c ^ sin b sin c cos A sin -a sin b sin c cos A = cos B cos C* sin B sin C cos a. sin a ctn b = sin C ctn B + cos a cos C. . 1X . /sin (s b} .sin (s c). !/ * /sins. sin (s Sin / -flf -*/ ; - ; COS/ \ sin ^.sin c . \ sin ^.sm c sin 8.sin( a) \ sin B. sin cos S. cos (S A) sin sin C "\cos(S B) .cos(S C) = a + -fc 29 = ^ + ^+C7. CIRCLES 45 CIRCLES Circular Arcs, Cords and Segments. Area A B C = A, Area O A B C =&. The meaning of L, H, C, r and u are shown by the figure. For a circle of unit radius, let /, h, c and a be the values corresponding to L, If, C and A respectively. ->^- c L = r r u 180 =-- 0.01745 r u = I r. / r // = r (l-cos A = ** f -sin u\ = 2 \ 180 ) arc 1 ^0.017453293 log arc 1 a 0.241877 2 arc 1' = 0.000290888 log arc 1' = 0.463726 4 arc 1"= 0.000004848 log arc 1" = 0.685575 6 1 radian = 57 17' 44.806" =* 57.2957795 I number radians in u degrees 46 CIRCLES Lengths of Circular Arcs for Unit Radius DEGREES MINUTES SECONDS U Length I U Length ' U Length / U Length I U Length 1 or or or or or Deg. Radians Deg. Radians Deg. Radians Min. Radians Sec. Radians 1 .017453 61 1.064651 121 2.111848 1 .000291 1 .0000048 2 .034907 62 1.082104 122 2.129302 2 .000582 2 .0000097 3 .052360 63 1.099557 123 2.146755 3 .000873 3 .0000145 4 .069813 64 1.117011 124 2.164208 4 .001164 4 .0000194 5 .087266 65 1.134464 125 2.181662 5 .001454 5 .0000242 6 .104720 66 1.151917 126 2.199115 6 .001745 6 .0000291 7 .122173 67 1.169371 127 2.216568 7 .002036 7 .0000339 8 .139626 68 1.186824 128 2.234021 8 .002327 8 .0000388 9 .157080 69 1.204277 129 2.251475 9 .002618 9 .0000436 10 .174533 70 1.221730 130 2.268928 10 .002909 10 .0000485 11 .191986 71 1.239184 131 2.286381 11 .003200 11 .0000533 12 .209440 72 1.256637 132 2.303835 12 .003491 12 .0000582 13 .226893 73 1.274090 133 2.321288 13 .003782 13 .0000630 14 .244346 74 1.291544 134 2.338741 14 .004072 14 .0000679 15 .261799 75 1.308997 135 2.356194 15 .004363 15 .0000727 16 .279253 76 1.326450 136 2.373648 16 .004654 16 .0000776 17 .296706 77 1.343904 137 2.391101 17 .004945 17 .0000824 18 .314159 78 1.361357 138 2.408554 18 .005236 .18 .0000873 19 .331613 79 1.378810 139 2.426008 19 .005527 19 .0000921 20 .349066 80 1.396263 140 2.443461 20 .005818 20 .0000970 21 .366519 81 1.413717 141 2.460914 21 .006109 21 .0001018 22 .383972 82 1.431170 142 2.478368 22 .006400 22 .0001067 23 .401426 83 1.448623 143 2.495821 23 .006690 23 .0001115 24 .418879 84 1.466077 144 2.513274 24 .006981 24 .0001164 25 .436332 85 1.483530 145 2.530727 25 .007272 25 .0001212 26 .453786 86 1.500983 146 2.548181 26 .007563 26 .0001261 27 .471239 87 1.518436 147 2.565634 27 .007854 27 .0001309 28 .488692 88 1.535890 148 2.583087 28 .008145 28 .0001357 29 .506145 89 1.553343 149 2.600541 29 .008436 29 .0001406 30 .523599 90 1.570796 150 2.617994 30 .008727 30 .0001454 31 .541052 91 1.588250 1b1 2.635447 31 .009018 31 .0001503 32 .558505 92 1.605703 152 2.652900 32 .009308 32 .0001551 33 .575959 93 1.623156 153 2.670354 33 .009599 33 .0001600 34 .593412 94 1.640609 154 2.687807 34 .009890 34 .0001648 35 .610865 95 1.658063 155 2.705260 35 .010181 35 .0001697 36 .628319 96 1.675516 156 2.722714 36 .010472 36 .0001745 37 .645772 97 1.692969 157 2.740167 37 .010763 37 .0001794 38 .663225 98 1.710423 158 2.757620 38 .011054 38 .0001842 39 .680678 99 1.727876 159 2.775074 39 .011345 39 .0001891 40 .698132 100 1.745329 160 2.792527 40 .011636 40 .0001939 41 .715585 101 1.762783 161 2.809980 41 .011926 41 .0001988 42 .733038 102 1.780236 162 2.827433 42 .012217 42 .0002U66 43 .750492 103 1.797689 163 2.844887 43 .012508 43 .0002085 44 .767945 104 1.815142 164 2.862340 44 .012799 44 .0002133 45 .785398 105 1.832596 165 2.879793 45 .013090 45 .0002182 46 .802851 106 1.850049 166 2.897247 46 .013381 46 .0002230 47 .820305 107 1.867502 167 2.914700 47 .013672 47 .0002279 48 .837758 108 1.884956 168 2.932153 48 .013963 48 .0002327 49 .855211 109 1.902409 169 2.949606 49 .014254 49 .0002376 50 .872665 110 1.919862 170 2.967060 50 .014544 50 .0002424 51 .890118 111 1.937315 171 2.984513 51 .014835 51 .0002473 52 .907571 112 1.954769 172 3.001966 52 .015126 52 .0002521 53 .925025 113 1 972222 173 3.019420 53 .015417 53 .0002570 54 .942478 114 1.'989675 174 3.036873 54 .015708 54 .0002618 55 .959931 115 2.007129 175 3.054326 55 .015999 55 .0002666 56 .977384 116 2.024582 176 3.071779 56 .016290 56 .0002715 57 .994838 117 2.042035 177 3.089233 57 .016581 57 .0002763 58 1.012291 118 2.059489 178 3.106686 58 .016872 58 .0002812 59 1.029744 119 2.076942 179 3.124139 59 .017162 59 .0002860 60 1.047198 120 2.094395 180 3.141593 60 .017453 60 .0002909 CIRCLES 47 Length of Cord, Rise and Area of Circular Segments for Unit Radius u De*. 1 2 3 Cord c .0175 .0349 .0524 Rise h .0000 .0002 .0003 Area a .00000 .00000 .00001 u Deg. 46 47 48 Cord C .7815 .7975 .8135 Rise h .0795 .0829 .0865 Area a .04176 .04448 .04731 4 5 6 .0698 .0872 .1047 .0006 .0010 .0014 .00003 .00006 .00010 49 50 51 .8294 .8452 .8610 .0900 .0937 .0974 .05025 .05331 .05649 7 8 9 .1221 .1395 .1569 .0019 .0024 .0031 .00015 .00023 .00032 52 53 54 .8767 .8924 .9080 .1012 .1051 .1090 .05978 .06319 .06673 10 11 12 .1743 .1917 .2091 .0038 .0046 .0055 .00044 .00059 .00076 55 56 57 .9235 ,9389 .9543 .1130 .1171 .1212 .07039 .07417 .07808 13 14 15 .2264 .2437 .2611 .0064 .0075 '.0086 .00097 .00121 .00149 58 59 60 .9696 .9848 1.0000 .1254 .1296 .1340 .08212 .08629 .09059 16 17 18 .2783 .2956 .3129 .0097 .0110 .0123 .00181 .00217 .00257 61 62 63 1.0151 1.0301 1.0450 .1384 .1428 .1474 .09502 .09958 .10428 19 20 21 .3301 .3473 .3645 .0137 .0152 .0167 .00302 .00352 .00408 64 65 66 1.0598 1.0746 1.0893 .1520 .1566 .1613 .10911 .11408 .11919 22 23 24 .3816 .3987 .4158 .0184 .0201 .0219 .00468 .00535 .00607 67 68 69 1.1039 1.1184 1.1328 .1661 .1710 .1759 .12443 .12982 .13535 25 26 27 .4329 .4499 .4669 .0237 .0256 .0276 .00686 .00771 .00862 70 71 72 1.1472 1.1614 1.1756 .1808 .1859 .1910 .14102 .14683 .15279 28 29 30 .4838 .5008 .5176 .0297 .0319 .0341 .00961 .01067 .01180 73 74 75 1.1896 .1.2036 1.2175 .1961 .2014 .2066 .15889 .16514 .17154 31 32 33 .5345 .5512 .5680 .0364 .0387 .0412 .01301 .01429 .01566 76 77 78 1.2313 1.2450 1.2586 .2120 .2174 .2229 .17808 .18477 .19160 34 35 36 .5847 .6014 .6180 .0437 .0463 .0489 .01711 .01864 .02027 79 80 81 1.2722 1.2856 1.2989 .2284 .2340 .2396 .19859 .20573 .21301 37 38 39 .6346 .6511 .6676 .0517 .0545 .0574 .02198 .02378 .02568 82 83 84 1.3121 1.3252 1.3383 .2453 .2510 .2569 .22045 .22804 .23578 40 41 42 .6840 .7004 .7167 .0603 .0633 .0664 .02767 .02976 .03195 85 86 87 1.3512 1.3640 1.3767 .2627 .2686 .2746 .24367 .25171 .25990 43 44 45 .7330 .7492 .7654 .0696 .0728 .0761 .03425 .03664 .03915 88 89 90 1.3893 1.4018 1.4142 .2807 .2867 .2929 .26825 .27675 .28540 48 CIRCLES Length of Cord, Rise and Area of Circular Segments for Unit Radius u Deg. 91 92 93 Cord C 1.4265 1.4387 1.4507 Rise h .2991 .3053 .3116 Area a .29420 .30316 .31226 u Deer- 136 137 138 Cord C 1.8544 1.8608 1.8672 Rise h .6254 .6335 .6416 Area a .83949 .85455 .86971 94 95 96 1.4627 1.4746 1.4863 .3180 .3244 .3309 .32152 .33093 .34050 139 140 141 1.8733 1.8794 1.8853 .6498 .6580 .6662 .88497 .90034 .91580 97 98 99 1.4979 1.5094 1.5208 .3374 .3439 .3506 .35021 .36008 .37009 142 143 144 1.8910 1.8966 1.9021 .6744 .6827 .6910 .93135 .94700 .96274 100 101 102 1.5321 1.5432 1.5543 .3572 .3639 .3707 .38026 .39058 .40104 145 146 147 1.9074 1.9126 1.9176 .6993 .7076 .7160 .97858 .99449 1.01050 103 104 105 1.5652 1.5760 1.5867 .3775 .3843 .3912 .41166 .42242 .43333 148 149 150 1.9225 1.9273 1.9319 .7244 .7328 .7412 1.02658 1.04275 1.05900 106 107 108 1.5973 1.6077 1.6180 .3982 .4052 .4122 .44439 .45560 .46695 151 152 153 1.9363 1.9406 1.9447 .7496 .7581 .7666 1.07532 1.09171 1.10818 109 110 111 1.6282 1.6383 1.6483 .4193 .4264 .4336 .47844 .49008 .50187 154 155 156 1.9487 1.9526 1.9563 .7750 .7836 .7921 1.12472 1.14132 1.15799 112 113 114 1.6581 1.6678 1.6773 .4408 .4481 .4554 .51379 .52586 .53807 157 158 159 1.9598 1.9633 1.9665 .8006 .8092 .8178 1.17472 1.19151 1.20835 115 116 117 1.6868 1.6961 1.7053 .4627 .4701 .4775 .55041 .56289 .57551 160 161 162 1.9696 1.9726 1.9754 .8264 .8350 .8436 1.22525 1.24221 1.25921 118 119 120 1.7143 1.7233 1.7321 .4850 .4925 .5000 .58827 .60116 .61418 163 164 165 1.9780 1.9805 1.9829 .8522 .8608 .8695 1.27626 1.29335 1.31049 121 122 123 1.7407 1.7492 1.7576 .5076 .5152 .5228 .62734 .64063 .65404 166 167 168 1.9851 1.9871 1.9890 .8781 .8868 .8955 1.32766 1.34487 1.36212 124 125 126 1.7659 1.7740 1.7820 .5305 .5383 .5460 .66759 .68125 .69505 169 170 171 1.9908 1.9924 1.9938 .9042 .9128 .9215 1.37940 1.39671 1.41404 127 128 129 1.7899 1.7976 1.8052 .5538 .5616 .5695 .70897 .72301 .73716 172 173 174 1.9951 1.9963 1.9973 .9302 .9390 .9477 1.43140 1.44878 1.46617 130 131 132 1.8126 1.8199 1.8271 .5774 .5853 .5933 .75144 .76584 .78034 175 176 177 1.9981 1.9988 1.9993 .9564 .9651 .9738 1.48359 1.50101 1.51845 133 134 135 1.8341 1.8410 1.8478 .6013 .6093 .6173 .79497 .80970 .82454 178 179 180 1.9997 1.9999 2.0000 .9825 .9913 1.0000 1.53589 1.55334 1.5706'J CIRCLES 1!) Areas and Circumferences of Circles Dia. Circ. Area Dia. Circ. Area Dia. Circ. Area 3-4 2.3562 .44179 3 9.4248 7.0686 1-64' .04909 .00019 49-64 2.4053 .46039 1-16 9.6211 7.3662 1-32 .09818 .00077 25-32 2.4544 .47937 1-8 9.8175 7.6699 3-64 .14726 .00173 51-64 2.5035 .49872 3-16 10.014 7.9798 1-16 .19635 .00307 13-16 2.5525 .51849 1-4 10.210 8.2958 5-64 .24545 .00479 53-64 2.6017 .53862 5-16 10.407 8.6179 3-32 .29452 .00690 27-32 2.6507 .55914 3-8 10.603 8.9462 7-64 .34363 .00939 55-64 2.6999 .58003 7-16 10.799 9.2806 1-8 .39270 .01227 7-8 2.7489 .60132 1-2 10.996 9.6211 9-64 .44181 .01553 57-64 2.7981 .62298 9-16 11.192 9.9678 5-32 .49087 .01917 29-32 2.8471 .64504 5-8 11.388 10.321 11-64 .53999 .02320 59-64 2.8962 .66746 11-16 11.585 10.680 3-16 .58905 .02761 15-16 2.9452 .69029 3-4 11.781 11.045 13-64 .63817 .03241 61-64 2.9944 .71349 13-16 11.977 11.416 7-32 .68722 .03758 31-32 3.0434 .73708 7-8 12.174 11.793 15-64 .73635 .04314 63-64 3.0925 ,76097 15-16 12.370 12.177 1-4 .78540 .04909 1 3.1416 .7854 4 12.566 12.566 17-64 .83453 .05542 1-16 3.3379 .8866 1-16 12.763 12.962 9-32 .88357 .06213 1-8 3.5343 .9940 1-8 12.959 13.364 19-64 .93271 .06922 3-16 3.7306 1.1075 3-16 13.155 13.772 5-16 .98175 .07670 1-4 3.9270 1.2272 1-4 13.352 14.186 21-64 1.0309 .08456 5-16 4.1233 1.3530 5-16 13.548 14.607 11-32 1.0799 .09281 3-8 4.3197 1.4849 3-8 13.744 15.033 23-64 1.1291 .10144 7-16 4.5160 1.6230 7-16 13.941 15.466 3-8 1.1781 .11045 1-2 4.7124 1.7671 1-2 14.137 15.904 25-64 1.2273 .11984 9-16 4.9087 1.9175 9-16 14.334 16.349 13-32 1.2763 .12962 5-8 5.1051 2.0739 5-8 14.530 16.800 27-64 1.3254 .13979 11-16 5.3014 2.2365 11-16 14.726 17.257 7-16 1.3744 .15033 3-4 5.4978 2.4053 3-4 14.923 17.721 29-64 1.4236 .16126 13-16 5.6941 2.5802 13-16 15.119 18.190 15-32 1.4726 .17257 7-8 5.8905 2.7612 7-8 15.315 18.665 31-64 1.5218 .18427 15-16 6.0868 2.9483 '15-16 15.512 19.147 1-2 1.5708 .19635 2 6.2832 3.1416 5 15.708 19.635 33-64 1.6199 .20880 1-16 6.4795 3.3410 1-16 15.904 20.129 17-32 1.6690 .22166 1-8 6.6759 3.5466 1-8 16.101 20.629 35-64 1.7181 .23489 3-16 6.8722 3.7583 3-16 16.297 21.135 9-16 1.7671 .24850 1-4 7.0686 3.9761 1-4 14.493 21.648 37-64 1.8163 .26248 5-16 7.2649 "4.2000 5-16 16.690 22.166 19-32 1.8653 .27688 3-8 7.4613 4.4301 3-8 16.886 22.691 39-64 1.9144 .29164 7-16 7.6576 4.6664 7-16 17.082 23.221 5-8 1.9635 .30680 1-2 7.8540 4.9087 1-2 17.279 23.758 41-64 2.0126 .32232 9-16 8.0503 5.1572 9-16 17.475 24.301 21-32 2.0617 .33824 5-8 8.2467 5.4119 5-8 17.671 24.850 43-64 2.1108 .35453 11-16 8.4430 5.6727 11-16 17.868 25.406 11-16 2.1598 .37122 3-4 8.6394 5.9396 3-4 18.064 25.967 45-64 2.2090 .38828 13-16 8.8357 6.2126 13-16 18.261 26.535 23-32 2.2580 .40574 7-8 9.0321 6.4918 7-8 18.457 27.109 47-64 2.3071 .42356 15-16 9.2284 6.7771 15-16 18.653 27.688 CIRCLES Areas and Circumferences of Circles Dia. Circ. Area Dia. Circ. Area Dia. Circ. Area 6 18.850 28.274 13 40.841 132.73 20 62.832 314.16 1-8 19.242 29.465 1-8 41.233 135.30 1-8 63.225 318.10 1-4 19.635 30.680 1-4 41.626 137.89 1-4 63.617 322.06 3-8 20.028 31.919 3-8 42.019 140.50 3-8 64.010 326.05 1-2 20.420 33.183 1-2 42.412 143.14 1-2 64.403 330.06 5-8 20.813 34.472 5-8 42.804 145.80 5-8 64.795 334.10 3-4 21.206 35.785 3-4 43.197 148.49 3-4 65.188 338.16 7-8 21.598 37.122 7-8 43.590 151.20 7-8 65.581 342.25 7 21.991 38.485 14 43.982 153.94 21 65.973 346.36 1-8 22.384 39.871 1-8 44.375 156.70 1-8 66.366 350.50 1-4 22.776 41.282 1-4 44.768 159.48 1-4 66.759 354.66 3-8 23.169 42.718 3-8 45.160 162.30 3-8 67.152 358.84 1-2 23.562 44.179 1-2 45.553 165.13 1-2 67.544 363.05 5-8 23.955 45.664 5-8 45.946 167.99 5-8 67.937 367.28 3-4 24.347 47.173 3-4 46.338 170.87 3-4 68.330 371.54 7-8 24.740 48.707 7-8 46.731 173.78 7-8 68.722 375.83 8 25.133 50.265 15 47.124 176.71 22 69.115 380.13 1-8 25.525 51.849 1-8 47.517 179.67 1-8 69.508 384.46 1-4 25.918 53.456 1-4 47.909 182.65 1-4 69.900 388.82 3-8 26.311 55.088 3-8 48.302 185.66 3-8 70.293 393.20 1-2 26.704 56.745 1-2 48.695 188.69 1-2 70.686 397.61 5-8 27.096 58.426 5-8 49.087 191.75 5-8 71.079 402.04 3-4 27.489 60.132 3-4 49.480 194.83 3-4 71.471 406.49 7-8 27.882 61.862 7-8 49.873 197.93 7-8 71.864 410.97 9 1-8 1-4 3-8 1-2 5-8 3-4 7-8 28.274 28.667 29.060 29.452 29.845 30.238 30.631 31.023 63.617 65.397 67.201 69.029 70.882 72.760 74.062 76.589 16 1-8 1-4 3-8 1-2 5-8 3-4 7-8 50.265 50.658 51.051 51.444 51.836 52.229 52.622 53.014 201.06 204.22 207.39 210.60 213.82 217.08 220.35 223.65 23 1-8 1-4 3-8 1-2 5-8 3-4 7-8 72.257 72.649 73.042 73.435 73.827 74.220 74.613 75.006 415.48 420.00 424.56 429.13 433.74 438.36 443.01 447.69 10 1-8 1-4 3-8 1-2 5-8 3-4 31.416 31.809 32.201 32.594 32.987 33.379 33.772 78.540 80.516 82.516 84.541 86.590 88.664 90.763 17 1-8 1-4 3-8 1-2 5-8 3-4 53.407 53.800 54.192 54.585 54.978 55.371 55.763 226.98 230.33 233.71 237.10 240.53 243.98 247.45 24 1-8 1-4 3-8 1-2 6-8 3-4 75.398 75.791 76.184 76.576 76.969 77.362 77.754 452.39 457.11 461.86 466.64 471.44 476.26 481.11 7-8 34.165 92.886 7-8 56.156 250.95 7-8 78.147 485.98 11 34.558 95 033 18 56.549 254.47 25 78.540 490.87 1-8 34.950 97.205 1-8 56.941 258.02 1-8 78.933 495.79 1-4 35.343 99.402 1-4 57.334 261.59 1-4 79.325 500.74 3-8 35.736 101.62 3-8 57.727 265.18 3-8 79.718 505.71 1-2 36.128 103.87 1-2 58.119 268.80 1-2 80.111 510.71 5-8 36.521 106.14 5-8 58.512 272.45 5-8 80.503 515.72 3-4 36.914 108.43 3-4 58.905 276.12 3-4 80.896 520.77 7-8 37.306 110.75 7-8 59.298 279.81 7-8 81.289 525.84 12 37.699 113.10 19 59.690 283.53 26 81.681 530.93 1-8 38.092 115.47 1-8 60.083 287.27 1-8 82.074 536.05 1-4 38.485 117.86 1-4 60.476 291.04 1-4 82.467 541.19 3-8 38.877 120.28 3-8 60.868 294.83 3-8 82.860 546.35 1-2 39.270 122.72 1-2 61.261 298.65 1-2 83.252 551.55 5-8 39.663 125.19 5-8 61.654 302.49 5-8 83.645 556.76 3-4 40.055 127.68 3-4 62.046 306.35 3-4 84.038 562.00 7-8 40.448 130.19 7-8 62.439 310.24 7-8 84.430 567.27 CIRCLES Areas and Circumferences of Circles 51 Dia. Circ. Area Dia. Circ. Area Dia. Circ. Area 27 84.823 572.56 30 94.248 706.86 33 103.673 855.30 1-8 85.216 577.87 1-8 94.640 712.76 1-8 104.065 861.7D 1-4 85.608 583.21 1-4 95.033 718.69 1-4 104.458 868.31 3-8 86.001 588.57 3-8 95.426 724.64 3-8 104.851 874.85 1-2 86.394 593.96 1-2 95.819 730.62 1-2 105.243 881.41 5-8 86.786 599.37 5-8 96.211 736.62 5-8 105.636 888.00 3-4 87.179 604.81 3-4 96.604 742.64 3-4 106.029 894.62 7-8 87.572 610.27 7-8 96.997 748.69 7-8 106.421 901.26 28 87.965 615.75 31 97.389 754.77 34 106.814 907.92 1-8 88.357 621.26 1-8 97.782 760.87 1-8 107.207 914.61 1-4 88.750 626.80 1-4 98.175 766.99 1-4 107.600 921.32 3-8 89.143 632.36 3-8 98.567 773.14 3-8 107.992 928.06 1-2 89.535 637.94 1-2 98.960 779.31 1-2 108.385 934.82 5-8 89.928 643.55 5-8 99.353 785.51 5-8 108.778 941.61 3-4 90.321 649.18 3-4 99.746 791.73 3-4 109.170 948.42 7-8 90.713 654.84 7-8 100.138 797.98 7-8 109.563 955.25 29 91.106 660.52 32 100.531 804.25 35 109.956 962.11 1-8 91.499 666.23 1-8 100.924 810.54 1-8 110.348 969.00 1-4 91.892 671.96 1-4 101.316 816.86 1-4 110.741 975.91 3-8 92.284 677.71 3-8 101.709 823.21 3-8 111.134 982.84 1-2 92.677 683.49 1-2 102.102 829.58 1-2 111.527 989.80 5-8 93.070 689.30 5-8 102.494 835.97 5-8 111.919 996.78 3-4 93.462 695.13 3-4 102.887 842.39 3-4 112.312 1003.8 7-8 93.855 700.98 7-8 103.280 848.83 7-8 112.705 1010.8 Areas of Circles Dia. Area Area in in in in. sq. in. sq. ft- Dia. Area Area in in in in sq. in- sq. ft. Dia. Area Area in in in in sq. in. sq. ft. Dia. Area Area in in in in. sq. in. sq. ft 1/4. .0491 .00034 % .1104 .00077 Vz .1963 .00136 3 A .4418 .00307 10 78.540 .54542 11 95.033 .65995 12 113.10 .78540 13 132.73 .92174 27 572.56 3.9761 28 615.75 4.2760 29 660.52 4.5870 30 706.86 4.9087 44 1520.5 10.560 45 1590.4 11.045 46 1661.9 11.541 47 1734.9 12.048 1 .7854 .00545 1'/ 4 1.2272 .00852 1'/ 2 1.7671 .01227 2 3.1416 .02182 14 153.94 1.0690 15 176.71 1.2272 16 201.06 1.3963 17 226.98 1.5762 31 754.77 5.2415 32 804.25 5.5851 33 855.30 5.9396 34 907.92 6.3050 48 1809.6 12.56$ 49 1885.7 13.095 50 1963.5 13.635 51 2042.S 14.186 21/2 4.9087 .03409 3 7.0686 .04909 31/2 9.6211 .06681 4 12.566 .08726 18 254.47 1.7671 19 283.53 1.9689 20 314.16 2.1817 21 346.36 2.4053 35 962.11 6.6813 36 1017.9 7.0686 37 1075.2 7.4667 38 1134.1 7.8757 52 2123.7 14.748 53 2206.2 15.321 54 2290.2 15.904 55 2375.8 16.499- 5 19.635 .13635 6 28.274 .19635 7 38.484 .26725 8 50.265 .34906 9 63.617 .44178 22 380.13 2.6398 23 415.48 2.8853 24 452.39 3.1416 25 490.87 3.4088 26 530.93 3.6870 39 1194.6 8.2958 40 1256.6 8.7265 41 1320.2 9.1681 42 1385.4 9.6208 43 1452.2 10.085 56 2463.0 17.104 57 2551.8 17.721 52 2642.1 18.348 59 2734.0 18.986 60 2827.4 19.635- FUNCTIONS OF THE NATURAL NUMBERS No. Square Cube Square Root Cube Root 1000 x Recip. No. Dia. Circum. Area 1 2 3 1 4 9 1 8 27 1.0000 1.4142 1.7321 1.0000 1.2599 1.4422 1000.000 500.000 333.333 3.142 6.283 9.425 0.7854 3.1416 7.0686 4 5 6 16 25 36 64 125 216 2.0000 2.2361 2.4495 1.5874 1.7100 1.8171 250.000 200.000 166.667 12.566 15.708 18.850 12.5664 19.6350 28.2743 7 8 9 49 64 81 343 512 729 2.6458 2.8284 3.0000 1.9129 2.0000 2.0801 142.857 125.000 111.111 21.991 25.133 28.274 38.4845 50.2655 63.6173 10 31 12 13 100 121 144 169 1000 1331 1728 2197 3.1623 3.3166 3.4641 3.6056 2.1544 2.2240 2.2894 2.3513 100.000 90.9091 83.3333 76.9231 31.416 34.558 37.699 40.841 78.5398 95.0332 113.097 132.732 14 15 16 196 225 256 2744 3375 4096 3.7417 3.8730 4.0000 2.4101 2.4662 2.5198 71.4286 66.6667 62.5000 43.982 47.124 50.265 153.938 176.715 201.062 17 18 19 289 324 361 4913 5832 6859 4.1231 4.2426 4.3589 2.5713 2.6207 2.6684 58.8235 55.5556 52.6316 53.407 56.549 59.690 226.980 254.469 283.529 20 21 22 23 400 441 484 529 8000 9261 10648 12167 4.4721 4.5826 4.6904 4.7958 2.7144 2.7589 2.8020 2.8439 50.0000 47.6190 45.4545 43.4783 62.832 65.973 69.115 72.257 314.159 346.361 380.133 415.476 24 25 26 576 625 676 13824 15625 17576 4.8990 5.0000 5.0990 2.8845 2.9240 2.9625 41.6667 40.0000 38.4615 75.398 78.540 81.681 452.389 490.874 530.929 27 28 29 729 784 841 19683 21952 24389 5.1962 5.2915 5.3852 3.0000 3.0366 3.0723 37.0370 35.7143 34.4828 84.823 87.965 91.106 572.555 615.752 660.520 30 31 32 33 900 961 1024 1089 27000 29791 32768 35937 5.4772 5.5678 5.6569 5.7446 3.1072 3.1414 3.1748 3 . 2075 33.3333 32.2581 31.2500 30.3030 94.248 97.389 100.531 103.673 706.858 754.768 804.248 855.299 34 35 36 1156 1225 1296 39304 42875 46656 5.8310 5.9161 6.0000 3.2396 3.2711 3.3019 29.4118 28.5714 27.7778 106.814 109.956 113.097 907.920 962.113 1017.88 37 38 39 1369 1444 1521 50653 54872 59319 6.0828 6J644 6.2450 3.3322 3.3620 3.3912 27.0270 26.3158 25.6410 116.239 119.381 122.522 1075.21 1134.11 1194.59 40 41 42 43 1600 1681 1764 1849 64000 68921 74088 79507 6.3246 6.4031 6.4807 6.5574 3.4200 3.4482 3.4760 3.5034 25.0000 24.3902 23.8095 23.2558 125.66 128.81 131.95 135.09 1256.64 1320.25 1385.44 1452.20 44 45 46 1936 2025 2116 85184 91125 97336 6.6332 6.7082 6.7823 3.5303 3.5569 3.5830 22.7273 22.2222 21.7391 138.23 141.37 144.51 3520.53 1590.43 1661.90 47 48 49 2209 2304 2401 103823 110592 117649 6.8557 6. 9282 7 . 0000 3.6088 3.6342 3.6593 21.2766 20.8333 20.4082 147.65 150.80 153.94 1734.94 1809.56 1885.74 FUNCTIONS OF THE NATURAL NUMBERS 5:: No. Square Cube Square Root Cube Root 1000 x Recip. No.=Dia. Circum. Area 50 51 52 53 2500 2601 2704 2809 125000 132651 140608 148877 7.0711 7.1414 7.2111 7.2801 3.6840 3.7084 3.7325 3.7563 20.0000 19.6078 19.2308 18.8679 157.08 160.22 163.36 166.50 1963.50 2042.82 2123.72 2206.18 54 55 56 2916 3025 3136 157464 166375 175616 7.3485 7.4162 7.4833 3.7798 3.8030 3.8259 18.5185 18.1818 17.8571 169.65 172.79 175.93 2290.22 2375.83 2463.01 57 58 59 3249 3364 3481 185193 195112 205379 7.5498 7.6158 7.6811 3.8485 3.8709 3.8930 17.5439 17.2414 16.9492 179.07 182.21 185.35 2551.76 2642.08 2733.97 60 61 62 63 3600 3721 3844 3969 216000 226981 238328 250047 7.7460 7.8102 7.8740 7.9373 3.9149 3.9365 3.9579 3.9791 16.6667 16.3934 16.1290 15.8730 188.50 191.64 194.78 197.92 2827.43 2922.47 3019.07 3117.25 64 65 66 4096 4225 4356 262144 274625 287496 8.0000 8.0623 8.1240 4.0000 4.0207 4.0412 15.6250 15.3846 15.1515 201.06 204.20 207.35 3216.99 3318.31 3421.19 67 68 69 4489 4624 4761 300763 314432 328509 8.1854 8.2462 8.3066 4.0615 4.0817 4.1016 14.9254 14.7059 14.4928 210.49 213.63 216.77 3525.65 3631.68 3739.28 70 71 72 73 4900 5041 5184 5329 343000 357911 373248 389017 8.3666 8.4261 8.4853 8.5440 4.1213 4.1408 4.1602 4.1793 14.2857 14.0845 13.8.889 13.6986 219.91 223.05 226.19 229.34 3848.45 3959.19 4071.50 4185.39 74 75 76 5476 5625 5776 405224 421875 438976 8.6023 8.6603 8.7178 4.1983 4.2172 4.2358 13.5135 13.3333 13.1579 232.48 235.62 238.76 4300.84 4417.86 4536.46 77 78 79 5929 6084 6241 456533 474552 493039 8.7750 8.8318 8.8882 4 . 2543 4.2727 4.2908 12.9870 12.8205 12.6582 241.90 245.04 248.19 4656.63 4778.36 4901.67 80 81 82 83 6400 6561 6724 6889 512000 531441 551368 571787 8.9443 9.0000 9.0554 9.1104 4.3089 4.3267 4.3445 4.3621 12.5000 12.3457 12.1951 12.0482 251.33 254.47 257.61 260.75 5026.55 5153.00 5281.02 5410.61 84 85 86 7056 7225 7396 592704 614125 636056 9.1652 9.2195 9.2736 4.3795 4.3968 4.4140 11.9048 11.7647 11.6279 263.89 267.04 270.18 5541.77 5674.50 5808.80 87 88 89 7569 7744 7921 658503 681472 704969 9.3274 9.3808 9.4340 4.4310 4.4480 4.4647 11.4943 11.3636 11.2360 273.32 276.46 279.60 5944.68 6082.12 6221.14 90 91 92 93 8100 8281 8464 8649 729000 753571 778688 804357 9.4868 9.5394 9.5917 9.6437 4.4814 4.4979 4.5144 4.5307 11.1111 10.9890 10.8696 10.7527 282.74 285.88 289.03 292.17 6361.73 6503.88 6647.61 6792.91 94 95 96 8836 9025 9216 830584 857375 884736 9.6954 9.7468 9.7980 4.5468 4.5629 4.5789 10.6383 10.5263 10.4167 295.31 298.45 301.59 6939.78 7088.22 7238.23 97 98 99 9409 9604 9801 912673 941192 970299 9.8489 9.8995 9.9499 4.5947 4.6104 4.6261 10.3093 10.2041 10.1010 304.73 307.88 311.02 7389.81 7542.96 7697.69 54 FUNCTIONS OF THE NATURAL NUMBERS No. Square Cube Square Root Cube Root 1000 x Recip. No.=Dia. Circum. Area 100 101 102 103 10000 10201 10404 10609 1000000 1030301 1061208 1092727 10.0000 10.0499 10.0995 10.1489 4.6416 4.6570 4.6723 4.6875 10.0000 9.90099 9.80392 9.70874 314.16 317.30 320.44 323.58 7853.98 8011.85 8171.28 8332.29 104 105 '106 10816 11025 11236 1124864 1157625 1191016 10.1980 10.2470 10.2956 4.7027 4.7177 4.7326 9.61538 9.52381 9.43396 326.73 329.87 333.01 8494.87 8659.01 8824.73 107 108 109 11449 11664 11881 1225043 1259712 1295029 10.3441 10.3923 10.4403 4.7475 4.7622 4.7769 9.34579 9.25926 9.17431 336.15 339.29 342.43 8992.02 9160.88 9331.32 110 111 112 113 12100 12321 12544 12769 1331000 1367631 1404928 1442897 10.4881 10.5357 10.5830 10.6301 4.7914 4.8059 4.8203 4.8346 9.09091 9.00901 8.92857 8.84956 345.58 348.72 351.86 355.00 9503.32 9676.89 9852.03 10028.7 114 115 116 12996 13225 13456 1481544 1520875 1560896 10.6771 10.7238 10.7703 4.8488 4.8629 4.8770 8.77193 8.69565 8.62069 358.14 361.28 364.42 10207.0 10386.9 10568.3 117 118 119 13689 13924 14161 1601613 1643032 1685159 10.8167 10.8628 10.9087 4.8910 4.9049 4.9187 8.54701 8.47458 8.40336 367.57 370.71 373.85 10751.3 10935.9 11122.0 120 121 122 123 14400 14641 14884 15129 1728000 1771561 1815848 1860867 10.9545 11.0000 11.0454 11.0905 4.9324 4.9461 4.9597 4.9732 8.33333 8.26446 8.19672 8.13008 376.99 380.13 383.27 386.42 11309.7 11499.0 11689.9 11882.3 124 125 126 15376 15625 15876 1906624 1953125 2000376 11.1355 11.1803 11.2250 4.9866 5.0000 5.0133 8.06452 8.00000 7.93651 389.56 392.70 395.84 12076.3 12271.8 12469.0 127 128 129 16129 16384 16641 2048383 2097152 2146689 11.2694 11.3137 11.3578 5.0265 5.0397 5.0528 7.87402 7.81250 7.75194 398.98 402.12 405.27 12667.7 12868.0 13069.8 130 131 132 133 16900 17161 17424 17689 2197000 2248091 2299968 2352637 11.4018 11.4455 11.4891 11.5326 5.0658 5.0788 5.0916 5.1045 7.69231 7.63359 7.57576 7.51880 408.41 411.55 414.69 417.83 13273.2 13478.2 13684.8 13892.9 134 135 136 17956 18225 18496 2406104 2460375 2515456 11.5758 11.6190 11.6619 5.1172 5.1299 5.1426 7.46269 7.40741 7.35294 420.97 424.12 427.26 14102.6 14313.9 14526.7 137 138 139 18769 19044 19321 2571353 2628072 2685619 11.7047 11.7473 11.7898 5.1551 5.1676 5.1801 7.29927 7.24638 7.19424 430.40 433.54 436.68 14741.1 14957.1 15174.7 140 141 142 143 19600 19881 20164 20449 2744000 2803221 2863288 2924207 11.8322 11.8743 11.9164 11.9583 5.1925 5.2048 5.2171 5.2293 7.14286 7.09220 7.04255 6.99301 439.82 442.96 446.11 449.25 15393.8 15614.5 15836.8 16060.6 144 145 146 20736 21025 21316 2985984 3048625 3112136 12.0000 12.0416 12.0830 5.2415 5.2536 5.2656 6.94444 6.89655 6.84932 452.39 455.53 458.67 16286.0 16513.0 16741.5 147 148 149 21609 21904 22201 3176523 3241792 3307949 12.1244 12.1655 12.2066 5.2776 5.2896 5.3015 6.80272 6.75676 6.71141 461.81 464.96 468.10 16971.7 17203.4 17436.6 FUNCTIONS OF THE NATURAL NUMBERS No. Square Cube Square Root Cube Root 1000 x Recip. No. Dia. Circum. Area 150 151 152 153 22500 22801 23104 23409 3375000 3442951 3511808 3581577 12.2474 12.2882 12.3288 12.3693 5.3133 5.3251 5.3368 5.3485 6.66667 6.62252 6.57895 6.53595 471.24 474.38 477.52 480.66 17671.5 17907.9 18145.8 18385.4 154 155 156 23716 24025 24336 3652264 3723875 3796416 12.4097 12.4499 12.4900 5.3601 5.3717 5.3832 6.49351 6.45161 6.41026 483.81 486.95 490.09 18626.5 18869.2 19113.4 157 158 159 24649 24964 25281 3869893 3944312 4019679 12.5300 12.5698 12.6095 5.3947 5.4061 5.4175 6.36943 6.32911 6.28931 493.23 496.37 499.51 19359.3 19606.7 19855.7 160 161 162 163 25600 25921 26244 26569 4096000 4173281 4251528 4330747 12.6491 12.6886 12.7279 12.7671 5.4288 5.4401 5.4514 5.4626 6.25000 6.21118 6.17284 6.13497 502.65 505.80 508.94 512.08 20106.2 20358.3 20612.0 20867.2 164 165 166 26896 27225 27556 4410944 4492125 4574296 12.8062 12.8452 12.8841 5.4737 5.4848 5.4959 6.09756 6.06061 6.02410 515.22 518.36 521.50 21124.1 21382.5 21642.4 167 168 169 27889 28224 28561 4657463 4741632 4826809 12.9228 12.9615 13.0000 5.5069 5.5178 5.5288 5.98802 5.95238 5.91716 524.65 527.79 530.93 21904.0 22167.1 22431.8 170 171 172 173 28900 29241 29584 29929 4913000 5000211 5088448 5177717 13.0384 13.0767 13.1149 13.1529 5.5397 5.5505 5.5613 5.5721 5.88235 5.84795 5.81395 5.78035 534.07 537.21 540.35 543.50 22698.0 22965.8 23235.2 23506.2 174 175 176 30276 30625 30976 5268024 5359375 5451776 13.1909 13.2288 13.2665 5.5828 5.5934 5.6041 5.74713 5.71429 5.68182 546.64 549.78 552.92 23778.7 24052.8 24328.5 177 178 179 31329 31684 32041 5545233 5639752 5735339 13.3041 13.3417 13.3791 5.6147 5.6252 5.6357 5.64972 5.61798 5.58659 556.06 559.20 562.35 24605.7 24884.6 25164.9 180 181 182 183 32400 32761 33124 33489 5832000 5929741 6028568 6128487 13.4164 13.4536 13.4907 13.5277 5.6462 5.6567 5.6671 5.6774 5.55556 5.52486 5.49451 5.46448 565.49 568.63 571.77 574.91 25446.9 25730.4 26015.5 26302.2 184 185 186 33856 34225 34596 6229504 6331625 6434856 13.5647 13.6015 13.6382 5.6877 5.6980 5.7083 5.43478 5.40541 5.37634 578.05 581.19 584.34 26590.4 26880.3 27171.6 187 188 189 34969 35344 35721 6539203 6644672 6751269 13.6748 13.7113 13.7477 5.7185 5.7287 5.7388 5.34759 5.31915 5.29101 587.48 590.62 593.76 27464.6 27759.1 28055.2 190 191 192 193 36100 36481 36864 37249 6859000 6967871 7077888 7189057 13.7840 13.8203 13.8564 13.8924 5.7489 5.7590 5.7690 5.7790 5'. 26316 5.23560 5.20833 5.18135 596.90 600.04 603.19 606.33 28352.9 28652.1 28952.9 29255.3 194 195 196 37636 38025 38416 7301384 7414875 7529536 13.9284 13.9642 14.0000 5.7890 5.7989 5.8088 5.15464 5.12821 5.10204 609.47 612.61 615 . 75 29559.2 29864.8 30171.9 197 198 199 38809 39204 39601 7645373 7762392 7880599 14.0357 14.0712 14.1067 5.8186 5.8285 5.8383 5.07614 5.05051 5.02513 618.89 622 04 625.18 30480.5 30790.7 31102.6 56 FUNCTIONS OF THE NATURAL NUMBERS No. Square Cube Square Root Cube Root 1000 x Recip. No.:=Dia. Circum. Area 200 201 202 203 40000 40401 40804 41209 8000000 8120601 8242408 8365427 14.1421 14.1774 14.2127 14.2478 5.8480 5.8578 5.8675 5.8771 5.00000 4.97512 4.95050 4.92611 628.32 631.46 634.60 637.74 31415.9 31730.9 32047.4 32365.5 204 205 206 41616 42025 42436 8489664 8615125 8741816 14.2829 14.3178 14.3527 5.8868 5.8964 5.9059 4.90196 4.87805 4.85437 640.88 644 . 03 647.17 32685.1 33006.4 33329.2 207 208 209 42849 43264 43681 8869743 8998912 9129329 14.3875 14.4222 14.4568 5.9155 5.9250 5.9345 4.83092 4.80769 4.78469 650.31 653.45 656.59 33653.5 33979.5 34307.0 210 211 212 213 44100 44521 44944 45369 9261000 9393931 9528128 9663597 14.4914 14.5258 14.5602 14.5945 5.9439 5.9533 5.9627 5.9721 4.76190 4.73934 4.71698 4.69484 659.73 662.88 666.02 669.16 34636.1 34966.7 35298.9 35632.7 214 215 216 45796 46225 46656 9800344 9938375 10077696 14.6287 14.6629 14.6969 5.9814 5.9907 6.0000 4.67290 4.65116 4.62963 672.30 675.44 678.58 35968.1 36305.0 36643.5 217 218 219 47089 47524 47961 10218313 10360232 10503459 14.7309 14.7648 14.7986 6.0092 6.0185 6.0277 4.60829 4.58716 4.56621 681.73 684.87 688.01 36983.6 37325.3 37668.5 220 221 222 223 48400 48841 49284 49729 10648000 10793861 10941048 11089567 14.8324 14.8661 14.899T 14.9332 6.0368 6.0459 6.0550 6.0641 4.54545 4.52489 4.50450 4.48430 691.15 694.29 697.43 700.58 38013.3 38359.6 38707.6 39057.1 224 225 226 50176 50625 51076 11239424 11390625 11543176 14.9666 15.0000 15.0333 6.0732 6.0822 6.0912 4.46429 4.44444 4.42478 703.72 706.86 710.00 39408.1 39760.8 40115.0 227 228 229 51529 51984 52441 11697083 11852352 12008989 15.0665 15.0997 15.1327 6.1002 6.1091 6.1180 4.40529 4.38596 4.36681 713.14 716.28 719.42 40470.8 40828.1 41187.1 230 231 232 233 52900 53361 53824 54289 12167000 12326391 12487168 12649337 15.1658 15.1987 15.2315 15.2643 6.1269 6.1358 6.1446 6.1534 4.34783 4.32900 4.31034 4.29185 722.57 725.71 728.85 731.99 41547.6 41909.6 42273.3 42638.5 234 235 236 54756 55225 55696 12812904 12977875 13144256 15.2971 15.3297 15.3623 6.1622 6.1710 6.1797 4.27350 4.25532 4.23729 735.13 738.27 741.42 43005.3 43373.6 43743.5 237 238 239 56169 56644 57121 13312053 13481272 13651919 15.3948 15.4272 15.4596 6.1885 6.1972 6.2058 4.21941 4.20168 4.18410 744.56 747.70 750 . 84 44115.0 44488.1 44862.7 240 241 243 57600 58081 58564 59049 13824000 13997521 14172488 14348907 15.4919 15.5242 15.5563 15.5885 6.2145 6.2231 6 . 2317 6.2403 4.16667 4.14938 4.13223 4.11523 753.98 757.12 760.27 763.41 45238.9 45616.7 45996.1 46377.0 244 245 246 59536 60025 60516 14526784 14706125 14886936 15.6205 15.6525 15.6844 6.2488 6.2573 6.2658 4.09836 4.08163 4.06504 766.55 769.69 772.83 46759.5 47143.5 47529.2 247 248 249 61009 61504 62001 15069223 15252992 15438249 15.7162 15.7480 15.7797 6.2743 6.2828 6.2912 4.04858 4.03226 4.01606 775.97 779.11 782.26 47916.4 48305.1 48695.5 FUNCTIONS OF THE NATURAL NUMBERS 57 No. Square Cube Square Root Cube Root 1000 x Recip. Circum. Area 250 251 252 253 62500 63001 63504 64009 15625000 15813251 16003008 16194277 15.8114 15.8430 15.8745 15.9060 6.2996 6.3080 6.3164 6.3247 4.00000 3.98406 3.96825 3.95257 785.40 788.54 791.68 794.82 49087.4 49480.9 49875.9 50272.6 254 255 256 64516 65025 65536 16387064 16581375 16777216 15.9374 15.9687 16.0000 6.3330 6.3413 6.3496 3.93701 3.92157 3.90625 797.96 801.11 804.25 50670.7 51070.5 51471.9 257 258 259 66049 66564 67081 16974593 17173512 17373979 16.0312 16.0624 16.0935 6.3579 6.3661 6.3743 3.89105 3.87597 3.86100 807.39 810.53 813.67 51874.8 52279.2 52685.3 260 261 262 263 67600 68121 68644 69169 17576000 17779581 17984728 18191447 16.1245 16.1555 16.1864 16.2173 6.3825 6.3907 6.3988 6.4070 3.84615 3.83142 3.81679 3.80228 816.81 819.96 823.10 826.24 53092.9 53502.1 53912.9 54325.2 264 265 266 69696 70225 70756 18399744 18609625 18821096 16.2481 16.2788 16.3095 6.4151 6.4232 6.4312 3.78788 3.77358 3.75940 829.38 832.52 835.66 54739.1 55154.6 55571.6 267 268 269 71289 71824 72361 19034163 19248832 19465109 16.3401 16.3707 16.4012 6.4393 6.4473 6.4553 3.74532 3.73134 3.71747 838.81 841.95 845.09 55990.2 56410.4 56832.2 270 271 272 273 72900 73441 73984 74529 19683000 19902511 20123648 20346417 16.4317 16.4621 16.4924 16.5227 6.4633 6.4713 6.4792 6.4872 3.70370 3.69004 3.67647 3.66300 848.23 851.37 854.51 857.66 57255.5 57680.4 58106.9 58534.9 274 275 276 75076 75625 76176 20570824 20796875 21024576 16.5529 16.5831 16.6132 6.4951 6.5030 6.5108 3.64964 3.63636 3.62319 860.80 863.94 867.08 58964.6 59395.7 59828.5 277 278 279 76729 77284 77841 21253933 21484952 21717639 16.6433 16.6733 16.7033 6.5187 6.5265 6.5343 3.61011 3.59712 3.58423 870.22 873.36 876.50 60262.8 60698.7 61136.2 280 281 282 283 78400 78961 79524 80089 21952000 22188041 22425768 22665187 16.7332 16.7631 16.7929 16.8226 6.5421 6.5499 6.5577 6.5654 3.57143 3.55872 3.54610 3.53357 879.65 882.79 885.93 889.07 61575.2 62015.8 62458.0 62901.8 284 285 286 80656 81225 81796 22906304 23149125 23393656 16.8523 16.8819 16.9115 6.5731 6.5808 6.5885 3.52113 3.50877 3.49650 892.21 895.35 898.50 63347.1 63794.0 64242.4 287 288 289 82369 82944 83521 23639903 23887872 24137569 16.9411 16.9706 17.0000 6.5962 6.6039 6.6115 3.48432 3.47222 3.46021 901.64 904.78 907.92 64692.5 65144.1 65597.2 290 291 292 293 84100 84681 85264 85849 24389000 24642171 24897088 25153757 17.0294 17.0587 17.0880 17.1172 6.6191 6.6267 6.6343 6.6419 3.44828 3.43643 3.42466 3.41297 911.06 914.20 917.35 920.49 66052.0 66508.3 66966.2 67425.6 294 295 296 86436 87025 87616 25412184 25672375 25934336 17.1464 17.1756 17.2047 6.6494 6.6569 6.6644 3.40136 3.38983 3.37838 923.63 926.77 929.91 67886.7 68349.3 68813.4 297 298 299 88209 88804 89401 26198073 26463592 26730899 17.2337 17.2627 17.2916 6.6719 6.6794 6.6869 3.36700 3.35570 3.34448 933.05 936.19 939.34 69279.2 69746.5 70215.4 FUNCTIONS OF THE NATURAL NUMBERS No. Sauare Cube Square Root Cube Root 1000 x Recip. No.=Dia. Circurn. Area 300 301 302 303 90000 90601 91204 91809 27000000 27270901 27543608 27818127 17.3205 17.3494 17.3781 17.4069 6.6943 6.7018 6.7092 6.7166 3.33333 3.32226 3.31126 3.30033 942.48 945.62 948.76 951.90 70685.8 71157.9 71631.5 72106.6 304 305 306 92416 93025 93636 28094464 28372625 28652616 17.4356 17.4642 17.4929 6.7240 6.7313 6.7387 3.28947 3.27869 3.26797 955.04 958.19 961.33 72583.4 73061.7 73541.5 307 308 309 94249 94864 95481 28934443 29218112 29503629 17.5214 17.5499 17.5784 6.7460 6.7533 6.7606 3.25733 3.24675 3.23625 964.47 967.61 970.75 74023.0 74506.0 74990.6 310 311 312 313 96100 96721 97344 97969 29791000 30080231 30371328 30664297 17.6068 17.6352 17.6635 17.6918 6.7679 6.7752 6.7824 6.7897 3.22581 3.21543 3.20513 3.19489 973.89 977.04 980.18 983.32 75476.8 75964.5 76453.8 76944.7 314 315 316 98596 99225 99856 30959144 31255875 31554496 17.7200 17.7482 17.7764 6.7969 6.8041 6.8113 3.18471 3.17460 3.16456 986.46 989.60 992.74 77437.1 77931.1 78426.7 317 318 319 100489 101124 101761 31855013 32157432 32461759 17.8045 17.8326 17.8606 6.8185 6.8256 6.8328 3.15457 3.14465 3.13480 995.88 999.03 1002.2 78923.9 79422.6 79922.9 320 321 322 323 102400 103041 103684 104329 32768000 33076161 33386248 33698267 17.8885 17.9165 17.9444 17.9722 6.8399 6.8470 6.8541 6.8612 3.12500 3.11527 3.10559 3.09598 1005.3 1008.5 1011.6 1014.7 80424.8 80928.2 81433.2 81939.8 324 325 326 104976 105625 106276 34012224 34328125 34645976 18.0000 18.0278 18.0555 6.8683 6.8753 6.8824 3.08642 3.07692 3.06748 1017.9 1021.0 1024.2 82448.0 82957.7 83469.0 327 328 329 106929 107584 108241 34965783 35287552 35611289 18.0831 18.1108 18.1384 6.8894 6.8964 6.9034 3.05810 3.04878 3.03951 1027.3 1030.4 1033.6 83981.8 84496.3 85012.3 330 331 332 333 108900 109561 110224 110889 35937000 36264691 36594368 36926037 18.1659 18.1934 18.2209 18.2483 6.9104 6.9174 6.9244 6.9313 3.03030 3.02115 3.01205 3.00300 1036.7 1039.9 1043.0 1046.2 85529.9 86049.0 86569.7 87092.0 334 335 336 111556 112225 112896 37259704 37595375 37933056 18.2757 18.3030 18.3303 6.9382 6.9451 6.9521 2.99401 2.98507 2.97619 1049.3 1052.4 1055.6 87615.9 88141.3 88668.3 337 338 339 113569 114244 114921 38272753 38614472 38958219 18.3576 18.3848 18.4120 6.9589 6.9658 6.9727 2.96736 2.95858 2.94985 1058.7 1061.9 1065.0 89196.9 89727.0 90258.7 340 341 342 343 115600 116281 116964 117649 39304000 39651821 40001688 40353607 18.4391 18.4662 18.4932 18.5203 6.9795 6.9864 6.9932 7.0000 2.94118 2.93255 2.92398 2.91545 1068.1 1071.3 1074.4 1077.6 90792.0 91326.9 91863.3 92401.3 344 345 346 118336 119025 119716 40707584 41063625 41421736 18.5472 18.5742 18.6011 7.0068 7.0136 7.0203 2.90698 2.89855 2.89017 1080.7 1083.8 1087.0 92940.9 93482.0 94024.7 347 348 349 120409 121104 121801 41781923 42144192 42508549 18.6279 18.6548 18.6815 7.0271 7.0338 7.0406 2.88184 2.87356 2.86533 1090.1 1093.3 1096.4 94569.0 95114.9 95662.3 FUNCTIONS OF THE NATURAL NUMBERS 59 No. Square Cube Square Root Cube Root 1000 x Recip. Circum. Area 350 351 352 353 122500 123201 123904 124609 42875000 43243551 43614208 43986977 18.7083 18.7350 18.7617 18.7883 7.0473 7.0540 7.0607 7.0674 2.85714 2.84900 2.84091 2.83286 1099.6 1102.7. 1105.8 1109.0 96211.3 96761.8 97314.0 97867.7 354 355 356 125316 126025 126736 44361864 44738875 45118016 18.8149 18.8414 18.8680 7.0740 7.0807 7.0873 2.82486 2.81690 2.80899 1112.1 1115.3 1118.4 98423.0 98979.8 99538.2 357 358 359 127449 128164 128881 45499293 45882712 46268279 18.8944 18.9209 18.9473 7.0940 7.1006 7.1072 2.80112 2.79330 2.78552 1121.5 1124.7 1127.8 100098 100660 101223 360 361 362 363 129600 130321 131044 131769 46656000 47045881 47437928 47832147 18.9737 19.0000 19.0263 19.0526 7.1138 7.1204 7.1269 7.1335 2.77778 2.77008 2.76243 2.75482 1131.0 1134.1 1137.3 1140.4 101788 102354 102922 103491 364 365 366 132496 133225 133956 48228544 48627125 49027896 19.0788 19.1050 19.1311 7.1400 7.1466 7.1531 2.74725 2.73973 2.73224 1143.5 1146.7 1149.8 104062 104635 105209 367 368 369 134689 135424 136161 49430863 49836032 50243409 19.1572 19.1833 19.2094 7.1596 7.1661 7.1726 2.72480 2.71739 2.71003 1153.0 1156.1 1159.2 105785 106362 106941 370 371 372 373 136900 137641 138384 139129 50653000 51064811 51478848 51895117 19.2354 19.2614 19.2873 19.3132 7.1791 7.1855 7.1920 7.1984 2.70270 2.69542 2.68817 2.68097 1162.4 1165.5 1168.7 1171.8 107521 108103 108687 109272 374 375 376 139876 140625 141376 52313624 52734375 53157376 19.3391 19.3649 19.3907 7.2048 7.2112 7.2177 2.67380 2.66667 2.65957 1175.0 1178.1 1181.2 109858 110447 111036 377 378 379 142129 142884 143641 53582633 54010152 54439939 19.4165 19.4422 19.4679 7.2240 7.2304 7.2368 2.65252 2.64550 2.63852 1184.4 1187.5 1190.7 111628 112221 112815 380 381 382 383 144400 145161 145924 146689 54872000 55306341 55742968 56181887 19.4936 19.5192 19.5448 19.5704 7.2432 7.2495 7.2558 7.2622 2.63158 2.62467 2.61780 2.61097 1193.8 1196.9 1200.1 1203.2 113411 114009 114608 115209 384 385 386 147456 148225 148996 56623104 57066625 57512456 19.5959 19.6214 19.6469 7.2685 7.2748 7.2811 2.60417 2.59740 2.59067 1206.4 1209.5 1212.7 115812 116416 117021 387 388 389 149769 150544 151321 57960603 58411072 58863869 19.6723 19.6977 19.7231 7.2874 7.2936 7.2999 2.58398 2.57732 2.57069 1215.8 1218.9 1221.1 117628 118237 118847 390 391 392 393 152100 152881 153664 154449 59319000 59776471 60236288 60698457 19.7484 19.7737 19-7990 19.8242 7.3061 7.3124 7.3186 7.3248 2.56410 2.55755 2.55102 2.54453 1225.2 1228.4 1232.5 1234.6 119459 120072 120687 121304 394 395 396 155236 156025 156816 61162984 61629875 62099136 19.8494 19.8746 19.8997 7.3310 7.3372 7.3434 2.53807 2.53165 2.52525 1237.8 1240.9 1244.1 121922 122542 123163 397 398 399 157609 158404 159201 62570773 63044792 63521199 19.9249 19.9499 19.9750 7.3496 7.3558 7.3619 2.51889 2.51256 2.50627 1247.2 1250.4 1253.5 123786 124410 125036 60 FUNCTIONS OF THE NATURAL NUMBERS No. Square Cube Square Root Cube Root 1000 x Recip. No.=: Dia. Circum. Area 400 401 402 403 160000 160801 161604 162409 64000000 64481201 64964808 65450827 20.0000 20.0250 20.0499 20.0749 7.3681 7.3742 7.3803 7.3864 2.50000 2.49377 2.48756 2.48139 1256.6 1259.8 1262.9 1266.1 125664 126293 126923 127556 404 405 406 163216 164025 164836 65939264 66430125 66923416 20.0998 20.1246 20.1494 7.3925 7.3986 7.4047 2.47525 2.46914 2.46305 1269.2 1272.3 1275.5 128190 128825 129462 407 408 409 165649 166464 167281 67419143 67917312 68417929 20.1742 20.1990 20.2237 7.4108 7.4169 7.4229 2.45700 2.45098 2.44499 1278.6 1281.8 1284.9 130100 130741 131382 410 411 412 413 168100 168921 169744 170569 68921000 69426531 69934528 70444997 20.2485 20.2731 20.2978 20.3224 7.4290 7.4350 7.4410 7.4470 2.43902 2.43309 2.4271S 2.42131 1288.1 1291.2 1294.3 1297.5 132025 132670 133317 133965 414 415 416 171396 172225 173056 70957944 71473375 71991296 20.3470 20.3715 20.3961 7.4530 7.4590 7.4650 2.41546 2.40964 2.40385 1300.6 1303.8 1306.9 134614 135265 135918 417 418 419 173889 174724 175561 72511713 73034632 73560059 20.4206 20.4450 20.4695 7.4710 7.4770 7.4829 2.39808 2.39234 2.38663 1310.0 1313.2 1316.3 136572 137228 137885 420 421 422 423 176400 177241 178084 178929 74088000 74618461 75151448 75686967 20.4939 20.5183 20.5426 20.5670 7.4889 7.4948 7.5007 7.5067 2.38095 2.37530 2.36967 2.36407 1319.5 1322.6 1325.8 1328.9 138544 139205 139867 140531 424 425 426 179776 180625 181476 76225024 76765625 77308776 20.5913 20.6155 20.6398 7.5126 7.5185 7.5244 2.35849 2.35294 2.34742 1332.0 1335.2 1338.3 141196 141863 142531 427 428 429 182329 183184 184041 77854483 78402752 78953589 20.6640 20.6882 20.7123 7.5302 7.5361 7.5420 2.34192 2.33645 2.33100 1341.5 1344.6 1347.7 143201 143872 144545 430 431 432 433 184900 185761 186624 187489 79507000 80062991 80621568 81182737 20.7364 20.7605 20.7846 20.8087 7.5478 7.5537 7.5595 7.5654 2.32558 2.32019 2.31481 2.30947 1350.9 1354.0 1357.2 1360.3 145220 145896 146574 147254 434 435 436 188356 189225 190096 81746504 82312875 82881856 20.8327 20.8567 20.8806 7.5712 7.5770 7.5828 2.30415 2.29885 2.29358 1363.5 1366.6 1369.7 147934 148617 149301 437 438 439 190969 191844 192721 83453453 84027672 84604519 20.9045 20.9284 20.9523 7.5886 7.5944 7.6001 2.28833 2.28311 2.27790 1372.9 1376.0 1379.2 149987 150674 151363 440 441 442 443 193600 194481 195364 196249 85184000 85766121 86350888 86938307 20.9762 21.0000 21.0238 21.0476 7.6059 7.6117 7.6174 7.6232 2.27273 2.26757 2.26244 2.25734 1382.3 1385.4 1388.6 1391.7 152053 152745 153439 154134 444 445 446 197136 198025 198916 87528384 88121125 88716536 21.0713 21.0950 21.1187 7.6289 7.6346 7.6403 2.25225 2.24719 2.24215 1394.9 1398.0 1401.2 154830 155528 156228 447 448 449 199809 200704 201601 89314623 89915392 90518849 21.1424 21.1660 21.1896 7.6460 7.6517 7.6574 2.23714 2.23214 2.22717 1404.3 1407.4 1410.6 156930 157633 158337 FUNCTIONS OF THE NATURAL NUMBERS No. Square Cube Square Root Cube Root 1000 x Recip. No- :Dia. Circum. Area 450 451 452 453 202500 203401 204304 205209 91125000 91733851 92345408 92959677 21.2132 21.2368 21.2603 21.2838 7.6631 7.6688 7.6744 7.6801 2.22222 2.21729 2.21239 2.20751 1413.7 1416.9 1420.0 1423.1 159043 159751 160460 161171 454 455 456 206116 207025 207936 93576664 94196375 94818816 21.3073 21.3307 21.3542 7.6857 7.6914 7.6970 2.20264 2.19780 2.19298 1426.3 1429.4 1432.6 161883 162597 163313 457 458 459 208849 209764 210681 95443993 96071912 96702579 21.3776 21.4009 21.4243 7.7026 7.7082 7.7138 2.18818 2.18341 2.17865 1435.7 1438.9 1442.0 164030 164748 165468 460 461 462 463 211600 212521 213444 214369 97336000 97972181 98611128 99252847 21.4476 21.4709 21.4942 21.5174 7.7194 7.7250 7.7306 7.7362 2.17391 2.16920 2.16450 2.15983 1445.1 1448.3 1451.4 1454.6 166190 166914 167639 168365 464 465 466 215296 216225 217156 99897344 100544625 101194696 21.5407 21.5639 21.5870 7.7418 7.7473 7.7529 2.15517 2.15054 2.14592 1457.7 1460.8 1464.0 169093 169823 170554 467 468 469 218089 219024 219961 101847563 102503232 103161709 21.6102 21.6333 21.6564 7.7584 7.7639 7.7695 2.14133 2.13675 2.13220 1467.1 1470.3 1473.4 171287 172021 172757 470 471 472 473 220900 221841 222784 223729 103823000 104487111 105154048 105823817 21.6795 21.7025 21.7256 21.7486 7.7750 7.7805 7.7860 7.7915 2.12766 2.12314 2.11864 2.11416 1476.5 1479.7 1482.8 1486.0 173494 174234 174974 175716 474 475 476 224676 225625 226576 106496424 107171875 107850176 21.7715 21.7945 21.8174 7.7970 7.8025 7.8079 2.10970 2.10526 2.10084 1489.1 1492.3 1495.4 176460 177205 177952 477 478 479 227529 228484 229441 108531333 109215352 109902239 21.8403 21.8632 21.8861 7.8134 7.8188 7.8243 2.09644 2.09205 2.08768 1498.5 1501.7 1504,8 178701 179451 180203 480 481 482 483 230400 231361 232324 233289 110592000 111284641 111980168 112678587 21.9089 21.9317 21.9545 21.9773 7.8297 7.8352 7.8406 7.8460 2.08333 2.07900 2.07469 2.07039 1508.0 1511.1 1514.2 1517.4 180956 181711 182467 183225 484 485 486 234256 235225 236196 113379904 114084125 114791256 2 .0000 .0227 .0454 7.8514 7.8568 7.8622 2.06612 2.00186 2.05761 1520.5 1523.7 1526.8 183984 184745 185508 487 488 489 237169 238144 239121 115501303 116214272 116930169 2 .0681 .0907 .1133 7.8676 7.8730 7.8784 2.05339 2.04918 2.04499 1530.0 1533.1 1536.2 186272 187038. 187805. 490 491 492 493 240100 241081 242064 243049 117649000 118370771 119095488 119823157 2 2 .1359 .1585 .1811 .2036 7.8837 7.8891 7.8944 7.8998 2.04082 2.03666 2.03252 2.02840 1539.4 1542.5 1545.7 1548.8 188574 189345 190117 190890 494 495 496 244036 245025 246016 120553784 121287375 - 122023936 2 .2261 .2486 .2711 7.9051 7.9105 7.9158 2.02429 2.02020 2.01613 1551.9 1555.1 1558.2 191665 192442 193221 497 498 499 247009 248004 249001 122763473 123505992 124251499 2 .2935 .3159 .3383 7.9211 7.9264 7.9317 2.01207 2.00803 2.00401 1561.4 1564.5 1567.7 194000 194782 195565 FUNCTIONS OF THE NATURAL NUMBERS No. Square Cube Square Root Cube Root 1000 x Recip. No. Dia. Circum. Area 500 501 502 503 250000 251001 252004 253CT09 125000000 125751501 126506008 127263527 22.3607 22.3830 22.4054 22.4277 7.9370 7.9423 7.9476 7.9528 2.00000 1.99601 1.99203 1.98807 1570.8 1573.9 1577.1 1580.2 196350 197136 197923 198713 504 505 506 254016 255025 256036 128024064 128787625 129554216 22.4499 22 4722 22.4944 7.9581 7.9634 7.9686 1.98413 1.98020 1.97628 1583.4 1586.5 1589.6 199504 200296 201090 507 508 509 257049 258064 259081 130323843 131096512 131872229 22.5167 22.5389 22.5610 7.9739 7.9791 7.9843 1.97239 1.96850 1.96464 1592.8 1595.9 1599.1 201886 202683 203482 510 511 512 513 260100 261121 262144 263169 132651000 133432831 134217728 135005697 22.5832 22.6053 22.6274 22.6495 7.9896 7.9948 8.0000 8.0052 1.96078 1.95695 1.95312 1.94932 1602.2 1605.4 1608.5 1611.6 204282 205084 205887 206692 514 515 516 264196 265225 266256 135796744 136590875 137388096 22.6716 22.6936 22.7156 8.0104 8.0156 8.0208 1.94553 1.94175 1.93798 1614.8 1617.9 1621.1 207499 208307 209117 517 518 519 267289 268324 269361 138188413 138991832 139798359 22.7376 22.7596 22.7816 8.0260 8.0311 8.0363 1.93424 1.93050 1.92678 1624.2 1627.3 1630.5 20992S 210741 211556 520 521 522 523 270400 271441 272484 273529 140608000 141420761 142236648 143055667 22.8035 22.8254 22.8473 22.8692 8.0415 8.0466 8.0517 8.0569 1.92308 1.91939 1.91571 1.91205 1633.6 1636.8 1639.9 1643.1 212372 213189 214008 214829 524 525 526 274576 275625 276676 143877824 144703125 145531576 22.8910 22.9129 22.9347 8.0620 8.0671 8.0723 1.90840 1.90476 1.90114 1646.2 1649.3 1652.5 215651 216475 217301 527 528 529 277729 278784 279841 146363183 147197952 148035889 22.9565 22.9783 23.0000 8.0774 8.0825 8.0876 1.89753 1.89394 1.89036 1655.6 1658.8 1661.9 218128 218956 219787 530 531 532 533 280900 281961 283024 284089 148877000 149721291 150568768 151419437 23.0217 23.0434 23.0651 23.0868 8.0927 8.0978 8.1028 8.1079 1.88679 1.88324 1.87970 1.87617 1665.0 1668.2 1671.3 1674.5 220618 221452 222287 223123 534 535 536 285156 286225 287296 152273304 153130375 153990656 23.1084 23.1301 23.1517 8.1130 8.1180 8.1231 1.87266 1.86916 1.86567 1677.6 1680.8 1683.9 223961 224801 225642 537 538 539 288369 289444 290521 154854153 155720872 156590819 23.1733 23.1948 23.2164 8.1281 8.1332 8.1382 1.86220 1.85874 1.85529 1687.0 1690.2 1693.3 226484 227329 228175 540 541 542 543 291600 292681 293764 294849 157464000 158340421 159220088 160103007 23.2379 23.2594 23.2809 23.3024 8.1433 8.1483 8.1533 8.1583 1.85185 1.84843 1.84502 1.84162 1696.5 1699.6 1702.7 1705.9 229022 229871 230722 231574 544 545 546 295936 297025 298116 160989184 161878625 162771336 23.3238 23.3452 23.3666 8.1633 8.1683 8.1733 1.83824 1.83486 1.83150 1709.0 1712.2 1715.3 232428 233283 234140 547 548 549 299209 300304 301401 163667323 164566592 165469149 23.3880 23.4094 23.4307 8.1783 8.1833 8.1882 1.82815 1.82482 1.82149 1718.5 1721.6 1724.7 234998 235858 236720 FUNCTIONS OF THE NATURAL NUMBERS 68 No. Square Cube Square Root Cube Root 1000 x Recip. No.^Dia. Circum. Area 550 302500 166375000 23.4521 8.1932 1.81818 1727.9 237583 551 303601 167284151 23.4734 8.1982 1.81488 1731.0 238448 552 304704 168196608 23.4947 8.2031 1.81159 1734.2 239314 553 305809 169112377 23.5160 8.2081 1.80832 1737.3 240182 554 306916 170031464 23.5372 8.2130 1.80505 1740.4 241051 555 308025 170953875 23.5584 8.2180 1.80180 1743.6 241922 556 309136 171879616 23.5797 8.2229 1.79856 1746.7 242795 557 310249 172808693 23.6008 8.2278 1.79533 1749.9 243669 558 311364 173741112 23.6220 8.2327 1.79211 1753.0 244545 559 312481 174676879 23.6432 8.2377 1.78891 1756.2 245422 560 313600 175616000 23.6643 8.2426 1.78571 1759.3 246301 561 314721 176558481 23.6854 8.2475 1.78253 1762.4 247181 562 315844 177504328 23.7065 8.2524 1.77936 1765.6 248063 563 316969 178453547 23.7276 8.2573 1.77620 1768.7 248947 564 318096 179406144 23.7487 8.2621 1.77305 1771.9 249832 565 319225 180362125 23.7697 8.2670 1.76991 1775.0 250719 566 320356 181321496 23.7908 8.2719 1.76678 1778.1 251607 567 321489 182284263 23.8118 8.2768 1.76367 1781.3 252497 568 322624 183250432 23.8328 8.2816 1.76056 1784.4 253388 569 323761 184220009 23.8537 8.2865 1.75747 1787.6 254281 570 324900 185193000 23.8747 8.2913 1.75439 1790.7 255176 571 326041 186169411 23.8956 8.2962 1.75131 1793.8 256072 572 327184 187149248 23.9165 8.3010 1.74825 1797.0 256970 573 328329 188132517 23.9374 8.3059 1.74520 1800.1 257869 574 329476 189119224 23.9583 8.3107 1.74216 1803.3 258770 575 330625 190109375 23.9792 8.3155 1.73913 1806.4 259672 576 331776 191102976 24.0000 8.3203 1.73611 1809.6 260576 577 332929 192100033 24.0208 8.3251 1.73310 1812.7 261482 578 334084 193100552 24.0416 8.3300 1.73010 1815.8 262389 579 335241 194104539 24.0624 8.3348 1.72712 1819.0 263298 580 336400 195112000 24.0832 8.3396 1.72414 1822.1 264208 581 337561 196122941 24.1039 8.3443 1.72117 1825.3 265120 582 338724 197137368 24.1247 8.3491 1.71821 1828.4 266033 583 339889 198155287 24.1454 8.3539 1.71527 1831.6 266948 584 341056 199176704 24.1661 8.3587 1.71233 1834.7 267865 585 342225 200201625 24.1868 8.3634 1.70940 1837.8 268783 586 343396 201230056 24.2074 8.3682 1.70648 1841.0 269703 587 344569 202262003 24.2281 8.3730 1.70358 1844.1 270624 588 345744 203297472 24.2487 8.3777 1.70068 1847.3 271547 589 346921 204336469 24.2693 8.3825 1.69779 1850.4 272471 590 348100 205379000 24.2899 8.3872 1.69492 1853.5 273397 591 349281 206425071 24.3105 8.3919 1.69205 1856.7 274325 592 350464 207474688 24.3311 8.3967 1.68919 1859.8 275254 593 351649 208527857 24.3516 8.4014 1.68634 1863.0 276184 594 352836 209584584 24.3721 8.4061 1.68350 1866.1 277117 595 354025 210644875 24.3926 8.4108 1.68067 1869.2 278051 596 355216 211708736 24.4131 8.4155 1.67785 1872.4 278986 597 356409 212776173 24.4336 8.4202 1.67504 1875.5 279923 598 357604 213847192 24.4540 8.4249 1.67224 1878.7 280S62 599 358801 214921799 24.4745 8.4296 1.66945 1881.8 281802 64 FUNCTIONS OF THE NATURAL NUMBERS No. Square Cube Square Root Cube Root 1000 x Recip. No.=Dia. Circum. Area 600 601 602 603 360000 361201 362404 363609 216000000 217081801 218167208 219256227 24.4949 24.5153 24.5357 24.5561 8.4343 8.4390 8.4437 8.4484 I 1 1 1 .66667 .66389 .66113 .65837 1885.0 1888.1 1891.2 1894.4 282743 2836S7 284631 285578 604 605 606 364816 366025 367236 220348864 221445125 222545016 24.5764 24.5967 24.6171 8.4530 8.4577 8.4623 1 1 1 .65563 .65289 .65017 1897.5 1900.7 1903.8 286526 287475 288426 607 ' 608 609 368449 369664 370881 223648543 224755712 225866529 24.6374 24.6577 24.6779 8.4670 8.4716 8.4763 1.64745 1.64474 1.64204 1906.9 1910.1 1913.2 289379 290333 291289 610 611 612 613 372100 373321 374544 375769 226981000 228099131 229220928 230346397 24.6982 24.7184 24.7386 24.7588 8.4809 8.4856 8.4902 8.4948 1 1 1 1 .63934 .63666 .63399 .63132 1916.4 1919.5 1922.7 1925.8 292247 293206 294166 295128 614 615 616 376996 378225 379456 231475544 232608375 233744896 24.7790 24.7992 24.8193 8.4994 8.5040 8.5086 1 1 1 .62866 .62602 .62338 1928.9 1932.1 1935.2 296092 297057 298024 617 618 619 380689 381924 383161 234885113 236029032 237176659 24.8395 24.8596 24.8797 8.5132 8.5178 8.5224 1 1 1 .62075 .61812 .61551 1938.4 1941.5 1944.6 298992 299962 300934 620 621 622 623 384400 385641 386884 388129 238328000 239483061 240641848 241804367 24.8998 24.9199 24.9399 24.9600 8.5270 8.5316 8.5362 8.5408 1 1 1 1 .61290 .61031 .60772 .60514 1947.8 1950.9 1954.1 1957.2 301907 302882 303858 304836 624 625 626 389376 390625 391876 242970624 244140625 245314376 24.9800 25.0000 25.0200 8.5453 8.5499 8.5544 1 1 1 .60256 .60000 .59744 1960.4 1963.5 1966.6 305815 306796 307779 627 628 629 393129 394384 395641 246491883 247673152 248858189 25.0400 25.0599 25.0799 8.5590 8.5635 8.5681 1 1 1 .59490 .59236 .58983 1969.8 1972.9 1976.1 308763 309748 310736 630 631 632 633 396900 398161 399424 400689 250047000 251239591 252435968 253636137 25.0998 25.1197 25.1396 25.1595 8.5726 8.5772 8.5817 8.5862 1 1 1 1 .58730 .58479 .58228 .57978 1979.2 1982.4 1985.5 1988.6 311725 312715 313707 314700 634 635 636 401956 403225 404496 254840104 256047875 257259456 25.1794 25.1992 25.2190 8.5907 8.5952 8.5997 1 I 1 .57729 .57480 .57233 1991.8 1994.9 1998.1 315696 316692 317690 637 638 639 405769 407044 408321 258474853 259694072 260917119 25.2389 25.2587 25.2784 8.6043 8.6088 8.6132 1 1 1 .56986 .56740 .56495 2001.2 2004.3 2007.5 318690 319692 320695 640 641 642 643 409600 410881 412164 413449 262144000 263374721 264609288 265847707 25.2982 25.3180 25.3377 25.3574 8.6177 8.6222 8.6267 8.6312 1 1 1 1 .56250 .56006 .55763 .55521 2010.6 2013.8 2016.9 2020.0 321699" 322705 323713 324722 644 645 646 414736 416025 417316 267089984 268336125 269586136 25.3772 25.3969 25.4165 8.6357 8.6401 8.6446 1 1 1 .55280 .55039 .54799 2023.2 2026.3 2029.5 325733 326745 327759 647 648 649 418609 419904 421201 270840023 272097792 273359449 25.4362 25.4558 25.4755 8.6490 8.6535 8.6579 1 1 1 .54560 .54321 .54083 2032.6 2035.8 2038.9 328775 329792 330810 FUNCTIONS OF THE NATURAL NUMBERS No. Square Cube Square Root Cube Root 1000 x Recip. No. Dia. Circum. Area 650 051 652 653 422500 423801 425104 426409 274625000 275894451 277167808 278445077 26.4951 25.5147 25.5343 25.5539 8.6624 8.6668 8.6713 8.6757 1.53846 1.53610 1.53374 1.53139 2042.0 2045.2 2048.3 2051.5 331831 332853 333876 334901 654 655 656 427716 429025 430336 279726264 281011375 282300416 25.5734 25.5930 25.6125 8.6801 8.6845 8.6890 1.52905 1.52672 1.52439 2054.6 2057.7 2060.9 335927 336955 337985 657 658 659 431649 432964 434281 283593393 284890312 286191179 25.6320 25.6515 25.6710 8.6934 8.6978 8.7022 1.52207 1.51976 1.51745 2064.0 2067.2 2070.3 339016 340049 341084 660 661 662 663 435600 436921 438244 439569 287496000 288804781 290117528 291434247 25.6905 25.7099 25.7294 25.7488 8.7066 8.7110 8.7154 8.7198 1.51515 1.51286 1.51057 1.50830 2073.5 2076.6 2079.7 2082.9 342119 343157 344196 345237 664 665 666 440896 442225 443556 292754944 294079625 295408296 25.7682 25.7876 25.8070 8.7241 8.7285 8.7329 1.50602 1.50376 1.50150 2086.0 2089.2 2092.3 346279 347323 348368 667 668 669 444889 446224 447561 296740963 298077632 299418309 25.8263 25.8457 25.8650 8.7373 8.7416 8.7460 1.49925 1.49701 1.49477 2095.4 2098.6 2101.7 349415 350464 351514 670 671 672 673 448900 450241 451584 452929 300763000 302111711 303464448 304821217 25.8844 25.9037 25.9230 25.9422 8.7503 8.7547 8.7590 8.7634 1.49254 1.49031 1.48810 1.48588 2104.9 2108.0 2111.2 2114.3 352565 353618 354673 355730 674 675 676 454276 455625 456976 306182024 307546875 308915776 25.9615 25.9808 26.0000 8.7677 8.7721 8.7764 1.48368 1.48148 1.47929 2117.4 2120.6 2123.7 356788 357847 358908 677 678 679 458329 459684 461041 310288733 311665752 313046839 26.0192 26.0384 26.0576 8.7807 8.7850 8.7893 1.47710 1.47493 1.47275 2126.9 2130.0 2133.1 359971 361035 362101 680 681 682 683 462400 463761 465124 466489 314432000 315821241 317214568 318611987 26.0768 26.0960 26.1151 26.1343 8.7937 8.7980 8.8023 8.8066 1.47059 1.46843 1.46628 1.46413 2136.3 2139.4 2142.6 2145.7 363168 364237 365308 366380 684 685 686 467856 469225 470596 320013504 321419125 322828856 26.1534 26.1725 26.1916 8.8109 8.8152 8.8194 1.46199 1.45985 1.45773 2148.8 2152.0 2155.1 367453 368528 369605 687 688 689 471969 473344 474721 324242703 325660672 327082769 26.2107 26 . 2298 26.2488 8.8237 8.8280 8.8323 1.45560 1.45349 1.45138 2158.3 2161.4 2164.6 370684 371764 372845 690 691 692 693 476100 477481 478864 480249 328509000 329939371 331373888 332812557 26.2679 26.2869 26.3059 26.3249 8.8366 8.8408 8.8451 8.8493 1.44928 1.44718 1.44509 1.44300 2167.7 2170.8 2174.0 2177.1 373928 375013 376099 377187 694 695 696 481636 483025 484416 334255384 335702375 337153536 26.3439 26.3629 26.3818 8.8536 8.8578 8.8621 1.44092 .43885 .43678 2180.3 2183.4 2186.5 378276 379367 380459 697 698 699 485809 487204 488601 338608873 340068392 341532099 26.4008 26.4197 26.4386 8.8663 8.8706 8.8748 .43472 .43266 .43062 2189.7 2192.8 2196.0 381553 382649 383746 FUNCTIONS OF THE NATURAL NUMBERS No. Square Cube Square Root Cube Root 1000 x Recip. No.=Dia. Circum. Area 700 701 702 703 490000 491401 492804 494209 343000000 344472101 345948408 347428927 26.4575 26.4764 26.4953 26.5141 8.8790 8.8833 8 . 8875 8.8917 1.42857 1.42653 1.42450 1.42248 2199.1 2202.3 2205.4 2208.5 384845 385945 387047 388151 704 705 706 495616 497025 498436 348913664 350402625 351895816 26.5330 26.5518 26.5707 8.8959 8.9001 8.9043 1.42045 1.41844 1.41643 2211.7 2214.8 2218.0 389256 390363 391471 707 708 709 499849 501264 502681 353393243 354894912 356400829 26.5895 26.6083 26.6271 8.9085 8.9127 8.9169 1.41443 1.41243 1.41044 2221.1 0094 222Y.4 392580 393692 394805 710 711 712 713 504100 505521 506944 508369 357911000 359425431 360944128 362467097 26.6458 26.6646 26.6833 26.7021 8.9211 8.9253 8.9295 8.9337 1.40845 1.40647 1.40449 1.40252 2230.5 2233.7 2236.8 2240.0 395919 397035 398153 399272 714 715 716 509796 511225 512656 363994344 365525875 367061696 26.7208 26.7395 26.7582 8.9378 8.9420 8.9462 1.40056 1.39860 1.39665 2243.1 2246.2 2249.4 400393 401515 402639 717 718 719 514089 515524 516961 368601813 370146232 371694959 26.7769 26.7955 26.8142 8.9503 8.9545 8.9587 1.39470 1.39276 1.39082 2252.5 2255.7 2258.8 403765 404892 406020 720 721 722 723 518400 519841 521284 522729 373248000 374805361 376367048 377933067 26.8328 26.8514 26.8701 26.8887 8.9628 8.9670 8.9711 8.9752 1.38889 1.38696 1.38504 1.38313 2261.9 2265.1 2268.2 2271.4 407150 408282 409415 410550 724 725 726 524176 525625 527076 379503424 381078125 382657176 26.9072 26.9258 26.9444 8.9794 8.9835 8.9876 1.38122 1.37931 1.37741 2274.5 2277.7 2280.8 411687 412825 413965 727 728 729 528529 529984 531441 384240583 385828352 387420489 26.9629 26.9815 27.0000 8.9918 8.9959 9.0000 1.37552 1.37363 1.37174 2283.9 2287.1 2290.2 415106 416248 417393 730 731 732 733 532900 534361 535824 537289 389017000 390617891 392223168 393832837 27.0185 27.0370 27.0555 27.0740 9.0041 9.0082 9.0123 9.0164 1.36986 1.36799 1.36612 1.36426 2293.4 2296.5 2299.6 2302.8 418539 419686 420835 421986 734 735 736 538756 540225 541696 395446904 397065375 398688256 27.0924 27.1109 27.1293 9.0205 9 . 0246 9.0287 1.36240 1.36054 1.35870 2305.9 2309.1 2312.2 423138 424293 425447 737 738 739 543169 544644 546121 400315553 401947272 403583419 27.1477 27.1662 27.1846 9.0328 9.0369 9.0410 1.35685 1.35501 1.35318 2315.4 2318.5 2321.6 426604 427762 428922 740 741 742 743 547600 549081 550564 552049 405224000 406869021 408518488 410172407 27.2029 27.2213 27.2397 27.2580 9.0450 9.0491 9.0532 9.0572 1.35135 1.34953 1.34771 1.34590 2324.8 2327.9 2331.1 2334.2 430084 431247 432412 433578 744 745 746 553536 555025 556516 411830784 413493625 415160936 27.2764 27.2947 27.3130 9.0613 9.0654 9.0694 1.34409 1.34228 1.34048 2337.3 2340.5 2343.6 434746 435916 437087 747 748 749 558009 559504 561001 416832723 418508992 420189749 27.3313 27.3496 27.3679 9.0735 9.0775 9.0816 1.33869 1.33690 1.33511 2346.8 2349.9 2353.1 438259 439433 440609 FUNCTIONS OF THE NATURAL NUMBERS 67 No. Square Cube Square Root Cube Root 1000 x Recip. No. Dia. Circum. Area 750 751 752 753 562500 564001 565504 567009 421875000 423564751 425259008 426957777 27.3861 27.4044 27.4226 27.4408 9.0856 9.0896 9.0937 9.0977 1.33333 1.33156 1.32979 1.32802 2356.2 2359.3 2362.5 2365.6 441786 442965 444146 445328 754 755 756 568516 570025 571536 428661064 430368875 432081216 27.4591 27.4773 27.4955 9.1017 9.1057 9.1098 1.32626 1.32450 1.32275 2368.8 2371.9 2375.0 446511 447697 448883 757 758 759 573049 574564 576081 433798093 435519512 437245479 27.5136 27.5318 27.5500 9.1138 9.1178 9.1218 1.32100 1.31926 1.31752 2378.2 2381.3 2384.5 450072 451262 452453 760 761 762 763 577600 579121 580644 582169 438976000 440711081 442450728 444194947 27.5681 27.5862 27.6043 27.6225 9.1258 9.1298 9.1338 9.1378 1.31579 1.31406 1.31234 1.31062 2387.6 2390.8 2393.9 2397.0 453646 454841 456037 457234 764 765 766 583696 585225 586756 445943744 447697125 449455096 27.6405 27.6586 27.6767 9.1418 9.1458 9.1498 1.30890 1.30719 1.30548 2400.2 2403.3 2406.5 458434 459635 460837 767 768 769 588289 589824 591361 451217663 452984832 454756609 27.6948 27.7128 27.7308 9.1537 9.1577 9.1617 1.30378 1.30208 1.30039 2409.6 2412.7 2415.9 462041 463247 464454 770 771 772 773 592900 594441 595984 597529 456533000 458314011 460099648 461889917 27.7489 27.7669 27.7849 27.8029 9.1657 9.1696 9.1736 9.1775 1.29870 1.29702 1.29534 1.29366 2419.0 2422.2 2425.3 2428.5 465663 466873 468085 469298 774 775 776 599076 600625 602176 463684824 465484375 467288576 27.8209 27.8388 27.8568 9.1815 9.1855 9.1894 1.29199 1.29032 1.28866 2431.6 2434.7 2437.9 470513 471730 472948 777 778 779 603729 605284 606841 469097433 470910952 472729139 27.8747 27.8927 27.9106 9.1933 9.1973 9.2012 1.28700 1.28535 1.28370 2441.0 2444.2 2447.3 474168 475389 476612 780 781 782 783 608400 609961 611524 613089 474552000 476379541 478211768 480048687 27.9285 27.9464 27.9643 27.9821 9.2052 9.2091 9.2130 9.2170 1.28205 1.28041 1.27877 1.27714 2450.4 2453.6 2456.7 2459.9 477836 479062 480290 481510 784 785 786 614656 616225 617796 481890304 483736625 485587656 28.0000 28.0179 28.0357 9.2209 9.2248 9.2287 1.27551 1.27389 1.27226 2463.0 2466.2 2469.3 482750 483982 485216 787 788 789 619369 620944 622521 487443403 489303872 491169069 28.0535 28.0713 28.0891 9 2326 9.2365 9.2404 1.27065 1.26904 1.26743 2472.4 2475.6 2478.7 486451 487688 488927 790 791 792 793 624100 625681 627264 628849 493039000 494913671 496793088 498677257 28.1069 28.1247 28.1425 28.1603 9.2443 9.2482 9.2521 9.2560 1.26582 1.26422 1.26263 1.26103 2481.9 2485.0 2488.1 2491.3 490167 49140D 492652 493897 794 795 796 630436 632025 633616 500566184 502459875 504358336 28.1780 28.1957 28.2135 9.2599 9.2638 9.2677 1.25945 1.25786 1.25628 2494.4 2497.6 2500.7 495143 496391 497641 797 798 799 635209 636804 638401 506261573 508169592 510082399 28.2312 28.2489 28.2666 9.2716 9.2754 9.2793 1.25471 1.25313 1.25156 2503.8 2507.0 2510.1 498892 500145 501399 68 FUNCTIONS OF THE NATURAL NUMBERS No. Square Cube Square Root Cube Root 1000 x Recip. No. Dia. Circum. Area 800 801 802 803 640000 641601 643204 644809 512000000 513922401 515849608 517781627 28.2843 28.3019 28.3196 28.3373 9.2832 9.2870 9.2909 9.2948 1.25000 1.24844 1 . 24688 1.24533 2513.3 2516.4 2519.6 2522.7 502655 503912 505171 506432 804 805 806 646416 648025 649636 519718464 521660125 523606616 28.3549 28.3725 28.3901 9.2986 9.3025 9.3063 1.24378 1.24224 1.24069 2525.8 2529.0 2532.1 507694 508958 510223 807 808 809 651249 652864 654481 525557943 527514112 529475129 28.4077 28.4253 28.4429 9.3102 9.3140 9.3179 1.23916 1.23762 1.23609 2535.3 2538.4 2541.5 511490 512758 514028 810 811 812 813 656100 657721 659344 660969 531441000 533411731 535387328 537367797 28.4605 28.4781 28.4956 28.5132 9.3217 9.3255 9.3294 9.3332 1.23457 1.23305 1.23153 1.23001 2544.7 2547.8 2551.0 2554.1 515300 516573 517848 519124 814 815 816 662596 664225 665856 539353144 541343375 543338496 28.5307 28.5482 28.5657 9.3370 9.3408 9.3447 1.22850 1.22699 1.22549 2557.3 2560.4 2563.5 520402 521681 522962 817 818 819 667489 669124 670761 545338513 547343432 549353259 28.5832 28.6007 28.6182 9.3485 9.3523 9.3561 1.22399 1 22249 1.22100 2566.7 2569.8 2573.0 524245 525529 526814 820 821 822 823 672400 674041 675684 677329 551368000 553387661 555412248 557441767 28.6356 28.6531 28.6705 28.6880 9.3599 9.3637 9.3675 9.3713 1.21951 1.21803 1.21655 1.21507 2576.1 2579.2 2582.4 2585.5 528102 529391 530681 531973 824 825 826 678976 680625 682276 559476224 561515625 563559976 28.7054 28.7228 28.7402 9.3751 9.3789 9.3827 1.21359 1.21212 1.21065 2588.7 2591.8 2595.0 533267 534562 535858 827 828 829 683929 685584 687241 565609283 567663552 569722789 28.7576 28.7750 28.7924 9.3865 9.3902 9.3940 1.20919 1.20773 1.20627 2598.1 2601.2 2604.4 537157 538456 539758 830 831 832 833 688900 690561 692224 693889 571787000 573856191 575930368 578009537 28.8097 28.8271 28.8444 28.8617 9.3978 9.4016 9.4053 9.4091 1.20482 1.20337 1.20192 1.20048 2607.5 2610.7 2613.8 2616.9 541061 542365 543671 544979 834 835 836 695556 697225 698896 580093704 582182875 584277056 28.8791 28.8964 28.9137 9.4129 9.4166 9.4204 1.19904 1.19760 1.19617 2620.1 2623.2 2626.4 546288 547599 548912 837 838 839 700569 702244 703921 586376253 588480472 590589719 28.9310 28.9482 28.9655 9.4241 9.4279 9.4316 1.19474 1.19332 1.19190 2629.5 2632.7 2635.8 550226 551541 552858 840 841 842 843 705600 707281 708964 710649 592704000 594823321 596947688 599077107 28.9828 29.0000 29.0172 29.0345 9.4354 9.4391 9.4429 9.4466 1.19048 1.18906 1.18765 1.18624 2638.9 2642.1 2645.2 2648.4 554177 555497 556819 558142 844 845 846 712336 714025 715716 601211584 603351125 605495736 29.0517 29.0689 29.0861 9.4503 9.4541 9.4578 1.18483 1.18343 1.18203 2651.5 2654.6 2657.8 559467 560794 562122 847 848 849 717409 719104 720801 607645423 609800192 611960049 29.1033 29.1204 29.1376 9.4615 9.4652 9.4690 1.18064 1.17925 1.17786 2660.9 2664.1 2667.2 563452 564783 566116 FUNCTIONS OF THE NATURAL NUMBERS No. Square Cube Square Root Cube Root 1000 x Recip. No.^Dia. Circum. Area 850 851 852 853 722500 724201 725904 727609 614125000 616295051 618470208 620650477 29.1548 29.1719 29.1890 29.2062 9.4727 9.4764 9.4801 9.4838 1 1 1 1 .17647 .17509 .17371 .17233 2670.4 2673.5 2676.6 2679.8 567450 568786 570124 571463 854 855 856 729316 731025 732736 622835864 625026375 627222016 29.2233 29.2404 29.2575 9.4875 9.4912 9.4949 1 1 1 .17096 .16959 .16822 2682 9 2686.1 2689.2 572803 574146 575490 857 858 859 734449 736164 737881 629422793 631628712 633839779 29.2746 29.2916 29.3087 9.4986 9.5023 9.5060 1 1 1 .16686 .16550 .16414 2692.3 2695.5 2698.6 576835 578182 579530 860 861 862 863 739600 741321 743044 744769 636056000 638277381 640503928 642735647 29.3258 29.3428 29.3598 29.3769 9.5097 9.5134 9.5171 9.5207 1 1 1 1 .16279 .16144 .16009 .15875 2701.8 2704.9 2708.1 2711.2 580880 582232 583585 584940 864 65 866 746496 748225 749956 644972544 647214625 649461896 29.3939 29.4109 29.4279 9.5244 9.5281 9.5317 1 1 1 .15741 .15607 .15473 2714.3 2717.5 2720.6 586297 587655 589014 867 868 869 751689 753424 755161 651714363 653972032 656234909 29.4449 29.4618 29.4788 9.5354 9.5391 9.5427 1 1 1 .15340 .15207 .15075 2723.8 2726.9 2730.0 590375 591738 593102 870 871 872 873 756900 758641 760384 762129 658503000 660776311 663054848 665338617 29.4958 29.5127 29.5296 29.5466 9.5464 9.5501 9.5537 9.5574 1 1 1 1 .14943 .14811 .14679 .14548 2733.2 2736.3 2739.5 2742.6 594468 595835 597204 598575 874 875 876 763876 765625 767376 667627624 669921875 672221376 29.5635 29.5804 29.5973 9.5610 9.5647 9.5683 1 1 I .14416 .14286 .14155 2745.8 2748.9 2752.0 599947 601320 602696 877 878 879 769129 770884 772641 674526133 676836152 679151439 29.6142 29.6311 29.6479 9.5719 9.5756 9.5792 1 1 1 .14025 .13895 .13766 2755.2 2758.3 2761.5 604073 605451 606831 880 881 882 883 774400 7"76161 777924 779689 681472000 683797841 686128968 688465387 29.6648 29.6816 29.6985 29.7153 9.5828 9.5865 9.5901 9.5937 1 1 1 1 .13636 .13507 .13379 .13250 2764.6 2767.7 2770.9 2774.0 608212 609595 610980 612366 884 885 886 781456 783225 784996 690807104 693154125 695506456 29.7321 29.7489 29.7658 9.5973 9.6010 9.6046 1 1 1 .13122 .12994 .12867 2777.2 2780.3 2783.5 613754 615143 616534 887 888 889 786769 788544 790321 697864103 700227072 702595369 29.7825 29.7993 29.8161 9.6082 9.6118 9.6154 1 1 1 .12740 .12613 .12486 2786.6 2789.7 2792.9 617927 619321 620717 890 891 92 893 792100 793881 795664 797449 704969000 707347971 709732288 712121957 29.8329 29.8496 29.8664 29.8831 9.6190 9.6226 9.6262 9.6298 1 1 1 I .12360 .12233 .12108 .11982 2796.0 2799.2 2802.3 2805.4 622114 623513 624913 626315 894 895 896 799236 801025 802816 714516984 716917375 719323136 29.8998 29.9166 29.9333 9.6334 9.6370 9.6406 I 1 1 .11857 .11732 .11607 2808.6 2811.7 2814.9 627718 629124 630530 897 898 899 804609 806404 808201 721734273 724150792 726572699 29.9500 29.9666 29.9833 9.6442 9.6477 9.6513 1 1 ] .11483 .11359 .11235 2818.0 2821.2 2824.3 631938 633348 634760 FUNCTIONS OF THE NATURAL NUMBERS No. Square Cube Square Root Cube Root 1000 x Recip. No.=Dia. Circum. Area 900 901 902 903 810000 811801 813604 815409 729000000 731432701 733870808 736314327 30.0000 30.0167 30.0333 30.0500 9.6549 9.6585 9.6620 9.6656 1.11111 1.10988 1.10865 1.10742 2827.4 2830.6 2833.7 2836.9 636173 637587 639003 640421 904 905 906 817216 819025 820836 738763264 741217625 743677416 30.0666 30.0832 30.0998 9.6692 9.6727 9.6763 1.10619 1.10497 1.10375 2840.0 2843.1 2846.3 641840 643261 644683 907 908 909 822649 824464 826281 746142643 748613312 751089429 30.1164 30.1330 30.1496 9.6799 9.6834 9.6870 1.10254 1.10132 1.10011 2849.4 2852.6 2855.7 646107 647533 648960 910 911 912 913 828100 829921 831744 833569 753571000 756058031 758550528 761048497 30.1662 30.1828 30.1993 30.2159 9.6905 9.6941 9.6976 9.7012 1.09890 1.09769 1.09649 1.09529 2858.8 2862.0 2865.1 2868.3 650388 651818 653250 654684 914 915 916 835396 837225 839056 763551944 766060875 768575296 30.2324 30.2490 30.2655 9.7047 9.7082 9.7118 1.09409 1.09290 1.09170 2871.4 2874.6 2877.7 656118 657555 658993 917 918 919 840889 842724 844561 771095213 773620632 776151559 30.2820 30.2985 30.3150 9.7153 9.7188 9.7224 1.09051 1.08932 1.08814 2880.8 2884.0 2887.1 660433 661874 663317 920 921 922 923 846400 848241 850084 851929 778688000 781229961 783777448 786330467 30.3315 30.3480 30.3645 30.3809 9.7259 9.7294 9.7329 9.7364 1.08696 1.08578 1.08460 1.08342 2890.3 2893 . 4 2896.5 2899.7 664761 666207 667654 669103 924 925 926 853776 855625 857476 788889024 791453125 794022776 30.3974 30.4138 30.4302 9.7400 9.7435 9.7470 1.08225 1.08108 1.07991 2902.8 2906.0 2909.1 670554 672006 673460 927 928 929 859329 861184 863041 796597983 799178752 801765089 30.4467 30.4631 30.4795 9.7505 9.7540 9.7575 1.07875 1.07759 1.07643 2912.3 2915.4 2918.5 674915 676372 677831 930 931 932 933 864900 866761 868624 870489 804357000 806954491 809557568 812166237 30.4959 30.5123 30.5287 30.5450 9.7610 9.7645 9.7680 9.7715 1.07527 1.07411 1.07296 1.07181 2921.7 2924.8 2928.0 2931.1 679291 680752 682216 683680 934 935 936 872356 874225 876096 814780504 817400375 820025856 30.5614 30.5778 30.5941 9.7750 9.7785 9.7819 1.07066 1.06952 1.06838 2934.2 2937.4 2940.5 685147 686615 688084 937 938 939 877969 879844 881721 822656953 825293672 827936019 30.6105 30.6268 30.6-131 9.7854 9.7889 9.7924 1.06724 1.06610 1.06496 2943.7 2946.8 2950.0 689555 691028 692502 940 941 942 943 883600 885481 887364 889249 830584000 833237621 835896888 838561807 30.6594 30.6757 30.6920 30.7083 9.7959 9.7993 9.8028 9.8063 1.06383 1.06270 1.06157 1.06045 2953.1 2956.2 2959.4 2962.5 693978 695455 696934 698415 944 945 946 891136 893025 894916 841232384 843908625 846590536 30.7246 30.7409 30.7571 9.8097 9.8132 9.8167 1.05932 1.05820 1.05708 2965.7 2968.8 2971.9 699897 701380 702865 947 948 949 896809 898704 900601 849278123 851971392 854670349 30.7734 30.7896 30.8058 9.8201 9.8236 9.8270 1.05597 1.05485 1.05374 2975.1 2978.2 2981.4 70435? 705840 707330 FUNCTIONS OF THE NATURAL NUMBERS 71 No. Square Cube Square Root Cube Root 1000 x Recip. No.=rDia. Circum. Area 950 951 952 953 902500 904401 906304 908209 857375000 860085351 862801408 865523177 30.8221 30.S383 30.8545 30.8707 9.S305 9.8339 9.8374 9.8408 1.05263 1.05152 1.05042 1.04932 2984.5 2987.7 2990.8 2993.9 708822 710315 711809 713306 954 955 956 910116 912025 913936 868250664 870983875 873722816 30.8869 30.9031 30.9192 9.8443 9.8477 9.8511 1.04822 1.04712 1.04603 2997.1 3000.2 3003.4 714803 716303 717804 957 958 959 915849 917764 919681 876467493 879217912 881974079 30.9354 30.9516 30.9677 9.8546 9.8580 9.8614 1.04493 1.04384 1.04275 3006.5 3009.6 3012.8 719306 720810 722316 960 961 962 963 921600 923521 925444 927369 884736000 887503681 890277128 893056347 30.9839 31.0000 31.0161 31.0322 9.8648 9.8683 9.8717 9.8751 1.04167 1.04058 1.03950 1.03842 3015.9 3019.1 3022:2 3025.4 723823 725332 726842 728354 964 965 966 929296 931225 933156 895841344 898632125 901428696 31.0483 31.0644 31.0805 9.8785 9.8819 9.8854 1.03734 1.03627 1.03520 3028.5 3031.6 3034.8 729867 731382 732899 967 968 969 935089 937024 938961 904231063 907039232 909853209 31.0966 31.1127 31.1288 9.8888 9.8922 9.8956 1.03413 1.03306 1.03199 3037.9 3041.1 3044.2 734417 735937 737458 970 971 972 973 940900 942841 944784 946729 912673000 915498611 918330048 921167317 31.1448 31.1609 31.1769 31.1929 9.8990 9.9024 9.9058 9.9092 1.03093 1.02987 1.02881 1.02775 3047.3 3050.5 3053 . 6 3056.8 738981 740506 742032 743559 974 975 976 948676 950625 952576 924010424 926859375 929714176 31.2090 31.2250 31.2410 9.9126 9.9160 9.9194 1.02669 1.02564 1.02459 3059.9 3063.1 3066.2 745088 746619 748151 977 978 979 954529 956484 958441 932574833 935441352 938313739 31.2570 31.2730 31.2890 9.9227 9.9261 9.9295 1.02354 1.02249 1.02145 3069.3 3072.5 3075.6 749685 751221 752758 980 981 982 983 960400 962361 964324 966289 941192000 944076141 946966168 949862087 31.3050 31.3209 31.3369 31.3528 9.9329 9.9363 9.9396 9.9430 1.02041 1.01937 1.01883 1.01729 3078.8 3081.9 3085.0 3088.2 754296 755837 757378 758922 984 985 986 968256 970225 972196 952763904 955671625 958585256 31 .3688 31.3847 31.4006 9.9464 9.9497 9.9531 1.01626 1.01523 1.01420 3091.3 3094.5 3097.6 760466 762013 763561 987 988 989 974169 976144 978121 961504803 964430272 967361669 31.4166 31.4325 31.4484 9.9565 9.9598 9.9632 1.01317 1.01215 1.01112 3100.8 3103.9 3107.0 765111 766662 768214 990 91 992 993 980100 982081 984064 986049 970299000 973242271 976191488 979146657 31.4643 31.4802 31.4960 31.5119 9.9666 9.9699 9.9733 9.9766 1.01010 1.00908 1.00806 1.00705 3110.2 3113.3 3116.5 3119.6 769769 771325 772882 774441 994 995 996 988036 990025 992016 982107784 985074875 988047936 31.5278 31.5436 31.5595 9.9800 9.9833 9.9866 1.00604 1.00503 1.00402 3122.7 3125.9 3129.0 776002 777564 779128 997 998 999 994009 996004 998001 991026973 994011992 997002999 31.5753 31.5911 31.6070 9.9900 9.9933 9.9967 1.00301 1.00200 1.00100 3132.2 3135.3 3138.5 780693 782260 78382k 72 FUNCTIONS OF THE NATURAL NUMBERS Fifth Roots and Fifth Powers Root Power Root Power Root Power 1 2 3 1 32 243 34 35 36 45435424 52521875 60466176 67 68 69 1350125101 145393356S 1564031349 4 5 6 1024 3125 7776 37 38 39 69343957 79235168 90224199 70 71 72 1680700000 1804229351 1934917632 7 8 9 16807 32768 59049 40 41 42 102400000 115856201 130691231 73 74 75 2073071593 2219006624 2373046875 10 11 ia 100000 161051 248832 43 44 45 147008443 164916224 184528125 76 77 78 2535525376 2706784157 2887174368 13 14 15 371293 537824 759375 46 47 48 205962976 229345007 254803968 79 80 81 3077056399 3276800000 3486784401 16 17 18 1048576 1419857 . 1889568 49 50 51 282475249 312500000 345025251 82 83 84 3707398432 3939040643 4182119424 19 20 21 2476099 3200000 4084101 52 53 54 380204032 418195493 459165024 85 86 87 4437053125 4704270176 4984209207 22 23 24 5153632 6436343 7962624 55 56 57 503284375 550731776 601692057 88 89 90 5277319168 5584059449 5904900000 25 26 27 9765625 11881376 14348907 58 59 60 656356768 714924299 777600000 91 92 93 6240321451 6590815232 6956883693 28 29 30 17210368 20511149 24300000 61 62 63 844596301 916132832 992436543 94 95 96 7339040224 7737809375 8153726976 31 32 33 28629151 33554432 39135393 64 65 66 1073741824 1160290625 1252332576 97 98 99 8587340257 903920796S 9509900499 Square Roots and Cube Roots of Fractions Fract- Sq. rt. Cu. rt. Fract. Sq. rt. Cu. rt. Fract. Sq. rt. Cu. rt. 1-3 2-3 1-4 3-4 .57735 .69336 .81650 .87358 .50000 .62996 .86603 .90856 3-7 4-7 5-7 6-7 .65465 .75395 .75593 .82983 .84515 .89390 .92582 .94991 1-9 2-9 4-9 5-9 .33333 .48075 .47140 .60571 .66667 .76314 .74536 .82207 1-6 5-6 1-7 2-7 .40825 .55032 .91287 .94104 .37796 .52276 .53452 .65863 1-8 3-8 5-8 7-8 .35355 .50000 .61237 .72112 .79057 .85449 .93541 .95647 7-9 1-12 5-12 7-12 .88192 .91964 .28868 .43679 .64550 .74690 .76376 .83555 CONVERSION OF UNITS CONVERSION OF UNITS Decimal Equivalents of Fractions of One Inch. Fract- Dec. Fract. Dec. Fract. Dec. Fract. Dec. 1-64 .015625 17-64 .265625 33-64 .515625 1-32 .03125 9-32 .28125 17-32 .53125 3-64 .046875 19-64 .296875 35-64 .546875 1-16 .0625 5-16 .3125 9-16 .5625 5-64 .078125 21-64 .328125 37-64 .578125 3-32 .09375 11-32 .34375 19-32 .59375 7-64 .109375 23-64 .359375 39-64 .609375 1-8 .125 3-8 .375 5-8 .625 9-64 .140625 25-64 .390625 41-64 .640625 5-32 .15625 13-32 .40625 21-32 .65625 11-64 .171875 27-64 .421875 43-64 .671875 3-16 .1875 7-16 .4375 11-16 .6875 13-64 .203125 29-64 .453125 45-64 .703125 7-32 .21875 15-32 .46875 23-32 .71875 15-64 .234375 31-64 .484375 47-64 .734375 1-4 .25 1-2 .5 3-4 .75 49-64 25-32 51-64 13-16 53-64 27-32 55-64 7-8 57-64 29-32 59-64 15-16 61-64 31-32 63-64 .765625 .78125 .796875 .8125 .828125 .84375 .859375 .875 .890625 .90625 .921875 .9375 .953125 .96875 .984375 Inches in Decimal Parts of i i Foot. In. 1-16 1-8 3-16 1-4 5-16 3-8 7-16 1 2 .0833 .1667 .00521 .0885 .1719 .01042 .0938 .1771 .01562 .0990 .1823 .02083 .1042 .1875 .02604 .1094 .1927 .03125 .1146 .1979 .03646 .1198 .2031 3 4 5 .2500 .3333 .4167 .2552 .3385 .4219 .2604 .3438 .4271 .2656 .3490 .4323 .2708 .3542 .4375 .2760 .3594 .4427 .2813 .3646 .4479 .2865 .3698 .4531 6 7 8 .5000 .5833 .6667 .5052 .5885 .6719 .5104 .5938 .6771 .5156 .5990 .6823 .5208 .6042 .6875 .5260 .6094 .6927 .5313 .6146 .6979 .5365 .6198 .7031 9 10 11 .7500 .8333 .9167 .7552 .8385 .9219 .7604 .8438 .9271 .7656 .8490 .9323 .7708 .8542 .9375 .7760 .8594 .9427 .7813 .8646 .9479 .7865 .8698 .9531 In. 1-2 9-16 5-8 11-16 3-4 13-16 7-8 15-16 1 2 .04167 .1250 .2083 .04687 .1302 .2135 .05208 .1354 .2188 .05729 .1406 .2240 .06250 .1458 .2292 .06771 .1510 .2344 .07292 .1563 .2396 .07812 .1615 .2448 3 4 5 .2917 .3750 .4583 .2969 .3802 .4635 .3021 .3854 .4688 .3073 .3906 .4740 .3125 .3958 .4792 .3177 .4010 .4844 .3229 .4063 .4896 .3281 .4115 .4948 6 7 8 .5417 .6250 .7083 .5469 .6302 .7135 .5521 .6354 .7188 .5573 .6406 .7240 .5625 .6458 .7292 .5677 .6510 .7344 .5729 .6563 .7396 .5781 .6615 .7448 9 10 11 .7917 .8750 .9583 .7969 .8802 .9635 .8021 .8854 .9688 .8073 .8906 .9740 .8125 .8958 .9792 .8177 .9010 .9844 .8229 .9063 .9896 .8281 .9115 .9948 74 CONVERSION OF UNITS U. S. AND BRITISH UNITS Lengths. U. S. and British. mil = .001 inch. foot = 12 inches. yard = 3 feet = 36 inches. rod, pole, or perch = 5% yards =16^ feet. furlong = 40 rods = 220 yards = 660 feet. 1 statute mile = 8 furlongs=320 rods =1760 yards = 5280 feet. 1 link=7.92 inches. 1 chain=100 links=66 feet=4 rods. 1 fathom=6 feet=8 spans=72 inches= Yi 2 o cable's length. 1 nautical mile (U. S. Coast and Geodetic Survey)=1.1516 statute miles=6080.26 feet. Areas. U. S. and British. 1 circular mil=area circle 1 mil diameter=. 0000007854 square inches. 1 square foot=144 square inches. 100 square feet=l square. 1 square yard=9 square feet=1296 square inches. 1 acre= 10 square chains=4 roods=160 square rods =4840 square yards=43560 square feet. 1 section=l mile square=640 acres. Volumes and Capacity. U. S. 1 cubic foot=1728 cubic inches. 1 cubic yard=27 cubic feet. 1 perch of masonry=24.75 cubic feet= 16 ^ feetXlK feetXl foot 1 gallon (U. S. liquid)=4 quarts = 8 pints = 231 cubic inches. 1 bushel = 8 gallons (Dry meas.)=2150.42 cubic inches = 1.24445 cubic feet. 1 barrel = 31.5 gallons = 4.2 11 cubic feet. Weights. U. S. and British. Troy. 1 pound = 12 ounces = 240 pennyweight = 5760 grains. Avoirdupois. 1 pound =16 ounces = 256 drams =7000 grains. 1 ton (gross) = 20 hundredweight (gross) =2240 pounds. 1 ton (net) = 20 hundredweight = 2000 pounds. CONVERSION OF UNITS 75 METRIC UNITS Length. 1 meter = 10 decimeters =100 centimeters = 1000 millimeters. 1 kilometer =10 hectometers = 100 decameters = 1000 meters. Area. 1 square meter or centiare = 100 square decimeters =10000 square centimeters. 1 hectare = 100 ares =10000 centiares. I Volume and Capacity. 1 cubic meter, or stere=1000 cubic decimeters = 1,000,000 cubic centimeters. 1 liter=l cubic decimeter = 10 deciliters = 100 centiliters. 1 kiloliter, or stere = 10 hectoliters = 100 decaliters = 1000 liters Weights. 1 gram =10 decigrams = 100 centigrams. 1 kilogram = 10 hectograms = 100 decagrams =1000 grams. 1 tonne, or metric ton =10 quintals = 1000 kilograms. Note Abbreviations for the English units are those in common use. The following are used for the metric: mm millimeters km kilometer mg milligram cm centimeter hect hectare eg centigram m meter g gram kg kilogram Gallons are U. S. liquid measure. Pounds are Avoirdupois. Calorie is gram-centigrade unit. In the logarithm column, negative characteristics have been increased by 10 and are shown in black type. 76 CONVERSION OF UNITS METRIC TO U. S. AND U. S. TO METRIC Lengths No. Mm. to 64th of In. Cm. to In. M. to Ft. Km. to Miles 64th of In. to Mm. In. to Cm. Ft. to M. Miles to Km. j 9 5197 3937 3 2808 0.62137 0.3969 2.54 0. 3048 1.60935 o 5 0394 0.7874 6 5617 1.24274 0.7938 5.08 0.6096 3.21869 3 7 .5590 1.1811 9 .8425 1.86411 1.1906 7.62 0. 9144 4.82804 10 0787 1.5748 13 1233 2.48548 1.5875 10.16 1. 2192 6.43739 >-, 12 .5984 1.9685 16 4042 3.10685 1.9844 12.70 1. 5240 8.04674 6 15 .1181 2.3622 19, ,6850 3.72822 2.3813 15.24 1 ,8288 9.65608 7 ft 17 20 .6378 .1574 2.7559 3.1496 22 26 ,9658 2467 4.34959 4.97096 2.7781 3.1750 17.78 20.32 2 1336 4384 11.26543 12.87478 9 22 .6771 3.5433 29 ,5275 5.59233 3.5719 22.86 2 ,74:i2 14.48412 Areas Sq.Cm- Sq. M. Sq. M. Hect. Sq. Km. Sq. In. Sq. ft. Sq. Y. Acres Sq. Mile No. to to to to to to to to to to Sq. In. Sq. Ft- Sq. Y. Acres Sq. Mile Sq. Cm. Sq. M. Sq. M. Hect. Sq. Km. 1 0.155 2 0.310 3 0.465 10.7639 21.5277 32.2916 1.196 2.392 3.588 2.471 0.3861 6.4516 0.0929 0.8361 0.4047 2.59 4 942 7722 12.9032 0.1858 1.6723 0.8094 5.18 7 413 1.1583 19.3549 0.2787 2.5084 1.2141 7.77 4 620 43 0555 4.784 9.884 1.5444 25.8065 0.3716 3.3445 1.6188 10.36 5 775 53 8193 5.980 12.355 1.9305 32.2581 0.4645 4.1806 2.0235 12.95 6 0^30 64:5832 7.176 14.826 2.3166 38.7097 0.5574 5.0168 2.4282 15.54 7 1 085 75.3471 8.372 17.297 2.7027 45.1614 0.6503 5,8529 2.8329 18.13 8 1240 86.1109 9.568 19.768 3.0888 51.6130 0.7432 6.6890 3.2376 20.72 9 1 395 96 8748 10.764 22.239 3.4748 58.0646 0.8361 7.5252 3.6423 23.31 Volumes Cu. Cm. Cu. M. Cu. M. Litres No. to to to to Cu. In. Cu. Ft. Cu. Yds. Gal. Cu. In. Cu. Ft. Cu. Yds. Gal- Gal. to to to to to Cu. M. Cu. M- Litres Cu. M. 1 .0610 35.314 2 .1220 70.629 3 .1831 105.943 1.308 2.616 3.924 .2642 264.17 .5283 528.34 .7925 792.51 4 2441 141.258 5.232 1.0567 1056.68 5 .3051 176.572 6.540 1.3209 1320.85 6 .3661 211.887 7.848 1.5850 1585.02 16 387 .02832 .7646 3.7854 .00379 32.774 .05663 1.5291 7.5709 .00757 49.161 .08495 2.2937 11.3563 .01136 65 549 .11327 3.0582 15.1417 .01514 8L936 .14159 3.8228 18.9272 .01893 98.323 .16990 4.5874 22.7126 .02271 CONVERSION OF UNITS METRIC TO U. S. AND U. S. TO METRIC Weights 77 No, M S . to Gr. G. to Ounce Kg. to Lb. Tonnes to Tons of 2000 Ib Gr. to M S . Ounce to G. Lb. to Kg. Tons of 2000 Ib. to Tonnes 1 2 3 .01543 .03086 .04630 .03527 .07055 .10582 2.2046 4.4092 6.6139 1.1023 2.2046 3.3069 64.799 129.598 194.397 28.350 56.699 85.049 .4536 .9072 1.3608 .9072 1.8144 2.7215 4 5 6 .06173 .07716 .09259 .14110 .17637 .21164 8.8185 11.0231 13.2277 4.4092 5.5116 6.6139 259.196 323.995 388.794 113.398 141.748 170.097 1.8144 2.2680 2.7215 3.6287 4.5359 5.4431 7 8 9 .10803 .12346 .13889 .24692 .28219 .31747 15.4324 17.6370 19.8416 7.7162 8.8185 9.9208 453.592 518.391 583.190 198.467 226.796 255.146 3.1751 3.6287 4.0823 6.3503 7.2575 8.1647 Pressures No. Kg. per Lin. M. to Lb. per Lin. Ft. Kg. per Sq, Cm. to Lb. per Sq. In. KS- per Sq. M. to Lb. per Sq, Ft. Tonnes per Sq. M. to Tons 20001b per Sq. Ft. Lb. per Lin. Ft. to Kg. per Lin. M. Lb. per Sq. In. to KS. per Sq. Cm. Lb. per Tons 20001b Sq. Ft. per Sq. Ft. to to Kg- per Tonnes Sq. M. perSq. M. 1 2 3 .6720 1.3439 2.0159 14.223 28.447 42.670 .2048 .4096 .6145 .1024 .2048 .3072 1.4882 2.9763 4.4645 .07031 .14061 21092 4.8824 9.7648 14.6472 9.765 19.530 29.294 4 5 6 2.6879 3.3598 4.0318 56.894 71.117 85.340 .8193 1.0241 1.2289 .4096 .5120 .6145 5.9526 7.4408 8.9290 .28123 .35153 .42184 19.5296 24.4120 29.2944 39.059 48.824 58.589 7 8 9 4.7038 5.3757 6.0477 99.564 113.787 128.011 1.4337 1.6385 1.8433 .7169 .8193 .9217 10.4171 11.9053 13.3934 .49215 .56245 .63276 34.1769 39.0593 43.9417 68.354 78.119 87.883 Miscellaneous G. per Cu. Cm. No. to Lbs. per Cu. Ft. K S . per Cu. M. to Lbs- per Cu. Ft. Kg.-M. to Ft.-Lb. Metric H. P. to U. S. H. P. Lbs. per Cu. Ft. to G. per Cu. Cm. Lbs. per Cu. Ft. to Kg. per Cu. M. 16.018 32.037 48.056 Ft.-lb. to Kg.-M. U. S. H. P. to Metric H. P. 1 2 3 62.43 124.86 187.28 .06243 .12486 .18728 7.233 14.466 21.699 .9863 1.9726 2.9589 .01602 .03204 .04806 .1383 .2765 .4148 1.0139 2.0Z77 3.0416 4 5 6 249.71 312.14 374.57 .24971 .31214 .37457 28.932 36.165 43.398 3.9453 4.9316 5.9179 .06407 .08009 .09611 64.073 80.092 96.110 .5530 .6913 .8295 4.0555 5.0694 6.0832 7 8 9 437.00 499.43 561.85 .43700 .49943 .56185 50.631 57.864 65.097 6.9042 7.8905 8.8769 .11213 .12815 .14416 112.129 128.147 144.165 .9678 1.1060 1.2443 7.0971 8.1110 9.1249 78 CONVERSION FACTORS To reduce N units of: To M units of: Multiply N by: Log Angles Min. rad. .00029089 6.46373 Deg. rad. 7T .017453 .005556 8.24187 7.74473 Rad. sec. min. deg. 7T 206265. 3437.75 57.2958 .31831 5.31443 3.53627 1.75812 9.50285 7T rad. 3.14159 0.49715 Lengths In. mil. ft. yd. cm. 1000. .083333 .027778 2.54 3.00000 8.92082 8.44370 0.40483 Ft. in. yd. chain (Gunter's) mile cm. m. 12. .33333 .015152 .00018939 30.48 .3048 1.07918 9.52288 8.18046 6.27737 1.48402 9.48402 Yd. in. ft. mile m. 36. 3. .00056818 .9144 1.55630 0.47712 6.75449 9.96114 Mile in. ft. yd. km 1 . 63360. 5280. 1760. 1.60935 4.80182 3.72263 3.24551 0.20665 Link (Gunter's) Chain (Gunter's) in. ft. 7.92 66. 0.89873 1.81954 Cm. in. ft. .3937 .032808 9.59517 8.51598 M. in. ft. yd. 39.37 3.2808 1.0936 1.59517 0.51598 0.03886 Km. ft. mile 3280.8 .62137 3.51598 9.79335 Areas ^ Cir. mils sq. mils sq. mm. .7854 .00050671 9.89509 6.70476 Sq. in. cir. mils sq. ft. sq. cm. 1273240. .0069444 6.4516 6.10491 7.84164 0.80967 Sq. ft. sq. in. sq. yd. sq. m. 144. .11111 .0929 2.15836 9.04576 8.96803 Sq. yd. sq. ft. sq. m. 9. .83613 95424 9.92227 CONVERSION FACTORS 79 To reduce N units of: To M units of: Multiply N by; Log. Areas Con't. Acres sq. rt. sq. yd. sq. miles sq. m. hectares 43560. 4840. .0015625 4046.87 .404687 4.63909 3.68485 7.19382 3.60712 9.60712 Sq. miles acres hectares sq. km. 640. 259. 2.59 2.80618 2.41330 0.41330 Sq. mm. cir, mils sq. in. 1973.52 .00155 3.29524 7.19033 Sq. cm. sq. in. sq. ft. .155 .0010764 9.19033 7.03197 Sq. m'. sq. ft. sq. yd. acres 10.7639 1.196 .0002471 1.03197 0.07773 6.39288 Hectares acres sq. miles 2.471 .003861 0.39288 7.58670 Sq. km. sq. miles .3861 9.58670 Volumes Cu. in. cu. ft. cu. cm. gal. liters .0005787 16.387 .004329 .016387 6.76246 1.21450 7.63639 8.21450 Cu. ft. cu. in. cu. yd. cu. cm. cu. m. gal. liters 1728. .03778 28317. .028317 7.4805 28.317 3.23754 8.56864 4.45205 8.45205 0.87393 1.45205 Cu. yd. cu. in. cu. ft. acre-ft. cu. m. 46656. 27. .0006198 .76456 4.66891 1.43136 6.79228 9.88341 Gal. cu. in. cu. ft. cu. cm 1 , liters cu. m. 231. .13368 3785.4 3.7854 .0037854 43560. 1613.33 325851. 1233.49 2.36361 9.12607 3.57812 0.57812 7.57812 Acre-ft. cu. ft. cu. yd. gal. cu. m. 4.63909 3.20772 5 . 51302 3.09114 Cu. cm. cu. in. cu. ft. gal. .061023 .000035314 .00026417 8.78550 5.54795 6.42188 Cu. m. cu. ft. cu. yd. gal. 35.3145 1.3079 264.17 1.54795 0.11659 2.42188 SI) CONVERSION FACTORS To reduce N units of : To M units of : Multiply N by: Log. Volumes Con 't. Liters cu. in. cu. ft. gal. 61.023 .0.35314 .2*6417 1.78550 8.51795 9.42188 Water Discharges Cu. ft. per sec. gal. per min. gal. per hr. gal. per day acre-ft. per day miner's in. 448.83 26930. 646317. 1.9835 40. 2.65208 4.43024 5.81045 0.29743 1.60206 Cu. ft. per min. Cu. ft. per hr. gal. per hr. gal. per day 448.83 179.53 2.65208 2.25414 Cu. ft. per day gal. per min. gal. per hr. .005195 .31169 7.71559 9.49372 Gal. per sec. cu. ft. per hr. acre-ft. per day 481.25 .26515 2.68237 9.42349 Gal. per min. cu. ft. per sec. cu. ft. per hr. acre-ft. per day miner's in. .002228 8.0208 .004419 .08912 7.34792 0.90422 7.64534 8.94998 Miner's in. cu. ft. per sec. cu. ft. per min. gal. per min. .025 1.5 11.22 8.39794 0.17609 1.05002 1000000 gal. per day cu. ft. per sec. gal. per min. miner's in. 1.547 694.4 61.88 0.18955 2.84164 1.79155 Acre-ft. per day cu. ft. per sec. gal. per min. miner's in. .5042 226.3 20.17 9.70257 2.35466 1.30463 Weights Gr. g. .064799 8.81157 Ounce g. 437.5 .0625 28.3496 2.64098 8.79588 1.45255 Lb. ounce Is. 16. 453.593 .453593 1.20412 2.65667 9.65667 Ton (2240 Ib.) Ton (2000 Ib.) metric ton metric ton 1.01605 .90719 0.00691 9.95770 G. gr. ounce Ib. 15.43235 .035274 .0022046 1.18843 8.54745 7.34333 Kg. Metric ton Metric ton Ib. ton (2000 Ib.) ton (2240 Ib.) 2.2046 1.1023 .98421 0.34333 0.04230 9.99309 CONVERSION FACTORS 81 To reduce N units of: To M units of: Multiply N by : Log. Weights per Unit Length Lb. per in. Lb. per ft. G. per cm. Kg. per m. g. per cm. kg. per m. Ib. per in. Ib. per ft. 17.858 1.48816 .056 .67197 1.25183 0.17265 8.74817 9.82735 Densities Lb. per cu. in. Lb. per cu. ft. G. per cu. cm. Kg. per cu. m. g. per cu. cm. kg. per cu. m. Ib. per cu. in. Ib. per cu. ft. 27.6797 16.018 .036128 .062428 1.44216 1.20462 8.55784 8.79538 Water Volumes to Welgfits Cu. in. ounces Ib. g. .578 .03613 16.387 9.76196 8.55784 1.21450 Cu. ft. Ib. kg. 62.428 28.317 1.79538 1.45205 Cu. yd. Ib. 1686. 3.22675 Cu. cm. ounces Ib. g. .03527 .0022046 1. 8.54745 7.34333 0.00000 Cu. m. Ib. 2205. 3.34333 Gal. Ib. kg. 8.345 3.785 0.92145 0.57812 Liters Ib. 2.205 0.34333 Water Weights to Volumes Ounces cu. in. cu. cm. 1.73 28.35 0.23804 1.45255 Lb. cu. in. cu. ft. cu. cm. gal. liters 27.68 .016018 453.59 .1198 .45359 1.44216 8.20462 2.65667 9.07855 9.65667 G. cu. in. .06102 8.78550 Kg. cu. ft. gal. .0353 .2642 8.54795 9.42188 Forces Lb.-wt. G.-wt. dynes dynes 444793. 980.6 5.64816 2.9914& Dynes Ib.-wt. g.- wt. 2.248xlO- 6 .00102 4.35184 7.00851 CONVERSION FACTORS To reduce N units of: To M units of: Multiply N by: Loar. Pressures Lb. per sq. in. cm. of mercury in. of mercury ft. of water kg. per sq. cm. atmospheres 5.1712 2.0359 2.3067 .070307 .068041 0.71359 0.30875 0.36298 8.84700 8.83277 Lb. per sq. ft. ft. of water kg. per sq. m. .016018 4.8824 8.20462 0.68863 Ft. of water Ib. per sq. in. Ib. per sq. ft. kg. per sq. cm. atmospheres in. of mercury cm. of mercury .43353 62.43 .03048 . 0295 .88261 2.2412 9.63702 1.79538 8.48402 B 46979 9.94577 0.35048 In. of mercury Ib. per sq. in. ft. of water kg. per sq. cm. atmospheres .49119 1.133 .034534 .03342 9.69125 O.C5423 8.53S25 8.52402 7 0.53297 H. P. -sec. kg.-m. calories 76.04 178.15 1.88104 2.25078 H. P.-hr. ft.-lb. B. T. U. 1980000. 2545. 6.2966T 3.4056D Calorie ft.-lb. kg.-m. joule B. T. U. 3.087 .42685 4.1857 .003968 0.48959 9.6302T 0.62182 7.59861 B. T. U. ft.-lb. kg.-m. watt-hr. H. P.-hr. calorie 778.1 107.56 .293 .0003929 252. 2.89098 2.0316S 9.46703 6.59431 2.40140' Power Ft.-lb. per sec. watts H. P. kw. metric H. P. 1.356 .001818 .001356 .001843 0.13235 7.25964 7.13235 7.26562 Kg.-m. per sec. watts H.P. metric H.P. 9.81 .0315 .01333 0.99167 8.11896 8.12494 Watts ft.-lb. per sec. kg.-m. per sec. H.P. metric H.P. .7373 .10194 .00134 .00136 9.86765 9.00833 7.12729 7.13327 Kw. ft.-lb. per sec. ft.-lb. per min. kg.-m. per min. H.P. metric H.P. B. T. U. per min. B. T. U. per hr. 737.3 44239. 6116.4 1.341 1.359 56.86 3411.7 2.86765 4.64580 3.78650 0.12729 0.13327 1.75481 3.53297 H. P. ft.-lb. per sec. ft.-lb. per min. watts kw. metric H.P. B. T. U. per min. B. T. U. per hr. 550. 33000. 746. .746 1.0139 42.416 2545. 2.74036 4.51851 2.87271 9.87271 0.00598 1.62753 3.4056&' M CONVERSION FACTORS ' To reduce N units of: To M units of: Multiply N by: ^Log. Power Con't. Metric H. P. ft.lb. per sec. kg.-m. per sec. watts kw. H.P. 542.47 75. 735.75 .73575 .9863 2.73438 1.87506 2.86673 9.86673 9.99402 B. T. U. per. min. kw. H.P. .01759 .02358 8.24519 8.37247 B. T. U. per hr. kw. H.P. .000293 .000393 6.46703 6.59431 Angular Velocities Rad. per sec. rev. per sec. rev. per min. .15916 9.5496 9.20182 0.97997 Rad. per min. rev. per sec. .002653 7.42367 Rev. per min. rad. per sec. deg. per sec. .10472 6. 9.02003 0.77815 Deg. per sec. rev. per min. .16667 9.22185 Linear Velocities Ft;- per sec. m. per min. Ft. per min. miles per hr. 18.288 .01136 1.26217 8.05552 Miles per hr. ft. per min. m. per min. 88. 26.82 1.94448 1.42850 M. per min. ft. per sec. miles per hr. .0547 .03728 8.73783 8.57150 Temperatures Deg. C. (deg. Fah. 32) (Deg-. Fah. 32) deg. C. 1.8 .55556 0.25527 9.74473 Electrostatic to Electromagnetic Potential volts Capacitv microfarads Quantity coulombs Current amperes Resistance ohms 300. l.llllxlO-fi 3.333xlO-io 3.333xlO~ 10 9.X10 11 2.47712 4.04575 0.52287 0.52287 11.95424 WIRE AND SHEET METAL GAUGES In Decimals of an Inch & c 1 CS J5 . J T3 "ft "c rt w 8 d ii! 6 6 El te S3 -M t) '& 2 he o c oj c .~ i" /2 Q) C ,0 .S CJ S 'fi o i i < w w * c rill III 1 c j /~2~~ 2.22144 0.34663 * 0.07093 8.85080 \/7r^2 1.25331 0.09806 ir~\/2g 0.39166 9.59291 %/^FiT 0.797889.90194 fy-$-^-Tf* q 0.4789 9.68027 S/- 1.46459 0.16572 A9 ^ ' 4.60115 0.66287 b ^:^ 0.001766 7 . 24704 7T ^ 7T tf 2^7T~ 1 84526 0.26606 CO K r 2.14503 0.33143 bZ - 1.9428 0.28843 NOTE: Negative characteristics of logarithms have been increased by 10 and are shown in black type. * Metric units HYPERBOLIC FUNCTIONS S7 HYPERBOLIC FUNCTIONS Formulae *sinhw= e ?-*~. cosh w= e '--' 2 tanh u sinh u _ e" e u cosh u e u -\-e u sinh ( M) = sinh u tanh ( w) = tanh u cosh 1 u =ln (w-j-N/w 2 1) cosh 2 w sinh 2 u = l cosh ( w) = -f- cosh u sinh- 1 u = In (w-f -v/w 2 -f 1 tanh 1 u= In Since cosh u -f- sinh u = e u and cosh u sinh u=e~ ", the value of e" and e~ u are readily obtained from the tables for sinh u and cosh u. * Read hyperbolic sine of u Tanh u tt=0 to u= 2.29 u 1 2 3 4 5 6 7 8 9 .1 .2 .3 0000 .0997 .1974 .2913 .0100 .1096 .2070 .3004 .0200 .1194 .2165 .3095 .0300 .1293 .2260 .3185 .0400 .1391 .2355 .3275 .0500 .1489 .2449 .3364 .0599 .1587 .2543 .3452 .0699 .1684 .2636 .3540 .0798 .1781 .2729 .3627 .0898 .1878 .2821 .3714 .4 .5 .6 .3800 .4621 .5370 .3885 .4700 .5441 .3969 .4777 .5511 .4053 .4854 .5581 .4137 .4930 .5649 .4219 .5005 .5717 .4301 .5080 .5784 .4382 .5154 .5850 .4462 .5227 .5915 .4542 .5299 .5980 . 7 .8 .9 .6044 .6640 .7163 .6107 .6696 .7211 .6169 .6751 .7259 .6231 .6805 .7306 .6291 .6858 .7352 .6352 .6911 .7398 .6411 .6963 .7443 .6469 .7014 .7487 .6527 .7064 .7531 .6584 .7114 .7574 1.0 .1 2 '.3 .7616 .8005 .8337 .8617 .7658 .8041 .8367 .8643 .7699 .8076 .8397 .8668 .7739 .8110 .8426 .8693 .7779 .8144 .8455 .8717 .7818 .8178 .8483 .8741 .7857 .8210 .8511 .8764 .7895 .8243 .8538 .8787 .7932 .8275 .8565 .8810 .7969 .8306 .8591 .8832 .4 .5 .6 .8854 .9052 .9217 .8875 .9069 .9232 .8896 .9087 .9246 .8917 .9104 .9261 .8937 .9121 .9275 .8957 .9138 .9289 .8977 .9154 .9302 .8996 .9170 .9316 .9015 .9186 .9329 .9033 .9202 .9342. .7 .8 .9 .9354 .9468 .9562 .9367 .9478 .9571 .9379 .9488 .9579 .9391 .9498 .9587 .9402 .9508 .9595 .9414 .9518 .9603 .9425 .9527 .9611 .9436 .9536 .9619 .9447 .9545 .9626 .9458 .9554 .9633 2.0 .1 2 .9640 .9705 .9757 .9647 .9710 .9762 .9654 .9716 .9767 .9661 .9722 .9771 .9668 .9727 .9776 .9674 .9732 .9780 .9680 .9738 .9785 .9687 .9743 .9789 .9693 .9748 .9793 .9699 .9753 .9797 HYPERBOLIC FUNCTIONS Sinh u = to t< = 4. .0000 .0100 .0200 .0300 .0400 .1 .1002 .1102 .1203 .1304 .1405 .2 .2013 .2115 .2218 .2320 .2423 .3 .3045 .3150 .3255 .3360 .3466 .4 .4108 .4216 .4325 .4434 .4543 .5 .5211 .5324 .5438 .5552 .5666 .6 .6367 .6485 .6605 .6725 .6846 .7 .7586 .7712 .7838 .7966 .8094 .8 .8881 .9015 .9150 .9286 .9423 .9 1.0265 1.0409 1.0554 1.0700 1.0847 1.0 1.1752 1.1907 1.2063 1.2220 1.2379 .1 1.3356 1.3524 1.3693 1.3863 1.4035 .2 1.5095 1.5276 1.5460 1.5645 1.5831 .3 1.6984 1.7182 1.7381 1.7583 1.7786 .4 1.9043 1.9259 1.9477 1.9697 1.9919 .5 2.1293 2.1529 2.1768 2.2008 2.2251 .6 2.3756 2.4015 2.4276 2.4540 2.4806 .7 2.6456 2.6740 2.7027 2.7317 2.7609 .8 2.9422 2.9734 3.0049 3.0367 3.0689 .9 3.2682 3.3025 3.3372 3.3722 3.4075 2.0 3.6269 3.6647 3.7028 3.7414 3.7803 .1 4.0219 4.0635 4.1056 4.1480 4.1909 .2 4.4571 4.5030 4.5494 4.5962 4.6434 .3 4.9370 4.9876 5.0387 5.0903 5.1425 .4 5.4662 5.5221 5.5785 5.6354 5.6929 .5 6.0502 6.1118 6.1741 6.2369 6.3004 .6 6.6947 6.7628 6.8315 6.9009 6.9709 .7 7.4063 7.4814 7.5572 7.6338 7.7112 .8 8.1919 8.2749 8.3586 8.4432 8.5287 .9 9.0596 9.1512 9.2437 9.3371 9.4315 3.0 10.018 10.119 10.221 10.324 10.429 .1 11.076 11.188 11.301 11.415 11.530 .2 12.246 12.369 12.494 12.620 12.747 .3 13.538 13.674 13.812 13.951 14.092 .4 14.965 15.116 15.268 15.422 15.577 .5 16.543 16.709 16.877 17.047 17.219 .6 18.285 18.470 18.655 18.843 19.033 .7 20.211 20.415 20.620 20.828 21.037 .8 22.339 22.564 22.791 23.020 23.252 .9 24.691 24.939 25.190 25.444 25.700 4.0 27.290 27.564 27.842 28.122 28.404 .1 30.162 30.465 30.772 31.081 31.393 .2 33.336 33.671 34.009 34.351 34.697 .3 36.843 37.214 37.588 37.966 38.347 .4 40.719 41.129 41.542 41.960 42.382 .5 45.003 45.455 45.912 46.374 46.840 .6 49.737 50.237 50.742 51.252 51.767 7 54.969 55.522 56.080 56.643 57.213 .8 60.751 61.362 61.979 62.601 63.231 .9 67.141 67.816 68.496 69.186 69.882 .0500 .0600 .0701 .0801 .0901 .1506 .1607 .1708 .1810 .1911 .2526 .2629 .2733 .2837 .2941 .3572 .3678 .3785 .3892 .4000 .4653 .4764 .4875 .4986 .5098 .5782 .5897 .6014 .6131 .6248 .6967 .7090 .7213 .7336 .7461 .8223 .8353 .8484 .8615 .8748 .9561 .9700 .9840 .9981 1.0122 1.0995 1.1144 1.1294 1.1446 1.1598 1.2539 1.2700 1.2862 1.3025 1.3190 1.4208 1.4382 1.4558 1.4735 1.4914 1.6019 1.6209 1.6400 1.6593 1.6788 1.7991 1.8198 1.8406 1.8617 1.8829 2.0143 2.0369 2.0597 2.0827 2.1059 2.2496 2.2743 2.2993 2.3245 2.3499 2.5075 2.5346 2.5620 2.589'> 2.6175 2.7904 2.8202 2.8503 2.8806 2.9112 3.1013 3.1340 3.1671 3.2005 3.2341 3.4432 3.4792 3.5156 3.5523 3.5894 3.8196 3.8593 3.8993 3.9398 3.9806 4.2342 4.2779 4.3221 4.3666 4.4117 4.6912 4.7394 4.7880 4.8372 4.8868 5.1951 5.2483 5.3020 5.3562 5.4109 5.7510 5.8097 5.8689 5.9288 5.9892 6.3645 6.4293 6.4946 6.5607 6.6274 7.0417 7.1132 7.1854 7.2583 7.3319 7.7894 7.8683 7.9480 8.0285 8.1098 8.6150 8.7021 8.7902 8.8791 8.9689 9.5268 9.6231 9.7203 9.8185 9.9177 10.534 10.640 10.748 10.856 10.966 11.647 11.764 11.883 12.003 12.124 12.876 13.006 13.137 13.269 13.403 14.234 14.377 14.522 14.668 14.816 15.734 15.893 16.053 16.214 16.378 17.392 17.567 17.744 17.923 18.103 19.224 19.418 19.613 19.811 20.010 21.249 21.463 21.679 21.897 22.117 23.486 23.722 23.961 24.202 24.445 25.958 26.219 26.483 26.749 27.018 28.690 28.979 29.270 29.564 29.862 31.709 32.028 32.350 32.675 33.004 35.046 35.398 35.754 36.113 36.476 38.733 39.122 39.515 39.913 40.314 42.808 43.238 43.673 44.112 44.555 47.311 47.787 48.267 48.752 49.242 52.288 52.813 53.344 53.880 54.422 57.788 58.369 58.955 59.548 60.147 63.866 64.508 65.157 65.812 66.473 70.584 71.293 72.010 72.734 73.465 HYPERBOLIC FUNCTIONS Cosh u u =0 to t<=4.99 1234 567 89 1.0000 1.0001 1.0002 1.0005 1.0008 .1 1.0050 1.0061 1.0072 1.0085 1.0098 .2 1.0201 1.0221 1.0243 1.0266 1.0289 .3 1.0453 1.0484 1.0516 1.0549 1.0584 .4 1.0811 1.0852 1.0895 1.0939 1.0984 .5 1.1276 1.1329 1.1383 1.1438 1.1494 .6 1.1855 1.1919 1.1984 1.2051 1.2119 .7 1.2552 1.2628 1.2706 1.2785 1.2865 .8 1.3374 1.3464 1.3555 1.3647 1.3740 .9 1.4331 1.4434 1.4539 1.4645 1.4753 1.0 1.5431 1.5549 1.5669 1.5790 1.5913 .1 1.6685 1.6820 1.6956 1.7093 1.7233 .2 1.8107 1.8258 1.8412 1.8568 1.8725 .3 1.9709 1.9880 2.0053 2.0228 2.0404 .4 2.1509 2.1700 2.1894 2.2090 2.2288 .5 2.3524 2.3738 2.3955 2.4174 2.4395 .6 2.5775 2.6013 2.6255 2.6499 2.6746 .7 2.8283 2.8549 2.8818 2.9090 2.9364 .8 3.1075 3.1371 3.1669 3.1972 3.2277 .9 3.4177 3.4506 3.4838 3.5173 3.5512 2.0 3.7622 3.7987 3.8355 3.8727 3.9103 .1 4.1443 4.1847 4.2256 4.2668 4.3085 .2 4.5679 4.6127 4.6580 4.7037 4.7499 .3 5.0372 5.0868 5.1370 5.1876 5.2388 .4 5.5569 5.6119 5.6674 5.7235 5.7801 .5 6.1323 6.1931 6.2545 6.3166 6.3793 .6 6.7690 6.8363 6.9043 6.9729 7.0423 .7 7.4735 7.5479 7.6231 7.6990 7.7758 .8 8.2527 8.3351 8.4182 8.5022 8.5871 .9 9.1146 S.2056 9.2976 9.3905 9.4844 3.0 10.068 10.168 10.270 10.373 10.476 .1 11.121 11.233 11.345 11.459 11.574 .2 12.287 12.410 12.534 12.660 12.786 .3 13.575 13.711 13.848 13.987 14.127 .4 14.999 15.149 15.301 15.455 15.610 .5 16.573 16.739 16.907 17.077 17.248 .6 18.313 18.497 18.682 18.870 19.059 .7 20.236 20.439 20.644 20.852 21.061 .8 22.362 22.586 22.813 23.042 23.273 .9 24.711 24.959 25.210 25.463 25.719 4.0 27.308 27.582 27.860 28.139 28.422 .1 30.178 30.482 30.788 31.097 31.409 .2 33.351 33.686 34.024 34.366 34.711 .3 36.857 37.227 37.601 37.979 38.360 .4 40.732 41.141 41.554 41.972 42.393 .5 45.014 45.466 45.923 46.385 46.851 .6 49.747 50.247 50.752 51.262 51.777 .7 54.978 55.531 56.089 56.652 57.221 .8 60.759 61.370 61.987 62.609 63.239 .9 67.149 67.823 68.505 69.193 69.889 1.0013 1.0018 1.0025 1.0032 1.0041 1.0113 1.0128 1.0145 1.0162 1.0181 1.0314 1.0340 1.0367 1.0395 1.0423 1.0619 1.0655 1.0692 1.0731 1.0770 1.1030 1.1077 1.1125 1.1174 1.1225 1.1551 1.1609 1.1669 1.1730 1.1792 1.2188 1.2258 1.2330 1.2402 1.2476 1.2947 1.3030 1.3114 1.3199 1.3286 1.3835 1.3932 1.4029 1.4128 1.4229 1.4862 1.4973 1.5085 1.5199 1.5314 1.6038 1.6164 1.6292 1.6421 1.6552 1.7374 1.7517 1.7662 1.7808 1.7956 1.8884 1.9045 1.9208 1.9373 1.9540 2.0583 2.0764 2.0947 2.1132 2.1320 2.2488 2.2691 2.2896 2.3103 2.3312 2.4619 2.4845 2.5073 2.5305 2.5538 2.6995 2.7247 2.7502 2.7760 2.8020 2.9642 2.9922 3.0206 3.0492 3.0782 3.2585 3.2897 3.3212 3.3530 3.3852 3.5855 3.6201 3.6551 3.6904 3.7261 3.9483 3.9867 4.0255 4.0647 4.1043 4.3507 4.3932 4.4362 4.4797 4.5236 4.7966 4.8437 4.8914 4.9395 4.9881 5.2905 5.3427 5.3954 5.4487 5.5026 5.8373 5.8951 5.9535 6.0125 6.0721 6.4426 6.5066 6.5712 6.6365 6.7024 7.1123 7.1831 7.2546 7.3268 7.3998 7.8533 7.9136 8.0106 8.0905 8.1712 8.6728 8.7594 8.8469 8.9352 9.0244 9.5791 9.6749 9.7716 9.8693 9.9680 10.581 10.687 10.794 10.902 11.011 11.689 11.806 11.925 12.044 12.165 12.915 13.044 13.175 13.307 13.440 14.269 14.412 14.556 14.702 14.850 15.766 15.924 16.084 16.245 16.408 17.421 17.596 17.772 17.951 18.131 19.250 19.444 19.639 19.836 20.035 21.272 21.486 21.702 21.919 22.139 23.507 23.743 23.982 24.222 24.466 25.977 26.238 26.502 26.768 27.037 28.707 28.996 29.287 29.581 29.878 31.725 32.044 32.365 32.691 33.019 35.060 35.412 35.768 36.127 36.490 38.746 39.135 39.528 39.925 40.X26 42.819 43.250 43.684 44.123 44.566 47.321 47.797 48.277 48.762 49.252 52.297 52.823 53.354 53.890 54.431 57.796 58.377 58.964 59.556 60.155 63.874 64.516 65.164 65.819 66.481 70.591 71.300 72.017 72.741 73.472 90 SOLUTION OF EQUATIONS Quadratic Equation 1. Algebraic Solution: *' +J>* + q = o, x= -p/2 J P\ _ ry ' 4 ax* + bx + c = o, * - - b yj^^c 2a 2. Solution by Slide Rule: Let the four scales on the rule be A, B, C and D; B and C being on the runner, and B and A the squares of C and D. The equation having been reduced to the form * 2 4- px +g=o, set the indicator at q on D and move the runner until the sum or difference of ^Number on D under 1 on C) and (Number on C under indicator) is equal to p. Then the two numbers are the required roots. . The following table has been arranged to facilitate the find- ing of the positive root since this one is usually of more import- ance than the negative. The column headed "Limit for positive root" shows whether the positive root (c) is greater or less than the limit, which is seen to be a number either known or easily found. It is best to set this limit on (C) under the indicator, and then move the runner to the proper setting by increasing or decreasing the numbers (on the runner under the indicator) according as c is greater or less than the limit. After the setting is made which satisfies the equation in the column headed "setting," the positive root is c, that is, appears under the indicator. It is well to remember that if 4q = p 2 , the roots are equal and if p 2 < 4q, the roots are imaginary. Form Setting Sign of Roots Limit for Larger Smaller Positive Root x 2 -f- px q = o p = d c -|- c < 7/p and p c and d are the two roots; d = number on D under 1 on C and c number on C under indicator. The equation x z -j- px + q = o has both roots negative and p = c + d. SOLUTION OF EQUATIONS 91 To illustrate the method, take the equation x 2 + 0.33 x 5.33 = Trying both limits, we find that the second one gives the lower, c V P or ^> $ Q an d the setting is p = b d. Then the positive roots are the roots of the quadratic x 2 yx -f- d = 0, and are imaginary if 7 < f 4q. Setting Sign of Limit for Roots Positive Root p = b d 1 pos. 2 neg. c >V P p = d -b 1 pos. 2 imag. c we arrive at another set of coefficients 1 + 17.3 -f 75.43 2.629 shown just below the second broken line. This process of adding successive figures to the root may be continued until zero remainder is reached or any required degree of approximation is obtained. If five or six decimal places are required, it is convenient to find only the first two or three by the above process, finding the remainder by the contracted method. In this case we will begin the contraction after having found the second place in the root. From the coefficient in the second column from the right, we drop one figure and from the third column we drop two. The fourth column we drop altogether, so that our coefficients appear 17 + 76.470 0.350503, as shown below the third broken line. Applying upon these coefficients the process em- ployed at the beginning, we find that 0.004 gives a negative re- mainder and 0.005 a positive one, and hence, that 4 is the next figure in the root. Continuing, we arrive at the coefficients 17 -f 76.606 0.044351. By dropping the last figure in the second column and the two in the third we have only the coefficients 76.60 and 0.044351 left. Dividing 0.044351 by 76.60 we find three more places, so that the root is 5.134579, which is correct to five places, the value to nine places being 5.134578725. The explanation is given for a cubic as this occurs more frequently in engineering problems, but it is plain that the pro- cess applies to an equation of any degree. 94 SOLUTION OF EQUATIONS Homer's Method Continued Homer's method is greatly facilitated by using the slide rule for performing the multiplications and, unless great accuracy is required, the root may be found at once, remembering any num- ber which will reduce the last remainder to zero is a root. For example, setting the slide at 5.13 and multiplying success- ively by 7.13 and 13.58, we obtain the small remainder 0.4 as shown below. 1+2 23.00 70.0 15^,3' + 5.13 +36.58 +69.6 + 7.13 +13.58 - 0.4 It will usually require but a few trials to locate the number which will give the smallest remainder. Newton's Method Let a = first approximation to root, b = (a + 5) = second approximation to root. Then 5 is found from =_/(> /() in whichy' () = first derivative of /(a). (b + 5') = third approximation in which ,'-_/.<*> f'(b) This process is carried on till the required degree of approx- imation is reached. By Plotting or by Use of Tables 1. By plotting. Separate equation into two parts, plot to scale each part and find value of variable for point of intersection. Example To find root of x' a + 7x sin x = Rewrite in form y = x 5 + 7x y = sin x Plot the two curves and find point of intersection. 2. By the use of tables. Example To find root of a tan x + e x = Find from table value of e x such that a + e* is equal to value of tan x as found from table of natural tangents. fir, SERIES n rn-r 1. (a 4- 6)- = a- + na-6 4-^T^ *"W + 2. 3. 4. 1.1 . 1 1-3 1.1.3.5 2.4.6-8 1.3-5 z s 1.3.5.7 4 X 5. e* = 6. sin*-* 3! + 5! 7 , - ^ 2T , X X- 7. cosx = l-- + --- INTEGRALS 1. I udu = uv I vdu. l5 Jx-rfz = |^p 16. / Vi = V( e*. 17 M. a -h 5. I a :r log a dx = a*. 6. I sin xdx = cos x, or versin x. 7. J cos xdx = sin x, or coversin x. 8. J sin xdx sin x x cos x. 9. J x cos xdx = cos x -h #sin x. 10 C *** = - ^ cosx m 2 r ' J sm m x "" w 1 sm m ~ ! x m 1 J 11 (* " (a) ] .... - V. However, it is not usual to consider the terms containing the coefficients A and B. SIMPSON'S RULE b J / (*) dx =- 3 [T + 2 (F 2 + F 4 + + r n _ 2 ) a + 4(r, + Y, + + r^_,) + Y n ] To apply this rule n must be an even number. PARABOLIC RULE 6 J EQUATIONS OF CURVES Logarithmic Spiral r = ae md m = ct n a = constant Catenary s = length of curve measured from vertex. * = c tan T = c sinh -2- c c = distance of vertex from x axis. H H = horizontal tension at vertex, e = weight per unit length. y* = c 2 + s 2 ) = c cosh 3 c c sinh c 100 PROBABLE ERROR PROBABLE ERROR Weighted Observations. If all the observations are con- sidered not to be of equal value, they are to be weighted to such an extent that the doubtful ones will be reduced to their proper values as regards the probable error. Any observation considered as being of maximum reli- ability is given a certain weight, as 10. The other observa- tions are weighted accordingly; as, for example, an observation taken under very unfavorable circumstances and hence quite doubtful might be given a weight of 2 or even 1, whereas an observation considered as being only slightly deficient in reli- ability would be weighted 9. The weighted mean = ^ w R 2 w in which w is the weight of any observation and R is the value of that observation. Then, if d = difference between R and the weighted mean n = number of observations e = probable error of any single observation]of weight unity E = probable error of the weighted mean 0.67449 - w - \ n 1 = 0.67449 _, n 1) 2 w [S wdr V Z* Unweighted Observations. If all the observations are taken to be of equal reliability, the above equations reduce to e =C v'~Z"d 2 and E= e -- = c V 2 & V n PROBABLE " ERROR 101 Values of C'and c.> n C c c 2 0.6745 0.4769 21 0.150$ 0.0329 40 0.10SO 0.0171 3 .4769 .2754 22 .1472 .0314 41 .1066 .0167 4 .3894 .1947 23 .1438 .0300 42 .1053 .0163 5 .3372 .1508 24 .1406 .0287 43 .1041 .0159 6 .3016 .1231 25 .1377 .0275 44 .1029 .0155 7 .2754 .1041 26 .1349 .0265 45 .1017 .0152 8 .2549 .0901 27 .1323 .0255 46 .1005 .0148 9 .2385 .0795 28 .1298 .0245 47 .0994 .0145 10 .2248 .0711 29 .1275 .0237 48 .0984 .0142 11 .2133 .0643 30 .1252 .0229 49 .0974 .0139 12 .2034 .0587 31 .1231 .0221 50 .0964 .0136 13 .1947 .0540 32 .1211 .0214 55 .0918 .0124 14 .1871 .0500 33 .1192 .0208 60 .0878 .0113 15 .1803 .0465 34 .1174 .0201 65 .0843 .0105 16 .1742 .0435 35 .1157 .0196 70 .0812 .0097 17 .1686 .0409 36 .1140 .0190 75 .0784 .0091 18 .1636 .0386 37 .1124 .0185 80 .0759 .0085 19 .1590 .0365 38 .1109 .0180 90 .0715 .0075 20 1547 .0346 39 .1094 .0175 100 .0678 .0068 SECTIONS AND SOLIDS Guldin's Rules. I. If a plane area revolve about an axis external to itself through any assigned angle, the volume of the solid generated will be equal to a prism whose base is the revolving area, and whose altitude is the length of the path described by the center of mass of the area. II. If the arc of a plane curve revolve about an external axis in its own plane through any assigned angle, the area of the surface generated will be equal to that of a rectangle, one side of which is the length of the revolving curve, and the other the length of the path described by its center of mass. The Prismoidal Formula Let A = area of the base of a prism, A t , A 2 , A m =the two end and middle areas of a prismoid, or of any of its elementary solids, h = altitude of the prismoid or elementary solid, V = its volume V = ft/6 (A, + 4 A m + A,) For a prism A l = A m = A 2 = A, V = ft/6 X 6 A = h A. For a wedge with parallel ends; A 2 = 0, A m = % A t , V = ft/6 ( A, + 2 A, ) = h A - For a cone or pyramid A, = 0, A m = % A x , ft/6 (A, + A,) o 102 SECTIONS AND SOLIDS Momencs of Inertia. The moment of inertia of a body about a given axis is the summation for all the particles. of the body of the product mass of particle X squire of its distance from the given axis; or sym- bolically, if dm is the elementary mass and r its distance from the axis, Moment of Inertia = j r 2 dm', the summation or integration being taken so as to include the total mass of the body. The fundamental properties of moments of inertia are the following: 1. The moment of inertia of a body about a given axis is the sum of the moments of inertia of its parts taken about the given axis. 2. If / is the moment of inertia of a body about an axis through its center of mass and /' its moment of inertia about any parallel axis, l f = I+r*M; where r is the distance between the two axes and M is the mass of the body. Axes lying in the surface of a thin lamina or plane section are known as equatorial axes; axes perpendicular to the lamina as polar axes. 3. For a thin lamina or plane section, the sum of the moments of inertia about any two equatorial axes at rightangles is equal to the moment of inertia about the polar axis passing through their intersection. Area, Center of Mass, Moments of Inertia, Radius of Gyration and Section Modulus of Sections E = area of section. G = radius of gyration for moment of inertia I. I = moment of inertia about an axis shown by dotted line. J = moment of inertia about an axis perpendicular to plane of section. Except where indicated by subscripts, I and J are taken about axes through the center of mass of the section or solid. If K = moment of inertia about the equatorial axis through the center of mass and perpendicular to the axis for I, then J = I + K. S = section modulus for I. x locates the center of mass. SECTIONS AND SOLIDS 103 Area, Center of Mass, Moments of Inertia, Radius of Gyration and Section Modulus of Sections. L E^bd-fyd,* f*(M*-kd') /z X = J=E(b*d +b J ~ xld 106 SECTIONS AND SOLIDS Area, Center of Mass, Moments of Inertia, Radius of Gyration and Section Modulus of Sections. =bd-hu, /= d 2 5- bd 3 -uh 3 6d ^U-. ~h d 6d ^i~\?i. E= d s+ ht JL sd*+ht 2 b-Q *=E ' y = d ~ X .6. A .6. JL -b- SECTIONS AND SOLIDS 107 Surface, Volume, Center of Mass and Moments of Inertia of Solids B = area of base. I = moment of inertia about axis I-I. J = moment of inertia about axis J-J. I and J are for axes through the center of mass. L = surface of solid^exclusive of base or bases. M = mass. V = volume. x locates the center of mass -5 V=Zabh 108 SECTIONS AND SOLIDS Surface, Volume, Center of Mass and Moments of Inertia of Solids SPHERE Radius r .a, a. 1= ^o Circular Crott Sectio Elliptical Cross Section. Axes and zb, b axts parallel to JJ. SECTIONS AND SOLIDS 109 Coefficients of Strength and Deflection for Steel Shapes In the tables the "coefficient of strength" is the safe load in pounds uniformly distributed, including weight of beam, for a steel beam one foot long. Hence, for any other span the safe uniform load in pounds is obtained by dividing the proper coefficient by the span in feet. Reduce other than uniform loads to equivalent uniform load by factor X, page 130 et seq. The " coefficient of deflection" is the deflection- in inches of a steel beam one foot long and loaded with 1000 pounds. For any other load and span, obtain the deflection in inches by multiplying the proper coefficient by the cube of the span in feet and the number of 1000-pound units in the load. Properties of Simple and Compound Section s Usually, only the moments of inertia for the axis 11 and 2 2 are required, but in case this quantity for some other axis is wanted it may be found from the tables as follows. 1. For an axis parallel to 1 1 or 22, find according to 2 page 102. 2. For an axis inclined to 1 1, find first the moment of inertia K about an axis through the center of mass (gravity) of the shape and parallel to the required axis. Then apply 2, page 102. For I-beams, bulb beams, channels and T-bars K = I cos 2 |8 + 1' sin 2 ft in which j8 is the angle which the inclined axis makes with the axis 1 1 For angles and Z-bars, K = I" cos 2 /3 + I'" sin 2 j8, i" = Ar"\ r = i + r -i" and /3 is the angle which the axis for K makes with the axis 33. For compound steel sections formed by combining simple shapes such as plates, angles, etc., the moment of inertia about an axis through its center of inasH will usually be required. The center of mass for symmetrical sections is at the center. For an unsymmetrical section, assume any convenient line as an axis, multiply the area of each simple shape by the distance of its center of mass from this axis and divide the sum of these products by the sum of the component areas. The quotient is the distance of the center of mass of the compound section from the assumed axis. The section modulus is the moment of inertia divided by the distance of the most remote point of the section from the axis about which the moment of inertia is taken. The radius of gyration is the square root of the quotient obtained by dividing the moment of inertia by the. area. 110 SECTIONS AND SOLIDS Properties of Cambria Standard I-Beams. 1 2 3 4 6 6 7 8 9 10 11 Kadius Radius Section T Beam. Weight Foot Area of Section. Thick- ness of Web. Width of Flange. Moment of Inertia Axis 1-1. Section Modulus Axis 1-1. of GTTE- tion Axis Moment of Inertia Axis 24 of Gyra- tion Alls Number. 1-1. 2-2. d A t b I S r I' r' Inches. Pounds. Sq.Ins. Inch. Inches. Inches.* Inchest Inches. Inches.* Inch. B 5 3 6.50 1.63 .17 2.33 2.5 1.7 1.23 .46 .53 i tt 6.5O 1.91 .26 2.42 2.7 1.8 .19 .53 .52 H tt 7.50 2.21 .36 2.52 2.9 1.9 .15 .60 .52 B 9 4 7.50 2.21 .19 2.66 6.0 3.0 .64 .77 .59 14 tt 8.50 2.5O .26 2.73 6.4 3.2 .59 .85 .58 ( tt 9.50 2.79 .34 2.81 6.7 3.4 .54 .93 .58 11 " 1O.5O 3.09 .41 2.88 7.1 3.6 .52 1.01 .57 BIS 5 9.75 2.87 .21 3.00 12.1 4.8 2.05 1.23 .65 M tt 12.25 3.6O .36 3.15 13.6 6.4 1.94 1.45 .63 It tt 14.75 4.34 .50 3.29 15.1 6.1 1.87 1.70 .63 B17 6 12.25 3.61 .23 3.33 21.8 7.3 2.46 1.85 .72 (i tt 14.75 4.34 .35 3.45 24.0 8.0i2.35 2.09 .69 it it 17.25 5.O7 .47 3.57 26.2 8.7 2.27 2.36 .68 B21 7 15.OO 4.42 .25 3.66 36.2 10.4 2.86 2.67 .78 ii 17.50 5.15 .35 3.76 39.2 11. 2 j 2.76 2.94 .76 ,i tt 20.00 5.88 .46 3.87 42.2 12.1 2.68 3.24 .74 B26 8 18.00 5.33 .27 4.00 56.9 14.2 3.27 3.78 .84 tt tt 20.25 5.96 .35 4.08 60.2 15.O3.18 4.04 .82 (< ** 22.75 6.69 .44 4.17 64.1 16.0 3.1 4.36 .81 II " 25.25 7.43 .53 4.26 68.O 17.0 3.O3 4.71 .80 B29 9 21.00 6.31 .29 4.33 84.9 18.9 3.67 5.16 .90 u 25.0O 7.35 .41 4.45 91.9 2O.43.54j 6.65 .88 it " 30.00 8.82 .57 4.61 1O1.9 22.6 13.40 6.42 .85 t 35.OO 10.29 .73 4.77 111.8 24.8 3.30 7.31 .84 B33 10 25.00 7.37 .31 4.66 122.1 24.4 4.07 6.89 .97 it 30.00 8.82 .45 4.8O 134.2 26.8 3.9O 7.65 .93 it *' 35.00 10.29 .60 4.95 146.4 29.3! 3.77 8.52 .91 t< tt 40.0O 11.76 .75 5.10 158.7 31.7 8.67 9.50 .90 B41 12 31.50 9.26 .35 5.00 215.8 36.0 4.83 9.50 1.01 < tt 35.00 10.29 .44 5.O9 228.3 38.O 4.71 10.07 .99 it tt 40.00 11.76 .56 5.21 245.9 41.0 4.57 10.95 .96 B53 15 42.00 12.48 .41 6.50 441.8 58.9 5.96 14.62 1.08 tt ii 45.00 60.00 13.24 14.71 .46 .56 5.55 6.65 455.8 483.4 6O.8 5.87 64.5 5.73 15.09 1.07 16.O4 1.O4 * tt ti tt t 55.00 16.18 60.00 17.65 .66 .75 5.75 611.01 68.1 6.62 5.84 5S8.6 1 71.8 ! 5.52 ! 17.06 1.O3 18.17 1.01 SECTIONS' AND SOLIDS Properties of Cambria Standard I-Beams. ill 12 13 | 14 15 1 16 1 Increase of Thickness of Web for each Pound Increase in Weight. Coefficient of Strength. Coefficient of Deflection. Section Number. B 5 ti t( For Fibre Stress of 16 000 Pounds per Square Inch for Buildings. For Fibre Stress of 12 500 Pounds per Square Inch for Bridges, Uniform Load. Center Load. f F F' N N' .098 17650 1914O 20710 13790 14950 16180 .OOO31253 .OOO28827 .00026644 .00050006 .00046124 .OOO4263O .074 31810 3389O 3598O 3807O 24850 2648O 2811O 29750 .00013009 .OOO122O9 .OOO115OO .OOO 10868 .00020815 .OO019535 .OOO184OO .OOO17389 B it YC .050 61590 581OO 64630 40300 45390 50490 .OOO06417 .O0005698 .00005122 .00010267 .OOOO9117 .OOO08195 B13 ti .049 77460 8527O 9311O 60520 6661O 72740 .00003561 .OOO03235 .00002963 .00005698 .OOOO 5 177 .00004741 B17 C( .042 110410 1194OO 12856O 86260 9329O 1OO43O .00002142 .OOOO 1980 .00001839 .OOOO3427 .OOO03168 .00002943 B21 M 1 it .037 151660 16O51O 17O97O 181480 118490 12540O 13357O 141740 .OOOO 1364 .OOOO 1289 .OOOO121O .00001140 .00002183 .OOOO2062 .OOOO 193 6 .00001825 B25 C( {( .033 201300 21793O 24146O 264990 157260 170260 18864O 207020 .O0000914 .OOOOO844 .OOOOO762 .OO000694 .00001462 .OOO0135O .OOOO121 9 .O0001110 B29 M tt .029 260470 28625O 31239O 33853O 2O350O 22363O 244O50 26448O .OO000635 .OOOOO578 .OO000530 .OO000489 .00001017 .OOOO0925 .00000848 .OOOO0782 B33 M .025 383670 4058OO 437170 29974O 317030 341 54O .OOOOO36O .OOOOO34O .O0000316 .OOOO0575 .OOOOO544 .OOOO0505 B41 el .020 628270 6483 1O 687530 72674O 765960 490840 5O649O 63713O 567770 69841O .00000176 .OOOOO17O .OOOOO1 61 .OOOOO 152 .OOOOO 144 .00000281 .OOOO0272 .OOOOO257 .00000243 .OOOOO231 B53 it i i< 112 SECTIONS AND SOLIDS Properties of Cambria Standard I-B earns. 1 2 3 4 5 6 7 8 1 9 | 10 11 Radius Radius Section Number. Depth of Beam. Weight J* r Foot. Area of Section. Thick- ness of Web. Width of Flange. Moment of Inertia Axis 1-1. Section Modulus Axis 1-1. of Gyra- tion Ails 1-1. Moment of Inertia Aiis2-2. of Gjra- tion Axis 2-2. d A t b I S r I' r' Inches. Pounds. Sq.Ins. Inch. Inches, Inches.* Inchest Inches. Inches.* Inch. B 65 18 65.0 15.93 .46 6.00 795.6 88.4 7.07 81.10ll.16 it tt 60.O 17.65 .56 6.1O 841.8 93.5 6.91 22.38 1.13 cc tt 65.O 19.12 .64 6.18 881.5 97.9 6.79 23.47 1.11 tt tt 70.O 20.59 .72 6.26 921.2 102.4 6.69 24.62 1.09 B 73 20 65.0 19.08 .50 e.25'1 169.5 117.0<7.83'27.86'1.21 tt tt 70.O 2O.59 .58 6.33 1219.8 122.0 7.70 29.04 1.19 tt tt 75.0 22.06 .65 6.4O 1268.8 126.9 7.58 30.25 1.17 B 89 24 80.O 23.32 .50 7.00 2087.2 173.9 9.46 42.86 1.36 tt 85.O 25.00 .57 7.07 2167.8 180.7 9.31 44.35 1.33 tt tt tt M 90.0 26.47 95.0 27.94 .63 .69 7.13 2238.4H86.5 9.2O 45.7O 1.31 7.19 23O9.O 192.4 9.09 47.10; 1.3O tt tt 100.O 29.41 .75 7.25 2379.6 198.3 8.99 48.55' 1.28 Properties of Cambria Bulb Beams. ntL 3. p 1 / 1l ^*^i 1 fl~ -J -. .^. ^^m.^M W JL 1 2 3 4 5 6 | 7 8 9 Depth Area Thickness Width Moment of Section Radius of Section of Beam. Weight of Section. ef Web. of Flange. Inertia Axis 1-1. Modulus Axis 1-1. Gyration Axis 1-1. Sumber. Foot. d A t b I 8 r Inches. Pounds. Sq. Ins. Inch. Inches. Inches.* Inches.3 Inches. B173 M 6 14.0 15.3 18.4 4.11 4.48 5.42 8 i 21.52 22.73 25.72 6.12 6.55 7.59 2.29 2.25 2.18 SECTIONS AND SOLIDS 113 Properties of Cambria Standard I-Beams. 12 13 I 14 | 15 I 16 1 Increase of Thickness of Web for each Pound Increase in Weight. Coefficient of Strength. Coefficient of Deflection. Section Number. For Fibre Stress of 16 000 Pounds per Square Inch for Buildings. For Fibre Stress of 12 500 Pounds per Square Inch for Bridges. Uniform Load. Center Load, f .016 F Jj N N' 942880 997680 104474O 1091800 736620 77944O 81620O 85297O .00000098 .00000092 .O0000088 .00000084 .00000156 .OOOOO 148 .OOOOO 141 .OOOO0136 B 66 ti i .015 1247490 1301110 1353400 974600 1O16490 105734O .00000066 .OOOOOO64 .00000061 .00000106 .OOOOO102 .00000098 B u 73 N .0123 1866310 192695O 19897OO 2062440 2115190 1449460 1505430 1554450 1603470 1652490 .00000037 .00000036 .OOOOO036 .OOOOO034 .OOOO0033 .00000060 .OOOOOO57 .00000056 .OCOOO054 .00000052 B 89 ct II (1 (( Properties of Cambria Bulb Beams. ^Q~--' 10 11 12 13 14 | 15 1 Distance of Center of Grarity from Outside of Flange. Increase of Thickness of Web for each Lb. Increase in Weight. Coef. of Strength. Coef. of Deflection. Section Number. Fibre Stress 16 000 Pounds per Sq. Inch for Buildings. Fibre Stress 12 500 Pounds per Sq. Inch for Bridges. Uniform Load. Center Load. X Inches. f F F N N' 2.49 2,53 2.61 .049 it 65320 6986O 80930 5103O 5458O 63230 .OOOO361 .O000341 .0000302 .OOO0577 .0000546 .0000483 B173 H 114 SECTIONS AND SOLIDS Properties of Cambria Standard Channels. -It- 1 2 3 4 6 6 ! 7 8 9 1O 11 12 Radius Radius Thick- Moment ^H 00 of Moment Section of Soction Num- Depth of Channel. Weight per Foot. Area of Section. ness of Web. Width of Flange. of Inertia Axis 1-1. Mod- ulus Axis 1-1. Gyra- tion Axis 1-1 of Inertia Axis 2-2. Mod- ulus Axis 2-2. Gyra- tion Axis 2-2. ber. (1 A t b I S p 1' S' T* Inches. Pounds. | Sq. Ins. Inch, i Inches. Inches. 4 Ins.3 Inches, Inches.* Ins.3 Inch. C 5 3 4.0oi 1.19 .17 1.41 1.6 1.1 1.17 .20 .21 .41 tt 6.00; 1.47 .26! 1.50 1.8 1.2 1.12 .26 .24 .41 44 u 6.OO 1.76 .3611.60 2.1 1.4 1.08 .31 .27 .42 C 9 4 - 6.25 1.55 .18 1.68 3.8 1.9 1.56 .32 .29 .45 6.25 1.84 .25 1.65 4.21 2..1 1.51 .38 .32 .45 44 tl 7.26 2.13 .33 1.73 4.6 2.3 1.46 .44 .35 -.46 C13 5 6.50 1.95 .19 1.75 7.4 3.0 1.95 .48 .38 .50 tt 9.00 2.65 .33 1.89 8.9 3.5 1.83 .64 .46 .49 44 44 11.50 3.38 .48 2.O4 10.4 4.2 1.75 .82 .54 .49 C17 6 8.00 2.38 .20 1.92 13.O 4.32.34 .70 .50 .54 44 10.50J 3.09 .32 2.04 15.1 6.O 2.21 .88 .57 .63 " 44 13.OOI 3.82 .44 2.16 17.3 5.82.13 1.07 .65 .63 it 44 15.50] 4.56 .56 2.28 19.5 6.5 ( 2.O7 1.28 .74 .53 C21 7 9.75 2.85 .21 2.O9 21.1 6.0 2.72 .98 .63 .59 12.25; 3.60 .32 2.20 24.2 6.9 2.59 1.19 .71 .57 tt 14.75! 4.34 .42 2.30 7.8 2.56 i 1.40 .79 .57 44 i 17.25 6.O7 .63 2.41 3O*2 8.62.44i 1.62 .87 .66 44 44 J19.75 6.81 .63 2.51 33.2 9.5 2.39 1.85 .96 .56 C25 8 11.25 3.35 .22 2.26 32.3 8.1 3.10 1.33 .79 .63 tt 13.75 4.04 .31 2.35 36.0 9.0 2.98 1.55 1 .87 .62 tt it 16.25 4.78 .40 2.44 39.9 10.0 2.89 1.78 .96 .61 i< it 18.75 5.51 .49 2.53 43.8 11. 2.82 2.01 1.02 .60 tt 44 21.25 6.25 .58 2.62 47.8,11.92.76 2.251.11 .60 C29 9 13.25 3.89 .23 2.43 47.3! 10.5 3.49 1.77 .97 .67 15.0O 4.41 .29 2.49 5O.9 11.3 3.40 1.951.03 .66 tt it 20.00 6.88 .45 2.66 60.8 13.5 3.21 2.45 1.19 .65 tt tt 25.00 7.35 .6112.81 70.7 15.7 3.10 2.98 1.36 .64 C33 10 15.00 4.46 .242.60 66.9 13.4 3.87 2.3O 1.17 .72 20.00 5.88 .3812.74 78.7:15.7 3.66^ 2.85 1.34 .70 tt tt' 25.00 7.35 .5312.89 91.0 18.2 3.521 3.40 1.50 .68 i tt 30.OOJ 8.82 .68'3.O4 103.220.6 3.42 3.99 1.67 .67 " 44 35.OO 10.29 .8213.18 115.523.1 3.35 4.66 1.87 .67 C41 12 20.50 6.03 .28 2.94 128.1 21.4 4.61 3.91 1.75 .81 25.OO 7.35 .39 3.05 144.0 24.0 4.43 4.53 1.91 .78 tt it 30.00 8.82 .51 3.17 161.626.9 4.28 6.212.09 .77 tt it 35.00 1O.29 .64 3.3O 179.3 29.9 4.17 6.902.27 .76 tt 40.00 11.76 .76 3.42 196.9,32.84.09 6.63 2.46 .75 Co 3 15 33.00 9.90 .403.40 312.641.7 5.62 8.23 3.16 .91 44 35.OO 1O.29 .43 3.43 319.942.7 5.57 8.48 3.22 .91 / tt || 40.0011.76 .52 3.52 347.5 46.3 6.44 9.39 3.43 .89 t 45.00.13,24 .62:3.62 375.1 50.O 5.32 1O.29 3.63 .88 44 ti 50.0OJ 14.71 55.00116.18 .72 3.72 402.7 53.7,5.23, 11.22 3.85 .82 3.82 430.2:57.4 5.16112.19 4.07 .87 .87 SECTIONS AND SOLIDS 115- Properties of Cambria Standard Channels. -It 13 14 15 16 17 18 1 Distance Increase of Coef. of Strength. Coef. of Deflection. of Center of Gravity from Thickness of Fibre Stress Web for each 16 000 Pounds 1 Fibre Stress 12 500 Pounds Uniform Center Section, 'Outside of Web. Pound Increase in Weight. oer So^. Inch for Buildings. : per Sq. Inch, for Bridges. load. Load. Number. X ~Inch7~ f Inches. F F' N N' .44 .098 1163O 9090 .OOO4743 .OOO7589J C 5 .44 13140 1O27O .OO04199 .OO06718 .46 14710 11490' .0003751! .0006001 M .46 .074 20230 15800 .0002046 .0003273 C 9 .46 22270 174OO .0001858 .0002973 .46 24360 19030 .OOO 1698 .0002717 " .49 .059 31640 24720 .0001046 .0001674 C13 .48 37860 29570 .OOOO875 .0001399: " .51 44390 34680 .0000746 .0001 193{ " .52 .049 4621O 36100 .O000597 .OOOO855 1 C17 .50 53750 42000 .O000513 .OOO0821! " .52 61600 48120 .0000448 .O000717 ll .55 69440 5425O .O000397 .OO00636! " .55 .042 64270 50210 .0000368 .O000588 C21 .53 73650 57540 .0000321! .OOOO7 14 .53 82740' 64690 .0000286, .0000457 " .55 91950 71840 .0000257 .OOO0411 if .58 101100 78990 .0000234 .O000374 .58 .037 86140 67300 .000024O .0000384 C25 .56 95990 75000 .0000216 .0000345 .56 106450 8317O .0000194 .0000311 u .57 116910 91340 .0000177 .0000283 <* .59 12737O 9951O .OOOO1 62 .O000260 " .61 .59 .033 11217O 120540 8763O 941 7O .0000164! .OO00262 .0000153 .OOOO244 C29 H .58 14407O 112550 .0000128 .O0002O4 " .62 167590 13093O .00001 1O 1 .0000176 " .64 .029 14268O 111470 .0000116 .0000186 C33 .61 167940 131210 .O000099 1 . .0000158 .62 194090 151630 .0000085 1 .O000136 .65 22023O 172060 .OOOOO75! .000012O " .69 246380 19248O .00000671 .O000107! " .70 .025 22775O 177930 .0000061 .00.00097 C41 .68 256000 20000O .OOOOO54 .0000086 .68 287370 224510 .0000048 .0000077 " .69 318750 249O20, .OOOO043 .O000069 " .72 35O120 273530 .0000039 .0000063 " .79 .020 444520 347280| .0000025 .0000040 C53 .79 455030 355500 .0000024 .0000039 ' .78 494250 38613O .OOOOO22 .0000036! .79 53347O 416770 .0000021 .O000033 * .80 672680 44741O .OOOOO19 .OO00031 * .82 611900 478050 .0000018; .0000029 116 SECTIONS AND SOLIDS Properties of Cambria Standard Angles. 1 2 3 4 5 6 7 8 Area Distance of Center of Moment Section Section Dimensions. Thickness. Weight per Foot. of Section. Gravity from Back of of Inertia Axis 1-1. Modulus Aus 1-1. Number. Leg. a x a t A x I S Inches. Inch. Pounds. Sq. Ins. Inch. Inchs.< Inchest A 6 s/x y Y* .6 .18 .23 .009 .017 44 it fc .9 .25 .25 .012 .024 A u 7 1 X 1 % .8 .24 .30 .022 .031 tt JL 1.2 .34 .32 .030 .044 tt 44 A 1.5 .44 .34 .037 .056 A 1% X IV y /e 1.1 .30 .36 .044 .049 tt t i ' 1.5 2.0 .44 .57 .38 .40 .061 .077 .071 .091 44 4 2.4 .69 .42 .090 .109 All IVx IV Y 1.3 .36 .42 .08 .072 2 4 2 1.8 .53 .44 .11 .104 ti t k 2.4 .69 .47 .14 .134 ti i ft 2.9 .84 .49 .16 .162 ti 11 % 3.4 .99 .51 .19 .188 it 44 -i* 3.9 1.13 .53 .21 .214 A13 jV x \y is 2.2 .63 .51 .18 .14 tt ' A 2.8 .82 .53 .23 .19 tt tt l* 3.4 1.00 .55 .27 .23 tt it S 4.0 1.18 .57 .31 .26 ti tt A 4.6 1.34 .59 .35 .30 44 it A 5.1 1.50 .61 .38 .33 A15 2 x-2 A 2.5 .72 .57 .27 .19 it M 3.2 .94 .59 .35 .25 ti it 4.O 1.16 .61 .42 .30 ti ii % 4.7 1.36 .64 .48 .35 ii . ti 7 5.3 1.56 .66 .54 .40 44 ti A 6.0 1.75 .68 .59 .45 A17 2% x 2V. fa 3.1 .91 .69 .55 .30 it ii y 4.1 1.19 .72 .70 .39 tt i $ 5.O 1.47 .74 .85 .48 " 11 % 5.9 1.74 .76 .98 .57 tt " l 6.8 7.7 2.00 2.25 .78 .81 1.11 1.23 .65 .72 tt 11 TB 8.5 2.50 .83 1.34 .80 A19 3 x 3 I/ 4.9 1.44 .84 1.24 .58 JL 6.1 1.78 .87 1.51 .71 f ii /* 7.2 2.11 .89 1.76 .83 t i J 8.3 2.44 .91 1.99 .95 it i y& 9.4 2.75 .93 2.22 1.07 tt i 3*8 10.4 3.06 .95 2.43 1.19 ti i 6^ 11.5 3.36 .98 2.62 1.30 " ' ii 12.5 3.66 l.OO 2,81 1.40 SECTIONS AND SOLIDS 117 Properties of Cambria Standard Angles. 9 1O 11 12 13 1 Radius of Gyration Axis 1-1. Distance of Center of Gravity from External Apex Least Moment of Inertia Axis 2-2. Section Modulus Axis 2-2. Least Radius of Gyration Axis 2-2. Section Number. r x" I" S" r" Inch. Inch. Inches.* Inches.3 Inch. .22 .33 .004 .011 .14 A 6 .22 .36 .005 .014 .14 14 .30 .42 .009 .021 .19 A 7 .SO .45 .013 .028 .19 a r// Number. Inch. Inch. Inches.* Inches.3 Inches. Inch. .60 .76 .51 .29 .79 .632 .43 A91 .59 .79 .65 .38 .78 .626 .42 .58 .81 .79 .47 .78 .620 .42 .58 .83 .91 .55 .77 .614 .42 .57 .85 1.03 .62 .76 .607 .42 .56 .88 1.14 .70 .75 .600 .42 .56 .90 1.24 .77 .75 .592 .42 .75 .91 .1.17 .56 .95 .684 .53 A93 .74 .93 1.42 .69 .94 .680 .53 .74 .96 1.66 .81 .93 .676 .52 .73 .98 1.88 .93 .92 .672 .52 .72 1.00 2.O8 1.O4 .91 .666 .52 .72 1.O2 2.28 1.15 .91 .661 .52 .71 1.04 2.46 1.26 .90 .655 .52 .74 1.11 1.80 .75 1.12 .506 .54 A95 .73 1.14 2.19 .93 1.11 .501 .54 .72 1.16 2.56 1.09 1.1O .496 .54 .71 1.18 2.91 1.26 1.O9 .491 .54 .70 1.2O 3.24 1.41 1.09 .486 .53 .70 1.23 3.55 1.56 1.O8 .480 .53 .69 1.25 3.85 1.71 1.07 .472 .53 .69 1.27 4.13 1.85 1.O6 .468 .53 i .68 1.29 4.40 1.99 1.O6 .461 .54 .90 1.O6 2.33 .95 1.10 .724 .63 A97 .90 1.08 2.72 1.13 1.O9 .721 .62 .89 1.10 3.1O 1.29 1.O8 .718 .62 .88 1.13 3.45 1.45 1.07 .714 .62 .87 1.15 3.79 1.61 1.07 .711 .62 .87 1.17 4.11 1.76 1.O6 .707 .62 .86 1.19 4.41 1.91 1.05 .703 .62 .85 1.21 4.7O 2.05 1.04 .698 .62 .85 1.23 4.98 2.20 1.O4 .694 .62 .84 1.25 5.24 2.33 1.03 .689 .63 .89 1.26 3.38 1.23 1.27 .554 .65 A99 .88 1.28 3.96 1.46 1.26 .551 .64 .87 1.30 4.52 1.68 1.25 .547 .64 .86 1.33 5.05 1.89 1.25 .543 .64 .86 1.35 5.55 2.O9 1.24 .538 .64 .86 1.37 6.03 2.3O 1.23 .534 .64 .84 1.39 6.49 2.49 1422 .529 .64 .84 1.42 6.93 2.68 1.22 .524 .64 .83 1.44 7.35 2.87 1.21 .518 .64 .83 1.46 7.75 3.05 1.20 .512 .64 122 SECTIONS AND SOLIDS Properties of Cambria Standard Angles. 1 2 3 4 6 6 7 8 Section Nuznbcr. Dimensions. Thickness. Weight per Foot. Area of Section. Distance of Center of Gravity from Back of Longer Leg. Moment of Inertia Axis 1-1. Section Modulus Axis 1-1. b x a t A x. I S Inches. Inch. Pounds. Sq. Ins. Inch. Inches.* Inches. 3 A101 5 x3 A 8.2 2.41 .68 1.75 .75 */ 9.8 2.86 .70 2.04 .89 44 44 Jr 11.3 3.31 .73 2.32 1.02 44 44 I/ 12.8 3.75 .75 2.58 1.15 44 41 I 9 9 14.3 4.19 .77 2.83 1.27 44 14 5/ 15.7 4.61 .80 3.06 1.39 " 4t i^ 17.1 5.03 .82 3.29 1.61 ti M *A 18.5 5.44 .84 3.51 1.62 tt 44 M 19.9 5.84 -.86 3.71 1.74 M " /S 21.2 6.24 .88 3.91 1.85 A103 5 x3V A 8.7 2.56 .84 2.72 1.02 $8 10.4 3.05 .86 3.18 1.21 44 44 A 12.O 3.53 .88 3.63 1.39 44 44 8 13.6 4.00 .91 4.05 1.56 44 14 15.2 4.47 .93 4.45 1.73 14 44 s 16.8 4.93 .95 4.83 1.90 4 44 i4 18.3 5.38 .97 5.20 2.06 44 44 % 19.8 5.82 1.00 5.55 2.22 44 44 9 21.3 6.25 1.02 5.89 2.37 44 44 / 22.7 6.68 1,04 6.21 2.52 44 * H 24.2 7.09 1.06 6.52 2.67 A 105 6 x Q% Ys 11.7 3.43 .79 3.34 1.23 13.5 3.97 .81 3.81 1.41 41 41 15.3 4.50 .83 4.25 1.59 44 44 17.1 5.03 .86 4-67 1.77 44 tt 5X 18.9 5.55 .88 5.08 1.94 44 44 H 20.6 6.06 .90 5.47 2.11 44 t I? 22.4 6.57 .93 5.84 2.27 44 I 13 24.0 7.06 .95 6.20 2.43 44 4 /a 25.7 7.55 .97 6.55 2.59 44 4 u 27.3 8.03 .99 6.88 2.74 " t 1 28.9 8.5O 1.01 7.21 2.90 A107 6 x4 3/ 12.3 3.61 .94 4.90 1.60 JC 14.3 4.19 .96 5.60 1.85 44 M I" 16.2 4.75 .99 6.27 2.08 44 44 '9 18.1 5.31 1.01 6.91 2.31 41 44 c/ 2O.O 5.86 1.03 7.52 2.54 44 tt if 21.8 6.41 1.06 8.11 2.76 If 14 k 23.6 6.94 1.O8 8.68 2.97 M 44 ii 25.4 7.47 1.10 9.23 3.18 M 4 i " 1 27.2 28.9 30.6 7.99 8.5O 9.0O 1.12 1.14 1.17 9.75 10.26 10.75 3.39 3.69 8.79 SECTIONS AND SOLIDS 123 Properties of Cambria Standard Angles. 9 1O 11 12 13 14 15 1 Radius of Gyration Axis 1-1. Distance of Center of Gravity from Back of Shorter Leg. Moment of Inertia Aiis 2-2. Section Modulus Aiis 2-2. Radius of Gyration Axi? 2-2. Tangent of Angle Least Radius of Gyration Aiis 3-3. Section Ki Vi r x' I' S' r' a r" ' Inch. Inches. Inches. 4 Inches/' Inch. Inch. .85 1.68 6.26 1.89 1.61 .368 .66 Alpl .84 1.70 7.37 2.24 1.61 .364 .65 .84 1.73 8.43 2.58 1.60 .361 .65 i .83 1.75 9.45 2.91 1.59 .357 .65 k .82 1.77 10.43 3.23 1.58 .353 .65 H .82 1.80 11.37 3.55 1.57 .349 .64 U .81 1.82 12.28 3.86 1.56 .345 .64 If .80 1.84 13.15 4.16 1.55 .340 .64 U .80 1.86 13.98 4.46 1.55 .336 .64 it .79 1.88 14.78 4.75 1.54 .331 .64 ( 1.03 1.02 1.59 1.61 6.60 7.78 1.94 2.29 1.61 1.60 .489 .485 g A 103 1.01 1.63 8.90 2.64 1.59 .482 .76 1C 1.01 1.66 9.99 2.99 1.58 .479 75 ti 1.00 1.68 11. 03 3.32 1.57 .476 .75 ' .99 1.70 12.03 3.65 1.66 .472 .75 " .98 1.72 12.99 3.97 1.5ft .468 .75 .98 1.75 13.92 4.28 1.55 .464 .75 1 .97 1.77 14.81 4.58 .54 .460 .75 * .96 1.79 15.67 4.88 .53 .455 .75 c .96 1.81 16.49 5.17 .53 ,451 .75 f .99 .04 12.86 3.24 .94 .350 .77 A105 .98 2.06 14.76 3.75 .93 .347 .76 4 .97 208 16.59 4.24 .92 .344 .76 t .96 2.11 18.37 4.72 .91 .341 .75 .96 2.13 2O.08 5.19 .90 .338 .75 1 .95 2.15 21.74 5.65 .89 .334 .75 t .94 2.18 23.34 6.10 .89 .331 .75 * .94 2.20 24.89 6.55 .88 .327 .75 * .93 2.22 26.39 6.98 .87 .323 .75 1 .93 2.24 27.84 7.41 .86 .320 .75 1 .92 2.26 29.15 7.8O 1.85 .317 .75 I 1.17 1.94 13.47 3.32 1.93 .446 .88 A1O7 1.16 1.96 15.46 3.83 1.92 .443 .87 1.15 1.99 17.40 4.33 1.91 .440 .87 < 1.14 2.01 19.26 4.83 1.9O .438 .87 M 1.13 2.03 21.07 5.31 1.9O .434 .86 1 1.13 2.O6 22.82 5.78 1.89 .431 .86 * 1.12 2.08 24.51 6.25 1.88 .428 .86 1.11 2.10 26.15 6.7O 1.87 .425 .86 1.11 2.12 27.73 7.16 1.86 .421 .86 ' 1.10 2.14 29.26 7.59 1.86 .418 .86 it 1.09 2.17 30.75 6.02 1.85 .414 86 " 124 SECTIONS AND SOLIDS Properties of Cambria T-Bars. EQUJLZ. X-EGS. 1 2-|3|4 6 6 7 8 9 Dimensions. Section Humber. Width of Flange. Depth of Bar. Thickness of Flange. Thickness of Stem. Weight per Foot. ire* of Section. Distance of Center of Gravity from Outside of Flange. Moment of Inertia Axis 1-1. b 1 8 to n' t to t' A x. I Inches. Inches. Inch. Inch. Pounds. Sq. Ins. Inch. Inches.* T 5 1 1 K** KoA 1.0 .27 .29 .02 T181 1% 1% A ' A A " A 1.4 .41 .33 .04 T183 1A IA A " K A " A 1.6 .45 .34 .05 T187 1/4 IK A " % A " X 1.7 .48 .36 .06 T189 1% 1% A " /4 A " /4 1.9 .55 .39 .08 T 37 T 39 2 2 2 2 & ' $ ^"S 3.7 4.4 1.07 1.28 .59 .61 .37 .43 T 41 2*4 2K /4 " A 54 " A 4.2 1.21 .68 .51 T 42 2^-1 A " 78 6.0 1.46 .67 .64 T 49 2/<2 2% A " % A " % 5.6 1.63 .73 .87 T 67 3 3 A " % A " % 6.8 1.99 .86 1.58 T 69 3 3 % " A I /* " A 7.9 2.31 .88 1.82 T 73 3 3 /1 " A 10.1 2.96 .93 2.27 T 97 Q% 3% % " A 3 J"A 9.3 2.74 .99 3.10 T108 4 4 %" A 3 /^ "A 10.9 3.19 1.12 4.54 UNEQUAL LEGS. T185 13* 1A AtoK A to A 1.5 .44 .29 .04 T 22 2K 1/4 A " A A" A 3.0 .86 .30 .08 T 27 2% 1/4 4.5 1.31 .43 .21 T 56 2>2 3 % "A %" 7.2 2.10 .95 1.72 T 62 2% 2 7.5 2.21 .74 .83 T 65 3 2% % " A ! % to A 7.2 2.11 .71 1.08 T 84 3 4 % " A ! Ys " A 9.3 2.74 1.27 4.12 T101 3% 4 % " A % " A 10.0 2.94 1.20 4.33 T120 4% 2% A " A A "% 8.0 2.29 .57 1.04 T138 4% 3^2 7 II is IB II 14.9 4.37 1.09 4.89 T140 4% 3/4 A "A 15.9 4.65 i 1.11 5.08 T169 5 3 13.6 3.99 ; .72 2.42 SECTIONS AND SOLIDS 125 Properties of Cambria T-Bars EQTCJAJL. LEGS. 10 ir 12 | 13 14 15 I 16 i 1 Coef. of Strength. Section Radius of Moment of Section Radiusof Modulus iiis 1-1. Gyration iiis 1-1 Inertia liis2-2. Modulus liis 2-2. Gyration Aiis 2-2. For Fibre Stress of 16 000 Lbs. a* For Fibre Stress of 12 500 Lbs. per Square Inch. Section Number. 8 r 1' S' r' F F' "incheaT Inch. Inches.* Inches.* Inch. .03 .30 .01 .02 .21 350 27O T 6 .05 .32 .02 .04 .25 560 44O T181 .06 .33 .03 .05 .26 630 490 T183 .07 .35 .03 .05 427 70O 55O T187 .08 .39 .05 .07 .29 89O 69O T189 .26 .59 .18 .18 .42 277O 2160 T 37 .31 .59 .23 .23 .42 3330 26OO T 39 .32 .66 .24 .21 .45 3440 2690 T 41 .40 .66 .32 .29 .47 43OO 336O T 42 .49 .74 .44 .35 .52 525O 410O T 49 .74 .90 .75 .50 .62 78 6O 6140 T 67 .86 .90 .92 .61 .64 918O 718O T 69 1.10 .88 1.2O .80 .64 1171O 915O T 73 1.23 1.O8 1.42 .81 .73 13140 1026O T 97 1.58 1.21 2.11 1.06 .83 1685O 13170 T108 TJTNTEQTJAX. LEGS. .05 .29 .03 .01 .28 5OO 39O T186 .00 .31 .28 .22 .58 93O 73O T 22 .16 .40 .47 .38 .60 170O 1320 T 27 .84 .91 .53 .42 .50 893O 698O T 56 .66 .62 .64 .46 .54 7020 5490 T 62 .60 .64 .90 .60 .66 6400 6000 T 65 |.*1 1.24 .90 .60 .58 1609O 12570 T 84 f.t4 1.23 1.42 .81 7Q 1647O 1286O T101 .54 .68 2.51 1.12 1.O5 576O 45OO T12O 2.03 1.06 3.68 1.64 .92 2161O 1688O T138 2.13 1.O5 3.73 1.66 .90 2269O 1772O T140 1.06 .78 5.42 2.17 1.17 11340 8860 T169 126 SECTIONS AND SOLIDS Properties of Cambria Z-Bars 1 234 5 6 7 | 8 9 1O 11 12 Thick- Radius . Radius Section Depth of Length of ness of Web V'ght Area of Moment of. Tn6rti& Section Modulus of Gyra- lion Moment of Inertia Section Mod- ulus of Gyra- tion Num- Bar. Legs. and per pXrtt Section. Axis 1-1. Axis 1-1. Axis Axis 2-2. Axis Axis ber. Legs. 1 001. 1-1. 2-2. 2-2. b a t j^ I $ r I' S' r' "Inches! Inches. 1 Inch. ! Lbs. SqTlnsT Indies/ iTcheZ 3 Ins. j Inches/ IM. Inch. Z 5 3 j &u ., x 6.7 1.97 2.87 1.92 1.21 2.81 1.10 1.19 " 2/4 8.4 2.48 3.64 2.38 1.21 3.64 1.40 1.21 Z 9 3 18 |B 3 4 9.7 2.86 3.85 2.57 1.16 3.92 1.57 1.17 tt H5 11.4 3.36 4.57 2.98 1.17 4.75 1.88.1.19 Z13 3* Jill I/ 12.5 3.69 4.59 3.06 1.12 4.85 1.99 1.15 2x4 ^ 14.2 4.18 6.26 3.43 1.12 5.68 ;2.30|1.17 Z21 4 3iV 8.2 2.41 6.28 3.14 1.62 4.23 1. 44 1 1.33 tt 4 A 3% 10.3 3.03 7.94 3.91 1.62 6.46 1. 84 i 1.34 tt 4k 3 1 3 - X8 12.4 3.66 9.63 4.67 1.62 6.77 2.26,1.36 Z25 4 3 f e A 13.8 4.05 9.66 4.83 1.64 6.73 2.37'l.29 tt 4 V 3 ? ? 15.8 4.66 11.18 5.50 1.55 7.96 2.77 1.81 " Si 17.9 5.27 12.74 6.18 1.55 9.263.19 1.32 Z29 4 3/B 5^ 18.9 5.55 12.11 6.05 1.48 8.733.18 1.25 " ftlz |! 2O.9 6.14 13.52 6.65 1.48 9.95 3.68 1.27 M 4/1 3^ 8 23.0 6.75 14.97 7.26 1.49 11.24 4.00 1.29 Z37 5 31^ ft 11.6 3.40 13.36 6.34 1.98 6.18 2.00 1.35 * g JL 3- 8 s 42 13.9 4.10 16.18 6.39 1.99 7.6ft 2.45 1.37 li 5}' wheel at center. 134 ATOMIC WEIGHTS AND DENSITIES Valency and Atomic Weight of Elements Element Symbol Atomic Atomic and Weierht Weight Valency H=l 0=16 Aluminum Al (3) 26.88 Antimony Sb (3) 119.23 Arsenic As (3) 74.4 Barium Ba (2) 136.3 Bismuth Bi (3) 206.83 Boron B (2) 10.91 Bromine Br (1) 79.32 Cadmium Cd (2) 111.50 Calcium Ca (2) 39.81 Carbon C (4) 11.90 Chlorine Cl (1) 35.17 Chromium Cr (2) (3) 51.68 Cobalt Co (2) (3) 58.53 Copper Cu (1) (2) 63.1 Fluorine F (1) 18.85 Gold Au (3) 195.62 Hydrogen H (1) 1.00 Iodine 1(1) 125.95 Iridium Ir (2) (4) 191.46 Iron Fe (2) (3) 55.44 Lead Pb (2) 205.24 Lithium Li (1) 6.97 Magnesium Mg (2) 24.17 Manganese Mn (2) (3) 54.55 Mercury Hg (1) (2) 198.4 Nickel Ni (2) (3) 58.23 Nitrogen N (3) 13.93 Oxygen O (2) 15.87 Palladium Pd (2) 105.15 Phosphorus P (3) 30.75 Platinum Pt (2) (4) 193.24 Potassium K (1) 38.84 Selenium Se (2) 78.57 Silicon Si (4) 28.17 Silver Ag (1) 107.07 Sodium Nad) 22.87 Strontium Sr (2) 86.9 Sulphur S (2) 31.8 Tellurium Te (2) 125.98 Tin Sn (2) 118.05 Zinc Zn (2) (4) 64.88 120.2 75.0 137.4 208.5 11.0 79.96 112.4 40.13 12.0 35.45 52.1 59.0 63.6 19.0 197.2 1.008 126.97 193.0 55.9 206.9 7.03 24.36 55.0 200.0 58.7 14.04 16.0 106.0 31.0 194. S 39.15 79.2 28.4 107.93 23.05 87.6 32.06 127.0 119.0 65.4 Densities of Substances Wt. of 1 cu. ft. of Water at 4C or 39.1Fz=62.4245 Ib. Air at 0C. and Aluminum Antimony Asphaltum Brass, cast Brass, rolled Brick, soft Brick, common Brick, hard Brick, pressed Brick, fire Grams per cu.cm. Pounds per cu.in .001293 2.67 6.76 1.4 8.1 8.4 1.6 1.79 2.0 2.16 2.3 .0963 .2439 .2917 .3031 Pounds per cu.ft. .0807 166.5 421.6 87.3 504 524 100 112 125 135 145 DENSITIES OF SUBSTANCES 135 Wt. of 1 cu. ft. of Water at 4 p Cor39.1F= 62.4245 Ib. Grams per cu. cm. Pounds per cu- in. Pounds per cu. ft. Cement, Rosendale, loose Cement, Portland Coal, anthracite .96 1.25 1.5 60 78 93.5 Coal, anthracite, loose Coal, bituminous Coal, bituminous, loose 1.3 53 84 48 Coke, loose Concrete, plain Concrete, reinforced 2.0 2.4 28 125 150 Copper, cast Earth, dry, loose Earth, dry, rammed 8.85 .3195 552 75 95 Glass Glass, common window Gold 2.6 2.52 19.26 .6949 162 157 1200.9 Granite Gravel Ice 2.6 1.75 .92 162 109 57.4 Iridium Iron, wrought Iron, cast 22.7 7.7 7.22 .8076 .2779 .2604 1396 480 450 Lead Limestone, marble Maple, dry 11.38 2.6 .4106 709.7 164 49 Masonry Mercury, 15 C. Mica 13.58 2.93 .490 125-165 846.8 183 Mud Nickel Oak, dry 8.8 .3175 115 548.7 60 Petroleum Pine, dry Fitch .878 1.15 54.8 25-45 72 Platinum Salt Sand 21.5 1.1 .7758 1347 68.6 90 to 110 Sandstone Silver Slate 2.4 10.51 2.8 .3791 150 655.1 175 Snow, fresh fallen Snow, moistened Soapstone 2.7 5 to 12 15 to 50 170 Steel Sulphur Tar 7.85 2.0 1 .2834 489.6 125 62.4 Tin* \vater, 4C. Water, 100 C. 7.35 1 .958 .2652 458.3 62.42 59.83 Water, sea water Zinc 1.028 7.0 .2526 64.08 437 136 HEAT Comparison of Centigrade and Fahrenheit. c F C F C F C F C F 40 40 17.8 4.4 40 27 80.6 49 120.2 39.4 39 17.2 1 5 41 27.2 81 49.4 121 39 38.2 17 1.4 5.6 42 27.8 82 50 122 38.9 38 16.7 2 6 42.8 28 82.4 j 50.6 123 38.3 37 16.1 3 6.1 43 28.3 83 51 123.8 38 36.4 16 3.2 6.7 44 28.9 84 51.1 124 37.8 36 15.6 4 7 44.6 29 84.2 51.7 125 37.2 35 15 5 7.2 45 30 86 52 125.6 37 34.6 14.4 6 7.8 46 29.4 85 52.2 126 36.7 34 14 6.8 8 46.4 30.6 87 52.8 127 36.1 33 13.9 7 8.3 47 31 87.8 53 127.4 36 32.8 13.3 8 8.9 48 31.1 88 53.3 128 35.6 32 13 8.6 9 48.2 31.7 89 53.9 129 35 31 12.8 9 9.4 49 32 89.6 54 129.2 34.4 30 12.2 10 10 50 32.2 90 ' 54.4 130 34 29.2 12 10.4 10.6 .51 32.8 91 55 131 33.9 29 11.7 11 11 51.8 33 91.4 55.6 132 33.3 28 11.1 12 11.1 52 33.3 92 56 132.8 33 27.4 11 12 2 11.7 53 33.9 93 56.1 133 32.8 27 10.6 13' 12 53.6 34 93.2 56.7 134 39 2 26 10 14 12.2 54 34.4 94 57 134.6 32 " 25.6 9.4 15 12.8 55 35 95 57.2 135 31.7 25 9 15.8 13 55.4 35.6 96 57.8 136 31.1 24 8.9 16 13.3 56 36 96.8 58 136.4 31 23.8 - 8.3 17 13.9 57 36.1 97 58.3 137 30.6 23 8 17.6 14 57.2 36.7 98 58.9 138 30 22 7.8 18 14.4 58 37 98.6 59 138.2 29.4 21 7.2 19 15 59 37.2 99 59.4 139 29 20.2 7 19.4 15.6 60 37.8 100 60 140 28.9 20 6.7 20 16 60.8 38 100.4 60.6 141 28.3 19 6.1 21 16.1 61 38.3 101 61 141.8 28 18.4 6 21 2 16.7 62 38.9 102 61.1 142 27.8 18 5.6 22 17 62.6 39 102.2 61.7 143 9 7 2 17 5 23 17.2 63 39.4 103 62 143.6 27 16.6 4.4 24 17.8 64 40 104 62.2 144 26.7 16 4 24.8 18 64.4 40.6 105 62.8 145 26.1 15 3.9 25 18.3 65 41 105.8 63 145.4 26. 14.8 3.3 26. 18.9 66. 41.1 106. 63.3 146. 25.6 14. 3. 26.6 19. 66.2 41.7 107. 63.9 147. 25. 13. 2.8 27. 19.4 67. 42. 107.6 64. 147.2 24.4 12. 2.2 28. 20. 68. 42.2 108. 64.4 148. 24. 11.2 2. 28.4 20.6 69. 42.8 109. 65. 149. 23.9 11. -j 7 29. 21. 69.8 43. 109.4 65.6 150. 23.3 10. 1.1 30. 21.1 70. 43.3 110. 66. 150.8 23. 9.4 1. 30.2 21.7 71. 43.9 111. 66.1 151. 22.8 9. 0.6 31. 22.^ 71.6 44. 111.2 66.7 152. 22.2 8. 32. 72. 44.4 112. 67. 152.6 22." 7.6 0.6 33. 2 2". 8 73. 45. 113. 67.2 153. 21.7 7. 1. 33.8 23. 73.4 45.6 114. 67.8 154. 21.1 6. 1.1 34. 23.3 74. 46. 114.8 68. 154.4 21. 5.8 1.7 35. 23.9 75. 46.1 115. 68.3 155. 20.6 5. 2. 35.6 24. 75.2 46.7 116. 68.9 156. 20. 4. 2.2 36. 24.4 76. 47. 116.6 69. 156.2 19.4 3. 2.8 37. 25. 77. 47.2 117. 69.4 157. 19. 2.2 3. 37.4 25.6 78. 47.8 118. 70. 158. 18.9 2. 3.3 38. 26. 78.8 48. 118.4 70.6* 159. 18 3 1. 3.9 39. 26.1 79. 48.3 119. 71. 159.S 18. 0.4 4. 39.2 26.7 80. 48.9 120. 71.1 160. HEAT 137 Comparison of Centigrade and Fahrenheit. c F C F C F C F C F 71.7 161. ! 93.9 201. 116.1 241. 138.3 281. 161. 321. S 72. 161.6 94. 201.2 116.7 242. 138.9 282. 161.1 322. 72.2 162. 94.4 202. 117. 242.6 139. 282.2 161.7 323. 72.8 163. 95. 203. 117.2 243. 139.4 283. 162. 323.6 /3. 163.4 95.6 204. 117.8 244. 140. 284. 162.2 324. 73.3 164. 96. 204.8 118. 244.4 140.6 285. 162.8 325. 73.9 165. 96.1 205. 118.3 245. 141., 285.8 163. 325.4 74. 165.2 96.7 206. 118.9 246. 141.1 286. ! 163.3 326. 74.4 166. 97. 206.6 119. 246.2 141.7 287. 163.9 327. 75. 167. 97.2 207. 119.4 247. 142. 287.6 164. 327.2 75.6 168. 97.8 208. 120. 248. 142.2 288. 164.4 328. 76. 168.8 98. 208.4 ; 120.6 249. 142.8 289. 165. 329. 76.1 169. 98.3 209. ; 121. 249.8 143. 289.4 165.6 330. 76.7 170. 98.9 210. 121.1 250. 143.3 290. 166. 330.8 77. 170.6 99. 210.2 121.7 251. 143.9 291. 166.1 331. 77.2 171. 99.4 211. 122. 251.6 144. 29121 166.7 332. 77.8 172. 100. 212. 122.2 252. 144.4 292! 167. 332.6 78. 172.4 100.6 213. ' 122.8 253. 145. 293. 167.2 333. 78.3 173. 101. 213.8 123. 253.4 145.6 294. 167.8 334. 78.9 174. 101.1 214. 123.3 254. 146. 294.8 168. 334.4 79. 174.2 101.7 215. ! 123.9 255. 146.1 295. 168.3 335. 79.4 175. 102. 215.6 . 124. 255.2 146.7 296. 168.9 336. 80. 176. 102.2 216. 124.4 256. 147. 296. 6 * 169. 336.2 80.6 177. 102.8 217. 125. 257. 147.2 297. 169.4 337. 81. 177.8 103. 217.4 ! 125.6 258. 147.8 298. 170. 338. 81.1 178. 103.3 218. 126. 258.8 148. 298 4 170.6 339. 81.7 179. 103.9 219. 126.1 259. 148.3 299^ 171. 339.8 82. 179.6 104. 219.2 126.7 260. 148.9 300. 171.1 340. 82.2 180. ! 104.4 220. 127. 260.6 149. 300.2 171.7 341. 82.8 181. 105. 221. 127.2 261. 149.4 301. - 172. 341.6 83. 181.4 105.6 222. 127.8 262. 150. 302. 172.2 342. 83.3 182. i 106. 222.8 128. 262.4 150.6 303. 172.8 343. 83.9 183. ' 106.1 223. 128.3 263. 151. 303.8 173. 343.4 84. 183.2 106.7 224. 128.9 264. 151.1 304. 173.3 344. 84.4 184. 107. 224.6 129. 264.2 151.7 305. 173.9 345. 85. 185. 107.2 225. 129.4 265. 152. 305.6 174. 345.2 85.6 186. W8 226. 130. 266. 152.2 306. 174.4 346. 6. 186.8 108. 226.4 130.6 267. 152.8 307. 175. 347. 86.1 187. | 108.3 227. 131. 267.8 153. 307.4 175.6 348. 86.7 188. ! 108.9 228. 131.1 268. 153.3 308. 176. 348.8 87. 188.6 109. 228.2 131.7 269. 153.9 309. 176.1 349. 87.2 189. 109.4 229. 132. 269.6 154. 309.2 176.7 350. 87.8 190. 110. 230. 132.2 270. 154.4 310. 177. 350.6 88. 190.4 110.6 231. 132.8 271. 155. 311. 177.2 351. 88.3 191. 111. 231.8 133. 271.4 155.6 312. 177.8 352. 88.9 192. Hl.l 232. 133.3 272. 156. 312.8 178. 352.4 89. 19-> 9. 111.7 233. 133.9 273. 156.1 313. 178.3 353. 89.4 193. 112. 233.6 134. 273.2 156.7 314. 178.9 354. 90. 194. H2.2 234. 134.4 274. 157. 314.6 179. 354.2 90.6 195. i 112.8 235. 135. 275. 157.2 315. 179.4 355. 91. 195.8 ! 113. 235.4 135.6 276. 157.8 316. 180. 356. 91.1 196. i H3.3 236. 136. 276.8 158. 316.4 180.6 357. 91.7 197. 113.9 237. 136.1 277. 158.3 317. 181. 357.8 92. 197.6 114. 237.2 136.7 278. 158.9 318. 181.1 358. 92.2 198. H4.4 238. 137. 278.6 159. 318.2 181.7 359. 92.8 199. 115. 239. 137.2 279. 159.4 319. 182. 359.6 93. 199.4 i 115.6 240. 137.8 280. 160. 320. 182.2 360. 200. 116. 240.8 138. 280.4 160.6 321. 182.8 361. 138 HEAT Coefficients of Expansion and Specific Heat. Solids. Coefficient x 10 7 1 F 1 C Aluminum 123 222 Antimony 63 113 Brass, cast 96 172 Brass, plate 105 189 Bronze 99 177 Bismuth 97 176 Cement, Portland 59 107 Concrete 80 143 Copper 89 160 Glass, flint 45 81 Glass, thermometer 50 90 Glass, hard 40 71 Gold 79 142 Ice Iridium 36 64 Iron, cast 56 100 Iron, wrought 65 117 Lead 157 283 Nickel 70 125 Plaster 92 166 Platinum 48 86 Platinum 85% } Iridium 15% f 45 82 Porcelain 20 36 Silver 108 194 Steel, cast 64 114 Steel, tempered 69 124 Sulphur Tin 116 209 Wood, pine 28 50 Zinc 141 253 Liquids (Cu. Expansion.) Alcohol 7780 14000 Glycerine Olive Oil 2780 4110 5000 7400 Petroleum 5530 10000 Mercury 1011 1820 Turpentine 5530 10000 Specific Heat .21 .05 \ .094 .093 .20 .032 .502 .130 .031 .11 .033 .055 .117 .18 .056 .094 .58 .58 .40 .50 .033 .42 The specific heat of water for t in degrees cent, is c = 1 + .00004 t + .0000009 t\ HEAT Density and Volume of Water. 139 Temp. C F Density Volume Temp. C F Density Volume 32 .99987 1.00013 60 140 .9833 1.0170 2 35.6 .99997 1.00003 80 176 .9719 1.0289 4 39.2 1.00000 1.00000 100 212 .9586 1.043-4 6 42.8 .99997 1.00003 120 248 .9435 1.0599 10 50. .99973 1.00027 140 284 .9263 1.0795 15 59. .99913 1.00087 160 320 .9076 1.1018 24 75.2 .99733 1.00268 180 356 .8875 1.1268 40 104 .99233 1.00773 200 392 .8662 1.1544 Melting and Boiling Points. At 760 mm. Mercury. Substance Melting Cent. or Freezing Boiling Fahr. Cent. Fahr. Alcohol, absolute Aluminum Ammonia gas Antimony 100 625 78 430 148 78.5 1157 108 33 806 173 27 Brass Bronze Copper Glass 900 900 1084 8001400 1652 1652 1983 14702500 Gold Iridium Iron, cast Iron, wrought 1064 1950 1150 1550 1945 3540 2100 2820 Lead Mercury Nickel Paraffine 327 39 1470 54 620 38.2 357 2680 129 300 675 570 Platinum Silver Steel Sulphur 1800 962 1350 115 3270 1760 2460 239 445 833 Sulphur Dioxide Tin Zinc 76 232 420 105 10 450 790 915 14 1680 Boiling Point of Water for various pressures in mm. of Mercury. Pressure. Cent Pressure. Temp. Cent. Pressure. Temp. Cent. 680 96.92 690 97.32 700 97.71 710 98.11 720 98.49 730 98.88 740 99.26 750 99.63 755, 99.82 760 100.00 765 770 780 790 800 100.18 100.37 100.73 101.09 101.44 140 HEAT Latent Heat. Latent Heat 760 mm. Mercury Substance Fusion Calories B. T. U. Vaporization Calories B . T. U Alcohol Lead Mercury 6 2.8 10.8 5. 210 378 62 111.6 Platinum Silver Sulphur 27 21 9 48.6 37.8 16.2 362 652 Tin Water (Ice) (Steam) Zinc 13 80.1 28 23.4 144.2 50.4 537 965. S Gases. Boyle's Law: Pv = constant. Charles' Law : P = P O (1 + '- ' \- r. for cent temp.; 273 P' (1 32\ = P / T' A ) 491.4 forFahr.temp. 273 t' 32 491 t = temp, in deg. cent., t' temp, in deg. Fahr. T = 273 + t and T' = 459.4 -r t' P = pressure at t or t', P = pressure at C, P' = pressure at 32 F. Combining these two laws gives the equation, P v = R T for metric units or Pv = E'T' for English; in which the gas constant R or R ] is inversely proportional to the density, or to the molecular weight m of the gas; if the molecular weight of oxygen is taken as 32, and r = volum e at 0C and v 1 = volume at 32 F, then R = 273 847 m and / t; 1 = 1541.5 491.4 m HEAT 141 Density and Constant of Gases. Gas Density Air = l * Weight of 1 of 1 cu. m. cu. ft. Constant Metric English units units It W Air Alcohol Ammonia Carbon Dioxide Hydrogen Oxygen Steam Sulphur Dioxide 1.000 1.590 .590 1.520 .0696 1.105 .623 2.213 1.295 2.059 .763 1.966 .090 1.430 .806 2.863 .8073 1.284 .476 1.227 .0562 .892 .503 1.787 29.26 18.4 49.6 19.25 420 26.5 47.0 13.2 53.27 33.5 90.27 35.03 764 48.2 85.54 24. 0.5 * At C = 32 F and 760 mm. mercury, or 14.697 Ib. per sq. in. Specific Heat of Gases. Gas Specific Constant Pressure Cp Heat Constant -vP- Volume Cv Cv Air Alcohol Ammonia Carbon Dioxide Hydrogen Oxygen Steam Sulphur Dioxide .238 .45 .53 .21 3.41 ' .217 .50 .15 .170 1.40 .40 .14 .41 .28 .16 .28 2.42 .40 .155 .40 .39 .28 .12 .25 Factor of Evaporation. The Factor of Evaporation F == - ~ ~ h 965.8 where H is the total heat of the steam at the given pressure and h is the heat of the liquid at the given temper- ature. In the following table, three decimals are given which is usually sufficient. If very great accuracy is required F must be found by the above formula, taking the values of H and h from the table for saturated steam. F may be found very accurately to four decimals by the following formula. Ft = F 32 0.001035 (t 32); in which F t is the factor for any given pressure of steam with feed water at t Fahr. temperature and jP 32 is the factor for the same steam pressure with feed water at 32 Fahr. 142 HEAT Factors of Evaporation. Gauge Pressure Lbs. 32 Temperature of Feed Water. Deg-. Fahr. 40 50 60 70 80 90 100 110 120 1.1873 1.179 1.169 1.158 1.148 1.138 1.127 1.117 1.107 1.096 10 1.1958 1.188 1.177 1.167 1.157 1.146 1.136 1.125 1.115 1.105 20 1.2019 1.194 1.183 1.173 1.163 1.152 1.142 1.132 1.121 1.111 30 1.2068 1.199 1.188 1.178 1.168 1.157 1.147 1.136 1.126 1.116 40 1.2107 1.203 1.192 1.182 1.171 1.161 1.151 1.140 1.130 1.120 50 1.2141 1.206 1.196 1.185 1.175 1.164 1.154 1.144 1.133 1.123 60 1.2172 1.209 1.199 1.188 1.178 1.167 1.157 1.147 1.136 1.126 70 1.2199 1.212 1.201 1.191 1.181 1.170 1.160 1.150 1.139 1.129 80 1.2224 1.214 1.204 1.193 1.183 1.173 1.162 1.152 1.142 1.131 90 1.2247 1.216 1.206 1.196 1.185 1.175 1.165 1.154 1.144 1.133 100 1.2269 1.219 1.208 1.198 1.188 1.177 1.167 1.156 1.146 1.136 110 1.2288 1.221 1.210 1.200 1.189 1.179 1.169 1.158 1.148 1.138 120 1.2307 1.223 1.212 1.202 1.191 1.181 1.171 1.160 1.150 1.140 130 1.2322 1.224 1.214 1.203 1.193 1.183 1.172 3.162 1.152 1.141 140 1.2341 1.226 1.215 1.205 1.194 1.184 1.174 1.164 1.153 1.143 150 1.2358 1.227' 1.217 1.207 1.196 1.186 1.176 1.165 1.155 1.145 160 1.2372 1.229 1.218 1.208 1.197 1.187 1.177 1.167 1.156 1.146 170 1.2386 1.230 1.220 1.210 1.199 1.189 1.179 1.168 1.158 1.147 180 1.2401 1.232 1.221 1.211 1.200 1.190 1.180 1.170 1.159 1.149 190 1.2415 1.233 1.223 1.212 1.202 1.192 1.181 1.171 1.160 1.150 200 1.2427 1.234 1.224 1.214 1.203 1.193 1.183 1.172 1.162 1.151 250 1.2485 1.240 1.230 1.219 1.209 1.199 1.188 1.178 1.168 1.157 300 1.2535 1.245 1.235 1.224 1.214 1.204 1.193 1.183 1.173 1.162 Gauge Pressure Lbs. 130 Temperature of Feed Water. 140 150 160 170 180 Deg. Fahr. 190 200 210 212 1.086 1.075 1.065 L.055 1.045 1.034 1.024 1.013 1.003 1.000 10 1.095 1.084 1 .074 1.063 1.053 1.043 1.032 1.022 1.012 1.009 20 1.101 1.090 1 .080 1.069 1.059 1.049 1.038 1.028 1.018 1.015 30 1.106 1.095 1 .085 1.074 1.064 1.054 1.043 1.033 1.023 1.020 40 1.110 1.099 1 .089 1.078 1.068 1.058 1.047 1.037 1.027 1.024 50 1.113 1.102 1 .091 1.081 1.070 1.060 1.050 1.039 1.029 1.027 60 1.116 1.105 1 .095 1.084 1.074 1.063 1.053 1.043 1.032 1.030 70 1.118 1.108 1 .098 1.087 1.077 1.066 1.056 1.045 1.035 1.033 80 1.121 1.110 1 .100 1.090 1.079 1.069 1.058 1.048 1.037 1.035 90 1.123 1.113 1 102 .092 1.081 1.071 1.060 1.050 1.040 1.038 100 1.125 1.115 1 .104 .094 1.083 1.073 1.063 1.052 1.042 1.040 110 1.127 1.117 1 .106 .096 1.085 1.075 1.065 1.054 1.044 1.042 120 1.129 1.119 1 108 .098 1.087 1.077 1.066 1.056 1.046 1.044 130 1.130 1 120 1 110 .100 1.089 1.079 1.068 1.058 1.047 1.045 140 1.132 1.122 1 111 .101 1.091 1.080 1.070 1.059 1.049 1.047 150 1.134 1.124 1 113 .103 1.092 1.082 1.071 1.061 1.051 1.049 160 1.136 1.125 1 115 .104 1.094 1.083 1.073 1.063 1.052 1.050 170 1.137 1.127 1 116 .106 1.095 1.085 1.074 1.064 1.053 1 051 180 1.138 1.128 1 118 ] .107 1.097 1.086 1.076 1.065 1.055 1.053 190 1.140 1.129 1 119 1.108 1.098 1.088 1.077 1.067 1.056 1.054 200 1.141 1.131 1 120 1.110 1.099 1.089 1.078 1.068 1.057 1.055 250 1.147 1.136 1 126 1.116 1.105 1.095 1.084 1.074 1.063 1.061 300 1.152 1.141 1 131 1.121 1.110 1.100 1.089 1.079 1.068 1.066 SATURATED STEAM 143 Properties of Saturated Steam Abso- lute Tcm- Press perature Lbs. Des. per Fah. Sq. in. Heat Volume Relative, Specific Water Cu. ft. at per Ib. ?9F. Density Wt. of 1 CU. ft. Lbs. Total of Liquid of Vapor- ization of Internal Work of External Work Entropy of Liquid 0.10 35.0 1092.6 3.0 1089.6 1033.6 56.0 .0061 188724 3022 .00033 0.15 45.5 1095.8 13.6 1082.3 1025.4 56.9 .0272 127023 2050 .00051 0.20 53.3 1098.2 21.4 1076.8 1019.3 57.5 .0426 97110 1555 .00064 0.30 64.7 1101.7 32.8 1068.9 1010.3 58.6 .0645 65760 1053 .00095 0.40 73.1 1104.2 41.2 1063.0 1003.7 59.3 .0804 50397 800.7 .00125 0.50 79.8 1106.2 47.9 1058.3 998.4 59.9 .0929 40411 647.1 .00155 0.60 85.5 1108.0 53.5 1054.5 994.1 60.4 .1031 33973 544.0 .00184 0.80 94.6 1110.8 62.6 1048.2 986.9 61.2 .1197 25842 413.8 .00242 1.00 102.0 1113.1 70.0 1043.0 981.1 61.9 .1329 20896 334.6 .00299 1.25 109.6 1115.4 77.6 1037.8 975.2 62.7 .1462 16549 265.0 .00369 150 115.9 1117.3 83.9 1033.4 970.1 63.3 .1573 14226 227.8 .00438 1.75 121.4 1119.0 89.5 1029.4 965.7 63.7 .1669 12303 197.0 .00508 2.0 126.3 1120.5 94.4 1026.1 961.9 64.2 .1754 10841 173.6 .00576 2.5 134.6 1123.0 102.8 1020.4 955.1 65.1 .1895 8780 140.6 .00711 3.0 141.6 1125.1 109.8 1015.3 949.5 65.8 .2013 7394 118.4 .00844 3.5 147.7 1127.0 116.0 1010.0 944.6 66.4 .2114 6395 102.4 .00976 4.0 153.1 1128.6 121.4 1007.2 940.4 66.8 .2203 5640 90.31 .01107 4.5 158.0 1130.1 126.4 1003.7 936.4 67.3 .2284 5039 80.70 .01239 5.0 162.3 1131.5 130.7 1000.8 933.1 67.7 .2353 4573 73.22 .01366 5.5 166.4 1132.7 134.8 997.9 929.8 68.1 .2419 4180 66.93 .01494 6.0 170.1 1133.8 138.6 995.2 926.7 68.5 .2480 3851 61.67 .01622 6.5 173.6 1134.9 142.1 992.8 924.0 68.8 .2535 3573 57.21 .01748 7.0 176.9 1135.9 145.4 990.5 921.4 69.1 .2587 3333 53.37 .01874 7.5 180.0 1136.8 148.5 988.3 918.9 69.4 .2636 3123 50.01 .02000 8.0 182.9 1137.7 151.5 986.2 916.5 69.7 .2682 2939 47.07 .02125 8.5 185.7 1138.6 154.3 984.3 914.4 70.0 .2725 2773 44.41 .02250 9.0 188.3 1139.4 156.9 982.5 912.4 70.1 .2766 2631 42.13 .02374 9.5 190.8 1140.1 159.4 980.7 910.3 70.4 .2805 2501 40.05 .02497 10.0 193.2 1140.9 161.9 979.0 908.4 70.6 .2842 2383 38.16 .02621 10.5 195.6 1141.6 164.3 977.3 906.5 70.8 .2878 2276 36.45 .02743 11.0 197.8 1142.3 166.5 975.8 904.8 71.0 .2912 2178 34.88 .02866 11.5 200.0 1142.9 168.7 974.2 903.0 71.2 .2946 2086 33.40 .02990 12.0 202.0 1143.6 170.7 972.9 901.5 71.4 .2976 2007 32.14 .03111 12.5 204.0 1144.2 172.7 971.5 900.0 71.5 .3007 1931 30.92 .03234 13.0 205.9 1144.7 174.6 970.1 898.4 71.7 .3035 1862 29.82 .03355 13.5 207.8 1145.4 176.7 968.6 896.8 71.9 .3066 1796 28.76 .03478 14.0 209.6 1145.8 178.3 967.5 895.5 72.0 .3091 1735 27.79 .03600 14.5 211.3 1146.1 180.1 966.3 894.2 72.2 .3117 1680 26.91 .03677 15 213.0 1146.9 181.8 965.1 892.6 72.5 .3143 1633 26.15 .03826 16 216.3 1147.9 185.1 962.8 890.0 72.8 .3192 1536 24.59 .04067 144 SATURATED STEAM Properties of Saturated Steam Abso- lute x Tem- Press. perature Lbs. Deg. Total per Fah. Sq. in. Heat of of Liquid Vapor- ization of Internal Work of External Work Entropy of Liquid Volume Relative. Specific. Water Cu. ft. at per Ib. 39F Density Wt. of 1 cu. ft. Lbs. 17 219.4 1148.9 188.3 960.6 887.6 73.0 .3238 1451 23 22 .04307 18 222.4 1149.8 191.3 958.5 885.3 73.2 .3282 1375 22.00 .04547 19 225.2 1150.7 194.1 956.6 883.2 73.4 .3324 1305 20.90 .04786 20 227.9 1151.5 196.9 954.6 881.0 73.6 .3363 1243 19.91 .05023 21 230.5 1152.3 199.5 952.8 879.0 73.8 .3401 1187- 19.01 .05259 22 233.1 1153.0 202.0 951.0 877.0 74.0 .3438 1137 18.20 .05495 23 235.5 1153.7 204.5 949.2 875.0 74.2 .3473 1090 17.45 .05731 24 237.8 1154.4 206.8 947.6 873.2 74.4 .3506 1047 16.76 .05966 25 240.0 1155.1 209.1 946.0 871.5 74.5 .3539 1007 16.13 .06199 26 242 2 1155.8 211.2 944.6 869.9 74.7 .3570 971.1 15.55 .06432 27 244.3 1156.5 213.4 943.1 868.2 74.9 .3600 936.7 15.00 .06666 28 246.3 1157.1 215.4 941.7 866.7 75.0 .3629 904.9 14.49 .06899 29 248.3 1157.7 217.4 940.3 865.1 75.2 .3657 876.2 14.03 .07130 30 250.3 1158.3 219.4 938.9 863.6 75.3 .3685 848.7 13.59 .07360 31 252.2 1158.8 221.3 937.5 862.0 75.5 .3712 823.1 13.18 .07590 32 254.0 1159.4 223.1 936.3 860.7 75.6 .3737 798.1 12.78 .07821 33 255.8 1159.9 224.9 935.0 859.2 75.8 .3762 775.0 12.41 .08051 34 257.5 1160.4 226.7 933.7 857.8 75.9 .3787 753.8 12.07 .08280 35 259.2 1161.0 228.4 932.6 856.6 76.0 .3811 733.8 11.75 .08508 36 260.9 1161.5 230.0 931.5 855.3 76.2 .3834 715.0 11.45 .08736 37 262.5 1162.0 231.7 930.3 854.0 76.3 .3856 697.0 11.16 .08964 38 264.1 1162.5 233.3 929.2 852.8 76.4 .3878 679.4 10.88 .09191 39 265.6 1163.0 234.8 928.2 851.7 76.5 .3900 663.2 10.62 .09417 40 267.1 1163.4 236.4 927.0 850.3 76.7 .3921 647.6 10.37 .09644 41 268.6 1163.9 237.9 926.0 849.2 76.8 .3942 632.6 10.13 .09869 42 270.1 1164.3 239.3 925.0 848.1 76.9 .3962 618.9 9.91 .1009 43 271.5 1164.8 240.8 924.0 847.0 77.0 .3982 605.1 9.69 .1032 44 272.9 1165.2 242.2 923.0 845.9 77.1 .4001 592.0 9.48 .1054 45 274.3 1165.6 243.6 922.0 844.8 77.2 .4020 580.2 9.29 .1077 46 275.7 1166.0 245.0 921.0 843.7 77.3 .4038 568.3 9.10 .1099 47 277.0 1166.4 246.3 920.1 842.7 77.4 .4056 556.4 8.91 .1122 48 278.3 1166.8 247.6 919.2 841.7 77.5 .4074 545.8 8.74 .1144 49 279.6 1167.2 248.9 918.3 840.7 77.6 .4092 535.2 8.57 .1166 50 280.9 1167.6 250.2 917.4 839.7 77.7 .4109 525.2 8.41 .1188 51 282.1 1168.0 251.5 916.5 838.7 77.8 .4126 515.8 8.26 .1211 52 283.3 1168.4 252.7 915.7 837.8 77.9 .4143 506.5 8.11 .1233 53 284.5 1168.7 253.9 914.8 836.8 78.0 .4160 497.7 7.97 .1255 54 285.7 1169.1 255.1 914.0 835.9 78.1 .4175 489.0 7.83 .1277 55 286.9 1169.4 256.3 913.1 834.9 ^ 78 2 .4191 480.9 7.70 .1299 56. 288.0 1169.8 257.5 912.3 834.0 78.3 .4207 472.7 7.57 .1321 SATURATED STEAM 145 Properties of Saturated Steam Abso- lute Tem- 3 ress. perature Lbs. Deg. per Fah. 3q. in Heat Entropy of Liquid Volume Relative. Specific. Density. Water Cu. ft. Wt. of at per Ib. 1 cu. ft. 39F. Lbs. Total of Liquid of Vapor- ization Internal External Work Work 57 289.2 1170.1 258.6 911.5 833.1 78.4 4222 464.6 7.44 .1344 58 290.3 1170.5 259.7 910.8 832.4 78.4 !4237 457.1 7.32 .1366 59 291.4 1170.8 260.8 910.0 831.5 78.5 .4252 450.3 7.21 .1387 60 292.5 1171.2 261.9 909.3 830.7 78.6 .4267 443.4 7.10 .1409 61 293.6 1171.5 263.0 90S.5 829.8 78'.7 .4281 436.5 6.99 .1431 62 294.7 1171.8 264.1 907.7 828.9 78.8 .4295 429.7 6.88 .1453 63 295.7 1172.1 265.2 906.9 828.0 78.9 .4309 423.4 6.78 .1475 64 296.7 1172.4 266.2 906.2 827.3 78.9 .4323 417.2 6.68 .1497 65 297.8 1172.7 267.2 905.5 826.5 79.0 .4337 410.9 6.58 .1519 66 298.8 1173.0 268.3 904.7 825.6 79.1 .4350 405.3 6.49 .1541 67 299.8 1173.3 269.3 904,0 824.8 79.2 .4363 399.7 6.40 .1562 68 300.8 1173.6 270.3 903.3 824.1 79.2 .4376 394.1 6.31 .1584 69 301.7 1173.9 271.2 902.7 823.4 79.3 .4389 389.1 6.23 .1606 70 302.7 1174.3 272.2 902.1 822.7 79.4 .4402 383.4 6.14 .1628 71 303.7 1174.6 273.2 901.4 821.9 79.5 .4415 378.4 6.06 .1649 72 304.6 1174.9 274.1 900.8 821.3 79.5 .4428 373.4 5.98 .1671 73 305.5 1175.2 275.1 900.1 820.5 79.6 .4440 369.1 5.91 .1693 74 306.5 1175.4 276.0 899.4 819.7 79.7 .4452 364.1 5.83 .1714 75 307.4 1175.7 276.9 898.8 819.1 79.7 .4464 359.7 5.76 .1736 76 308.3 1176.0 277.8 898.2 818.4 79.8 .4476 355.3 5.69 .1757 77 309.2 1176.2 278.7 897.5 817.6 79.9 .4487 351.0 5.62 .1779 78 310.1 1176.5 279.6 896.9 817.0 79.9 .4499 346.6 5.55 .1801 79 310.9 1176.8 280.5 896.3 816.3 80.0 .4511 342.8 5.49 .1822 80 311.8 1177.0 281.4 895.6 815.5 80.1 .4522 338.5 5.42 .1843 81 312.7 1177.3 282.3 895.0 814.9 80.1 .4534 334.7 5.36 .1865 82 313.5 1177.6 283.2 894.4 814.2 80.2 .4545 331.0 5.30 .1886 83 314.4 1177.8 284.1 893.7 813T.4 80.3 .4557 327.2 5.24 .1908 84 315.2 1178.1 285.0 893.1 812.8 80.3 .4568 323.5 5.18 .1930 85 316.0 1178.3 285.8 892.5 812.1 80.4 .4579 319.7 5.12 .1951 86 316.8 1178.6 286.7 891.9 811.5 80.4 .4590 316.6 5.07 .1973 87 317.6 1178.8 287.5 891.3 810.8 80.5 .4601 312.9 5.01 .1994 88 318.4 1179.1 288.4 890.7 810.2 80.5 .4612 309.8 4.96 .2016 89 319.2 1179.3 289.2 890.1 809.5 80.6 .4622 306.6 4.91 .2037 90 320.0 1179.6 290.0 889.6 808.9 80.7 .4633 303.5 4.86 .2058 91 320.8 1179.8 290.8 889.0 808.3 80.7 .4643 300.4 4.81 .2080 92 321.6 1180.0 291.6 888.4 807.6 80.8 .4653 297.3 4.76 .2101 93 322.4 1180.3 292.4 887.9 807.1 80.8 .4663 294.1 4.71 .2122 94 323.1 1180.5 293.2 887.3 806.4 80.9 .4673 291.0 4.66 .2144 95 323.9 1180.7 294.0 886.7 805.8 80.9 .4683 288.5 4.62 .2165 96 324.6 1181.0 294.8 886.2 805.2 81.0 .4693 285.4 4.57 .2186 146 SATURATED STEAM Properties of Saturated Steam Abso- lute Tem- Press. peratur* Lbs. Deg. per Fah. Sq. in. Heat Volume Relative. Specific. Density. Water Cu. ft. Wt. of at per Ib. 1 cu. ft 39Ft Lb.. Total of Liquid of Vapor- ization of Internal Work of Entropy External of Work Liquid 97 325.4 1181.2 295.6 885.6 804.6 81.0 .4703 282.9 4.53 .2208 98 326.1 1181.4 296.4 885.0 803.9 81.1 .4713 280.4 4.49 9 >>() 99 326.9 1181.6 297.1 884.5 803.4 81.1 .4723 277.3 4.44 .'2250 100 327.6 1181.9 297.9 884.0 802.8 81.2 .4733 274.8 4.40 .2271 101 328.3 1182.1 298.6 883.5 802.3 81.2 .4743 272.3 4.36 .2293 102 329.0 1182.3 299.4 882.9 801.6 81.3 .4753 269.8 4.32 .2314 103 329.7 1182.5 300.1 882.4 801.1 81.3 .4762 267.3 4.28 .2335 104 330.4 1182.7 300.9 881.8 800.4 81.4 .4771 264.8 4.24 .2356 105 331.1 1182.9 301.6 881.3 799.9 81.4 .4780 262.9 4.21 .2378 106 331.8 1183.1 302.3 880.8 799.3 81.5 .4790 260.4 4.17 .2399 107 332.5 1183.4 303.0 880.4 798.9 81.5 .4799 257.9 4.13 .2420 108 333.2 1183.6 303.8 879.8 798.2 81.6 .4808 256.1 4.10 .2441 109 333.9 1183.8 304.5 879.3 797.7 81.6 .4817 253.6 4.06 .2462 110 334.6 1184.0 305.2 878.8 797.1 81.7 .4826 251.7 4.03 .2484 111 335.2 1184.2 305.9 878.3 796.6 81.7 .4835 249.2 3.99 .2505 112 335.9 1184.4 306.6 877.8 796.1 81.7 .48"43 247.3 3.96 .2526 113 336.5 1184.6 307.3 877.3 795.5 81.8 .4852 245.4 3.93 .254^ 114 337.2 1184.8 308.0 876.8 795.0 81.8 .4860 242.9 3.89 .2568 115 337.9 1185.0 308.7 876.3 794.4 81.9 .4869 241.1 3.86 .2589 116 338.5 1185.2 309.4 875.8 793.9 81.9 .4877 239.2 3.83 .2610 117 339.1 1185.4 310.0 875.4 793.5 81.9 .4886 237.3 3.80 .2631 118 339.8 1185.6 310.7 874.9 792.9 82.0 .4894 235.4 3.77 .2653 119 340.4 1185.8 311.4 874.4 792.4 82.0 .4903 233.6 3.74 .2674 120 341.0 1186.0 312.0 874.0 791.9 82.1 .4911 231.7 3.71 .2695 121 341.7 1186.2 312.7 873.5 791.4 82.1 .4919 229.8 3.68 .2715 122 342.3 1186.3 313.3 873.0 790.8 82.2 .4927 227.9 3.65 .2736 123 342.9 1186.5 314.0 872.5 790.3 82.2 .4935 226.7 3.63 .2757 124 343.5 1186.7 314.6 872.1 789.9 82.2 .4943 224.8 3.60 .2779 125 344.1 1186.9 315.2 871.7 789.4 82.3 .4951 223.0 3.57 .2800 126 344.7 1187.1 315.9 871.2 788.9 82.3 .4959 221.7 3.55 .2820 127 345.3 1187.3 316.5 870.8 788.4 82.4 .4967 219.8 3.52 .2841 128 345.9 1187.4 317.1 870.3 787.9 82.4 .4974 218.0 3.49 .2862 129 346.5 1187.6 317.7 869.9 787.5 82.4 .4982 216.7 3.47 .2883 130 347.1 1187.8 318.4 869.4 786.9 82.5 .4990 214.8 3.44 .2904 131 347.7 1188.0 319.0 869.0 786.5 82.5 .4997 213.6 3.42 .2925 132 348.3 1188.2 319.6 868.6 786.1 82.5 .5005 211.7 3.39 .2946 133 348.9 1188.4 320.2 868.2 785.6 82.6 .5012 210.5 3.37 .2967 134 349.5 1188.5 320.8 867.7 785.1 82.6 .5020 209.2 3.35 .2988 - 135 350.0 1188.7 321.4 867.3 784.7 82.6 .5027 207.3 3.32 .3009 136 350.6 1188.9 322.0 866.9 784.2 82.7 .5035 206.1 3.30 .3030 SAT CRATED STEAM 147 Properties of Saturated Steam Abso- ute Tem- Heat Volume 3 ress perature of of of of Entropy Relative. Specific. Density. Lbs. per Deg. Fah. Total Liquid Vapor- ization Internal Work External Work of Liquid Water at Cu. ft. per Ib. Wt. of 1 cu. ft. Sq. in 39F. Lb. 137 351.2 1189.0 322.6 866.4 783.7 82.7 .5042 204.8 3.28 .3051 38 351.7 1189.2 323.2 866.0 783.3 82.7 .5049 203.0 3.25 .3072 139 352.3 1189.4 323.8 865.6 782.8 82.8 .5056 201.7 3.23 .3092 140 352.9 1189.5 324.4 865.1 782.3 82.8 .5064 200.5 3.21 .3113 141 353.4 1189.7 325.0 864.7 781.9 82.8 .5071 199.2 3.19 .3134 142 353.9 1189.9 325.6 864.3 781.4 82.9 .5078 198.0 3.17 .315& 4:j 354.5 1190.1 326.1 864.0 781.1 82.9 .5085 196.7 3.15 .3176 144 355.0 1190.2 326.7 863.5 780.6 82.9 .5092 195.5 3.13 .3197 .45 355.6 1190.4 327.2 863.2 780.2 83.0 .5099 194.2 3.11 .3218 ,4j 356.1 1190.6 327.8 862.8 779.8 83.0 .5106 193.0 3.09 .3239- .47 356.7 1190.7 328.3 862.4 779.4 83.0 .5113 191.7 3.07 .3259- .48 357.2 1190.9 328.9 862.0 778.9 83.1 .5119 190.5 3.05 .3280- '.4! 357.7 1191.0 329.4 861.6 778.5 83.1 .5126 189.2 3.03 .3300 :50 358.3 1191.2 330.0 861.2 778.1 83.1 .5133 188.0 3.01 .3321 151 358.8 1191.4 330.5 860.9 777.7 83.2 .5140 186.7 2.99 .3342: L52 359.3 1191.5 331.1 860.4 777.2 83.2 .5146 185.5 2.97 .336a 5:j 359.8 1191.7 331.6 860.1 776.9 83.2 .5153 184.2 2.95 .3384 360.3 1191.8 332.2 859.6 776.3 83.3 .5160 183.6 2.94 .3405- 360.9 1192.0 332.7 859.3 776.0 83.3 .5166 182.3 2.92 .342$ i5t; 361.4 1192.2 333.3 858.9 775.6 83.3 .5173 181.1 2.90 .3447 57 361.9 1192.3 333.8 858.5 775.2 83.3 .5179 179.8 2.88 .3467 362.4 1192.5 334.3 858.2 774.8 83.4 .5186 179.2 2.87 .3488 59 362.9 1192.7 334.9 857.8 774.4 83.4 .5192 178.0 2.85 .3509* 60 363.4 1192.8 335.4 857.4 774.0 83.4 .5198 176.7 2.83 .3530 fil 363.9 1193.0 335.9 857.1 773.7 83.4 .5205 176.1 2.82 .3551 .62 364.4 1193.1 336.4 856.7 773.2 83.5 .5211 174.9 2.80 .357 63 364.9 1193.3 337.0 856.3 772.8 83.5 .5217 173.6 2.78 .3593 64 365.4 1193.4 337.5 855.9 772.4 83.5 .5224 173.0 2.77 .3614 65 365.9 1193.6 338.0 855.6 772.0 83.6 .5230 171.7 2.75 .3635 66 366.4 1193.7 338.5 855.2 771.6 83.6 .5236 171.1 2.74 .3655 67 366.8 1193.9 339.0 854.9 771.3 83.6 .5242 169.9 2.72 .3675- 68 367.3 1194.0 339.5 854.5 770.9 83.6 .5248 169.2 2.71 .3695 69 367.8 1194.2 340.0 854.2 770.5 83.7 .5254 168.0 2.69 .3716 70 368.3 1194.3 340.5 853.8 770.1 83.7 .5260 167.4 2.68 .3737 71 368.8 1194.4 341.0 853.4 769.7 83.7 .5266 166.1 2.66 .3758 72 369.2 1194.6 341.5 853.1 769.4 83.7 .5272 165.5 2.65 .3778 73 369.7 1194.7 342.0 852.7 768.9 83.8 .5278 164.3 2.63 .3799 74 370.2 1194.8 342.5 852.3 768.5 83.8 .5284 163.6 2.62 .3820 75 370.6 1195.0 343.0 852.0 768.2 83.8 .5290 162.4 2.60 .3841 76 371.1 1195.1 343.5 851.6 767.8 83.8 .5296 161.8 2.59 .3862 148 SATURATED STEAM Properties of Saturated Steam Abso- lute Tem- Heat Volume Press. Lbs. perature Deg. Total of Liquid of Vapor- of Internal of External Entropy of Relative. Water Specific. Cu. ft. Density. Wt. of per Fah. ization Work Work Liquid at per Ib. 1 cu. ft. Sq.in 39F. Lbs. 177 371.6 1195.3 344.0 851.3 767.5 83.8 .5302 160.5 2.57 .3883 178 372.0 1195.4 344.4 851.0 767.1 83.9 .5308 159.9 2.56 .3904 179 372.5 1195.6 344.9 850.7 766.8 83.9 .5313 159.3 2.55 .3925 180 373.0 1195.7 345.4 850.3 766.4 83.9 .5319 158.0 2.53 .3945 181 373.4 1195.9 345.9 850.0 766.1 83.9 .5325 157.4 2.52 .3966 182 373.9 1196.0 346.4 849.6 765.6 84.0 .5331 156.7 2.51 .3987 183 374.3 1196.1 346.8 849.3 765.3 84.0 .5336 155.5 2.49 .4008 184 374.8 1196.2 347.3 848.9 764.9 84.0 .5342 154.9 2.48 .4029 185 375.2 1196.4 347.8 848.6 764.6 84.0 .5347 154.3 2.47 .4049 186 375.7 1196.5 348.2 848.3 764.3 84.0 .5353 153.6 2.46 .4070 187 376.1 1196.6 348.7 847.9 763.8 84.1 .5359 152.4 2.44 .4090 188 376.6 1196.8 349.2 847.6 763.5 84.1 .5364 151.8 2.43 .4111 189 377.0 1196.9 349.6 847.3 763.2 84.1 .5370 151.1 2.42 .4132 190 377.4 1197.1 350.1 847.0 762.9 84.1 .5375 150.5 2.41 .4153 191 377.9 1197.2 350.5 846.7 762.5 84.2 .5381 149.9 2.40 .4174 192 378.3 1197.3 351.0 846.3 762.1 84.2 .5386 148.7 2.38 .4194 193 378.7 1197.4 351.4 846.0 761.8 84.2 .5391 148.0 2.37 .4215 194 379.2 1197.6 351.9 845.7 761.5 84.2 .5397 147.4 2.36 .4236 195 379.6 1197.7 352.4 845.3 761.1 84.2 .5402 146.8 2.35 .4257 196 380.0 1197.8 352.8 845.0 760.8 84.2 .5408 146.1 2.34 .4278 197 380.5 1198.0 353.3 844.7 760.4 84.3 .5413 144.9 2.32 .4298 198 380.9 1198.1 353.7 844.4 760.1 84.3 .5418 144.3 2.31 .4318 199 381.3 1198.2 354.1 844.1 759.8 84.3 .5423 143.6 2.30 .4338 200 381.7 1198.4 354.6 843.8 759.5 84.3 .5429 143.0 2.29 .4359 201 382.1 1198.5 355.0 843.5 759.1 84.4 .5434 142.4 2.28 .4379 202 382.6 1198.6 355.4 843.2 758.8 84.4 .5439 142.8 2 27 .4399 203 383.0 1198.8 355.9 842.9 758.5 84.4 .5444 142.2 2^26 .4420^ 204 383.4 1198.9 356.3 842.6 758.2 84.4 .5449 140.5 2 25 .4441^ 205 383.8 1199.0 356.8 842.2 757.8 84.4 .5454 139.9 2.24 .4461 206 384.2 1199.1 357.2 841.9 757.4 84.5 .5459 139.3 2.23 .4482 207 384.6 1199.3 357.6 841.7 757.2 84.5 .5465 138.6 2 22 .4503 208 385.0 1199.4 358.0 841.4 756.9 84.5 .5470 138.0 2i21 .4524 209 385.5 1199.5 358.5 841.0 756.5 84.5 .5475 137.4 2.20 .454 210 385.9 1199.6 358.9 840.7 756.2 84.5 .5480 136.8 2.19 .456o 211 386.3 1199.8 359.3 840.5 756.0 84.5 .5485 136.1 2.18 .458 212 386.7 1199.9 359.7 840.2 755.6 84.6 .5489 135.5 2.17 .460 213 387.1 1200.0 360.1 839.9 755.3 84.6 .5494 134.9 2.16 .462 214 387.5 1200.1 360.6 839.5 754.9 84.6 .5499 134.3 2.15 .464 215 387.9 1200.2 361.0 839.2 754.6 84.6 .5504 133.6 2.14 .466 216 388.3 1200.4 361.4 839.0 754.4 84.6 .5509 133.0 2.13 .469 217 388.7 1200.5 361.8 838.7 754.1 84.6 .5514 132.4 2.12 .4711 21 S 389.1 1200.6 362.2 838.4 753.8 84.6 .5519 131.8 2.11 .4731 SATURATED STEAM 149 Properties of Saturated Steam Abso- lute Tem- ^ (eat Vo urne Pr^ss. Lbs. per Sq. in perature Deg. Fah. Total of Liquid of Vapor- ization of Internal Work of External Work Entropy of Liquid Relative. Water 39 a V Specific. Cu. ft. per Ib. Density. Wt. of 1 cu. ft. Lbs. 219 389.4 1200.7 362.6 838.1 753.4 84.7 .5524 131.1 2.10 .4751 220 389.8 1200.8 363.0 837.8 753.1 84.7 .5529 131.1 2.10 .4772 221 390.2 1201.0 363.5 837.5 752.8 84.7 .5533 130.5 2.09 .4792 '222 390.6 1201.1 363.9 837.2 752.5 84.7 .5538 129.9 2.08 .4813 223 391.0 1201.2 364.3 836.9 752.2 84.7 .5543 129.3 2.07 .4834 224 391.4 1201.3 364.7 836.6 751.9 84.7 .5548 128.6 2.06 .4855 225 391.8 1201.4 365.1 836.3 751.6 84.7 .5553 128.0 2.05 .4876 :226 392.2 1201.6 365.5 836.1 751.3 84.8 .5557 127.4 2.04 .4896 227 392.5 1201.7 365.9 835.8 751.0 84.8 .5562 126.8 2.03 .4917 228 392.9 1201.8 366.3 835.5 750.7 84.8 .5567 126.2 2.02 .4939 229 393.3 1201.9 366.7 835.2 750.4 84.8 .5571 126.2 2.02 .4959 230 393.7 1202.0 367.1 834.9 750.1 84.8 .5576 125.5 2.01 .4979 231 394.1 1202.1 367.5 834.6 749.8 84.8 .5581 124.9 2.00 .5000 L':J2 394.4 1202.2 367.9 834.3 749.5 84.8 .5585 124.3 1.99 .5021 2!i;! 394.8 1202.4 368.3 834.1 749.2 84.9 .5590 123.7 1.98 .5041 2:; 4 395.2 1202.5 368.6 833.9 749.0 84.9 .5594 123.7 1.98 .5062 235 395.6 1202.6 369.0 833.6 748.7 84.9 .5599 123.0 1.97 .5082 2;;t; 395.9 1202.7 369.4 833.3 748.4 84.9 .5603 122.4 1.96 .5103 237 396.3 1202.8 369.8 833.0 748.1 84.9 .5608 121.8 1.95 .5123 : L'i'.N 396.7 1202.9 370.2 832.7 747.8 84.9 .5612 121.2 1.94 .5144 23!t 397.0 1203.0 370.6 832.4 747.5 84.9 .5617 121.2 1.94 .5165 240 397.4 1203.2 371.0 832.2 747.3 84.9 .5621 120.5 1.93 .5186 241 397.8 1203.3 371.3 832.0 747.0 85.0 .5626 119.9 1.92 .5206 24_ 398.1 1203.4 371.7 831.7 746.7 85.0 .5630 119.3 1.91 .5226 24.", 398.5 1203.5 372.1 831.4 746.4 85.0 .5635 119.3 1.91 .5247 244 398.8 1203.6 372.5 831.1 746.1 85.0 .5639 118.7 1.90 .5268 - 5 399.2 1203.7 372.8 830.9 745.9 85.0 .5643 118.0 1.89 .5289 2 i; 399.6 1203.8 373.2 830.6 745.6 85.0 .5648 117.4 1.88 .5311 L , 7 399.9 1203.9 373.6 830.3 745.3 85.0 .5652 116.8 1.87 .5332 24 N 400.3 1204.0 374.0 830.0 745.0 85.0 .5656 116.8 1.87 .5353 2 !< 400.6 1204.1 374.3 829.8 744.8 85.0 .5661 116.2 1.86 .5373 250 401.0 1204.2 374.7 829.5 744.5 85.0 .5665 115.5 1.85 .5393 260 404.5 1205.3 378.4 826.9 741.7 85.2 .5707 111.2 1.78 .5601 270 407.8 1206.3 381.9 824.4 739.2 85.2 .5748 107.4 1.72 .5809 280 411.1 1207.3 385.3 822.0 736.7 85.3 .5787 103.7 1.66 .602 290 414.3 1208.3 388.6 819.7 734.3 85.4 .5826 100.5 1.61 .622 300 417.4 1209.3 391.9 817.4 732.0 85.4 .5863 96.8 1.55 .644 310 420.4 1210.2 395.0 815.2 729.8 85.4 .5898 94.3 1.51 .664 t325 424.8 1211.5 399.6 811.9 726.4 85.5 .5950 89.9 1.44 .696 350 431.8 1213.1 406.7 806.4 721.1 85.3 .6028 83.7 1.34 .749 375 438.5 1215.5 413.7 801.7 716.4 85.2 .6107 78.1 1.25 .802 400 444.8 1217.6 420.4 797.3 712.1 85.2 .6181 73.1 1.17 .855 150 VAPORS OF AMMONIA AND SULPHUR DIOXIDE Saturated Vapor of Ammonia. Heat Temp. Pressure of of of Volume Density Deg. Fabr. Lbs. per sq. in. Total in Liquid Vapor- ization Internal Work External Work Entropy Cu. ft. per Ib. Lbs. per cu. ft. 25 15.43 530.9 62.1 593.0 543.2 49.8 .133 17.1 .0585 15 20.48 537.7 51.5 589.2 538.5 50.7 .110 13.5 .0741 5 26.41 544.0 40.7 584.7 533.2 51.5 .085 10.5 .0952 5 33.71 549.0 29.9 578.9 526.7 52.2 .062 8.40 .119 15 42.57 553.0 18.9 571.9 519.2 52.7 .039 6.72 .149 25 53.35 555.7 7.9 563,6 510.4 53.2 .016 5.46 .183 30 59.44 556.8 . 2.3 559.1 505.6 53.5 .005 4.91 .204 32 61.87 557.3 0.0 557.3 503.6 53.7 .000 4.69 .213 40 72.91 558.2 11.1 547.1 493.2 53.9 .022 4.02 .249 50. 89.18 558.5 20.5 538.0 484.0 54.0 .041 3.30 .303 60 108.08 558.0 32.1 525.9 471.7 54.2 .069 2.74 .365 70 129.83 556.4 43.5 512.9 458.6 54.3 .084 2.29 .437 80 154.73 554.3 55.4 498.9 444.8 54.1 .107 1.83 .546 90 182.82 551.3 67.4 483.9 430.0 53.9 .128 1.63 .613 100 214.45 548.0 79.6 468.4 414.8 53.6 .150 1.39 .719 Saturated Vapor of Sulphur Dioxide. Heat Temp. Deg. Fahr. Pressure Lbs. per sq. in. Total in Liquid of Vapor- ization of Internal Work of External Work Entropy Volume Cu. ft. per Ib. Density Lbs. per cu.ft. 25 15 5 4.98 6.82 8.96 159.4 160.4 161.5 17.1 14.4 11.3 176.5 174.8 172.8 163.1 161.1 158.8 13.4 13.7 14.0 .0370 .0306 .0241 14.5 10.8 8.40 .0690 .0926 .119 5 11.81 162.5 8.2 170.7 156.4 14.3 .0176 6.67 .150 15 15.22 163.1 5.1 168.2 153.7 14.5 .0111 5.15 .194 25 19.21 163.5 2.0 165.5 150.9 14.6 .0046 4.12 .243 30 21.61 163.6 0.5 164.1 149.4 14.7 .001:; 3.70 .270 32 40 22.48 26.90 163.6 163.7 0.0 2.7 163.6 161.0 148.8 146.1 14.8 14.9 .0000 .0052 3.56 3.00 .281 .333 50 33.30 163.6 6.0 157.6 142.7 14.9 .0117 2.43 .412 60 40 83 163.4 9.4 154.0 139.1 14.9 .0182 1.99 .503 70 49.65 162.9 12.8 150.1 135.2 14.9 .0247 1.64 .611 80 59.74 162.2 16.2 146.0 131.1 14.9 .0312 1.35 .739 90 71 41 16l!2 19.8 141.4 126.6 14.8 .0377 1.13 .888 100 84.63 160.2 23.4 136.8 122.2 14.6 .0442 .943 1.06 MAGNETIC CONSTANTS 151 Magnetic Properties of Iron and Steel H = magnetizing force = 1.258 ampere turns per cm.= .495 ampere turns per inch. B = kilo-lines per square centimeter. B"= kilo-lines per square inch. A ampere turns per centimeter length. A"= ampere turns per inch length. Permeability /A = B-=-H. (Values given by Sheldon and Foster for avei'age metal.) first quality H A A" Sheet Metal B B" Cast Iron B B" Wrought Iron B B" CaSt Steel B B 1 10 20 30 40 7.95 15.90 23.85 31.80 20.2 40.4 60.6 80.8 14.3 15.6 16.2 16.6 92.2 100.7 104.5 107.1 4.3 5.7 6.5 7.1 27.7 36.8 41.9 45.8 13.0 14.7 15.3 15.7 83. 94. 98. 101. 8 fr i 2 11.5 13.8 14.9 15.5 74.2 89.0 96.1 100.0 50 60 70 39.75 47.70 55.65 101.0 121.2 141.4 16.9 17.3 17.5 109.0 111.6 112.9 7.6 8.0 8.4 49.0 51.6 53.2 16.0 16.3 16.5 103. 105. 106. 2 2 5 16.0 16.5 16.9 103.2 106.5 109.0 80 90 100 63.65 71.60 79.50 161.6 181.8 202.0 17.7 18.0 18.2 114.1 116.1 117.3 8.7 9.0 9.4 56.1 58.0 60.6 16.7 16.9 17.2 107. 109. 110. 8 9 17.2 17.4 17.7 110.0 112.2 114.1 150 200 250 119.25 159.0 198.8 303.0 404.0 505.0 19.0 19.6 20.2 122.7 126.5 130.2 10.6 11.7 12.4 68.3 75.5 80.0 18.0 18.7 19.2 116. 120. 123. 1 8 9 18.5 19.2 19.7 119.2 123.9 127.1 300 400 600 238.5 318.0 477.0 606.0 808.0 1212 20.7 21.0 21.5 133.5 135.0 138.0 13.2 85.1 19.7 127. 1 20.1 129.6 800 1000 1200 637.0 795 954 1616 2020 2424 21.8 22.0 22.3 140.0 142.0 144.0 * .- Loss Due to Hysteresis Loss in ergs per cubic centimeter is L = *? B 1<(! in which 77 is a constant depending upon the iron. Values of tj: Very soft iron wire .002 Soft annealed cast steel .008 Very thin soft sheet iron .0015 Soft machine steel Thin good sheet iron .003 Cast steel Thick sheet iron .0033 Cast iron Ordinary sheet iron .004 Hardened cast steel .025 Transformer cores .003 152 ELECTRICAL, CONSTANTS Specific Resistances and Temperature Coefficients R= r (1 + a f) Ohms per R = resistance at r C cir. mil-foot r = resistance at o C at C Specific Resistance Temperature Microhms per Coefficient a. cubic centimeter Divide by 10 5 Aluminum 17.38 2.889 390 Carbon, graphite j 2400 ( 42000 Carbon, arc light 4000 Constantine 307 51 1 Copper, annealed 9.44 1.570 388 Copper, hard 9.64 1.603 German silver 125.0 20.76 28 to 44 Gold 2.07 ' 365 la. la., hard 302 50.2 1.1 la. la., soft 283 47.1 0.5 Iron 58.0 9.64 453 Iron, wire 57.2 9.50 Iron, telegraph wire 90.0 150 Lead 118 19.6 387 Manganin 287 47.5 1 Mercury 94.34 88 Nickel 12.35 500 Nickeline No. I, hard 262 43.6 7.6 Nickeline No. I, soft 245 40.7 7.7 Nickeline No. II, hard 204 33.9 16.8 Nickeline No. II, soft 194 32.3 18.1 Platinum 54.03 8.98 247 Silver, annealed 1.49 377 Silver, hard 1.62 Steel, wire 95 15.8 390 Dilute H N O 3 30% H 2 S 4 5% H 2 S O 4 30% 129 X 10 4 486 X 10 4 137 X 10 4 [ Liquids H 2 S 4 80% 918 X 10 4 fat 18 C Zn S O 4 24% 214 X 10 5 \ Water 26.5X 10 8 ; V Megohms Benzine 14 X 10 6 Ebonite 28 X 10 9 Glass, 20C 91 X 1Q 6 Glass, 200C 22.7 Gutta-percha, 24C 4.5 X 10 8 Mica 84 X 10 6 Paraffine 34 X 10 9 Paraffine, oil sc 8 X 10 6 Shellac 9 X 10 9 Wood Tar 167 X 10 7 ELECTRICAL CONSTANTS 153 Specific Inductive Capacities. Specific Specific Medium. Inductive Medium. Inductive Capacity. Capacity. Air, 760 mm. 1.0 Mica 6.64 Alcohol 26.0 Paraffine, solid 1.96-2.3 Beeswax 1.8 Paraffine, oil 1.92 Ebonite 2.2-3 2 Petroleum 2.05 Glass, light flint 6.72 Shellac 2.7-3.7 Glass, dense flint 7.38 Sulphur 2.8-3.9 Glass, hard crown 6.96 Turpentine 2 2 Glass, plate 5.8-8.5 Vacuum !999 Gutta-percha 2.5 Water 76.0 Kerosene 2-2.5 Rate of Electrolytic Deposition. Element. Mg. per Coulomb. Element. Mg. per Coulomb. Aluminum Copper Gold Hydrogen Lead Mercury .0935 .3282 .0679 .0104 .1072 1.038 Nickel Oxygen Platinum Silver Tin Zinc .3043 .0831 1.0095 1.1186 .6097 .3365 WIRES AND CABLES Fusing of Wires. : V I = ad ' in which / is the fusing current in amperes, d is the diameter of the wire and a is a constant, given as follows for various materials with d measured either in inches, centimeters or millimeters. Constant a. Material In. Cm. Mm. Alloy, lead (2) tin (1) Aluminum 1318 7585 325.5 1873 10.3 59.2 Copper German silver 10244 5230 1530 1292 80.0 40.8 Iron Lead 3148 1379 777.4 340.6 24.6 10.8 Platinum Tin 5172 1642 1277 405.5 40.4 12.8 154 WIRES AND CABLES Copper Wire. To obtain resistance at 30C 40 C 50C 60C 70 C C 80C Multiply values in it i hie by 1.036 1.072 1.108 1.144 1.180 1-216 No. B.&S. Dia. inches Area Cir. Mils Ohms at 20 Q C 1,000 ft- Mile Wt. Ibs. 1,000 ft. Bare Mile 3381.8 2682.2 2126.8 1687.0 Safe Cap. W. P. Amp. R. C, 210 177 150 127 0000 000 00 .460 .4096 .3648 .3249 211600 167800 133100 105500 .0489 .0617 .0778 .0981 .258 .326 .411 .518 640.5 508.0 402.8 319.5 312 262 L'20 185 1 2 3 .2893 .2576 .2294 83690 66370 52630 .1237 .1560 .1967 .653 .824 1.039 253.3 200.9 159.3 1337.4 1060.8 841.1 156 131 110 107 90 76 4 5 6 .2043 .1819 .1620 41740 33100 26250 .2480 .3128 .3944 1.309 1.651 2.082 126.4 100.2 79.46 667.4 529.6 419.5 92 77 65 65 54 46 7 8 9 .1443 .1285 .1144 20820 16510 13090 .4973 .6271 .7908 2.626 3.311 4.175 63.02 49.98 39.63 332.7 263.9 209.2 46 33 10 11 12 .1019 .09074 .08081 10380 8234 6530 .9972 1.257 1.586 5.265 6.640 8.374 31.43 24.93 19.77 165.9 131.6 104.4 23 24 17 13 14 15 .07196 .06408 .05707 5178 4107 3257 1.999 2.521 3.179 10.555 13.311 16.785 15.68 12.43 9.86 82.9 65.6 52.0 16 12 16 17 18 .05082 .04526 .04030 2583 2048 1624 4.009 5.055 6.374 21.168 26.690 33.655 7.82 6.20 4.92 41.3 32.7 26.0 8 5 6 3 19 20 21 .03589 .03196 .02846 1288 1022 810 8.038 10.14 12.78 42.441 53.539 67.478 3.90 3.09 2.45 20.6 16.3 12.9 22 23 24 .02535 .02257 .02010 642 509 404 16.12 20.32 25.63 1.95 1.54 1.22 25 26 27 .01790 .01594 .01420 320 254 201 32.31 40.75 51.38 .97 .769 .610 28 29 30 .01264 .01126 .01003 160 127 100 64.79 81.70 103.0 .484 .384 .304 31 32 33 .00893 .00795 .00708 80 63 50 129 9 163.8 206.6 .241 .191 .152 34 35 36 .00631 .00562 .00500 40 32 25 260.5 328.4 414.2 .120 .096 .076 37 38 39 40 .00445 .00397 .00353 .00315 20 16 12 10 522.2 658.5 830.4 1047.0 .060 .048 .OS'S .030 WIRES AND CABLES Aluminum Wire. 155 To obtain resistance at Multiply values in ta* ble by 30C 1.036 40C 1-072 50C 1.109 60C 1.145 70C 80C 1.181 1.217 No. B.&S- Dia. Bare inches Area Cir. Mils Ohms at 20 Q C 1,000 ft. Mile Wt. Ibs. 1,000 ft. Bare Safe Cap. Amp. Mile W. P. R. C. 0000 000 00 .460 .4096 .3648 .3249 211600 167800 133100 105500 .0804 .1014 .1278 .1610 .425 .535 .675 .850 192.0 .152.3 120.8 95.9 1014.0 804.0 637.7 506.2 262 177 220 150 185 127 156 107 1 2 3 .2893 .2576 .2294 83690 66370 52630 .2032 .2562 .3230 1.073 1.353 1.705 76.0 60.2 47.8 401.1 318.0 252.2 131 90 110 76 92 65 4 5 6 .2043 .1819 .1620 41740 33100 26250 .4075 .5140 .6480 2.152 2.714 3.421 37.9 .30.0 23.8 200.0 158.6 125.8 77 54 65 46 55 39 7 8 9 .1443 .1285 .1144 20820 16510 13090 .8171 1.050 1.300 4.314 5.544 6.864 18.9 15.0 11.9 99.8 79.1 62.7 39 28 10 11 12 .1019 .09074 .08081 10380 8234 6530 1.640 2.068 2.605 8.659 10.92 13.75 9.42 7.47 5.93 49.8 39.4 31.3 27 20 19 14 13 14 15 .07196 .06408 .05707 5178 4107 3257 3.284 4.250 5.111 17.34 22.44 26.99 4.71 3.73 2.96 24.8 19.7 15.6 13 10 16 17 18 .05082 .04526 .04030 2583 2048 1624 6.591 8.315 10.49 34.80 43.80 59.39 2.34 1.86 1.47 12.4 9.83 7.78 7 5 4 2.5 19 20 21 .03589 .03196 .02846 1288 1022 810 13.60 16.66 21.03 71.81 87.96 1.17 .929 .737 6.18 4.91 22 23 24 .02535 .02257 .02010 642 509 404 26.51 33.42 42.15 .581 .464 .367 25 26 27 .01790 .01594 .01420 320 254 201 53.17 67.05 84.50 .291 .230 .183 28 29 30 .01264 .01126 .01003 160 ' 127 100 106.5 134.4 169.3 .144 .116 .091 31 32 33 .00893 .00795 .00708 80 63 50 208.6 269.5 339.6 .072 .057 .045 34 35 36 .00631 .00562 .00500 40 32 25 428.5 540.0 681.0 .036 .029 .023 37 38 39 40 .00445 .00397 .00353 .00315 20 16 io 858.9 1084 1366 1722 .018 .014 .011 .009 156 WIRES AND CABLES Copper Cables. SizeB.&S. Dia. Bare orCir. Mils inches Ohms at 20C 1.000 ft. Mile Wt. Ibs. 1,000 ft- Bare Mile Safe Cap. Amp. W. P. R. C. 8 6 5 .147 .180 .209 .575 .362 .293 3.040 1.912 1.550 57 85 112 301 449 591 46 65 77 33 46 54 4 3 2 .234 .263 .295 .234 .185 .147 1.236 .980 .776 140 178 224 739 940 1183 92 110 131 65 76 90 1 00 .325 .378 .425 .129 .0972 .0772 .681 .513 .407 255 338 426 1346 1785 2249 156 185 220 107 127 150 000 0000 250000 .475 .524 .568 .0618 .0478 .0414 .326 .252 .218 532 650 790 2809 3432 4171 262 .312 350 177 210 235 300000 350000 400000 .637 .680 .735 .0340 .0295 .02,57 .179 .156 .136 949 1092 1224 5010 5766 6463 400 450 500 270 300 330 500000 600000 750000 .820 .900 1.020 .0204 .0172 .0138 .108 .0909 .0727 1550 1874 2331 8184 9895 10208 590 680 800 390 450 525 800000 900000 1000000 1.037 1.096 1.157 .0129 .0115 .0103 .0683 .0605 .0543 2462 2815 3138 12999 14863 16579 840 920 1000 550 600 650 1250000 1500000 2000003 1.296 1.412 1.652 .00828 .00685 .00517 .0437 .0362 .0273 3831 4681 6237 20228 24716 32932 1185 1360 1670 750 850 1050 Aluminum Cables. Cir. Mils Dia. Bare inches Ohms at 20 C 1,000 ft. Mile Wt. Ibs. 1,000 ft. Bare Mile Safe Cap. Amp W. P. R. C. 250000 300000 350000 .590 .630 .679 .0668 .0557 .0477 .353 .294 .252 230 276 322 1215 1458 1701 294 340 378 198 228 252 400000 450000 500000 .728 .770 .819 .0417 .0371 .0334 .220 .196 .176 368 414 460 1924 2187 2430 420 457 495 277 302 327 550000 600000 650000 .855 .891 .927 .0304 .0278 .0257 .161 .147 .136 506 552 598 2673 2916 3159 532 570 604 352 378 399 700000 750000 800000 .963 .999 1.035 .0239 .0223 .0209 .126 .118 .110 644 690 736 3402 3645 3888 638 672 686 420 441 462 850000 900000 950000 1000000 1.062 1.092 1.125 1.152 .0196 .0186 .0176 .0167 .103 .0982 .0929 .0882 782 828 874 920 4131 4374 4617 4860 729 772 806 840 483 504 525 546 WIRES AND CABLES Cotton Covered Wire. 157 No. B. &S. Dia. Inches. Single Double Triple No. B.&S. Dia. Inches. Single Double Triple 0000 .480 17 .050 .053 .057 000 .430 18 .045 .048 .052 00 .385 19 .040 .044 .048 .339 .343 20 .036 .040 .044 1 .303 .307 21 .032 .036 .040 2 .272 .276 22 .029 .033 .037 3 .242 .247 23 .027 .031 .035 4 .211 .216 .220 24 .024 .028 .032 5 .189 .194 .198 25 .022 .026 .030 6 .169 .174 .178 26 .020 .024 7 .151 .156 .160 27 .018 .022 8 .136 .141 .145 28 .017 .021 9 .121 .126 .130 29 .015 .019 10 .108 .112 .116 30 .014 .018 11 .097 .101 .105 31 .013 .017 12 .087 .091 .095 32 .012 .016 13 .078 .082 .086 33 .011 .015 14 .070 .074 .078 34 .0103 .0143 15 .063 .067 .071 35 .0096 .0136 16 .056 .059 .063 36 .0085 .0120 The B. & S. Gauge The diameter of any wire of number n is _ 0.3249 " 1.123- and the area in circular mils a n = a 1.261" These equations give rise to the following approximate ones for slide rule calculation. The right hand side of the equation gives the entire logarithm, but only the decimal part is to be set off on the L scale on the back of the slide rule. The decimal point will probably be known from experience, but if there is any doubt it may be determined from the integral part (characteristic) of the logarithm by the rules on page 4 10 n 20 log d n log a, (diameter, inches) n (area, cir. mils) log r log W r . 10 n 10 (resistarce, ohms, copper) - _ (weight per 1000 ft., Ibs. copper; 158 TRANSMISSION LINE CONSTANTS Formulae. The two following tables are for one wire and the neutral of a three phase circuit, or for one wire of a single phase cir- cuit. For both wires of a single phase circuit, the reactances and impedances as given must be doubled and capacities and charging currents divided by two. But for charging currents, it must be observed that for single phase the e. m. f. is that be- tween wires and for three phase between one wire and the neutral. Reactance per mile, one cycle x = F 0.00465 log f a \ + 0.0019 1 ohms, ' Capacity per mile, C = - 038 M^ microfarads; log 2 a/d- in which a = distance between wires and d = diameter of wire, the wires being arranged at the vertices of an equiangular triangle. Charging current / = 2 w n E C 10 ~ fi amperes, where n = frequency in cycles per second and E = e. in. f. between one wire and neutral. The table on page 160 is for two wires of a single phase cir- cuit at 25 cycles per second. Reactance, Capacity and Charging Current. (Per mile one wire three phase at 1 cycle per sec. and E - 10000 v.) Reactance a/d X Capacity C Charging Current 1 Reactance a/d x Capacity C Charging Current 1 25 30 35 .00841 .00877 .00908 .02298 .02187 .02112 .001444 .001374 .001327 150 175 200 .01202 .01233 .01260 .01572 .01531 .01497 .000988 .000962 .000940 40 45 50 .00935 .00959 .00978 .02047 .01994 .01947 .001286 .001253 .001223 250 300 350 .01305 .01342 .01373 .01443 .01402 .01369 .000906 .000880 .000860 60 70 80 .01017 .01048 .01075 .01877 .0181(5 .01767 .001179 .001141 .001110 400 450 500 .01400 .01424 .01445 .01342 .01318 .01298 .000843 .000828 .000815 90 100 125 .01098 .01120 .01165 .01730 .01693 .01624 .001087 .001064 .001020 600 700 800 .01482 .01513 .01540 .01265 .01238 .01216 .000794 .000777 .000764 TRANSMISSION LINE CONSTANTS 159 Capacity, Charging Current, Reactance and Impedance. (Per mile one wire three phase at 60 cycles per sec. and E ~ 10000 v.) No. B. &S. 12 Distance Between 18 24 36 Wires, Inches. 48 60 72 84 0000 C .0227 .0206 .0193 .0178 .0168 .0161 .0157 .0152 1 .0855 .0776 .0728 .0670 .0634 .0608 .0591 .0572 X .509 .557 .593 .642 .677 .705 .726 .745 *z .575 .618 .651 .696 .728 .754 .774 .792 **z .668 .705 .734 .774 .803 .827 .845 .861 000 C .0220 .0200 .0189 .0174 .0164 .0158 .0153 .0149 1 .0830 .0756 .0711 .0654 .0620 .0595 .0577 .0562 X .522 .572 .607 .656 .691 .718 .740 .759 z .622 .664 .695 .738 .769 .794 .814 .831 z .720 .790 .815 .852 .879 .901 .918 .934 00 C .0214 .0195 .0184 .0169 .0161 .0155 .0150 .0146 1 .0807 .0737 .0692 .0640 .0606 .0585 .0566 .0551 X .538 .586 .622 .671 .706 .732 .754 .773 z .686 .724 .753 .794 .824 .846 .866 .882 z .872 .902 .926 .960 .984 1.003 1.020 1.033 C .0208 .0191 .0179 .0166 .0158 .0152 .0147 .0144 1 .0785 .0720 .0676 .0627 .0594 .0572 .0554 .0541 X .551 .601 .633 .684 .719 .747 .769 .787 z .769 .806 .830 .870 .897 .920 .938 .953 z 1.026 1.053 1.072 1.103 1.125 1.143 1.157 1.170 1 C .0203 .0186 .0175 .0162 .0155 .0149 .0144 .0141 1 .0765 .0702 .0660 .0612 .0584 .0561 .0544 .0531 X .565 .614 .649 .699 .733 .761 .783 .801 z .881 .913 .937 .972 .997 1.018 1.035 1.045 z 1.229 1.251 1.269 1.295 1.314 1.330 1.342 1.354 2 C .0198 .0181 .0172 .0159 .0151 .0146 .0142 .0138 .0746 .0685 .0649 .0600 .0572 .0550 .0534 .0522 X .579 .629 .663 .713 .747 .774 .796 .815 z 1.031 1.060 1.080 1.112 1.134 1.152 1.167 1.180 z 1.492 1.516 1.530 1.553 1.570 1.584 1.593 1.604 3 C .0193 .0177 .0168 .0157 .0148 .0143 .0139 .0136 1 .0726 .0668 .0632 .0591 .0560 .0540 .0524 .0512 X .594 .643 .678 .727 .762 .788 .810 .828 z 1.224 1.248 1.267 1.294 1.314 1.329 1.342 1.355 z 1.836 1.858 1.863 1.882 1.896 1.907 1.916 1.923 4 C .0188 .0173 .0164 .0153 .0146 .0140 .0137 .0134 1 .0710 .0654 .0619 .0576 .0549 .0530 .0515 .0504 X .608 .656 .692 .741 .776 .803 .825 .843 z 1.481 1.501 1.517 1.540 1.557 1.571 1.582 1.593 z 2.270 2.284 2.295 2.310 2.331 2.338 2.345 5 C .0184 .0169 .0161 .0150 .0143 .0138 .0134 .0131 1 .0692 .0640 .0606 .0567 .0539 .0521 .0506 .0495 X .623 .671 .706 .755 .789 .816 .839 .857 z 1.866 1.883 1.896 1.915 1.930 1.940 1.950 1.957 z 2.828 2.840 2.849 2.861 2.871 2.878 2.885 2.890 6 C .0180 .0166 .0157 .0147 .0140 .0136 .0132 .0129 1 .0676 .0626 .0594 .0554 .0530 .0512 .0497 .0487 X .633 .685 .719 .768 .803 .830 .853 .871 z 2.251 2.265 2.274 2.293 2.305 2.313 2.323 2.330 z 3.541 3.547 3.553 3.564 3.571 3.578 3.583 3.587 * For copper. ** For aluminum. Other quantities are the same for either copper or aluminum. 160 TRANSMISSION LINE CONSTANTS Capacity, Charging Current, Reactance and Impedance. (Per mile two wires single phase at 25 cycles per sec. and E = 10000 v.) No. B. &S. 12 Distance Between Wires, Inches- 18 24 36 48 60 72 84 0000 C .0113 .0103 .0097 .0089 .0084 .0081 .0078 .0076 I .0178 .0162 .0152 .0140 .0132 .0127 .0123 .0119 X .424 .464 .494 .535 .564 .588 .605 .621 *Z .684 .709 .729 .757 .778 .796 .809 .821 ** Z .962 .981 .994 1.015 1.030 1.045 1.055 1.064 000 C .0110 .0100 .0094 .0087 .0082 .0079 .0077 .0075 I .0173 .0157 .0148 .0136 .0129 .0124 .0120 .0117 X .435 .477 .506 .546 .576 .598 .616 .632 z .804 .827 .844 .870 .888 .903 .916 .926 z 1.172 1.189 1.200 1.218 1.231 1.243 1.251 1.259 00 C .0107 .0098 .0092 .0085 .0080 .0078 .0075 .0073 I .0167 .0154 .0144 .0133 .0126 .0122 .0118 .0115 X .448 .487 .518 .559 .588 .610 .628 .644 z .962 .980 .996 1.017 1.033 1.046 1.056 1.066 z 1.444 1.456 1.467 1.482 1.494 1.502 1.509 1.516 C .0104 .0096 .0090 .0083 .0079 .0076 .0074 .0072 .0161 .0150 .0141 .0131 .0124 .0119 .0115 .0113 X .458 .500 .527 .569 .599 .622 .640 .655 z 1.164 1.181 1.194 1.211 1.226 1.238 1.247 1.266 z 1.790 1.800 1.809 1.823 1.831 1.838 1.845 1.851 1 C .0101 .0093 .0088 .0081 .0077 .0074 .0072 .0070 I .0160 .0146 .0138 .0128 .0122 .0117 .0113 .0111 X .471 .512 .541 .582 .611 .634 .652 .668 z 1.430 1.445 1.454 1.470 1.482 1.493 1.500 1.507 z 2.228 2.238 2.246 2.256 2.264 2.270 2.276 2.280 2 C .0099 .0092 .0086 .0080 .0076 .0073 .0071 .0069 I .0156 .0143 .0135 .0125 .0119 .0115 .0111 .0109 X .483 .524 .552 .594 .622 .645 .664 .679 z 1.758 1.780 1.788 1.802 1.810 1.818 3.825 1.830 z is.790 2.800 2.806 2.814 2.820 2.826 2.830 2.832 3 C .0097 .0089 .0084 .0078 .0074 .0072 .0070 .0068 I .0151 .0139 .0132 .0123 .0117 .0113 .0109 .0107 X .495 .536 .565 .605 .634 .656 .675 .690 z 2.197 2.205 2.213 2.2^6 2.234 2.240 2.246 2.251 z 3.501 3.511, 3.518 3.524 2.529 3.534 3.538 3.535 4 C .0094 .0087 .0082 .0076 .0073 .0070 .0068 .0067 I .0148 .0136 .0129 .0120 .0114 .0110 .0107 .0105 X .506 .546' .576 .618 .647 .668 .687 .703 z 2.745 2.753 2.760 2.770 2.776 2.781 2.786 2.790 z 4.400 4.405 4.410 t 4.417 4.420 4.422 4.424 4.426 5 C .0092 .0085 .0080 .0075 .0072 .0069 .0067 .0066 I .0144 .0133 .0126 .0118 .0112 .0108 .0105 .0103 X .518 .559 .588 .628 .657 .680 .697 .714 z 3.559 3.564 3.569 3.574 3.579 3.583 3.587 3.591 z 5.544 5.548 5.552 5.555 5.558 5.561 5.564 5.567 6 C .0090 .0083 .0078 .0074 .0070 .0068 .0066 .0065 I .0141 .0130 .0124 ,0115 .0110 .0107 .0104 .0102 X .527 .571 .599 .641 .669 .692 .710 .725 z 4.353 4.357 4.361 4.365 4.369 4.373 4.377 4.381 z 6.980 6.982 6.984 6.987 6.990 6.993 6.996 6.998 For copper. ** For aluminum. HYDRAULICS 161 HYDRAULICS Immersed Rectangle. Surface Level Fig. 1 The center of pressure for an immersed rectangle, e. g., a dam, is given by 9 fl * rl ? > 91* _ 7 3 1 4 a, a, 7 _ - " ' The pressure P = 7 J pounds where 7 is the weight of a cubic foot of water and A is the area of the rectangle in square feet. Impact due to Jet. The impact due to a jet is P = JL Q( f v) (1 cosa) 9 in which Q is the quantity per second striking the bucket or vane, 7 is the weight per unit volume of the fluid, c is the velocity of the jet, y. the velocity of the vane in the direction of c and a is the angle through which the jet is deflected by the 162 HYDRAULICS Orifices. The coefficients of flow for an orifice are a '= coefficient of contraction. < = coefficient of velocity. c = coefficient of efflux, f = coefficient of resistance. The average values of these coefficients for an orifice in a thin plate are: o= .62, 0=.98, i J and for weirs with full end contractions, Q = 3.33 (6 0.2/1.) [(h + h, ) ~f~ _ j h 2 ] where /tj = - = 0.01555 # 2 , v being the velocity of approach. 2 9 For Smith's formula, where /jj is the same as above, and n = 1.4 for contracted weirs and n 1 K for weirs without contractions. Table XI gives values of h 2 For rough estimates of weirs 'Table XII may be used. Flow Through Pipes The general formula for the loss of head in pipes is in which h is the loss of head in feet, v is the velocity in feet per second and is a coefficient. The loss of head by friction is ' 7, -X lv * ^0\ where I is the length of the pipe in feet, d is the diameter in feet and X is a coefficient depending upon d and v. Tables XIII and XV give values of X for cast iron and wood stave pipes and Table XIV gives the loss of head per 1000 feet of pipe for cast iron when d and v are known, or when d and the discharge in cubic feet per minute are given. HYDRAULICS Flow Through Pipes Continued 165 Fig. 5 Fig. 6 For elbows f = .9457 sin 2 V 2 5 + 2.047 sin 4 y% 8. /3 For curvesf= $"' - 90 f for pipes of circular section; f" for rectangular section. Tables XVI and XVII give values of the above coefficients. Flow of Water in Open Channels For the flow of water in ditches, flumes, etc., the general formula is v c -j/ rs, where v is the mean velocity in feet per sec., r the hydraulic mean radius or depth = a = areaof cross section and g . g P wetted perimeter sine of the angle of slope = h/l. The coefficient c is calculated by the aid of Kutter's formula; 41.6 + 1.811 + .00281 1 + (41.6 + 122281) 8 V r In this formula, r and s have the same meaning as above, and n is a constant depending on the roughness of the surface. If sides and bottom of channel are lined with Well planed timber ................................... n= .009 Neat cement (also for very smooth iron pipes) ........ n = .010 Plaster of 1 sand and 1 cement (or smooth iron pipes). n = .011 Rough lumber (ordinary iron pipes) ............... . . .n = .012 Ashlar or brickwork ........................ .......... n = .013 Rubble ............................................... n = .017 Canals in very firm gravel .................... ......... n = .020 Canals and weirs in moderately good order ............. n = .025 Canals in bad order ................................... n = .035 Table XVIII gives values of rand c \/ r for various values of 166 HYDRAULICS Table I. Coefficients of Efflux for Circular Vertical Orifices. HAMILTON SMITH, JR. Head in Feet 0.02 0.04 Diameter of Orifice in Feet 0.07 0.1 0.2 0.6 1 0.4 .637 .624 .618 0.6 .655 .630 .618 .613 .601 .593 0.8 .648 .626 .615 .610 .601 .594 .590 1.0 .644 .623 .612 .608 .600 .595 .591 1.5 .637 .618 .608 .605 .600 .596 .593 2 .632 .614 .607 .604 .599 .597 .595 3 .627 .611 .604 .603 .599 .598 .597 4 .623 .609 .603 .602 .599 .597 .596 6 .618 .607 .602 .600 .598 .597 .596 8 .614 .605 .601 .600 .598 .596 .596 10 .611 .603 .599 .598 .597 .596 .595 20 .601 .599 .597 .59& .596 .596 .594 50(?) .596 .595 .594 .594 .594 .594 .593 100 (?) .593 .592 .592 .592 .592 .592 .592 Table 11. Coefficients of Efflux for Square Vertical Orifices. HAMILTON SMITH. JR. Head in Feet Side of Square in Feet 0.02 0.04 0.07 0.1 0.2 0.6 1 0.4 .643 .628 .621 0.6 .660 .636 .623 .617 .605 .598 0.8 .652 .631 .620 .615 .605 .600 .597 1.0 .648 .628 .618 .613 .605 .601 .599 1 F .641 .622 .614 .610 .605 .602 .601 2 .637 .619 .612 .608 .605 .604 .602 3 .632 .616 .609 .607 .605 .604 .603 4 .628 .614 .608 .606 .605 .603 .602 6 .623 .612 .607 .605 .604 .603 .602 8 .619 .610 .606 .605 .604 .603 .602 10 .616 .608 .605 .604 .603 .602 .601 20 .606 .604 .602 .602 .602 .601 .600 50(?) .602 .601 .601 .600 .600 .599 .599 100(?) .599 .598 .598 .598 .598 .598 .598 HYDRAULICS 167 Table III Imperfect Contraction CnCo Cn-C 'o Co Co n C R n c R .05 .007 .009 .55 .161 .178 .10 .or4 .019 .60 .189 .208 .15 .023 .030 .65 .223 .241 .20 .034 .042 .70 .260 .278 .25 .045 .056 .75 .303 .319 .30 .059 .071 .80 .351 .365 .35 .075 .088 .85 .408 .416 .40 .092 .107 .90 .471 .473 .45 .112 .128 .95 .546 .537 .50 .134 .152 1.00 .631 .608 C Circular , R Rectangular Openings. Table IV Coefficient (c) for Conical Orifices 5 11 22 45 67 90 112 135 157 180 C .96 .92 .90 .75 .68 .63 .60 .58 .54 .54 Table V Coefficients for Conical Tubes Angle of Cone C a 00' .829 .829 1.000 1 36 .866 .867 1.000 4 . 10 .912 .910 1.000 7 52 .930 .932 .998 10 20 .938 .951 .986 13 24 .946 .963 .983 16 36 .938 .971 .966 21 00 .919 .972 .945 29 58 .895 .975 .918 48 50 .847 .984 .861 168 HYDRAULICS Table VI Discharge Coefficients for Rectangular Weirs FRANCIS' FORMULA. EffecHve head // Coefficient c Effective head h Coefficient c 0.5 and over 0.4 0.3 0.25 3.33 3.337 3.353 3.368 0.2 0.15 0.1 0.06 3.338 3.430 3.528 3.750 Table VII Discharge Coefficients for Rectangular Weirs SMITH'S COEFFICIENTS FOR WEIRS WITHOUT END CONTRACTIONS Effective head h 2 Length of Weir in Feet. 3457 10 19 0.1 0.659 0.658 0.658 0.657 0.15 0.652 0.649 0.647 .645 .645 .644 .643 0.2 .645 .642 .641 .638 .637 .637 .635 0.25 .641 .638 .636 .634 .633 .632 .630 0.3 .639 .636 .633 .631 .629 .628 .626 0.4 .636 .633 .630 .628 .625 .623 .621 0.5 .637 .633 .630 .627 .624 .621 .619 0.6 .638 .634 .630 .627 .623 .620 .618 0.7 .640 .635 .631 .628 .624 .620 .618 0.8 .643 .637 .633 .629 .625 .621 .618 0.9 .645 .639 .635 .631 .627 .622 .619 1.0 .648 .641 .637 .633 .628 .624 .619 1.2 .646 .641 .636 .632 .626 .620 1.4 .644 .640 .634 .629 .622 1.6 .647 .642 .637 .631 .623 Table VIM Discharge Coefficients for Rectangular Weirs SMITH'S COEFFICIENTS FOR WEIRS WITH END CONTRACTIONS Effective Length of Weir in Feet. head h 0.66 1 2 3 5 10 19 0.1 0.632 0.639 0.646 0.652 0.653 0.655 0.656 0.15 .619 .625 .634 .638 .640 .641 .642 0.2 .611 .618 .626 .630 .631 .633 .634 0.25 .605 .612 .621 .624 .626 .628 .629 0.3 .601 .608 .616 .619 .621 .624 .625 0.4 .595 .601 .609 .613 .615 .618 .620 0.5 .590 .596 .605 .608 .611 .615 .617 0.6 .587 .593 .601 .605 .608 .613 .615 0.7 .590 .598 .603 .606 .612 .614 0.8 .595 .600 .604 .611 .613 0.9 .592 .598 -603 .609 .612 1.0 .590 .595 .601 .608 .611 1.2 .585 .591 .597 .605 .610 1.4 .580 .587 .594 .602 .609. 1.6 .582 .591 .600 .607 HYDRAULICS 169 Table IX. Discharge Coefficient C for Rectangular Weirs. From Smith's Coefficients for Weirs without end contractions. Effective Length of Weir in Feet. head h 2 3 4 5 7 10 19 0.1 3.526 3.520 3.520 3.515 0.15 3.488 3.472 3.461 3.451 3.451 3.445 3.440 0.2 3.450 3.435 3.429 3.413 3.408 3.408 3.397 0.25 3.429 3.413 3.403 3.392 3.386 3.381 3.371 0.3 3.418 3.403 3.386 3.376 3.365 3.360 3.349 0.4 3.403 3.386 3.371 3.360 3.344 3.333 3.322 0.5 3.408 3.386 3.371 3.354 3.338 3.322 3.312 0.6 3.413 3.392 3.371 3.354 3.333 3.317 3.306 0.7 3.424 3.397 3.376 3.360 3.338 3.317 3.306 0.8 3.441 3.408 3.386 3.365 3.344 3.322 3.306 0.9 3.451 3.418 3.397 3.375 3.354 3.328 3.312 1.0 3.467 3.429 3.408 3.386 3.360 3.338 3.312 1.2 3.456 3.429 3.403 3.381 3.349 3.317 1.4 3.445 3.424 3.392 3.365 3.328 1.6 3.461 3.435 3.408 3.376 3.333 Table X. Discharge Coefficient C for Rectangular Weirs. From Smith's Coefficients for Weirs with end contractions. Effectiv< l Length of Weir in Feet head //, 0.66 1 2 3 5 10 19 0.1 3.381 3.419 3.456 3.488 3.494 3.504 3.510 0.15 3.312 3.344 3.392 3.413 3.424 3.429 3.435 0.2 3.269 3.306 3.349 3.371 3.376 3.386 3.392 0.25 3.237 3.274 3.322 3.338 3.349 3.360 3.365 0.3 3.215 3.253 3.296 3.312 3.322 3.338 3.344 0.4 3.183 3.215 3.258 3.280 3.290 3.306 3.317 0.5 3.156 3.189 3.237 3.253 3.269 3.290 3.301 0.6 3.140 3.172 3.215 3.237 3.253 3.280 3.290 0.7 3.130 3.156 3.199 3.226 3.242 3.274 3.285 0.8 3.183 3.215 3.231 3.269 3.280 0.9 3.167 3M99 3.226 3.258 3.274 1.0 3.156 3.183 3.215 3.253 3.269 1.2 3.130 3.162 3.194 3.237 3.264 1.4 3.103 3.140 3.178 3.221 3.258 1.6 3.114 3.162 3.210 3.247 :i Table XI. Values of h 2 for h = to 2.95 h 05 10 15 20 25 30 35 40 45 .0112 .0316 .0581 .0894 .1250 .1643 .2071 .2530 .3019 1 1.0000 1.0759 1.1537 1.2332 1.3145 1.3975 1.4822 1.56.S6 1.6565 1 ,7460 2 2.8284 2.9352 3.0432 3.1525 3.2631 3.3750 3.4881 3.6025 3.7181 3.8349 50 55 60 65 70 75 80 85 90 95 .3536 .4079 .4648 .5240 .5857 .6495 .7155 .7837 .8538 .9259 1 1.8371 1.9297 2.0238 2.1195 2.2165 2.3150 2.4150 2.5163 2.6190 2.7230 2 3.9529 4.0720 4.1924 4.3139 4.4366 4.5604 4.6853 4.8114 4.9385 5.0668 170 HYDRAULICS Table XII Weir Measurement PELTON WATER WHEEL CO. Giving Cubic Feet of Water per minute that will flow over a Weir 1 inch wide and from % to 20% inches deep. Inches '/s !/4 % !/2 % % 7/8 .0 .00 .01 .05 .09 .14 .19 .26 .32 1 .40 .47 .55 .64 .73 .82 .92 1.02 2 1.13 1.23 1.35 1.46 1.58 1.70 '1.82 1.95 3 2.07 2.21 2.34 2.48 2.61 2.76 2.90 3.05 4 3.20 3.35 3.50 3.66 3.81 3.97 4.14 4.30 5 4.47 4.64 4.81 4.98 5.15 5.33 5.51 5.69 6 5.87 6.06 6.25 , 6.44 6.62 6.82 7.01 7.21 7 7.40 7.60 7.80 8.01 8.21 8.42 8.63 8.83 8 9.05 9.26 9.47 9.69 9.91 10.13 10.35 10.57 9 10.80 11.02 11.25 11.48 11.71 11.94 12.17 12.41 10 12.64 12.88 13.12 13.36 13.60 13.85 14.09 14.34 11 14.59 14.84 15.09 15.34 15.59 15.85 16.11 16.36 12 16.62 16.88 17.15 17.41 17.67 17.94 18.21 18.47 13 18.74 19.01 19.29 19.56 19.84 20.11 20.39 20.67 14 20.95 21.23 21.51 21.80 22.08 22.37 22.65 22.94 15 23.23 23.52 23.82 24.11 24.40 24.70 25.00 25.30 16 25.60 25.90 26.20 26.50 26.80 27.11 27.42 27.72 17 28.03 28.34 28.65 28.97 29.28 29.59 29.91 30.22 18 30.54 30.86 31.18 31.50 31.82 32.15 32.47 32.80 19 33.12 33.45 33.78 34.11 34.44 34.77 35.10 35.44 20 35.77 36.11 36.45 36.78 37.12 37.46 37.80 38.15 HYDRAULICS 171 Table XIII Coefficients of Flow (X) of Water in Clean Iron Pipes Under Pressure Velocity Diameters ft. per !/ 2 " 1" 2" 3" 4" 6" 8" 10" 12" 14" sec. .0417' .0833' .1667' .2500' .3333' .5000' . 6667' .8333' 1.000' 1.167' .1 .0480 .0456 .0432 .0422 .0404 .0370 .0344 .0320 .0304 .0292 .3 .0443 .0428 .0396 .0380 .0360 .0336 .0312 .0296 .0279 .0268 .5 .0418 .0398 .0364 .0354 .0340 .0317 .0296 .0283 .0268 .0256 .7 .0402 .0372 .0348 .0335 .0322 .0303 .0285 .0272 .0260 .0248 1. .0381 .0353 .0330 .0317 .0306 .0289 .0274 .0262 .0250 .0241 1.5 .0356 .0332 .0312 .0299 .0289 .0274 .0261 .0249 .0239 .0231 2. .0340 .0317 .0301 .0288 .0279 .0264 .0253 .0242 .0233 .0225 2.5 .0327 .0308 .0292 .0280 .0271 .0257 .0246 .0236 .0228 .0220 3. .0317 .0300 .0284 .0273 .0265 .0252 .0242 .0232 .0224 .0217 3.5 .0308 .0292 .0277 .0267 .0259 .0247 .0237 .0228 .0221 .0214 4. .0300 .0285 .0272 .0262 .0255 .0243 .0233 .0225 .0217 .0211 5. .0287 .0274 .0263 .0254 .0247 .0236 .0227 .0220 .0213 .0207 6. .0276 .0266 .0255 .0248 .0241 .0231 .0223 .0216 .0210 .0203 7. .0268 .025& .0249 .0243 .0236 .0227 .0219 .0212 .0207 .0201 8. .0261 .0253 .0244 .0238 .0232 .0224 .0216 .0210 .0205 .0199 10. .0250 .0245 .0237 .0232 .0226 .0219 .0212 .0206 .0201 .0196 12. .0244 .0239 .0232 .0228 .0223 .0216 .0210 .0204 .0199 .0194 16. .0235 .0232 .0226 .0220 .0218 .0211 .0205 .0200 .0194 .0191 20. .0231 .0228 .0223 .0219 .0214 .0208 .0201 .0197 .0192 .0188 Velocity Diameters ft. per 16" 18" 20" 24" 30" 36" 44" 48" 54" 60" sec. 1.333' 1 . 500' 1 . 667' 2.000' 2.500' 3.000' 3.667 4.000' 4.500' 5.000' 0.1 .0280 .0268 .0260 .0246 .3 .0256 .0248 .0240 .0225 .5 .0244 .0236 .0229 .0212 .0194 .0177 .0160 .0153 .0144 .0136 .7 .0238 .0229 .0221 .0207 .0190 .0174 .0158 .0152 .0143 .0136 1. .0232 .0224 .0216 .0202 .0186 .0171 .0156 .015*0 .0142 .0135 1.5 .0224 .0216 .0209 .0196 .0182 .0168 .0154 .0148 .0140 .0134 2. .0218 .0211 .0204 .0193 .0179 .0166 .0153 .0147 .0140 .0134 2.5 .0213 .0207 .0201 .0190 .0177 .0165 .0152 .0146 .0139 .0132 3. .0210 .0204 .0198 .0187 .0175 .0164 .0151 .0145 .0138 .0132 3.5 .0207 .0201 .0196 .0186 .0174 .0163 .0150 .0145 .0138 .0132 4. 5. .0206 .0201 !om .0194 .0191 .0184 .0182 .0173 .0171 .0162 .0161 .0150 .0149 .0145 .0144 .0137 .0137 .0131 .0131 6 .0198 .0193 .0189 .0180 .0169 .0160 .0148 .0143 .0136 .0130 7. .0196 .0191 .0187 .0178 .0168 .0158 .0147 .0142 .0135 .0130 8. .0194 .0190 .0186 .0177 .0167 .0157 .0147 .0142 .0135 .0129 10. .0192 .0188 .0184 .0176 .0166 .0156 .0146 .0141 .0134 .0128 12. .0190 .0186 .0182 .0174 .0164 .0155 .0145 .0140 .0133 .0128 16. .0187 .0182 .0180 .0172 .0162 .0154 .0143 .0139 .0132 .0126 20. .0184 .0181 .0177 .0170 .0161 .0152 .0142 .0138 .0130 .0126 172 HYDRAULICS TabJe XIV Loss of Head by Friction in Clean Iron Pipes Dia- Area Velocity in Feet per Second. meter sq. 1 ' M/2 2 3 in. ft. Q F Q F Q F Q F 1 . 0055 .33 7.00 .50 . 12.9 .66 24.2 .99 49.3 2 .0218 1.31 3.16 1.96 6.55 2 62 11.3 3.92 23.6 3 .0491 2 95 2.00 4.42 4.20 5! 89 7.15 8.84 15.1 4 .0893 5.23 1.47 7.84 3.04 10.5 5.18 15.7 11.1 6 .1963 11.8 .93 17.7 1.92 23.5 3.27 35.3 7.00 8 .3491 20.9 .63 31.3 1.37 41.8 2.36 62.8 5.08 10 .5454 32.7 .50 49.0 1.05 65.4 1.80 98.2 3.89 12 .7854 47.1 .40 70.6 .83 94.2 1.44 141 3.14 14 1.069 64.1 .33 96.1 .69 128 1.20 192 2.60 16 1.396 83.7 .29 126 .59 167 1.13 251 2.21 18 1.768 106 .24 159 .50 212 .87 318 1.90 20 2.182 131 .21 195 .44 261 .76 393 1.66 24 3.142 188 .16 282 .34 377 .60 565 1.31 30 4.909 294 .12 441 .25 589 .44 883 .98 36 7.069 424 .091 636 .20 848 .34 1272 .77 44 10.56 634 .068 951 .15 1267 . 26 1900 .58 48 12.57 754 .060 1131 .13 1508 .23 2262 .51 54 15.90 954 .050 1431 .11 1909 .19 2863 .43 60 19.64 1178 .043 1767 .094 2356 .17 3534 .37 Dia - Area Velocity in Feet per Second met* ;r sq- 4 5 6 7 in. ft. Q F Q F Q F Q F 1 .0055 1.32 83.8 1.65 127 1.98 180 2.31 240 2 .0218 5.23 39.0 6.54 61.2 7.85 85.2 9.16 114 3 .0491 11.7 26.0 14.7 39.4 17.7 55.1 20.5 73.7 4 .0893 20.9 19.0 21 2 28.8 31.4 40.4 36.7 53.9 6 .1963 47.1 12.1 58.9 18.2 70.6 25.9 82.4 34.5 8 .3491 83.7 8.31 104 13.2 126 18.7 146 25.0 10 .5454 131 6.70 163 10.3 196 14.5 229 19.3 12 .7854 188 5.37 235 8.26 283 11.8 330 15.7 14 1.069 257 4.49 321 6.89 385 9.74 449 13.1 16 1.396 335 3.84 419 5.86 502 8.34 586 11.2 18 1.768 424 3.27 530 5.08 636 7.11 742 9.65 20 2.182 523 2.89 654 4.42 785 6.3^ 916 8.51 24 3.142 753 2.28 942 3.54 1131 5 'V - 1320 6.76 30 4.909 1177 1.72 1472 2.65 1767 3.80 2062 5.11 36 7.069 1696 1.34 2120 2.10 2544 2.97 2969 4.03 44 10.56 2534 1.01 3168 1.58 3801 2.26 4435 3.04 48 12.57 3015 .89 3760 1.40 4524 2.00 5278 2.70 54 15.90 3816 .75 4771 1.18 5725 1.69 6680 2.28 60 19.64 4712 .65 5890 1.02 7068 1.46 8246 1.98 Q Discharge in cubic feet per minute. F Head loss in feet per 1000 ft of pipe. Friction in riveted pipe = 1.2F. HYDRAULICS 173 Table XV Coefficients of Flow (X) of Water in Wood Stave Pipes Under Pressure Velocity ft- per sec. 14" 1.167' 16" 1.333' Diameters 18" 20" 1.500' 1.667' 22" 1.833' 24" 2' 30" 2.5' .2 .5 .7 .0379 .0229 .0214 .0415 . 0232 .0216 .0494 .0243 .0219 .0262 .0229 .0281 .0237 . 0301 .0247 .0398 .0294 1.0 1.5 2.0 .0206 .0202 .0201 .0206 .0201 .0199 .0207 .0200 .0198 .0210 .0200 .0198 .0213 .0201 .0197 .0218 .0202 .0197 .0239 0210 .0199 2.5 3.0 3.5 .0201 .0200 .0200 .0199 .0198 .0198 .0197 .0196 .0196 .0196 .0195 .0195 .0195 .0194 .0193 .0194 .0193 .0192 .0194 .0192 .0190 4.0 5.0 6.0 .0199 .0199 .0199 .0198 .0197 .0197 .0196 .0195 .0195 .0195 .0194 .0194 .0193 .0192 .0192 .0192 .0191 .0191 .0189 .0188 .0187 7.0 8.0 10.0 .0199 .0199 .0199 .0197 .0197 .0197 .0195 .0195 .0195 .0194 .0194 .0194 .0192 .0192 .0192 .0191 .0190 .0190 .0187 .0187 .0187 12.0 16.0 20.0 .0199 .0199 .0199 .0197 .0197 .0197 .0195 .0195 .0195 .0194 .0194 .0194 .0192 .0192 .0192 .0190 .0190 .0190 .0186 .0186 .0186 Velocity ft. per sec. 36" 3' 42" 3.5' Diameters 48" 54" 4' 4.5' 60" 5' 66" 5.5' 72" 6' .5 .7 1.0 .0499 .0344 .0261 . .0398 .0285 .0360 .0260 .0259 .0203 .0250 .0200 .0239 .0197 .0222 .0190 1.5 2.0 2.5 .0217 .0202 .0195 .0225 .0204 .0194 .0207 .0188 .0180 .0173 .0162 .0158 .0173 .0164 .0160 .0173 .0166 .0163 .0174 .0168 .0166 3.0 3.5 4.0 .0191 .0189 .0187 .0188 .0185 .0183 .0175 .0172 .0170 .0155 .0154 .0152 .0157 .0156 .0155 .0161 - .0160 .0159 .0164 .0163 .0163 5.0 6.0 7.0 .0185 .0184 .0184 .0181 .0179 .0179 .0168 .0167 .0166 ~?0151 .0150 .0150 .0154 .0154 .0153 .0158 .0157 .0157 .0162 .0162 .0162 8.0 10.0 12.0 .0183 .0183 .0183 .0178 .0177 .0177 .0166 .0165 .0165 .0150 .0150 .0149 .0153 .0152 .0152 .0157 .0157 .0157 .0161 .0165 .0161 16.0 20.0 .0183 .0182 .0177 .0177 .0165 .0164 .0149 .0149 .0152 .0152 .0156 .0156 .0161 .0161 3 7-i HYDRAULICS Table XVI Coefficient of Friction for Elbows 5 r 20 .046 40 .139 60 .364 80 .740 90 .984 100 1.260 120 1.861 140 2.4S1 Table XVII Coefficient of Friction for Curves d/r .4 .6 .8 1 1.2 1.4 1.6 1.8 2 r .138 .158 .206 .294 .440 .661 .977 1.408 1.979 r .14 .18 .25 .40 .64 1.02 1.55 2.27 3.23 Circular Section. " Rectangular Section. Table XVIII. Kutter's Coefficients, n - 0.009 1 in 20000 *J r c c\/r 1 in c 10000 ft/r 1 in c 5000 c v x r 1 in c 2500 c v/r 1 in 1000 c cv'r A 93 37 106 42 114 46 119 48 123 49 .5 108 54 120 60 128 64 133 67 137 68 .6 121 73 132 79 140 84 145 87 147 88 .7 133 93 143 100 150 105 154 108 156 109 .8 142 114 152 122 158 126 162 129 164 131 .9 151 136 160 144 165 149 168 151 170 153 1.0 159 159 167 167 171 171 174 174 175 175 1.2 173 207 178 214 181 218 183 220 184 221 1.5 188 283 191 287 193 289 193 290 194 291 2.0 208 415 206 413 205 411 205 410 205 409 2.5 221 553 217 541 214 535 213 532 212 529 3.0 231 694 224 673 220 660 218 654 217 650 n - 0.010 v/r 1 in 20000 c c\/r 1 in c 10000 rv/r 1 in c 5000 9 .7 .8 .9 104 112 120 72 90 108 112 120 127 78 96 114 118 125 131 82 100 118 121 128 134 85 102 120 123 130 136 86 104 122 1.0 1.2 1.5 126 138 153 126 166 229 133 143 155 133 172' 233 137 146 156 137 175 235 139 148 157 139 177 236 140 149 158 140 178 237 2.0 2.5 3.0 171 184 193 341 459 ' 580 169 179 187 339 448 560 169 177 183 337 442 549 168 176 181 337 439 543 168 175 180 336 437 539 n = 0.012 1 in 20000 1 in 10000 1 in 5000 1 in 2500 1 in 1000 Vr c C \/ T c c\/r c cVr c cVr c c Vr .4 .5 .6 63 74 84 25 37 50 71 83 92 28 41 55 77 88 98 31 44 59 81 92 101 32 46 61 84 95 ,103 34 47 62 .7 .8 .9 93 101 108 65 81 97 101 108 114 70 86 103 106 113 119 74 90 107 109 115 121 76 92 109 111 117 123 78 94 110 1.0 1.2 1.5 114 126 140 114 151 209 120 130 142 120 156 212 124 133 143 124 159 214 126 134 144 126 161 216 127 135 144 127 162 216 2.0 2.5 3.0 157 169 179 314 423 536 156 165 173 311 413 518 155 163 169 '310 408 507 155 162 167 309 405 501 154 161 166 308 403 498 n = 013 1 in 20000 1 in 10000 1 in 5000 1 in 2500 1 in 1000 \/r c C V T c cVr c c\/ r c cV r c c\/ r A .5 .6 57 67 76 23 33 46 64 74 83 26 37 50 69 80 88 28 40 53 73 83 92 29 41 55 75 85 94 30 43 56 .8 1.0 1.2 92 104 115 73 104 138 98 110 119 78 110 143 102 113 122 82 113 146 105 115 123 84 115 148 107 ' 117 124 85 117 149 1.5 2.0 2.5 128 145 157 193 290 393 130 144 153 196 288 384 132 143 151 197 286 378 132 .143 150 198 286 375 133 142 149 199 285 373 3.0 167 500 161 482 157 471 155 466 154 462 176 HYDRAULICS Kutter's Coefficients Con't. n * 0.017 1 in 20000 1 in 10000 1 in oOOO 1 in 2500 1 in 1000 Vr c cVr c cV r c cVr c cVr c c V r A .5 .6 40 47 54 16 24 32 45 52 59 18 26 35 48 56 63 19 28 38 50 58 65 20 29 39 52 60. 67 21 30 40 .8 1.0 1.5 66 77 97 53 77 145 71 81 98 57 81 148 74 83 99 59 83 149 76 85 100 61 85 150 78 86 100 62 86 151 2.0 3.0 112 132 223 394 111 126 221 379 110 123 220 370 110 122 219 365 110 121 219 362 n = 0.020 1 in 20000 1 in 10000 1 in 5000 1 in 2500 1 in 1000 Vr c cVr c cV r c cV r C cVr c c Vr A .6 .8 32 44 55 13 26 44 36 48 58 14 29 47 39 51 61 15 31 49 41 53 63 16 32 50 42 54 64 17 33 51 1.0 1.5 2.0 64 82 95 64 123 190 67 83 94 67 125 189 69 84 94 69 126 188 71 84 94 71 127 187 72 85 93 72 127 187 3.0 4.0 114 127 342 506 109 119 328 475 107 114 320 458 105 112 315 448 104 110 312 442 n = 0.025 1 in 20000 1 in 10000 1 in 5000 1 in 2500 1 in 1000 Vr c cVr c cVr c c V r c cVr c cV r A .6 1.0 24 34 49 10 20 49 26 36 52 11 52 29 39 54 11 23 54 30 40 55 12 24 55 31 41 55 12 25 55 1.5 2.0 3.0 65 77 94 97 153 281 66 76 90 99 152 270 67 76 87 100 151 263 67 75 86 100 151 259 67 75 85 101 150 256 4.0 106 423 99 396 95 380 93 371 91 366 n - 0.035 1 in 20000 1 in 10000 1 in 5000 1 in 2500 1 in 1000 Vr c *Vr c cV r c cVr c cV r c cV r. .5 1.0 2.0 19 34 55 9 34 110 21 35 55 10 35 109 ''2 36 54 11 36 109 23 37 54 11 37 108 38 54 12 108 3.0 4.0 70 81 209 322 67 75 201 302 65 72 195 289 64 70 192 282 63 69 190 277 INDEX Aluminum cables, 156 wire. 155 Ammonia vapor, 150 Angles conversion of, 78 functions of, 31,40 steel, 116 Angular velocities, 84 Arcs, circular, 45, 46 Areas conversion of, 78 of circles, 49, 51, 52 of segments, 45, 47 of sections, 102 Atomic weights, 134 Beams mechanics of, 129 table, 131 T, 124 I, 110 bulb, 112 Bending moment, 129, 131 Boiling points, 139 Boyle's law. 140 Bulb beams, 112 Cables, 156 Capacity carrying, 154, 155, 156 specific inductive. 153 of transmission lines, 158 of wires, 154, 155 of cables. 156 units of. 76 Catenary, 99 Center of mass, 102, 107 of pressure, 161 Centigrade to Fahrenheit, 136 Channels flow in. 165 steel, 114 Charging current, 158 Charles' law, 140 Circles, 49, 51, 52 Circular- arcs. 45. 46 cords, 45, 47 segments. 45, 47 Circumferences of circles, 49, 52 Coefficients of contraction, 167 for curves, 174 of deflection, 109 of efflux, 166 of elasticity, 128 for elbows, 175 of expansion, 138 of flow, 162 Kutter's, 174 for orifices. 166. 167 for pipes, 171. 173 of strength, 109 temperature. 152 for weirs, 168, 169, 170 Common logarithms, 3. 10 Columns, 128 Compound rolled shapes, 109 Compression, 128 Contraction partial, 162 Conversion factors, 78 Copper cables, 156 wire, 154 Cords of circles, 45, 47 Cosine hyperbolic, 89 trigonometric, 31 Cotangent trigonometric, 31 Cotton-covered wire, 57 Crane traveling, 133 Cube roots of numbers, 52 of fractions, 72 Cubes of numbers, 52 Cubic equation, 91, 92 Curves plane, 99 flow through, 165, 17-4 Cycloid, 99 Decimals of an inch, 73 of a foot, 73 Definite integrals, 98 Deflection of beams, 109, 131 Density of gases, 141 of substances, 134 of water, 139 units of, 81 Deposition electric, 153 Descartes' rule. 92 Discharges water, 80 Ditches flow in, 165 Elbows. 165, 174 Efflux, 162, 166 Electrolytic deposition, 153 Electrostatic to electromagnet- ic. 84 Elements chemical, 134 Equations of curves, 99 solution of, 90 Error probable, 100 Evaporation factors of, 142 Expansion coefficients of. 138 Factors of evaporation. 141. 142 Fahrenheit to Centigrade. 136 Fifth roots and powers, 72 Flow coefficients of, 162 in channels. 165 in elbows, 165, 174 through orMfices, 162 in pipes, 164 over weirs, 163 Flumes flow in, 165 Forces units, 81 178 INDEX Fractions of an inch, 73 of a foot, 73 roots of, 72 Francis' formula, 163 Fusing of wires, 153 Gas constants, 141 Gases, 140 Gauges B. & S., 157 wire and sheet metal, 85 Greek letters, 2 Guldin's rules, 101 Gyration radius of, 102, 109 Heat units, 82 Horner's method, 92 Hyperbolic functions, 87 Hysteresis, 151 I-beams, 110 Immersed rectangle, 161 Impedance, 158 Imperfect contraction, 162, 167 Inches decimals of, 73 Inertia moments of, 102, 109 Integrals, 96, 98 Jet impact of, 161 Kutter's coefficients, 174 formula, 165 Moments of inertia, 102, 107, 109 Multiples of if , e and g, 86 Natural logarithms, 3, 6, 8 sine, cosine, etc., 31 Newton's method, 94 Oblique triangles, 43 Orifices, 162, 166, 167 Partial contraction, 162 Pipes, 164. 171, 172, 173 Plane triangles, 43 Power units, 83 Powers of T , e and g, 86 Pressure center of, 161 units of. 82 Prismoidal formula, 101 Probable error, 100 Properties of rolled angles, 116 rolled channels, 114 I-beams, 110 rolled shapes, 109 T-beams, 124 bulb beams, 112 Z-bars, 126 copper cables, 156 copper wires, 154 aluminum cables, 156 aluminum wires, 155 Latent heat, 140 Law Boyle's, 140 Charles', 140 of B. & S. gauge, 157 Length of arcs, 45, 46 of cords, 45, 47 units of, 78 Linear velocities, 84 Logarithmic spiral, 99 Logarithms - common, 3, 10 natural, 3, 6, 8 Loss in curves, 165 in elbows. 165 of head, 164 by hysteresis, 151 Magnetic properties, 151 Materials- strength of, 128 density of, 134 Melting points, 139 Metric units, 75 Minutes to degrees, 40 Radius of gyration, 102, 109 Rate of electrolytic deposition, Reactance, 158 Reactions of beams, 129, 131 Reciprocals of numbers, 52 Resistance of aluminum cables, 156 of aluminum wires, 155 of copper cables, 156 of copper wires, 154 specific, 152 Rise of segments, 45, 47 Saturated vapor ammonia. 150 steam. 143 sulphur dioxide. 150 Section modulus, 102, 109 Segments. 45, 47 Series, 95 Shear, 128. 129 Sheet metal gauge, 85 Sine- hyperbolic. 88 trigonometric. 31 Single phase lines, 160 INDEX 170 Slide rule- solution of equations by, 90 on wire table, 157 Smith's formula, 163 Specific- heats, 138, 141 inductive capacity, 153 resistances, 152 Spherical triangles, 44 Square roots r of fractions, 72 of numbers, 52 Squares of numbers, 52 Steam tables, 143 Sulphur dioxide vapor of, 150 T-beams, 124 Tangent- hyperbolic, 87 trigonometric, 31 Temperature coefficients, 152 units of, 84 Tension, 128 Three phase lines, 158 Transmission lines, 158 Triangles plane, 42, 43 spherical, 44 Trigonometric formulae, 41 Units U. S. and British, 74 Metric, 75 conversion of, 76, 78 Valency, 134 Vapor of ammonia. 150 of steam, 143 of sulphur dioxide, 150 Velocities- angular, 84 linear, 84 Volumes of solids, 107 of water, 81 units of, 74, 75, 76, 79 Water- boiling point of, 139 density of, 139 discharges, 80 flow of, 162 pressure of, 161 weights and volumes, 81 Weights- Metric, 75 of materials, 134 U. S. and British, 74 units of, 74, 75, 76, 80 Weirs, 163, 168, 169, 170 Wire- aluminum, 155 copper, 154 cotton-covered, 157 fusing of, 153 gauges, 85 Work units of, 82 Z-bars, 126 180 NOTLS NOTES 181 182 NOTES NOTES 183 184 NOTES NOTES 185 186 NOTES NOTES 1*7 NOTES NOTES 189 190 NOTES NOTFS 191 192 NOTES NOTP:S 193 194 NOTES NOTES 195 - o. ffichter's Improved ^ JJrawing Instruments The new modern patterns have received the highest approv- als of the entire Mechanical world Every genuine Richter is guaranteed if they are stamped. E.D.CO Avoid imitations and old patterns by refusing to buy with- out this stamp. EUGEWE DIETZGEJV CO. 16>-28 First St., San Francisco NEW ORLEANS CHICAGO SOLE AGENTS FOR RICHTER DRJUVIXG I X S T RV M E X T S NEW YORK TORONTO AN INITIAL FINE OF 25 CENTS OCT 3 ,342 LD 21-100m- 7/40 (6936s) 342232 UNIVERSITY OF CALIFORNIA LIBRARY