THE CALDWELL COLLECTION THE GIFT OF THE FAMILY OF GEORGE CHAPMAN CALDWELL TO THE DEPARTMENT OF CHEMISTRY whose senior Professor he was from J 868 to 1903. 723 C-a, 1\ 13 ItTTTT/if Cornell University Library QD 65.D61 Chemical arithmetic.Pt. 1. 3 1924 000 057 707 Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924000057707 CHEMICAL ARITHMETIC. PART I. Published by William Hodge & Co., Glasgow. Williams & Norgaie, London and Edinburgh. CHEMICAL ARITHMETIC. PART I. A COLLECTION OF TABLES, MATHEMATICAL, CHEMICAL, AND PHYSICAL, FOR THE USE OF CHEMISTS AND OTHERS. BY W. DITTMAR, LL.D., F.R.SS. Lond. and Edin., PROFESSOR OF CHEMISTRY IN THE GLASGOW AND WEST OF SCOTLAND TECHNICAL COLLEGE. GLASGOW^ WILLIAM HODGE & CO., 26 BOTHWELL STREET. 1890. <2- \All rights reserved?^ PREFACE. This book, when completed, will consist of two parts. The first is a collection of tables and formulae intended for the use of Chemists and others ; the second an exposition of the applications of arithmetic to the solution of chemical problems. Part I., which is embodied in the present volume, may be viewed as a new edition of my " Tables to facilitate Chemical Calculations" but the changes and additions are so numerous that it is more correct to call it a new book. The general idea by which I have been guided in its compilation is to try to offer to the scientific as well as to the technical Chemist, a fairly complete collection of what he needs of mathematical auxiliaries, and of chemical and physical constants, in the ordinary routine of his laboratory work, and that without going further. This book, accord- ingly, does not pretend to compete with works of the order of Clarke's " Constants of Nature " or even of Landolt and Bornstein's " Tabellen," nor is it intended to supplant books which, like Lunge and Hurter's " Pocket-Book," address themselves to specialists in this or that branch of technical chemistry. From the majority of the now numerous works of a similar tendency it differs chiefly in this, that it is not a mere compilation but, essentially, an original production. There are, indeed, a few tables, and a few only, in it which were simply copied from other books, but amongst these transferences there is, I trust, not one which could be called a piracy. What I have said applies to some extent to even the logarithm tables given between pages 2 and 47. The logarithms themselves were, of course, simply transcribed from standard works, but each of the tables exhibits certain original features, which, I venture to hope, will be recognized as improvements on the ordinary system. VI PREFACE. The Five-Place Table differs from others in this, that it is continued up to the numerus 14 999, which does away with the troublesome interpolations in connection with numeri between 10 000 and 15 000; and the interpolations generally are greatly facilitated by a novel arrangement of the marginal tables of proportional parts (" PP's ") which consists in this, that instead of the increment in the number, A num., that of the logarithm, A log., is taken as the independent variable, and that every integer value of A log. presents itself in the respective column. In seeking for the logarithm of a number not given directly by the table, find the nearest tabular A n, and adopt the value A log. standing opposite to it. If the number for a given logarithm has to be found, any value A log. that can possibly present itself is found in the second PP-column, and opposite to it stands the corresponding An calculated to one decimal-place. Thus all interpolations can be done at sight, and this with the highest attainable precision. The table, indeed, may be said to combine the advantages of a logarithm and an " anti-logarithm '' table with this important difference in its favour, that the precision of the number found is the same throughout the whole of the cycle. (In ordinary five-place anti-logarithm tables the numeri are given to five places only, which obviously does not do justice to any mantissa lying between .000 00 and about .6). The graphic representa- tion of the PP's for the logarithmic intervals 44 to 29 (on page 19) is an experiment, the practical issue of which remains to be seen. All I can say, meanwhile, is that the diagrams have been tested by a series of critical readings and been found to be sufficiently correct. Quick arithmeticians may perhaps rather use page 18 with these graphic PP's than the part of the table which goes from 10 000 to 15 000. The use of the diagrams, of course, is optional, and, at the worst, they stand in nobody's way. In the arrangement of the table I have followed Bremiker in so far that, like him, I let every page begin with an integer multiple of 50. Another improvement of Bremiker's, which is that for every logarithm the four last figures are given in its tabular locus, I could not have adopted without enlarging my pages, which I did not feel inclined to do, the less so as the advantage of Bremiker's arrangement, which is to reduce the number of changes in the separated figures, is not so great in a table going from 2 000 to 1 5 000 as it is in one which stops at 10 000. I preferred to adopt the usual system, which is to separate only the first two figures, i.e., print each set of these only once where it appears for the first time in the "o "-column. Of the many devices which have been invented for showing where the change comes in, that of Sang is undoubtedly the best. It consists in the substitution of a " notka " ♦ for any o appearing as a third figure in the PREFACE. Vll respective line-section. I accordingly have adopted it, and, as in a five-place table the notka may appear in only one or two of the cases where it would be needed (if the third figure were a o), I have caused the whole of the respective set of entries to be printed in " black type." To pass to the Four-Place Table. I originally intended to reproduce that single-entry table which stands at the beginning of my " Tables to facilitate," and was the more inclined to do so, as this table, in the course of a number of years' constant use, had proved very convenient ; but, with the less size of page adopted for the present edition, it would have taken up too many pages, and, on this account, have been less handy. After some hesitation I decided upon rejecting it and substitut- ing an ordinary double-entry table, taking care however to continue this to 2000, and to introduce my own system of PP's, although these, for the interval from the num. 200 to the num. 500, demand a whole page for themselves. I do not approve of that now popular kind of four-place table in which the PP's for A num. = 1, 2, 3, 4, . . . 9, are given in a line with respective line of ten logarithms, and calculated, as integers, on the assumption that, even for the numerical interval 1000 to 1100, the ratio - is constant. With such a table the uncertainty of a logarithm A num. found by interpolation rises to ± .000 25, corresponding to an error in the number equal to about i/i6ooth of its value, while in a properly constructed four-place table the corresponding uncertainties are .000 1 arid 1 /4000th. In addition to the five- and four-place, I give two Three-Place Logarithm Tables, distinguished as " No. 1 and No. 2." Table No. 1 is a single-entry table constructed on precisely the same principle as the now abolished four-place table of the " Tables to facilitate " was ; in No. 2, all three-place mantissse from .000 to .999 are enumerated in column II., and, opposite to each, stands the respective numerus calculated to four figures. It is, indeed, nothing more or less than an ordinary four-place anti-logarithm table with the variables reversed, and, of course, may be used in this capacity. But it is meant to be used as a three-place logarithm table demanding no interpolation. As may be seen at a glance, any single logarithm taken out of the table cannot be wrong by more than ± .000 5, corresponding to i/8ooth of the value of the respective number, hence the table, in point of precision, is fully at a par with the majority of ordinary analytical determinations. In a book which, like the present, contains an indexed five-place table, both the four- and the three-place table might appear to be Vlll PREFACE. superfluous. A good many years ago, when I had just realised the idea of indexing the five-place table which I was then in the habit of using, I thought so myself, but I have come to see that I was mistaken. For anyone who cares it is easy to put the matter to the test. All he needs to do is to carry out (say) fifty calculations with four-place logarithms, once with our four-place table, and once with our five-place one used as four-place. In most hands the five-place will come off second best. I am not so sure if our four-place table used as a three- place would not work more expeditiously than our three-place No. 2 ; the latter, however, will always retain its value as an occasionally useful four-place anti-logarithm table. Our three-place table No. 1 will not perhaps appear in the next edition of this book; but it so happened that it was printed and corrected before No. 2 was thought of, and as it fills only two pages, and is useful for teaching purposes, I allowed it to stand. The Reciprocal Table is simply a reprint of the one which formed part of the old edition; but, in correcting the proofs, instead of merely following that edition, I re-read all the reciprocals with Barlow's table, and, I am glad to say, found no error. A four-place table of reciprocals is a little ahead of a four-place table of logarithms in point of precision, and it is obviously the more convenient table of the two whenever the reciprocal is wanted for its own sake. Now, such cases occur nowhere more frequently than in practical chemistry. To give only one example, and there are plenty of analogous cases. Supposing we have found that 1 CC of standard solution A neutralises x CC of solution B, and we want to know the number of CC of A which corresponds to 1 CC of B. The reciprocal table gives the answer quite directly, and on this account more exactly than the logarithm table does. The table of values of the Probability-Integral and my own tables for the Conversion of Decimals into Integer-ratios will be welcome, if not to many, at least to a minority of my confreres ; at anyrate, neither of the two needs apologize for its appearance in the book. The "Formula Values" and '■'■Analytical Factors" have all been calculated expressly for this book on the basis of the table of atomic weights given on page 1. Each calculation has been done at least twice, and this by different calculators working independently of each other. The proofs were read, finally, not with the manuscript after which they had been set up, but with the records of the respective calculations. Either table might, of course, have been spun out to any length, but I thought I must, in this case more than in any other, confine myself to what is likely to be of service to the Chemist in his PREFACE. IX every-day practice. In my opinion the tables as they now stand are probably a little too complete. The factors for gas-volumetric deter- minations are based on Regnault's direct determinations of the respective gas-densities and coefficients of expansion ; not that I thought, thereby, to add to the real precision of the factors as such, but it so happened that at the time the respective constants had been already calculated for the section on gasometry, and I therefore had no occasion to use the customary approximate formulae. The section on the "Metric and British Systems of Units" is based on Mr. Chaney's well-known tables in card form ; but the majority of precisely those factors, which I deem to be particularly useful to the Chemist, have been calculated by myself from the fundamental data supplied by the tables referred to. The factors for the conversion of Pounds Sterling into Marks or Francs, and vice versa, and those for the mutual conversion of the price in Shillings or Pounds of the pound Avoirdupois and the price of the Kilogramme in Marks or Francs, etc., will be welcome to all those who, like myself, are in the habit of pro- curing their chemicals largely from German or French houses. In the section on the " Relation in certain Solutions between Percentage of Dissolved Matter and Specific Gravity " I confined myself to the few cases which are of importance to the general Chemist. A complete collection of the respective tables would, by themselves, have filled a larger volume than the present. Two of the tables on sulphuric acid are calculated by myself after Bineau's direct determinations, and a similar remark applies to the tables on aqueous ethyl-alcohols, which latter are based on Mendelejeff's determinations. The two tables for the conversion of percentages of alcohol by volume into percentages by weight, and vice versa, meet a want felt, I am sure, by many Chemists besides myself, and will, I hope, be found useful. Of the several sections of this book those on " Calibration " and " Gasometry " are, I believe, most entitled to the claim of originality. The tables giving the vapour-tensions of water and alcohol, and those embodying Bunsen's coefficients of absorption of gases are, as far as I can remember just now, while writing this preface, the only portions of these sections which have been copied from other books. All the rest is original, so far at least as arrangement is concerned. In the table giving the " Physical Constants of some Gases according to Regnault," the fundamental values for the gas-densities (i.e. the weight of a litre of air and the specific gravities on the air-scale of the other gases included) were taken directly from Lasch's memoir on his re-calculation of Regnault's determinations ; the coefficients of expansion were copied X PREFACE. from Landolt and Bornstein's " Tabellen," and Baron Wrede's specific gravity of carbonic oxide from Berzelius' "Jahresbericht." For all the rest of the entries I am myself responsible. In the section of " Gasometry" I reproduce my own methods for formulating the relations in perfect gases between weight and volume, and for the calculation and interpretation of gas-analyses, which I expounded many years ago in my article " Analyse, volunietrische, fur Gase" in Fehling's "Handworterbuch." Neither of the two has, as far as I can see, been much heeded by Chemists outside of my own Labora- tory. But, as it so happens that two chemical philosophers of world- wide fame have, however unconsciously, reproduced one of them the one, and the other the other of my notions, and have found it worth while to publish them in scientific journals, I venture to hope that my methods will now become more popular. From the fact that I have left my ideas shut up in "Handworterbuch" for so many years, it is clear that /am far from attaching to them more than a certain amount of didactic value. I must not conclude without tendering my cordial thanks to Messrs. James Robson, William Cullen, J. B. Henderson, and J. F. Ness (all Assistants or Senior Students in my Laboratory), for the unselfish and zealous way in which they have assisted me in the calculation of the formula-values and analytical factors, and in the compilation of the logarithm tables. In order, however, to do full justice to Mr. Robson, I must acknowledge, specially, that he cheerfully assisted me in the reading of all the proofs of the logarithm tables — a kind of work which, it is true, is not difficult, but tiresome to a degree of which only he who has tried it can form an adequate conception. Having thus come back to the subject of the logarithm tables, I may state that we always used Sang's excellent seven-place table as our final standard, and, of course, did not rest before we were absolutely convinced that every one of our own entries agreed with the corresponding one in that classical work. In all those cases, however, where the seven-place logarithm ends in 50, we used Schron's table for seeing whether the 50 is a genuine 50 or stands as an approximation for 49 plus more than half-a-unit of the seventh place. The diagrams on pages 19 and 164 are the work of Mr. Ness, who constructed them most carefully by means of a little dividing-engine of my own contriving, which we are in the habit of using for the graduation of burettes, etc. Mr. Ness's drawings were photo-zincographed by Messrs. Ford & Wall of London, and thus made available for the printing press. PREFACE. XI I am fully convinced that my tables of logarithms, and also my table of reciprocals, are free of errors ; but I am not sufficiently bold to venture upon a similar statement in regard to the tables generally, although I have done my best towards making them all correct. Who- ever may detect errors will earn my gratitude by communicating them to me. W. DITTMAR. Anderson's College, Glasgow, July, 1890. TABLE OF CONTENTS. PAGE Preface, v-xi Atomic Weights, i Three-place Logarithms, Table No. i, 2 Three-place Logarithms, Table No. 2, 4 Four-place Logarithms, 8 Table for the Calculation of Chemical Formula;, 1 3 Reciprocals, 14 Five-place Logarithms, 18 Decimal- Values F of all Integer-Ratios x :y up to F = .5 and y = 60, 48 Values of the Probability-Integral, 5 1 Formula- Values and their Logarithms, 52 Analytical Factors and their Logarithms, 61 Empirical Factors for the Chloroplatinate Method, 67 Empirical Factors for the determination of Sugars by Fehling's Method, 68 Factors for Gas- Volumetric Determinations, 69 The Metric and British Systems of Units, 7 1 Factors in connection therewith, 7 2 Conversion of Statements of Composition, 74 Barometric Readings, 75 Values of Gold Coins according to the weight of Gold con- tained in them by Law, - 76 Conversion of Prices, 76 Hydrometer Scales, 78 Specific Gravities corresponding to n degrees Baume\ 79 XIV CONTENTS. PAGE Relations in a Number of Solutions betiveen Specific Gravity and Percentage of Solutum. Sulphuric Acid, 80 Hydrochloric Acid, 86 Hydrobromic and Hydriodic Acids, 87 Nitric Acid, 88 Caustic Potash and Soda, 89 Aqueous Ammonia, 90 Ethyl- Alcohol ; relation between Specific Gravity and Percentage, 91 To pass from Percents by Weight to Percents by Volume, 95 To pass from Percents by Volume to Percents by Weight, 96 Change of Standard, 97 Specific Gravities of Aqueous Methyl-Alcohols at 0° and at 15.56°, 98 Corrections of Weighings. Formulas, 101 Weight of a litre of Air of 0° to 30°, 103 Reduction of Specific Gravities. Solids. — Formulae and Table for the Reduction to the Vacuum, 104 Change of Standard, 106 Tables giving the Densities of Water at different temperatures, referred to that of Water of 4 , 107 Same Densities referred to Water of 0°, 15°, 15.56°, and 17.5°, 108 To pass from Water of 4° to Water of 15° as a Standard, and vice versa, 109 Liquids. — Reduction to the Vacuum, formulas, 109 Reduction to the Vacuum, Table, in Change of Standard, 112 Correction for the Thermic Expansion of the Body under Examination, - 114 The Author's Differential Method, 115 Calibration. Geometric Method, - 117 Eudiometers, 118 CONTENTS. XV PAGE Titrimetric Apparatus, - 119 Gauging in Absolute Units, with Water, 1 2 1 Do. do., with Mercury, - 124 Expansion of Glass according to Recknagel, 126 Gasometry. Generalities, 127 Table for the Reduction of Water Pressure to Mercury Pressure, 128 Logarithms of 1 +.000 181 43 t, 129 Values of k, 1 +kt and Log. (1 +kt), 130 Relative Value of Gravity at a Number of Stations, 131 Tension of Vapour of Water, 0° to 30.9°, 133 Do. do. 98° to 102°, 134 Do. do. 30° to 100°, 135 Boiling Points of Water under Higher Pressures, 135 Tension of Alcohol from 0° to 30°, 136 Table of the Physical Constants of some Gases according to Regnault, 137 Weight in Grammes of One Litre of Air, Nitrogen, Hydrogen, Carbonic Oxide, Carbonic Acid, at t° and P millimetres, 138 Perfect Gases. General Rules and Formula?, 139 Values of Constants for the Reduction of Gas- Weight and Gas- Volume to each" other, 140 Tables for the Reduction of Gasometric Constants to one another, 142 Gas Analysis. Measurement of Gas Quanta, 144 Notes on Method of Combustion, 146 Combustion Constants of some Gases, 147 The Method of Combustion considered as a method of Proximate Analysis, 149 Gas Absorption. Bunsen's Law, and Formulae based thereon, 150 Composition of Air and Absorption of its Components by Water 153 XVI CONTENTS. PAGE Table of the Coefficients of Absorption by Water of Oxygen, Nitrogen, Carbonic Acid, and Air, 154 Absorption of Air by Sea Water, 155 Coefficients of Absorption of some Gases in Water or Alcohol, 156 Absorption in Water of Ammonia, Sulphurous Acid, and Hydro- chloric Acid, 158 Thermometry. Corrections to be added to the readings of a mercury Thermometer to obtain the corresponding readings of the Air Thermometer, 160 Correction for the Lower Temperature of the Out-standing Thread of Mercury, 161 Thermometer Scales, relations, 162 Table for the Reduction of the Readings of the Three Scales to one another, 163 Diagram showing the Relations between the Fahrenheit and Cen- tigrade Scales, 164 ATOMIC WEIGHTS. = 16 After Lothar Meyer and Seubert's calculations. Exceptions : according to authorities quoted in the notes. Atomic Weight. Atomic Weight. Name of Element. Name of Element. Symbol. Value. Symbol. Value. Aluminium, Al 27.10 Molybdenum Mo 96.2 Antimony, - Sb 119.9 Nickel, Ni 58.7 Arsenic, As 7S-°9 Niobium, Nb 93-9 Barium, Ba 137.20 Nitrogen, - N 14.046 Beryllium, - Be 9.1 Osmium, 7 Os I 9 I -S Bismuth, 1 - Bi 208.0 Oxygen, O 16. Boron, B 10.9 Palladium, - Pd 106.6 Bromine, Br 79-952 Phosphorus, P 31.04 Cadmium, - Cd 112. Platinum, 8 - Pt 195-5 Caesium, Cs !33-o Potassium, - K 39- r 36 Calcium, Ca 40.02 Rhodium, - Rh i°4-3 Carbon, C 12.00 Rubidium, - Rb 85-4 Cerium, Ce I4I-5 Ruthenium, Ru 103.8 Chlorine, CI 35-454 Scandium, - Sc 44.1 Chromium, 3 Cr 5 2 - I 3 Selenium, - Se 79.07 Cobalt, Co 58.7 Silicon, 9 - - Si 28.40 Copper, Cu 63-34 Silver, Ag 107.93 Didymium, - Di 145-4 Sodium, - Na 23-053 Erbium, - E 166.4 Strontium, - Sr 87.52 Fluorine, F 19.1 Sulphur, - - S 32.06 Gallium, Ga 70.1 Tantalum, - Ta 182.7 Gold, 4 Au 197.22 Tellurium, - Te 128.0 Hydrogen, - H 1.0024 Thallium, - Tl 204.2 Indium, - - In "3-7 Thorium, Th 232.5 Iridium, - - Ir 193.0 Tin, Sn 117. 6 Iodine, I 126.85 Titanium, 10 - Ti 48.08 Iron, - Fe 56.02 Tungsten, - W 184.0 Lanthanum, La 138.9 Uranium, - U 240.5 Lead, - - Pb 206.9 Vanadium, V 51.2 Lithium, 6 - Li 6.89 Yttrium, - - Y 89.8 Magnesium, 2 Mg 24-37 Ytterbium, - Yb 173- Manganese, 6 Mn 55-° Zinc, 11 Zn 65-37 Mercury, Hg 200.3 Zirconium, - Zr 90.6 1 and 2 Marignac, 1884. 3 Siewert, as calculated by Clarke. * Mean of determinations by Thorpe and Laurie, 1887, and Kriiss, 1887. 6 W. Dittmar, analyses of the Carbonate ; R.S.E. Trans. 1889. 6 Dewar and Scott ; Marignac. 7 Seubert, 1888. 8 Dittmar and MacArthur, 1886-7. 9 Thorpe and Young, 1887. W Thorpe, Roy. Soc. Proc, 1883. n Mean of determinations by Marignac and by Baubigny, 1884. THREE-PLACE LOGARITHMS— Table No. i. N. Log. P.P. N. Log. P.P. N. Log. P.P. N. Log. P.P. 100 .OCX) 150 .176 200 .301 250 •398 101 .004 151 .179 201 •3°3 251 .400 102 .009 152 .182 202 •3°5 252 .401 103 .013 153 .185 203 • 3°7 253 •403 104 .017 154 .188 204 .31° 254 • 405 105 .021 155 .190 205 .312 255 .407 106 .025 5 156 • 193 206 •314 256 .408 107 .029 An Al 157 .196 207 .316 257 .410 108 •033 •°37 158 .199 .201 208 209 • 3i3 .320 258 .412 •413 109 0-2 I 159 259 0-4 2 110 .041 6 3 160 .204 210 .322 260 .415 111 .045 0-8 4 161 .207 211 •324 261 .417 112 .049 10 S 162 .210 212 .326 262 .418 113 •°53 163 .212 213 .328 263 .420 114 ■OS7 164 .215 3 214 •33° 2 264 .422 2 115 .061 165 .217 An Al 215 •332 An Al 265 •423 An Al 116 .064 .068 166 .220 216 !-334 •336 266 •425 .427 117 167 .223 0-7 I 217 0-5 I 267 0-5 [ 118 .072 168 .225 0-3 2 218 •338 1-0 2 268 .428 1-0 2 119 .076 169 .228 10 3 219 •340 269 •43° 120 .079 170 .230 220 •342 270 ■43i 121 .083 171 •233 221 •344 271 •433 122 .086 4 172 .236 222 •346 272 ■435 123 .090 An Al 173 .238 223 •348 273 •43° 124 ■093 174 .241 224 ■35° 274 •438 0-25 I 125 .097 0-50 2 175 •243 225 •352 275 •439 126 .100 0-75 3 176 .246 226 •354 276 .441 127 .104 1-00 4 177 .248 227 •356 277 •442 128 .107 178 .250 228 •358 278 ■444 129 .111 179 •253 229 .360 279 .446 130 .114 180 ■255 2 230 .362 1 280 •447 1 131 .117 181 .258 An Al 231 ■364 An Al 281 •449 An Al 132 . 121 182 .260 232 233 •365 •367 282 283 •450 ■452 133 .124 183 .262 0-5 I 10 I 1-0 I 134 .127 184 • 265 1-0 2 234 •369 284 •453 135 .130 185 .267 235 •371 285 •455 136 ■134 186 .270 236 •373 286 •45° 137 •137 3 187 .272 237 •375 287 •458 138 .140 An Al 188 .274 238 ■377 288 •459 139 ■143 189 .276 239 •378 289 .461 0-3 I 140 .146 0-7 2 190 .279 240 .380 290 .462 141 .149 1-0 3 191 .281 241 .382 291 .464 142 .152 192 .283 242 •384 292 •465 .467 143 •I5S 193 .286 243 .386 293 144 .158 194 .288 244 •387 294 .468 145 .161 195 .290 245 ■389 295 .470 .471 •473 •474 .476 146 .164 196 .292 246 •39i 296 147 .167 197 .294 247 •393 297 148 .170 198 .297 248 •394 298 149 • 173 199 •299 249 •396 299 150 .176 200 .301 | 250 •398 300 •477 THREE-PLACE LOGARITHMS— Table No. t. 5 -place N. Log. P.P. N. Log. P.P. N. Log. P.P. N. Log. P.P. 300 ■477 400 .602 500 .699 750 ■ 875 302 .480 402 .604 505 •703 755 .878 304 •483 404 .606 510 .708 760 .881 306 .486 406 .609 515 .712 765 .S84 30S .4S9 408 .611 520 .716 770 .886 310 .491 410 .613 525 .720 775 .8S9 312 •494 412 .615 530 .724 s 780 .892 314 •497 414 .617 535 .728 An Al 785 ■895 316 .500 416 .619 540 •732 790 .898 318 .502 418 .621 545 • 736 1-0 2-0 I 2 795 .900 320 .505 420 .623 550 .740 3-0 3 800 •9°3 322 .50S 422 .625 555 ■744 4-0 4 805 .906 324 .511 424 .627 560 • 748 5-0 5 810 .908 326 •513 426 .629 565 ■752 815 .911 328 .516 3 428 .631 3 570 .756 820 .914 3 330 •519 An Al 430 •633 An Al 575 .760 825 .916 An Al 332 .521 •524 432 434 •635 ■637 580 ■763 .767 830 .919 .922 334 0-7 I 07 1 585 835 1-7 I 336 .526 1-3 2 436 • 639 1-3 2 590 .771 840 .924 3-3 2 338 .529 2 3 438 .641 2-0 3 595 ■775 845 .927 5 3 340 •S3i 440 ■643 600 .778 850 .929 342 •534 442 .645 605 .782 855 ■932 344 •537 444 .647 610 •785 4 860 •934 346 •539 446 .649 615 • 7S9 An Al 865 •937 348 •542 448 .651 620 .792 870 .940 1-3 1 350 •544 450 •653 625 .796 2-5 2 875 .942 352 • 547 452 •655 630 •799 3-8 3 880 •944 354 •549 454 •657 635 •803 5-0 4 885 •947 356 •551 456 .659 640 .S06 890 •949 358 •554 458 .661 645 .810 895 •952 360 •556 2 4G0 .663 2 650 .813 900 •954 2 362 •559 An Al 462 .665 An 1 Al 655 .816 905 •957 An Al 364 .561 .563 464 .667 .668 060 820 910 •959 .961 366 10 I 466 10 1 065 '.823 915 2-5 I 368 .566 2-0 2 468 .670 2-0 z 670 .826 920 .964 5 2 370 .568 470 .672 675 .829 925 .966 372 •57i 472 .674 680 ■833 930 .968 374 ■573 474 .676 685 .836 3 935 .971 376 •575 476 .678 690 ■839 An Al 940 •973 378 •577 478 .679 695 .842 945 •975 1-7 1 380 .580 480 .681 700 .845 3 3 2 950 • 978 1 382 .582 482 .683 705 .848 5-0 3 955 .980 ! 384 .584 484 .685 710 .851 960 .982 1 386 .587 486 .687 715 • 854 965 .985 1 388 •589 488 .688 720 •857 970 .987 390 •59i 490 .690 725 .860 975 .989 392 •593 492 .692 730 .863 980 .991 394 •595 494 .694 735 .866 985 • 993 396 .598 496 .695 740 .869 990 .996 398 .600 498 .697 1 745 .872 995 .998 400 .602 500 .699 1 750 ■875 1000 .000 J THREE-PLACE LOGARITHMS— Table No. 2. N. Log. N. Log. N. Log. N. Log. N. Log. 100 .000 112-2 .050 125-9 .100 141-3 .150 158-5 .200 100-2 .001 112-5 .051 126-2 .101 141-6 .151 158-9 .201 100-5 .002 112-7 ■052 126-5 .102 141-9 .152 159-2 .202 100-7 .003 113 •053 126-8 .103 142-2 •153 159-6 .203 100-9 .004 113 2 .054 127-1 .104 142-6 • 154 160 .204 101-2 .005 113-5 •05S 127-4 .105 142-9 •155 160-3 .205 101-4 .006 113-8 .056 127-6 .106 143 2 .156 160-7 .206 101-6 .007 114-0 .057 127-9 .107 143-5 •157 161-1 .207 101-9 .008 114-3 .058 128-2 .108 143-9 .158 161-4 .208 1021 .009 114-6 .059 128-5 .109 144-2 •159 161-8 .209 102-3 .010 114-8 .060 128-8 .110 144-5 .160 162-2 .210 102-6 .Oil 115-1 .061 129-1 .111 144-9 .161 162-6 .211 102-8 .012 115-3 .062 129-4 .112 145-2 .162 162-9 .212 103-0 .013 115-6 .063 129-7 •"3 145-5 .163 163-3 .213 103-3 .014 115-9 .064 1300 .114 145-9 .164 163-7 .214 103 5 .015 1161 .065 130-3 ■"S 146-2 .165 1641 .215 103-8 .016 116-4 .066 130-6 .116 146-6 .166 164-4 .216 104-0 .017 116-7 .067 130-9 .117 146-9 .167 164-8 .217 104-2 .018 116-9 .068 131-2 .118 147-2 .168 165-2 .218 104-5 .019 117-2 .069 131-5 .119 147-6 .169 165-6 .219 104-7 .020 1175 .070 131-8 .120 147-9 .170 166-0 .220 105-0 .021 117-8 .071 132-1 .121 148-3 .171 166-3 .221 105-2 .022 118-0 .072 132-4 .122 148-6 .172 166-7 .222 105-4 .023 118-3 ■073 132-7 .123 148-9 •173 167-1 .223 105-7 .024 118-6 .074 133-0 .124 149-3 .174 167-5 .224 105-9 .025 118-9 •o7S 133-4 .125 1496 •175 167-9 .225 106-2 .026 1191 .076 133-7 .126 150 .176 168-3 .226 106-4 .027 119-4 .077 134-0 .127 150-3 .177 168-7 .227 106-7 .028 119-7 .078 134-3 .128 150-7 .178 169-0 .228 106-9 .029 119-9 .079 134-6 .129 151-0 .179 169-4 .229 107-2 .030 120-2 .080 134-9 .130 151-4 .180 169-8 .230 107-4 .031 120-5 .081 135-2 • 131 151-7 .181 170-2 • 231 107-6 .032 120-8 .082 135-5 .132 152-1 .182 170-6 .232 107 9 •033 121-1 .083 135-8 •133 152-4 .183 171-0 •233 108 1 •°34 121-3 .084 136-1 • 134 152-8 .184 171-4 •234 108-4 ■o3S 121-6 .085 136-5 • i3S 153-1 .185 171-8 •235 108-6 .036 121-9 .086 136-8 .136 153-5 .186 172-2 .236 108-9 .037 122-2 .087 137-1 •137 153-8 .187 172-6 •237 109-1 .038 122-5 .088 137-4 .138 154-2 .188 173 .238 109-4 ■039 122-7 .089 137-7 •139 154-5 .189 173-4 •239 109-6 .040 123-0 .090 138-0 .140 154-9 .190 173-8 .240 109-9 .041 123-3 .091 138-4 .141 155-2 .191 174-2 .241 110-2 .042 123-6 .092 13S-7 .142 155-6 .192 174-6 .242 110-4 •043 123-9 •093 139 •143 156-0 •193 175-0 •243 110-7 .044 124-2 .094 139-3 .144 156-3 .194 175-4 .244 110-9 ■045 124-5 • 095 139-6 • 145 156-7 •195 175-8 •245 111-2 .046 124-7 .096 1400 .146 157-0 .196 176-2 .246 111-4 .047 125-0 .097 140-3 .147 157-4 .197 176-6 .247 111-7 .048 125-3 .098 140-6 .148 157-8 .198 177-0 .248 111-9 .049 125-6 .099 140-9 .149 158-1 .199 177-4 .249 1 112-2 .050 125-9 .IOO 141-3 .150 158-5 .200 177-8 .250 THREE-PLACE LOGARITHMS— Table No. 2. N. Log. N. Log. N. Log. N. Log. N. Log. 177-8 .250 199-5 .300 223-9 •35° 251-2 .400 281-8 .450 178-2 •251 200-0 .301 224-4 ■351 251-8 .401 282-5 •45i 178-6 .252 200-4 .302 224-9 ■352 252-3 .402 283-1 • 452 179-1 •253 200-9 •3°3 225-4 •353 252-9 ■ 403 283-8 •453 179-5 •254 201-4 • 304 225-9 •354 253-5 .404 284-4 •454 179-9 •255 201-8 •3°5 226-5 • 355 254-1 •4°5 285-1 •455 180-3 .256 202-3 .306 227-0 • 356 254-7 .406 285-8 .456 180-7 • 257 202-8 • 3°7 227-5 •357 255-3 .407 286-4 ■457 181-1 .258 203-2 .308 228-0 •358 255-9 .408 287-1 .458 181-6 .259 203-7 •309 228-6 •359 256-4 .409 2877 •459 182-0 •260 204-2 .310 229-1 .360 257-0 .410 288-4 .460 182-4 •261 204-6 •3" 229-6 .361 257-6 .411 289-1 .461 182-8 ■262 205-1 .312 230-1 .'362 258-2 .412 289-7 .462 183-2 .263 205-6 ■313 230-7 •363 258-8 •413 290-4 .463 183-7 .264 206-1 •3M 231-2 •364 259-4 .414 291-1 .464 1841 .265 206-5 •31S 231-7 •365 260-0 .415 291-7 .465 184-5 .266 207-0 .316 232-3 .366 260-6 .416 292-4 .466 184-9 .267 207-5 .317 232-8 • 367 261-2 .417 293-1 .467 185-4 .268 208-0 .318 233-3 .368 261-8 .418 293-8 .468 185-8 .269 208-4 •319 233-9 ■369 262-4 .419 294-4 .469 186-2 .270 208-9 .320 234-4 • 370 263-0 .420 295-1 .470 186-6 .271 209-4 .321 235-0 •37i 263-6 .421 295-8 •47i 187-1 .272 209-9 .322 235-5 ■ 372 264-2 .422 296-5 ■ 472 187-5 •273 210-4 •323 236-0 ■373 264-9 •423 297-2 •473 187-9 .274 210-9 ■324 236-6 ■374 265-5 •424 297-9 • 474 188-4 ■275 211-3 ■32S 237-1 •375 266-1 .425 298-5 •475 188-8 .276 211-8 .326 237-7 .376 266-7 .426 299-2 • 476 189-2 .277 212-3 • 327 238-2 • 377 267-3 .427 299-9 •477 189-7 .278 212-8 .328 238-8 .378 267-9 .428 300-6 .478 190-1 .279 213-3 ■329 239-3 •379 268-5 .429 301-3 • 479 190-5 .280 213-8 •330 239-9 .380 269-2 •43° 302-0 .480 191-0 .281 214-3 ■331 240-4 .381 269-8 ■431 302-7 .481 191-4 .282 214-8 •332 241-0 .382 270-4 •432 303-4 .482 191-9 .283 215-3 ■333 241-5 •383 271-0 •433 304-1 • 483 192-3 .284 215-8 •334 242-1 •384 271-6 •434 304-8 .4S4 192-8 .285 216-3 • 335 242-7 .385 272-3 •435 305-5 •485 1932 .286 216-8 •336 243-2 .386 272-9 ■43° 306-2 .486 193 6 .287 217-3 •337 243-8 •387 273-5 •437 306-9 .487 194-1 .288 217-8 •338 244-3 .388 274-2 •438 307-6 .488 194-5 .289 218-3 •339 244-9 •389 274-8 • 439 308-3 •489 195-0 .290 218-8 •34° 245-5 •39° 275-4 .440 309-0 .490 195-4 .291 219-3 •34i 246-0 •39i 276-1 .441 309-7 .491 195-9 .292 219-8 •342 246-6 •392 276-7 .442 310-5 .492 196-3 ■293 220-3 •343 247-2 ■393 277-3 •443 311-2 ■493 196-8 .294 220-8 •344 247-7 •394 278-0 •444 311-9 ■494 197-2 .295 221-3 •345 248-3 •395 278-6 ■445 312-6 •495 1977 .296 221-8 • 346 248-9 ■396 279-3 .446 3133 .496 198-2 .297 222-3 ■347 249-5 •397 279-9 ■447 314-1 •497 198-6 .298 222-8 ■ 348 250-0 •398 280-5 •448 314-8 .498 1991 .299 223-4 • 349 250-6 •399 281-2 •449 315-5 •499 199-5 .300 223-9 •35° 251-2 .400 281-8 .450 316-2 .500 THREE-PLACE LOGARITHMS— Table No. 2. N. Log. N. Log. N. Log. N. Log. N. Log. 316-2 .500 354-8 ■55° 398-1 .600 446-7 .650 501-2 .700 317-0 .501 355-6 •55i 399-0 .601 447-7 .651 502-3 .701 317-7 .502 356-5 •552 3999 .602 448-7 .652 503-5 .702 318-4 •5°3 357 '3 •553 400 9 .603 449-8 • 653 504-7 ■703 319-2 .504 358-1 •554 401-8 .604 450-8 •654 505-8 .704 319-9 .505 358-9 •555 402-7 .605 451-9 .655 507-0 .705 320-6 .506 359-7 •556 403-6 .606 452-9 .656 508-2 .706 321-4 •5°7 360-6 •557 404-6 .607 453-9 .657 509-3 .707 322-1 .508 361-4 ■558 405-5 .608 455-0 .658 510-5 .708 322-8 •5°9 362-2 •559 406-4 .609 456-0 ■659 511-7 .709 323 6 .510 363-1 .560 407-4 .610 457-1 .660 512-9 .710 324-3 .511 363 9 .561 408-3 .611 458-1 .661 514-0 .711 325-1 .512 364-8 .562 409-3 .612 459-2 .662 515-2 .712 325-8 ■5i3 365-6 •563 410-2 .613 460-3 .663 516-4 •713 326-6 .514 366-4 .564 4111 .614 461-3 .664 517-6 .714 327-3 •515 367-3 .565 412-1 .615 462-4 .665 518-8 •715 328-1 .516 368-1 .566 413 .616 463-4 .666 520-0 .716 328-9 ■517 369 .567 414-0 .617 464-5 .667 521-2 .717 329-6 .518 369-8 .568 415-0 .618 465-6 .668 522-4 .718 330 4 •519 370-7 ■569 415 9 .619 466-7 .669 523-6 .719 331-1 .520 371-5 •570 416-9 .620 467-7 .670 524-8 .720 331-9 .521 372-4 •57i 417-8 .621 468-8 .671 526-0 .721 332-7 .522 373-3 • 572 418-8 .622 469-9 .672 527-2 .722 333-4 •523 374-1 •573 419-8 .623 471-0 • 673 528-4 •723 334-2 •S24 375-0 •574 420-7 .624 472-1 .674 529-7 •724 335 .525 375-8 •575 421-7 .625 473-2 .675 530-9 .725 335-7 .526 376-7 .576 422-7 .626 474-2 .676 532-1 .726 336-5 •527 377-6 •577 423-6 .627 475-3 .677 533-3 .727 337-3 .528 378-4 .578 424-6 .628 476-4 .678 534-6 .728 338-1 .529 379 3 •579 425-6 .629 477-5 .679 535-8 •729 338-8 •53° 380-2 .580 426-6 .630 478-6 .680 537-0 •73° 339-6 ■S3i 381-1 .581 427-6 .631 479-7 .681 538-3 •73i 340-4 •532 381-9 .582 428-5 .632 480-8 .682 539-5 •732 341-2 •533 382-8 .583 429-5 ■633 481-9 .683 540-8 ■733 312 ■534 383-7 .584 430-5 •634 483-1 .684 542-0 •734 342-8 •535 384-6 .58S 431-5 .635 484-2 .685 543-3 •735 343 6 ■536 385-5 .586 432-5 .636 485-3 .686 544-5 ■ 736 344-3 •537 386-4 .587 433 5 .637 486-4 .687 545-8 •737 345-1 •538 387-3 .588 434-5 .638 487-5 .688 547-0 •738 345-9 •539 388-2 •589 435-5 •639 488-7 .689 548-3 ■739 346-7 •540 389 • 590 436-5 .640 489-8 .690 549-5 .740 347-5 •54i 389-9 •59i 437-5 .641 490-9 .691 550-8 .741 348-3 ■542 390-8 •592 438-5 .642 492-0 .692 552-1 .742 349-1 •543 391-7 •593 439-5 •643 493-2 •693 653-4 •743 349-9 '544 392-6 •594 440-6 .644 494-3 .694 554-6 ■744 350-8 •545 393-6 •595 441-6 .645 495-5 .695 555-9 •745 351-6 •546 394-5 •596 442-6 .646 496-6 .696 557-2 .746 352-4 • 547 395-4 ■597 443-6 .647 497-7 .697 558-5 •747 353-2 •548 396-3 .598 444-6 .648 498-9 .698 559-8 •748 354-0 •549 397-2 •599 445-7 .649 500-0 .699 561-0 •749 354-8 •550 398-1 .600 446-7 •650 501-2 .700 562-3 •7So THREE-PLACE LOGARITHMS— Table No. 2. N. Log. N. Log. N. Log. N. Log. N. Log. 562-3 • 750 631-0 .800 707-9 .850 794-3 .900 891-3 .950 563 6 •7Si 632-4 .801 709-6 .851 796-2 .901 893-3 •95i 564-9 • 752 633-9 .802 711-2 .852 798-0 .902 895-4 .952 566-2 •753 635-3 .803 712-9 .853 799-8 •903 897-4 •953 567-5 •754 636-8 .804 714-5 .854 801-7 .904 899-5 •954 568-9 •755 638-3 .805 716-1 ■855 803-5 •9°5 901-6 •955 570-2 • 756 639-7 .806 717-8 .856 805-4 .906 903-6 •956 571-5 • 757 641-2 .807 719-4 •857 807-2 .907 905-7 •957 572-8 • 758 642-7 .808 721-1 .858 809-1 .908 907-8 •958 574-1 ■759 644-2 .809 722-8 .859 811-0 .909 909-9 •959 575-4 .760 645-7 .810 724-4 .860 812-8 .910 912-0 .960 576-8 .761 647-1 .811 726-1 .861 814-7 .911 914-1 .961 578-1 .762 648-6 .812 727-8 ' .862 816-6 .912 916-2 .962 579-4 • 763 650-1 .813 729-5 .863 818-5 •913 918-3 •963 580-8 .764 651-6 .814 731-1 .864 820-4 .914 920-4 .964 582-1 ■765 653-1 .815 732-8 .865 822-2 •915 922-6 .965 583-4 .766 654-J .816 734-5 .866 824-1 .916 924-7 .966 584-8 .767 656-1 .817 736-2 .867 826-0 .917 926-8 .967 586-1 .768 657-7 .818 737-9 .868 827-9 .918 929 .968 587-5 .769 659-2 .819 739-6 .869 829-9 .919 931-1 .969 588-8 • 770 660-7 .820 741-3 .870 831-8 .920 933-3 .970 590-2 .771 662-2 .821 743-0 .871 833-7 .921 935-4 .971 591-6 .772 663-7 .822 744-7 .872 835-6 .922 937-6 .972 592-9 •773 665-3 .823 746-4 •873 837-5 •923 939-7 •973 594-3 •774 666-8 .824 748-2 •874 839-5 .924 941-9 •974 595-7 •775 668-3 .825 749-9 •875 841-4 •925 944-1 •975 597-0 .776 669-9 .826 751-6 .876 843-3 .926 946-2 .976 598-4 • 777 671-4 .827 753 4 •877 845-3 .927 948-4 • 977 599-8 • 778 673-0 .828 755-1 .878 847-2 .928 950-6 .978 601-2 • 779 674-5 .829 756-8 .879 849-2 .929 952-8 •979 602-6 .780 676-1 .830 758-6 .880 851-1 ■93° 955-0 .980 603 9 .781 677-6 .831 760-3 .881 853-1 •93i 957-2 .981 605-3 .782 679-2 .832 762-1 .882 855-1 ■ 932 959-4 .982 606-7 •783 680-8 •833 763-8 .883 857-0 ■933 961-6 •983 608-1 .784 682-3 •834 765-6 .884 859-0 •934 963-8 •984 609-5 .785 683-9 .835 767-4 • 88 S 861-0 •935 966-1 •985 610-9 .786 685-5 .836 769-1 .886 863-0 •936 968-3 .986 612-4 .787 687-1 • 837 770-9 .887 865-0 • 937 970-5 .987 613 8 .788 688-7 .838 772-7 .888 867-0 .938 972-7 .988 615-2 .789 690-2 •839 774-5 .889 869-0 ■939 975-0 .989 616-6 .790 691-8 .840 776-2 .890 871-0 .940 977-2 .99° 618-0 .791 693-4 .841 778-0 .891 873-0 .941 979-5 .991 619-4 .792 695-0 .842 779-8 .892 875-0 .942 981-7 .992 620-9 •793 696-6 ■843 781-6 .893 877-0 •943 984-0 •993 622-3 •794 698-2 .844 783-4 .894 879-0 •944 986-3 • 994 623-7 ■795 699-8 •845 785-2 .895 881-0 •945 988-6 ■ 995 625-2 .796 701-5 .846 787-0 .896 883-1 .946 990-8 .996 626-6 ■797 703-1 .847 788-9 .897 885-1 •947 993-1 •997 628-1 •798 704-7 .848 790-7 .898 887-2 .948 995-4 •998 629-5 •799 706-3 .849 792-5 ■899 889-2 •949 997-7 •999 6310 ■800 707-9 .850 794-3 .900 891-3 •950 1000 .000 FOUR-PLACE LOGARITHMS. N. L. o 1 2 3 4 5 6 7 8 9 A P.P. 0000 3010 4771 6021 6990 7782 8451 9031 95 tl 1 .oooo 0414 0792 "39 1461 1 761 2041 2304 2553 2788 2 .3010 3222 3424 361; 3802 3979 4150 43i4 4472 4624 3 •4771 4914 5051 5185 5315 5441 5563 5682 5798 59" 22 4 5 .6021 .6990 6128 7076 6232 7160 6335 7243 6435 7324 6532 7404 6628 7482 6721 7559 6812 7634 6902 7709 An Al 5 I 6 .7782 7853 7924 7993 8062 8129 8i95 8261 8325 8388 0-9 2 7 ■8451 8513 8573 8633 8692 8751 8808 8S65 8921 8976 1-4 3 8 •9031 9085 9138 9191 9243 9294 9345 9395 9445 9494 1-8 4 9 •9542 9590 9638 9685 9731 9777 9823 9868 9912 9956 2 - 3 5 6 10 .0000 0043 0086 0128 0170 0212 0253 0294 0334 0374 40 2-7 11 .0414 0453 0492 0531 0569 0607 0645 0682 0719 0755 37 3-2 7 12 .0792 0828 0864 0899 0934 0969 1004 1038 1072 1 106 33 3 6 8 13 ■1 139 "73 1206 1239 1271 1303 1335 1367 1399 1430 3i 41 9 14 .1461 1492 1523 1553 1584 1614 1644 1673 1703 1732 29 4-5 10 15 .1761 1790 1818 1847 1875 1903 I93i 1959 1987 2014 27 5-0 11 16 .2041 2068 2095 2122 2148 2175 2201 2227 2253 2279 25 5-5 12 17 .2304 2330 2355 2380 2405 2430 2455 2480 2504 2529 24 5-9 13 18 ■2553 2577 2601 2625 264S 2672 2695 2718 2742 2765 23 6-4 H 19 .2788 2810 2833 2856 2878 2900 2923 2945 2967 2989 21 6*8 15 16 20 .3010 3032 3054 3075 3096 3118 3139 3160 3181 3201 21 7-3 21 .3222 3243 3263 3284 3304 3324 3345 3365 3385 3404 20 7-7 17 22 ■3424 3444 3464 3483 3502 3522 3541 356o 3579 3598 19 8-2 18 23 .3617 3636 3655 3674 3692 37" 3729 3747 3766 3784 18 8-6 19 24 .3802 3820 3838 3856 3874 3892 3909 3927 3945 3962 17 9'1 25 ■3979 3997 4014 4031 4048 4065 4082 4099 4116 4133 17 9-5 21 26 •4150 4166 4183 4200 4216 4232 4249 4265 4281 4298 16 10 22 27 ■43H 4330 4346 4362 4378 4393 4409 4425 4440 4456 16 28 .4472 44S7 4502 45i8 4533 4548 4564 4579 4594 4609 iS 29 .4624 4639 4654 4669 4683 4698 4713 4728 4742 4757 14 30 •4771 4786 4800 4814 4829 4843 4857 4S71 4886 4900 14 TC 31 .4914 4928 4942 4955 4969 4983 4997 5011 5024 5038 13 An Al 32 .5051 5065 5°79 5092 5105 5119 5132 5H5 5159 5172 13 33 ■5185 5198 5211 5224 5237 5250 5263 5276 5289 5302 13 - 7 I 34 •5315 5328 5340 5353 5366 5378 5391 5403 5416 5428 13 1-3 2 35 ■5441 5453 5465 5478 5490 5502 5514 5527 5539 5551 12 2-0 2-7 3 4 36 •5563 5575 5587 5599 56" 5623 5635 5647 5658 5670 12 37 .5682 5694 5705 5717 5729 5740 5752 5763 5775 5786 12 3-3 5 g 38 .5798 5809 5821 5832 5843 5855 5866 5877 5888 5899 12 4'0 39 • 59" 5922 5933 5944 5955 5966 5977 5988 5999 6010 11 4-7 7 40 .6021 6031 6042 6053 6064 6075 6085 6096 6107 6117 11 5 3 6'0 8 9 41 •.6128 6138 6149 6160 6170 6180 6191 6201 6212 6222 10 42 .6232 6243 6253 6263 6274 6284 6294 6304 6314 6325 10 6-7 7-3 8-0 10 43 •6335 6345 6355 6365 6375 6385 6395 6405 6415 6425 10 44 •6435 6444 6454 6464 6474 6484 6493 6503 6513 6522 10 12 45 ■6532 6542 6551 6561 6571 6580 6590 6599 6609 6618 10 8-7 9-3 13 46 .6628 6637 6646 6656 6665 6675 6684 6693 6702 6712 9 r 4 47 .6721 6730 6739 6749 6758 6767 6776 6785 6794 6803 9 ift-n 48 .6812 6821 6830 6839 6848 6857 6866 6875 6884 6893 9 LV U ij 49 .6902 691 1 6920 6928 6937 6946 6955 6964 6972 6981 9 50 .6990 6998 7007 7016 7024 7033 7042 7050 7059 7067 9 PARTES PROPORTIONALES. 21 20 19 18 17 16 An Al An Al An Al An Al Anl Al An Al 0-5 I 0-5 I 0-5 I 6 I 6 I 06 I 1-0 2 1-0 2 11 2 11 2 1-2 2 1-2 2 1-4 3 1-5 3 1-6 3 1-7 3 1-8 3 1-9 3 1-9 4 2 4 21 4 2-2 4 2-4 4 2-5 4 2-4 5 25 5 2-6 S 2-8 S 2-9 S 3-1 S 2 9 6 3 6 3 2 6 33 6 3 5 6 3-8 6 3 3 7 3-5 7 3-7 7 3 9 7 4-1 7 4.4 7 3-8 8 4-0 8 4-2 8 4-4 8 4-7 8 5-0 8 4-3 9 4-5 9 4-7 9 5-0 9 5-3 9 5-6 9 4-8 IO 5-0 10 5 3 10 5-6 10 5-9 10 62 10 \ 5-2 II 5-5 11 5-8 11 61 11 6 5 11 6-9 11 5-7 12 6-0 12 63 12 6-7 12 7-1 12 7-5 12 6-2 13 6 5 13 6 8 13 7-2 13 7-6 13 8-1 13 6-7 14 7-0 14 7-4 14 7-8 14 8-2 14 8-7 14 7-1 iS 7-5 15 7-9 IS 8-3 15 8-8 IS 9-4 16 7 6 16 8-0 16 8-4 16 8-9 16 9-4 16 10 8-1 17 8-5 17 8-9 i7 9-4 17 10 17 8-6 18 9 18 9-5 18 10 18 9-0 19 9-5 19 10 19 9-5 20 10 20 10 21 14 13 12 11 9 8 An Al An Al An Al An Al An Al An Al 0-7 I 0-8 I 0-8 I 09 I 11 I 1-3 I 1-4 2 1-5 2 1-7 2 1-8 2 2-2 2 2-5 2 21 3 2-3 3 2-5 3 2-7 3 33 3 3-8 3 2-9 4 31 4 3-3 4 3-6 4 4.4 4 5-0 4 3 6 5 3-8 S 4-2 5 4-5 S 5-6 5 6-3 S 4-3 6 4-6 6 5-0 6 5-5 6 67 6 7-5 6 5-0 7 5-4 7 5-8 7 6-4 7 7-8 7 8-8 7 j 5-7 8 6-2 8 6-7 8 7-3 8 8-9 8 10 8 6-4 9 6 9 9 7-5 9 8-2 9 10 9 7-1 10 7-7 10 8-3 10 9-1 10 7-9 11 8-5 11 9-2 11 10 11 8-6 12 9-2 12 10 12 9-3 13 10 13 10 14 10 FOUR-PLACE LOGARITHMS. N. L.o 1 2 3 4 5 6 7 8 9 P.P. 50 .6990 6998 7007 7016 7024 7033 7042 7050 7059 7067 51 .7076 7084 7093 7101 7110 7118 7126 7135 7143 7152 52 .7160 7168 7177 7185 7193 7202 7210 7218 7226 7235 n 53 •7243 7251 7259 7267 7275 7284 7292 7300 7308 73i6 An Al 54 •7324 7332 7340 7348 7356 7364 7372 738o 7388 7396 55 .7404 7412 7419 7427 7435 7443 745i 7459 7466 7474 11 2-2 I 2 56 .7482 7490 7497 7505 7513 7520 7528 7536 7543 7551 3-3 3 4 57 •7559 7566 7574 7582 7589 7597 7604 7612 7619 7627 4-4 58 ■7634 7642 7649 7657 7664 7672 7679 7686 7694 7701 59 .7709 7716 7723 7731 7738 7745 7752 7760 7767 7774 5-6 5 60 .7782 7789 7796 7803 7810 7818 7825 7832 7839 7846 6-7 7-8 8-9 6 7 8 61 ■7853 7860 7868 7375 7882 7889 7896 7903 7910 7917 62 .7924 7931 7938 7945 7952 7959 7966 7973 7980 7987 10 9 C3 • 7993 8000 8007 8014 8021 8028 803S 8041 8048 8055 64 .8062 8069 8075 8082 8089 8096 8102 8109 8116 8122 65 .8129 8136 8142 8149 8156 8162 8169 8176 8182 8189 8 66 .8195 8202 8209 8215 8222 8228 8235 8241 8248 8254 67 68 .8261 •8325 8267 8331 8274 8338 8280 8344 8287 8351 8293 8357 8299 8363 8306 8370 8312 8376 8319 8382 A n LA 1 1-3 2-5 69 .8388 8395 8401 8407 8414 8420 8426 8432 8439 8445 2 70 .8451 8457 8463 8470 8476 8482 8488 8494 8500 8506 3-8 5-0 3 71 ■8513 8519 8525 8531 8537 8543 8549 8555 8561 8567 4 72 ■8573 8579 8585 8591 8597 8603 8609 8615 8621 8627 6 3 7-5 8-8 73 •8633 8639 8645 8651 8657 8663 8669 8675 8681 8686 5 6 7 74 .8692 8698 8704 8710 8716 8722 8727 8733 8739 8745 75 .8751 8756 8762 8768 8774 8779 8785 8791 8797 8802 10 8 76 .8808 8814 8820 8825 8831 8837 8842 8848 8854 8859 77 .8865 8871 8876 8882 8887 8893 8899 8904 8910 8915 7S .8921 8927 8932 8938 8943 8949 8954 8960 8965 8971 79 .8976 8982 8987 8993 8998 9004 9009 9015 9020 9025 An / Al 80 .9031 9036 9042 9047 9053 9058 9063 9069 9074 9079 1*4 81 .9085 9090 9096 9101 9106 9112 9117 9122 9128 9133 2 - 9 82 .9138 9H3 9149 9154 9159 9165 9170 9175 9180 9186 4-3 57 3 83 .9191 9196 9201 9206 9212 9217 9222 9227 9232 9238 84 •9243 9248 9253 9258 9263 9269 9274 9279 9284 9289 4 85 .9294 9299 9304 93°9 9315 9320 9325 9330 9335 9340 7-1 8-6 10 5 6 7 86 •9345 9350 9355 9360 9365 9370 9375 9380 9385 9390 87 •9395 9400 9405 9410 9415 9420 9425 943° 9435 9440 88 • 9445 9450 9455 9460 9465 9469 9474 9479 9484 9489 89 • 9494 9499 9504 9509 9513 9518 9523 9528 9533 9538 90 •9542 9547 9552 9557 9562 9566 9571 9576 958i 9586 6 91 .9590 9595 9600 9605 9609 9614 9619 9624 9628 9633 A n 1 ■> A 92 .9638 9643 9647 9652 9657 9661 9666 9671 9675 9680 1-7 3-3 5-0 93 .9685 9689 9694 9699 9703 9708 9713 9717 9722 9727 I 94 ■9731 9736 9741 9745 9750 9754 9759 9763 9768 9773 2 3 95 •9777 9782 9786 9791 9795 9800 9805 9809 9814 9818 6-7 4 96 .9823 9827 9832 9836 9841 9845 9850 9854 9859 9863 8-3 in 97 .9868 9872 9877 9881 9886 9890 9894 9899 9903 9908 5 c 98 .9912 9917 992i 9926 993° 9934 9939 9943 9948 9952 1\J u 99 •9956 9961 9965 9969 9974 9978 9983 9987 9991 9996 100 .0000 0004 0009 0013 0017 0022 0026 0030 0035 0039 FOUR-PLACE LOGARITHMS. ii P.P. N. L.o 1 2 3 4 5 6 7 8 9 100 .oooo 0004 0009 0013 0017 0022 0026 0030 0035 0039 1 101 .0043 0048 0052 0056 0060 0065 0069 0073 0077 0082 1 102 .0086 0090 0095 0099 0103 0107 OIII 01 16 0120 0124 : 103 .0128 0133 oi37 0141 0145 0149 oi54 0158 0162 0166 104 .0170 0175 0179 0183 0187 0191 0195 0199 0204 0208 5 Al 105 .0212 0216 0220 0224 0228 0233 0237 0241 0245 0249 A n 106 .0253 0257 0261 0265 0269 0273 0278 0282 0286 0290 2 4-0 6-0 107 .0294 0298 0302 0306 0310 0314 0318 0322 0326 0330 I 108 .0334 0338 0342 0346 035o 0354 0358 0362 0366 0370 2 3 109 •0374 0378 0382 0386 0390 0394 0398 0402 0406 0410 8-0 4 110 .0414 0418 0422 0426 0430 0434 0438 0441 0445 0449 10 5 111 • 0453 0457 0461 0465 0469 0473 0477 0481 0484 0488 112 .0492 0496 0500 0504 0508 0512 0515 0519 0523 0527 113 •°S3i 0535 0538 0542 0546 0550 0554 0558 0561 0565 114 .0569 0573 0577 0580 0584 0588 0592 0596 0599 0603 115 .0S07 061 1 0615 0618 0622 0626 0630 0633 0637 0641 116 .0645 0648 0652 0656 0660 0663 0667 0671 0674 0678 117 .0682 0686 0689 0693 0697 0700 0704 0708 0711 0715 118 .0719 0722 0726 0730 0734 0737 0741 0745 0748 0752 An 2l 119 ■0755 0759 0763 0766 0770 0774 0777 0781 0785 0788 2-5 5-0 7-5 10 120 .0792 0795 0799 0803 0806 0810 0813 0817 0821 0824 I 121 .0828 0831 0835 0839 0842 0846 0849 0853 0856 0860 2 3 122 .0864 0867 0871 0874 0878 0881 0885 0888 0892 0896 123 .0399 0903 0906 0910 0913 0917 0920 0924 0927 0931 4 124 ■0934 0938 0941 0945 0948 0952 0955 0959 0962 0966 125 .0969 0973 0976 0980 0983 0986 0990 0993 0997 1000 126 .1004 1007 IOII 1014 1017 1021 1024 1028 103 1 1035 127 .1038 1041 1045 1048 1052 1055 1059 1062 1065 1069 128 .1072 1075 1079 1082 1086 1089 1092 1096 1099 1 103 129 .1106 1 109 i"3 1116 1119 1 123 1 126 1 129 "33 1136 A n 3 Al 130 • 1 139 "43 1 146 1 149 "53 1156 "59 1 163 1 166 1 169 131 • "73 1176 1 179 1 183 1 186 1 189 "93 1 196 "99 1202 3 3 6-7 10 ~ 132 .1206 1209 1212 1216 1219 1222 1225 1229 1232 1235 2 133 .1239 1242 1245 1248 1252 1255 1258 1261 1265 1268 3 134 .1271 1274 1278 1281 1284 1287 1290 1294 1297 1300 135 • 1303 1307 1310 1313 1316 1319 1323 1326 1329 1332 136 •1335 1339 1342 1345 1348 i35i 1355 1358 1361 1364 137 .1367 1370 1374 1377 1380 1383 1386 1389 1392 1396 138 • 1399 1402 1405 1408 1411 1414 1418 1421 1424 1427 139 .1430 1433 1436 1440 1443 1446 1449 1452 1455 1458 140 .1461 1464 1467 1471 1474 1477 1480 1483 i486 1489 An Al 141 .1492 1495 1498 1 501 1504 1508 i5" 1514 1517 1520 142 •1523 1526 1529 1532 1535 1538 1541 '544 1547 1550 5-0 10 143 • 1553 1556 1559 1562 1565 1569 1572 1575 1578 1581 I 2 144 .1584 1587 .1590 1593 1596 1599 1602 1605 1608 1611 145 .1614 1617 1620 1623 1626 1629 1632 1635 1638 1641 146 .1644 1647 1649 1652 1655 1658 1661 1664 1667 1670 147 • i673 1676 1679 1682 1685 1688 1691 1694 1697 1700 148 ■ 1703 1706 1708 1711 1714 1717 1720 1723 1726 1729 149 • 1732 1735 1738 1741 1744 1746 1749 1752 1755 1758 J 150 .1761 1764 1767 1770 1772 1775 1778 1781 1784 1787 FOUR-PLACE LOGARITHMS. N. L.o i 2 3 4 5 6 7 8 9 P.P. 150 .1761 1764 1767 1770 1772 I77S 1778 1781 1784 1787 151 .1790 1793 1796 1798 1801 1804 1807 1810 1813 1816 152 .1818 1821 1824 1827 1830 1833 1836 1838 1841 1844 153 •1847 1850 1853 1855 1858 1861 1864 1867 1870 1872 154 .1875 1878 188 1 1884 1886 1889 1892 1895 1898 1901 155 .1903 1906 1909 1912 1915 1917 1920 1923 1926 1928 156 •1931 1934 1937 1940 1942 1945 1948 1951 1953 1956 157 .1959 1962 1965 1967 1970 1973 1976 1978 1981 1984 158 .1987 1989 1992 1995 1998 2000 2003 2006 2009 201 1 159 .2014 2017 2019 2022 2025 2028 2030 2033 2036 2038 3 160 .2041 2044 2047 2049 2052 2055 2057 2060 2063 2066 An Al 161 .2068 2071 2074 2076 2079 2082 20S4 2087 2090 2092 3 - 3 I 162 .2095 2098 2101 2103 2106 2109 2111 21 14 2117 2119 6-7 2 163 .2122 2125 2127 2130 2133 2135 2138 2140 2143 2146 10 1 164 .2148 2151 2154 2156 2159 2162 2164 2167 2170 2172 ±u j 165 •2175 2177 2180 2183 2185 2188 2191 2193 2196 2198 166 .2201 2204 2206 2209 2212 2214 2217 2219 2222 2225 167 .2227 2230 2232 2235 2238 2240 2243 2245 2248 2251 168 .2253 2256 2258 2261 2263 2266 2269 2271 2274 2276 169 .2279 2281 2284 2287 2289 2292 2294 2297 2299 2302 170 .2304 2307 2310 2312 231S 2317 2320 2322 2325 2327 171 •2330 2333 2335 2338 2340 2343 2345 2348 2350 2353 172 •2355 2358 2360 2363 2365 2368 2370 2373 2375 2378 173 ■ 2380 2383 2385 2388 2390 2393 2395 2398 2400 2403 174 .2405 2408 2410 2413 2415 2418 2420 2423 2425 2428 2 175 .2430 2433 2435 2438 2440 2443 2445 2448 2450 2453 An Al 176 .2455 2458 2460 2463 2465 2467 2470 2472 2475 2477 5-0 x 177 .2480 2482 2485 2487 2490 2492 2494 2497 2499 2502 10 178 .2504 2507 2509 2512 2514 2516 2519 2521 2524 2526 179 .2529 2531 2533 2536 2538 2541 2543 2S45 2548 2550 180 •2553 2555 2558 2560 2562 2565 2567 2570 2572 2574 181 •2577 2579 2582 2584 2586 2589 2591 2594 2596 2598 182 .2601 2603 2605 2608 2610 2613 2615 2617 2620 2622 183 .2625 2627 2629 2632 2634 2636 2639 2641 2643 2646 184 .2648 2651 2653 2655 2658 2660 2662 2665 2667 2669 185 .2672 2674 2676 2679 2681 2683 2686 2688 2690 2693 186 .2695 2697 2700 2702 2704 2707 2709 2711 2714 2716 187 .2718 2721 2723 2725 2728 2730 2732 2735 2737 2739 188 .2742 2744 2746 2749 2751 2753 2755 27S8 2760 2762 T 189 190 .2765 .2788 2767 2790 2769 2792 2772 2794 2774 2797 2776 2799 2778 2801 2781 2804 2783 2806 2785 2808 An Al 10 191 .2810 2813 2815 2817 2819 2822 2824 2826 2828 2831 I 192 •2833 2835 2838 2840 2842 2844 2847 2849 2851 2853 193 .2856 2858 2860 2862 2865 2867 2869 2871 2874 2876 194 .2878 2880 2882 2885 2887 2889 2891 2894 2896 2898 195 .2900 2903 2905 2907 2909 291 1 2914 2916 2918 2920 196 .2923 2925 2927 2929 2931 2934 2936 2938 2940 2942 197 .2945 2947 2949 2951 2953 2956 2958 2960 2962 2964 198 .2967 2969 2971 2973 2975 2978 2980 2982 2984 2986 199 .2989 2991 2993 2995 2997 2999 3002 3004 3006 3008 200 .3010 3012 3015 3017 3019 3021 3023 3025 3028 3030 13 TABLE FOR THE CALCULATION OF CHEMICAL FORMULA. Taking x and y as symbols for integers less than 31, the table gives directly the decimal values F of all ratios x :y up to x : y = 0.5. For the approximate reduction of a given value F to an integer-ratio, if F < 0.5, find (if possible) that tabular F which does not differ from F by more than you can (or must) toleiate. If F > 0.5, but less than 1, operate upon 1 - F , and subtract the corresponding x' : y' from / ; y'. If F = a + f where a = 1, 2, 3 .... 5 find the x" :y" corresponding to f and say : a + f = a + (x 1 ' : y") = (ay" + x") :y". Or else operate upon the reciprocal l/F . A table going up to y = 60 is given on pp. 48 et seqq. F. x :y. F. x :y. F. x : y. F. x : y. •°3333 1 : 30 •1333 2 : 15 .2609 6 : 23 .3810 8 : 21 .03448 1 : 29 .1364 3 = 22 .2632 5 : 19 .3846 5 13 •03571 1 : 28 •1379 4 : 29 .2667 4 : 15 .3889 7 18 .03704 1 : 27 .1429 1 :7 .2692 7 : 26 •3913 9 23 .03846 1 : 26 .1481 4 = 27 .2727 3 = 11 •3929 11 28 .04000 1 =25 .1500 3 = 20 •2759 8 : 29 .4000 2 5 .04167 1 : 24 •!539 2 : 13 .2778 5:18 .4074 11 27 .04348 1 : 23 •1579 3 = 19 .2800 7 = 25 .4091 9 22 .04546 1 : 22 .1600 4 : 25 .2857 2 : 7 .4118 7 17 .04762 1 : 21 .1667 1 :6 .2917 7 : 24 .4138 12 29 .05000 1 : 20 .1724 5 = 29 .2941 5 : 17 .4166 5 12 .05263' 1 : 19 •1739 4 = 23 .2963 8 : 27 .4211 8 19 •o55S6 1 : 18 •1765 3 = 17 .3000 3 = 10 •4231 11 26 .05882 1 : 17 .1786 5 :28 •3043 7 = 23 .4285 3 7 .06250 1 : 16 .1818 2 : 11 •3077 4 = 13 •4333 13 3° .06667 1 : 15 .1852 5 : 27 •3103 9 = 29 •4348 10 23 .06897 2 : 29 •1875 3 : 16 •3125 5:16 •4375 7 16 •07143 1 : 14 .1905 4 : 21 •3158 6 : 19 .4400 11 25 .07407 2 : 27 .1923 5 : 26 .3182 7 : 22 •4444 4 9 .07692 1 : 13 .2000 1 =5 .3200 8:25 •4483 13 29 .08000 2 : 25 .2069 6 : 29 .3214 9 : 28 .4500 9 20 •08333 1 : 12 .2083 5 : 2 4 •3333 1:3 •4545 5 11 .08696 2 : 23 .2105 4 = 19 •3448 10 : 29 •4583 11 24 .09091 1 : 11 .2143 3 = 14 .3462 9 : 26 .4615 6 13 .09524 2 : 21 .2174 5 = 23 ■3478 8:23 •4643 13 28 .1000 1 : 10 .2222 2 : 9 •35oo 7 : 20 .4667 7 15 .1034 3 •• 29 ■ 2273 5 : 22 •3529 6 : 17 .4706 8 17 •i°53 2 : 19 .2308 3 : 13 •357i 5 = 14 •4737 9 19 .1071 3 : 28 •2333 7 : 30 .3600 9 = 25 .4762 10 21 .1111 1 : 9 ■2353 4 = 17 • 3 6 36 4 : 11 •4783 11 23 .1154 3 : 26 .2381 5 : 21 .3667 11 : 30 .4800 12 25 .1176 2 : 17 .2400 6 : 25 .3684 7 = 19 .4815 13 27 .1200 3 : 25 .2414 7 : 29 •3704 10 : 27 .4828 '4 29 .1250 1 : 8 .2500 1 : 4 •375° 3 = 8 .5000 1 2 .1304 3 = 2 3 •2593J 7 : 27 • 3793 11 : 29 14 RECIPROCALS. i 1 1 : "• n P.P. n. n P.P. n. n P.P. 1000 1 0000 1100 9091 1200 8333 1002 9980 1102 9074 1202 8319 1004 9960 1104 9058 1204 8306 An A- 1006 9940 I 1106 9042 1206 S292 n 1008 1010 9921 9901 An A- n 1108 1110 9025 9009 An ■2 A- n 2 1208 1210 8278 8264 1 •3 •4 I 2 3 •2 2 1012 9881 •4 4 1112 8993 •5 4 1212 8251 •6 4 1014 9862 •6 6 1114 8977 7 6 1214 8237 7 5 1016 9843 ■8 8 1116 8961 10 8 1216 8224 •9 6 1018 9823 10 10 1118 8945 1-2 10 1218 8210 10 7 1-2 12 1-5 12 1-2 8 1020 9804 1-4 14 1120 8929 17 14 1220 8197 1-3 9 1022 978s 17 16 1122 8913 2-0 16 1222 8183 1-5 10 1024 9766 1-9 18 1124 8897 2-2 18 1224 8170 1-6 11 1026 9747 21 20 1126 8881 2-5 20 1226 8157 1-8 12 1028 9728 1128 8865 1228 8143 1-9 2-1 13 1030 9709 1130 8850 1230 8130 H 1032 1034 9690 9671 1132 1134 8834 8818 1232 1234 8117 8104 1036 9653 1136 8803 1236 8091 1038 9634 1138 8787 1238 8078 1 An A- 1040 96iS An 1 A- 1140 8772 An 1 A~ 1240 8065 ? 1042 9597 9579 n 1142 8757 8741 n 1242 8052 8039 •2 I 1044 •2 2 1144 •3 2 1244 •3 2 1046 9560 •4 4 1146 8726 ■5 4 1246 8026 ■5 3 1048 9542 7 6 1148 8711 •8 6 1248 8013 •6 4 ■9 8 1-1 8 •8 5 1050 9524 11 10 1150 8696 1-3 10 1250 8000 •9 6 1052 9506 1-3 12 1152 868 1 1-6 !2 1252 7987 1-1 7 1054 9488 1-5 14 1154 8666 1-9 H 1254 7974 l - 3 8 1056 9470 1-8 16 1156 8651 2-1 16 1256 7962 1"4 9 1058 9452 2-0 18 1158 8636 2-4 18 1258 7949 1*6 17 10 2-2 20 27 20 11 1060 9434 1160 8621 1260 7937 1'9 12 1062 9416 1162 8606 1262 7924 2 13 1064 9398 1164 8591 1264 791 1 1066 1068 938i 9363 1166 1168 8576 8562 1266 1268 7899 7886 1070 9346 1170 8547 1270 7874 1 1072 9328 1 1172 8532 I 1272 7862 An A- n 1074 93i 1 An A- 1174 8518 An A- 1274 7849 1076 9294 9276 n 1176 8503 8489 n 1276 1278 7837 7825 ■2 I 1078 •2 2 1178 •3 2 •3 •5 2 •5 4 •6 4 3 1080 9259 7 6 1180 8475 ■8 6 1280 78i3 7 4 1082 9242 •9 8 1182 8460 11 8 1282 7800 •8 5 1084 9225 1-2 10 1184 8446 1-4 10 1284 7788 ro 6 1086 9208 1-4 12 1186 8432 17 12 1286 7776 1-2 7 1088 9191 1-6 14 1188 8418 2-0 14 1288 7764 1'3 8 1-9 16 2-2 16 1'6 9 1090 9174 2-1 18 1190 8403 2-5 18 1290 7752 1-6 10 1092 9158 2-4 20 1192 8389 2-8 20 1292 7740 1-8 11 1094 9141 1194 8375 1294 7728 2-0 12 1096 9124 1196 8361 1296 7716 2-1 '3 1098 9107 1198 8347 1298 7704 1100 9091 1200 8333 1300 7692 RECIPROCALS. iS n. i P.P. n. 1 P.P. n. 1 p.p. ; n n n 1300 7692 1400 7H3 1514 6605 An A- 1302 7680 1402 7133 1517 6592 - n 1304 7669 1 1404 7123 1 1521 6575 •2 I 1306 76S7 An A- 1406 7112 An A- 1524 6562 •5 2 1308 7645 - 1408 7102 - " 1528 6545 ■7 3 •2 I •2 I 1-0 4 1310 7634 •3 2 1410 7092 •4 2 1531 6532 1-2 5 1312 7622 •5 3 1412 7082 •6 3 1535 651S 1-4 6 1314 7610 •7 4 1414 7072 ■8 4 1538 6502 1-7 7 1316 7S99 ■9 5 1416 7062 10 5 1542 64S5 1-9 S 1318 7S»7 10 6 1418 7052 1-2 6 1545 6472 21 9 1-2 7 1-4 7 2 '4 10 1320 7576 1-4 8 1420 7042 1-6 8 1549 6456 2-6 11 1322 7564 1-6 9 1422 7032 1-8 9 1552 6443 29 12 1324 7553 1-7 10 1424 7022 2-0 10 1556 6427 3-1 13 1326 7541 1-9 11 1426 7013 2-2 11 1560 6410 3 3 14 1328 7530 2-1 12 1428 7003 1563 6398 3 6 15 1 3-8 16 1330 7519 1 1430 6993 1567 6382 4-1 17 1332 1334 75o8 7496 1432 1434 6983 6974 1570 1574 6369 6353 1336 7485 1436 6964 1578 6337 An A- 1338 7474 1438 6954 1581 6-?2i; n An A- v j j ■3 I 1340 7463 — 1440 6944 An A- 1585 6309 •5 2 1342 7452 •2 I 1442 6935 n 1589 6293 •8 3 1344 7440 •4 2 1444 6925 •2 I 1592 6281 1-0 4 1346 7429 ■5 3 1446 6916 ■4 2 1596 6266 1-3 5 1348 7418 ■7 4 1448 6906 •6 3 1600 6250 1-5 6 ■9 5 •8 4 1-8 7 1350 7407 11 6 1450 6897 1-1 5 1603 6238 2-1 8 1352 7396 1-3 7 1452 6887 13 6 1607 6223 2 3 9 1354 7386 1-5 8 1454 6878 1-5 7 1611 6207 2-6 10 1356 7375 1-6 9 1456 6868 1-7 8 1614 6196 2-8 11 1353 7364 1-8 10 1458 6859 1-9 9 1618 6180 3-1 12 2'0 11 21 10 3 3 13 1360 7353 2-2 12 1460 6849 1622 6165 3 6 14 1362 7342 1462 6840 1626 6150 3 9 15 1364 7331 1464 6831 1629 6i39 41 16 1366 1368 7321 7310 1466 1468 6821 6812 1 1633 1637 6124 6109 __ An A- n I 1370 7299 1470 6803 1641 6094 An A- 1372 1374 7289 7278 1472 6793 6784 •2 I 1644 6083 6068 n 1 An A- 1474 •4 2 1648 •3 I 1376 1378 7267 7257 n 1476 6775 6766 ■7 3 1652 6053 •6 2 •2 •4 I 2 1478 •9 1-1 4 5 1656 6039 •8 l'l 3 4 1380 7246 •6 3 1480 6757 1-3 6 1660 6024 1-4 5 1382 7236 •8 4 1483 6743 1-6 7 1663 6013 1-7 6 1384 7225 10 5 1486 6729 1"8 8 1667 5999 1-9 7 1386 7215 11 6 1489 6716 2-0 9 1671 5984 2-2 8 1388 7205 1-3 1-5 7 8 1493 6698 2-2 2-4 10 11 1675 5970 2 -5 2-8 9 10 1390 7194 1-7 9 1496 6684 2-7 12 1679 5956 31 11 1392 7184 1-9 10 1500 6667 2 9 13 1683 5942 3 3 12 1394 7174 2-1 11 1503 6653 3'1 14 1687 5928 3-6 13 1396 7163 1507 6636 3 - c 15 1690 5917 3-9 14 | 1398 7153 1510 6623 36 3-8 16 17 1694 5903 4 '2 15 1400 7143 1 1514 6605 4-0 18 1698 5889 i6 RECIPROCALS. n. I n P.P. u. 1 n P.P. n. 1 n P.P. 1700 5882 1905 5249 2155 4640 1702 S87S 1910 5236 I 2160 4630 1706 5862 An A- 1915 5222 An A- 2165 4619 An X A - 1710 1714 5848 - 5834 •3 n I 1920 1925 5208 - 5«95 •4 n I 2170 2175 4608 4598 n ■5 I 1718 5821 ■6 ■9 2 3 1930 5181 ■8 11 2 3 2180 4587 1-0 1-4 2 3 4 5 6 1722 5807 1-2 4 1935 5168 1-5 4 2185 4577 1-9 1726 5794 1-5 5 1940 5155 1-9 5 2190 4566 2-4 1730 5780 1-8 6 1945 5Hi 2-3 6 2195 4556 2 9 1734 5767 2-1 7 1950 5128 2-7 7 2200 4545 3 4 7 8 2-4 8 3-0 8 3 '9 1738 5754 2-7 9 1955 5"5 3-4 9 2205 4535 4-3 9 10 1742 5741 3 10 1960 5102 3-8 10 2210 4525 4-8 1746 5727 3 3 11 1965 5089 42 11 2215 4515 5-3 11 1750 57H 3'6 12 1970 5076 4-5 12 2220 4505 5-8 T7. 1754 57oi 3-9 13 1975 5063 49 13 2225 4494 1758 5688 1980 5051 2230 4484 1762 1766 5675 5663 1985 1990 5038 5025 2235 2240 4474 4464 1770 5650 1 1995 5013 2245 4454 1774 1778 5637 5624 An •3 ■6 1-0 1-3 1-6 1-9 A - n I 2000 2005 5000 4988 An I A- n 2250 2255 4444 4435 1 •4 •8 1-2 1-7 2-1 An A~ 1782 5612 2 2010 4975 I 2260 4425 1786 1791 5599 5583 3 4 2015 2020 4963 4950 2 3 2265 2270 4415 4405 •5 1-0 I 2 1795 557i 5 6 2025 4938 4 5 2275 4396 1-6 2 - l 3 4 5 1799 5559 2-3 2-6 2 9 3-2 7 8 2030 4926 2-5 2-9 33 3-7 6 2280 4386 2-6 1803 5546 2035 4914 7 8 2285 4376 3 1 6 1807 5534 9 2040 4902 2290 4367 3 6 7 1811 5522 10 2045 4890 9 2295 4357 4-2 8 1816 5507 3 "6 3 9 11 12 2050 4878 4"1 4-5 10 11 2300 4348 4-7 5-2 9 10 1820 1824 5495 5482 4-2 4-5 4-9 13 14 2055 2060 4866 4854 5 5-4 12 13 2305 2310 4338 4329 1828 5470 l 5 2065 4843 2315 4320 1832 1837 5459 5444 2070 2075 4831 4819 2320 2325 43io 4301 1841 5432 1 2080 4808 2330 4292 1845 542o An A~ 2085 4796 1 2335 4283 1849 5408 •3 •7 1-0 2090 4785 An A- n 2340 4274 1 1854 1858 5394 5382 I 2 3 2095 2100 4773 4762 •4 ■9 I 2 2345 2350 4264 4255 An A" D ■6 I 1862 5371 1-4 1-7 4 2105 4751 1-3 3 2355 4246 1-1 2 1866 5359 5 6 2110 4739 1"8 4 2360 4237 l - 7 3 1871 5345 2'1 2-4 2115 4728 2 "2 2-7 5 6 2365 4228 2-2 4 1875 5333 7 8 9 2120 4717 2370 4219 2'S 5 1879 5322 2 '8 3-1 2125 47o6 3"1 3-6 7 8 2375 421 1 3-4 3-8 6 7 1884 53°8 3'5 10 2130 4695 4-0 9 2380 4202 4-5 8 1888 5297 3 - 8 11 2135 4684 4'£ 10 2385 4193 5-C 9 1892 5285 4 '2 4-5 4-9 12 2140 4673 4"J 5-4 11 2390 4184 5"{ 10 1897 5271 13 2145 4662 12 2395 4175 1901 5260 14 2150 4651 2400 4167 1905 5249 2155 4640 1 2405 4118 RECIPROCALS. i7 1 P.P. 1 — — 1 n. — n. — P.P. n. — P.P. n n n 2405 4158 2655 3766 2905 3442 2410 4149 2660 3759 2910 3436 2415 4141 2665 3752 2915 3431 2420 4132 An I A- a 2670 3745 2920 3425 2425 4124 2675 3738 An 1 A~ 2925 3419 , •6 1'2 An A- 2430 4"5 I 2 3 4 5 6 7 8 9 2680 3731 n 2930 3413 n 2435 4107 1-8 2-4 3 36 4-2 4-8 5-4 6'0 2685 3724 •7 I 2935 3407 •9 I 2440 4098 2690 3717 1-5 2 2940 3401 1-7 2 2445 4090 2695 37" 2-2 3 2945 3396 2-6 3 2450 4082 2700 3704 2-9 4 2950 339o 3-5 4 2455 4073 2705 3697 3 6 4-4 5 6 2955 3384 4-3 5-2 5 6 2460 4065 2710 3690 51 7 2960 3378 61 7 2465 4057 2715 36S3 2965 3373 2470 4049 2720 3676 2970 3367 2475 4040 2725 3670 2975 336i 2480 4032 2730 3663 2980 3356 2485 2490 4024 4016 2735 2740 3656 3650 2985 2990 335° 3344 2495 4008 2745 3643 2995 3339 2500 4000 2750 3636 3000 3333 2505 3992 2755 3630 3005 3328 2510 3984 An A- 2760 3623 An I 3010 3322 , 2515 3976 11 2765 3617 A - a 3015 33i7 An A- 2520 3968 •6 1 2770 3610 •8 1'5 I 2 3020 33" n 2525 3960 1-3 2 2775 3604 3025 3306 ■9 I 2530 3953 1-9 2'6 3 4 2780 3597 2-3 31 3 9 4-6 5-4 3 4 5 6 7 3030 33oo 1-8 2-8 2 3 2535 3945 3-2 5 2785 3591 3035 3295 3-7 4 2540 3937 3-8 6 2790 3584 3040 3289 4-6 5 2545 3929 4-5 7 2795 3578 3045 3284 5 5 6 2550 3922 51 8 2800 357i 3050 3279 2555 3914 2805 3565 3055 3273 2560 3906 2810 3559 3060 3268 2565 3899 2815 3552 3065 3263 2570 2575 3891 3883 2820 2825 3546 3540 3070 3075 3257 3252 2580 3876 2830 3534 3080 3247 2585 3868 2835 3527 3085 3241 An A- 2590 3861 2840 352i 3090 3236 n 2595 3854 1 2845 3515 , 3095 3231 10 I 2600 3846 An •7 1-4 2-1 2-7 3 4 4-1 4-8 5-5 A~ n 2850 35°9 An A- n 3100 3226 1-9 2-9 2 3 2605 3839 I 2 3 2855 35°3 •8 I 3105 3221 3 9 4 2610 3831 2860 3497 1-6 2 3110 3215 4-9 5 2615 3824 2865 3490 2-5 3 3115 3210 5-8 6 2620 3817 4 5 6 2870 34S4 33 4 3120 3205 6-8 7 2625 3810 2875 3478 4-1 5 3125 3200 7-8 8 4-9 6 8-8 9 2630 3802 7 8 2880 3472 5-7 7 3130 3195 9-7 10 2635 3795 2885 3466 3135 3190 10-7 11 2640 3788 2890 3460 3140 3185 11-7 12 2645 378i 2895 3454 3150 3175 2650 3774 2900 3448 3160 3165 2655 3766 1 2905 3442 3162-3 3162.3 i8 FIVE-PLACE LOGARITHMS. N. Log o 1 2 3 4 S 6 7 8 9 A P.P. 100 oo ooo °43 087 130 173 217 260 3°3 346 389 43 101 432 475 518 561 604 647 689 732 775 817 43 44 102 86o 903 945 988 ♦30 ♦72 "5 iS7 199 242 42 103 oi 284 326 368 410 452 494 536 578 620 662 4i N L 104 703 745 787 828 870 912 953. 995 ♦36 ♦78 4i 0— — 105 02 119 160 202 243 284 325 366 407 449 490 41 1 — 106 53i 572 612 653 694 735 776 816 857 898 40 — 5 107 938 979 ♦ 19 ♦60 100 141 181 222 262 302 40 108 °3 342 3S3 423 463 5o3 543 583 623 663 703 40 2 — —10 109 743 782 822 862 902 941 981 ♦21 ♦60 100 39 3 110 04 139 179 218 258 297 336 376 415 454 493 39 Z"' 5 111 532 57i 610 650 689 727 766 805 844 883 39 4— 112 922 961 999 ♦38 ♦77 "5 iS4 192 231 269 39 - 113 05 308 346 385 423 461 500 538 576 614 652 38 ~ 114 690 729 767 805 843 881 918 956 994 ♦32 38 5 — 115 06 070 108 145 183 221 258 296 333 37i 408 38 6- r s5 116 446 483 521 558 595 633 670 707 744 781 38 _ 117 819 856 893 93° 967 ♦04 ♦41 ♦78 X J S ISI 37 / — r*° 118 07 188 225 262 298 335 372 408 445 482 5i8 37 - 119 555 591 628 664 700 737 773 809 846 882 36 8 — =-tf 120 918 954 990 ♦27 ♦63 ♦99 135 171 207 243 36 - - 121 08 279 314 35° 386 422 458 493 5 2 9 565 600 36 9 — r4° i 122 636 672 707 743 778 814 849 884 920 9SS 36 123 991 ♦26 ♦61 ♦96 132 167 202 237 272 307 35 10 — 1 = 44 124 09 342 377 412 447 482 517 552 587 621 656 35 125 691 726 760 795 830 864 899 934 968 ♦03 34 126 10 037 072 106 140 175 209 243 278 312 346 34 36 127 380 4i5 449 483 Si7 SSi 585 619 653 687 34 128 721 755 789 823 857 890 924 958 992 ♦25 34 N L 129 11 059 093 126 160 193 227 261 294 327 361 33 — — 130 394 428 461 494 528 56i 594 628 661 694 33 2 131 727 760 793 826 860 893 926 959 992 ♦24 33 — s 132 12 057 090 123 156 189 222 254 287 320 352 33 133 385 418 450 483 S i6 548 58i 613 646 678 32 2 — 134 710 743 775 808 840 872 90s 937 969 ♦01 32 3 —10 135 .J3 033 066 098 130 162 194 226 258 290 322 32 136 354 386 418 450 481 513 545 577 609 640 32 4 — - ■ 137 672 704 735 767 799 •830 862 893 92S 956 32 —15 138 988 ♦ 19 ♦Si ♦82 114 145 176 208 239 270 3i 139 ,14 3°i 333 364 395 426 457 489 520 55i 582 31 5 — so 140 613 644 675 706 737 768 799 829 860 891 3i 6- 141 922 953 983 ♦ 14 ♦45 ♦76 106 137 168 198 3i 142 15 229 259 290 320 351 381 412 442 473 503 3i 7 — -45 143 534 564 594 625 655 685 715 746 776 806 3° -■ 144 836 866 897 927 957 987 ♦17 ♦47 ♦77 107 3° 8 — -. 145 16 137 167 197 227 256 286 3i6 346 376 406 29 - -30 146 435 465 495 524 554 584 613 643 673 702 3° 9 — Z 147 732 761 791 820 850 879 909 938 967 997 29 - z i5 148 17 026 056 085 114 143 173 202 231 260 289 30 10— 149 3i9 348 377 406 435 464 493 522 SSi 580 29 N. Log 1 2 3 4 5 6 7 8 9 A P.P. PARTES PROPORTIONALES. l 9 43 5 — 6 — 7- 8- 9- 10- — 25 -35 42 i- 2- 3— — 20 6-M5 -r -35 s— P -40 41 1 0- 1- 2- s- 4—; 5 6 7— -30 -35 -40 40 .-« 6-= 39 5 4— s 6 -25 -30 -40 10 -25 -30 38 2 3 4 5 M -35 9— -23 -25 —30 -35 37 -25 -30 ^35 100 •000 150 •176 35 34 33 32 31 30 29 N L N L N L N L N L N L N L — u — V ,0 — — u — U — ■ -U — — — — u — — — - ~ - ,— 1 — 1 — — 1 — 1 — 1 — 1— 1 1 — - —5 —5 —5 - -5 - -5 -5 -5 2 — 2 — - 2 — - 2 — 2 — 2 — 2 — - _ — - - 3 — — 10 3 — — 10 3 — —19 3 — — 10 3 — —10 3 — - — lfl 3— — 13 4 — 4 — — 4— - 4— 4 — 4 — 4— — —i5 —15 -15 __ ~ 5 — 5— 5 — 5 — —15 5— —15 5 — —15 £ — -15 6- — 20 6- — 20 6- — 20 6- —2d 6- 6- 6- — 20 — 20 7 — —25 7— ~ 25 7— r 25 7 — 1 7— 7— 7— — 20 8 — 8 — a— 8 — r 25 8 — -~-5 8 — 8 — -25 - —30 ~ 3 ° ~ — ~ — 25 9 — S — 9 — —3° 9— 9 — 9 — 9 — — 30 — ~ 10- - 35 10 — 10— 10 — 10 — —i" 10 — — ^0 10 — 1300 •118 L400 FIVE-PLACE LOGARITHMS. N. Log o 1 2 3 4 5 6 7 8 9 A P.P. 150 17 609 638 667 696 725 754 782 811 840 869 29 151 898 926 955 984 ♦ 13 ♦41 ♦70 ♦99 127 156 28 152 18 184 213 241 270 298 327 355 384 412 441 28 153 469 498 526 554 583 611 639 667 696 724 28 154 752 780 80S 837 865 893 921 949 977 ♦OS 28 155 19 033 061 0S9 117 H5 173 201 229 257 285 27 156 312 340 368 396 424 45i 479 507 535 562 28 157 590 618 645 673 700 728 756 783 8n 838 28 158 866 893 921 948 976 ♦03 ♦30 ♦58 ♦85 112 28 159 20 140 167 194 222 249 276 3°3 33° 358 385 27 160 412 439 466 493 520 548 575 602 629 656 27 161 683 710 737 763 79o 817 844 871 898 925 27 162 9S2 978 ♦05 ♦32 ♦59 ♦85 112 139 165 192 27 163 21 219 245 272 299 325 352 378 405 43i 458 26 29 164 484 5" 537 564 590 617 643 669 696 722 26 An Al 165 748 775 801 827 854 880 906 932 958 985 26 •3 •7 I 2 166 22 on 037 063 089 »5 141 167 194 220 246 26 l'O 3 167 272 298 324 35o 376 401 427 453 479 505 26 1"4 4 168 53i 557 583 608 634 660 686 712 737 763 26 1-7 5 169 789 814 840 866 891 917 943 968 994 ♦ 19 26 2-1 2'4 6 7 170 23 045 070 096 121 147 172 198 223 249 274 26 2-8 31 8 9 171 300 325 35o 376 401 426 452 477 502 528 25 172 553 578 603 629 654 679 704 729 754 779 26 3-4 10 173 805 830 855 880 90S 930 955 980 ♦05 ♦30 25 3-8 4'1 11 174 24 055 080 i°5 130 155 180 204 229 254 279 25 4-5 4-8 13 175 304 329 353 378 403 428 452 477 502 527 24 x 4 176 55i 576 601 625 650 674 699 724 748 773 24 6 '2 5*5 15 16 177 797 822 846 871 895 920 944 969 993 ♦18 24 5-9 x 7 178 25 042 066 091 "5 139 164 188 212 237 261 24 6-2 18 179 285 310 334 358 382 406 431 455 479 503 24 6-6 *9 180 527 55i 575 600 624 648 672 696 720 744 24 6-9 7-2 20 21 181 768 792 816 840 864 888 912 935 959 983 24 7'6 7-9 8'3 22 182 26 007 031 055 079 102 126 150 174 198 221 24 2 3 183 245 269 293 316 34o 364 387 411 435 458 24 2 4 184 482 505 529 553 576 600 623 647 670 694 23 8-6 9-0 25 26 185 717 74i 764 788 811 834 858 881 905 928 23 9-3 9-7 27 28 186 95i 975 998 ♦21 ♦45 ♦68 ♦91 114 138 l6l 23 10 29 187 27 184 207 231 254 277 300 323 346 37o 393 23 188 416 439 462 485 508 53i 554 577 600 623 23 189 646 669 692 7i5 738 761 784 807 830 852 23 190 875 898 921 944 967 989 ♦ 12 ♦35 ♦58 ♦81 22 191 28 103 126 149 171 194 217 240 262 285 3°7 23 192 330 353 375 398 421 443 466 488 5" . 533 23 193 556 578 601 623 646 668 691 7i3 735 758 22 194 780 803 825 847 870 892 914 937 959 981 22 195 29 003 026 048 070 092 "5 137 159 181 203 23 196 226 248 270 292 3H 336 358 380 403 425 22 197 447 469 491 5'3 535 557 579 601 623 645 22 198 667 688 710 732 754 776 798 820 842 863 22 199 N. 885 907 929 951 973 994 ♦ 16 ♦38 ♦60 ♦81 22 Log 1 2 3 4 5 6 7 8 9 A P.P. PARTES PROPORTIONALES. 28 An Al ■4 i •7 2 1-1 3 1-4 4 1-8 S 2'1 6 2-5 7 2D 8 3-2 9 3-6 IO 3-» II 4-3 12 4-6 13 5'0 14 5-4 is 57 16 6-1 17 fi-4 18 6-8 19 7-1 20 7-5 21 7-9 22 8-2 21 8't) 24 8-9 2=; !C3 26 9-6 27 10 28 27 An Al •4 1 '7 2 l'l 3 15 4 1-9 S 2-2 6 2-6 7 SMI 8 33 9 3'7 10 4-1 11 4-4 12 4'8 11 6-2 14 B'6 11 5'9 16 6'3 17 fi-7 t8 7-0 19 7'4 20 7-8 21 8-1 22 8-5 = 3 8'9 24 93 2S 9 Mi 26 10 27 26 An Al 1-2 1-5 1-9 2-3 2-7 31 3-5 4'2 4'6 5-0 5 '4 6'8 6-2 6'6 6-9 7-3 77 8-1 8'5 9-6 10 25 An Al 1-2 1-6 2-0 2-4 2-8 3-2 3'6 40 4-4 4-8 5-2 5-6 6-0 6'4 6-8 7-2 7-6 8-0 8-4 8-8 9-2 9'6 24 23 22 21 An Al An Al An Al An Al •4 1 •4 1 ■6 1 ■5 1 •8 2 ■9 2 ■9 2 1-0 2 1-3 3 1-3 3 14 3 1-4 3 1-7 4 1-7 4 1-8 4 1-9 4 2-1 5 2-2 5 2-3 5 2-4 S 2-5 6 2'6 6 2-7 6 2-9 6 2'9 7 3-0 7 3'2 7 3-3 7 3-3 8 3-5 8 3 MS 8 3-8 8 3-8 9 3-9 9 4-1 9 4-3 9 4-2 10 4-3 10 4'5 10 4-8 10 4-6 11 4'8 11 5 Ml 11 5'2 11 5-0 12 6-2 12 6-5 12 6-7 12 6-4 x 3 67 13 6'9 13 62 13 5-8 14 6-1 14 6-4 14 6'7 14 6'3 15 6-5 15 6-8 IS 7-1 15 6-7 16 7'0 16 7-3 16 7'6 16 7-1 17 7-4 17 7-7 ! 7 8-1 17 7'5 18 7'8 18 8-2 18 8-6 18 7-9 19 8-3 19 8-6 19 9MJ 19 8-3 20 8'7 20 91 20 9-5 20 8-S 21 9-1 21 9-5 21 10 21 9 '2 22 9-6 22 10 22 9-6 23 10 23 10 24 FIVE-PLACE LOGARITHMS. N. Log o 1 2 3 4 5 6 7 8 9 A P.P. 200 3° 103 125 146 168 190 211 233 255 276 298 22 201 320 34i 363 384 406 428 449 47i 492 5H 21 202 535 557 578 600 621 643 664 685 707 728 22 203 75° 771 792 814 835 856 878 899 920 942 21 204 963 984 ♦06 ♦27 ♦48 ♦69 ♦91 112 133 154 21 22 21 205 3i 175 387 197 408 218 239 260 281 302 323 345 366 21 An Al An Al 206 429 450 47i 492 513 534 555 576 21 ■5 •9 •6 1-0 207 597 618 639 660 681 702 723 744 765 785 21 2 2 208 806 827 848 869 890 911 931 952 973 994 21 1-4 3 1-4 3 209 32 015 °35 056 077 098 118 139 160 181 201 21 1-8 4 1-9 4 210 222 243 263 284 3°5 325 346 366 387 408 20 2-3 27 5 6 2-4 2-9 5 6 211 428 449 469 490 510 53i 552 572 593 613 21 3-2 7 3-3 7 212 634 654 675 695 7i5 736 756 777 797 818 20 3-6 4-1 8 3-S 4-3 8 213 838 858 879 899 919 940 960 9S0 ♦01 ♦21 20 9 9 214 33 041 062 082 102 122 143 163 183 203 224 20 4-5 6-0 TO II 4-8 5-2 10 11 215 244 264 284 304 325 345 365 385 405 425 20 6-5 6 "9 12 6-7 6"2 12 216 445 465 486 506 526 546 566 586 606 626 20 6-4 I 3 14 67 *3 T 4 217 646- 666 686 706 726 746 766 786 806 826 20 218 846 866 885 905 925 945 965 985 ♦05 ♦25 19 6-8 7'3 15 16 7-1 7'6 IS 16 219 34 044 064 084 104 124 143 163 183 203 223 19 7-7 8-2 17 18 8-1 8-6 17 18 220 242 262 282 301 321 34i 36l 3S0 400 420 19 8-6 19 9-0 19 221 439 459 479 498 5i8 537 557 577 596 616 19 9'1 20 9-5 222 635 655 674 694 713 733 753 772 792 811 19 9-5 21 10 21 223 830 850 869 889 908 928 947 967 986 ♦05 20 10 22 224 35 025 044 064 0S3 102 122 141 160 1S0 199 19 225 218 238 257 276 295 315 334 353 372 392 19 226 411 430 449 468 488 507 526 545 564 583 20 227 603 622 641 660 679 698 717 736 755 774 19 228 793 813 832 851 870 889 908 927 946 965 19 229 984 ♦03 ♦21 ♦40 ♦59 ♦78 ♦97 116 I3S 154 19 20 19 An Al An Al 230 36 173 192 211 229 248 267 286 305 324 342 19 231 361 380 399 418 436 455 474 493 5" 53o 19 •5 1 •5 I 232 549 568 586 605 624 642 661 680 698 717 19 1"0 1"5 2 1*1 1*6 2 233 736 754 773 791 810 829 847 866 884 903 19 2-0 3 4 21 3 4 234 922 940 959 977 996 ♦14 ♦33 ♦51 ♦70 ♦88 19 2-6 5 2-6 5 235 37 i°7 125 144 162 181 199 218 236 254 273 18 3-0 3-5 6 7 3-2 3-7 6 7 236 291 310 328 346 365 383 401 420 438 457 18 4-0 8 4-2 8 237 475 493 5» 530 548 566 585 603 621 639 19 4-5 9 4-7 9 238 658 676 694 712 73i 749 767 785 S03 822 18 5*0 5 '3 239 840 858 876 894 912 93i 949 967 985 ♦03 18 6-6 6-0 11 12 6-8 6 3 11 12 240 38 021 039 057 075 093 112 130 148 166 184 18 6-6 7-0 13 6-8 7-4 '3 241 202 220 238 256 274 292 310 328 346 364 18 *4 "4 242 382 399 417 435 453 47i 489 507 525 543 18 7-5 15 7-9 15 243 56i 578 596 614 632 650 668 686 703 721 18 8-0 16 8-4 16 244 739 757 775 792 810 828 846 863 881 899 18 8*5 9-0 9 6 17 18 "9 8'9 9-5 10 17 18 '9 245 917 934 952 970 987 ♦05 ♦23 ♦41 ♦58 ♦76 18 246 39 °94 in 129 146 164 182 199 217 235 252 18 10 20 247 270 287 305 322 340 358 375 393 410 428 17 248 445 463 480 498 515 533 550 568 585 602 18 249 620 637 655 672 690 707 724 742 759 777 17 N. Log 1 2 3 4 5 6 7 8 9 A P.P. FIVE-PLACE LOGARITHMS. 23 P.P. N. Log 1 2 3 4 5 6 7 8 9 A « 17 250 39 794 811 829 846 863 881 898 915 933 95° 17 An Al An Al 251 967 985 ♦02 ♦19 ♦37 ♦54 ♦71 . ^88 106 123 17 252 40 140 157 175 192 209 226 243 261 278 295 17 •6 11 1-7 I ■6 1-2 1-8 I 253 312 329 346 364 381 398 415 432 449 466 17 3 3 254 483 500 5i8 535 552 569 586 603 620 637 17 2-2 4 2-4 4 255 654 671 688 705 722 739 756 773 790 807 17 2*8 3 3 5 6 2 '9 3 '5 5 6 256 824 841 S58 875 892 909 926 943 960 976 17 3-9 7 4-1 7 257 993 ♦ 10 ♦27 ♦44 ♦61 ♦78 ♦95 "I 128 I4S 17 4-4 8 4-7 8 258 41 162 179 196 212 229 246 263 280 296 313 17 6'0 9 6-3 9 259 33° 347 363 380 397 414 430 447 464 481 16 5-6 61 10 11 5-9 6-5 10 ir 260 497 5H 53i 547 564 58i 597 614 631 647 17 6-7 12 7-1 12 261 664 681 697 714 73i 747 764 780 797 814 16 7 "2 7-8 '3 14 7-6i 3 8-2i4 1 262 263 830 996 847 ♦ 12 863 ♦29 880 ♦45 896 ♦62 913 ♦78 929 946 ♦95 I" 963 127 979 144 17 16 8-3 15 s-s'is 264 42 1 60 177 193 210 226 243 259 275 292 308 17 8'9 16 9-4i6 9-4 10 17 18 10 17 265 325 34i 357 374 390 406 423 439 455 472 16 266 488 504 521 537 553 570 586 602 619 635 16 267 6Si 667 684 700 716 732 749 765 78i 797 16 268 813 830 846 862 878 894 911 927 943 959 16 16 15 269 975 991 ♦oS ♦24 ♦40 ♦56 ♦72 ^88 104 120 16 in'A An Al | 270 43 136 152 169 185 201 217 233 249 265 281 16 . •6 1-2 1'9 1 2 3 4 •7 1-3 2-0 1 2 3 4 271 272 297 457 313 473 329 489 345 505 361 521 377 537 393 409 553 569 425 584 441 600 16 16 2-5 2-7 273 616 632 648 664 680 696 712 727 743 759 16 3-1 5 3-3 5 274 775 791 807 823 838 854 870 886 902 917 16 3-8 4'4 6 7 4-0 4-7 6 7 275 933 949 965 981 996 ♦ 12 ♦28 *44 ♦59 ♦75 16 5-0 8 5-3 8 276 44 091 107 122 138 154 170 185 201 217 232 16 5-6 9 6-0 9 277 248 264 279 295 3" 326 342 358 373 389 15 6-2 10 67 10 278 404 420 436 45i 467 483 498 514 529 545 15 6-9 11 7-3 11 279 560 576 592 607 623 638 654 669 685 700 16 7-5 12 8-0 12 8-1 8 '7 13 87 9-3 J 3 280 716 731 747 762 778 793 809 824 840 855 16 T 4 H 281 871 S86 902 917 932 948 963 979 994 ♦ 10 15 9'4 15 10 15 282 45 025 040 056 071 086 102 »7 133 148 163 16 10 16 1 283 179 194 209 225 240 255 271 286 301 3i7 15 284 332 347 362 378 393 408 423 439 454 469 15 285 484 500 515 53° 545 561 576 59i 606 621 16 14 Al 286 637 652 667 682 697 712 728 743 758 773 15 An 287 788 803 818 834 849 864 879 894 909 924 15 ■7 288 939 954 969 984 ♦00 ♦ 15 ♦30 *45 ♦60 ♦75 IS Vt 2-1 2 3 289 46 090 105 120 135 150 165 180 195 210 225 15 2-9 4 290 240 255 270 285 300 315 330 345 359 374 15 3-6 5 291 389 404 419 434 449 464 479 494 5°9 523 15 4-3 6 292 538 553 568 583 598 613 627 642 657 672 15 5'0 6-7 6-4 7 8 9 293 687 702 716 73i 746 761 776 790 805 820 15 294 835 850 864 879 894 909 923 938 953 967 15 7-1 7-9 8 '6 10 295 982 997 ♦ 12 ♦26 ♦41 ♦56 ♦70 ^85 100 114 15 11 296 47 129 144 159 173 188 202 217 232 246 261 15 9-3 13 297 276 290 305 319 334 349 363 378 392 407 15 10 14 298 422 436 451 465 480 494 509 524 538 553 14 299 N. 567 582 596 611 625 640 654 669 683 698 14 A P. P. Log 1 2 3 4 5 6 7 8 9 24 FIVE-PLACE LOGARITHMS. N. Log o 1 2 3 4 5 6 7 8 9 A P.P. 300 47 712 727 74i 756 770 784 799 813 828 842 15 301 857 871 885 900 914 929 943 958 972 986 15 15 302 48 001 015 029 044 058 073 087 IOI 116 130 14 An Al 303 144 159 173 187 202 216 230 244 259 273 H 304 287 302 316 33° 344 359 373 387 401 416 14 '7 1-3 1 2 j 305 430 444 458 473 487 501 515 53° 544 558 14 2'0 2-7 3 4 306 572 586 601 6i5 629 643 657 671 686 700 14 307 714 728 742 756 770 785 799 813 827 841 14 3*3 4*0 5 6 308 855 869 883 897 911 926 940 954 968 982 14 47 7 309 996 ♦ 10 ♦24 ♦38 ♦52 ♦66 ♦80 ♦94 108 122 14 6-3 6-0 8 9 310 49 136 150 164 178 192 206 220 234 248 262 14 6-7 10 311 276 290 3°4 3i8 332 346 360 374 388 402 13 7'3 11 312 415 429 443 457 471 485 499 513 527 54i 13 8-0 87 9 '3 12 313 SS4 568 582 596 610 624 638 651 665 679 14 13 14 314 693 707 721 734 748 762 776 790 803 817 14 10 315 831 845 S59 872 8S6 900 914 927 941 955 14 316 969 9S2 996 ♦ 10 ♦24 ♦37 ♦5i ♦65 ♦79 ♦92 14 317 50 106 120 133 147 161 174 188 202 215 229 14 318 243 256 270 284 297 3" 325 338 352 365 14 319 379 393 406 420 433 447 461 474 488 5°i 14 14 320 SiS 529 542 556 569 583 596 610 623 637 '4 An Al 321 651 664 678 691 705 718 732 745 759 772 14 ■7 1 322 786 799 813 826 840 853 866 880 893 907 13 1*4 2*1 2- 323 920 934 947 961 974 987 ♦01 ♦ 14 ♦28 ♦41 14 2-9 3 4 324 Si 055 068 081 095 108 121 '35 148 162 175 13 3 '6 5 325 188 202 215 228 242 255 268 282 295 308 14 4-3 5'0 6 7 8 326 322 335 348 362 375 388 402 415 428 441 14 57 327 4SS 468 481 495 508 521 534 548 56i 574 13 6-4 9 328 587 601 614 627 640 654 667 680 693 706 14 7'1 329 720 733 746 759 772 786 799 812 825 838 13 7-9 11 330 8 S 1 865 878 891 904 917 93° 943 957 970 13 9-3 10 13 331 983 996 ♦09 ♦22 ♦35 ♦48 ♦61 ♦75 ♦88 IOI 13 T 4 I 332 S 2 »4 127 140 153 166 179 192 205 218 231 13 333 244 257 270 284 297 310 323 336 349 362 13 334 375 388 401 414 427 440 453 466 479 492 12 335 5°4 517 53° 543 556 569 582 595 608 621 13 13 12 336 634 647 660 789 673 802 686 8i5 699 827 711 840 724 853 737 866 750 879 13 An Al An Al 337 763 776 13 •8 1-6 1 -8 2 1-7 338 892 905 917 93° 943 956 969 982 994 ♦07 13 2 339 53 020 o33 046 058 071 084 097 no 122 135 13 2-3 31 3 2-6 4 3-3 3 4 340 148 161 173 186 199 212 224 237 250 263 12 3'8 5 4-2 6 6-0 5 6 341 275 288 301 3H 326 339 352 364 377 390 13 4 6 342 403 415 428 441 453 466 479 491 5°4 5i7 12 5-4 7 6-8 7 343 529 542 555 567 580 593 605 618 631 643 '3 6-2 6-9 8 6-7 9 7-5 8 344 656 668 681 694 706 719 732 744 757 769 13 9 7-7 10 8-3 10 345 782 794 807 820 832 845 857 870 882 895 13 8-5 11 9-2 11 346 908 920 933 945 958 970 983 995 ♦08 ♦20 13 92 12 1U 10 I13 12 347 54 033 045 058 070 083 095 10S 120 133 145 13 348 158 170 183 195 208 220 233 245 258 270 13 349 283 295 307 320 332 345 357 37° 382 394 13 N. Log 1 2 3 4 5 6 7 8 9 A P.P. FIVE-PLACE LOGARITHMS. 25 N. Log 1 2 3 4 5 6 7 8 9 A P.P. 350 54 407 419 432 444 456 469 481 494 506 518 13 351 53i 543 555 568 580 593 605 617 630 642 12 352 654 667 679 691 704 716 728 74i 753 765 12 353 777 790 802 814 827 839 851 864 876 888 12 13 354 900 913 925 937 949 962 974 986 998 ♦ 11 12 An Al 355 55 023 °35 047 060 072 084 096 108 121 133 12 •s 1-6 1 2 356 145 157 169 182 194 206 218 230 242 255 12 23 3 357 267 279 291 303 315 328 340 352 364 376 12 3'1 4 358 388 400 413 425 437 449 46! 473 485 497 12 3-8 5 359 509 522 534 546 558 570 582 594 606 618 12 4-6 6-4 6 7 360 630 642 654 666 678 691 703 715 727 739 12 6-2 6'9 8 9 361 75i 763 775 787 799 811 823 835 847 859 12 362 871 883 895 907 919 931 943 955 967 979 12 77 10 363 991 ♦03 ♦15 ♦ 27 ♦38 ♦50 ♦62 ♦74 ♦86 ♦98 12 8 "5 9 - 2 11 364 56 no 122 '34 146 1 S 8 170 182 194 205 217 12 10 '3 365 229 241 253 265 277 289 301 312 324 336 12 366 348 360 372 384 396 407 419 431 443 455 12 367 467 478 490 502 514 526 538 549 561 573 12 368 585 597 608 620 632 644 656 667 679 691 12 369 703 714 726 738 750 761 773 785 797 808 12 370 820 832 844 855 867 879 891 902 914 926 An 1Z Al 371 937 949 961 972 984 996 ♦08 ♦ 19 ♦3i ♦43 372 57 054 066 078 089 IOI ii3 124 136 148 '59 •8 1'7 1 373 171 183 194 206 217 229 241 252 264 276 2-5 3 374 287 299 310 322 334 345 357 368 380 392 3 3 4 375 403 415 426 438 449 461 473 484 496 So7 4-2 B'O 5 6 376 5i9 53° 542 553 565 576 588 600 611 623 6-8 7 377 634 646 657 669 680 692 7°3 715 726 738 6'7 8 378 749 761 772 784 795 807 818 830 841 852 7'5 9 379 864 875 887 898 910 921 933 944 955 967 8'3 9 "2 10 380 978 990 ♦01 ♦ 13 ♦24 ♦35 ♦47 ♦58 ♦70 ♦81 10 12 381 58 092 104 "5 127 138 149 161 172 184 195 382 206 218 229 240 252 263 274 286 297 309 383 320 33i 343 354 365 377 388 399 410 422 384 433 444 456 467 478 490 50J 512 524 535 385 546 557 569 580 59i 602 614 625 636 647 386 659 670 681 692 704 715 726 737 749 760 11 387 771 782 794 805 816 827 838 850 861 872 An Al 388 883 894 906 917 928 939 950 961 973 984 •9 1 389 995 ♦06 ♦ 17 ♦28 ♦40 ♦ 5i ♦62 ♦73 ♦84 ♦95 1-8 2-7 2 390 59 106 118 129 140 »5' 162 '73 184 '95 207 3 6 3 4 391 218 229 240 251 262 273 284 295 306 318 4-6 5 392 329 340 35' 362 373 384 395 406 4'7 428 6-5 6 393 439 450 461 472 483 494 506 5i7 528 539 6 '4 7'i 7 g 394 55° 561 572 583 594 605 616 627 638 649 8-2 9 395 660 671 682 693 704 715 726 737 748 759 9-1 10 396 770 780 791 802 813 824 835 846 857 868 10 IX 397 879 890 901 912 923 934 945 956 966 977 398 988 999 ♦ 10 ♦21 ♦32 ♦43 ♦54 ♦65 ♦76 ♦86 399 60 097 108 119 130 141 152 163 173 184 195 N. Log 1 2 3 4 5 6 7 8 9 A P.P. 26 FIVE-PLACE LOGARITHMS. N. Log o 1 2 3 4 5 6 7 8 9 A P.P. 400 6o 206 217 228 239 249 260 271 282 293 304 IO 401 314 325 336 347 358 369 379 39° 401 412 II 402 423 433 444 455 466 477 487 498 509 520 II 403 53i 54i 552 563 574 584 595 606 617 627 II 404 638 649 660 670 681 692 703 713 724 735 II 405 746 756 767 778 78S 799 810 821 831 842 II 406 8S3 863 874 885 895 906 917 927 938 949 IO 11 407 408 959 61 066 970 077 981 087 991 098 ♦02 109 ♦ 13 119 ♦23 130 ♦34 140 ♦45 151 ♦55 162 II IO •9 1 i 409 172 183 194 204 215 225 236 247 257 268 IO 1-8 2-7 2 3 410 278 289 300 310 321 33i 342 352 363 374 IO 3-6 4 411 384 395 405 416 426 437 448 458 469 479 II 4-5 5 412 490 500 5" 521 532 542 553 563 574 584 II 5-5 6 413 595 606 616 627 637 648 658 669 679 690 IO 6'4 7-3 7 g 414 700 711 721 73i 742 752 763 773 784 794 II 8-2 9 415 805 815 826 836 847 857 868 878 888 899 IO 9-1 10 IO 416 909 920 930 941 95i 962 972 982 993 ♦03 II 417 62 014 024 034 045 o55 066 076 086 097 107 II 418 118 128 138 149 159 170 180 190 201 211 IO 419 221 232 242 252 263 273 284 294 3°4 3i5 IO 420 325 335 346 356 366 377 387 397 408 418 IO 421 428 439 449 459 469 480 490 500 5" 521 IO 10 422 423 53i 634 542 644 552 655 562 665 572 675 583 685 593 696 603 706 613 716 624 726 IO II An Zil 1 I 424 737 747 757 767 778 788 798 808 818 829 IO 2 3 2 425 839 849 859 870 880 890 900 910 921 93i IO 4 3 4 426 941 95i 961 972 982 992 ♦02 ♦ 12 422 ♦33 IO 6 5 427 63 043 053 063 073 083 094 104 114 124 134 IO 6 6 428 144 155 165 175 185 195 205 215 225 236 IO 7 8 9 7 8 9 429 246 256 266 276 286 296 306 317 327 337 IO 430 347 357 367 377 387 397 407 417 428 438 IO 10 10 431 448 458 468 478 488 498 5°S 5i8 528 538 IO 432 548 558 568 579 589 599 609 619 629 639 IO 433 649 659 669 679 689 699 709 719 729 739 IO 434 749 759 769 779 789 799 809 819 829 839 IO 435 849 859 869 879 889 899 909 919 929 939 IO 9 436 949 959 969 979 988 998 ♦08 ♦ 18 ♦28 ♦38 IO An Al 437 64 048 058 068 078 088 098 10S 118 12S 137 IO 438 147 157 167 177 187 197 207 217 227 237 9 l'l 2 '2 1 439 246 256 266 276 286 296 306 3i6 326 335 IO 3-3 4 'i 3 440 345 355 365 375 385 395 404 414 424 434 IO 5-6 6-7 4 441 444 454 464 473 483 493 5°3 513 523 532 IO 5 6 442 542 552 562 572 582 59i 601 611 621 631 9 7-8 7 443 640 650 660 670 680 689 699 709 719 729 9 8-9 3 444 738 748 758 768 777 787 797 S07 816 826 IO 10 9 ; 445 836 846 856 865 875 885 895 904 914 924 9 446 933 943 953 963 972 982 992 ♦02 ♦ 11 ♦21 IO 447 65 031 040 050 060 070 079 089 099 10S 118 IO 448 128 137 147 157 167 176 186 196 205 215 IO 449 225 234 244 254 263 273 283 292 302 312 9 N. Log 1 2 3 4 5 6 7 8 9 A P.P. FIVE-PLACE LOGARITHMS. 27 N. Log 1 2 3 4 5 6 7 8 9 A P.P. 450 65 321 33i 34i 350 360 369 379 389 398 408 IO 451 418 427 437 447 456 466 475 485 495 504 IO 452 514 523 533 543 552 562 57i 581 59i 600 IO 453 610 619 629 639 648 658 667 677 686 696 IO 454 706 715 725 734 744 753 763 772 782 792 9 455 801 811 820 830 839 849 858 868 877 887 9 456 896 906 916 925 935 944 954 963 973 982 IO A n Al 457 992 ♦01 ♦ 11 ♦20 ♦30 ♦39 ♦49 ♦58 ♦68 ♦77 IO 458 66 087 096 106 »5 124 134 143 153 162 172 9 1 I 459 181 191 200 210 219 229 238 247 257 266 IO 2 8 2 3 460 276 285 295 304 314 323 332 342 35i 361 9 4 4 461 37o 380 389 398 408 4i7 427 43 6 445 455 9 6 5 ! 462 464 474 483 492 502 5" 521 53° 539 549 9 6 7 8 6 463 558 567 577 586 596 605 614 624 633 642 IO 7 ; 8 464 652 661 671 680 689 699 708 717 727 736 9 9 9 465 745 755 764 773 783 792 801 811 S20 S29 IO 10 IO 466 839 84S 857 867 876 885 894 904 913 922 IO 467 932 941 95° 960 969 978 987 997 ♦06 ♦ 15 IO 468 67 025 034 043 052 062 071 080 0S9 099 108 9 469 117 127 136 145 154 164 173 182 191 201 9 « 470 210 219 228 237 247 256 265 274 284 293 9 471 302 3" 321 33° 339 348 357 367 376 385 9 Q 472 394 403 413 422 43i 440 449 459 468 477 9 An Al 473 486 495 504 514 523 532 54i 55o 560 569 9 474 578 587 59° 605 614 624 633 642 651 660 9 11 2-2 I 2 475 669 679 688 697 706 7i5 724 733 742 752 9 33 4-4 3 4 476 761 770 779 788 797 806 815 825 834 843 9 477 852 861 870 879 888 897 906 916 925 934 9 6'6 6-7 7-8 5 478 943 952 961 970 979 988 997 ♦06 ♦ 15 ♦24 IO 6 479 68 034 043 052 061 070 079 088 097 106 "5 9 8 9 10 8 480 124 133 142 151 160 169 178 187 196 205 IO 481 215 224 233 242 251 260 269 278 287 296 9 482 3°5 314 323 332 34i 35° 359 368 377 386 9 483 395 404 413 422 43i 440 449 458 467 476 9 484 485 494 502 5" 520 529 538 547 55° 565 9 485 574 583 592 601 610 619 628 637 646 655 9 486 664 673 681 690 699 708 717 726 735 744 9 8 487 753 762 771 780 789 797 806 8i5 824 833 9 A n L\ 1 488 842 851 860 869 878 886 895 904 9i3 922 9 1-8 I 489 93' 940 949 958 966 975 984 993 ♦02 ♦ 11 9 2-5 38 2 3 490 69 020 028 037 046 055 064 073 082 090 099 9 6-0 4 491 108 117 126 135 144 152 161 170 179 188 9 6-3 5 492 197 205 214 223 232 241 249 258 267 276 9 7-5 6 493 285 294 302 3" 320 329 338 346 355 364 9 8'S 10 7 R 494 373 38i 39° 399 408 417 425 434 443 452 9 495 461 469 478 487 496 5°4 513 522 531 539 9 496 548 557 566 574 583 592 601 609 618 627 9 497 636 644 653 662 671 679 688 697 705 714 9 498 723 73 2 740 749 758 767 775 7S4 793 801 9 499 810 819 827 836 845 854 862 871 880 888 9 N. Log 1 2 3 4 5 6 7 8 9 A p.r. 28 FIVE-PLACE LOGARITHMS. N. Log o 1 2 3 4 5 6 7 8 9 A P.P. 500 69 897 906 914 923 932 940 949 958 966 975 9 501 984 992 «oi »io ♦18 ♦27 ♦36 ♦44 «53 462 8 502 70 070 079 088 096 105 114 122 131 140 148 9 503 157 165 174 183 191 200 209 217 226 234 9 504 243 252 260 269 278 286 295 303 312 321 8 505 329 338 346 355 364 372 381 389 398 406 9 9 506 4i5 424 432 441 449 458 467 475 484 492 9 An'Al 507 5°i 509 518 526 535 544 552 561 569 578 8 1 508 586 595 603 612 621 629 638 646 655 663 9 l-l 2-2 3 3 1 509 672 680 689 697 706 714 723 731 740 749 8 3 510 757 766 774 783 791 800 808 817 825 834 8 4 '4 4 511 842 851 859 868 876 885 893 902 910 919 8 6"6 6 7 5 6 512 927 935 944 952 961 969 978 986 995 *03 9 7-8 7 ' 513 71 012 020 029 037 046 054 063 071 079 088 8 8-9 8 : 514 096 i°5 113 122 130 139 147 155 164 172 9 10 9 515 181 189 198 206 214 223 231 240 248 257 8 516 265 273 282 290 299 3°7 315 324 332 34i 8 517 349 357 366 374 383 391 399 408 416 425 8 518 433 441 45° 458 466 475 483 492 500 508 9 519 517 525 533 542 55o 559 567 575 584 592 8 520 600 609 617 625 634 642 650 659 667 675 9 521 684 692 700 709 717 725 734 742 75° 759 8 8 522 767 775 784 792 800 809 817 825 834 842 8 An Al 523 850 858 867 875 883 892 900 908 917 925 8 - — 1 524 933 941 950 958 966 975 983 991 999 ^08 8 1"3 2 5 1 2 525 72 016 024 032 041 049 057 066 074 082 090 9 3 '8 6-0 3 4 526 099 107 US 123 132 140 148 156 165 173 8 6-3 7'5 527 181 189 198 206 214 222 230 239 247 255 8 5 6 528 263 272 280 288 296 304 313 321 329 337 9 8'8 7 529 346 354 362 370 378 387 395 403 4" 419 9 10 8 530 428 436 444 452 460 469 477 485 493 501 8 531 5°9 518 526 534 542 550 558 567 575 583 8 532 59i 599 607 616 624 632 640 648 656 665 8 533 673 681 689 697 705 7i3 722 730 738 746 8 534 754 762 770 779 787 795 803 811 819 827 8 535 835 843 852 860 868 876 884 892 900 908 8 536 916 925 933 94i 949 957 965 973 981 989 8 7 537 997 ♦06 ♦14 422 ♦ 30 ♦ 38 ♦46 ♦54 462 »7o 8 An Al 538 73 078 086 094 102 111 119 127 135 143 151 8 539 159 167 175 «83 191 199 207 215 223 231 8 V4 2-9 1 2 540 239 247 255 263 272 280 288 296 304 312 8 i-'c 3 4 541 320 32S 33° 344 352 360 368 376 384 392 8 542 400 408 416 424 432 440 448 456 464 472 8 7*' 8-1 5 6 543 480 488 496 504 512 520 528 536 544 552 8 10 7 544 560 568 576 584 592 600 608 616 624 632 8 545 640 648 656 664 672 679 687 695 703 7» 8 546 719 727 735 743 751 759 767 775 783 79i 8 547 799 807 815 823 830 838 846 854 862 870 8 548 878 886 894 902 910 918 926 933 94i 949 8 549 957 965 973 981 989 997 ♦05 ♦ 13 *20 »28 8 N. Log 1 2 3 4 5 6 7 8 9 A P.P. FIVE-PLACE LOGARITHMS. 29 N. Log 1 2 3 4 5 6 7 8 9 A P.P. 550 74 036 044 052 060 068 076 084 092 099 107 8 551 »5 123 131 139 147 •55 162 170 178 186 8 552 194 202 210 218 225 233 241 249 257 265 8 553 273 280 288 296 304 312 320 327 335 343 8 554 351 359 367 374 382 390 398 406 414 421 8 555 429 437 445 453 461 468 476 484 492 500 7 556 5°7 5i5 523 53' 539 547 554 562 57o 578 8 557 586 593 601 609 617 624 632 640 648 656 7 558 663 671 679 687 695 702 710 718 726 733 8 559 741 749 757 764 772 780 788 796 803 811 8 560 819 827 834 842 850 858 865 873 88 1 889 7 561 896 904 912 920 927 935 943 950 958 966 8 562 974 981 9S9 997 ♦05 ♦ 12 ♦20 ♦28 ♦35 ♦43 8 8 563 75 05i 059 066 074 082 089 097 105 "3 120 S An Al 564 128 136 143 151 159 166 174 182 189 197 8 1*3 565 205 213 220 228 236 243 251 259 266 274 8 2-5 3 '8 2 566 282 289 297 305 312 320 328 335 343 351 7 5-0 3 4 567 358 366 374 38l 389 397 404 412 420 427 8 568 435 442 45° 458 465 473 481 488 496 5°4 7 G'3 7.5 5 6 569 5" 519 526 534 542 549 557 5<-5 572 580 7 8-8 7 570 587 595 603 610 618 626 633 641 648 656 8 10 8 571 664 671 679 686 694 702 709 717 724 732 8 572 740 747 755 762 770 778 785 793 800 808 7 573 8i5 823 831 838 846 853 861 868 876 884 7 574 891 899 906 914 921 929 937 944 952 959 8 575 967 974 982 989 997 ♦05 ♦ 12 ♦20 ♦27 ♦35 7 576 76 042 050 057 065 072 080 0S7 095 103 no 8 577 118 125 «33 140 148 '55 163 170 17S 185 8 578 193 200 208 215 223 230 238 245 253 260 8 579 268 275 283 290 298 305 313 320 328 335 8 580 343 35° 358 365 373 380 388 395 403 410 8 7 581 418 425 433 440 448 455 462 470 477 485 7 An Al 582 492 500 507 5i5 522 530 537 545 552 559 8 583 567 574 582 5S9 597 604 612 619 626 634 7 1'4 2'C 1 584 641 649 656 664 671 678 686 693 701 708 8 43 5-7 7-1 8'6 3 585 716 723 730 738 745 753 760 768 775 782 8 4 586 790 797 805 812 819 827 834 842 849 856 8 5 6 587 864 871 879 886 893 901 908 916 923 930 8 10 7 588 938 945 953 960 967 975 9S2 989 997 ♦04 8 589 77 012 019 026 034 041 048 056 063 070 078 7 590 085 093 100 107 "5 122 129 137 144 151 8 591 159 166 173 181 188 195 203 210 217 225 7 592 232 240 247 254 262 269 276 283 291 298 7 593 305 313 320 3 2 7 335 342 349 357 364 37i 8 594 379 386 393 401 408 415 422 43° 437 444 8 595 452 459 466 474 481 488 495 503 510 517 8 596 525 532 539 546 554 561 568 576 583 590 7 597 597 605 612 619 627 634 641 648 656 663 7 598 670 677 685 692 699 706 7H 721 728 735 8 599 743 750 757 764 772 779 786 793 801 808 7 N. Log 1 2 3 4 5 6 7 8 9 A P.P. 3° FIVE-PLACE LOGARITHMS. N. Log o 1 2 3 4 5 6 7 8 9 A P.P. 600 77 815 822 830 837 844 851 859 866 873 880 7 601 887 895 902 909 916 924 93i 938 945 952 8 602 960 967 974 981 988 996 ♦03 ♦ 10 ♦ 17 ♦25 7 603 78 032 039 046 °53 061 068 075 082 089 097 7 604 104 in 118 125 132 140 147 154 161 168 8 605 176 183 190 197 204 211 219 226 233 240 7 606 247 2 54 262 269 276 283 290 297 305 312 7 8 607 319 326 333 34° 347 355 362 369 376 383 7 An Al 608 390 393 405 412 419 426 433 440 447 455 7 609 462 469 476 483 490 497 504 512 519 526 7 1*3 2-5 X 2 610 S33 540 547 554 56i 569 576 583 590 597 7 3-8 5-0 3 4 611 604 611 618 625 633 640 647 654 661 668 7 612 675 682 689 696 704 711 7i8 725 732 739 7 6*3 7 '5 5 6 613 746 753 760 767 774 781 789 796 803 810 7 8-8 7 614 817 824 831 838 845 852 859 866 873 880 8 10 3 615 888 895 902 909 916 923 930 937 944 951 7 616 958 965 972 979 986 993 ♦00 ♦07 ♦ 14 ♦21 8 617 79 029 036 043 050 057 064 071 078 085 092 7 618 099 106 "3 120 127 '34 141 148 155 162 7 619 169 176 183 190 197 204 211 218 225 232 7 620 239 246 253 260 267 274 281 288 295 302 7 621 309 316 323 33° 337 344 35i 358 365 372 7 622 379 386 393 400 407 414 421 428 435 442 7 7 623 449 456 463 470 477 484 491 498 505 5" 7 An A I 624 5I 8 525 532 539 546 553 560 567 574 581 7 1-4 1 625 588 595 602 609 616 623 630 637 644 650 7 2-9 4'3 2 3 626 657 664 671 678 685 692 699 706 713 720 7 51 4 627 727 734 74i 748 754 761 768 775 782 789 7 7"1 628 796 803 810 817 824 831 837 844 851 858 7 8-8 5 6 629 865 872 879 886 893 900 906 913 920 927 7 10 7 630 934 941 948 955 962 969 975 982 989 996 7 631 80 003 010 017 024 030 037 044 051 058 065 7 632 072 079 085 092 099 106 113 120 127 134 6 633 140 147 154 161 168 175 182 188 195 202 7 634 209 216 223 229 236 243 250 257 264 271 6 635 277 284 291 298 305 312 318 325 332 339 7 636 346 353 359 366 373 380 387 393 400 407 7 637 414 421 428 434 441 448 455 462 468 475 7 6 638 639 482 489 557 496 564 502 509 516 584 523 530 598 536 604 543 6u 7 An Al : 55° 57° 577 59i 7 1-7 I 640 618 625 632 638 645 652 659 665 672 679 7 3-3 5-0 2 3 641 686 693 699 706 713 720 726 733 740 747 7 6-7 4 ; 642 754 760 767 774 78i 787 794 801 808 814 7 643 821 828 835 841 848 855 862 868 875 882 7 8*3 10 5 644 889 895 902 909 916 922 929 936 943 949 7 645 956 963 969 976 983 990 996 ♦03 ♦ 10 ♦ 17 6 646 81 023 030 037 043 050 057 064 070 077 084 6 647 090 097 104 in 117 124 131 137 144 151 7 648 158 164 171 178 184 191 198 204 211 218 6 649 224 231 238 245 251 258 265 271 278 285 6 N. Log 1 2 3 4 5 6 7 8 9 A P.P. FIVE-PLACE LOGARITHMS. 3i N. Log 1 2 3 4 5 6 7 8 9 A P.P. 650 81 291 298 305 311 3i8 325 33i 338 345 35i 7 651 358 365 37i 378 385 39i 398 405 411 418 7 652 425 43i 438 445 45i 458 465 47i 478 485 6 653 491 498 5°5 5" 518 525 53i 538 544 5Si 7 654 558 564 57i 578 584 59i 598 604 611 617 7 655 624 631 637 644 65" 657 664 671 677 684 6 656 690 697 704 710 717 723 73° 737 743 75o 7 657 757 763 770 776 783 790 796 803 809 816 7 658 823 829 836 842 849 856 862 S69 875 882 7 659 889 895 902 908 915 921 928 935 941 948 6 660 954 961 968 974 981 987 994 ♦00 ♦07 ♦H 6 661 82 020 027 o33 040 046 053 060 066 073 079 7 7 662 086 092 099 105 112 119 125 132 138 145 6 An il 663 151 158 164 171 178 184 191 197 204 210 7 1-4 1 664 217 223 230 236 243 249 256 263 269 276 6 2-9 4'3 2 665 282 289 295 302 308 315 321 328 334 34i 6 6-7 3 4 666 347 354 360 367 373 380 387 393 400 406 7 7-1 5 667 413 419 426 432 439 445 452 458 465 47i 7 8-6 6 668 478 484 491 497 504 510 5i7 523 53o 536 7 10 7 669 543 549 556 562 569 575 582 588 595 601 6 670 607 614 620 627 633 640 646 653 659 666 6 671 672 679 685 692 698 705 7" 718 724 73o 7 672 737 743 750 756 763 769 776 782 789 795 7 673 802 808 814 821 827 834 840 847 853 860 6 674 866 872 879 885 892 898 905 911 918 924 6 675 93° 937 943 95° 956 963 969 975 982 988 7 676 995 ♦01 ♦08 ♦ 14 ♦ 20 ♦27 ♦33 ♦4° ♦46 ♦52 7 677 83 059 065 072 078 085 091 097 104 no 117 6 678 123 129 136 142 149 155 161 168 174 181 6 679 187 193 200 206 213 219 225 232 238 245 6 680 251 257 264 270 276 283 289 296 302 308 7 681 315 321 327 334 340 347 353 359 366 372 6 682 378 385 39i 39S 404 410 4i7 423 429 436 6 6 683 442 448 455 461 467 474 480 487 493 499 7 An Al 684 506 512 5i8 525 531 537 544 55o 556 563 6 1-7 I 685 569 575 582 588 594 601 607 613 620 626 6 3"3 6-0 1 3 686 632 639 645 651 658 664 670 677 683 689 7 6'7 4 687 696 702 708 7i5 721 727 734 740 746 753 6 8-3 5 688 759 765 771 778 784 790 797 S03 809 816 6 10 6 689 822 828 835 841 847 853 860 866 872 879 6 690 885 891 897 904 910 916 923 929 935 942 6 691 948 954 960 967 973 979 985 992 998 ♦04 7 692 84 on 017 023 029 036 042 048 055 061 067 6 693 o73 080 086 092 098 i°5 in 117 123 130 6 694 136 142 148 155 161 167 173 180 186 192 6 695 198 205 211 217 223 230 236 242 248 255 6 696 261 267 273 280 286 292 298 3°5 311 317 6 697 323 33o 336 342 348 354 361 367 373 379 I 698 386 392 398 404 410 4i7 423 429 435 442 6 699 448 454 460 466 473 479 485 491 497 504 6 N. Log 1 2 3 4 5 6 7 8 9 A P.P. 32 FIVE-PLACE LOGARITHMS N. Log 01234 56789 A P.P. 700 84 510 516 522 528 535 54i 547 553 559 §66 6 701 572 578 584 590 597 603 609 615 621 628 6 702 634 640 646 652 658 665 671 677 683 689 7 703 696 702 708 714 720 726 733 739 745 75* 6 704 757 763 77° 776 782 788 794 800 807 813 6 705 819 825 831 837 844 850 856 862 868 874 6 706 880 887 893 899 905 911 917 924 930 936 6 707 942 948 954 960 967 973 979 985 99i 997 6 7 708 85 003 009 016 022 028 034 040 046 052 058 7 An Al 709 065 071 077 083 089 095 101 107 114 120 6 1-4 1 710 126 132 138 144 150 156 163 169 175 181 6 2-9 4-3 2 3 711 187 193 199 205 211 217 224 230 236 242 6 57 4 712 248 254 260 266 272 278 285 291 297 303 6 7-1 8 11 713 309 315 321 327 333 339 345 352 358 364 6 5 6 714 370 376 382 388 394 400 406 412 418 425 6 10 7 715 43 1 437 443 449 455 461 467 473 479 485 6 716 491 497 503 509 516 522 528 534 540 546 6 717 552 558 564 570 576 582 588 594 600 606 6 718 612 618 625 631 637 643 649 655 661 667 6 719 673 679 685 691 697 703 709 715 721 727 6 720 733 739 745 75* 757 763 769 775 78i 788 6 721 794 800 806 812 818 824 830 836 842 848 6 722 854 860 866 872 878 884 890 896 902 908 6 6 723 914 920 926 932 938 944 950 956 962 968 6 An Al 724 974 980 986 992 998 ♦04 *lo »i6 422 428 6 1-7 j 725 86 034 040 046 052 058 064 070 076 082 088 6 33 5-0 2 3 726 094 100 106 112 118 124 130 136 141 147 6 6-7 4 727 153 159 l6 5 171 177 183 189 195 201 207 6 728 213 219 225 231 237 243 249 255 261 267 6 8 '3 10 5 729 273 279 285 291 297 303 308 314 320 326 6 730 332 338 344 35o 356 362 368 374 380 386 6 731 392 398 404 410 415 421 427 433 439 445 6 732 451 457 463 469 475 481 487 493 499 504 6 j 733 510 516 522 528 534 540 546 552 558 564 6 734 570 576 581 587 593 599 605 611 617 623 6 735 629 635 641 646 652 658 664 670 676 682 6 736 688 694 700 705 711 717 723 729 735 741 6 E 737 747 753 759 764 77o 776 782 788 794 800 6 A r Al 738 806 812 817 823 829 835 841 847 853 859 5 739 864 870 876 882 888 894 900 906 911 917 6 2' 4- ) 1 ) 2 740 923 929 935 941 947 953 958 964 970 976 6 6- 8' ) 3 3 4 741 982 988 994 999 405 ♦11 417 »23 429 »35 5 742 87 040 046 052 058 064 070 075 081 087 093 6 10 5 743 099 105 m 116 122 128 134 140 146 151 6 744 157 163 169 175 181 186 192 198 204 210 6 745 216 221 227 233 239 245 251 256 262 268 6 746 274 280 286 291 297 3°3 309 3 J 5 320 326 6 747 332 338 344 349 355 361 367 373 379 384 6 748 390 396 402 408 413 419 425 431 437 442 6 749 448 454 460 466 47 « 477 483 489 495 5oo 6 N. Log 01234 56789 A P. P. FIVE-PLACE LOGARITHMS. 33 N. Log o i 2 3 4 56789 A P.P. 750 751 752 753 754 87 506 512 518 523 529 564 570 576 581 587 622 628 633 639 645 679 683 691 697 7°3 737 743 749 754 76o 535 541 547 552 558 593 599 604 610 616 651 656 662 668 674 708 714 720 726 731 766 772 777 783 789 6 6 5 6 6 755 756 757 758 759 79S 800 806 812 818 852 858 864 869 875 910 915 921 927 933 967 973 978 984 99° 88 024 030 036 041 047 823 829 835 841 846 881 887 892 898 904 938 944 950 955 961 996 ioi *07 ^13 »i8 053 058 064 070 076 6 6 6 6 5 760 761 762 763 764 081 087 093 098 104 138 144 150 156 161 195 201 207 213 218 252 258 264 270 275 309 315 321 326 332 no 116 121 127 133 167 173 178 184 190 224 230 235 241 247 281 287 292 298 304 338 343 349 355 360 5 5 5 5 6 An 6 Al 1-7 3-3 1 765 766 767 768 769 366 372 377 383 389 423 429 434 440 446 480 485 491 497 502 536 542 547 553 559 593 598 604 610 615 395 400 406 412 417 45i 457 463 468 474 508 513 519 525 530 564 570 576 581 587 621 627 632 638 643 6 6 6 6 6 6-0 6-7 8'3 10 3 4 5 6 770 771 772 773 774 649 655 660 666 672 705 711 717 722 728 762 767 773 779 784 818 824 829 835 840 874 880 885 891 897 677 683 689 694 700 734 739 745 7 50 756 790 795 801 807 812 846 852 857 863 868 902 908 913 919 925 5 6 6 6 5 775 776 777 778 779 930 936 941 947 953 986 992 997 403 409 89 042 048 053 059 064 098 104 109 115 120 154 159 165 170 176 958 964 969 975 981 ♦14 «2o »25 »3i «37 070 076 081 087 092 126 131 137 143 148 182 187 193 198 204 5 5 6 6 5 780 781 782 783 784 209 215 221 226 232 265 271 276 282 287 321 326 332 337 343 376 382 387 393 398 432 437 443 448 454 237 243 248 254 260 293 298 304 310 315 348 354 360 365 371 404 409 415 421 426 459 465 47° 476 481 5 6 5 6 6 An 2-0 4-0 5 Al I 2 785 786 787 788 789 487 492 498 504 509 542 548 553 559 564 597 603 609 614 620 653 658 664 669 675 708 713 719 724 730 515 520 526 531 537 570 575 581 586 592 625 631 636 642 647 680 686 691 697 702 735 74i 746 752 757 5 5 6 6 6 8-C 10 4 5 790 791 792 793 794 763 768 774 779 785 818 823 829 834 840 873 878 883 889 894 927 933 938 944 949 982 988 993 998 ♦ch 790 796 801 807 812 845 851 856 862 867 900 905 911 916 922 955 96o 966 971 977 ♦09 *I5 »2o ^26 »3i 6 6 5 5 6 795 796 797 798 799 90 037 042 048 053 059 091 097 102 108 113 146 151 157 162 168 200 206 211 217 222 255 260 266 271 276 064 069 075 080 086 119 124 129 135 140 173 179 184 189 195 227 233 238 244 249 282 287 293 298 304 5 6 5 6 5 N. Log 01234 56789 A P.P. 34 FIVE-PLACE LOGARITHMS. N. Log o 1 2 3 4 5 6 7 8 9 A P.P. 800 90 3°9 314 320 325 331 336 342 347 352 358 5 801 363 369 374 380 335 390 396 401 407 412 5 802 417 423 428 434 439 445 45° 455 461 466 6 803 472 477 482 488 493 499 5°4 509 515 520 6 804 526 531 536 542 547 553 558 563 569 574 6 805 580 585 590 596 601 607 612 617 623 628 6 806 634 639 644 650 655 660 666 671 677 682 5 807 687 693 698 703 709 7H 720 725 730 736 5 808 741 747 752 757 763 768 773 779 784 789 6 809 795 800 806 811 816 822 827 832 838 843 6 810 849 854 859 865 870 875 881 886 891 897 5 811 902 907 913 918 924 929 934 940 945 950 6 812 956 961 966 972 977 982 988 993 998 ♦04 5 A t 813 91 009 014 020 025 030 036 041 046 052 057 5 A T\ Al 814 062 068 073 078 084 089 094 100 105 no 6 1-7 3 3 5-0 I 815 116 121 126 132 137 142 148 153 158 164 5 2 3 816 169 174 180 185 190 196 201 206 212 217 5 6*7 4 817 222 228 233 238 243 249 254 259 265 270 5 8-3 5 818 27S 281 286 291 297 302 307 312 3i8 323 5 10 6 819 328 334 339 344 35° 355 360 365 37i 376 5 820 38i 387 392 397 403 408 413 418 424 429 5 821 434 440 445 450 455 461 466 47i 477 482 5 822 487 492 498 5°3 508 5>4 519 524 529 535 5 823 540 545 55i 556 56i 566 572 577 582 587 6 824 593 598 603 609 614 619 624 630 635 640 5 825 645 651 656 661 666 672 677 682 687 693 5 826 698 703 709 7i4 719 724 730 735 740 745 6 827 75i 756 761 766 772 777 782 787 793 798 5 828 803 808 814 819 824 829 834 840 845 850 5 829 855 861 866 871 876 882 887 892 897 903 5 830 908 913 918 924 929 934 939 944 950 955 5 831 960 965 971 976 981 986 991 997 ♦02 ♦07 5 832 92 012 018 023 028 033 038 044 049 054 o59 6 5 833 834 065 070 075 080 085 091 096 148 101 106 in 6 An Al 117 122 127 132 137 H3 153 158 163 6 2-0 1 835 169 174 179 184 189 195 200 205 210 215 6 4-0 6-0 2 3 836 221 226 231 236 241 247 252 257 262 267 6 8-0 4 837 273 278 283 288 293 298 3°4 309 314 319 5 10 838 324 330 335 340 345 350 355 36i 366 37i 5 AU J 839 376 38l 387 392 397 402 407 412 418 423 5 840 428 433 438 443 449 454 459 464 469 474 6 841 480 485 490 495 500 505 5» 5i6 521 526 5 842 S3i 536 542 547 552 557 562 567 572 578 5 843 583 588 593 598 603 609 614 619 624 629 5 844 634 639 645 650 655 660 665 670 675 681 5 845 686 691 696 701 706 711 716 722 727 732 5 846 737 742 747 752 758 763 768 773 778 783 5 847 788 793 799 804 809 814 819 824 829 834 6 848 840 845 850 855 860 865 870 875 881 886 5 849 891 896 901 906 911 916 921 927 932 937 5 .*. Log 1 2 3 4 5 6 7 8 9 A p.p. FIVE-PLACE LOGARITHMS. 35 N. Log o 1 2 3 4 5 6 7 8 9 A P. P. 850 92 942 947 952 957 962 967 973 978 983 988 5 851 993 998 ♦03 ♦08 ♦ 13 ♦18 ♦24 ♦29 ♦34 ♦39 5 852 93 °44 049 054 059 064 069 075 0S0 085 090 5 853 °95 100 105 no »5 120 125 131 136 141 5 854 146 151 156 161 166 171 176 1S1 186 192 5 855 197 202 207 212 217 222 227 232 237 242 5 856 247 252 258 263 268 273 278 283 288 293 5 857 298 303 308 313 318 323 328 334 339 344 5 858 349 354 359 364 369 374 379 384 389 394 5 859 399 404 409 414 420 425 43° 435 440 445 5 An Al 860 450 455 460 465 470 475 480 485 490 495 5 1'7 1 861 500 5°5 5io 515 520 526 53i 536 541 546 5 3'3 2 862 5Si 556 56i 566 571 576 58i 586 59i 596 5 6-0 67 3 863 601 606 611 6l6 621 626 631 636 641 646 5 4 864 6 S i 656 661 666 671 676 682 687 692 697 5 8-3 10 5 865 702 707 712 717 722 727 732 737 742 747 5 866 752 757 762 767 772 777 782 787 792 797 5 867 802 807 812 817 822 827 832 837 842 847 5 868 852 857 862 867 872 877 882 887 892 897 5 869 902 907 912 917 922 927 932 937 942 947 5 870 952 957 962 967 972 977 982 987 992 997 5 871 94 002 007 012 017 022 027 032 037 042 047 5 872 052 057 062 067 072 077 082 086 091 096 5 873 IOI 106 in 116 121 126 131 136 141 146 5 5 874 ISI 156 161 166 171 176 181 186 191 196 5 An Al 875 201 206 211 216 221 226 231 236 240 245 5 2-0 4-0 1 3 876 250 255 260 265 270 275 280 285 290 295 5 6-0 3 877 300 305 310 315 320 325 33° 335 340 345 4 8-0 4 878 349 354 359 364 369 374 379 384 389 394 5 10 c 879 399 404 409 414 419 424 429 433 438 443 5 880 448 453 458 463 468 473 478 483 488 493 5 881 498 5°3 5°7 512 517 522 527 532 537 542 5 882 547 552 557 562 567 57i 576 581 586 591 5 883 596 601 606 611 616 621 626 630 635 640 5 884 645 650 655 660 665 670 675 680 685 689 5 885 694 699 704 709 7H 719 724 729 734 738 5 886 743 748 753 758 763 768 773 778 783 787 5 A. 887 792 797 802 807 812 817 822 827 832 836 5 An Al 888 841 846 851 856 861 866 871 876 880 885 5 889 890 895 900 9°5 910 915 919 924 929 934 5 2-5 6-0 1 2' 890 939 944 949 954 959 963 968 973 978 983 5 7-6 10 3 A 891 988 993 998 ♦02 ♦07 ♦ 12 ♦ 17 ♦22 ♦27 ♦32 4 892 95 036 041 046 051 056 061 066 071 075 080 5 893 085 090 095 100 105 109 114 119 124 129 5 894 134 139 143 148 153 158 163 168 173 177 5 895 182 187 192 197 202 207 211 216 221 226 5 896 231 236 240 245 250 255 260 265 270 274 5 897 279 284 289 294 299 3°3 308 313 318 323 5 898 328 332 337 342 347 352 357 361 366 37i 5 899 376 381 386 390 395 400 405 410 415 419 5 N. Log 1 2 3 4 5 6 7 8 9 A P.P. 36 FIVE-PLACE LOGARITHMS. N. Log o 1 2 3 4 5 6 7 8 9 A P.P. 900 95 424 429 434 439 444 448 453 458 463 468 4 901 472 477 482 487 492 497 501 506 5" 5i6 5 902 521 525 530 535 540 545 5SO S54 559 564 5 903 569 574 578 583 588 593 598 602 607 612 5 904 617 622 626 631 636 641 646 650 655 660 5 905 665 670 674 679 684 689 694 698 703 708 5 906 7i3 718 722 727 732 737 742 746 751 756 5 907 761 766 770 775 780 785 789 794 799 804 5 908 809 813 818 823 828 832 837 842 847 852 4 909 856 861 866 871 875 880 885 890 895 899 5 910 904 909 914 918 923 928 933 938 942 947 5 911 952 957 9 6 i 966 971 976 980 98S 990 995 4 912 999 ♦04 K>9 +14 ♦19 ♦23 ♦28 ♦33 ♦38 ♦42 5 913 96 047 052 057 061 066 071 076 080 085 090 5 5 914 095 099 104 109 114 118 123 128 133 137 5 An Al 915 142 147 152 156 161 166 171 175 180 185 5 2-0 4-0 6'0 I 916 190 194 199 204 209 213 218 223 227 232 5 2 917 237 242 246 251 256 261 265 270 275 280 4 8-0 4 918 284 289 294 298 3o3 308 3i3 3i7 322 327 5 919 332 336 341 346 350 355 360 365 369 374 5 10 5 ; 920 379 384 388 393 398 402 407 412 417 421 5 921 426 431 435 440 445 450 454 459 464 468 5 922 473 478 483 487 492 497 5 °J 506 5ii 515 5 923 520 525 53o 534 539 544 548 553 558 562 5 924 567 572 577 581 586 591 595 600 605 609 5 925 614 619 624 628 633 638 642 647 652 656 S 926 661 666 670 675 680 685 689 694 699 703 5 927 708 713 717 722 727 731 736 74i 745 75° 5 928 755 759 764 769 774 778 783 788 792 797 5 929 802 806 811 816 820 825 830 834 839 844 4 930 848 853 858 862 867 872 876 881 886 890 5 931 895 900 904 909 914 918 923 928 932 937 5 932 942 946 951 956 960 965 970 974 979 984 4 4 933 934 988 993 997 »02 ♦07 ♦11 058 ♦16 063 ♦21 ♦25 ♦30 5 AnlAl 97 °35 039 044 049 053 067 072 077 4 2-5 1 935 081 086 090 095 100 104 109 114 118 123 5 5-0 7-6 2 3 936 128 132 137 142 146 151 155 160 165 169 S 10 4 937 i74 179 183 188 192 197 202 206 211 216 4 938 220 225 230 234 239 243 248 253 257 262 S 939 267, 271 276 280 285 290 294 299 3°4 308 5 940 313 317 322 327 33i 336 34° 345 35° 354 5 941 359 364 368 373 377 382 387 39i 396 400 5 942 405 410 414 419 424 428 433 437 442 447 4 943 451 456 460 465 470 474 479 483 488 493 4 944 497 502 506 511 5i6 520 525 529 534 539 4 945 % 3 548 552 557 562 566 571 575 580 58S 4 946 589 594 598 603 607 612 617 621 626 630 S 947 635 640 644 649 653 658 663 667 672 676 s 948 681 685 690 695 699 704 708 713 717 722 5 949 727 731 736 740 745 749 754 759 763 768 4 N. Log 1 2 3 4 5 6 7 8 9 A P.P. FIVE-PLACE LOGARITHMS. 37 N. Log o 1 2 3 4 5 6 7 8 9 A P.P. 950 97 772 777 782 786 791 795 800 804 809 813 5 951 818 823 827 832 836 841 845 850 855 859 5 952 864 868 873 877 882 886 891 896 900 90S 4 953 909 914 918 923 928 932 937 941 946 950 5 954 955 959 964 968 973 978 982 987 991 996 4 955 98 000 005 009 014 019 023 028 032 037 041 5 956 046 050 055 059 064 068 o73 078 082 087 4 957 091 096 100 105 109 114 118 123 127 132 5 958 137 141 146 150 155 159 164 168 173 177 5 959 182 186 191 195 200 204 209 214 218 223 4 960 227 232 236 241 245 250 254 259 263 268 4 961 272 277 281 286 290 295 299 3°4 308 313 5 962 318 322 327 33i 336 340 345 349 354 358 5 963 363 367 372 376 38i 385 390 394 399 403 5 6 964 408 412 417 421 426 43° 435 439 444 448 5 An Al 965 453 457 462 466 471 475 480 484 489 493 5 2-0 4'0 X 966 498 502 5°7 5» 5i6 520 525 529 534 538 5 60 3 967 543 547 552 556 56i 565 570 574 579 583 5 8-0 4 968 588 592 597 601 605 610 614 619 623 628 4 10 969 632 637 641 646 650 655 659 664 668 673 4 5 970 677 682 686 691 695 700 704 709 713 717 5 971 722 726 73i 735 740 744 749 753 758 762 5 972 767 771 776 780 784 789 793 798 802 807 4 973 811 816 820 825 829 834 838 843 847 851 5 974 856 860 865 869 874 878 S83 887 892 896 4 975 900 905 909 914 918 923 927 932 936 941 4 976 945 949 954 958 963 967 972 976 981 985 4 977 989 994 998 ♦03 ♦07 ♦ 12 ♦16 ♦21 ♦25 ♦29 5 978 99 034 038 043 047 052 056 061 065 069 074 4 979 078 083 087 092 096 100 ios 109 114 118 5 980 123 127 131 136 140 145 149 154 158 162 5 981 167 171 176 180 185 189 193 198 202 207 4 982 211 216 220 224 229 233 238 242 247 251 4 4 983 255 260 264 269 273 277 282 286 291 295 5 An Al 984 300 304 308 313 317 322 326 33° 335 339 5 2-5 1 i 985 344 348 352 357 361 366 37o 374 379 383 5 6'0 7-5 3 3 986 388 392 396 401 405 410 414 419 423 427 5 10 4 987 432 436 441 445 449 454 458 463 467 47i 5 988 476 480 484 489 493 498 502 506 5ii 515 5 989 520 524 528 533 537 542 546 550 555 559 5 990 564 568 572 577 58i 585 590 594 599 603 4 991 607 612 616 621 625 629 634 638 642 647 4 992 651 656 660 664 669 673 677 682 686 691 4 993 695 699 704 708 712 717 721 726 73° 734 5 994 739 743 747 752 756 760 765 769 774 778 4 995 782 787 791 795 800 804 808 813 817 822 4 996 826 830 835 839 843 848 852 856 861 865 5 997 870 874 878 883 887 891 896 900 904 909 4 998 913 917 922 926 93o 935 939 944 948 952 5 999 957 961 965 970 974 978 983 987 991 996 4 \T Log 1 2 3 4 5 6 7 8 9 A p.p. 3» FIVE-PLACE LOGARITHMS. N. Log o 1 2 3 4 5 6 7 8 9 A P.P. 1000 oo ooo 004 009 013 017 022 026 030 o35 039 4 1001 043 048 052 056 061 065 069 074 078 082 5 1002 087 091 095 100 104 108 "3 117 121 126 4 1003 130 134 139 143 147 152 156 160 165 169 4 1004 173 178 182 186 191 195 199 204 208 212 5 1005 217 221 225 230 234 238 243 247 251 255 5 1006 260 264 268 273 277 281 286 290 294 299 4 1007 303 307 312 316 320 325 329 333 337 342 4 1008 346 35° 355 359 363 368 372 376 381 385 4 1009 389 393 398 '402 406 411 415 419 424 428 4 1010 432 436 441 445 449 454 458 462 467 47i 4 1011 47S 479 484 488 492 497 501 505 5°9 514 4 1012 5i8 522 527 531 535 540 544 548 552 557 4 1013 561 565 570 574 578 582 587 59i 595 600 4 5 1014 604 608 612 617 621 625 629 634 638 642 5 An Al 1015 647 651 655 659 664 668 672 677 681 685 4 2-0 4-0 60 z 1016 689 694 698 702 706 711 715 719 724 728 4 3 1017 732 736 741 745 749 753 758 762 766 771 4 8-0 4 1018 77S 779 783 788 792 796 800 805 809 813 4 1019 817 822 826 830 834 839 843 847 852 856 4 10 5 1020 860 864 869 873 877 881 886 890 894 898 5 1021 903 907 911 9i5 920 924 928 932 937 941 4 1022 945 949 954 958 962 966 971 975 979 983 5 1023 988 992 996 ♦00 ♦05 ♦09 ♦ 13 ♦ 17 ♦22 ♦26 4 1024 01 030 034 038 043 047 051 055 060 064 068 4 1025 072 077 081 085 089 094 098 102 106 in 4 1026 «5 119 J23 127 132 136 140 144 149 153 4 1027 157 161 166 170 174 178 182 187 191 195 4 1028 199 204 208 212 216 220 225 229 233 237 5 1029 242 246 250 254 258 263 267 271 275 280 4 1030 284 288 292 296 301 305 309 3i3 317 322 4 1031 326 33° 334 339 343 347 35i 355 360 364 4 1032 368 372 376 381 385 389 393 397 402 406 4 4 1033 410 414 418 423 427 43i 435 439 444 448 4 An Al 1034 452 456 460 465 469 473 477 481 486 490 4 2-5 1 1035 494 498 5°2 507 5" 5i5 5'9 523 528 532 4 6 - 7-5 2 3 1036 536 540 544 549 553 557 561 565 569 574 4 10 4 1037 578 582 586 590 595 599 603 607 611 616 4 1038 620 624 628 632 636 641 645 649 653 657 5 1039 662 666 670 674 678 682 687 691 695 699 4 1040 703 708 712 716 720 724 728 733 737 741 4 1041 745 749 753 758 762 766 770 774 778 783 4 1042 787 791 795 799 803 808 812 816 820 824 4 1043 828 833 837 841 845 849 853 858 862 866 4 1044 870 874 878 883 887 891 895 899 903 907 5 1045 912 916 920 924 928 932 937 941 945 949 4 1046 953 957 961 966 970 974 978 982 986 991 4 1047 995 999 ♦03 ♦07 ♦ 11 ♦ 15 ♦20 ♦24 ♦28 ♦32 4 1048 02 036 040 044 049 o53 057 061 065 069 073 5 1049 078 082 086 090 094 098 102 107 in 115 4 N. Log 1 2 3 4 5 6 7 8 9 A P.P. FIVE-PLACE LOGARITHMS. 39 N. Log o 1 2 3 4 5 6 7 8 9 A P.P. 1050 02 119 123 127 131 135 140 144 148 152 156 4 1051 160 164 169 173 177 181 185 189 193 197 5 1052 202 206 210 214 218 222 226 230 235 239 4 1053 243 247 251 255 259 263 268 272 276 280 4 1054 284 288 292 296 301 305 309 313 317 321 4 1055 325 329 333 338 342 346 35° 354 358 362 4 1056 366 37i 375 379 383 387 391 395 399 403 4 1057 407 412 416 420 424 428 432 436 440 444 5 1058 449 453 457 461 465 469 473 477 481 485 5 1059 490 494 498 502 506 510 5H 518 522 526 5 All Al 1060 S3i 535 539 543 547 55i 555 559 563 567 5 20 1 1061 572 576 580 584 588 592 596 600 604 608 4 4-0 9 1062 612 617 621 625 629 633 637 641 645 649 4 6-0 3 1063 653 657 661 666 670 674 678 682 686 690 4 8'0 4 1064 694 698 702 706 710 7i5 719 723 727 731 4 10 S 1065 73S 739 743 747 75i 755 759 763 768 772 4 1066 776 780 784 78S 792 796 800 804 808 812 4 1067 816 821 825 829 833 837 841 845 849 853 4 1068 857 861 865 869 873 877 882 886 890 894 4 1069 898 902 906 910 914 918 922 926 930 934 4 1070 938 942 946 95i 955 959 963 967 971 975 4 1071 979 983 987 991 995 999 ♦°3 ♦07 ♦ 11 ♦IS 4 1072 03 019 024 028 032 036 040 044 048 052 056 4 1073 060 064 068 072 076 080 084 oSS 092 096 4 1074 1075 100 141 104 145 109 149 »3 153 117 157 121 161 125 165 129 169 133 173 137 177 4 4 An 4 Al 2-5 x 1076 181 185 1S9 193 197 201 205 209 214 218 4 6-0 2 1077 222 226 230 234 238 242 246 250 254 258 4 7'5 3 1078 262 266 270 274 278 282 286 290 294 298 4 10 4 1079 302 306 310 314 3i8 322 326 33° 334 338 4 1080 342 346 350 354 358 362 366 371 375 379 4 1081 383 387 391 395 399 403 407 411 415 419 4 1082 423 427 431 435 439 443 447 451 455 459 4 1083 463 467 47i 475 479 483 487 491 495 499 i 1084 5°3 507 5ii 515 519 523 527 53' 535 539 4 1085 543 547 551 555 559 563 567 571 575 579 4 1086 583 587 591 595 599 603 607 611 615 619 4 1087 623 627 631 635 639 643 647 6 5I 655 659 4 3 1088 663 667 671 675 679 683 687 691 695 699 4 An Al 1089 703 707 711 715 719 723 727 731 735 739 4 3-3 1 1090 743 747 7Si 755 759 763 767 771 775 778 4 67 10 2 3 1091 782 786 790 794 798 802 806 810 814 818 4 1092 822 826 830 834 838 842 846 850 854 858 4 1093 862 866 870 874 878 882 886 890 894 898 4 1094 902 906 910 914 918 922 926 93° 933 937 4 1095 941 945 949 953 957 961 965 969 973 977 4 1096 981 985 989 993 997 ♦01 ♦05 ♦09 ♦ 13 ♦ 17 4 1097 04 021 025 029 033 036 040 044 048 052 056 4 1098 060 064 068 072 076 080 084 088 092 096 4 1099 100 104 108 112 116 120 123 127 131 135 4 N. Log 1 2 3 4 5 6 7 8 9 A p. p. : 4 o FIVE-PLACE LOGARITHMS. N. Log o 1 2 3 4 5 6 7 8 9 A P.P. 1100 04 139 143 147 151 155 159 163 167 171 175 4 1101 179 183 187 191 195 198 202 206 210 214 4 1102 218 222 226 230 234 238 242 246 250 254 4 1103 258 261 265 269 273 277 281 285 289 293 4 1104 297 301 305 309 313 3i7 321 324 328 332 4 1105 336 340 344 348 352 356 360 364 368 372 4 1106 376 379 383 3S7 391 395 399 403 407 411 4 1107 415 419 423 427 43o 434 438 442 446 45o 4 1108 454 458 462 466 47o 474 477 481 48S 489 4 1109 493 497 50i 5°5 509 513 517 521 524 S28 4 1110 532 536 540 544 548 552 556 560 564 567 4 1111 S7i 575 579 583 587 591 595 599 603 607 3 1112 610 614 618 622 626 630 634 638 642 646 4 1113 650 653 657 661 665 669 673 677 681 685 4 1114 689 692 696 700 704 708 712 716 720 724 3 4 An Al 1115 727 73i 735 739 743 747 75i 755 759 763 3 2-5 6'0 1116 766 770 774 778 782 786 790 794 798 801 4 1117 805 809 813 817 821 825 829 833 836 840 4 7-6 3 1118 844 848 852 856 860 864 867 871 875 879 4 10 4 1119 883 887 891 895 899 902 906 910 914 918 4 1120 922 926 930 933 937 941 945 949 953 957 4 1121 961 964 968 972 976 980 984 988 992 995 4 1122 999 ♦03 ♦07 ♦ 11 ♦ 15 ♦ 19 ♦23 ♦26 ♦30 ♦34 4 1123 05 038 042 046 050 053 o57 061 065 069 073 4 1124 077 080 084 088 092 096 100 104 108 in 4 1125 "5 119 123 127 131 135 138 142 146 150 4 1126 154 158 162 165 169 173 177 181 185 189 3 1127 192 196 200 204 208 212 216 219 223 227 4 1128 231 2 35 239 242 246 250 254 258 262 266 3 1129 269 273 277 281 28S 289 292 296 300 304 4 1130 308 312 316 319 323 327 33i 335 339 342 4 1131 346 35o 354 358 362 365 369 373 377 38i 4 3 1132 385 388 392 396 400 404 408 411 415 419 4 An Al 1133 423 427 43i 434 438 442 446 450 454 457 4 1134 461 465 469 473 477 480 484 488 492 496 4 3 '3 67 1 2 1135 500 503 507 5ii 5i5 Si9 523 S26 53o 534 4 10 3 1136 538 542 545 549 553 557 561 565 568 572 4 1137 576 580 584 588 59i 595 599 603 607 610 4 1138 614 618 622 626 629 633 637 641 645 649 3 1139 652 656 660 664 668 671 675 679 683 687 3 1140 690 694 698 702 706 710 713 717 721 725 4 1141 729 732 73° 740 744 748 75i 755 759 763 4 1142 767 770 774 778 782 786 789 793 797 801 4 1143 805 808 812 816 820 824 827 831 835 839 4 1144 843 846 850 854 858 862 86 5 869 873 877 4 1145 881 884 888 892 896 900 903 907 911 915 3 1146 918 922 926 930 934 937 941 945 949 953 3 1147 956 960 964 968 971 975 979 983 987 990 4 1148 994 998 402 ♦06 ♦09 ♦ 13 ♦ 17 ♦21 ♦24 ♦28 4 1149 06 032 036 O4O 043 047 051 055 058 062 066 4 N. Log 1 2 3 4 5 6 7 8 9 A P.P. FIVE-PLACE LOGARITHMS. 4i N. Log 1 2 3 4 56789 A P.P. 1150 06 070 074 077 081 085 089 092 096 100 104 4 1151 108 in 115 119 123 126 130 134 138 141 4 1152 '45 149 1 S3 IS7 160 164 168 172 175 179 4 1153 183 187 190 194 198 202 206 209 213 217 4 1154 321 224 228 232 236 239 243 247 251 254 4 1155 258 262 266 269 273 277 281 285 288 292 4 1156 296 300 303 307 3" 315 318 322 326 330 3 1157 333 337 34i 345 348 352 356 360 363 367 4 1158 371 375 378 382 386 390 393 397 401 405 3 1159 408 412 416 420 423 427 43i 435 438 442 4 1160 446 450 453 457 461 465 468 472 476 479 4 1161 483 487 49i 494 498 502 506 509 513 5:7 4 1162 521 524 528 532 536 539 543 547 55 1 554 4 1163 558 562 565 569 573 577 580 584 588 592 3 1164 595 599 603 606 610 614 618 621 625 629 4 4 An Al 1165 633 63 6 640 644 648 651 655 659 662 666 4 1166 670 674 677 681 685 688 692 696 700 703 4 2'5 6-0 X 1167 707 711 715 718 722 726 729 733 737 741 3 7-5 3 1168 744 748 752 755 759 763 767 770 774 778 3 10 4 1169 78i 785 789 793 796 800 804 807 811 815 4 1170 819 822 826 830 833 837 841 845 848 852 4 1171 856 859 863 867 871 874 878 882 885 889 4 1172 893 896 900 904 908 911 915 919 922 926 4 1173 93° 934 937 94i 945 948 952 956 959 963 4 1174 967 971 974 978 982 985 989 993 996 ♦00 4 1175 07 004 007 on 015 019 022 026 030 033 037 4 1176 041 044 048 052 056 °59 063 067 070 074 4 1177 078 081 085 089 092 096 100 103 107 in 4 1178 115 118 122 126 129 133 137 140 144 148 3 1179 151 155 159 162 166 170 173 177 181 185 3 1180 188 192 196 199 203 207 210 214 218 221 4 1181 225 229 232 236 240 243 247 251 254 258 4 3 1182 262 265 269 273 276 280 284 287 291 295 3 An Al 1183 298 302 306 309 313 317 320 324 328 332 3 1184 335 339 343 346 350 354 357 361 365 368 4 3 - 3 6-7 I 2 1185 372 375 379 383 386 390 394 397 401 405 3 10 3 1186 408 412 416 419 423 427 43° 434 438 441 4 1187 445 449 452 456 460 463 467 471 474 478 4 1188 482 485 489 493 496 500 504 507 511 515 3 1189 518 522 525 529 533 536 540 544 547 551 4 1190 555 558 562 566 569 573 577 580 584 588 3 1191 59i 595 598 602 606 609 613 617 620 624 4 1192 628 631 635 639 642 646 649 653 657 660 4 1193 664 668 671 675 679 682 686 690 693 697 3 1194 700 704 708 711 715 719 722 726 730 733 4 1195 737 74° 744 748 75i 755 759 762 766 769 4 1196 773 777 78o 784 788 791 795 799 802 806 3 1197 809 813 817 820 824 828 831 835 838 842 4 1198 846 849 853 857 860 864 867 871 875 878 4 j 1199 882 886 889 893 896 900 904 907 911 915 3 1 N# Log 0123 4 56789 A P.P. I 42 FIVE-PLACE LOGARITHMS. N. Log 01234 56789 A P.P. 1200 1201 1202 1203 1204 07 918 922 925 929 933 954 958 962 965 969 990 994 998 k>i ♦os 08 027 030 034 037 041 063 066 070 073 077 936 940 943 947 951 972 976 980 983 987 ♦09 m ^16 419 »23 045 048 052 055 059 081 084 088 091 095 3 3 4 4 4 1205 1206 1207 1208 1209 099 102 106 no 113 135 138 142 146 149 171 174 178 182 185 207 210 214 217 221 243 246 250 253 257 117 120 124 128 131 153 156 160 164 167 189 192 196 200 203 225 228 232 235 239 261 264 268 271 275 4 4 4 4 4 1210 1211 1212 1213 1214 1215 1216 1217 1218 1219 279 282 286 289 293 314 318 322 325 329 35° 354 357 361 365 386 390 393 397 400 422 425 429 433 436 458 461 465 468 472 493 497 500 504 5o8 529 533 536 540 543 565 568 572 575 579 600 604 607 611 615 296 300 304 307 311 332 336 340 343 347 368 372 375 379 382 404 408 411 415 418 440 443 447 450 454 3 3 4 4 4 An 4 Al 475 479 483 486 490 511 515 518 522 525 547 55° 554 558 S61 583 586 590 593 597 618 622 625 629 632 3 4 4 3 4 2-5 5-0 7-5 10 1 a 3 4 1220 1221 1222 1223 1224 636 640 643 647 650 672 675 679 682 686 707 711 714 718 721 743 746 75° 753 757 778 782 785 789 792 654 657 661 664 668 689 693 696 700 704 725 728 732 736 739 760 764 767 771 775 796 799 803 807 810 4 3 4 3 4 1225 1226 1227 1228 1229 814 817 821 824 828 849 853 856 860 863 884 888 892 895 899 920 923 927 930 934 955 959 962 966 969 831 835 838 842 846 867 870 874 877 881 902 906 909 913 916 938 94i 945 948 952 973 976 980 983 987 3 3 4 3 4 1230 1231 1232 1233 1234 991 994 998 »oi fos 09 026 029 033 036 040 061 065 068 072 075 096 100 103 107 no 132 135 139 142 146 «0S »I2 tI5 «I9 »22 O43 047 05O 054 O58 079 082 086 0§9 093 114 117 121 124 I2§ 149 153 156 ISO I63 4 3 3 4 4 An 3 Al 3-3 6-7 I 2 1235 1236 1237 1238 1239 167 170 174 177 181 202 205 209 212 216 237 240 244 248 251 272 276 279 283 286 307 311 314 318 321 184 188 191 I95 I98 219 223 226 230 233 255 258 262 265 269 290 293 297 300 304 325 328 332 335 339 4 4 3 3 3 10 3 1240 1241 1242 1243 1244 342 346 349 353 356 377 381 384 388 39i 412 416 419 423 426 447 45i 454 458 461 482 486 489 493 496 360 363 367 370 374 395 398 402 405 409 43° 433 437 44° 444 465 468 472 475 479 499 5°3 5°6 510 513 3 3 3 3 4 1245 1246 1247 1248 1249 517 520 524 527 531 552 555 559 562 566 587 590 594 597 601 621 625 628 632 635 656 660 663 , 667 670 534 538 54i 545 548 569 573 576 580 583 604 608 611 614 618 639 642 646 649 653 674 677 681 684 688 4 4 3 3 3 N. ( Log 01234 56789 A P.P. FIVE-PLACE LOGARITHMS. 43 N. Log o 1 2 3 4 5 6 7 8 9 A P.P. 1250 09 691 694 698 701 7o5 708 712 715 719 722 4 1251 726 729 733 736 740 743 747 75° 753 757 3 1252 760 764 767 771 774 778 781 785 788 792 3 1253 79S 799 802 806 809 812 816 819 823 826 4 1254 830 833 837 840 844 847 851 854 857 861 3 1255 864 868 871 875 878 882 885 889 892 896 3 1256 899 902 906 909 913 916 920 923 927 930 4 1257 934 937 940 944 947 95i 954 958 961 965 3 1258 968 972 975 978 982 985 989 992 996 999 4 1259 10 003 006 009 013 016 020 023 027 030 034 3 1260 037 041 044 047 OSI 054 058 061 065 068 4 1261 072 075 078 082 085 089 092 096 099 102 4 1262 106 109 "3 116 120 123 127 130 133 137 3 1263 140 144 147 151 154 158 161 164 168 171 4 1264 I7S 178 182 185 188 192 195 199 202 206 3 4 An Al 1265 209 212 216 219 223 226 230 233 237 240 3 2*5 — : 1266 243 247 250 254 257 261 264 267 271 274 4 5-0 2 1267 278 281 285 288 291 295 298 302 3°5 3°9 3 7-6 3 1268 312 315 3i9 322 326 329 332 336 339 343 3 10 4 1269 346 35° 353 35° 360 363 367 37o 374 377 3 1270 380 384 387 391 394 397 401 404 408 411 4 1271 415 418 421 425 428 432 435 438 442 445 4 1272 449 452 456 459 462 466 469 473 476 479 4 1273 483 486 490 493 496 500 503 5o7 510 5'4 3 1274 517 520 524 527 53i 534 537 54i 544 548 3 1275 SSi 554 558 56i 565 568 57i 575 578 582 3 1276 S8S 588 592 595 599 602 605 609 612 616 3 1277 619 622 626 629 633 636 639 643 646 650 3 1278 653 656 660 663 667 670 673 677 680 684 3 1279 687 690 694 697 701 704 707 711 714 718 3 1280 721 724 728 73i 735 738 74i 745 748 752 3 1281 755 758 762 765 768 772 775 779 782 785 4 3 1282 789 792 796 799 802 806 809 813 816 819 4 An AI 1283 823 826 829 833 836 840 843 846 850 853 4 S'3 6-7 in 1284 857 860 863 867 870 873 877 880 884 887 3 2 1285 890 894 897 900 904 907 911 914 917 921 3 U) j 1286 924 927 931 934 938 941 944 948 951 954 4 1287 958 961 965 968 971 975 978 981 9S5 988 4 1288 992 995 998 402 ♦°5 ♦08 ♦ 12 ♦ 15 ♦ 19 +22 3 1289 11 025 029 032 035 039 042 046 049 052 056 3 1290 059 062 066 069 072 076 079 083 086 089 4 1291 093 096 099 103 106 109 "3 116 120 123 3 1292 126 130 133 136 140 143 146 150 153 156 4 1293 160 163 167 170 « 73 177 180 183 187 190 3 1294 193 197 200 203 207 210 214 217 220 224 3 1295 227 230 234 237 240 244 247 250 254 257 4 1296 261 264 267 271 274 277 281 284 287 291 3 1297 294 297 301 304 307 3" 314 3i7 321 324 3 1298 327 33i 334 338 34i 344 348 35i 354 358 3 1299 361 3 6 4 368 371 374 378 38i 384 388 39i 3 N. Log 1 2 3 4 S 6 7 8 9 A P.P. ,44 FIVE-PLACE LOGARITHMS. N. Log o 1 2 3 4 5 6 7 8 9 A P.P. 1300 « 394 398 401 404 408 411 414 418 421 424 4 1301 428 43i 434 438 441 444 448 45i 454 458 3 1302 461 464 468 47i 474 478 481 484 488 491 3 1303 494 498 Soi 5°4 508 5» 5'4 5i8 521 524 4 1304 S28 53i 534 538 54i 544 548 55i 554 558 3 1305 56l 564 568 57i 574 578 58i 584 588 59i 3 1306 594 598 601 604 608 611 614 618 621 624 4 1307 628 631 634 638 641 644 647 651 654 657 4 1308 661 664 667 671 674 677 681 684 687 691 3 1309 694 697 701 704 707 711 7i4 717 720 724 3 1310 727 73° 734 737 740 744 747 75° 754 757 3 1311 760 764 767 770 774 777 780 783 787 790 3 1312 793 797 800 803 807 810 813 817 820 823 3 1313 826 830 833 836 840 843 846 850 853 856 4 1314 860 863 866 869 873 876 879 883 886 889 4 4 An Al 1315 893 896 899 902 906 909 912 916 919 922 4 2*5 1316 926 929 932 935 939 942 945 949 952 955 4 5-0 2 1317 959 962 965 968 972 975 978 982 985 988 4 7-6 3 1318 992 995 998 ♦01 ♦OS ♦08 ♦ u ♦ 15 ♦ 18 ♦21 3 10 4 1319 12 024 028 031 034 038 041 044 048 051 054 3 1320 057 061 064 067 071 074 o77 080 084 087 3 1321 090 094 097 100 103 107 no 113 117 120 3 1322 123 126 130 133 136 140 143 146 149 153 3 1323 156 159 163 166 169 172 176 179 182 186 3 1324 189 192 195 199 202 205 208 212 215 218 4 1325 222 225 228 231 235 238 241 245 248 251 3 1326 254 258 261 264 267 271 274 277 281 284 3 1327 287 290 294 297 300 303 307 310 313 317 3 1328 320 323 326 33° 333 336 339 343 346 349 3 1329 352 356 359 362 366 369 372 375 379 382 3 1330 385 388 392 395 398 401 405 408 411 415 3 1331 418 421 424 428 43i 434 437 441 444 447 3 3 1332 450 454 457 460 463 467 470 473 476 480 3 An Al 1333 483 486 490 493 496 499 S03 506 5°9 512 4 1334 516 519 522 525 529 532 535 538 542 545 3 3 '3 6-7 1 2 1335 548 55i 555 558 56i 564 568 571 574 577 4 10 3 1336 58i 584 587 59° 594 597 600 603 607 610 3 1337 613 616 620 623 626 629 633 636 639 642 4 1338 646 649 652 655 659 662 665 668 672 675 3 1339 678 681 685 688 691 694 698 701 704 707 3 1340 710 714 717 720 723 727 730 733 736 740 3 1341 743 746 749 753 756 759 762 766 769 772 3 1342 775 778 782 785 788 791 795 798 801 804 4 1343 808 811 814 817 821 824 827 830 833 837 3 1344 840 843 846 850 853 856 859 863 866 869 3 1345 872 875 879 882 885 888 892 895 898 901 4 1346 905 908 911 914 917 921 924 927 930 934 3 1347 937 940 943 946 95o 953 956 959 963 966 3 1348 969 972 975 079 982 985 988 992 995 998 3 1 1349 13 001 004 008 on 014 017 021 024 027 030 3 1 N. Log 1 2 3 4 5 6 7 8 9 A P.P. FIVE-PLACE LOGARITHMS. 45 N. Log 01234 56789 A P.P. 1350 13 °33 037 040 043 046 049 053 056 059 062 4 1351 066 069 072 075 078 082 085 088 091 094 4 1352 098 101 104 107 III 114 117 120 123 127 3 1353 130 133 136 139 143 146 149 152 155 159 3 1354 162 165 168 171 175 178 181 184 188 191 3 1355 194 197 200 204 207 210 213 216 220 223 3 1356 226 229 232 236 239 242 245 248 252 255 3 1357 258 261 264 268 271 274 277 280 284 287 3 135S 290 293 296 300 303 306 309 312 316 319 3 1359 322 325 328 332 33s 338 341 344 348 351 3 1360 354 357 360 363 367 37o 373 376 379 383 3 1361 386 389 392 39s 399 402 405 408 411 415 3 1362 418 421 424 427 430 434 437 440 443 446 4 1363 45° 453 456 459 462 466 469 472 475 478 3 1364 481 485 488 491 494 497 5oi 504 5°7 510 3 4 An Al 1365 513 516 520 523 526 529 532 536 539 542 3 1366 545 548 55i 555 558 561 564 567 570 574 3 2*. 6-0 z 2 1367 577 580 583 586 59o 593 596 599 602 605 4 7-6 3 1368 609 612 615 618 621 624 628 631 634 637 3 10 4 1369 640 644 647 650 653 656 659 663 666 669 3 1370 672 675 678 682 685 688 691 694 697 701 3 1371 704 707 710 713 716 720 723 726 729 732 3 1372 735 739 742 745 748 75i 754 758 761 764 3 1373 767 770 773 777 780 783 786 789 792 796 3 1374 799 802 805 808 811 814 818 821 824 827 3 1375 830 833 837 840 843 846 849 852 856 859 3 1376 862 865 868 871 874 878 881 884 887 890 3 1377 893 897 900 903 906 909 912 915 919 922 3 1378 925 928 931 934 938 941 944 947 950 953 3 1379 956 960 963 966 969 972 975 978 982 985 3 1380 988 991 994 997 ♦00 ♦04 K>7 410 «I3 *i6 3 1381 14 019 023 026 029 032 035 038 041 045 048 3 3 1382 051 054 057 060 063 067 070 073 076 079 3 An Al 1383 082 085 088 092 095 098 101 104 107 no 4 1384 114 117 120 123 126 129 132 136 139 142 3 3'3 6-7 z 2 1385 14S 148 151 154 158 161 164 167 170 173 3 10 3 1386 176 179 183 186 189 192 195 198 201 205 3 1 1387 208 211 214 217 220 223 226 230 233 236 3 1 1388 239 242 245 248 251 255 258 261 264 267 3 1389 270 273 276 280 283 286 289 292 295 298 3 1390 301 305 308 311 314 317 320 323 326 330 3 1391 333 336 339 342 345 348 35i 355 358 361 3 1392 364 367 370 373 376 380 383 386 389 392 3 1393 395 398 4 01 4°4 408 411 414 417 420 423 3 1394 426 429 433 436 439 442 445 448 451 454 3 1395 457 461 464 467 47o 473 476 479 482 485 4 1396 489 492 495 498 501 504 507 510 513 517 3 1397 520 523 526 529 532 535 538 54i 545 548 3 1398 55i 554 557 560 563 566 569 572 576 579 3 1399 582 585 588 s 9I 594 597 600 603 607 610 3 N. Log 01234 56789 A P.P. | 4 6 FIVE-PLACE LOGARITHMS. N. Log o 1 2 3 4 5 6 7 8 9 A P.P. 1400 14 613 616 619 622 625 628 631 635 638 641 3 1401 644 647 650 653 656 659 662 666 669 672 3 1402 675 678 681 684 687 690 693 696 700 703 3 1403 706 709 712 715 718 721 724 727 73i 734 3 1404 737 740 743 746 749 752 755 758 761 765 3 1405 768 771 774 777 780 783 786 789 792 795 4 1406 799 802 805 808 811 814 817 820 823 826 3 1407 829 832 836 839 842 845 848 851 854 857 3 1408 860 863 866 S70 873 876 879 882 885 888 3 1409 891 894 897 900 903 907 910 913 916 919 3 1410 922 925 928 93i 934 937 940 943 947 950 3 1411 953 956 959 962 965 968 971 974 977 980 3 1412 983 987 990 993 996 999 ♦02 ♦05 ♦08 ♦ 11 3 1413 15 014 017 020 023 027 030 033 036 039 042 3 1414 o4S 048 051 054 057 060 063 066 070 073 3 4 An Al 1415 076 079 082 0S5 088 091 094 097 100 103 3 2 - 6 1416 106 109 112 116 "9 122 125 128 131 134 3 6-0 2 1417 137 140 143 146 149 152 155 158 161 165 3 7-5 3 1418 168 171 174 177 180 183 186 189 192 195 3 10 4 1419 198 201 204 207 210 214 217 220 223 226 3 1420 229 232 235 238 241 244 247 250 253 256 3 1421 259 262 266 269 272 275 278 281 284 287 3 1422 290 293 296 299 302 3o5 308 3" 314 3i7 3 1423 320 324 3 2 7 33° 333 336 339 342 345 348 3 1424 35i 354 357 360 363 366 369 372 375 378 3 1425 38i 385 388 391 394 397 400 403 406 409 3 1426 412 415 418 421 424 427 43° 433 436 439 3 1427 442 445 448 452 455 458 461 464 467 470 3 1428 473 476 479 482 485 488 491 494 497 500 3 1429 S03 506 5°9 512 515 5i8 521 524 528 53i 3 1430 534 537 540 543 546 549 552 555 558 56i 3 1431 564 567 570 573 576 579 582 58S 588 59i 3 3 1432 1433 594 625 597 628 600 631 603 634 606 637 609 640 612 643 616 646 619 649 622 652 3 An Al 3 1434 655 658 661 664 667 670 673 676 679 682 3 3*3 61 z 2 1435 685 688 691 694 697 700 703 706 709 712 3 10 3 1436 715 718 721 725 728 73i 734 737 740 743 3 1437 746 749 752 755 758 761 764 767 770 773 3 1438 776 779 782 785 788 791 794 797 800 803 3 1439 806 809 812 815 818 821 824 827 830 833 3 1440 836 839 842 845 848 851 854 857 860 863 3 1441 866 869 872 875 878 881 884 887 891 894 3 1442 897 900 903 906 909 912 915 918 921 924 3 1443 927 930 933 936 939 942 945 948 951 954 3 1444 957 960 963 966 969 972 975 978 981 984 3 1445 987 990 993 996 999 ♦02 ♦05 ♦08 ♦11 ♦ 14 3 1446 16 017 020 023 026 029 032 035 038 041 044 3 1447 047 050 053 056 059 062 065 068 071 074 3 1448 077 080 083 086 089 092 095 098 IOI 104 3 1449 107 no «3 116 119 122 125 128 131 134 3 N. Log 1 2 3 4 5 6 7 8 9 A P.P. FIVE-PLACE LOGARITHMS. 47 N. Log o 1 2 3 4 5 6 7 8 9 A P.P. 1460 16 137 140 143 146 149 152 155 158 161 164 3 1451 167 170 173 176 179 182 185 188 191 194 3 1452 197 200 203 206 209 212 215 218 221 224 3 1453 227 230 233 236 239 242 244 247 250 253 3 1454 256 259 262 265 268 271 274 277 280 283 3 1455 286 289 292 295 298 301 304 307 310 313 3 1456 316 319 322 325 328 331 334 337 340 343 3 1457 346 349 352 355 358 361 364 367 37o 373 3 1458 376 379 382 385 388 391 394 397 400 403 3 1459 406 409 411 414 417 420 423 426 429 432 3 1460 435 438 441 444 447 450 453 456 459 462 3 1461 46S 468 471 474 477 480 483 486 489 492 3 1462 495 498 Soi 504 507 5io 513 516 5i8 521 3 1463 524 527 53° 533 536 539 542 545 548 55i 3 1464 554 557 560 563 566 569 572 575 578 581 3 3 1465 584 587 59o 593 596 599 602 605 607 610 3 An Al 1466 6i3 616 619 622 625 628 631 634 637 640 3 3-3 X 1467 643 646 649 652 655 658 661 664 667 670 3 6-7 3 1468 673 676 679 681 684 687 690 693 696 699 3 10 3 1469 702 705 708 711 714 717 720 723 726 729 3 1470 732 735 738 74i 744 747 749 752 755 758 3 1471 761 764 767 770 773 776 779 7S2 785 788 3 1472 791 794 797 800 803 806 808 811 814 817 3 1473 , 820 823 826 829 832 835 838 841 844 847 3 1474 850 853 856 859 862 864 867 870 873 876 3 1475 879 882 885 888 891 894 897 900 903 906 3 1476 909 912 915 917 920 923 926 929 932 935 3 1477 938 941 944 947 95° 953 956 959 962 965 2 1478 967 970 973 976 979 982 985 988 991 994 3 1479 997 ♦00 ♦03 ♦06 ♦09 ♦ 11 ♦ 14 ♦ '7 ♦20 ♦23 3 1480 17 026 029 032 035 038 041 044 047 050 o53 3 1481 056 058 061 064 067 070 073 076 079 082 3 1482 085 088 091 094 097 099 102 105 108 in 3 2 1483 1484 114 143 117 146 120 149 123 152 126 155 129 158 132 161 135 164 138 167 140 170 3 3 An Al 5-0 X 1485 173 176 178 181 184 187 190 193 196 199 3 10 2 1486 202 205 208 211 214 216 219 222 225 228 3 1487 231 234 237 240 243 246 249 252 254 257 3 1488 260 263 266 269 272 275 278 281 284 287 2 1489 289 292 295 298 301 3°4 3°7 310 313 316 3 1490 319 322 324 327 33° 333 336 339 342 345 3 1491 348 351 354 357 359 362 365 368 371 374 3 1492 377 380 383 386 389 391 394 397 400 403 3 1493 406 409 412 4i5 418 421 423 426 429 432 3 1494 435 438 441 444 447 45° 452 455 458 461 3 1495 464 467 470 473 476 479 482 484 487 490 3 1496 493 496 499 5°2 505 508 5il 5i3 516 519 3 1497 522 525 528 53i 534 537 54° 542 545 548 3 1498 55i 554 557 560 563 566 569 57i 574 577 3 1499 580 583 586 589 592 595 598 600 603 606 3 N. Log 1 2 3 ' 4 5 6 7 8 9 A P.P. 48 TABLE giving the Decimal- Values F of all Integer- Ratios x :y UP TO F = 0.5 AND y = 60. F. x:y. F. x :y. F. x :y. F. x :y. .016 667 1 :6o .040000 1:25 .075 000 3:40 .ill 11 1:9 .016 949 i:59 .040 816 2:49 .075 472 4 = 53 .113 21 6 = 53 .017 241 1:58 .041 667 I : 24 .076 923 1:13 .11364 5:44 .017 544 1=57 •042 SS3 2:47 .078 431 4:51 .114 29 4:35 .017 857 1:56 .043 478 I :23 .078 947 3:38 ■"5 38 3:26 .018 182 i:55 .044 444 2=45 .080 000 2 :25 .11628 5=43 .018 519 i:54 .045 455 I :22 .081 081 3:37 .11667 7:60 .018 868 i=S3 .046 512 2:43 .081 633 4:49 .11765 2 : 17 .019 231 1:52 .047 619 1 :2i •083 333 I : 12 .11864 7:59 .019 608 1:51 .048 780 2 141 .084 746 5 = 59 .11905 5:42 .020000 1:50 .050 000 1 : 20 .085 106 4 = 47 .12000 3:25 .020 408 1 :49 .050 847 3 = 59 .085 714 3 = 35 . 120 69 7:58 .020 833 1 :48 .051 282 2:39 .086 207 5:58 .121 21 4 = 33 .021 277 1:47 .051 724 3:58 .086 957 2:23 .12195 5:41 .021 739 1 146 .052 632 1:19 .087 719 5=57 .12245 6:49 .022 222 1 =45 •o53 571 3:56 .088 235 3 = 34 .122 81 7 = 57 .022 727 1 :44 •054 054 2:37 .088 889 4:45 . 125 00 1:8 .023 256 1:43 .054 545 3 = 55 .089 286 5:56 .127 27 7 = 55 .023 810 I : 42 •°55 556 1:18 .090 909 1 : II .127 66 6:47 .024 390 I 141 .056 604 3:53 .092 593 5:54 .128 21 5 = 39 .025 000 I :40 •057 143 2:35 .093 023 4 = 43 • 129 03 4:31 .025 641 I :39 .057 692 3:52 •093 75o 3:32 .12963 7:54 .026 316 1:38 .058 824 1:17 .094 340 5 = 53 ■13043 3:23 .027 027 1=37 .060 000 3:5o .095 238 2 : 21 .131 58 5:38 .027 778 I 136 .060 606 2:33 .096 154 5:52 .132 08 7:53 .028 571 1=35 .061 224 3:49 .096 774 3:31 •133 33 2:15 .029 412 1=34 .062 500 1 :l6 .097 561 4:41 .13462 7:52 .030 303 1 =33 .063 830 3:47 .098 039 5=51 •135 14 5:37 .031 250 1 :32 .064 516 2:31 .10000 I : 10 •135 59 8:59 .032 258 1:31 .065 217 3:46 .101 69 6 = 59 ■ 13636 3 = 22 •033 333 1 :30 .066 667 1:15 . 102 04 5=49 •13725 7 = 5i .033 898 2:59 .067 797 4 = 59 . 102 56 4 = 39 •137 93 4:29 •034 483 1 129 .068 182 3:44 •10345 3:29 .13889 5:36 .035 088 2:57 .068 966 2 : 29 .10417 5:48 •139 53 6:43 •035 714 1 :28 .069 767 3=43 . 105 26 2 : 19 .140 00 7:50 .036 364 2:55 .070 175 4 = 57 . 106 38 5=47 •140 35 8:57 • 037 037 1 : 27 .071 429 1 = 14 .107 14 3:28 . 142 86 1:7 .037 736 2:53 .072 727 4:55 .108 u 4:37 •H5 45 8 = 55 .038 462 1 : 26 .073 171 3:41 .108 70 5:46 ■14583 7:48 .039 216 2 = 51 .074 074 2 127 . 109 09 6:55 .14634 6 :4i DECIMAL-VALUES. 49 F. x:y. F. x:y. F. x :y. F. x:y. . 147 06 5:34 .191 49 9=47 .236 36 13:55 .282 05 11:39 .148 is 4:27 .19231 5 :26 .236 84 9:38 .282 61 13:46 .14894 7=47 . 192 98 ":57 .237 29 14:59 .283 02 15:53 .15000 3 :20 •i93 55 6:31 .238 10 5:21 .285 71 2:7 .15094 8 = 53 .19444 7:36 •239 13 II : 46 .288 14 17:59 .151 52 5 = 33 .195 12 8:41 .24000 6 125 .288 46 15:52 .152 17 7:46 • 19565 9:46 • 240 74 13:54 .288 89 13:45 •152 54 9:59 .196 08 10 :5I .241 38 7:29 .289 47 11:38 •15385 2:13 .19643 11 :56 .242 42 8 = 33 .290 32 9:31 •155 17 9:58 .20000 1=5 .243 24 9 = 37 .290 91 16:55 .15556 7=45 •203 39 12:59 .243 90 10 141 .291 67 7:24 •IS625 5:32 .203 70 11 :54 .24444 11:45 .292 68 12 : 41 .15686 8:51 .204 08 10:49 .244 90 12:49 .293 10 17:58 • 15789 3:19 .204 55 9:44 .245 28 13:53 .294 12 5:17 .15909 7:44 .205 13 8:39 .245 61 H:57 •295 45 13:44 .16000 4:25 .205 88 7:34 .25000 1 :4 .296 30 8:27 .160 71 9:56 .206 90 6 : 29 .254 24 15:59 ■297 30 11 :37 .161 29 5=31 •207 55 11 =53 •254 55 H:55 •297 87 14:47 .162 16 6:37 .208 33 5=24 •254 9° 13:51 .298 25 17:57 .162 79 7=43 .209 30 9 = 43 •255 32 12:47 .30000 3: 10 .16327 8:49 .210 53 4:19 •255 81 11 :43 .301 89 16:53 .16364 9 = 55 .211 54 11 : 52 .25641 10:39 ■ 302 33 13:43 .16667 1 :6 .212 12 7:33 •257 14 9 = 35 • 303 03 10:33 .16949 10:59 .212 77 10:47 .258 06 8:31 •303 57 17:56 .16981 9 = 53 .214 29 3:i4 .258 62 15:58 •304 35 7:23 .170 21 8:47 .215 69 11:51 .259 26 7:27 .30508 18:59 • 170 73 7:41 .216 22 8:37 .260 00 13:50 .305 56 11 :36 •171 43 6 = 35 .216 67 13 :6o .260 87 6:23 .306 12 15:49 .17241 5 = 29 •217 39 5:23 .261 90 11 :42 •307 69 4:i3 .17308 9:52 .218 18 12:55 .263 16 5:i9 .309 09 17:55 ■173 9i 4:23 .218 75 7:32 .264 15 14:53 •309 52 13:42 .17500 7:40 .21951 9:41 .264 71 9:34 •3io 34 9:29 •175 44 10:57 .220 00 II :50 .265 31 13:49 .3" 11 14:45 .17647 3:17 .220 34 13 = 59 .266 67 4:15 .312 50 5:16 .177 78 8:45 .222 22 2:9 .267 86 15:56 •313 73 16:51 .17857 5:28 .224 14 13:58 .268 29 II :4I •31429 ":35 • 179 49 7 = 39 .224 49 11 :49 .269 23 7 :26 .31481 17:54 .180 00 9:50 .225 00 9:40 .270 27 10:37 •315 79 6 : 19 .181 82 2 : 11 .225 81 7:31 .270 83 13:48 .316 67 19 :6o •18333 11 :6o .226 42 12:53 .271 19 16:59 •317 07 13:41 .18367 9:49 .227 27 5 :22 .272 73 3 = 11 .318 18 7 :22 .184 21 7:38 .228 07 13:57 ■274 51 14:51 •319 15 15:47 .185 19 5:27 .228 57 8:35 .275 00 11 :40 .32000 8 :25 .18605 8:43 .229 17 11 :48 •275 86 8:29 .320 75 17:53 .186 44 11 :59 .230 77 3:13 .276 60 13:47 • 321 43 9 :28 .18750 3:16 .232 14 13:56 .277 78 5:18 .322 03 .322 58 19 = 59 .18868 10:53 .232 56 10:43 • 279 07 12:43 10:31 .189 19 7 = 37 •233 33 7:30 .280 00 7:25 •323 53 ":34 .18966 11:58 .234 04 II :47 .280 70 16:57 • 324 32 12:37 .190 48 4 : 21 •235 29 4:17 .281 25 9:32 .325 00 13:40 5° DECIMAL-VALUES. F. x:y. F. x:y. F. x:y. F. x :y. •325 58 14:43 .372 88 22:59 ■419 35 13:31 .464 29 13:28 .326 09 15:46 •375 00 3:8 .42000 21 : 50 .465 12 20:43 •326 S3 16:49 •377 36 20:53 .421 05 8:19 .465 52 27:58 .326 92 17:52 ■377 78 17:45 .422 22 19:45 .466 67 7:i5 .327 27 18:55 .378 38 14:37 .423 08 II : 26 .468 09 22:47 • 327 59 19:58 •379 3i II : 29 •423 73 25:59 .468 75 15 = 32 •333 33 1 = 3 .38000 19:50 .424 24 H:33 ■469 39 23:49 • 338 98 20:59 .380 95 8:21 .42500 17:40 •470 59 8:17 •339 29 19:56 .381 82 21:55 .425 53 20:47 •471 70 25:53 •339 62 18:53 ■382 35 13:34 •425 93 23:54 .472 22 17:36 .34000 17:50 .382 98 18:47 .428 57 3:7 •472 73 26:55 •340 43 16:47 •383 33 23 :6o •431 03 25:58 .473 68 9:19 •34° 9i 15=44 .38462 5:i3 •43i 37 22:51 •474 58 28:59 .341 46 14:41 .385 96 22:57 .431 82 19:44 .47500 19:40 .342 11 13:38 .386 36 17:44 •432 43 16:37 .476 19 10:21 .342 86 12:35 .387 10 12:31 •433 33 13:30 •477 27 21 :44 •343 75 11 :32 .387 76 19:49 •433 96 23:53 .478 26 11 123 •344 83 10 :29 .388 89 7:18 •434 78 10:23 •479 17 23:48 •345 45 19:55 .38983 23:59 •435 90 17:39 .48000 12 :25 •346 15 9 : 26 .390 24 16 :4I •436 36 24:55 .480 77 25:52 •346 94 17:49 •391 30 9:23 ■437 50 7 : 16 .481 48 13:27 •347 83 8:23 •392 16 20:51 .438 60 25:57 .482 14 27:56 .348 84 15:43 .392 86 11 :28 ■ 439 02 18:41 .482 76 14:29 .35000 7 :20 •393 94 13:33 .440 00 11 :25 •483 33 29 :6o .350 88 20:57 • 394 74 15:38 .44068 26:59 .483 87 15:31 •351 35 13:37 • 395 35 17:43 .441 18 15:34 .484 85 16:33 •351 85 19:54 ■395 83 19:48 .441 86 19:43 • 485 7i 17:35 •352 94 6:17 ■ 39623 21:53 .44231 23:52 .486 49 18:37 •354 17 17:48 ■396 55 23:58 •444 44 4:9 .487 18 19:39 • 354 84 11 :3' .400 00 2:5 .44643 25:56 .487 80 20 141 •355 56 16:45 ■403 5i 23:57 .446 81 21 :47 ■488 37 21 :43 •355 93 21 : 59 .403 85 21 : 52 •447 37 17:38 .488 89 22:45 •357 14 5:14 .404 26 19:47 .44828 13:29 .489 36 23:47 ■358 49 19:53 .404 76 17:42 .44898 22:49 .489 80 24:49 •358 97 14:39 .405 41 15:37 .45000 9 :20 .490 20 25:51 .36000 9:25 .406 25 13:32 .450 98 23:51 ■490 57 26:53 .361 11 13:36 .406 78 24:59 .451 61 14:31 •490 91 27:55 .361 70 17:47 .407 41 11 :27 •452 38 19:42 .491 23 28:57 .36207 21:58 .408 16 20:49 •452 83 24:53 ■491 53 29:59 .363 64 4:11 .409 09 9 :22 •454 55 5 = 11 .50000 I :2 .365 38 19:52 .410 26 16:39 .456 14 26:57 .365 85 15:41 .410 71 23:56 •456 52 21 146 .366 67 11 :3° .411 76 7:17 •457 14 16:35 ■367 35 18:49 .413 04 19:46 •457 63 27:59 .36842 7:19 •413 79 12 : 29 •458 33 II : 24 .369 56 17:46 .414 63 17:41 •459 46 17:37 • 370 37 10 :27 .415 09 22:53 .460 00 23:50 •371 43 13:35 .416 67 5:12 .461 54 6:13 .372 09 16:43 .418 18 23 = 55 .462 96 25:54 •372 55 19:51 .418 60 18:43 .463 41 19:41 5i Values, P, of the " Probability-Integral " in function of x/r, where x is a stated error-limit, and t the "probable error."* Explanatory Example. — Case of iooo similar determinations. Of the iooo errors (taken irrespective of +), 500 do not exceed r units. Then: 54 don't exceed o. 1 r; 289 don't exceed 0.55 r; 993 don't exceed 4 r, &c. X r P. 1 2 3 4 5 6 7 8 9 A 0.0 .000 .005 .011 .016 022 .027 .032 .038 ■043 .048 6 0.1 .054 •°59 .065 .070 075 .081 .086 .091 .097 .102 5 0.2 .107 •"3 .118 .123 129 ■ 134 ■139 .145 .150 •155 5 o-3 .160 .166 .171 .176 181 .187 .192 .197 .202 .208 5 0.4 .213 .218 .223 .228 233 .239 .244 •249 • 254 •259 5 0-5 .264 .269 ■ 274 .279 284 .289 .294 .299 ■3°4 • 309 5 0.6 •3i4 ■ 3'9 •324 •329 334 •339 •344 •349 •354 •358 5 0.7 •363 .368 •373 ■378 382 ■387 •392 •397 .401 .406 5 0.8 .411 .415 .420 .424 429 ■ 434 ■438 •443 ■447 •452 4 0.9 .456 .461 •465 .470 474 .478 ■483 .487 .491 .496 4 1.0 .500 •5°4 .509 •513 SI £ .521 •525 •53° •534 .538 4 1.1 •542 .546 •550 •554 558 .562 .566 .570 ■574 • 578 4 1.2 .582 .586 .589 •593 597 .601 .605 .608 .612 .616 3 i-3 .619 .623 .627 .630 634 .638 .641 •645 .648 .652 3 1.4 •655 .658 .662 .665 669 .672 •675 .679 .682 .685 3 i-5 .688 .692 .695 .698 701 .704 .707 .710 ■713 .717 3 1.6 .720 .723 .726 .728 73i • 734 • 737 .740 •743 .746 3 i-7 • 749 •75i ■754 • 757 759 .762 .765 .768 .770 •773 2 1.8 • 775 • 778 .780 .783 785 .788 .790 • 793 •795 ■ 798 2 i-9 .800 .802 .805 .807 809 .812 .814 .816 .818 .821 2 2.0 .823 .825 .827 .829 831 •833 .835 .837 •839 .841 2 2.1 •843 .845 .847 .849 851 •853 •855 •857 •859 .860 2 2.2 .862 .864 .866 .867 869 .871 •873 •874 .876 .878 1 2-3 •879 .881 .882 .884 886 .887 .889 .890 .892 •893 2 2.4 •895 .896 .897 .899 900 .902 .903 .904 .906 .907 1 2-5 .908 .910 .911 .912 913 •915 .916 .917 .918 .919 1 2.6 .921 .922 •923 •924 925 .926 .927 .928 .929 •93° 1 2.7 ■931 •932 •933 •934 935 •936 •937 .938 •939 .940 1 2.8 .941 •942 •943 •944 945 •945 .946 •947 .948 •949 1 2.9 •95° •950 •951 •952 953 ■953 •954 •955 •956 .956 1 3- •957 .964 .969 •974 978 .982 •98S .987 .990 • 992 7 to 1 4- •993 •994 •995 .996 997 .998 •998 •999 •999 •999 1 „o 5- •9993 9994 9996 •9997 •! )997 .9998 •9998 •9999 •9999 •9999 1 „o * Abridged from a table in Merriman's Method of the least Squares, LondoD, Macmillan ; where the values P are given to 4 decimals. 5 2 FORMULA-VALUES F of a Number of Substances and Radicals, and their Logarithms. The Characteristics of the Logarithms are omitted. Name or Formula. Aluminium, Al A1 2 3 A1 2 (S0 4 ) 3 A1 2 K 2 (S0 4 ) 4 + 2 4 H 2 A1 2 (NH 4 ) 2 (S0 4 ) 4 +2 4 H 2 - • A1 2 P 2 8 Antimony, Sb Sb 2 3 Sb 2 4 (SbO)K.C 4 H 4 6 + |H 2 Sb 2 S 3 Arsenic, As - As 2 O s As 2 5 AS0O3 As0 4 Mg(NH 4 ) As 2 O r Mg 2 As0 4 H 2 K Barium, Ba- BaO Ba(OH) 2 - Ba(OH) 2 + 8H 2 BaCl 2 BaCl 2 + 2H 2 BaN 2 8 BaS0 4 BaC0 3 Bismuth, Bi Bi 2 O s BiOCl Bi(N0 3 ) 3 . 5 H 2 BiO(N0 3 ) F(0 = i6.) Log. F 27-10 433° 102-20 0095 342.38 5345 948-83 9772 906-67 9574 244-28 3879 119-9 0788 287-8 •459i 303-8 4826 332-0 5211 336-0 5*63 75-09 8756 198-18 2971 230-18 3621 246-36 3916 181-52 2589 310-92 4926 180-23 2558 137.20 13735 153-20 18526 170-20 2335° 315-24 49866 208-11 31829 244-12 38760 261-29 -41712 233-26 36784 197-20 29491 208-0 3181 464-00 6665 259-45 4141 484-16 6850 286-0 45 6 4 FORMULA-VALUES. 53 Name or Formula. F(0=i6.) Log. F. Boron, B - 10-9 •0374 BjO, - - - 69-8 •8439 Na,O.B<0,- - 201-7 • 3°47 Na20.B 4 6 .ioH 2 - - 381-8 •5818 BF 4 K - 126-4 .1017 Bromine, Br - 79-955 .90283 HBr - - - - 80-95 .90822 AgBr- 187-88 .27388 KBr - - 119-09 .07588 KBr0 3 • 167-09 •22295 Br-Cl • - 44-498 .64834 Cadmium, Cd - . - 112-0 •0492 CdO - 128-0 .1072 CdS - 144-06 •1585 CdS0 4 208-06 .3182 Calcium, Ca - 40-02 .60228 CaO - 56-02 •74834 CaC0 3 - 100-02 .00009 CaC 2 4 - 128-02 .10728 CaC 2 4 .H 2 (110°) 146-02 .16444 CaS0 4 - 136-08 ■13379 CaS0 4 .2H 2 - - - 172-09 • 23576 CaCl 2 - - 110-93 •°45°5 CaCl 2 .6H 2 - - 218-96 •34037 Carbon, C - - Multipl es by n. 12-00 .07918 n nC n nC n nC n nC 11 12 13 14 15 16 17 18 19 132 144 156 168 180 192 204 216 228 21 22 23 24 25 26 27 28 29 252 264 276 288 300 312 324 336 348 31 32 33 34 35 36 37 38 39 372 384 396 408 420 432 444 456 468 41 42 43 44 45 46 47 48 49 492 5°4 516 528 ; 540 552 5 6 4 576 588 54 FORMULA-VALUES. Name or Formula. Carbon — Continued. CO ... co 2 C 2 4 H 2 (Oxalic Acid) C 2 4 H 2 + 2 H 2 C 2 4 .(NH 4 ) 2 + H 2 C 4 H 6 6 (Tartaric Acid) C 5 H 10 O 6 (Cellulose, Starch) y<2 (C 12 H 22 O n ) (Cane Sugar, Anhydrous Milk Sugar, Maltose) C 6 H 12 6 (Glucose, Crystallised Milk Sugar) Chlorine, Cl 2CI 3CI • - 4CI 5CI - - - 6C1 ... 7 C1 ... 8C1 9CI - HC1 - - Cl 2 -0 - - AgCl- KC1 - NaCl - • KCIO3 KC10 4 Cyanogen, NC ; see Nitrogen. Chromium, Cr Cr 2 3 Cr 2 7 K 2 Cr0 3 - - Cobalt, Co - CoO F(0 = i6.) Log. F. 28-00 .44716 44-00 ■6434S 90-00 ■95424 126-01 .10041 142-12 .15266 150-01 .17612 162-02 .20957 171-03 •23307 180-03 • 2 5535 35-454 •54967 70-91 .85071 106-36 .02678 141-82 •i5i74 177-27 .24864 212-72 .32781 248-18 ■39477 283-63 •45 2 75 319-09 •50391 36-46 .56182 54-91 ■73965 143-38 .15649 74-59 .87268 58-507 .76721 122-59 .08846 138-59 - -14173 52-13 - .7171 152-26 .1826 294-53 .4691 100-13 .0006 58-7 - .7686 74-7 •8733 FORMULA-VALUES. 55 Name or Formula. F(0=i6.) Log. F. COBALT- —Continued. CoSO, t • - 154-8 .1898 Co 2 P 2 O v - 291-5 .4646 Oopper, Cu . 63-34 .80168 CuO - 79-34 .89949 Cu 2 142-68 •15436 Cu 2 S 158-74 .20069 CuSO t • - 159-40 .20249 CuSO i5H 2 249-42 •39693 Cu.NCS 121-45 .08440 Fluorine, F . 19-10 .2810 CaF 2 • • 78-22 •8933 AlF 3 . 3 NaF 210-86 •3240 Gold, Au (Thorpe and Laurie), 197-32 ■295!7 „ Au (Kruss). Ann. 238, 2 72, yeai 1887, 197-11 .29471 Hydrogen, H 1-0024 .00104 Multiples. 2 .0048; 3.0072 j 4.0096; 5.0120 ; 6.0144; 7- 0168; 8.0192; 9.0216; 10.0240. Water , H 2 18-005 •25539 Multiples. n nH 2 Log. n nH 2 Log. 1 18.005 25539 16 288.080 •45951 2 36.010 55642 17 306.085 .48584 ! 3 54-oi5 73251 18 324.090 .51067 4 72.020 85745 19 342.095 •53415 5 90.025 95436 20 360.100 •55642 6 108.030 03354 21 378-105 •57761 7 126.035 10049 22 396.110 •59782 8 144.040 15848 23 414.115 .61712 9 162.045 20964 24 432.120 .63560 LO 180.050 25539- 25 450-125 ■65333 11 198.055 29679 26 468.130 .67037 12 216.060 33457 27 486.135 .68676 13 234.065 36934 28 504.140 •70255 14 252.070 40152 29 522.145 .71779 15 270.075 43148 30 540.150 •73251 56 FORMULA-VALUES. Name or Formula. F(0=i6.] Log. F. Iodine, I - - 126-85 - .10329 n = 2 3 1 4 5 nl = 253.70 380.5S 1 507.40 634.: '■5 HI - 127-85 .10670 Agl 234-78 .37066 AgI0 3 - 282-78 •45145 KI - - 165-99 .22008 Cul - - 190-19 .27919 I-Cl- - 91-396 .96093 Iron, Fe . 56-02 •74834 FeO - 72-02 ■85745 Fe 2 3 160-04 • 20423 FeCl 2 126-93 • 10356 FeCl 3 162-38 • 21053 FeCl 3 .6H 2 (Small ' bellow Crystals) 270-41 .43202 FeS0 4 .7H 2 - 278-12 .44423 Fe(NH 4 ) 2 S 2 8 .6Aq. - - 392-28 •59360 Lead, Pb . 206-9 •31576 PbO 222-9 .34811 Pb0 2 - - 238-9 .37822 PbS 239-0 .37840 PbS0 4 303-0 .48144 PbCr0 4 323-0 .50920 ; PbCl 2 277-8 •44373 Pb(N0 3 ) 2 331-0 ■S 1 ^ Pb(C 2 H 3 2 ) 2 . 3 H 2 378-9 •57853 Li = = 6.89. Li = 7.00. F. Log. F. F. Log. F. Lithium, Li 6-89 .8382 7-00 ■8451 Li 2 29-78 .4739 30-00 •4771 Li 2 0. 3 H 2 83-80 .9232 84-02 .9244 Li 2 C0 3 73-78 .8679 74-00 .8692 Li 2 S0 4 109-84 .0408 110-06 .0416 j Li 3 P0 4 115-71 .0634 116-04 .0646 ■ FORMULA-VALUES. 57 Name or Formula. Magnesium, Mg MgO MgS0 4 MgS0 4 .7H,0 MgCl 2 .6H 2 Mg 2 P 2 T Mg(NH 4 )P0 4 Mg (NH 4 )P0 4 .6H 2 Manganese, Mn- MnO- Mn 3 4 Mn 2 O s Mn0 2 MnS MnS0 4 MnCl 2 Mn0 4 K Mercury, Hg HgO HgCl- HgCl 2 HgS - Nickel, Ni NiO - NiS NiS0 4 Nitrogen, N n nN 28.092 3 42.138 NH 3 NH 4 (NH 4 ) 2 (NH 4 ) 2 S0 4 NH 4 C1 F(0 = i 6.) Log. F. 24-37 .3869 40-37 .6061 120-43 .0807 246-47 .3918 203-31 .3082 222-82 .3480 137-47 .1382 245-50 .3901 55-00 .74036 71-00 .85126 229-00 -35984 158-00 .19866 87-00 Q3952 87-06 -93982 151-06 -17915 125-9] .10006 158-14 - .19904 200-3 - .30168 216-3 •33506 235-75 .37245 271-2] L -43331 232-36 .36616 58-7 .7686 74-7 - -8733 90-8 .9581 154-8 .1898 14.046 .14755 4 5 56.184 70.230 17-05 3 .23180 18-05 6 .25662 52-11 1 .71694 132-17 .12113 53-51 - .72844 5» FORMULA-VALUES. Name or Formula. Nitrogen — Continued. NH 4 N0 3 PtCl 6 (NH 4 ) 2 N 2 8 NA HNO s KN0 3 NaNO s AgN0 3 NC NC.H AgNC NCS.K NCS.NH 4 Phosphorus, P PA Ag s P0 4 BiP0 4 P 2 7 Mg 2 "2^8^ a 3 P 2 8 CaH 4 Platinum, Pt PtClg PtCl 6 K 2 PtCl 6 Na 2 PtCl 6 (NH 4 ) 2 Potassium, K 2K 3K 4 K K 2 KHO K 2 C0 3 KHCO s KC1 F(0 = t6.) Log. F. 80-102 .90364 444-33 .64771 76-09 .88133 108-09 •03379 63-05 .79969 101-18 .00510 85-10 .92993 169-98 .23040 26-046 •41574 27-048 .43214 133-976 .12703 97-242 •98785 76-162 .88174 31-04 .49192 142-08 •i5 2 53 418-83 .62204 303-04 48144 222-82 34795 310-14 49156 234-11 36942 195-5 29115 408-22 61089 486-49 68707 454-33 65737 444-33 64771 39-136 59258 78-272 89361 117-408 06970 156-544 19464 94-27 97437 56-14 74927 138-27 14073 100-14 00061 74-59 87268 FORMULA-VALUES. 59 Name or Formula. Potassium — Continued. KC10 S KC10 4 KNO s K 2 S0 4 - - KH 2 P0 4 K 2 C 2 4 .H 2 (Oxalate) KH.C 4 H 4 0« (Bitartrate) K 4 .Fe(NC)„ + 3H 2 (Yellow Prussiate) Silicon, Si Si0 2 Silver, Ag 2Ag - ... 3Ag ... 4Ag Ag 2 AgCl AgBr - Agl - - Ag 2 S Ag 2 S0 4 AgN0 3 Ag(NC) - - Ag(NCS) Sodium, Na 2Na 3Na 4Na Na 2 - NaOH Na 2 C0 3 Na 2 C0 8 +ioH 2 NaHC0 3 NaCl - NaN0 8 Na 2 S0 4 F(0 = i6.) Log. F. 122-59 .08846 138-59 •14*73 101-18 .00510 174-33 •24137 136-18 .13411 184-28 .26548 188-15 •2745° 422-85 .62619 28-40 •4533 60-40 .7810 107-93 •°33!4 215-86 •33417 323-79 .51026 431-72 .63520 231-86 •36523 143-38 .15649 187-88 ■27388 234-78 .37066 247-92 ■39431 311-92 .49404 169-98 .23040 133-98 .12704 166-04 .22021 23-053 .36273 46-106 .66376 69-159 •83985 92-212 .96479 62-11 .79316 40.06 .60271 106-11 .02576 286-16 .45661 84-06 .92454 58-507 .76721 85-10 .92993 142-17 .15281 5o FORMULA-VALUES. Name or Formula. Sodium — Continued. Na 2 S0 4 +ioH 2 Na 2 S 2 3 + 5H 2 Na 2 HP0 4 +i2H 2 - Na 3 PC- 4 - Strontium, Sr SrO - SrCO s - ... SrS0 4 Sr(N0 3 ) 2 - - - Sulphur, S n= 2 3 4 nS = 64.i2 96.18 128.24 S0 3 S0 4 S0 4 H 2 S0 4 H 2 + yj H 20 (Marignac's Acid) S0 4 Ba Tin, Sn SnO Sn0 2 - ... SnCl 2 SnCl 2 +2H 2 (Crystals, Marignac) Uranium, U U0 2 u 3 o 8 - uo 3 *(U0 2 )(N0 3 ) 2 + 6H 2 *(U0 2 )(C 2 H 3 2 ) 2 + 4 H 2 F(0 = i6.) 322-22 248-25 358-21 164-20 87-52 103-52 147-52 183-58 211-61 32-06 S 160.3 80-06 96-06 98-06 99-56 233-26 117-6 133-6 149-6 188-5 224-5 240-5 272-5 849-5 288-5 504-6 462-5 * Corresponding to l / z P2O5 =71.04 in the process of titration. Log. F. • S°8lS •39489 ■55414 •21537 .94211 .01502 .16885 .26383 •32554 •50596 6 192.36 .90342 .98254 .99149 .99808 .36784 0704 1258 1749 2753 35 12 3811 4354 9292 4601 7029 6651 Zinc, Zn ZnO ZnS ZnS0 4 ZnS0 4 +7H 2 65-37 81-37 97-43 161-43 287-47 .8154 .9105 .9887 .2080 .4586 6i ANALYTICAL FACTORS and their LOGARITHMS. The Characteristics of the Logarithms are omitted. Given Wanted Multiply by f a b f = b -r a | Logf Aluminium. A1 2 3 Al 2 •5303 ■7245 A1 2 3 - A1 2 (S0 4 ) 3 3-350 •S2SI Antimony. Sb 2 4 2 Sb- •7893 •8973 Sb 2 4 Sb 2 3 •9473 •976s Sb 2 S 3 Sb 2 - •7137 •8535 Arsenic. As 2 S 3 2As •6096 .7850 1 AS 2 Sg As 2 3 •8044 ■9°S5 As 2 S 3 As 2 5 •9343 •97°S As 2 7 Mg 2 - 2AS ■4830 .6840 As 2 7 Mg 2 As 2 3 •6374 .8044 As 2 7 Mg 2 As 2 5 •7403 .8694 2 As0 4 MgNH 4 As 2 7 Mg 2 •8564 •9327 Barium. BaS0 4 Ba •58818 - •76951 BaS0 4 BaO •65678 .81742 BaC0 3 BaO •77688 •8903S BaO Ba - •89556 .95209 Bismuth. Bi 2 3 2Bi- •8965 .9526 BiOCl - Bi •8017 .9040 2 BiOCl Bi 2 3 •8942 •9SI4 Bromine. AgBr Br •42554 .62894 AgBr BrH •43088 •63435 Br -CI Br 1-79675 ■25449 j Br -CI AgBr 4-22227 ■62555 62 ANALYTICAL FACTORS. Given Wanted Multiply by f a b f=b-fa | Log f Calcium. CaO Ca •71439 85393 CaS0 4 - CaO •41167 6i455 \ CaC0 3 CaO •56009 74826 Chlorine. AgCl CI •24727 393i6 AgCl C1H •25426 40527 2AgCl - (Cl 2 -0)- •19147 28211 Cl 2 o •22564 35342 Ag CI •32849 51652 Chromium. Cr 2 3 Cr 2 7 K 2 1-9344 28654 Cr 2 3 2CrO s 1-3152 11901 Cr 2 3 6Fe- 2-2075 34391 6Fe Cr 2 3 •45299 65609 Cr 2 3 6FeS0 4 7H 2 10-960 03980 Cobalt. CoO Co •7858 8953 Co CoO 1-273 1047 CoS0 4 - Co •3793 579° Co 2 P 2 7 2 Co •4028 .6051 Co 2 P 2 O r 2 CoO •5125 .7097 Copper. CuO Cu ■7983 .9022 Cu CuO 1-2526 .0978 CuNCS CuO •6533 .8151 CuNCS Cu •5215 •7i73 Cu 2 S 2CuO •9996 .9998 Cu 2 S sCu- •7980 .9020 Cyanogen. Ag NC- •24132 .38260 Ag NCH •25061 •39899 \ ANALYTICAL FACTORS. 63 Given Wanted Multiply by f a b f = b -r a | Logf Cyanogen — Continued. NCAg- - NC- •19441 - .28871 2NCH - AgN0 3 - ■ 3-1422 - •49723 Iodine. Agl ■ . I - •54029 .73263 I -CI - - I - 1-3879 .14236 I -CI - Agl • 2-5688 ■40973 Cul - - I •66696 .82410 Iron. Fe 2 3 - aFe- •70007 .84514 Fe 2 3 2FeO •90002 •95425 Fe FeS0 4 7H 2 4-9647 •69589 Fe 2 3 2 FeS0 4 7H 2 3-4756 •54io3 FeO Fe •77784 .89089 Fe - - FeO 1-28561 .10911 2Fe - - •14281 •15475 Lead. PbO - . Pb - •92822 .96765 ! pb - - PbO 1-07733 •03235 PbS0 4 PbO •73574 .86673 PbS0 4 - Pb ■68293 •83438 PbCl 2 Pb •74476 .87202 PbS PbO •93279 .96979 PbS - Pb •86583 •93744 Lithium (if Li = 6.89) Li 2 S0 4 Li 2 •2711 ■4332 Li 3 P0 4 1.5LLO - ■3861 .5866 Li 2 2Li - •4627 •6653 Lithium (if Li = 7.00) Li 2 S0 4 Li 2 - •2726 •4355 Li 3 P0 4 1.5LLO •3878 .5886 Li 2 2Li •4667 .6690 6 4 ANALYTICAL FACTORS. Given Wanted Multiply by r a b f = b -=- a | Log 1 Magnesium. MgO Mg- •6037 7808 \ Mg MgO 1-6565 2192 MgS0 4 - - MgO •3352 5253 P 2 6 2MgO 2MgO ■3624 559i Manganese. MnO Mn- •7746 8891 Mn - MnO 1-2909 1109 Mn 3 4 3MnO •9301 9685 ; MnS0 4 MnO •4700 6721 MnS - MnO •8155 9114 MnO •2254 3S 2 9 MnO 4-438 6471 ' 2C0 2 - Mn0 2 - •9886 995° | zC0 2 - •1818 .2596 | Nickel. NiO Ni - •7858 •8953 Ni NiO - 1-273 .1047 | I NiS0 4 - Ni - •3793 •579° | Nitrogen. N NH S - 1-21410 .08425 NH 3 - - N - •82366 •9IS7S zNH 3 - (NH 4 ) 2 1-52790 .18410 (NH 4 ) a O - 2 NH 3 •65449 .81591 NH 4 C1 NH 3 •31869 •5°337 2 NH 4 C1 (NH 4 ) 2 ■48693 .68747 2NH3 (NH 4 ) 2 S0 4 3-87526 .58830 so 3 - 2NH3 •42601 .62942 (NH 4 ) 2 S0 4 2NH3 ■25805 .41170 PtCl 6 (NH 4 ) 2 2N •063223 - .80087 PtCl 6 (NH 4 ) 2 2NH3 •076758 .88512 PtCl 6 (NH 4 ) 2 (NH 4 ) 2 ■11728 .06922 ANALYTICAL FACTORS. 65 Given a, Wanted b Multiply by f f = b -r a | Log f Nitrogen — Con tinned. Pt 2N •14369 • 15744 Pt 2NH3 •17446 .24169 2NH3 N 2 6 3-1693 .50096 N 2 5 2NH3 •31553 .49904 NA 2 N0 3 H 1-16657 .06691 2HN0 3 N 2 5 •85722 •93309 Oxygen. 2C0 2 O •1818 .2596 Mn 2 O r K 2 5x0 ■2529 .4030 SxO Mn 2 ? K 2 3-954 •597o Mn 2 7 K 2 3x0 •1518 .1812 3x0 Md 2 O y K 2 6-589 .8188 Phosphorus. PA 2P •43694 .64042 P 2 7 Mg 2 2P •2786 •445° P 2 7 Mg 2 P 2 5 •6376 .8046 ■^W-^S P 2 5 •4581 .6610 ?A ^•fi&^s 2-1829 •3390 p 2 o 5 P 2 8 Ca.H 4 1-6477 .2169 Phosphomolybdate of Ammonia I Phosph. Pentoxide, -03794 ■5791 Phosphomolybdate of Ammonia ) Elementary J Phosphorus, •01658 •2195 JPZatimim. PtCl 6 (NH 4 ) 2 Pt •43998 .64344 Pt PtCl 6 (NH 4 ) 2 - 2-2728 •35656 PtCl 6 K 2 Pt •40185 .60407 Pt PtCl 6 K 2 - - 2-4885 •39593 i Potassium. PtCl 6 K 2 - 2K •16089 .20653 PtCl 6 K 2 K 2 ■19378 .28730 J 66 ANALYTICAL FACTORS. Given Wanted Multiply by f a b f =b-r a | Logf Potassium — Continued. PtCl 6 K 2 2KCI •30664 .48663 Pt K 2 •48221 .68324 I Pt 2KCI •76307 .88256 2KCI - - K 2 •63193 .80067 KC1 - K •52468 .71990 K 2 S0 4 - • - K 2 •54076 •733 01 K 2 S0 4 - . • 2K •44898 .65223 K 2 - 2K •83028 .91922 2K - - K 2 - - 1-20442 .08078 1 K 2 - K 2 C0 3 - 1-4667 ■16635 K,CO a - K 2 - •68179 - •8336S Silver. AgCl - - • - Ag - - •75273 .87664 Ag - - AgCl - - 1-32849 .12336 2Ag - Ag 2 ■ 1-07412 •0310S Sodium. 2NaCl - - Na 2 •53076 .72490 NaCl - Na •39402 •59552 Na 2 S0 4 Na 2 •43686 .64034 Na 2 S0 4 - - 2Na •32431 .51096 ; Na 2 - - 2Na •74238 .87062 2Na - - Na 2 1-34703 .12938 Na 2 - Na 2 C0 3 1-70846 .23261 Na 2 C0 8 - Na 2 •58532 .76739 Na 2 C0 3 - - 2Na •43453 .63802 Sulphur. BaS0 4 - - - S - •13744 .13812 BaS0 4 - - S0 3 •34322 ■53558 BaS0 4 - - so 4 ■41182 .61470 BaS0 4 - H 2 S0 4 •42041 .62367 30 8 H 2 S0 4 1-2249 .08810 H 2 S0 4 so 3 ■81640 .91190 ANALYTICAL FACTORS. 67 Given Wanted b Multiply by f f = b -r a | Log f Sn •7861 •8955 SnO •8931 •95°9 Zn - •8034 .9049 ZnO 1-2447 .0951 ZnO ■8352 .9218 Zn •6710 .8267 Tin. SnO, Sn0 2 Zinc. ZnO Zn ZnS ZnS EMPIRICAL FACTORS. I. — Factors in connection with the Chloroplatinate Method for determining Potassium and Ammonium, according to w. dlttmar and john m'arthur.* (a) Fitikcner's Method as modified by W. D. and J. MA. Given. Pt Pt Pt PtCl 6 K 2 Pt PtCl 6 K 2 Wanted. f. Log. f. 2KC1 •76084 •88129 K 2 •48080 .68196 2K ■39920 .60119 (b) Tatlock's Method. - 2KCI •30627 .48610 2KCI** ■76016 .88090 K 2 ■19354 .28678 2K •16069 .20600 PtCl 6 K 2 ** Refers to the KC1 in the Ca\oro^\aimz.te-precipitate. In all other cases under (a) and (b) and [c) the factor gives the KC1, etc., in the substance analysed. (c) Determination of Ammonia given as Sal^Ammoniac. PtCl 6 (NH 4 ) 2 PtCl 6 (NH 4 ) 2 PtCl (NH 4 ) 2 Pt Pt Pt •2389 •07614 ■06271 •5459 2NH3 -17397 2N -14331 * Trans. R.S. Ed.; Vol. 33, Part 2, p. 633. 2NELCI 2NH3 2N 2NH 4 C1 37822 88159 79733 737H 24048 15626 68 EMPIRICAL FACTORS. II. — Determination of Sugars by means of Fehling's Solution. General Modus Operandi. — The sugar is brought into the form of a ^ to I per cent, solution. This is poured into «»-diluted boiling Fehling's Solution, and the boiling continued for, in general, two minutes ; in the case of milk sugar for six minutes. Supposing now the copper to be just completely precipitated ; then, according to Soxhlet, the following relations hold : GrammesofSugar CC. of Fehling Sugar per 1 CC. per 1 Gramme of Name of Sugar. per 1 Gramme of Fehling, in Metallic Copper of Sugar. Milligrammes. in the precipitate. Dextrose, 210-4 4-753 ■5403 Lsevulose, 199-4 5-144 •5848 Invert-Sugar, 202-4 4-941 •5617 Galactose, 195-7 5-110 •5809 Milk Sugar, calcu- "1 lated as > 148-0 6-756 ■7680 C12H24U12 J Maltose, C 12 H 22 O n , 128-5 7-780 •8844 In exact analysis these factors must be accepted only as first approximations, and the exact factors be determined ex temp, with known weights of the respective sugars, under the precise con- ditions of the analysis. (See Dittmar's Exercises in Quantitative Analysis, p. 258.) Equivalents of some Fatty Acids and Fats. If one molecule of fatty acid = Ac, and one molecule of the cor- responding fat = 3 F, then : Ac + £C 3 H 8 3 - H 2 = F, or Ac+i2.67 = F. Acid. Butyric Caproic - Caprylic - Capric - Laurie - Palmitic - Linoleic - Stearic Oleic Ricinoleic Formula. C 4 H s 2 C 6 H 12 2 CsHi 6 2 CioH 2 o0 2 QiaH^Og Oi6H S2 2 ^-18^86^2 ^'18-"-S4^-'2 QlsHgdOg Ac. F. 88.02 IOO.69 116.03 128.70 144.04 156.71 172.05 184.72 200.06 212.73 256.08 268.75 252.07 264.74 284.09 296.76 282.08 298.08 294-75 310.75 6 9 GAS VOLUMETRIC DETERMINATIONS. The Gas obtained is supposed to be measured in cc. (Milli-litres), at t°C and a dry pressure of P Millimeters of Mercury of o°C. " Weight" means weight in Milligrammes. Nitrogen. Given. Wanted. Multiply by. Log. of Constant. * Volume of Nitrogen. Weight of Nitrogen. ■45089* x P 272-63 + t .65407 - I Weight of Ammonia, NH 3 . •54743* x P 272-63 + t •73833-1 Weight of Nitric Acid, N 2 6 . 1-7349* xP 272-63 + t .23929 Weight of Urea,§ CON 2 H 4 . •9647* x P 272-6 + t .9844-1 § In the ordinary hypobromite process for the determination of Urea in Urine, only 91 per cent, of the nitrogen of the Urea are set free. Our factor applies to the Nitrogen -volume as corrected 'by multiplication with 100-^91 = 1.099. Carbonic Acid. Given. Wanted. Multiply by. Log. of Constant.* Volume of Carbonic Acid, Weight of Carbonic Acid, C0 2 . •70174* x P 269-54 +t •84618 -1 7o Nitric Acid. Given. Wanted. Multiply by. Log. of Constant.* Volume of Nitric Oxide. Weight of Nitrous Acid, N 2 8 . ■61102* xP 273 + t .78606-1 Weight of Nitric Acid, N 2 5 . ■86798* x P 273 + t •93851 - 1 Weight of Ammonia, NH 3 . •27387* xP 273 + t ■437SS Weight of Radical N0 2 . ■73950* x P 273 + t .86894-1 Weight of Radical N0 3 . •99646* x P 273 + t .99846 - 1 Oxygen obtained in Lunge's Method for the Analysis of Hypochlorites. Given. Wanted. Multiply by. Log. of Constant.* Volume of Oxygen obtained. Weight of Oxygen in Hypochlorite. •25619* xP 272-18 + t .40857-1 Weight of Active Chlorine. 1-1354* xP 272-18 + t ■OSS 1 * 7i THE METRIC AND THE BRITISH SYSTEMS OF UNITS. The Metric System. Its Fundamental Units are : The Metre, the Litre, and the Gramme. Original Definitions : i Metre = the ten-millionth part of the distance from the North Pole to the Equator, measured along the Meridian of Paris, i Litre = i Cubic-Decimetre, iooo Grammes = i Kilogramme = Mass of r Cubic-Decimetre of Water of + 4°C. The present de-facto definitions are as follows : — The " Metre " is the Length, at o°C, of a bar of Iridio-Platinum, kept in Paris as the " Mitre Prototype." The " Kilogramme " is the Mass of a block of Iridio-Platinum, preserved in Paris as the "Kilogramme Prototype.'" The " Litre " is the Volume at + 4°C, of a quantity of Water, which, in Vacuo, balances a true Kilogramme. The original standard metre was made after the "Toise du Perou," and its length, at o°C, adjusted tc o. 5 13 074 07* of the length of this Toise at its standard temperature. Log.* = .710 18007- 1. Hence 1 Toise =1.949 03631 Metres. Log. =.289 819 93. According to Bessel's calculation the length of the quadrant of a meridian is 10 000 855.76 Metres as thus defined. Log. =7.000 037 16. The Subsidiary Units are derived from the fundamental by Multi- plication or Division with 10, 100, 1000, &c. Their names are formed by combining those of the fundamental Units with the following pre- fixes : — Deci- for 0.1 of, Centi- for 0.01 of, Milli- for 0.001 of; Deka- for 10 times, Hecto- for 100 times, Kilo- for 1000 times, Myria- for 10000 times. Thus, for instance, "Millimetre" means 0.001 Metre; " Hectolitre" means 100 Litres, &c. Appendix. For the measurement of Liquids a more convenient unit than the Cubic-Centimetre or Litre is The " Fluidgramme," which term was introduced by the Author long ago* as designating the volume, at i5°C, of a quantum of Water, the uncorrected weight of which, as determined in Air of i5°C and 760mm pressure by means of brass weights, is one Gramme. Professor de Koninck of Liege has adopted the Author's unit, but proposed to use the name of " Mohr" for 1000 Fluidgrammes, and the name of " Milli-Mohr" for the Fluidgramme itself. What, in commercial volu- metric apparatus figures as "Cubic Centimetre" or "Litre" comes, as a rule, nearer to our Fluidgramme and Mohr respectively than to the ostensible units. The British Units. Of Length.— -The Fundamental Unit is the "Yard," defined by reference to a material Standard. 1 Mile =1760 Yards; 1 Yard = 3 feet = 36 Inches. * Article : Analyse volumetrische fur Fliissigkeiten, etc., in Fehling's Hand- worterbuch der Chemie. 72 Of Volume. — The Fundamental Unit is the •' Gallon " denned as being the Volume, at 62° Fahrenheit of the quantity of Water which, in Air of 62° F. and 30 inch pressure, balances 10 true Pounds Avoir- dupois of Brass. Of Mass. — Fundamental (nominal) Unit: The "Grain" denned by reference to a block of Platinum preserved in London as representing the Weight in Vacuo {i.e., the Mass) of " one Pound Avoirdupois " or 7000 Grains. There are three sets of Units of Weight, viz. : — (1) The Avoirdupois Set. 1 Ton = 20 Hundredweights (Cwt.) = 2240 Pounds (Lb.) = 16 x 2240 = 36 840 Ounces (Oz.) 1 Lb., by definition, = 7000 Grains, hence 1 Ounce = 437-5 Grains. (2) The Troy Set. 5760 Grains = 1 Pound = 12 Ounces. 1 Ounce = 20 Penny-weights = 20 x 24 = 480 Grains. (3) Apothecaries Weight. In former times the Troy Ounce was used in Medicine and Pharmacy as unit of Weight. It was divided into 8 Drachmas = 8x3 = 24 Scruples = 24x20 = 480 Grains. The present Units are : The Avoirdupois Ounce ( = 437.5 Grains) and the Grain. It is, however, permitted to use the names and symbols of " Drachma " and " Scruple " in their old sense, i.e., as designating 60 Grains and 20 Grains respectively. Note that there is only one kind of Grain in the British System. The words " Pound " and " Ounce," however, have each two different meanings. Apothecaries Fluid Measures. To define these let us introduce the term " Fluid-Grain " as meaning the volume of 1 Grain of Water at 62°F, weighed out, without correc- tion, in Air of this temperature and 30 inch pressure. 1 Fluid-Ounce = 43 7. 5 Fluid-Grains = 8 Fluid-Drachmas = 8 x 60 = 480 Minims. Hence 1 Fluid-Drachma = 54.6875 Fluid-Grains; and 1 Minim = 0.91 1458 Fluid-Grains. Relations of British Units to one another. 1 Ton = 32 666-7 Ounces Troy. Log. = 4.5i4 105. 1 Pound Avoirdupois = 14-583 3 Ounces Troy. Log. = 1.163 857. 1 Ounce Avoirdupois = -911 458 Ounces Troy. Log. = .959 737 - 1. 1 Ounce Troy = 1'097 14 Ounces Avoirdupois. Log. = .040 263. 1 Gallon, by definition, = 10 Fluid-Pounds = 160 Fluid-Ounces = 70000 Fluid-Grains. By experiment, 1 Gallon = 277.274 Cubic Inches. 73 In Air of 62° F. and 30 inch pressure. 1 Cubic Foot of Water weighs 997-71 Ounces Avoirdupois. L °g- = 3-999 °°4- 1 Cubic Inch of Water weighs 252-458 Grains. Log. = 2.402 189. Other Relations. 1 Metre = 1 -093 633 Yards, Log. = .038 872, 1 Metre = 3-280 899 Feet. Log.^515 993. 1 Metre = 39-370 79 Inches. Log. = 1.595 173. 1 Yard= -914 383 5 Metres. Log. = .961 128 - 1. 1 Foot = -304 794 5 Metres. Log. = .484 007-1. 1 Inch = 25-399 54 Millimetres. Log. = 1.404 827. 1 Cubic Decimetre = 61-027 05 Cub. Inch. Log. = 1.785 522. 1 Cubic Metre = 35-316 58 Cubic Feet. Log. = 1.547 979. 1 Cubic Inch = 16-386 18 Cubic Centimetres. Log. = 1.214 478. 1 Cubic Foot = 28-315 31 Cubic Decim. Log. = 1.452 022. 1 Litre = -998 081 " Mohr." Log- = . 999 166- 1. 1 Mohr = 1-001 923 Litres. Log. =.000 834. *i Litre = -219 981 2 Gallons. Log. =.342 386 - 1. *i Litre = 35-197 00 Fluid Ounces. Log. = 1.546 506. *i " Mohr " = 35-264 7 Fluid Ounces. Log. = 1.547 340. *i Gallon = 4-545 85 Litres. Log. =.657 615. *i Fluid-Ounce = 28-357 Fluid Gramme. Log. = 1.452 660. 1 Kilogramme = 2-204 621 Pounds Avoirdupois. Log. =.343 334. 1 Kilogramme = 35-273 94 Ounces Avoirdupois. Log. = 1.547 454. 1 Kilogramme = 32-150 7 Ounces Troy. Log. = 1.507 190. 1 Gramme = 15*432 35 Grains. Log. = 1. 188 432. 1 Gramme = -032 150 7 Ounces Troy. Log. =.507 190 - 2. 1 Pound Avoirdupois = 453-593 Grammes. Log. = 2.656 666. 1 Ounce Avoirdupois = 28-349 54 Grammes. Log. = r.452 546. 1 Ounce Troy = 3L103 50 Grammes. Log. = 1.492 810. 1 Ton = l 016-05 Kilogrammes. Log. = 3.006 915. * Calculated by the Author from the original definition of the " Imperial Gallon," the details of which were kindly furnished by Mr. H. J. Chaney, of the Standard Office. One Gallon, by definition, is the volume at 62T of a quantum of water, which balances ten brass pound-weights (true in vacuo), in air of 62T and 30 inch (red. to o°C) pressure, and two-thirds suturated with moisture. The Spec. Gravity of the brass is supposed to be = 8.143. By Act of Parliament, however, 1 Gallon is equal to 4. 543458 Litres. This number is based upon the experimentally ascertained weight of the cubic inch in grains, and the relation of the cubic decimetre to the cubic inch. 74 CONVERSION OF STATEMENTS OF COMPOSITION. Given. Wanted. Multiply by f. Log. f = Milligrammes per Mohr. Grains per Gallon. •07 .845 098 - 2 Grains per Gallon. Milligrammes per Mohr. 14-286 = 1004-7 1.154902 Milligrammes per Litre. Grains per Gallon. •070 135 .845932-2 Grains per Gallon. Milligrammes per Litre. 14-258 3 1. 154 068 Ounces Troy per Ton. Milligrammes per Kilo. 30-612 2 1.485 895 Milligrammes per Kilo. Ounces Troy per Ton, •Q32 666 7 •514 105 -2 75 BAROMETRIC READINGS. To pass from Inches to Millimeters, and vice versa. Inches. 28.0 28.1 28.2 28.3 28.4 28.5 28.6 28.7 28.8 28.9 29.0 29.1 29.2 29-3 29.4 29-5 29.6 29.7 29.8 29.9 30.0 30.1 30.2 30-3 30.4 Mm. 711.19 7I3-73 716.27 718.81 721.35 723.89 726.43 728.97 73I-5 1 734-os 736.59 739-13 741.67 744.21 746.75 749.29 75I.83 754-37 756-9I 759-45 761.99 764-53 767.07 769.61 772.15 Inches. 30-5 30.6 30.7 30.8 3°-9 31.0 Mm. 774.69 777.23 779-77 782.31 784.85 787.39 P.P. Inch. Mm. .01 • 254 .02 .508 •03 .762 .04 1. 016 •OS 1.270 .06 I.524 .07 1.778 .08 2.032 .09 2.286 P.P. Mm. Inch 4 2,5 .0039 .0079 .0118 .0157 .0197 .0236 .0276 ■0315 •0354 •0394 ■°433 .0472 .0512 •0551 .0591 .0630 .0669 .0709 .0748 .0787 .0827 .0866 .0906 •0945 .0984 7 6 APPENDIX. Values of Gold Coins according to the weight of Gold contained in them by law.* * Calculated after data given in Bremiker's 5-place Logarithm Table, edited by A.. Kallius, 1887. i Mark = -979 030 Shilling. 1 Shilling = 1-021 42 Marks. 1 Mark = 048 951 5 Pound. 1 Pound = 20-428 4 Marks. 1 Dollar = 4.109 80 Shillings. 1 Shilling = •243 321 Dollars. 1 Dollar = •205 490 Pound. 1 Pound = 4-866 42 Dollars. 1 Franc = ■793 014 Shilling. 1 Shilling = 1-261 01 Francs. 1 Franc = •039 650 7 Pound. 1 Pound =25.220 2 Francs. Log. = .990 796- 1. Log: = .009 204. Log. = .689 766-2. Log. = 1.310 234. Log. = .613 821. Log. = .386 179- 1. Log. = .312 791 -1. Log. =.687 209. Log. = .899 281. Log. =.100 719. Log. =.598 251-2. Log. = 1.401 749. * The corresponding factors for commercial purposes are liable to fluctuations ; hence the factors in the following table are given to only 4 figures ; the Logarithms given, as far as they go, are in exact accordance with the data of the first table. CONVERSION OF PRICES. Given. Wanted. Multiply by f. f= Log. f= Price of Pound in Shillings. Price of Kilogramme in Marks. 2-252 •35 2 54 Price of Kilogramme in Marks. Price of Pound in Shillings. 0.444 1 .647 46-1 Price of Ounce in Pence. Price of 100 Grammes in Marks. •300 2 •477 48-1 77 CONVERSION OF TRICES— Continued. Given. Wanted. Multiply by f. f= Log. f= Price of ioo Grammes in Marks. Price of Ounce in Pence. 3-331 •522 S 1 Price of Pound in Shillings. Price of Kilogramme in Francs. 2-780 •444 05 Price of Kilogramme in Francs. Price of Pound in Shillings. •359 7 •555 95-i Price of Ounce in Pence. Price of ioo Grammes in Francs. ■370 7 .568 99-1 Price of ioo Grammes in Francs. Price of Ounce in Pence. 2-698 •43 1 0I Value of Pence in Decimals of the Shilling. Pence. Shilling. Pence. Shilling. Pence. Shilling. Pence. Shilling. # .04 3^ .29 6y 2 •54 9% •79 1 .08 4 •33 7 •58 10 •83 i# .12 4% •37 1% .62 ioj4 .88 2 •17 5 .42 8 .67 11 .92 2^ .21 S# .46 8 J / 2 •7i n J / 2 .96 3 •25 6 •5° 9 •75 12 1. 00 For the conversion of a sum stated in j£ s. d. into Pounds and Decimals of the Pound, proceed as shewn in the following example : ^3 17s. 9d. =^3 + 8 Florins + 1 Shilling 4- 36 Farthings = £{3 + 0.8 + 0.05 4- .036)^3.886. (1 Farthing being very nearly = 0.001 Pound) and, conversely, ,£3.886 reads £3 + (8.5 Florin = 17/) + (36 Farthings = 9&) = £3 17s. gd. 78 Hydrometer Scales. (n = Hydrometer reading ; S = Specific Gravity.) Hydrometer after Liquids heavier than Water. Liquids lighter than Water. (i) Twaddle ; 60° F. <>_ n + 200 200 or n = 20o(S- 1) Does not exist. *(2) Gay-Lussac, Centigrade ; 4°C§ S- I0 ° 100 — n e_ I0 ° 100 + n *( 3 ) Baume" 10° R=i2°5C. g _ 145.88 145.08 -n s _ 145-88 145.88 + (n- 10) *(4) Baume" i 4 °R=i 7 °5C. g _ 146.78 146.78 -n o_ 146.78 14678 + ^-10) *(5) Brix; Official Prussian Hydrometer i2°5R = i5°62C. Q _ 400 s- 4 °° 400 -l-n 400 -n *(6) Beck io"R = i2°5C. s- I7 ° 170-n s- I7 ° 170 + n * According to Gerlach ; Dingler's Polytechnisches Journal, vol. 176, p. 444 (1865) ; vol. 181, p. 358 (1866) : vol. 198 p. 313 (1870). § A very inconvenient temperature. We do not know whether it is being adhered to by Instrument Makers. 79 By means of first formula under (4). Gerlach calculated the following Table :* Specific Gravities corresponding to n Degrees Baume at i 4 °.oR=i7 .sC. n Spec. Grav. n Spec. Grav. n Spec. Grav. 1. 000 25 I.205 5° I-5I7 1 I.007 26 I-2I5 51 T -532 2 1. 014 27 1.225 52 1-549 3 1. 021 28 I.236 53 I-565 4 1.028 29 1.246 54 1.582 5 i-°35 30 1-257 55 1-599 6 1043 31 I.268 56 1. 617 7 1.050 32 1.279 57 "•635 8 1-058 33 1.290 58 1-653 9 1.065 34 1. 301 59 1.672 10 1.073 35 I-3I3 60 1. 691 11 1.081 36 I-325 61 1. 711 12 1.089 37 1-337 62 i-73i 13 1.097 33 1-349 63 i-752 14 1. 105 39 1.362 64 1-773 iS 1. 1 14 40 1-375 65 i-795 16 1. 122 41 1.388 66 1. 817 17 1. 131 42 1. 40 1 67 1.840 18 1. 140 43 1.414 68 1.863 19 1. 149 44 1.428 69 1.887 20 1.158 45 1.442 70 1. 912 21 1. 167 46 1.456 7i 1-937 22 1. 176 47 1.471 72 1.963 23 1.186 48 1.486 73 1.989 24 i-i95 49 1. 501 74 2.017 * Dingler's Polytechnisches Journal, vol. 198, p. 316 (1870). The original gives the specific gravity to 4 decimals. 8o RELATIONS IN A NUMBER OF AQUEOUS SOLUTIONS BETWEEN SPECIFIC GRAVITY AND PERCENTAGE OF SOLUTUM. I. Sulphuric Acid. Tables " A " and " B " were deduced (by the Author) from Bineau's DIRECT DETERMINATIONS by a combination of algebraic and graphical interpolation. These tables go up to Spec. Grav. 1.7 only. The subsequent tables for stronger acids are transcriptions from the Originals quoted. A. Percentage p of H 2 S0 4 in function of true* Specific Gravity S at 0°, taken in reference to Water of o° as a Standard. * " True" means : reduced to the Vacuum. 0S0 1 2 3 4 5 6 7 8 9 1-00 0.00 0.14 0.28 0.42 0.56 0.70 0.83 0.97 1. 11 1.25 1-01 1-39 i-53 1.67 1.81 1.94 2.08 2.22 2.36 2.50 2.63 1-02 2.78 2.91 3-05 3-19 3-32 3-46 3.60 3-74 3-87 4.01 1-03 4. IS 4.29 4.42 4.56 4.69 4.83 4-97 5.10 5- 2 4 5-37 1-04 5-52 S-65 5-79 5-92 6.06 6.20 6-33 6.47 6.60 6.74 1-05 6.87 7.01 7.14 7.28 7.41 7-55 7.68 7.82 7-95 8.09 1-06 8.22 8.36 8.49 8.62 8.76 8.89 9-03 9.16 9.29 9-43 1-07 9-56 9.69 9-83 9.96 10.09 10.23 10.36 10.49 10.62 10.76 1-08 10.89 11.02 11. 16 11.29 11.42 »-55 11.68 11.82 "•95 12.08 1-09 12.21 12.34 12.47 12.61 12.74 12.87 13.00 I3.I3 13.26 13-39 1-10 '3-53 13-66 13-79 13.92 14.05 14.18 14-31 14.44 I4.57 14.70 1-11 14.83 14.96 15.09 15.22 15-35 15.48 15.61 15-74 15.87 16.00 1-12 16.12 16.25 16.38 16.51 16.64 16.77 16.90 17-03 17.16 17.29 113 17.41 17-54 17.67 17.79 17.92 18.05 18.18 18.31 18.43 18.56 1-14 18.69 18.81 18.94 19.07 19.19 19.32 19-45 19.58 19.70 19.83 1-15 19.96 20.08 20.21 20.33 20.46 20.59 20.71 20.84 20.96 21.09 1-16 21.22 21.34 21.47 21.59 21.72 21.84 21-97 22.09 22.22 22.34 1-17 22.47 22.59 22.72 22.84 22.96 23.09 23.21 23-34 23.46 23-58 1-18 23.71 23-83 23-95 24.08 24.20 24.32 24-45 24-57 24.69 24.81 1-19 24.94 25.06 25.19 25-31 25-43 25-55 25.67 25.80 25.92 26.04 1-20 26.16 26.28 26.41 26.53 26.65 26.78 26.90 27.02 27.14 27.27 1-21 27-39 27.51 27.63 27.76 27.88 28.00 28.12 28.24 28.37 28.49 1-22 28.61 28.73 28.85 28.97 29.09 29.22 29-34 29.46 29.58 29.70 1-23 29.82 29.94 30.06 30.18 30.30 30.43 30.55 30.67 3°-79 30.91 1-24 31-03 3I-I5 31.27 31.40 3i-52 31.64 31.76 31.88 32.01 32-13 1-25 32.25 32-37 32-49 32.60 32.72 32.84 32.96 33-oS 33-19 33-31 1-26 33-43 33-55 33-67 33-79 33-91 34-03 34-14 34.26 34-38 34-50 1-27 34.62 34-74 34-85 34-97 35-09 35-21 35-32 35-44 35-56 35-67 1-28 35-79 35-91 36.02 36.14 36-25 36.37 36.49 36.60 36.72 36.83 1-29 36-95 37-07 37-18 37.30 37-42 37-54 37-65 37-77 37.89 38.00 1-30 38.12 38-23 38.35 38.46 38.57 38.69 38.80 38.91 39.02 39-14 1-31 39-25 39-36 39-47 39-59 39-70 39.8i 39-92 40.03 40.15 40.26 1-32 40-37 40.48 40.60 40.71 40.82 40.94 41.05 41.16 41.27 41-39 1-33 41.50 41.61 41.72 41.83 41.94 42.06 42.17 42.28 42.39 42.50 1-34 42.61 42.72 42.83 42.94 43-05 43-17 43-28 43-39 43-50 43.61 0S0 J 1 2 3 4 5 6 7 8 9 8i 0S0 1-35 1 2 3 4 5 6 7 8 9 1 43-72 43-83 43-94 44.04 44.15 44.26 44-37 44.48 44.58 44.69 1-36 44.80 44.91 45.01 45.12 45-23 45-34 45-44 45-55 45.66 45-76 1-37 45-87 45-98 46.08 46.19 46.29 46.40 46-51 46.61 46.72 46.82 1-38 46.93 47.04 47.14 47.25 47-35 47.46 47.56 47.67 47-77 47-88 1-39 47.98 48. oS 48. 18 48.29 48-39 48.49 48.59 48.69 48.80 48.90 J 1-40 49.00 49.10 49.21 49-31 49.41 49.52 49.62 49.72 49.82 49-93 1-41 50.03 5°-'3 50.23 50.33 50.43 50.54 50.64 50.74 50.84 5°-94 1-42 51.04 51.14 51.24 51-34 51-44 51-54 51.64 51-74 51.84 51-94 1-43 52.04 52-14 52.24 52-33 52-43 52-53 52.63 52-73 52.82 52-92 144 53.02 53-12 53-22 53-31 53-41 53-51 53-6i 53-71 53-80 53-90 1-45 54.00 54.10 54-19 54-29 54-38 54.48 54-58 54.67 54-77 54.86 1-46 54.96 55.06 55-15 55-25 55-34 55-44 55-53 55-63 55-72 55-82 1-47 55-91 56.00 56.10 56.19 56.29 56.38 56.47 56.57 56.66 56.76 1-48 56.85 56-95 57-04 57-14 57-23 57-33 57-43 57-52 57.62 57-71 1-49 57.81 57-9Q 58. 00 5S.09 58.19 58.28 58.37 58.47 58.56 58.66 1-50 58-75 58.84 5S.93 59.02 59.11 59.20 59.29 59.38 59-47 59.56 1-51 59-65 59-74 59-83 59.92 60.01 60.10 60.19 60.28 60.37 60.46 1-52 60.55 60.64 60.73 60. S2 60.91 61.00 61. oS 61.17 61.26 6i-35 1-53 61.44 6i-53 61.62 61.71 61.80 61. S9 61.97 62.06 62.15 62.24 154 62-33 62.42 62.51 62.60 62.69 62.78 62.86 62.95 63.04 63-I3 1-55 63.22 63-31 63.40 63.48 63-57 63.66 63-75 63.84 63.92 64.01 1-56 64.10 64.19 64.27 64.36 64.45 64.54 64.62 64.71 64.80 64.88 1-57 64.97 65.06 65.14 65-23 65-3I 65.40 65.49 65-57 65.66 65.74 1-58 65.83 65.92 66.01 66.09 66.18 66.27 66.36 66.45 66.53 66.62 1-59 66.71 66.80 66.88 66.97 67.05 67.14 67.23 67-31 67.40 67.48 1-60 67.57 67.66 67.74 67.83 67.92 68.00 6S.09 68.18 68.27 68.35 1-61 68.44 68.53 68.61 68.70 68.78 68.87 68.95 69.04 69.12 69.21 1-62 69.29 69-37 69.46 69.54 69.63 69.71 69.79 69. 8S 69.96 70.05 1-63 70.13 70.22 70.30 7o.39 70.47 70.56 70,64 7o.73 70.81 70.90 1-64 70.98 71.06 71.14 71-23 7i-3i 71-39 71-47 71-55 71.64 71.72 1-65 71.80 71.88 71-97 72.05 72.14 72.22 72.30 72.39 72.47 72.56 1-66 72.64 72.72 72.S1 72.89 72.98 73.06 73-14 73-23 73-31 73-40 167 73-48 73-57 73-65 73-74 73-82 73-91 73'99 74.08 74.16 74-25 1-68 74-33 74.41 74.50 74-58 74.66 74-75 74-83 74-91 74-99 75.0S 1-69 75.16 75-24 75-33 75-41 75-5o 75-58 75-66 75-75 75-83 75.92 1-70 76.00 76.08 76.16 76.24 76.32 76.41 76.49 76.57 76.65 76.73 1-71 76.81 76.89 76.97 77-o5 77-13 77.22 77-30 77-38 77.46 77-54 1-72 77.62 77.70 77-78 77-85 77-93 78.01 78.09 78.17 78.24 78.32 oSo 1 2 3 4 5 6 7 8 9 H 2 S0 4 x 0-8164 SO3X 1.2249 = S0 S . Log. = .9119- 1. H 2 S0 4 . Log. = .0881. 82 AUXILIARY TABLES. Table (1). For the reduction of a specific gravity "S" to the Vacuum; the weighings being supposed to be made at 15° and 760 mm. pressure. The reduced value (S) is less, than the value S found, by the quantity given in the table as A ' (S) = S-A'; S = (S)+A' s A' S A' 1. 00 Nil. 1.49 .000 6 1.08 .000 1 i-S7 .000 7 1.16 .000 2 1.65 .000 8 1.24 .000 3 i-73 .000 9 i-33 .000 4 1.82 .001 1.41 .000 s 1.90 .001 1 For the given S substitute the nearest tabular S and adopt the correction given opposite to it. The result is right within ± .000 05. Table (2). To pass from 15 S to S, i.e., from water of 15 to water of 0° as a standard. S = 15 S - A " and 16 S = S + A " s A" S A* (1.00 .OOO 711) 0.98 .OOO 7 i-55 .OOI I i-i3 .000 8 1.69 .001 2 1.27 .000 9 1.83 .OOI 3 ..4. .001 1.97 .OOI 4 83 Table (3). *Dependence of the Specific Gravity of an Acid on its own Temperature. To pass from C S to c S t subtract A'" per degree of tetnperature. (Up to t= 15°) A'" is a function of S, but independent of the tempera- ture " c " of the water which serves as a standard. Example: S - S t = 1B S • 15^t The Table gives the values of 1000 x A'", i.e., the changes in S if the sp. gr. of water be taken as = 1000. So Second Decimal in S. 3 4 5 6 05 .09 38 .40 59 -6o 70 .71 79 -8o 86 .86 88 .89 9i • 13 .16 .20 •23 27 •3° •33 • •44 .46 .48 ■So 53 •54 •56 • .61 •63 .64 •65 66 .67 .67 •73 •73 •74 •75 75 .76 •77 • .80 .81 .81 .82 83 .84 •85 • .86 .87 •87 .87 87 .88 .88 . .89 .89 .90 .91 9i .91 .91 . 35 57 68 78 85 83 9i Ex. — For S=i.34, the correction is 0.001x0.74=0.00074 for 1° and consequently 5x0. 00074 = o. 003 7 for 5°. Examples to show how the above Tables are used. I. 1 vol. of water of 15° weighs 1.0000. The same volume of a sulphuric acid of 10° weighs 1.5000 in air of 15" and 760mm. To find the percentage of H 2 S0 4 , apply the following corrections : — (1) To reduce to the vacuum, add (by Table 1), - .000 6 (2) To pass from 15 S 10 to S 10 , add (by Table 2), -.001 1 (3) To pass from S 10 to S , add (by Table 3), + .008 6 Sum of corrections, + .006 9 Hence S = 1.5069; and hence, by Table A, p = 59.37; or rather 59.4 because the second place is utterly uncertain. II. The ^-corrected value for 16 S 15 is 1.5. Find p. In this case we have (1) for the vacuum-correction, -.000 6 (2) To pass from 16 S 15 to S 15 , - .001 1 (3) To pass from S 15 to S , +.012 9 Sum of corrections, +.011 2 Hence S = 1.5112 ; and thence, by Table A, p = 59-76 ; or rather 59.8. For cases like the second, the following Table B gives p directly or by an easy interpolation : — * Derived from a few data of Bineau's by graphic interpolation. His numbers an nnlv to the third place. 8 4 B. Relation between 15 S 15 (not reduced to the Vacuum) and Percentage p of H 2 S0 4 . 15S15 P Diff. AS for a P = i% 16S15 P Diff. AS for Ap=i% 1-00 0.00 1-35 44.78 1. II •0090 1-01 1.46 1.46 •0068 1-36 45.86 1.08 •0093 1-02 2.92 1.46 •0068 1-37 46-93 1.07 •0093 1-03 4-35 i-43 •0070 1-38 47-99 1.06 •0094 1-04 5.80 i-45 •0069 1-39 49.04 1.05 •0095 1-05 7.21 1. 41 •0071 1-40 50.06 1.02 •0098 1-06 8.62 1.41 •0071 1-41 5!-°9 1.03 •0097 1-07 10.02 1.40 •0071 1-42 52.09 1. 00 •0100 1-08 11. 41 i-39 •0072 1-43 53-°9 1. 00 •0100 1-09 12.77 1.36 •0074 1-44 54-°7 O.98 •0102 1-10 14.14 i-37 •0073 1-45 55-°5 0.98 •0102 1-11 15-49 i-35 •0074 1-46 56.00 °-95 ■0105 1-12 16.82 i-33 •0075 1-47 56.95 °-95 •0105 1-13 18.16 i-34 •0075 1-48 57.88 °-93 •0108 1-14 19.47 i-3i •0076 1-49 58.83 0-95 •0105 1-15 20.77 1.30 •0077 1-50 59-76 o-93 •0108 1-16 22.06 1.29 •0078 1-51 60.66 0.90 •0111 1-17 23-34 1.28 •0078 1-52 61.55 0.89 •0112 1-18 24.60 1.26 •0079 1-53 62.44 0.89 ■0112 1-19 25.86 1.26 •0079 1-54 63-33 0.89 •0112 1-20 27.10 1.24 •0081 1-55 64.22 0.89 ■0112 1-21 28.34 1.24 ■0081 1-56 65.09 0.87 •0115 1*22 29.58 1.24 •0081 1-57 65.96 0.87 •0115 1-23 30.81 1.23 •0081 1-58 66.81 0.85 •0118 1-24 3 2 -°3 1.22 •0082 1-59 67.69 0.88 •0114 1-25 33-27 1.24 •0081 1-60 68-55 0.86 ■0116 1-26 34-45 1.18 ■0085 1-61 69.42 0.87 •0115 1-27 35-65 1.20 •0083 1-62 70.26 0.84 ■0119 1-28 36.83 1. 18 •0085 1-63 71.10 0.84 •0119 1-29 37-99 1.16 •0086 1-64 7i-95 0.85 ■0118 1-30 39-17 1.18 ■0085 1-65 72.76 0.81 ■0123 1-31 40.30 113 •0088 1-66 73.60 0.84 •0119 1-32 41.42 1. 12 •0089 1-67 74-44 0.84 ■0119 1-33 42.56 1. 14 •0088 1-68 75- 2 9 0.85 •0118 1-34 43-67 1. 11 •0090 1-69 76.12 0.83 •0120 1-70 76.96 0.84 •0119 Examples. — To 8=1.546 corresponds the percentage 63-33 + 0.6x0.89 = 63.33 + 0.53 = 63.86; or rather 63.9. To the percentage 60.0 (as the S for 59.76 is 1.5) corresponds S = 1. 50 + . 24 x. 0111 = 1. 50 + . 0027 = 1. 5027; or rather 1.503. 85 C. Relation in Sulphuric Acid between Specific Gravity S or degrees Baume" (°B), and Percentage of H 2 S0 4 ; according to Bineau (Ann. de Chim. et de Phys. (3) vol. 26, page 125; Year 1849.) Percents. Percents. °B Spec. Grav. Diff. H 2 SO4 if Temp, of Acid=o° Diff. H 8 SO4 if Temp, of Acid = 15° Diff. 59 1. 691 75-2 76.3 60 1. 711 .020 76.9 i-7 78.0 i-7 61 1.732 .021 78.6 i-7 79.8 1.8 62 1-753 .021 80.4 1.8 81.7 1.9 63 1-774 .021 82.4 2.0 83-9 2.2 64 1.796 .022 84.6 2.2 86.3 2.4 65.0 1. 819 .023 87.4 2.8 89-5 3-2 65-5 1.830 .Oil 89.I i-7 91.8 2-3 65.8 1-837 .007 90.4 i-3 94-5 2.7 66.0 1.842 .005 91-3 0.9 1 00.0 5-5 66.2 1.846 .004 9 2 -5 1.2 66.4 1.852 .006 95-° 2-5 66.6 1-857 .005 100.0 5-° D. Specific Gravities of most highly Concentrated Sulphuric Acids, according to G. Lunge and Naef. Temperature of Acid =15°, that of Water = 4°. a. Direct Observations. Percents. of H2SO4 90.20 91.48 92.83 94.84 95-97 97.70 98-39 98.66 99-47 100.00 4S15 1.8195 1. 8271 1-8334 1.8387 1.8406 1-8413 1.8406 1.8409 1-8395 1.8384 b. Working Table, by Interpolation from a. Percents. of Spec. Grav. Diff. H2SO4 4S15 90 1.8185 — 91 1. 8241 .0056 92 1.8294 •oo53 93 I-8339 .0045 94 1.8372 •0033 95 I.8390 .0018 96 I.8406 - .0016 97 1. 8410 .0004 98 1. 8412 .0002 99 I.8403 - .0009 100 1.8384 -.0019 The above Table of Lunge's and the following Tables on Hydro- chloric Acid and Nitric Acid are extracted from Lunge and Hurter's " Alkali Makers' Pocket-Book":— 86 II. Hydrochloric Acid. (a) Calculated after Kolb's determinations. isSis Percents. of HC1. I6S15 Percents. of HC1. I.OO O.00 1. 10 20.06 I.OI 2.12 1. 11 22.06 1. 02 4.11 1. 12 24.05 1.03 6.11 i-i3 26.04 1.04 8.10 1. 14 28.04 1.05 10.09 i-iS 30-03 1.06 12.09 1. 16 32.02 1.07 14.08 1.17 34.02 1.08 16.07 1. 18 36.01 1.09 18.07 1. 19 38.01 1. 10 20.06 1.20 40.00 Approximately : Ap = 200 x A S, and A S = 0.005 x A p j or percentages of HC1 are equal to the number of degrees Twaddle. The following Table, which is based on determinations made in my laboratory by Mr. Kling, covers only a small interval, but, as far as it goes, probably affords a higher degree of precision than Kolb's : — Table for finding the Percentage p of HC1 from the Specific Gravity 15 S at io° to 20°. (b) Calculated by W. Dittmar, after determinations by Arch. Kling. ForAt=i°, ForAt = l', 15S15 p Ap AS = 15S15 P Ap AS = (0.000 1 X ) (0.0001 X ) 1.085 16.96 3-o 1. 100 I9.85 .19 3-5 1.086 17.16 .20 3-o I.IOI 20.04 .19 3-6 1.087 17-35 .19 3-o 1. 102 20.24 .20 3-6 1.088 17-54 .19 3-i 1-103 2O.43 •19 3-6 1.089 17-73 .19 3-i 1. 104 20.62 .19 3-7 1.090 17-93 .20 3-2 1-105 20.8l .19 3-7 1. 091 18.12 •19 3-2 1. 106 2I.OI .20 3-8 1.092 18.31 .19 3-2 1. 107 21.20 .19 3-8 1.093 18.50 •19 3-3 1. 108 2I.39 ■19 3-8 1.094 18.70 .20 3-3 1. 109 2I.58 .19 3-9 1.095 18.89 •19 3-3 1. no 2I.78 .20 3-9 1.096 19.08 •19 3-4 I. Ill 21.97 .19 3-9 1.097 19.27 .19 3-4 1. 112 22.l6 .19 4.0 1.098 19.47 .20 3-5 1. 113 22 -35 .19 4.0 1.099 19.66 .19 3-5 1. 114 22 -55 .20 4.1 Water of 15° is the standard for S throughout; the values S are not reduced to the Vacuum. 87 III. Hydrobromic Acid. According to Topsoe; Ber. der Deutschen Chem. Gesellschaft, 1870, 404. t tS t %ofHB 14° i-°55 7.67 14° i-°75 10.19 1 4° 1.089 11.94 1 4° 1.097 12.96 1 4° 1. 118 15-37 14° 1. 131 16.92 O 14 1. 164 20.65 13° 1.200 24-35 1 3° 1.232 27.62 *3° i-253 29.68 i3° 1.302 33-84 i3° i-335 36.67 1 3° 1-349 37.86 13° 1.368 39-13 13° 1-419 43.12 13 I-43I 43-99 *3° 1.438 44.62 1 4° I-45 1 45-45 13° 1.460 46.09 14 1.485 47.87 14 1.490 48.17 IV. Hydriodic Acid. According to Topsoe ; Ber. der Deutschen Chem. Gesellschaft, 1870, 403. 13-5 i3-5° 1 3° !3-5° i3-5° i3-5° 13-8° 13-8° 13-5° !3-5° 13° 13° 13° 13° o 13 13° 13° i3-5° 1 3° 12-5° o 14 13-7" o 13 12.5° 13-7' 12° tS, %ofHI. 1. 017 2.29 1.052 7.02 1.077 10.15 1.095 12.21 1. 102 13.09 1. 126 15-73 1. 164 19.97 1. 191 22.63 1.225 25.86 1.254 28.41 1.274 30.20 1.309 33-°7 1-347 36.07 1.382 38.68 I-4I3 4°-45 i-45i 43-39 1.487 45-7i 1.528 48.22 1.542 49-13 i-573 5°-75 1.603 52.43 1.630 53-93 1.674 s 6 -^ 1.696 57.28 1.703 57-42 1.706 57-64 1.708 57-74 88 V. Nitric Acid. After determinations by Kolb. Percents 1 wSl6 p of HNO3 DifF. AS for AP=I% isSis P Diff. AS for AP=I% 1-00 0.00 1-30 47-57 i-53 •0065 1-01 i-7S 1-75 ■0057 1-31 49.10 J -53 •0065 1-02 3-5° i-75 •0057 1-32 50-63 i-53 •0065 1-03 5-25 i-75 •0057 1-33 52.24 1.61 ■0062 1-04 6-95 1.70 •0059 1-34 53-94 1.70 •0059 1-05 8-59 1.64 •0061 1-35 55-64 1.70 ■0059 1-06 10.23 1.64 •0061 1-36 57-42 1.78 •0056 1-07 11.88 1.65 •0061 1-37 59.21 1.79 •0056 1-08 I3-S2 1.64 ■0061 1-38 61.00 1.79 •0056 1-09 15.16 1.64 •0061 1-39 62.95 i-95 •0051 1-10 16.80 1.64 •0061 1-40 64.90 i:95 •0051 1-11 i8.35 i-55 •0065 1-41 67.12 2.22 •0045 1-12 19.89 i-54 •0065 1-42 69-34 2.22 •0045 1-13 21.44 i-55 •0065 1-43 71-83 2.49 •0040 1-14 22.98 1-54 •0065 1-44 74-59 2.76 ■0036 1-15 24-53 i-55 •0065 1-45 77-36 2.77 ■0036 1-16 26.08 !-55 •0065 1-46 80.13 2.77 •0036 1-17 27.62 1-54 •0065 1-47 82.90 2.77 •0036 1-18 29.17 i-55 •0065 1-48 85.66 2.76 •0036 1-19 30.71 1-54 •0065 1-49 88.43 2.77 •0036 1-20 32.26 i-55 •0065 1-50 91.20 2.77 •0036 1-21 33-79 J-53 •0065 1-51 94-13 2-93 ■0034 1-22 35-32 i-53 •0065 1-52 97.06 2-93 ■0034 1-23 36.85 i-53 •0065 1-53 100.00 2.94 •0034 1-24 38.38 i-53 •0065 1-25 39.91 i-53 •0065 1-26 41.44 i-53 ■0065 1-27 42.97 i-53 •0065 1-28 44-5° i-S3 •0065 1-29 46.04 i-54 .0065 2HI JO3X 0-8572 = N 2 6 ; Log. = .9331 - 1. N 2 5 x 1-1666 = 2HNO s ; Log. = 1.066 9- 8 9 VI. Specific Gravities of Caus- tic Potash Leys, at 15°. (Calculated by Gerlach from Determina- tions by Tunnermann and by H. Schiff.) Per- cents ofKHO o 1 2 3 4 5 6 10 11 12 13 14 iS 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 3i 32 33 34 Spec. Gr. 1. 000 1.009 1. 017 1.025 I-033 1. 041 1.049 1.058 1.065 1.074 1.083 1.092 1.101 I. no 1. 119 1. 128 i-i37 1. 146 i-i55 1. 166 1.177 1. 188 1.198 1.209 1.220 1.230 1. 241 1.252 1.264 1.276 1.288 1.300 1.311 i-3 2 4 i-336 Per- cents ofKHO Spec. Gr. 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 5° 5i 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 1-349 1. 361 1-374 1-387 1.400 1. 412 1-425 1.438 1 "45° 1.462 1-475 1.488 1.499 1.511 1-525 i-539 I-55 2 i-565 i-578 1.590 1.604 1. 618 1.630 1.642 i-655 1.667 1.681 1.695 i-7o5 1. 718 1.729 1.740 i-754 1.768 1.780 2KHO x o Log.= K 2 Ox 1. 19 T r\rr = 70 1-790 8396 = K 2 0; 9241 - 1. io = 2KHO; riica. VII. Specific Gravities of Caustic Soda Leys, at 15". (Calculated by Gerlach from Determina- tions by H. Schiff.) Per- cents of NaOH 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 3° 3 1 32 33 34 Spec. Gr. I. OOO 1. 012 1.023 i-°35 1.046 1.058 1.070 1. 081 1.092 1. 103 1.115 1. 126 i-i37 1. 148 i-i59 1. 170 1. 181 1. 192 1.202 1. 213 1.225 1.236 1.247 1-.258 1.269 1.279 1.290 1.300 1. 310 1. 321 r-332 1-343 1-353 1-363 1-374 Per- centsofl Spec. Gr. NaOH 1. 000 35 1 1. 012 36 2 1.023 37 3 1-035 38 4 1.046 39 5 1-058 40 6 1.070 41 42 43 44 45 46 47 48 49 5° 5 1 5 2 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 2NaOHx 0.7752: Log. = .8894 Na 2 0x 1.2900=2 Log. = .1106. 1.384 i-395 1.405 i-4i5 1.426 1-437 1-447 i-457 1.468 1.478 1.488 1.499 1-509 i-5i9 i-5 2 9 i-54o i-55o 1.560 i-57o 1.580 i-59i 1. 601 1. 611 1.622 1-633 1 '643 1.654 1.664 1.674 1.684 i-695 1-705 i-7i5 1.726 1-737 1.748 = Na 2 0; — 1. NaOH; go VIII. Specific Gravities of Aqueous Ammonias at 15", Water of 15° = 1. According to Lunge and Wiernik * 16S16 ( Per- cents ofNH 3 Dne Litre of 15° contains jrammes ofNH 3 Correction of isS for + 1° in the temp of the Aqueous Ammonia. ** 15S16 Per- cents ofNH 3 One Litre of 15° contains Grammes ofNH 3 Correction of «S for + i° in the temp of the Aqueous Ammonia. ** 1. 000 0.00 0.0 0.00018 0.940 IS-63 146.9 0.00039 0.998 o.45 4-5 0.00018 0.938 16.22 I52-I 0.00040 0.996 0.91 9-i O.OOOI9 0.936 16.82 157-4 0.00041 0.994 i-37 13.6 O.OOOI9 °-934 17.42 162.7 0.00041 0.992 1.84 18.2 O.OOO20 0.932 18.03 168.1 0.00042 0.990 2.31 22.9 0.00020 0.930 18.64 173-4 0.00042 0.988 2.80 27.7 0:00021 0.928 I9-2S 178.6 0.00043 0.986 3-3° 32-5 0.00021 0.926 19.87 184.2 0.00044 0.984 3.80 37-4 0.00022 0.924 20.49 189.3 0.00045 0.982 4-3° 42.2 0.00022 0.922 21.12 194.7 0.00046 0.980 4.80 47.0 0.00023 0.920 2i-75 200.1 0.00047 0.978 5-3° 51.8 0.00023 0.918 22.39 205.6 0.00048 0.976 5.80 56.6 0.00024 0.916 23-03 210.9 0.00049 0.974 6.30 61.4 0.00024 0.914 23.68 216.3 0.00050 0.972 6.80 66.1 0.00025 0.912 24-33 221.9 0.00051 0.970 7-3i 70.9 0.00025 0.910 24.99 227.4 0.00052 0.968 7.82 75-7 0.00026 0.908 25- 6 5 232.9 0.00053 0.966 8-33 80.5 0.00026 0.906 26.31 238.3 0.00054 0.964 8.84 85.2 0.00027 0.904 26.98 243-9 0.00055 0.962 9-35 89.9 0.00028 0.902 27.65 249.4 0.00056 0.960 9.91 9S-i 0.00029 0.900 28.33 255-o 0.00057 0.958 10.47 100.3 0.00030 0.898 29.01 260.5 0.00058 0.956 11.03 105.4 0.00031 0.896 29.69 266.0 0.00059 °-954 11.60 110.7 0.00032 0.894 3°-37 27i-5 0.00060 0.952 12.17 "5-9 0.00033 0.892 3i-o5 277.0 0.00060 0.950 12.74 121. 9.00034 0.890 3i-75 282.6 0.00061 0.948 i3-3i 126.2 0.00035 0.888 32-50 288.6 0.00062 0.946 13.88 i3!-3 0.00036 0.886 33-25 294.6 0.00063 0.944 14.46 136.5 0.00037 0.884 34- 10 3°i-4 0.00064 0.942 i5-°4 i4i-7 0.00038 0.882 34-95 3°8-3 0.00065 * Zeitschrift fiir Angewandte Chemie, i£ Professor Lunge. ** For the interval 13 to 15°. Heft 7. — Kindly communicated by 9i IX. Table giving the Relation in Aqueous Ethyl-Alcohol between Specific Gravity at is°C. on the one hand, and the Percentage by Weight of Real Alcohol, or the Specific Gravity at any Temperature from io° to 20 on the other. Calculated after Mendelejeff's Determinations, by W. Dittmar. Water of i5°C. = 1000. Diff. in Spec. Diff. in Spec. Spec. Grav. at 15.° Percent- age Ap for AS = Grav. for Spec. Grav. at 15.° Percent- age Ap for AS = Grav. for 15S16 P- I Ap = I At=I° lb$15 P- 1 Ap= 1 At=i° IOOO 0.00 0.144 975 I7-25 .84 I.19 •33° 999 °-53 ■53 I.89 .146 974 18.08 •83 1.20 •348 998 1.06 •53 1.89 .147 973 18.91 •83 I.20 ■367 997 1. 61 •55 I.82 .150 972 19-73 .82 1.22 •387 996 2.16 •55 I.82 ■152 971 20.54 .81 I.23 •4°5 995 2-73 •57 i-75 ■154 970 21-35 .81 I.23 .421 994 3-3° •57 i-75 • J 57 969 22.13 •78 1.28 •434 993 3-9i .61 1.64 .161 968 22.89 .76 I.32 •45° 992 4-5i .60 1.67 .166 967 23-65 •76 1.32 .466 991 5.12 .61 1.64 .170 966 24-39 •74 i-35 .480 990 5-77 •65 i-54 .174 965 25-I3 •74 i-35 .496 989 6-43 .66 i-52 .180 964 25.84 •7i 1.41 .510 988 7.09 .66 1.52 .186 963 26.53 .69 i-45 .522 987 7.78 .69 i-45 .192 962 27.23 .70 i-43 •535 986 8.49 •7i 1.41 .199 961 27.90 .67 1.49 •548 985 9.20 •7i 1.41 .206 960 28.55 •65 i-54 .560 984 9.96 .76 1.32 .214 959 29.21 .66 1.52 •57° 983 10.72 .76 1.32 .223 958 29.84 •63 i-59 .581 982 11.50 .78 1.28 .231 957 30.46 .62 1. 61 •593 981 12.28 .78 1.28 .242 95 6 31.09 •63 i-59 .604 980 13.10 .82 1.22 • 254 955 31.69 .60 1.67 .614 979 13.92 .82 1.22 .267 954 32-3° .61 1.64 •623 978 14.74 .82 1.22 .281 953 32.88 •58 1.72 .632 977 iS-58 .84 1. 19 .296 952 33-45 •57 I -75 .641 976 16.41 •83 1.20 •3" 95i 34-03 •58 1.72 ■650 92 ETHYL-ALCOHOL. Diff. in Spec. Diff. in Spec. Spec. Grav. at 15.° Percent- age Ap for AS = Grav. for Spec. Grav. at 15." Percent- age Ap for AS = Grav. for 15S15 P- I Ap= 1 At=I° 15S15 P- I Ap = I At=i° 95° 34-59 •56 1.79 •657 915 51.68 •45 2.22 .800 949 35-15 •56 1.79 .666 914 s 2 -^ •45 2.22 .801 948 35-7° ■55 1.82 •673 913 52.58 •45 2.22 .802 947 36.25 •55 1.82 .680 912 53-02 •44 2.27 .803 946 36.78 •53 1.89 .687 911 53-47 •45 2.22 .804 945 37-31 •53 1.89 •694 910 53-91 •44 2.27 .806 644 37-84 •53 1.89 .700 909 54-35 •44 2.27 .807 943 38.36 •52 1.92 .707 908 54.80 •45 2.22 .808 942 38.88 •52 1.92 .712 907 55-24 •44 2.27 .809 941 39-39 ■5i 1.96 .718 906 55-68 •44 2.27 .810 940 39-9° •5i 1.96 •723 9°5 56.12 •44 2.27 .811 939 40.40 ■5° 2.00 .727 904 56.56 •44 2.27 .812 938 40.90 •5° 2.00 •733 9°3 57.00 •44 2.27 .813 937 41.40 •5° 2.00 •737 902 57-43 ■43 2-33 .813 936 41.89 •49 2.04 .741 901 57-87 •44 2.27 .813 935 42.38 •49 2.04 .746 900 58.31 •44 2.27 .814 934 42.86 .48 2.08 •75o 899 58.74 •43 2-33 .815 933 43-35 •49 2.04 •753 898 59.18 •44 2.27 .816 932 43-83 .48 2.08 •756 897 59.61 •43 2-33 .817 93 1 44-3° •47 2.13 .760 896 60.05 •44 2.27 .817 93° 44-78 .48 2.08 •763 895 60.48 •43 2-33 .817 929 45-25 •47 2.13 .767 894 60.91 •43 2-33 .818 928 45-73 .48 2.08 .770 893 6i-35 •44 2.27 .819 927 46.20 •47 2.13 •773 892 61.78 •43 2-33 .819 926 46.66 .46 2.17 •774 891 62.21 •43 2-33 .820 925 47-13 •47 2.13 •778 890 62.64 •43 2-33 .820 924 47-59 .46 2.17 .781 889 63.07 •43 2-33 .820 9 2 3 48.05 .46 2.17 •783 888 63-5° •43 2-33 .820 922 48.51 .46 2.17 .786 887 63-93 •43 2-33 .821 921 48.96 •45 2.22 .788 886 64.36 •43 2-33 .822 920 49.42 .46 2.17 ■79° 885 64.79 •43 2-33 .822 919 49.88 .46 2.17 .792 884 65.21 .42 2.38 .823 918 5°-33 •45 2.22 •794 883 65.64 •43 2-33 .824 917 50.78 •45 2.22 .796 882 66.07 •43 2-33 .825 916 5!- 2 3 .45 2.22 45 | | .798 881 66.49 .42 2.38 .826 ETHYL-ALCOHOL. 93 Diff. in Spec. Diff. in Spec. Spec. Grav. Percent- Ap for Grav. for Spec. Grav. Percent- age Ap for Grav. for at IS." age AS = at 15.° AS = 15S15 P- I Ap= 1 At=I° tfSl6 P- I Ap = I At=l° 880 66.92 ■43 2-33 .825 845 81.46 .40 2.50 .854 879 67-34 .42 2.38 .826 844 81.86 .40 2.50 •855 878 67.77 •43 2-33 .826 843 82.26 •40 2.50 .856 877 68.20 •43 2-33 .827 842 82.66 .40 2.50 •857 876 68.62 .42 2.38 .828 84I 83.06 .40 2.50 .858 875 69.04 .42 2.38 .829 84O 83.46 .40 2.50 •859 874 69.46 .42 2.38 .830 839 83.86 .40 2.50 .860 873 69.88 .42 2.38 .830 838 84.26 .40 2.50 .860 872 70.30 .42 2.38 .830 837 84.65 •39 2.56 .860 871 7°-73 •43 2-33 .831 836 85-05 .40 2.50 .861 870 7I-I5 .42 2.38 .832 835 85-44 •39 2.56 .862 869 7i-57 .42 2.38 •833 834 85-83 •39 2.56 .862 868 71.99 .42 2.38 •834 833 86.2 2 •39 2.56 .862 867 72.40 .41 2.44 •835 832 86.61 ■39 2.56 .863 866 72.82 .42 2.38 .836 831 87.00 •39 2.56 .863 865 73-24 .42 2.38 .836 83O 87.38 ■38 2.63 .864 864 73.66 .42 2.38 •837 829 87.77 ■39 2.56 ■865 863 74.08 .42 2.38 .838 828 88.15 ■38 2.63 .865 862 74-49 .41 2.44 •839 827 88.53 •38 2.63 .865 861 74.90 .41 2-44 .840 826 88.91 •38 2.63 .865 860 75-32 .42 2.38 .840 825 89.29 •38 2.63 .865 859 75-73 .41 2.44 .840 824 89.67 ■38 2.63 .866 858 76.15 .42 2.38 .841 823 90.05 •38 2.63 .866 857 76.56 .41 2.44 .842 822 90.42 •37 2.70 .866 856 76.97 •4i 2-44 .842 821 90.79 •37 2.70 .866 855 77-38 .41 2.44 .844 820 91.16 •37 2.70 .865 854 77.79 .41 2-44 •845 819 91-53 •37 2.70 .864 853 78.21 .42 2.38 .846 8l8 91.89 •36 2.78 .864 852 78.62 .41 2.44 •847 817 92.26 ■37 2.70 • 863 851 79.02 .40 2.50 .848 8l6 92.62 •36 2.78 .862 850 79-43 .41 2.44 .848 815 92.98 •36 2.78 .862 849 79.84 .41 2.44 .849 814 93-34 •36 2.78 .861 848 80.24 .40 2.50 .850 813 93-7° ■36 2.78 .861 847 80.65 .41 2-44 .852 812 94-°5 •35 2.86 .860 846 81.06 .41 2.44 •853 8ll 94.41 ■36 2.78 ■859 94 ETHYL-ALCOHOL. Diff. in Spec. 1 Diff. in Spec. Spec. Grav. at 15.° Percent- age Ap for AS = Grav. for Spec. Percent . Grav. arro $2 aS = Grav. tor at 15.° isSis P- 1 Ap= 1 At=I° I6S15 P- 1 Ap = I At=I° 810 94.76 ■35 2.86 •859 800 98.16 •33 3-03 .850 809 95-11 •35 2.86 •859 799 98.49 •33 3-03 .849 808 95-45 ■34 2.94 •858 798 98.81 •32 313 .848 807 95.80 35 2.86 •857 797 99.14 •33 3-°3 .847 806 96.15 •25 2.86 .856 796 99.46 •32 3-i3 .846 805 96.48 •33 3-°3 .856 795 99.78 •32 3-13 .845 804 96.82 •34 2.94 •855 794 100 10 •32 3-i3 .844 803 97.16 •34 2.94 .854 794-32 100 802 97-49 •33 3-o3 .852 801 97-83 •34 2.94 .851 Referring to water of 4°C. as = 100 000, the specific gravity 4 S, of absolute alcohol, according to Mendelejeff, is 4 S t = 80 625 - 83.4 t - 0.029 t 2 ; hence (by Mendelejeff's computation) we have for the Specific Gravities of Absolute Alcohol — t 0° 5° 10° iS° 20° 25° 30° *St 806 25 802 07 79788 793 67 789 45 785 22 780 96 In regard to Change of Standard, see p. 97. Note. — The specific gravities in our table, though given to only 3 decimals, are correct arithmetically, to within less than .000 02. Thus, for instance, the specific gravity of 87 per cent. Alcohol is .831 00 + .000 02 at the worst. 95 Table X.* Ethyl> Alcohol; to pass from Percents of Alcohol by Weight, p , to Percents by Volume (p), at is°C. Percents by Percents by Percents by Diff. Diff. Diff. Weight Volume Weight Volume Weight Volume P (p) P (p) P (P) O 0.00 35 41.83 1. 12 70 76.91 .89 I 1.26 I.26 : 36 42.94 I. II 71 77.80 •89 2 2.51 1-25 37 44-05 I. II 72 78.68 .88 3 3-76 I.25 38 45-15 1. 10 73 79-55 .87 4 5.00 I.24 39 46.24 1.09 74 80.42 •87 5 6.24 I.24 40 47-33 I.09 75 81.27 •85 6 7.48 I.24 41 48.41 1.08 76 82.13 .86 7 8.71 I.23 42 49.48 I.07 77 82.97 .84 8 9-94 I.23 43 50.55 1.07 78 83.81 .84 9 11. 16 1.22 44 51.61 1.06 79 84.64 •83 10 12.39 I.23 45 52.66 1.05 80 85.47 •83 ii 13.61 1.22 46 53-71 I.05 81 86.28 .81 12 14-83 1.22 47 54-75 I.04 8? 87.09 .81 13 16.04 1. 21 48 55-78 I.03 83 87.90 .81 14 17.25 1. 21 49 56.81 I.03 84 88.69 •79 IS 18.46 1. 21 50 57-83 1.02 85 89.47 • 78 16 19.67 1. 21 51 58.85 1.02 86 90.25 .78 17 20.87 I.20 52 59-85 I. OO 87 91.02 •77 18 22.07 I.20 53 60.86 I. OI 88 91.78 .76 19 23.27 1.20 54 61.85 ■99 89 92.52 ■74 20 24.47 1.20 55 62.84 • 99 90 93.26 •74 21 25.66 1. 19 56 63.82 .98 91 93-99 •73 22 26.84 1. 18 57 64.80 •98 92 94-71 • 72 23 28.02 1. 18 58 65-77 •97 93 95.42 • 7i 24 29.20 I.l8 59 66.73 .96 94 96.11 .69 25 30.38 I.l8 60 67.69 .96 95 96.79 .68 26 3i-5S 1. 17 61 68.64 •95 96 97.46 .67 27 32.71 I.I6 62 69.59 •95 97 98.12 .66 28 33-87 1. 16 63 70.52 •93 98 98.76 .64 29 35.02 I-I5 64 71.46 •94 99 99-39 •63 30 36.17 i-i5 65 72.38 ■92 100 100.00 .61 31 37-32 i»i5 66 73-30 .92 32 38-4S 1.13 67 74.21 ■9i 33 39-58 i-i3 68 75.12 •9i 34 40.71 1-13 69 76.02 .90 Calculated by W. Dittmar. 9 6 fable XL* To pass from Percents by Volume at is°C, (p). to Percents by Weight p. Percents by Diff. Percents by Diff. Percents by Diff. Volume Weight Volume Weight Volume Weight (p) P (P) P (p) P O.OO 35 28.98 •87 70 62.44 I.06 I 0.80 .80 36 29-85 -87 71 63-5I I.07 2 i-59 •79 37 30.72 .87 72 64.59 I.08 3 2-39 .80 38 31-60 .88 73 65.68 I.09 4 3.20 .81 39 32.48 .88 74 66.77 I.09 5 4.00 .80 40 33-37 .89 75 67.87 1,10 6 4.81 .81 41 34.26 .89 76 68.98 I. II 7 5.62 .81 42 35-i6 .90 77 70.10 1. 12 8 6-43 .81 43 36.05 .89 78 71.23 i-i3 9 7.24 .81 44 36.96 ■91 79 72.37 1. 14 IO 8-05 .81 45 37-87 .91 80 73-52 1. 15 ii 8.87 .82 46 38.78 •9i 81 74.68 1. 16 12 9.68 .81 47 39-70 .92 82 75-85 1.17 13 10.50 .82 48 40.62 .92 83 77-03 1.18 14 11.32 .82 49 4i-55 •93 84 78.23 1.20 IS 12.14 .82 5o 42.49 •94 85 79-43 1.20 16 12.97 •83 5i 43-43 •94 86 80.65 1.22 17 13-79 .82 52 44-37 •94 87 81.88 1.23 18 14.62 •83 53 45-32 •95 88 83-I3 1.25 19 15-45 .83 54 46.28 .96 89 84.40 1.27 20 16.27 .82 55 47.24 ■96 90 85.68 1.28 21 17.11 .84 56 48.21 •97 9i 86.98 1.30 22 17-94 •83 57 49.19 •98 92 88.30 1.32 23 18.77 •83 58 50.17 .98 93 89.64 i-34 24 19.61 .84 59 5i.i5 .98 94 91.01 i-37 25 20.45 .84 60 52-15 1. 00 95 92.41 1.40 26 21.29 .84 61 53-15 1. 00 96 93.84 1-43 27 22.13 .84 62 54.15 1. 00 97 95-31 i-47 28 22.98 ■85 63 55-i6 1. 01 98 96.82 i-5i 29 23-83 •85 64 56.18 1.02 99 98.38 1.56 30 24.68 ■85 65 57-21 1.03 100 100.00 1.62 31 25-53 •85 66 58.24 1.03 32 26.39 .86 67 59.28 1.04 33 27.25 .86 68 60.33 1.05 34 28.11 .86 69 61.38 1.05 * Calculated by W. Dittmar. 97 CHANGE OF STANDARD. To pass from Water of t' to Water of t" degrees as a Standard, apply the formulae or tables given on pp. 108, 109, 113. But for t = 4°, 15°, 1 5 "56°, the following table is more convenient. To pass from Water of To pass from Water of 4° to Water of 15° or 15.56°, add on the correction given for the nearest tabular S. 15° to Water of '15.56° To Water of 15° S = Correction. To Water of i5-S<5° S = Add on .000 08 to the given value S. The error is less than .000 01 ■71 •83 •95 1.07 .000 7 .000 8 .000 9 .001 ■75 .86 •97 1.08 For the corresponding inverse changes subtract the correction given. Tables IX., X., XI., are deduced from Mendelejeff's determinations, but not quite directly. Landolt and Bornstein's " Tabellen" include (on their page 151), a table which, proceeding from percent to percent, gives the specific gravities isSjs to five decimal places. According to its heading the table is " basirt auf Mendelejeff's Formeln. " From this table I calculated the entries p of my table IX. to, originally, three decimals, by simple interpolation. At the end, the third decimals in the values p were cut off; and in order to see how this and the neglecting of second differences had affected the results, I calculated the specific gravities for all integer p's from my table as it stood (and now stands in this book), and compared the results with the original specific gravities in L. and B.'s tables. The difference amounted to nil in 51, to + I in 44, and to + 2 in five cases ; Water = 100 000. The values — - I calculated from data found in Mendelejeff's original memoirs in Pogg. Annalen, vol. 138; pp. 103 and 230. Tables X. and XI. are calculated directly from the entries in L. and B.'s table. METHYL ALCOHOL. According to Dittmar and Fawsitt,* the specific gravity of Methyl- Alcohol at temperatures from 0° to 64.7 is in accordance with equation : 4 S t = .810 18 - 90.53 t - .085 057 t 2 Their specific gravities for Aqueous Methyl-Alcohols are given in the following table. * Royal Society of Edinburgh Transactions, vol. 33, part II., p. 523, and pp. 531-533. Year 1886-7. 9 8 DlTTMAR AND FAWSITT'S TABLE OF THE SPECIFIC GRAVITIES OF Aqueous Methyl-Alcohols at o° and at is.°s6 G; Water of + 4° = ioo ooo. I. — From o to 30 per cent, of CH 4 0. 4S0 - iSt = at + bt> ; from t = o to t = 20°. Per- Sp. Gravity Diff. b Sp. Gravity Diff. centage. at 0° C. at IS.°S6 C. 999 87 - 6.0 + •705 999 07 1 998 06 -181 5-4 .694 997 29 -178 2 996 31 175 4.8 .681 995 54 175 3 994 62 169 3-9 .670 993 82 172 4 992 99 163 3-o •659 992 14 168 S 991 42 157 — 2.2 .648 990 48 166 6 989 90 152 1.2 •634 988 93 155 7 988 43 147 0.2 .621 987 26 167 8 987 01 142 4- 0.9 .609 985 69 157 9 98S 63 138 2.1 •596 984 14 155 10 984 29 134 + 3-3 .581 982 62 152 n 982 99 130 48 •569 981 11 151 12 981 71 128 6.2 •552 979 62 149 13 980 48 123 7.8 •536 978 14 148 14 979 26 122 9-5 •5i9 976 68 146 15 978 06 120 + 11. .500 975 23 145 16 976 89 117 12.5 .480 973 79 144 17 975 73 Il6 14-5 .461 972 35 144 18 974 59 114 16.2 .440 97° 93 142 19 973 46 "3 18.3 .420 969 50 143 20 972 33 "3 + 20.0 •398 968 08 142 21 971 20 "3 22.2 •373 966 66 142 22 970 07 "3 24-3 •35° 965 24 142 23 968 94 "3 26.4 .321 963 81 143 24 967 80 114 29.0 .291 962 38 143 25 966 65 "5 + 3I-3 .261 960 93 145 26 965 49 116 33-8 .230 959 49 144 27 964 30 119 36.0 .191 958 02 147 28 963 10 120 38.8 •151 956 55 147 29 961 87 123 41.1 .106 955 °6 149 3° 960 57 130 44.0 .063 953 55 151 METHYL-ALCOHOL. II. — From 30 to 100 per cent. 4S0 - 4St = a t ; from t = o to t = 20° 99 Per- Specific Gravity Din". Specific Gravity Diff. centage. at 0° C. at 15-56° C. 30 960 57 -130 + 44-36 953 67 31 959 21 136 45.66 952 II -156 32 957 83 138 46.93 95° 53 158 33 956 43 140 48.17 948 94 159 34 955 °° 143 49-39 947 32 162 35 953 54 146 5o-S8 945 67 165 36 952 04 l5o 5i-75 943 99 168 37 95° 5i 153 52.89 942 28 171 38 948 95 156 54.01 940 55 173 39 947 34 161 55-i° 938 77 178 40 945 7i 163 56.16 936 97 180 4i 944 00 171 57.20 935 10 187 42 942 39 161 58.22 933 35 175 43 940 76 163 59.20 93i 55 180 44 939 " 165 60.17 929 75 180 45 937 44 167 61.10 927 93 182 46 935 75 169 62.01 926 10 183 47 934 03 172 62.90 924 24 186 48 932 29 174 63.76 922 37 187 49 93° 52 177 64.60 920 47 190 5° 928 73 179 65.41 918 55 192 5i 926 91 182 66.19 916 61 194 52 925 07 184 66.95 9 J 4 65 196 53 923 20 187 67.68 912 67 198 54 921 30 190 68.39 910 66 201 55 919 38 192 69.07 908 63 203 56 917 42 196 69.72 9°6 57 206 57 9J5 44 198 7°-35 904 50 207 58 9i3 43 201 70.96 902 39 211 59 9" 39 204 7i-54 900 26 213 60 909 17 222 71.96 897 98 228 61 907 06 211 72-37 895 80 2l8 62 904 92 214 72.91 893 58 222 63 902 76 2l6 73-45 891 33 225 64 900 56 220 73-98 889 05 228 100 METHYL- ALCOHOL. II- — From 30 to 100 per cent. — Continued. Per- centage Specific Gravity at o° C. Diff. a Specific Gravity at 15.56° C. Diff. 65 898 35 - 221 74-51 886 76 - 229 66 896 II 224 75-°5 884 43 233 67 893 84 227 75-57 882 08 235 68 891 54 230 76.10 879 7° 238 69 889 22 232 76.62 877 14 256 70 886 87 235 77-14 874 87 227 7i 884 70 230 77.66 872 62 225 72 88a 37 2 33 78.18 870 21 241 73 880 03 234 78.69 867 79 242 74 877 67 236 79.20 865 35 244 75 875 3° 237 79.71 862 90 245 76 872 90 240 80.22 860 42 248 77 870 49 241 80.72 8 57 93 249 78 868 06 243 81.23 855 42 25 1 79 865 61 245 8i.73 852 90 252 80 863 14 247 82.22 850 35 2 55 81 860 66 248 82.72 847 79 256 82 858 16 250 83.21 745 21 258 83 855 64 252 83.70 842 62 259 84 853 10 2 54 84.19 840 01 261 85 850 55 255 84.67 837 38 263 86 847 98 257 85.16 834 73 265 87 845 39 259 85.64 832 07 266 88 842 78 261 86.12 829 38 269 89 840 15 263 86.59 826 68 270 90 837 51 264 87.07 823 96 272 9i 834 85 266 87-54 821 23 2 73 92 832 18 267 88.01 818 49 274 93 829 48 270 88.48 815 72 277 94 826 77 271 88.94 812 93 279 95 824 04 273 89.40 810 13 280 96 821 29 275 89.86 807 31 282 97 818 53 276 90.32 804 48 283 98 815 76 277 90.78 801 64 284 99 812 95 281 91-23 798 76 1 288 100 1 810 15 280 91.68 795 89 287 CORRECTIONS OF WEIGHINGS. (i.) Absolute Weighings. — If we want the weight x of an object in terms of a fixed upon unit, for instance in terms of the true Gramme, we need a set of standards, which have their correct weights in vacuo, or, at least an ordinary set combined with a correction-table which gives the true weights in vacuo. Supposing the weight of the object in air of the density $ *is P Grammes, i CC of the object weights S grm., and i CC of the material which the weight-standards are made of weights S grm., we have for the weights in air the equality : x P * - -g-(5 = P - ~-<5, whence rather 10.999 7. io6 Change of Standard. — Supposing the specific gravity of a body is = S, water = r, what is the specific gravity x in reference to another liquid, whose specific gravity on the water scale is = d? Answer: We have, for the relative weights of equal volumes of Water New Liquid Body the values, i d S g Obviously, x = - and conversely, if x be given, S = dx. Supposing, for instance, the specific gravity of a salt, taken in reference to alcohol of 0.800 specific gravity had been found = 4. 12, the specific gravity in reference to water is = 4. 12 x 0.800 = 3.296. In exact specific gravity determinations the temperature of the water, which of course, with the method of immersion, is that of the solid under operation, must be noted. Supposing the specific gravity had been determined at, and consequently, in reference to water of, t°, what is it in reference to water of t' degrees ? Rosetti's table (given on page 107) affords the answer. This table gives the specific gravities of water of 0°, i°, 2 , &c, in reference to that of water of 4° as = 1 ; in other words the weight of 1 CC of water of t°. Taking w' w" w'", &c. as representing Rosetti's numbers for t'° t"° t'"°, &c, we have for the relative weights of equal volumes of Water of 4 Water of t'° Body under Exam. \ w' x and supposing the specific gravity of the body in reference to water of t'° to have been found = S', obviously x = w'S'; but the same relation holds for any value of t, hence we have, for any two temperatures t' and t" the equation w' S' = w" S", because either product is the "weight of 1 CC at that temperature." Substituting 1 - e' for w' and 1 - e" for w", we have, 1 - e" + e' + terms in ± ee, eee, etc. S' I — £ S" I - e S" = t I — e S' it I - e = I - e' + e" + terms in ± ee, eee, etc. where these higher terms may be cancelled, so that we have : S' = S" + (e'-e")S", and S" = S'-(e'-e")S' Supposing S' in one case, and S" in another, to have the same value, the correction for passing from t' to t" is the same numerically, as the one for passing from t" to t'. 107 Table IV. Giving the Density, w, of Water at t°; that at 4° = 1 ACCORDING TO ROSETTI. (Pogg. Annalen , • Erganzungsband V. / . 268, year 1871). . Density* Diff. t Density Diff. . Density Diff. w w w - 5° -999 298 3°° •995 77 65° • 980 74 -4 455 + 157 3i 47 -3° 66 19 -55 -3 59° + 135 32 17 3° 67 . 979 64 55 -2 703 + 113 33 •994 85 32 68 08 56 -1 797 + 94 34 52 33 69 97851 57 871 + 74 35 18 34 70 977 94 57 + 1 928 + 57 36 •993 83 35 7i 36 58 2 969 + 4i 37 47 36 72 976 77 59 3 99 1 + 22 38 10 37 73 18 59 4 1. 000 000 + 009 39 •992 73 37 74 975 58 60 5 -999 990 — 10 40 35 38 75 97498 60 6 970 20 41 .991 97 38 76 38 60 7 933 37 42 58 39 77 973 77 61 8 886 47 43 18 40 78 16 61 9 824 62 44 .990 78 40 79 972 55 61 10 747 77 45 37 4i 80 97i 94 61 " 655 92 46 .989 96 41 81 32 62 12 549 106 47 54 42 82 970 70 62 13 43° 119 48 10 44 83 07 63 14 299 131 49 .988 65 45 84 969 43 64 15 160 139 5° 19 46 85 96879 64 16 002 158 5i .087 72 47 86 15 64 17 .998 841 161 5 2 25 47 87 967 51 64 18 654 183 53 .986 77 48 88 966 87 64 19 460 194 54 29 48 89 22 65 20 259 201 55 .985 81 48 90 965 56 66 21 047 212 56 34 47 9i 964 90 66 22 .997 828 219 57 .984 86 48 92 23 67 23 601 227 58 37 49 93 963 56 67 24 367 234 59 .983 88 49 94 962 88 68 25 120 247 60 38 5° 95 19 69 26 .996 866 254 61 .982 86 52 96 961 49 70 27 603 263 62 34 52 97 960 79 70 28 331 272 63 .981 82 52 98 08 7i 29 051 280 64 28 54 99 •959 37 7i 30 .995 765 286 65 .980 74 54 100 .958 66 7i At 60° Fahr., w = -999 073 6. At 14° Reaumur, w = -998 749 5. * For most purposes the 6th place may be cancelled. At I5.5°> for instance, A w = .000 005 corresponds to only = 0.03° in temperature. io8 From Rosetti's numbers we have calculated the specific gravities of water of from o to 25°, in reference to water of 15°, (15.56 = 60° F.) and (17.5° = 14° R) respectively, and embodied the results in the following tables, along with Rosetti's own results for water of 0° as a standard. Table V. giving the Specific Table VI. giving the Specific Gravities 01 WATER OF t° Gravities OF WATEB OF t° REFERRED to water OF REFERRED TO WATER OF (iSf = 6o- F.) or (1 7-5° = OR 15° AS=I 14° R) as = 1. t 0° 15° Diff. t iS-S6° 17-5° Diff. 0° 1. 000 000 i. 000 712 0° 1. 000 798 1. 001 123 I °57 769 + 57 I 855 180 + 57 2 098 810 41 2 896 221 41 3 120 832 22 3 918 243 22 4 129 841 9 4 927 252 9 5 119 831 - 10 5 917 242 -10 6 099 ' 811 20 6 897 222 20 7 062 774 37 7 860 i8 S 37 8 015 727 47 8 813 138 47 9 •999 953 665 62 9 7Ji 076 62 10 976 588 77 10 674 1. 000 999 77 11 784 496 92 11 582 907 92 12 678 390 106 12 476 800 106 13 559 271 119 13 357 681 119 14 429 139 131 14 226 55° 131 IS 289 000 140 15 087 411 139 16 131 .999 842 158 16 •999 928 253 158 17 .998 970 681 161 17 767 092 161 18 782 494 188 18 580 •999 904 188 19 588 299 195 19 386 710 194 20 388 099 200 20 185 •999 509 201 21 176 .998 886 213 21 • 998 972 297 212 22 .997 956 667 220 22 753 077 220 23 730 440 227 23 526 .998 850 227 24 495 206 234 24 292 616 234 25 249 ■997 959 247 25 044 368 248 With these two tables before one, the reduction of a specific gravity to water 0°. 15°, 15.56°, or 17.5°, becomes very easy. Supposing, for instance, a specific gravity, taken in reference to water of 18°, is 18 S; the specific gravity in reference to water of 15°, is 15 S = 18 S x .999 494 = (1 -.000 506) S 18 . 109 For the sake of brevity we will use symbols of the form b S a for designating the specific gravity of a body at a", taken in reference to water of b° as = unity. In the determination of specific gravities 15" and 4° are convenient standard temperatures to adopt, the former for working, the latter for the registration of results. According to Rosetti's table the relation between 4 S and S 18 is, = 4 S + 4 S x .000840, and i 5 S = jS — 15 S x 0.000840. Table (VII.) To pass from £ to U S, add \ the Correction _ To pass from 15 S to 4 S subtract 1 s Correction. S Correction. 1. 00 .000 84 1. 19 .001 13.10 .oil 2.38 .002 14.29 .012 3-57 .003 15.48 .013 4.76 .004 16.67 .014 5-95 • 005 17.86 .015 7.14 .006 I9-OS .016 8-33 .007 20.24 .017 9-5 2 .008 21-43 .018 10.71 .009 22.62 .019 11. 91 .OIO 23.81 .020 Example. — Given 15 S = 20.102; find 4 S. Adopt the "correction" given for the nearest tabular S, which is 20.24. The " correction " given is .017 hence 4 S = 20.102 -.017 = 20.085, to within ±.0005. If a higher degree of precision is wanted, better calculate the correction by multiplying the given S into 0.000 84. Our Three-Place Logarithm Table No. 2 affords sufficient exactitude. LIQUIDS. Suppose the specific gravity of a liquid to be determined, in one case by the plunger-method, and in another by the bottle-method, and the results to be as follows : (Plunger.) The plunger weighs: (1) In air of the density S, P' grm. (2) in water of the temperature t', P" grm., and (3) In the Liquid under examination, at t" degrees, P'" grm. The uncorrected specific gravity of no /p' _ p'" _ j\ the liquid at t"°, taken in reference of water of t'° is ' , _ p t . (P) = P|.i - A. (t -t)],ift >t. A = }i y is the coefficient of linear expansion of the respective kind of glass, which, however, for ordinary purposes, needs not be determined expressly but put down at .000 0086 from 0° to about 30° ) , , } See Recknagels Table, page 126. and at .000 0092 from o to 100 . j ° > c o The correction amounting to only 0.92 mm. per metre per 100 degrees, may as a rule be neglected. Correction for the Temperature of the Mercury. If our mercury- column measures P mm. at t°, it exerts the same pressure as a column of mercury of 0°, P mm. high, where P - P (1 + kt) k stands for the mean coefficient of expansion of mercury from 0° to t°. According to Levi, k, for t = 30, is .000 181 43. Table II. The Logarithms of (1 + .000 181 43 t), up to 30 . t Log. (i+kt) t Log. (i+kt) t Log. (i+kt) p. P. •000 000 10 .000 787 20 .001 573 At A log. VI. I 079 n 866 21 652 .1 8 2 158 12 945 22 73° .2 16 3 236 13 .001 023 23 809 •3 24 4 315 14 102 24 887 •4 3i S 394 i5 180 25 965 •5 39 6 473 16 259 26 .002 044 .6 47 7 55* 17 338 27 122 •7 55 8 630 18 416 28 201 .8 63 9 709 19 495 29 279 •9 7i 10 787 20 573 3° 357 1.0 79 i3° Table III. Values of k, i + kt and Log. (i + kt) for a few HIGHER TEMPERATURES, ACCORDING TO LEVI. t 10 000 k. 1 + kt. Log. (1 + kt). Diff. Logs. P. P. for Logs. A t A log. 30° 40 50 60 70 80 90 100 150 200 1-8143 1. 8149 1.8x57 1.8165 1.8174 1. 8184 1.8195 1.8207 1. 8281 1.8378 1.005 443 1.007 260 1.009 °78 1. 010 900 1. 012 722 1.014 547 1. 016 376 1. 018 207 1.027 421 1-036 755 .002 357 .003 142 .003 925 .004 708 .005 490 .006 272 .007 054 .007 836 785 783 783 782 782 782 782 1° 2 3 4 5 6 7 8 6 10 78 157 235 3i3 39i 470 548 626 704 783 (3) Correction for the Gravity at the Place of Observation. Taking the gravity at lat. 45° and sea-level as unity, that for latitude , 133-9 8 „ 170.8 1 3 „ 192. 1 4 ,, 144.0 9 „ 175-8 14 „ !95-5 5 » 152.2 10 „ 180.3 136 Table IX. Giving the Tension of Absolute Alcohol from t = 0° to t = 30° Calculated after Regnault s determinations by L. Carius. Bunseris Gasometrische Methoden. t Tensions. Diff. t Tensions. Diff. Mm.* Mm. 12.73 15° 32.60 1.91 I I3-6S .92 16 34.62 2.02 2 14.60 •95 17 36-77 2-15 3 !S-59 •99 18 39-°5 2.28 4 16.62 1.03 19 41-45 2.40 5 17.70 1.08 20 44.00 2-55 6 18.84 1. 14 21 46.69 2.69 7 20.04 1.20 22 49-54 2.85 8 21.31 1.27 23 52.54 3.00 9 22.66 i-35 24 55-7° 3.16 10 24.08 1.42 25 59-°3 3-33 11 25-59 i-5i 26 62-53 3-5° 12 27.19 1.60 27 66.22 3-69 !3 28.89 1.70 28 70.02 3-8o 14 30.69 1.80 29 74.09 4.07 3° 78.41 4-3 2 * Millimetres of mercury of o°C in Paris. The original proceeds by tenth-degrees ; but a glance at the numbers shows that the values between t and t + 1 were obtained by rectilinear interpolation. For the gases enumerated in the following Table X. the exact relations between the weight D of a litre in grammes on the one hand, and the temperature t, the pressure P in millimetres of mercury of 0° and the gravity at the place of observation on the other, may be formulated thus; taking (D Regnault) as a symbol for the D for t = 0° and P = 760 mm. of 0° in Regnault's laboratory, A P D = (D Regnault) x — P — x— ( I+<5 ); or D = Const. A + t (i + a) By means of this formula Table XI. has been calculated. In this Table the " logarithms " given are the logarithms of the constants. J 37 J3 r-? CO t O o M-t lasgow and inburg (G = 100 945 8 ON ■* ro M Th cc ■* o MD On «3 ■* m vO go 6 ON co vO ON CO Oi CO in CO X' r Sp ater o m N ■* -< 1-1 6 ^ l-H -SpJ'K. fM ,^^ So J 1 co 00 |_! 00 t— 1 in Gram d to tha = 0° and ra ON co VO CO CO "J") 1^ l-H N ■* N p M On > " HI 1-1 H M . w >* *j- co 1^. CO CO 00 j-i CO "1 3- .tl JJ — Jl co ON vO ON ON M r^. HH tof I L ravity re: = 1000), ercury o: CO CO p On > M " l-H hH 0" i m ,_j TH ^_ o> 00 to ^D to CO Th r^. -ao^s . (2 1-4 rt <" ri w co co ON LO ON CO en 1^ vO HH 5j ON N *j-i CO : LT) 1^. > £ N HI O p b IN On CO ri ho I- 1 *^ N ro O r~ "cT" u-> co VO w 8 cs r^ r^ T3 II ■*■ ON O p On q !> Is S g £ o >, ■* ^o ^d- ►H 6 ro N K 1-1 ^* C 2 73 t-H O l-H O O 8 r-H 8 4 IN vN IN O o 8 00 O OO CI v» rf in >o MD O On CO 11 vo ro w r-* r-- O " ~ d O 6 ^O 1-1 1 *J ^ xpan istan abou «l B II vO IN h- J H- 1 < 1^- IN VO -W P3 )H S * g 8 S B * * Coeffici sion Pres 760 S NO ro 00 On O O O 00 MD GO vO On O ,_; vO vO >o vO VO r-. CO CO ro ro ro CO ^ ^- : - H a z < hH 1— I O ' ■ V, f s s O '3 O '0 8 ^ O < V „ c N >» rt d < "l X a u CJ o.m a, 0^3 m a o C 0-, ft, 1-1 w c -s w ^ .- >, HI l_ (> S dirt +1 3 hn O a rf JJ m -U ■TJ 3 ra ^ M o. C •a 3 J fl> ^J HH .d (S o a . cJJ O 138 Table XI. Weight in Grammes of One Litre of Gas if measured at t° and a Pressure of P Mm. of Mercury of o°. According to Regnault's direct determinations. Name of Gas. Regnault's Laboratory. Edinburgh-Glasgow. Air free of C0 2 and H 2 0. ■463 579 * ° D/y 272.44 + t Log. = .666 124 - 1 .463 865 £ 272.44 + t Log. = .666 392 - 1 Oxygen. ■ 512 oiq 272.16 + t Log. = .709 286 - 1 •5*2 334 Log. 272.16 + t 709 554 - 1 Nitrogen. .4150 1583 ^ 3 6 272.61 + t Log. = .653 775-1 .450 861 Log. = 272.16 + t ■ 6 54 043 Hydrogen. .032 192 5 273.13 + t Log. = .507 755 - 2 .032 212 4 273.13 + t Log. = .508 023 - 2 Carbonic Oxide.'* .448 861 272.57 + t Log. = .652 112 - 1 .449 1^8 W 6 272.57 + t Log. = .652 380 Carbonic Acid. •7°i 33 J -^ ' •" 269.55 + t Log. = .845 923 - 1 .701 763 ' •* 269.55 + t Log. = .846 191 - 1 * Calculated after Baron Wrede's value of the specific gravity, Air= I. At temperatures from somewhere below o°C to ioo°C, and pres- sures not exceeding one atmosphere, the formulae given in this Table afford a closer approximation to the truth, than the general equation for "perfect gases" given as No. VII. on page 140. At temperatures above 100°, it is probably the reverse, in general at least. 139 Rules concerning " Perfect " Gases. No gas is a "perfect gas" in an absolute sense, but any gas, if at a moderate pressure P its temperature be raised sufficiently above its point of condensation at P, becomes "perfectly gaseous" which, in practice, means that within a certain area of combinations of temperature and pressure, which includes at least all pressures from P downwards and all temperatures from t° upwards, it obeys the law ; VP T T _ = Q; orV= -Q; orP = Q- I. V stands for the volume at the pressure P and the "absolute temperature," i.e., the temperature counted from the absolute zero, which, in practice T is T = 273 + t. The ratio — is called the " disgregatwn " of the gas, Q is a constant which may be defined, arithmetically, as the volume of the gas at T = i° and P = 1 mm., or as the pressure of the gas at V = 1 and T = 1 ; or physically, as the volume which the gas occupies when- ever the disgregation T : P = 1, it being understood that the combina- tion T, P falls within the area of perfect gaseousness ; or as the pressure which the gas exhibits whenever T = V, subject to the qualification named. The equation, in the case of a mixed gas, holds for any one of the components, in the sense of the following explanation : — (Mixed Gases.) Given a mixture of the (unitary or themselves mixed) gases I., II., III., . . . . N, and supposing, at a given constant temperature t°, v 1( v 2 , v 3 , v 4 , . . . . are the partial volumes of I., II., etc., i.e., the volumes which they would occupy at t° and P mm. if they were floating upon one another in layers, while p x , p 2 , p 8 , . . . . are the partial pressures, meaning the pressures which the components severally would exert, if each of them filled the V units of space by itself (at t°), we have, as well established propositions, Vj + v 2 + v 3 +....= V II. Pi + P2 + Ps + ■ ■ ■ • = p HI., and for any one of the components, V P - = - IV., v p whence v x : v 2 : v 3 . . . . = p x : p 2 : p 3 . . . . IVa. If we define the " specific gravity" of a gas-species as meaning the weight of a given volume, measured by the weight of the same volume of some standard gas of the same disgregation, then by eq. I., the specific gravity S is independent of P and t. As shown by Avogadro, S is a function of only the molecular weight M thus : S' : S" : S'" .... = M' : M" : M'" . . and consequently, S' = kM'; S" = kM"; S'" = k M'", .... V. 140 where k is a constant which depends only on the units chosen for S and M. Relation between Weight and Volume. According to Regnault, one litre of oxygen, if measured off in Paris at 0° and under a pressure of 760 mm. of mercury of 0°, weights 1.429 803 grammes. (Log. = .155 276.) Hence at the pressure of P mm. and the absolute temperature T we have for the weight of one litre of a gas of the molecular weight M the value D = —■ 1.429 803 x 273 M t x yfo x o; VI. The corresponding value for any other place than Paris is found by multiplying^ the expressions by (1 + = QxJ_ ,,{,-* (,--,)} Supposing the barometer to stand at B mm. at t°, we have for P" the value B - (h), and as V" = (V), for the true quantity of the mixture of steam and gas V " = V ' L ^ '-J Notes on the Method of Combustion. — Avogadro's law enables us to read the volumetric composition of a gas in its formula. Thus, for instance, a glance at the formula of methylamine CH 6 N = 31, shows to us that if H 2 grammes* of hydrogen occupy one reduced volume, i e., represent one gasometric unit of quantity, N 2 grammes of nitrogen do the same, etc., and that, consequently, one volume of methylamine con- tains 2.5 H 2 = 2.5 volumes of hydrogen, 0.5 N 2 = 0.5 volume of nitrogen, and 0.5 C 2 = "half a volume of carbon-gas," meaning the quantity of carbon contained in one volume of carbonic acid, or marsh-gas, or one- carbon gas generally. The volume of oxygen required for the combustion of one volume of a named gas-species, the contraction involved in the combustion, the volume of C0 2 produced, etc., are easily seen from the respective chemical equation. Example : C 2 H 4 + 3 2 = 2 C0 2 + 2 H 2 ; the reduced volumes are 1 3 2 2 Hence contraction = c = (i + 3)-2 = 2 * It is hardly necessary to point out that, in the present connection, H2 grammes of hydrogen mean only the weight of what, at the adopted standard disgregation, is one reduced volume or one unit of hydrogen ; and similarly in similar cases. 147 Yet, in practical gas-analysis, the following Table will be found con- venient. Part A. of the Table reads as follows :— Every i volume of the gas named in column I. requires for its combustion s volume of oxygen ; the contraction involved = c volume, the volume of steam produced is w, that of the C0 2 = k, that of the nitrogen = n. In the second part B. (which refers to gases combustible by hydrogen) h stands for the volume of hydrogen required for the combustion of i volume of gas. Table XIV. Combustion-Constants of a Number of Gases. A. — Gases Combustible by Oxygen. u. k. V w. n. Hydrogen, H 2 , Carbonic oxide, CO, Methyl-aldehyde, CH 2 0, Ammonia, NH 3 , °-5 i 1.25 1 1 °-5 °-5 1 °-75 1 1 !-5 ' Methylamine, CH 5 N, Cyanogen, N 2 C 2 , Hydrocyanic acid, NCH, Marsh Gas, CH 4 , - !-75 °-75 2 1 2 1 1 2.25 2 I -25 2 2-5 °-5 2 I Acetylene, C 2 H 2 , - Ethylene, C 2 H 4 , Ethane, C 2 H 6 , Propylene, C 3 H 6 , - Propane, C 3 H 8 , Oxide of methyl, C 2 H 6 0, Benzol, C 6 H 6 , i-5 2 2-5 2-5 3 2 2-5 . + £. 4 2 2 2 3 3 2 6 2-5 3 3-5 4-5 5 3 7-5 I 2 3 3 4 3 3 Gas, CaH^=i vol., U, , 4- B 4 2 Gas, Ca H^ Oy= i vol., B y 1 + --- 4 2 a a + Z-L- 4 2 2 B. — Gases Combustible by Hydrogen. c. h. w. n. Oxygen, 2 , Ozone, 3 , - Nitrous oxide, N 2 0, Nitric Oxide, NO,* 3 4 1 i-5 2 3 1 1 2 3 1 1 O I * The entries for NO have a purely theoretical significance, because this gas does not explode with hydrogen. 148 The Method of Combustion is a method of ultimate analysis, which, in the case of gases of class A., consists in this, that, starting with a known quantity of the gas to be analysed, we add a known sufficient quantity of oxygen, fire the mixture with an electric spark, and, besides measuring the product of combustion, determine the carbonic acid, the nitrogen, and, if necessary, the total water present in it. Before explaining the arithmetic of the subject, let us adopt the formula a C 2 . P H 2 . y 2 . S N 2 = i volume as expressing the as yet unknown composition of the gas, by saying that every unit (reduced volume) of the gas contains a units of carbon, j3 units of hydrogen, y units of oxygen, 8 units of nitrogen-gas, potentially of course ; the state of combination of the several elements we will suppose to be unknown. In this case we must determine the following gas-quantities :— Units or reduced volumes. (o) That of the sample under analysis, V (i) That of the mixture of gas and oxygen, Vj After firing. (2) That of the product, measuring cold and deducting the tension of H 2 from P, as usual, V 2 (3) That of the total product, measuring at a sufficiently high temperature to convert the steam into a perfect gas, V 8 After removing the C0 2 . (4) That of the C0 2 -free residue, V 4 The residue (4) which consists of nitrogen and oxygen, must be analysed, either by absorbing the oxygen with pyrogallate and measuring the nitrogen, or by firing the gas with a sufficiency of measured hydrogen, and determining the contraction ; one-third of it is the quantity of the oxygen. Suppose (5) The nitrogen found amounts to V 6 Taking K, |£, W as representing the quantity of carbonic acid, nitrogen and steam produced, respectively, we easily see that, W = V s - V 2 ; whence /? = ^ W, (1) » = V 6 (2) K = V 2 -V 4 ;and « = ^f (3) 149 The oxygen is calculated as follows : — Let S denote that part of the added oxygen, which, conjointly with the oxygen of the substance, served to burn the carbon and the hydrogen, and S r denote the surplus oxygen added, so that S + S r = S = total oxygen added, then, V T = V + S + S r V 2 = g + K + S r V 1 -V 2 = C = V + S -i;-K; (4) and, as K = V 2 - V 4 ; V 1 - V 4 = V + S - f ( S ) Now it was obviously the oxygen-sum S + Vy which produced the C0 2 and the H 2 0, hence, S + Vy = K + ^W; or y = JL(K + 0.5 W - S ) (6) C, K and fjC being known, S can be calculated from eq. (4) or eq. (5). If nitrogen and oxygen be absent, we need determine only K and C, because eq. (4) becomes C = V + S - K ; and, as S - K =_LV /? p v On the other hand, whenever oxygen is present W must be deter- mined, unless such determination is rendered superfluous by independent determinations by other methods than the method of combustion. Supposing for instance the gas to be analysed is a C 2 . /? H 2 + e (CO) = 1 volume, the determination of only C and K does not enable us to calculate the three unknown quantities a, ft, e ; we must add the deter- mination of W, unless we know that there is CO, and by determining it by absorption with cuprous chloride, eliminate e as an unknown quantity. Sometimes the determination of W in addition to C and K is the only method for determining the oxygen. Supposing for instance we even know that one of the components is ethylene, C 2 H 4 , or oxide of methyl C 2 H 6 O = C 2 H 4 + H 2 0, there is no other way for determining the latter component. In order now to show how far the method of combustion goes as an indirect method of proximate quantitative analysis, let us assume, in the first instance, we had to deal with a mixture of gases I., II., III., etc., all falling under the general formula a C 2 . ft EI 2 . y 2 . S N 2 = 1 molecule, and that the special formulse of I., II., etc./ were known to us. To find the per-unitages x' x" x'", etc., of the several components, we might begin by determining the coefficients a, ft, y, S, for the given mixture, and then establish equations between these on the one hand and the special values a a . . . . / r p . . • • Y y" thus : a = a'x' + a"x" + a'"x'".. . &c, P = P* + i8"x" + j8'"x"'.. .&c, 7 = y'x' + y"x" + y"'x'".. . &c, <5 = "J C) v > " / one hand, and the special values c', c" . . . . k' , k" . . . . «', «" . w', w", .... on the other, thus : k = k'x' + k"x"...&c, - la. c = c'x' + c"x" . . . &c, lla. w = w'x' + w"x"... &c, Ilia. M = K'x' + M"x"... &C, IVa. ! = X ' + X"...&C, Ya. and solve these equations in respect to x', x", etc. Gas Absorption. Bunsen's Law may be explained by reference to the following fictitious experiment : — A glass vessel of v + h volumes capacity is charged with a mechanical mixture of the gases I., II., III., etc., i5i h volumes of a gas-free liquid are now introduced, so that the mixed gas is compressed into v volumes and its (dry) pressure raised to P mm. at t°, which temperature from now remains constant. Gas and liquid are now shaken together until absorptiometric equilibrium is established. The effect is, that the gas-space is saturated with the vapour of the liquid, while the (dry-gas) pressure in it is reduced (from P) to a less value p, because, of each of the gas-components I., II., III., etc., a portion q has passed into solution, and only the rest r remained in the gas-space. The general law of distribution is complex, but, if the solvent is water or alcohol, if none of the gases act chemically upon, or are abundantly soluble in, the liquid, if the pressure P is less than (say) i atmosphere and t lies between o° and (about) 30° C, then according to Bunsen, the quantity of any one of the several gases I., II., III., etc., which is in solution, is governed by the equation q = h (3 - I. where -k stands for the partial pressure of this component in the gas-rest. j3 and q are of course of the same denomination : if q means milli- grammes, [3 means milligrammes likewise, but we will agree upon measuring all gas-quanta by volume at (virtually) t = 0° and 1 mm. pressure, so that " q '' units of gas means q times the quantity which fills unit of volume at t = 0° and P = 1 mm., or at the disgregation 273, and on the basis of this stipulation, with Bunsen, call (3 the " coefficient of absorption " of the gas at t°. (3, in virtue of an as yet unknown law, increases, when the temperature falls, and is in general, greater for alcohol than for water. Assuming now that every one volume of the original mixed gas contains m' volume of I., m" of II., m'" of III., etc. (in the sense in which volume of air contains 0.21 volume of oxygen and 0.79 of nitrogen), it is clear beforehand, that the corresponding per-unitages in the gas-rest and those in the dissolved gas have different values. Let those for the dissolved gas be s' s" s'" .... and those for the gas-rest n' n" n'" .... In discussing our equation let us begin with a limit-case. I. v is so very much greater than h, that the process of absorption involves no appreciable change of composition in the gas-mixture. In this case the dissolved portions of the several individual gases are q' = p h (m' j3') ; q" = p h (m" /3") + p h (m'" f3'"), etc.; the total quantity of dissolved gas is Q = ph(m' l 8' + m" f3" + m'" £"', etc.,) II. and the per-unitage in it of any one. of the components is s = ^= m fi-- III. Q 2m/3 *5» II. We will now return to our original conception and assume h : v to be finite. In this case, as quite generally tt = n p, eq. I. reads q = n p (h (3) . IV. and for the undissolved rest of the respective gas we have r = v n p x Z3 — which we will abbreviate into 273 + t r = v n p V. hence q + r = n p (v + /3 h) .. VI. But q + r is a function of P and m thus : q + r = v m P hence v m P = n p (v + p h) VII. or n = m / P , V V, Vila, p (v + P h) P. _ n(v + /3h) p mv By summing up the series of special equations of which Vila, is the general form, we have n' + n" + n'". . . = 1 =— /. m ' + "2! . A VIII — - I - g m whence p " ' - ^h IX. p By substituting this for the — in eq. Vila, we have m 1 , fih Y X. v o where ' 2 ' stands for the sum which figures in eq. IX. n P From Vila, we see that — depends only on the ratio — but m p is independent of the absolute values of P and p. The total quantity of gas dissolved is Q = P v - p v , whence Q = v P(i-S), XI. Q -=- h p is what, in the case of a unitary gas, would be the coefficient of absorption; in the case of mixed gas, however, the coefficient — we will call it A — is variable, being Q_ i-S A_ hp _ h. - - XIa. Vn 153 By equations IV. and Vila, we have, for any one of the components, q =h/3(J^-\ - XII. \v + /3h^ hence s = -^- = -. f 1 " — = XIII. Q (v o + 0h)(i-S) Composition of Air and Absorption of its Components by Water. For laboratory purposes the mean composition of dry air may be assumed to be By Volume. By Weight. Oxygen, 20.90 2 3-i4 Nitrogen, 79. 10 76.86 100.00 100.00 Carbonic acid, 0.03 0.046 100.03 100.046 If one volume of water of t° is shaken with constantly renewed portions of a mixture containing m 1 of oxygen, m 2 of nitrogen, and m 3 of carbonic acid per unit volume until absorptiometric equilibrium is established and the dry-gas pressure at the end is p mm., the water dissolves of Oxygen, m x p /?, volumes. Nitrogen, m 2 p /3 2 „ Carbonic acid, m 3 p /3 3 „ measured at 0° and 1 mm. dry pressure, the /3's being the " coefficients of absorption." The three absorbed gas volumes, if measured at p mm., occupy nij/?!, m 2 /? 2 , m 3 /3 3 volumes. Hence, at any pressure, unit volume of the absorbed gas contains : r\ m l Pi Oxygen, n : Nitrogen, n 2 m l& + m 2 / S 2 m 2/ S 2 + m. .ft m 1 /3 1 + m 2 /3 2 m 3 ^ 3 + m £ ;ft Carbonic acid, n, — m 1 /3 1 + m 2 /J 2 + m 3 /J s The following coefficients of absorption were determined, those of nitrogen and oxygen by Dittmar* (absorption experiments with air at 1 atm. pressure), that of carbonic acid by Bunsen (by experiments with the pure gas at varying pressures from 1 atm. downwards) : — * Challenger Memoirs, vol. of " Physics and Chemistry," pp. 172 and 173. 154 Table XV. Coefficients of Absorption by Water of Oxygen. Nitrogen. Carbonic Acid. Air, free from C0 2 . Symbols, p 1 & A, t 1000 /?! Diff. 1000 P2 Diff. 1000 /? 3 Diff. 1000 A Diff. 100%* » o° 49-03 24.40 1797 29-54 34-69 I 47.70 -i-33 23.78 -.62 1721 -76 28.78 -.76 ■65 2 46.45 1.25 23.21 •57 1648 73 28.06 .72 .60 3 45-25 1.20 22.65 •56 1579 69 27-37 .69 •55 4 44.II 1. 14 22.12 •53 1513 66 26.72 .65 • 5i 5 43-°3 1.08 21.62 .50 H50 63 26.09 •63 •47 6 41.99 1.04 21.14 .48 1390 60 25.50 •59 .42 7 41.00 •99 20.69 •45 1334 56 24-93 •57 •38 8 40.06 •94 20.25 •44 I28l 53 24-39 .56 •33 9 39-15 ■91 19-83 .42 1231 5o 23.87 •52 .28 10 38.28 .87 19-43 .40 II85 46 23-37 •50 .24 ii 37-45 •83 19.05 •38 1 142 43 22.89 . 4 8 •19 12 36.65 .80 18.68 ■37 1 102 40 22.43 .46 •15 13 35-88 •77 18.32 -36 IO65 37 21.99 •44 .10 H 35-15 •73 17.98 ■34 1032 33 21.57 .42 .06 IS 34-44 •7i 17-65 •33 1002 30 21.16 .41 .01 16 33-75 .69 17-34 •3i 975 27 20.77 •39 33-97 17 33-09 .66 17-03 •3i 952 23 20.39 •38 .92 18 32.46 •63 16.74 .29 932 20 20.03 •36 .87 19 31.84 .62 16.46 .28 9i5 17 19.67 •36 •83 20 31-25 •59 16.19 •27 901 14 19-33 •34 • 78 21 30.68 •57 15-92 .27 19.01 •32 • 74 22 3°- 1 3 •55 15.67 •25 18.69 •32 .69 23 29-59 •54 15.42 •25 18.38 •31 ■65 24 29.07 ■52 15.18 • 24 18.08 •30 .60 25 28.57 .50 H-95 •23 17.80 .28 .56 26 28.09 .48 14-73 .22 17.52 .28 •5i 27 27.62 •47 14.51 .22 17.25 .27 ■47 28 27.16 .46 14.30 .21 16.99 .26 .42 29 26.72 • 44 14.10 .20 16.73 .26 •37 3° 26.29 •43 13.90 .20 16.49 .24 ■33 3 1 25.87 .42 13.70 .20 16.25 .24 .28 32 25-47 .40 13-52 .18 16.01 .24 .24 33 25.07 .40 13-34 .18 15-79 .22 .19 34 24.69 .38 13.16 .18 15-57 .22 ■15 35 24.32 •37 12.99 -17 I5-36 .21 .10 40 22.60 12.20 14-37 32.88 45 21.09 II.50 13-5° •65 5° 19-75 10.88 12.73 .42 i55 The coefficient of absorption of air given means the sum, A = o. 209 /8j + 0.791 /3 2 , and consequently refers (in theory) to 00 volume of air used per 1 volume of water. From the above we see (as an example) that 1000 cc. of water of is°C. when saturated at 15° and a barometric pressure of B + t* mm. with normal air, absorbs quantities of the three components which, when measured dry at B mm. pressure and o°C, amount : I. The oxygen to 1000 fi 1 x .209 = 7.196 II. The nitrogen to 1000 /S 2 x 791 = 13.961 (I. + II.) The two conjointly to 1000 A = 21.16 III. The carbonic acid to 0.3 /3 8 = 0.30 Total, 21.46 Percentage of oxygen in I. + II. = 100 n x ** = 34.01. Absorption of Air by Sea Water.§ In the case Sea Water, the coefficients A, a, /?, and n^ assume different values A', a, f3', n\, governed by the equations : x = -92731 39 + t 100 n\ = 34.40 — 0.031 1 t. f3\ = A' n\ for the Oxygen. j3' 2 = A' n' 2 for the Nitrogen. From these equations we have : Table XVa. Values of A and n x for Sea Water. t -5° o 5 10 15 20 25 30 35 The corresponding Table in the Memoir (p. 175), goes from degree to degree, and includes the values A n2 = ^2 and Ihe ratios ni : n2 * T stands for the tension of steam at 15 . ** The student will observe that from pp. 153 to 155 the symbols ni and n 2 have a different meaning from that assigned to n' and n" on pp. 151 and 152. § Challenger Memoirs; Physics and Chemistry, vol. I., pp. 173 and 174. 1000 A 100 ni 27.27 34-56 23.78 .40 21.08 .24 18.92 .09 17.17 - 33-93 15-72 .78 14.49 .62 13-44 ■47 12.53 •3i iS6 Table XVI. Coefficients of Absorption of some Gases in Water or Alcohol. Taken from Bunsen, Gasometrische Methoden, Ed. II., pp. 384 to 387. Nitrogen Hydrogen Carbonic Carbonic Nitric Nitrous Oxide in Alcohol. Alcohol. Alcohol. VJX1QL in Water. Alcohol. Water. Alcohol. a .1263 .0693 4-33° .03287 .3161 I-305 4.178 1 .1259 .0691 4-237 .03207 .3126 1. 261 4.109 2 •125s .0690 4.147 •03131 •3093 1. 217 4.041 3 .1251 .0688 4-OS9 ■°3°57 .3060 i-i75 3-974 4 .1248 .0687 3-974 ,02987 .3029 "35 3-909 5 .1244 .068 S 3.891 .02920 .2999 1.095 3-844 6 .1241 .0684 3.811 .02857 .2969 1.058 3-78i 7 .1237 .0683 3-733 .02796 .2941 1. 021 3-7I9 8 .1234 .0681 3-657 .02739 .2913 0.986 3-659 9 .1231 .0680 3-584 .02686 .2887 0.952 3-599 10 .1228 .0679 3-5*4 .02635 .2861 0.920 3-54i n .1225 .0677 3-446 .02588 .2836 0.889 3-484 12 .1222 .0676 3-38i •02544 .2813 0.859 3.428 J 3 .1219 .0675 3-3i8 .02504 .2790 0.830 3-373 14 .1217 .0674 3-257 .02466 .2769 0.803 3-32o 15 .1214 .0673 3-199 •02432 .2748 0.778 3.268 16 .1212 .0671 3-144 .02402 .2728 o-754 3.217 i7 .1210 .0670 3.091 •02374 .2709 0.731 3.167 18 .1208 .0669 3.040 .02350 .2692 0.709 3-H9 19 .1206 .0668 2.992 .02329 •2675 0.689 3.071 20 .1204 .0667 2.947 .02312 .2659 0.670 3- 02 5 21 .1202 .0666 2.903 .2644 0-653 2.981 22 .1201 .0665 2.863 .2631 0.636 2-937 23 24 .1199 .0664 2.825 .2618 0.626 2.894 .1198 .0663 2.789 .2606 0.608 2.853 iS7 Table XVI. — Continued. t Marsh Gas in Ethylene in Sulphuretted Hydro- gen in Water. Alcohol. Water. Alcohol. Water. Alcohol. o° •0545 °-5 2 3 O.2563 3-595 4-371 17.89 I •OS33 0.520 0.2473 3-538 4.287 17.24 2 .0522 0.517 0.2388 3.482 4.205 16.61 3 .0510 0.514 0.2306 3.428 4.124 I5.98 4 .0499 0.5 1 1 0.2227 3-375 4.044 15-37 5 .0489 0.509 0.2153 3-323 3-965 14.78 6 .0478 0.506 0.2082 3-273 3.887 14.19 7 .0467 o-5°3 0.2018 3.224 3.810 13.62 8 ■°457 0.501 0.1952 3-!77 3-735 13.07 9 .0447 0.498 0.1893 3-i3i 3.660 12.52 IO ■o437 o-495 0.1837 3.086 3-586 II.99 ii .0428 o.493 0.1786 3-o43 3-5I3 11.48 12 .0418 0.490 0.1737 3.001 3-442 10.97 13 .0409 0.488 0.1693 2.960 3-371 10.48 14 .0400 0.485 0.1652 2.921 3-30I 10.00 is .0391 0.483 0.1615 2.883 3-233 9-54 16 .0382 0.480 0.1583 2.846 3-165 9.09 17 •°374 0.478 O.I553 2. 811 3-099 8.65 18 .0366 0.476 0.1528 2-777 3-o33 8.23 19 •°358 0-473 0.1506 2-744 2.969 7.81 20 •°3S° 0.471 0.1488 2.713 2.905 7.42 21 2.683 2.843 7-o3 22 2 -655 2.782 6.66 2 3 2.628 2.722 6.30 24 2.602 2.662 5-96 i5» In the following cases the coefficient of absorption is constant from o° to 2o° or 24° : — Name of Gas. Solvent. Constant Value of /S. Interval of Temp. Hydrogen, Oxygen, Carbonic Oxide, Water Alcohol Alcohol 0.01930 0.2840 O.2044 to 24 0° to 20° O tO 20 The following three Tables give the solubilities in water of some gases which do not obey Bunsen's law : — Table XVII. Ammonia and Sulphurous Acid according to Th. H. Sims.* " G " signifies the number of grammes of gas absorbed by 1 gramme of water if the " dry "pressure of the gas = 760 mm. " Vol'' stands for the volume of G grammes of (dry) gas at 0° and 760 mm. Ammonia. Sulphurous Acid. t G t G t G Vol. .899 53° .274 8° .168 58.7 4 .809 56 .256 12 .142 49.6 8 .724 60 .238 16 .121 42.2 12 .646 64 .220 20 .104 3 6 -4 16 ■578 68 .202 24 .092 32-3 20 •Si8 72 .186 28 .083 28.9 24 .467 76 .170 32 •o73 25-7 28 .426 80 •154 36 .065 22.8 3 2 •393 84 .138 40 .058 20.4 36 ■363 88 .122 44 •°53 18.4 40 •338 92 .106 48 .047 16.4 44 •31S 96 .090 5° •o45 15.6 48 .294 100 .074 * Jahresb. for 1861, pp. 54 and 56. 159 Table XVIII. Hydrochloric Acid according to H. E. Roscoe AND W. DlTTMAR.* In this Table, G stands for the number of grammes of gas absorbed at t° by one gramme of water, if the barometer stands at 760 mm. t G t G t G t G 0° .825 16° .742 32 .665 48° .603 4 .804 20 .721 36 .649 5 2 •589 8 •783 24 .700 40 •633 56 •575 12 .762 28 .682 44 .618 60 .561 Aqueous hydrochloric acid of any strength, if boiled down briskly in a flask or retort (not in an open basin) under a given pressure P, loses or gains strength until its percentage of HC1 has reached a certain value p which depends on P, or rather more directly on the boiling point determined by P. The residual acid then, on continued boiling, remains constant in composition. For P = 0.760 metres, involving the boiling point 110°, p is very exactly equal to 20.24. The following values p for other pressures are not so exactly determined. For P = 0.1 0.5 1.0 1.5 2.0 2.5 metres. p = 22.9 21. 1 19.7 19.0 18.5 18.0 * Liebig's Ann. vol. 112, p. 327; Jahresb. for 1859, pp. 103 and 104. i6o THERMOMETRY. Table I. Correction to be "added" to the reading of a Mercury (Centigrade) Thermometer to obtain the corresponding indications of the alr thermometer.* Mercury Thermometer. Correction. I. A. B. C. o° 0.00 5 -.01 IO -•°3 *5 -.07 20 - .11 25 — .12 3° -.12 35 -.10 40 -.08 54 -.04 73 -.06 82 + .04 100 0.00 110° - -05 + .02 + .02 120 - .12 + .04 + .05 130 — .20 + .09 4- .09 140 - .29 + .16 + -15 r 5° - .40 + -25 + .20 160 - -52 + -33 + .26 170 - -65 + -35 + .32 180 - .80 + .34 + -37 190 - 1. 01 + -32 + -35 200 -1.25 4- .27 + -3° 210 -i-53 + .18 4- .25 220 -1.82 4- .08 4- .20 230 - 2.16 - .02 + -i5 240 -2-55 - -14 + .10 250 -3.00 - .26 - -°5 260 -3-44 - -39 - .20 270 -3-9° - -5° - -38 280 -4.48 - -63 - -52 290 -5.10 - .88 - .80 300 -5-72 - 1. 21 -1.08 * Taken from Landolt and Bornstein's Tables. I. Refers to thermometers made of modern Thiiringian glass. Authority : Grunmach, Metronomische Beitr'age, herausgegeben von der kaiserlichen Normal- Aichungs-Commission ; No. 3, page 54 ; year 1881. A. Results obtained by Regnault, with a thermometer, made of crystal from Choisy-le-Roy, containing about 34 per cent, of oxide of lead. B. Regnault's result with a thermometer made of " verre ordinaire.'' C. Mean of observations made by Crafts, with 15 thermometers, namely 7 each from Baudin and Alvergniat with about 18 per cent, of oxide of lead, and one from Muller (Geisler's successor) in Bonn, made of German soda-glass. After Crafts ; Comptes Rendus, vol. 95, page 836 ; year 1882. i6i Table II. Correction to be applied to Thermometer-Readings in boiling-point Determinations and similar work, for the Lower Temperature of the Thread of Mercury outside the medium. T = Corrected temperature. t = Temperature as read. t' = Mean temperature of the outside-part of the thread of mercury. n = Length of the outside-thread in degrees. The correction, according to T. E. Thorpe, (Journ. Chem. Soc. vol. 37, p. 160 ; year 1880) is A = 0.000 143 (t - t') n. Log. 000 143 = .155-4. Three-place logarithms suffice. Multiples of 0.000 143- I .000 143 2 .000 286 3 .000 429 4 .000 572 5 .000 715 6 .000 858 7 .001 001 8 .001 144 9 .001 287 From a recent experimental investigation of E. Rimbach's (Ber. d. deutschen chem. Gesellschaft, year 1889, p. 3072), it appears that the true relations between A on the one hand and t, t' and n on the other demand, for their formulation, a more complex equation than the above, and in it the constants depend on the construction of the thermometer. They have for instance certain values for rod-thermometers (consisting of a plain thick capillary tube with the graduation on this tube) and others for "enclosed" thermometers, in which the stem of the thermometer is a thin capillary thread enclosed in a wide thin-walled envelope. Rimbach gives very complete correction-tables for these two kinds of thermometers, based upon extensive experiments with instruments made of ii Jenaer Glas." According to Rimbach, Thorpe's formula, if applied to relativity long threads of mercury, gives corrections, less than the truth. 162 Thermometer Scales. The construction of the three scales known as the Centigrade or Celsius scale, the Fahrenheit scale, and the Reaumur scale respectively is seen from the following statement : — Centigrade. Reaumur. Fahrenheit. Melting point of Ice, o° o° 32 Boiling point of Water under a pressure of 760 mm. of Mer- cury of o°C,* 100° 80° 212° Number of degrees between the two fixed points, 100 80 180 In reducing a temperature from one of the three scales to another, imagine a thermometer carrying the respective two scales beside each other, note that, for differences of temperature, S°C = 4 °R = 9°F. and the rest follows by itself. * Reduced to lat. 45° and Sea- Level, strictly speaking. In Great Britain how- ever the general practice is to define "2I2°F" as the temperature at which water boils, when the pressure of the atmosphere is equal to that of 30 inches ( = 762 mm. ) of mercury of 32T. , at the place of observation. 163 Table III. Equivalent Readings. F. C. R. F. C. R. o -17* -Hi 284 140 112 Equal - 4 - 20 -16 293 H5 116 Differences of + 5 + 14 -is - 10 - 12 - 8 302 3" 150 155 120 124 Temperature. AF AC aR + 23 - 5 - 4 320 160 128 + 3 2 329 165 132 41 + 5 + 4 338 170 136 1 2 •56 1. 11 •44 .89 5° 10 8 347 175 140 3 1.67 i-33 59 15 12 356 180 144 4 2.22 1.7S 68 20 16 365 185 148 5 6 2.78 3- 33 2.22 2.67 77 25 20 374 190 152 86 3° 24 383 195 156 7 8 3-89 4.44 3-" 3-56 95 35 28 392 200 160 9 5.00 4.00 104 40 32 401 205 164 "3 45 36 410 210 168 122 5° 40 419 215 172 AC aF AR 131 55 44 428 220 176 140 60 48 437 225 180 149 65 52 446 230 184 I 2 1.8 3-6 5-4 .8 1.6 158 70 56 455 235 188 3 2.4 167 75 60 464 240 192 4 7.2 3-2 176 80 64 473 245 196 5 9.0 4.0 185 85 68 482 250 200 194 90 72 491 255 204 203 95 76 5°° 260 265 208 AR aC AF 212 100 80 S°9 212 221 i°5 no "5 84 88 92 5i8 527 536 270 275 280 216 230 239 220 224 1 2 3 1.25 2.50 3-75 2.25 4. 50 6-75 248 120 96 545 285 228 4 5-°° 9.00 257 125 100 554 290 232 266 130 104 5 6 3 295 236 275 135 108 572 300 240 164 The following Diagram representing a Double Thermometer Scale, Fahrenheit degrees on the left side and Centigrade on the right, competes with Table III. as far as these two scales are concerned; but, to those who have learned to guess tenths of degrees, offers the advantage of giving the desired result without interpolation — apart from the reading error, which however needs not exceed o.i°C or 0.2° Fahr. 'Below. ■uemux. riuouiq Xo bouituv . 1 Above, oouinc Y F c F c F c F C F c F C F C 58 — — 50 32 — -0 122 — -50 212- -100 302- —150 392 — -200 482- —250 - - - - - - 5J 67-f — 55 23-5 —-5 113 -5 -45 203-5 -95 2935: —145 3835; -195 4735; —245 _ _ _ _ _ _ — - - - - - - - 76 T — 60 14 ~E - 10 104-5 -40 194-5 — 00 284-5 —140 3745; -190 464^ —240 — - — - _ - 1 - - - - - - ~ _ _ 35 ~ -65 +5 -5 - 15 95-5 -35 185 -5 -65 275~ -135 3655; -185 4555; -255 - - - - - - ~ - - 94-f - 70 -4 ~E -20 66-5 -30 176-5 -80 2665; -130 3565; -180 446-5 -230 103^ - 75 13 -5 -25 77-5 — 25 167-5 -75 2575: — 125 3475; — 175 4375; -225 - - if- — -t- - j^ - j; - - 71: — _ -L - - - .5 - 112-5 -80 22-5 -30 68-5 -20 158 — -70 248— — 120 558— -170 426^ —220 * JZ - 4>sJI - '1 - il - - - — - - SI— - n — - - - - 121-5 -85 31-5 -35 59 -5 -15 149 5] -65 239^ -115 5295: -165 4495; -215 j: _: - _2 - - ~ - _n - _z — - - 150 — — 90 40- —.40 50 ~ — 10 140- — 60 250— -110 320^ — 160 410 5; -210 - -5 ~ -_ 55. 5; - 159-5 — 95 4S-5 — 45 41 5: — 5 131-5 -55 221- -105 311 5; -155 401 5; -205 -5 -5 51 -5 -5 _ 5; _ -5 -5 -5 -5 51 - 5! - l^fld— looi -sa-^-so 1 32-3- 122^-50 212 J- 100 302^-150 392-2— 200 i65 CORRECTIONS AND ADDITIONS. Atomic Weights, page i. Lithium. — Since the publication of my memoir I have caused Mr. J. F. Ness to try and check my result by analyses of the chloride. His results so far agree better with Li = 7.00 than with my number, 6.89. We are engaged in fresh analyses of the carbonate by means of an improved method in order to explain the discrepancy. Meanwhile analyses had better be calculated after the value 7.00. Hydrogen. — Since the completion of the MS. for this volume I undertook, conjointly with Mr. J. B. Henderson, a critical experimental enquiry on Dumas' and on Erdmann and Marchand's determinations of the constant H 2 : O, which ultimately developed into a re-determination of its value. Our principal results are as follows : — ( 1 ) Dumas' results, if taken conjointly with Erdmann and Marchand's and properly in- terpreted, assign to the ratio O : H the value 15.996, or practically, 16.00. But (2) Dumas, it seems, forgot to reduce his water-weights to the vacuum, while Erdmann and Marchand lost some of their water by employing only chloride of calcium as a dehydrator for the gas which left their water-receptacle. Allowing for the respective errors we arrived at values lying at about 1.004 an( i 1.006 respectively. (3) Our own (final) series of (ten) syntheses gave values lying between 1.0073 an d 1.0114; the mean of the ten values was 1. 0091 18; probable error of a single observation = ± 0.00087 ; of tne mean = ± .00027. But i although all the ten experiments were successful, we had reason to attach greater weight to certain seven of the ten results than to the other three. These seven results varied from H = 1.00888 to 1.00949 ; mean of the seven numbers = 1.008808. But all our numbers as given so far are liable to a small correction for the unavoidable presence of occluded hydrogen in the reduced copper produced. The correction which we applied was based upon experimental determinations of the occluded hydrogen in a previous series. Referring to our memoir for details, we confine ourselves here to giving the number which we adopted finally as being, in our opinion, the most probable. It is H = 1.008473. But ours is only one of quite a series of recent attempts to fix the value of this constant ; the following table gives the names and results of our principal predecessors. 1 66 TAKING = 1 6. I. Lord Rayleigh found H = 1.006 92 II. Cooke and Richards 1.008 25 III. W. A. Noyes 1.007 x 7 IV. Dittmar and Henderson ; their adopted number 1.008 47 IVa. D. & H.; the mean of the ten experi- ments corrected - 1.008 78) V. E. H. Keiser 1.003 J 8 The first four results, as we see, agree very fairly with one another ; their mean, 1.00702, indeed, does not differ from any of the numbers I. to IV. by more than its possible error; but Keiser's value is so far below any of the rest that either it or they must be infected with an unobserved error. In these circumstances one feels inclined to adopt that mean value 1.00702 as being, at present, probably the closest obtainable approximation to the truth. Considering, however, that of the four methods used, Dittmar and Henderson's is by far the easiest of execution, I prefer to adopt their result, which practically may be put down as H = 1-0085 as the most probable value. But, as I do so, I ought, in strictness, to correct all those of my formula-values, analytical factors, and gasometric constants, which depend on the atomic weight of hydrogen. In practice, however, it suffices to apply the correction to those numbers whose inherent uncertainty is not greater than the correction itself, and these form only a small minority. All the necessary corrections are found below under their respective headings. Values of the "Probability-Integral," &c, page 51. As a supplement thereto, we will give the formulae for the calcula- tion of the probable error "r" of a single determination in, and of the probable error r of the mean of, a series of direct determinations of an unknown quantity. If the differences between the mean x and the individual results x', x", x'" . . . are x'-x = v'; x"-x = v"; x'"-x = v'", etc., and the number of the determinations is n, we have /^ 2 "- - Log. Const. = .8290 - 1 and r , ' - ± .6 7 4S \/^ 167 or approximately, r= + - 8 453 2 v \/n (n- 1) Log. Const. = .9270 - 1 and Jn as before. In the approximate formula, 2 v stands for the arithmetical sum of the numbers v. Thus, for instance, if v' = - 1, v" = - 2, v'" = + 5, then 2 v = 8, and not + 2. Since the values r and r , whether calculated by one of the formulae or the other, cannot be more than guesses at naturally indeterminate quantities, it is permissible to substitute n - o. 5 for Jn (n - 1 ). Examples. V3 * 2 = 2-449; *A * 3 = 3-4 6 4; "Js * 4 = 4-472; \/6 x 5 = 5.477. The approximation, of course, becomes the better the greater n. Thus, V13 x 12 = 12.490; J20 x 19 = 19.494. Table of Square-Roots. n Vn n ■v/n n Jn n Jn 1 1. 0000 26 5.0990 5i 7.1414 76 8.7178 2 1.4142 27 5.1962 52 7.2111 77 8-775° 3 I-732I 28 5-29I5 53 7.2801 78 8.8318 4 2.0000 29 5-3852 54 7-3485 79 8.8882 5 2.2361 3° 5-4772 55 7.4162 80 8.9443 6 2-4495 3i 5-S678 56 7-4833 81 9.0000 7 2.6458 32 5- 6 569 57 7-5498 82 9-°554 8 2.8284 33 5-7446 58 7.6158 83 9. 1 104 9 3.0000 34 5-831° 59 7.6811 84 9.1652 10 3.1623 35 5-9i6i 60 7.7460 85 9.2195 11 3.3166 36 6.0000 61 7.8102 86 9.2736 12 3.4641 37 6.0828 62 7.8740 87 9-3274 13 3.6056 38 6.1644 63 7-9373 88 9.3808 14 3-7417 39 6.2450 64 8.0000 89 9.4340 15 3-873° 40 6.3246 65 8.0623 90 9.4868 16 4.0000 41 6.4031 66 8.1240 91 9-5394 i7 4-1231 42 6.4807 67 8.1854 9i 9-59I7 18 4.2426 43 6-5574 68 8.2462 93 9-6437 J 9 4-3S89 44 6.6332 69 8.3066 94 9.6954 20 4.4721 45 6.7082 70 8.3666 95 9.7468 21 4.5826 46 6.7823 7i 8.4261 96 9.7980 22 4.6904 47 6.8557 72 8-4853 97 9.8489 23 4-795 8 48 6.9282 73 8.5440 98 9.8995 24 4.8990 49 7.0000 74 8.6023 99 9.9499 25 5.0000 5° 7.0711 75 8.6603 100 10.0000 1 68 FORMULA VALUES. Hydrogen, page 55. The corrected values are : Hydrogen, F(0=i6.) Log. F. 1-0085 .00368 Multiples. 1.0085; 2-0170; 3.0255; 4.0340; 5.0425; 6.0510; 7.0595; 8.0680; 9.0765; 10.0850. Water, H 2 18-017 .25568 Values of n x 18.017. n Multiple. Log. Multiple.* n Multiple. Log. Multiple.* 1 18.02 • 2 557S 16 288.27 45980 2 36-03 .55666 17 306.29 48613 3 54-05 .73280 18 324-3 1 51096 4 72.07 •85775 19 342.32 53443 5 90.09 •95468 20 360.34 S5671 6 108.10 •03383 21 378.36 5779o 7 126.12 .10078 22 396.37 59810 8 144.14 .15878 23 414-39 61741 9 162.15 .20992 24 432.41 63590 10 180.17 .25568 25 450-43 65363 11 198.19 .29708 26 468.44 67065 12 216.20 •33486 27 486.46 68705 13 234.22 .36962 28 504.48 70284 14 252.24 .40181 29 522.49 71808 15 270.26 •43178 30 540.51 73280 * The " Logg." are the logarithms of the multiples as they stand in the table, not the logarithms of the exact multiples of 18.017. Lithium, page 56. Adopt the values given under Li = 7.00. Nitrogen, page 57. The corrections embodied in the following numbers are very slight ; yet we thought we had better effect them. For H = 1 "0085, we have: — NH 3 NH 4 (NH 4 ) 2 NH 4 C1 F. Log. F. 17-071 .23226 18-080 .25720 52-160 • -71734 53-534 .72863 169 ANALYTICAL FACTORS. Lithium, page 63. Adopt the numbers given under Li = 7.00. Nitrogen, page 64. For H = 1.0085, we have: — a b b :a Log. f N NH 3 1-21536 08471 NH 3 N 0-82280 9iS 2 9 2NH 3 (NH 4 ) 2 1-52774 18405 (HN 4 ) 2 2 NH 3 0-65456 8i59S NH 4 C1 NH, 0-31888 50363 2 NH 4 C1 (NH 4 ) 2 0-48717 68768 PtCl 6 (NH 4 ) 2 2NH3 0-07683 88553 PtCl 6 (NH 4 ) 2 (NH 4 ) 2 0-11738 06958 Pt 2 NH 3 0-17464 24214 Pt (NH 4 ) 2 0-26680 42619 EMPIRICAL FACTORS, page 67. (c) Determination of Ammonia given as Sal-Ammoniac. For H = 1-0085, we have: — Given. Wanted. f. Log. f. PtCl 6 (NH 4 ) 2 2NH 4 C1 0-2389* PtCl 6 (NH 4 ) 2 2NH 3 0-07618 PtCl 6 (NH 4 ) 2 2N 0-06268 Pt 2 NH 4 C1 0-5459* Pt 2NH 3 ' 0-17408 Pt 2N 0-14323 * Independent of the value assumed for H, because directly determined .37822 .88185 •797 r 4 •737" .24074 •i5 6 °3 THE METRIC AND BRITISH SYSTEM OF UNITS. Other Relations, page 73. Additional entry : — 1 grain = 0.064 799 grammes. Log. = .811 568 - 2. 170 CORRECTIONS OF WEIGHINGS. Table I., page 103. This table is inconvenient, the intervals being too large to admit of mental interpolation ; we therefore here substitute another edition, which virtually gives the values of D for every tenth of a degree. Weight, in Grammes, of i Litre of (Dry) Air containing 0.04 % by volume of Carbonic Acid, supposing the air to be measured at f C and 760 mm. of mercury of 0° C ; page 103. Weight of Diff Weight of Diff t 1 Litre D. (Neg.) P. p. t 1 Litre D. (Neg.) P.P. At AD o 1-2943 1.2267 At AD " *5 I 2 I.2895 I.2848 48 47 O.I 0.2 °-3 5 9 H 16 17 1.2225 1. 2182 42 43 0.1° 0.2 o-3 4 8 '3 3 I.2802 46 0.4 19 18 1. 2140 42 0.4 17 4 r-2755 47 0.5 0.6 23 28 19 1.2099 4 1 o-S 0.6 21 25 5 1.2709 46 0.7 0.8 33 37 20 1.2057 42 0.7 0.8 29 33 6 1.2664 45 0.9 42 21 1. 2016 4i 0.9 38 7 8 1. 2618 1-2573 46 45 1.0 46.4 22 23 I-I975 I-I935 4i 40 1.0 4i-7 At AD At AD 9 1.2529 44 24 1-1895 40 0.1° 4 0.1° 4 10 1.2484 45 0.2 9 25 1-1855 40 0.2 8 11 12 1.2440 1.2397 44 43 °-3 0.4 0.5 •3 17 22 26 27 1-1815 1. 1776 40 39 o-3 0.4 o-5 16 20 13 1-2353 44 0.6 26 28 1. 1736 40 0.6 24 14 1. 2310 43 0.7 0.8 3i 35 29 1 1. 1697 39 0.7 0.8 28 32 15 1.2267 43 0.9 1.0 39 43-7 3° 1-1659 38 0.9 1.0 35 39-4 D at 76o±p = |D at 760 J- |i± -P_| = |d at 760 j |i±.ooi 315 8 p| Multiples of 0.00 1 3158. I .001 316 2 .002 632 3 .003 947 4 .005 263 5 .006 579 6 .007 895 7 .009 211 8 .010 526 9 .011 842 10 .013 158 i7i Logarithmic Table. t Log. D. Diff. (Neg.) P.P. t Log. D. Diff. (Neg.) P.P. At A log. At A log. o .11202 15 .08875 I 2 .11043 .10885 159 158 0.1° 0.2 0.3 16 31 47 16 17 .08724 •08573 !5! 151 0.1° 0.2 0.3 15 30 45 3 .10727 158 0.4 63 18 .08424 149 0.4 60 4 .10569 158 0.5 0.6 79 94 19 .08274 I50 0.5 0.6 75 90 5 .10413 156 0.7 0.8 no 126 20 .08126 I48 0.7 0.8 105 120 6 .10256 157 0.9 142 21 .07977 149 0.9 134 7 8 .10101 ■09945 155 156 1.0 157-3 22 23 .07829 .07682 148 147 1.0 149.4 At A IOC. At A log. 9 .09791 154 24 .07535 J 47 O.I° IS 0.1° 15 IO .09637 I.S4 0.2 31 25 .07389 146 0.2 29 ii .09483 154 0-3 0.4 46 61 26 .07243 146 0-3 0.4 44 58 12 .09330 153 0.5 76 27 .07098 145 °-5 73 13 .09178 152 0.6 92 28 •06953 145 0.6 87 14 .09026 I5 2 0.7 0.8 107 122 29 .06809 144 0.7 0.8 102 116 15 .08875 151 0.9 1.0 i3» 152.9 3° .06665 144 0.9 1.0 131 145-3 For the value D' corresponding to the pressure P' mm. we have : — Log. D' = Log. D + Log. P' 4- .119 19 - 3. GASOMETRY. Correction for the Gravity at the place of Observation, page 130. For the calculation of the relative gravity at a place from its latitude

. NO ,0 en Q O o ^~ H_rt O O o m Os Os ON 't N a S° II q q i-i M °o u-i ON 00 m II °o "lb M o on o 8- q CI o to ■*■ y3 ON c/i 3 P 1-4 is R r^. in w o ■a CO 1-1 o IT) O o o o fO ON o on ON ■*• w •SB .J q q ss .- o £ ii o o CO in O CO PSJS o on q o ON q ON 00 q On M PO ON > •w' «* C — ! rt J) LO o ro 00 \£ 00 ON CO 1^ ON On o o ON ON ■ rt 00 ON * M q q " _o c 'C 0) * 4J bjQ„ Tj" NO M NO rX O M 00 NO NO NO /*> t -S II 1-1 ** M 00 00 ON ro o lA in in •* a ffi « " " " * S o a o r^ ■* -O.S aj no •* CO J^ 3-3 1" O o O ■* q q o ■* l-H §° o |_ NO t^ 00 2 o 00 NO II NO NO N o o 'o )H ON On ON , "O . , , , . , < (A vT ►J s c £> o _o S S5 1 C O c 1 '_ ' 5 ti o is < w C3 o « Q ' c o :- QJ T3 C OJ a < S a C M O T3 C ^ 5 i S 3 ij~ ■a s "3 C q's 0) « >, OJ « 4J >, « pq « « Q f4 ff g « .1 ^ G c O ^J R rt o fi G rt rG X hfl CJ 3 O .a >N cq "75 Table for the Reduction of Gasometric Constants to one ANOTHER. For 2 : H 2 = 16 : 1.0085, we have :— A. — Given: The Molecular Weight, O = 16; page 142. Wanted. Formula. (2) Specific Gravity, S H ; Hydrogen = 1 S H = 0.495 79 M. Log. = .695 29 - 1 B. — Wanted: The Molecular Weight, O = 16; page 143. Given. Formula. (2) Specific Gravity, S H ; Hydrogen = 1 M = 2.017 S H j Log. = .304 71 D. — Change of Standard for Specific Gravity S ; page 144. Given S in reference to Wanted S in reference to Multiply by f f Log. f Oxygen =16 Hydrogen = 1 O.991 57 .99632-1 Air = 1 Hydrogen = 1 14-349 1.15684 Hydrogen = 1 Air = 1 0.069 689 .843 17-2 Hydrogen = 1 Oxygen =16 1.008 5 .003 68 OTHER WORKS BY THE SAME AUTHOR. I.— Report on the Composition of Oeean Water : Part of "The Physics and Chemistry of the Voyage of H. M.S. Challenger." Part I. Printed for Her Majesty's Stationery Office. Sold by Adam and Charles Black, and Douglas & Foulis, Edinburgh ; and others. 1884. II.— A Manual of Qualitative Chemical Analysis : Douglas & Foulis, Edinburgh, 1876. Tables belonging thereto, same publishers, 1877. III.— Analytical Chemistry : A Series of Laboratory Exercises, constituting an Elementary Course of Qualitative Analysis. W. & R. Chambers, Edinburgh ; 2nd edition; 1886. IV.— Exercises in Quantitative Chemical Analysis : With a Short Treatise on Gas Analysis. William Hodge & Co., Glasgow ; Williams & Norgate, London and Edinburgh. 1887. EXERCISES IN THE PREPARATION OF ORGANIC COMPOUNDS. By Prof. Emil Fischer, Wiirzburg. Translated, with permission of the Author, from the second German edition, by Archibald Kling, F.I.C., with a Preface by W. Dittmar, LL.D., F.R.S.S. L. and E. William Hodge lS; Co. , Glasgow ; Williams & Norgate, London and Edinburgh. ■ ■"■ ■' ■ ■' ' ' r