MATHEMATICAL TABLES CONSISTING OF LOGS AND COLOGS OF NUMBERS FROM 1 TO 40,000 ALSO, TRIGONOMETRICAL FUNCTIONS AND THEIR With Subsidiary Tables COMPILED BY E. CHAPPELL, B.Sc., A.C.G.I., Assoc.M.Inst.C.E. Late Lecturer in Engineering Science at the City and Guilds' (Engineering) College, NEW YORK: 25 Park Place D. VAN NOSTRAND COMPANY LONDON: 38 Soho Square, W. W. & R. CHAMBERS, LIMITED EDINBURGH: 339 High Street THE genius and labours of Napier, Briggs, and Vlacq during the early part of the seventeenth century resulted in the production of what may perhaps be termed the most wonderful piece of work in the whole history of Mathematics —namely, a Table of Logarithms. At that period the only Mathematical Sciences were Astronomy, Navigation, and Surveying. In these the new tables could be used to reduce the necessary calculations to the simplest possible form-that is to say, Addition and Subtraction were substituted for Multiplication and Division. It has been well said that the invention of logarithms has added years to the life of the astronomer. For two hundred years, roughly speaking, it is probable that no one found these tables inadequate for the calculation of any problem contemplated by contemporary science. In the nineteenth century, however, other sciences-notably Electricity and Thermo-Dynamics-became mathematical; and one fundamental difference between the new sciences and the old is that whereas, in the latter, numbers had only to be raised to simple powers-merely squared in the majority of cases— fractional indices occur in all branches of the former. Consequently, it cannot be said that the tables of three hundred years ago reduce the calculations of to-day to the simplest possible form. Their defect is that processes of Involution and Evolution are somewhat tedious with any but the simplest indices; whereas, by the use of tables adapted to the processes in question, modern calculations can also be reduced to the simplest form. The lolog and illolog tables now published for the first time are intended to accomplish for the modern scientist what the original logarithm tables did for the scientist of the seventeenth century, although they are but an obvious development of Napier's unparalleled work. The compilation of such tables was decided upon because of the great assistance they would give to the author in his own work. Their publication is due to a firm conviction that many scientists and others will find them equally useful. It may be well to point out that the calculations of modern Experimental Science are all based on Measurement. In the great majority of cases measurement of length is necessary, and this can only be carried out with great accuracy if the most refined methods are employed. As we necessarily start from data of limited accuracy, the Theory of Error must be employed to find to what extent the result can be relied on. The number of figures that can justifiably be retained in a result is quite small, even though the ordinary processes of arithmetic. might, in some cases, give an infinite number. It is for this reason that the Engineer and the Applied Scientist make use of logarithm tables to but few decimal places; in fact, as few as will enable the results to be as accurate as the data allow. The advantage of tables to few places is not only that fewer figures have to be written down, added, or subtracted, but there is less page-turning in finding values. These remarks have been made in explanation of the comparatively small number of figures in some of the tables in this book, which might cause the Pure Mathematician to doubt their sufficiency. The number of figures given is thought to be sufficient to meet almost all the requirements of many branches of Applied Science. If it should be found that the present tables are inadequate in any respect, no pains will be spared to augment them. It seems a matter for some regret that Napier gave his new function such an awkward name, the last three letters of its anglicised form being quite unpronounceable. This arrangement of letters cannot be justified etymologically, because in the Greek word ȧpoμós there is a syllabic division in the middle of the three consonants complained of. In addition to its being unpronounceable, the word is too long for repeated use. Most users of logarithms realise this, and avoid the difficulty by calling them 'Logs.' Since this contraction is already in general use, and very suitable as the basis of a systematic series of names, it is proposed that the word 'log' should be regularised, and freed from any taint of slang. It is next proposed that the log of the log of a number should be called the 'Lolog' of that number. Again, to facilitate the finding of results, the scientist very often employs tables in which the log is the argument. These tables have been somewhat clumsily called 'Anti-logarithm Tables.' In this book it is proposed to adopt the systematic name 'Illog,' which may be regarded as a contracted form of Anti-log.' In order to make the lolog tables as convenient as possible to use, they are accompanied by their inverse tables, which, to conform to the old nomenclature, should be called 'Anti-anti-logarithms' or 'Anti-lologs;' but it is proposed that they should be called 'Illologs.' We thus have two pairs of names- -Logs and Lologs, Illogs and Illologs. These names will not, of course, carry their meaning on the surface, but will strongly suggest it to those to whom the terms have once been defined. In constructing the tables, the utmost care has been taken to make them reliable a very difficult matter, owing to the fact that the lolog of unity is infinite. It was somewhat disconcerting to observe the erratic variation of the lolog differences, which was traced to the approximate last figure of sevenfigure logs. Nowhere in the calculation of the tables have less than seven-figure logs been used; for a large portion of them eight figures were employed, and in the neighbourhood of unity ten figures. The number of figures to which the calculated values have been reduced for publication represents greater accuracy on the whole, it is thought, than can be obtained when performing Involution and Evolution in the ordinary way with seven-figure logs, even though one deludes one's self that one has a six-figure result. In an entirely new work it is, of course, quite impossible to foresee exactly the best arrangement and the magnitude of the intervals at which values should be given. The author, therefore, will be greatly indebted to any one, having actually used the new tables, who will favour him with suggestions for their improvement. The tables of Logs, Cologs, Illogs, Trigonometrical Functions and their Logs have been added in order to make the book complete in itself. The unusual arrangement of these tables has been adopted with a view to convenience in use, and the intervals have been made sufficiently small to reduce interpolation to a minimum. E. C. INTRODUCTION. I.-LOGS. THE log of a number is the index to which another number, called the base, must be raised in order to be equal to the original number. Thus, if AB=C, B is the log of C to the base A. This is sometimes written : B=loga C. If AB-C and AD=E, then B=loga C and D=loga E. Now Cx E=AB × AD=AB+D, therefore B+D=loga (C × E). But Blog C and D=log, E, therefore loga (CE) = loga C+loga E. Hence, if a table is constructed, giving the logs of all numbers, it can be used to simplify multiplication. To multiply C by E, their logs must first be found in the table and then added. The sum so obtained is the log of the desired result, which can be found from the same table. C Suppose that instead of the product, the quotient is required. E or loga ()=loga C-log1 E. C Hence, to find the quotient subtract the log of E from the log of C, and the difference is the log of the desired result, which can then be found from the table. If it is required to find the value of C", a somewhat similar method can be used : This is in the same form as the previous case considered, from which it follows that From these four examples of the simple arithmetical processes, it will be seen that a table of logs can be used to simplify multiplication and division to addition and subtraction; and involution and evolution to multiplication and division. Apart from practical convenience, a table of logs to any base would answer the purpose, but there are two distinct advantages gained by using 10 as the base. One advantage is that the first figure of any log can be written down at once without using the table, for :— |