Logarithmic Tables Research Articles

Mathematics of Growth—I

THE first three articles in this series discussed the squares method for fitting a straight line trend to a series of sales or earnings figures. It was suggested that the most useful method would be to fit a straight line trend to the logarithms of the individual items in each series. The antilog of this trendline would then give the annual percentage growth trend. While this annually compounded growth trend is a convenient measure for most applications, it does not lend itself as an eflicient measure when one wishes to measure several components of growth. Fortunately, annually compounded growth rates can be converted into a form which will enable the analyst to work such problems. In the last issue of this Journal, Associate Editor John Fountain's article Growth Figured the Simple Way presented a table of continuously compounded growth rates which are faster to use than a book of annually compounded interest rates (perhaps 15 seconds to locate the growth rate rather than 30 seconds with a book or a lot longer if you don't have the book and are working with logarithms). His table was a table of natural logarithms with different headings for the columns to make them easier to read. The article included a delightful exposition of the case for regarding continuous compounding as a more realistic concept of a company's growth pattern than annual compounding. Does a company really grow once a year in an annual jump? . . . No. A company is like a living thing that grows not in annual or monthly spurts, but continuously, every hour, every minute, every second. There is not a great deal of difference between annual growth rates and continuously compounded growth rates -at least not enough to cause one to undergo much of a change in his thought pattern. For example, if a savings and loan association paid 5% interest compounded only once a year, a $100 account would grow to $105.00 at the end of the year. If interest were calculated at the midyear as well as the close, the year-end value would be $105.06. If interest were compounded monthly, it would grow to $105.10 and if compounding was continuous, every second of every minute of every day in the year, the account would grow to $105.12 by the end of the year. Since this isn't very startling, the reader might be wondering what are the practical advantages, if any, of continuous compounding. The advantages come when one is analyzing the components of a company's growth. The reason is that continuous growth rates are additive in nature, while annually compounded growth rates are multiplicative in nature. One of the first practical applications of this difference was Daniel Murphy's reports when he was associated with Faulkner, Dawkins & Sullivan. To illustrate, let us use a hypothetical case in which a company has a 10% improvement in each of the three major factors of earnings growth-sales growth, pretax profit margins, and the tax rate. For this purpose it is more convenient to use the retention rate, the percent of pretax earnings carried down to net income. In Table 4.1 below, we have listed the annually compounded growth rates and also the continuously compounded growth rates:

Priority Correction to "Unbiased Estimation of the Standard Deviation"

In a recent issue of this journal, Edward E. Cureton [1] published a table of correction factors, k, which allow unbiased estimation of the standard deviation of a normal population using the sample standard deviation. Excerpts from the following table which I have recently calculated will appear in [2]. The table was calculated using double precision arithmetic and the table of logarithms of the gamma function given in [3]. The table is in a form preferred by many statisticians and is more complete than Cureton's. If n is the sample size, x2 = (X X)2, and you are given lx2 S n n-I (1)

More on Unbiased Estimation of the Standard Deviation

In a recent issue of this journal, Edward E. Cureton [1] published a table of correction factors, k, which allow unbiased estimation of the standard deviation of a normal population using the sample standard deviation. Excerpts from the following table which I have recently calculated will appear in [2]. The table was calculated using double precision arithmetic and the table of logarithms of the gamma function given in [3]. The table is in a form preferred by many statisticians and is more complete than Cureton's. If n is the sample size, x2 = (X X)2, and you are given lx2 S n n-I (1)

An Exercise in Ratio Estimation

Substitute (2) and (11) into (1) to find equation (8). Table 1 below gives the probability for the circular case for d = 1(1)30, N = 2(1)35. For example, in problem (a), d 2, N = 18, D = 365; from Table 1, the probabili-ty -that there are at least two girls with the same or adjacent birthdays is 0.730. To evaluate probabilities no-t included in Table 1, use tables of logarithms of fac-torials as illustrated in Mosteller [3, p. 47]; for certaini problems one might use approxiimations similar to those in Feller [1, p. 32]. For example, equation (7) gives the solutioin to problem (b), (N = 18, D = 365),

New Tendencies in Economic History

When I was at the lycée, I listened with passion to the lessons of my history professor and I eagerly enlarged upon his lessons by doing a great deal of reading. On the other hand, I disliked mathematics. For me, equations, theorems and logarithms were no more than cumbersome baggage for the historian I aspired to become. Worse than that, it was an obstacle which inevitably I had to surmount in order to obtain my baccalauréat. Having surmounted the obstacle, I simbolically sold my table of logarithms, for had I not freed myself forever of that which for too long had been a nightmare?… Now, about twelve years later, without my ever having deviated from my original course, I find myself once again confronted with mathematics. Of the same sort? Certainly not. What I am now confronted with is infinitely more complicated and beyond my comprehension as I lack the basis which I erstwhile ignored. What then has happened? Why is it that the historian that I became in the meantime now needs mathematics today?

A Critical Table of Two-Decimal Logarithms

A Critical Table of Two-Decimal Logarithms

USE OF THE AREA IN FORMATION OF THE CONDITION EQUATIONS FOR TRILATERATION AND MIXED FIGURES

Abstract Corrections to observed quantities in trilateration nets have always been obtained by satisfying certain geometric conditions, e.g. a side condition or angle condition. This paper introduces the area condition as a further geometric condition which could be used conveniently to obtain the necessary corrections.It is only required to have one redundant observation for each simple geometric figure, which can always be obtained even in ordinary observations.The large quantities which usually appear when areas are considered are avoided by using logarithmic tables in a convenient way.

R.S.M. Tables No. 8: Tables of Natural and Common Logarithms.

R.S.M. Tables No. 8: Tables of Natural and Common Logarithms.

Tables of Natural and Common Logarithms to 110 Decimals. By W. E. Mansell. Edited by A. J. Thompson. (Royal Society Math. Tables, Vol. 8.) Pp. xviii, 95. 40s. 1964. (Published for the Royal Society at the University Press, Cambridge)

Tables of Natural and Common Logarithms to 110 Decimals. By W. E. Mansell. Edited by A. J. Thompson. (Royal Society Math. Tables, Vol. 8.) Pp. xviii, 95. 40s. 1964. (Published for the Royal Society at the University Press, Cambridge) - Volume 50 Issue 372

TABLES OF BINARY LOGARITHMS, UNCERTAINTY FUNCTIONS, AND BINARY LOG FUNCTIONS.

Three sets of 4-place tables are presented, one set on each of two facing pages: binary logarithms (log2 n) of 3-digit numbers, uncertainty functions (– p log2 p) and binary log functions (log2 1/ p) for values of p from .001 to .999.

Legibility of mathematical tables: the Babbage experiments of 1827

In 1827, Charles Babbage published his Table of Logarithms in London. In the preparation of the first edition, the author, together with the printer, carried out one of the first systematic investigations into the factors which make printed pages both easily legible and optically endurable. In the preface, Babbage summarizes the results of his studies and experiments with sizes and arrangements of type as well as with the color and texture of paper. That the Babbage experiments and conclusions are valid is readily apparent when one compares his book with similar volumes printed more than one hundred years later.

Tables of Natural and Common Logarithms to 110 Decimals

Tables of Natural and Common Logarithms to 110 Decimals

Tables of Natural and Common Logarithms to 110 Decimals.

Tables of Natural and Common Logarithms to 110 Decimals.

Numerical evaluation of the elliptic integral of the third kind

1. C.-B. LING, of values of the integrals fl (xm/sinhvx) dx and fl (x/coshPx) dx, J. Math. and Phys., v. 31, 1952, pp. 58-62. MR 13, 690. (Two misprints are here noted. p. 59, the subscript of s' in the last equation of (8) should be m 2p + 2r. Again, on p. 60, the exponent of 2 in the first equation of (9) should be p + 1.) 2. C.-B. LING & C. W. NELSON, On evaluation of Howland's integrals, Annals of Academia Sinica, no. 2, part 2, 1955, pp. 45-50. 3. C. W. NELSON, A Fourier integral solution for the plane-stress problem of a circular ring with concentrated radial loads, J. Appl; Mech., v. 18, 1951, pp. 173-182. MR 12, 880. 4. C. W. NELSON, Tables of Howland's and related integrals, Math. Comp., v. 15, 1961, pp. 12-18. MR 22 S 10203. 5. J. W. L. GLAISHER, Numerical values of the series 1 1/3n + 1/5n 1/7n + 1/9n , Messenger of Math., v. 42, 1912, pp. 35-49. 6. J. W. L, GLAISHER, of 1 i 2 n + 3 4t 4 + etc. and 1 + 3+ 5n + 7-n + etc. to 32 places of decimals, Quart. J. Pure and Appl. Math., v. 45, 1914, pp. 141-152. 7. H. T. DAVIS, Tables of Higher Functions, Vol. 2, Principia Press, Bloomington, Indiana, 1955. 8. J. PETERS & J. STEIN, Mathematical tables, Appendix of Peters' Ten-place Logarithmic Tables, Vol. 1, Ungar, New York, 1957. 9. A. H. R. GRIMSEY, On the accumulation of chance effects and the Gaussian frequency distribution, Philos. Mag., v. 36, 1945, pp. 294-295. MR 1, 311. 10. L. S. GODDARD, The accumulation of chance effects and the Gaussian frequency distribution, Philos. Mag., v. 36, 1945, pp. 428-433. MR 7, 311. 11. B. BUTLER, On the evaluation of fJ (sin t/t)m dt by the trapezoidal rule, Amer. Math. Monthly, v. 67, 1960, pp. 566-569. 12. K. HARUMI, S. KATSURA & J. W. WRENCH, JR., Values of (2/ir) fJ (sin t/t) dt, Math. Comp., v. 14, 1960, p. 379. MR 22 N 12737. 13. R. G. MEDHURST & J. H. ROBERTS, Evaluation of the integral (2/0r) fo (sin t/t)m. cos bt dt, Math. Comp., v. 19, 1965, pp. 113-117.

Table of Common Logarithms and their Squares

Table of Common Logarithms and their Squares

Natural bases of logarithms

Most Elementary Expositions of logarithms include the information that there are tables of logarithms to the base 10 and to the base e. The student who is told this finds the first bit of information perfectly understandable and the second perfectly mystifying.

On the Distribution of First Significant Digits

Introduction. It has been noticed by astute observers that well used tables of logarithms are invariably dirtier at the front than at the back. Upon reflection one is led to inquire whether there are more physical constants with low order first significant digits than high. Actual counts by Benford [2] show that not only is this the case but that it seems to be an empirical truth that whenever one has a large body of physical data, Farmer's Almanac, Census Reports, Chemical Rubber Handbook, etc., the proportion of these data with first significant digit n or less is approximately log,o(n + 1). Any reader formerly unaware of this peculiarity will find an actual sampling experiment wondrously tantalizing. Thus, for example, approximately 0.7 of the physical constants in the Chemical Rubber Handbook begin with 4 or less (log,o(4 + 1) = 0.699). This is to be contrasted with the widespread intuitive evaluation Aths. At least two books call attention to this peculiarity, Furlan [6] and Wallis [18], but to my knowledge there are only five published papers on the subject, Benford [2], Furry et al [7], [9], Gini [8], and Herzel [11]. The first consists of excellent empirical verifications and a discussion of the implied distribution of 2nd, 3rd, -.significant digits. The second and third put forth the thesis that the distribution of significant digits should not depend markedly on the underlying distribution, and the authors present numerical evaluations for a range of underlying distributions in support of their contention. The fourth maintains that explanation is to be sought in empiric considerations. The fifth considers three different urn models; each yields a distribution of initial digits which the author compares with log,o(n + 1). This paper is a theoretical discussion of why and to what extent this so called abnormal law must hold. The flavor of the results is, I think, conveyed in the following remarks. (i) The only distribution for first significant digits which is invariant under scale change of the underlying distribution is log,o(n + 1). Contrary to suspicion this is a non-trivial mathematical result, for the variable n is discrete. (ii) Suppose one has a horizontal circular disc of unit circumference which is pivoted at the center. Let the disc be given a random angular displacement o where oo < 0 < oo. If the final position of the disc mod one is called so, i.e.,

Tables of Natural Logarithms for Arguments between Five and Ten to Sixteen Decimal Places.

Tables of Natural Logarithms for Arguments between Five and Ten to Sixteen Decimal Places.

A two-and-one-half-place logarithm table

Approximating log x by finding integers m and n such that xm is approximately equal to 10n.

Table of Natural Logarithms for Arguments between Five and Tei to Sixteen Decimal Places. Pp. 506. $4.00. 1958. (U.S. Governmen Printing Office, Washington D.C.)

Table of Natural Logarithms for Arguments between Five and Tei to Sixteen Decimal Places. Pp. 506. $4.00. 1958. (U.S. Governmen Printing Office, Washington D.C.)