Logarithmic Tables Research Articles

Work for the Hairdressers: The Production of de Prony's Logarithmic and Trigonometric Tables

A summary account is given of the large set of logarithmic and trigonometric tables produced at the end of the 18th century under the direction of Gaspard Riche de Prony. Although they were completed in 1801, their size made publication a costly task; and it was never done, despite the fact that printing was started more than once and various efforts were made over the years to find finance. Eventually a reduced edition of some tables appeared in 1891. In addition to the history some details are given of the mathematical methods used in the compilation of the tables. An appendix considers the possible influence of this project upon Babbage's ideas on computers and tables.

Charles Babbage's Table of Logarithms (1827)

In 1827 Charles Babbage published his Table of Logarithms of the Natural Numbers, from 1 to 108,000. His logarithms were generally considered to be the most accurate in his day and were reprinted on numerous occasions, well into the 20th century. This paper describes Babbage's motivation for producing the tables, and the measures taken to ensure their accuracy. An assessment is given of Babbage's contribution to the art of table making.

Wroński's Canons of Logarithms

As everyone knows, the electronic computer and the hand calculator have made tables of logarithms all but obsolete as practical aids to calculation. The history of these tables, however, glistens with examples of remarkable diligence and ingenuity. There may be no more striking example of cleverness in this area than Wrofiski's canons of logarithms. This does not seem too strong a claim to make for a scheme that made it possible to put a seven-place table of common logarithms, in readable type, on a single page less than 17 cm by 22 cm in size.

Table errata: G. W. Spenceley, R. M. Spenceley and E. R. Epperson, Smithsonian Logarithmic Tables to Base 𝑒 and Base 10, The Smithsonian Institution, Washington, D.C., 1952; reprinted with corrections, 1960

Table errata: G. W. Spenceley, R. M. Spenceley and E. R. Epperson, Smithsonian Logarithmic Tables to Base 𝑒 and Base 10, The Smithsonian Institution, Washington, D.C., 1952; reprinted with corrections, 1960

Edmund Gunter (1581–1626)

The year 1981 marks the fourth centenary of the birth of Edmund Gunter who, during his relatively short life, played a dominant role in England in the advancement of navigation. Born in Hertfordshire, of Welsh parentage, Gunter was educated at Westminster School before entering Christchurch College, Oxford, where he graduated BA in 1603; MA in 1606; and BD in 1615, in which year he was presented to the living at St George's, Southwark.As early as 1603 Gunter had written an account of a ‘New Projection of the Sphere’ which was circulated in manuscript among some of his mathematical acquaintances. This appears to have gained for him the friendship of Henry Briggs, the first Gresham Professor of Geometry and a brilliant mathematician best known for having been first to suggest a table of logarithms to base 10 – ‘common logarithms’, as they are now known.

When Are Logarithms Used?

With the widespread use of pocket calculators, no one would use a table of logarithms to multiply or raise a number to a power. Of course, the calculator raises numbers to arbitrary powers by using logarithms internally, but why should students learn logarithms if they don't plan to design calculators? Certainly the presence of the log key on many calculators will do away entirely with the need for tables of logarithms. But when is the log key to be used?

Computers—Work in progress

The history of computers, particularly digital ones, dates from the first quarter of the 17th century (see ref. 1 for review). The first known machine was built by Wilhelm Schickard, a professor at Tubingen and a friend of Kepler's. Interestingly, this occurred at the same time that Napier invented logarithms. The device was built, but it, as well as the copy for Kepler, and the inventor himself were destroyed by the fires and plagues of the Thirty Years' War. The next machine, copies of which still exist, was built by Pascal and was described in Diderot's Encylopedie. This device became an important part of a desk calculator designed and constructed by Leibniz who said: Also the astronomers surely will not have to continue to exercise the patience which is required for computation. It is this that deters them from computing or correcting tables, from the construction of Ephemerides, from working on hypotheses, and from discussions of observations with each other. For it is unworthy of excellent men to lose hours like slaves in the labor of calculation which could safely be relegated to anyone else if machines were used. This enunciation by Leibniz of a purpose for automatic computing is a memorable one. Science, or at least mathematical astronomy, had advanced sufficiently by his time so that calculation was a real burden and it was recognized to some extent how this burden could be lightened. Certainly Kepler, using tables of logarithms he himself calculated based on Napier's schema, did extensive calculations in order to produce his Rudolphine Tables. The time for the digital principle, however, had still not come, and even by the early part of the 19th century the Nautical Almanac was being calculated by groups of humans all busy making separate and independent calculations with attendant errors. The situation was so bad by 1823 that Charles Babbage, one of the founders of the Royal Astronomical Society in 1820, set out to create a digital device to construct tables by a method, certainly well-known to Newton, called subtabulation. In this method, one first calculates a comparatively few entries in a table by hand a priori, and then the entries lying between them are filled in by systematic interpolation using essentially only additions and subtractions. For various reasons this was a propitious time in English history to attempt to automate computation. Trevelyan (2) tells

A method for computing addition tables in<tex>GF(p^n)</tex>(Corresp.)

Conway showed that a table of Zech's logarithms is useful to perform addition in GF <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(p^{n})</tex> when the elements are represented as powers of a primitive element. The Zech's logarithm <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Z(x)</tex> of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</tex> is defined by the equation <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\alpha^{z(x)}=\alpha^{x} + 1</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\alpha</tex> is a primitive element, zero is written as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\alpha^{\ast}</tex> , and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x=\ast,O,1, \cdots ,p^{n}-2</tex> . A simple algorithm for making a table of Zech's logarithms is presented.

Radiation Protection in the Radiologic and Health Sciences

This small book was written primarily as an introduction to radiation protection for medical health physicists, radiology technicians, and physicians in residency training. The practicing physician will similarly find it useful, especially if he is concerned with problems of radiation exposure. The topics covered in succinct form include general aspects of radiation protection (elementary physics, units, detectors, meters, monitors, and maximum permissible dose), protection from radionuclides (working habits, legal matters, and internal dosimetry), and protection against external radiation (barriers and external dosimetry). The newly proposed Systeme International units, such as the becquerel, gray, and sievert, are used throughout the book. A useful glossary and good index are included. The brief table of natural logarithms is poorly reproduced from another publication. The concise presentation is enhanced by sample problems and by references to more extensive source books, both in the field of radiation protection and in radiobiology. The authors have not

Student Construction of a Table of Common Logarithms

Student Construction of a Table of Common Logarithms

Binary logarithms for computing integral and non-integral roots and powers

This paper suggests a method of utilizing binary logarithms to compute the power, root, or any exponential of a number. It is an extension of Mitchell's (1962) technique. Due to the approximation of the binary logarithms, there will be errors in the calculations of the results. However, using this method, no logarithmic tables are required. Approximations to binary logarithms are very easy to generate by simple shifting and counting. Evaluating power, roots, and any exponent of a number is reduced to simple addition and subtraction operations. Finally the approximation error might be reduced hy adding a fixed number to correct the binary logarithms.

Decision Trees with Continuous Distribution

Decision trees are often used as a convenient way to visualize, and then solve, a decision problem. A standard problem approached in this way is a decision as to whether or not to sample, followed by a decision whether or not to engage in an activity. This approach has been limited to a small number of sample outcomes to be practical. This paper develops an approach which permits a continuous of sample outcomes to be used. A new distribution, the parabolic distribution is introduced as a flexible way of representing the judgemental (probabilistic) beliefs of the assessor. This new is compared with the beta distribution. An algorithm, which is easily programmed and is practical for hand computation with a calculator and logarithm tables, is developed for the complete solution of a decision tree problem using this skewed parabolic distribution. An example is given. It is concluded that the skewed parabolic makes use of continuous distributions practical for decision trees, and has advantages over the beta for this purpose.

11.2. Johannes Kepler and the development of mathematics

The name of Kepler heads the list of those associated with the development of a whole new period in mathematics—the period of variable quantities. At the very beginning of the seventeenth century his investigations of the laws of planetary motion led him into the difficult and important area of integral calculation. Though he was not able to perform integrations as such, he treated the new concept in terms of the area bounded by a curve, and developed procedures that come near to the modern methods of numerical integration. Nor was this all; he developed these methods and used them in his Nova Stereometria to determine the volume of various quite complicated solids of revolution. The questions of differential calculation were not ignored either: in the same book Kepler deals with extrema and the isoperimeter problem. In his work on optics, Kepler was the first to introduce into geometry the principle of continuity and the concept of a point at infinity—an important step towards the creation of projective geometry. His work on the theory of polygons and polyhedra was also noteworthy, particularly his discovery of the regular star-shaped polyhedra and of the space-filling property of these and other regular convex solids. The very scope of the calculations which he performed in his many years of search for the planetary laws is sufficient to earn Kepler his place as the greatest master of calculation of his times. While carrying out this arduous task he turned his creative ability to the perfection of methods and aids that would speed up the work. He was among the first to work out an original theory of logarithm construction, and he prepared logarithmic tables not unlike those of Napier, though not identical with them. He was indeed the first to use logarithmic calculation in important astronomical researches. His tables outlived those of Bürgi and Napier, and were reissued in 1700. In the 12,700 pages of Kepler manuscript material preserved in the archives of the USSR Academy of Sciences it can be seen that Kepler was not only helper but very probably initiator in the invention of the first calculating machine by his friend, the Tübingen Professor W. Schickard. In the same manuscript collection his German text on the logarithmic calculation, his “Weinvisierbüchlein”, and other writings all show his substantial contribution to the development of German-language scientific literature. Kepler's views on the nature of mathematical concepts are reflected in his manuscript (also in the Academy's collection) “De quantitatibus Libelli”. Parts of this also supplement our knowledge of Kepler's contribution to the development of infinitesimal methods and other problems.

Discovering and Constructing Logarithm Tables and a Slide Rule by Simple Arithmetic

Summary Logarithm tables and a slide rule can be constructed to one‐figure accuracy in a few moments of simple calculation by an individual. A group of students with the aid of tables of roots and powers can quickly achieve two‐digit accuracy. By using compound‐interest tables, three‐digit accuracy may be achieved without undue effort. No knowledge of infinite series or other advanced topics is necessary, and indeed experimental search for a way of multiplying by addition can be carried through successfully without the logarithm concept or as an introduction to it. The paper sketches some possible classroom procedures and gives examples of the devices that may be discovered by the student.

A Numerical Approach to Natural Logarithms

Of these three approaches, only the last possesses the virtue of presenting In as a logarithm from the outset. All three approaches are, of course, mathematically valid. The choice between them depends, therefore, on their suitability for class room presentation. The selection of one particular mode of introduction of this topic depends on the teacher, the class, and the aim of the course being taught. This article is the result of dissatisfaction with all three approaches. The author's teaching experience in this area has been largely drawn from two fields: the educa tion of engineering undergraduates and the late secondary instruction of the dis advantaged. Both these groups experience motiva tional difficulties with the three standard approaches listed above. Approach num ber three, initially the most promising, suffers from the difficulty that condition (6) appears somewhat arbitrary. In searching for a possible new ap proach, the author was guided by the re flection that Napierian logarithms in fact preceded historically the common (base 10) logarithms of Briggs. It is, indeed, a common misconception that Napier discovered natural logarithms. The term Napierian, still used in some table-books as a synonym for natural, no doubt adds to the confusion. It is not the point of this article to describe the history of logarithms. This is already available in the works of Cajori (1913), Boyer (1968, pp. 342-44), Cairns (1928), and others. It is, however, fair to record that the approach described here arose largely from my own reading of these works, although it duplicates none of them. The present approach is numerical to a point, but the computational aspect is used in tandem with the more theoretical view that it motivates. It begins with the Napierian idea that before a table of logarithms can be used, it must first be constructed. The actual construction of a logarithmic table is a problem in numerical analysis. Napier attacked it by constructing a table of powers. To take a suitable base and find its successive powers is much easier than to seek to calculate logarithms to a previously designated base.1

The two-place logarithm table. An aid to understanding and use of logarithms

Presents a two-place logarithm table that serves to aid the understanding and use of logarithms.

Gaussian Logarithms and Navigation

Gaussian logarithms are designed to facilitate finding the common logarithm of a sum or difference of two numbers whose common logarithms are known. The object of a table of Gaussian logarithms, sometimes known as Addition and Subtraction Logarithms, is to give log (a±b) by single entry when log a and log b are known.

The ruin probability in a special case

It is fantastic how the computer has changed our attitude to numerical problems. In the old days when our numerical tools were paper, pencil, desk calculator and logarithm tables we had to stay away from formulas and methods which led to too lengthy calculations. A consequence is that we have a tendency to think of numerical analysis in terms of the classical tools. If we go back to the results of earlier writers it seems, however, very likely that many results and formulas developed by them which had earlier a theoretical interest only could nowadays be applied successfully in numerical analysis.As an example I take the ruin probability ψ(x). The Laplace transform of ψ(x) is given by the following expressionwhere c &gt; 1. In fact (c — 1) is equal to the “security loading”. The function p(y) is equal to the Laplace transform of the claim distribution. We assume that the mean claim amount is equal to one, i.e. p′(0) = — 1.In his book from 1955 [1] Cramer points out that this formula will be more easy to handle if the claim distribution is an exponential polynomial. In this case we havewhereCramér's results are given on pages 81-83 in his book. We reproduce them here with a slight change of notations only.

Historically Speaking—: Identification of Napier's Inequalities

The discovery of logarithms by John Napier (1550-1617) is a well known facet in the history of mathematics. His singular accomplishment in defining the logarith mic function of a real variable by providing a numerical description of it, over a wide range of its argument, at small intervals and to several (decimal) places, antedated by many yeara the development of funda mental concepts which the modern stu dent regards as necessary to achieve even the same limited goals. Napier success fully bridged, solely in regard to this function, these lacunae in the mathematical knowledge of bis day. It has long been of interest to identify the concepts which he intuitively invoked. This is not done, it should be clearly said, with any idea of assigning to him some kind of priority for them, but merely in the interests of a elearer appreciation of the ingenuity he displayed and the power of his methods. Two inequalities that he obtained are the key to his numerical resolution of the problem and his consequent table of logarithms. The analytical identification of these inequalities appears to have been overlooked. Before exhibiting this identi-fication we shall speak of the fundamental role that these inequalities played. In the interests of intelligibility we first recollect a few familiar facts regarding Napier's formulation of the problem.

20. Positions and Motions of Minor Planets, Comets and Satellites

The Report in the part dealing with the minor planets and satellites has been compiled at the Institute of Theoretical Astronomy in collaboration of S. G. Makover (minor planets) and V. A. Shor (satellites). Contributions to this report have been received from S. Arend, H. Debehogne, J. Dommanget, P. Herget, S. Herrick, G. M. Jannini, B. G. Marsden, V. V. Michkovitch, B. Milet, Z. M. Pereyra, B. Popovič, E. Rabe, E. Roemer, J. Schubart, W. Strobel, C. Torres, H. Wood. The part on comets has been prepared by E. Roemer.The main feature of minor planet research during this triennium has been the further application of modern electronic computing machines to the practice of computing. The time has gone when the computations of orbits and ephemerides were performed with the aid of logarithmic tables or with desk calculators. Therefore it is no wonder that all computational work in the field of the motion of minor planets is nowadays concentrated in a limited number of institutions provided with powerful up-to-date computing techniques, like the following.(1)Institute of Theoretical Astronomy, Leningrad has used till recently a computer BESM-2 and now has at its disposal a new more powerful computer BESM-4.(2)The Observatory of Cincinnati has used till recently a powerful computer NORC. Unfortunately this machine has been nowadays dismantled, but the observatory can perform some kinds of computations with an IBM-360 computer.(3)Latvian State University, Riga, has made a large amount of computations with the computer BESM-2.(4)University of California, Los Angeles, makes a large amount of computations with its own computer.