Mathematical Tables Research Articles

On Squeezing Tables

We live in days when more than ordinary care is taken to extract from jam-pots, gardens, dust-bins and ourselves the maximum of productivity; and in such times why should mathematical tables be exempt from the general squeeze? To all of us, I suppose, there come moments when we say “For this calculation I wish I had tables giving more figures”: such cases occur especially when anything is found as the difference of two much larger quantities, in which a small percentage-error may appear greatly magnified in their difference (weighing the ship’s cat by the displacements before and after she fell overboard). I shall outline here and attempt to justify the least laborious that I know of methods by which the inevitable approximation-errors can be cut down to a much smaller mean than is involved in using the crude values as printed.

On Squeezing Tables

We live in days when more than ordinary care is taken to extract from jam-pots, gardens, dust-bins and ourselves the maximum of productivity; and in such times why should mathematical tables be exempt from the general squeeze? To all of us, I suppose, there come moments when we say “For this calculation I wish I had tables giving more figures”: such cases occur especially when anything is found as the difference of two much larger quantities, in which a small percentage-error may appear greatly magnified in their difference (weighing the ship’s cat by the displacements before and after she fell overboard). I shall outline here and attempt to justify the least laborious that I know of methods by which the inevitable approximation-errors can be cut down to a much smaller mean than is involved in using the crude values as printed.

Table of Fourier Coefficients

Journal of Mathematics and PhysicsVolume 22, Issue 1-4 p. 136-147 Article Table of Fourier Coefficients† Arnold N. Lowan, Arnold N. LowanSearch for more papers by this authorJack Laderman, Jack LadermanSearch for more papers by this author Arnold N. Lowan, Arnold N. LowanSearch for more papers by this authorJack Laderman, Jack LadermanSearch for more papers by this author First published: April 1943 https://doi.org/10.1002/sapm1943221136Citations: 1 †The results reported here were obtained during 1940-41 by the Mathematical Tables Project of the Work Projects Administration operated under the sponsorship of the National Bureau of Standards, Dr. Lyman C. Briggs, Director. AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume22, Issue1-4April 1943Pages 136-147 RelatedInformation

Errors in Mathematical Tables

I AM now trying to consolidate for publication, partly in a new American National Research Council quarterly, and partly in a book entitled“A Computer's Guide to Mathematical Tables”, the work that I have done during the past twenty years on compiling lists of errors in mathematical tables. I should be glad to be told of any known errors in tables, or to have references to published lists. News of unpublished tables would also be appreciated.

Guide to Tables in the Theory of Numbers

THE U.S. National Research Council has embarked on the ambitious venture of producing a series of reports dealing with mathematical tables and aids to computation in every branch of mathematics, as well as in allied and associated subjects. All the reports are to take fully into account what is important in current research, and are to be written in such a way as to be intelligible for the most part to specialists in other fields. The reports are prepared primarily for scholars and others active in scientific work. Guide to Tables in the Theory of Numbers By Derrick Henry Lehmer. (Bulletin of the National Research Council, No. 105: Division of Physical Sciences, Committee on Mathematical Tables and Aids to Computation, Report 1: Report of the Sub-Committee on Section F, Theory of Numbers.) Pp. xiv + 177. (Washington, D.C.: National Academy of Sciences, 1941.) 2.50 dollars.

Mathematical Tables in Reports of the B.A.A.S.

Mathematical Tables in Reports of the B.A.A.S.

Errors in Mathematical Tables

I AM now trying to consolidate for publication, partly in a new American National Research Council quarterly, and partly in a book entitled“A Computer's Guide to Mathematical Tables”, the work that I have done during the past twenty years on compiling lists of errors in mathematical tables. I should be glad to be told of any known errors in tables, or to have references to published lists. News of unpublished tables would also be appreciated.

Radiologic PhysicsRadiologic Physics. By Weyl Charles, Reid Warren S.Jr., and O’Neill Dallett B., Moore School X-ray Laboratory, Moore School of Electrical Engineering and University of Pennsylvania. With a Foreword by Eugene P. Pendergrass, M.D., Director of the Department of Radiology, University of Pennsylvania. A volume of 459 pages, illustrated with 158 figures, 25 tables and an appendix containing many mathematical

Radiologic PhysicsRadiologic Physics. By Weyl Charles, Reid Warren S.Jr., and O’Neill Dallett B., Moore School X-ray Laboratory, Moore School of Electrical Engineering and University of Pennsylvania. With a Foreword by Eugene P. Pendergrass, M.D., Director of the Department of Radiology, University of Pennsylvania. A volume of 459 pages, illustrated with 158 figures, 25 tables and an appendix containing many mathematical

British Association for the Advancement of Science Mathematical Tables

THIS work is based on a table compiled by Glaisher about 1873, giving the first twelve powers of the first thousand integers. Only one copy of Glaisher's table is known to have survived, and it came into the possession of the British Association Tables Committee in 1935. It is not known how Glaisher computed his table, but it is so accurate that, apart from one missing digit, no error has been found in it. British Association for the Advancement of Science Mathematical Tables Vol. 9: Tables of Powers giving Integral Powers of Integers. Initiated by J. W. L. Glaisher. Extended by W. S. Bickley, C. E. Gwyther, J. C. P. Miller, E. J. Ternouth, on behalf of the Committee for the Calculation of Mathematical Tables. Pp. xii + 132. (Cambridge: At the University Press, 1940.) 15s. net.

Announcement of the project for the computation of mathematical tables

Announcement of the project for the computation of mathematical tables

The National Research Council Committee on Mathematical Tables and Aids to Computation

The National Research Council Committee on Mathematical Tables and Aids to Computation

The National Research Council Committee on Mathematical Tables and Aids to Computation

The National Research Council Committee on Mathematical Tables and Aids to Computation

Engineering

SUBSTANTIALLY the same publication in the form in which it has appeared for a number of years, “Kempe's Engineer's Year-Book” has been issued in its 1940 edition. Though once again in its forty-six years' history it has been published in time of war, this fact has not changed its form or restricted the great range of the information it presents. A few changes there have been, but these are in the nature of those developments which must necessarily take place in a work of this kind in order to keep it up to date and to maintain its usefulness at a maximum. The section dealing with the costs of engineering works has therefore been excised, for at the time of preparation it was apparent that the prospects of war negatived its value for so long as conditions should remain abnormal. The space thus released has been occupied with some of the mathematical tables which were discarded a few years ago so that the bulk of the volume might be limited.

Book Reviews

Abstract Business Cycles: A Theoretical, Historical, and Statistical Analysis of the Capitalist Process, by Joseph A. Schumpeter. New York: McGraw-Hill Book Company, Inc. 1939. Vol. I—xvi, 448 pp. Vol. II—ix, 449–1095 pp. $10.00. Reviewed by Oskar Morgenstern Turning Points in Business Cycles, by Leonard P. Ayres. New York: The Macmillan Company. 1939. xiii, 214 pp. $2.75. Reviewed by J. Philip Wernette Statistical Method from the Viewpoint of Quality Control, by Walter A. Shewhart with the editorial assistance of W. Edwards Deming. Washington, D. C.: Department of Agriculture Graduate School. 1939. ix, 155 pp. Reviewed by Joseph Manuele Mathematical Analysis for Economists, by R. G. D. Allen. New York and London: The Macmillan Company. London School of Economics and Political Science, Studies in Statistics and Scientific Method, No. 3. 1938. xv, 548 pp. $9.00. Reviewed by Theodore O. Yntema Mathematical Tables, Volume VII, The Probability Integral, by W. F. Sheppard. New York: The Macmillan Company. Cambridge: Cambridge University Press. Published for the British Association for the Advancement of Science. 1939. xi, 34 pp. $2.50. Reviewed by Helen M. Walker Statistik als ein Erkenntnismittel der Wirtschaftspolitik, by Fritz Huhle. Jena, Germany: Gustav Fischer. 1938. vi, 158 pp. RM 6. Reviewed by Eugen Altschul Consumer Expenditures in the United States, Estimates for 1935–36. Washington, D. C.: National Resources Committee. 1939. ix, 195 pp. 50 cents. Reviewed by Elizabeth W. Gilboy Studies in Income and Wealth, Volume Three, by the Conference on Research in National Income and Wealth. New York: National Bureau of Economic Research. 1939. xxiii, 479 pp. $3.50. Reviewed by Dwight B. Yntema Normal and Abnormal International Capital Transfers, by Marco Fanno. Minneapolis, Minnesota: University of Minnesota Press. November, 1939. xi, 120 pp. $2.00. Reviewed by Clifford L. James International Monetary Organisation, by Michael A. Heilperin. Paris: League of Nations, International Institute of Intellectual Co-operation. New York: Columbia University Press. 1939. 61 pp. 75 cents. Reviewed by James D. Paris Golden Avalanche, by Frank D. Graham and Charles R. Whittlesey. Princeton, New Jersey: Princeton University Press. 1939. xv, 228 pp. $2.50. Reviewed by R. J. Saulnier Marketing Burned-Clay Products, by A. Hamilton Chute. Columbus, Ohio: The Ohio State University. June, 1939. xx, 374 pp. Reviewed by J. F. Pyle Price Maintenance in Knoxville, Tennessee, under the Tennessee Fair Trade Act of 1937, by Charles W. Lewis. Knoxville: University of Tennessee. November, 1939. 64 pp. Reviewed by Arend E. Boer The Economics of Building, by Herbert W. Robinson. London: P. S. King & Son, Ltd. 1939. xii, 162 pp. 10s. 6d. Reviewed by Rosalind Tough The Economics of Business Enterprise, by Walter Rautenstrauch. New York: John Wiley & Sons, Inc. London: Chapman & Hall, Ltd. 1939. xiv, 445 pp. $4.00. Reviewed by Robert W. Burgess Capitalism the Creator, the Economic Foundations of Modern Industrial Society, by Carl Snyder. New York: The Macmillan Company. 1940. xii, 473 pp. $3.75. Reviewed by Willford I. King Consumer Credit and Economic Stability, by Rolf Nugent. New York: Russell Sage Foundation. 1939. 420 pp. $3.00. Reviewed by John H. Cover A Statistical Study of Credit Unions in New York, by Francis E. Wilcox. Chicago: The University of Chicago Press. 1940. ix, 91 pp. $1.00. Reviewed by Joseph L. Snider De Svenska Privatanställda, en Sociologisk Studie, by Fritz Croner. Stockholm: Kooperativa Förbundet Bokförlag. 1939. 474 pp. Kr. 12:-. Reviewed by Walter A. Lunden Labor in the United States, Basic Statistics for Social Security, by W. S. Woytinsky. New York: Social Science Research Council. A Report Prepared for the Committee on Social Security. 1938. xxii, 333 pp. $3.50. Reviewed by Royal E. Montgomery Old-Age Security, Social and Financial Trends, by Margaret Grant. New York: Social Science Research Council. A Report Prepared for the Committee on Social Security. 1939. xiii, 261 pp. $2.50. Reviewed by Abraham Epstein Predicting Success or Failure in Marriage, by Ernest W. Burgess and Leonard S. Cottrell, Jr. New York: Prentice-Hall, Inc. 1939. xxiii, 472 pp. $3.25. Reviewed by Agnes M. H. Byrnes New Facts on Mental Disorders, Study of 89,190 Cases, by Neil A. Dayton. Springfield, Illinois: Charles C Thomas. 1940. xxxiv, 486 pp. $4.50. Reviewed by Harold F. Dorn

Mathematical Tables. VII. The Probability Integral. By W F Sheppard. Completed and edited by the British Association committee for the calculation of mathematical tables. Pp. xi, 34. 8s. 6d. 1939. (Cambridge University Press, for the British Association)

Mathematical Tables. VII. The Probability Integral. By W F Sheppard. Completed and edited by the British Association committee for the calculation of mathematical tables. Pp. xi, 34. 8s. 6d. 1939. (Cambridge University Press, for the British Association)

Mathematical Table

THE Works Progress Administration of the City of New York is undertaking extensive computation of mathematical tables. The work is sponsored by Dr. L. J. Briggs, director of the National Bureau of Standards, and is intended to give employment to clerical workers in New York. Several tables, including Bessel functions for complex arguments, exponential and trigonometric integrals, and the error function are completed or in active progress. Of the tables under consideration, the most note-worthy is a table of elliptic functions for complex arguments so designed as to cover a period parallelogram in each case.

Concise School Physics

THIS is the third and final part of “A Concise School Physics”, which is also available in a single volume. It carries the story well up to matriculation stage and contains 144 pages of copiously illustrated descriptive matter, about 34 pages of questions and answers, and four of mathematical tables. The general treatment follows good conventional practice and the writing is very condensed, so that the book reads like copious notes of excellent lectures supported by adequate laboratory work. The author, avoiding educational fads, has obviously striven to make his difficult story as straightforward as possible for beginners. The book is amazingly cheap at the price and it should prove very useful in class work which is being directed by a capable teacher.

The Compulallon of Dimensions in Planigraphy with Mathematical Instruments

BY planigraphy we mean the visualization, localization, or mensuration of objects occupying specific planes in the radiograph. Certain standard conditions, namely, target film distance and target shift distance are easily established. Graphical methods of planigraphy have been presented by Kaufman (1, 2, and 3). We feel that while this method is adequate and accurate, a more simple and more rapid method could be applied. When standard conditions are used for each case, mathematical tables for computation as well as certain well known precision instruments such as the pantograph and plani-meter are applicable. The derivation of tables and application of mathematical instruments will be presented. Kaufman (1, 2, and 3) has given in three articles a very clear and detailed description of the methods of planigraphy and has shown methods of graphical reconstruction of the planes produced. In this paper we wish to outline methods of localization and plane production which are not dependent on graphical methods, and to show the application of the use of mathematical instruments to this problem. It is assumed that the reader is fully conversant with the descriptions of Kaufman so that the methods and relationships produced in planigraphy will not be reviewed here. The degree of spread of a point in space is determined by the target shift and the height, h, of the point above the plate. Kaufman has drawn graphs to illustrate this relationship. For purposes of computation the computed values for shift, 5, for three target shifts, 7.5, 15, and 22.5 em., are recorded for planes at intervals of 2 em. above the plate. II, the tube height, is taken as 80 em. As an aid in interpolation the value ΔS/Δh for each pair of planes is listed. Numerically, this is equal to the change in shift per centimeter in the region and corresponds analytically to the slope ds/dh for the shift height curve or standard depth curve of Kaufman. Since the beams from the x-ray tube diverge, the image on the plate is increased by projection by a factor which is equal to H/(H-h). Consequently, all linear measurements on the plate must be multiplied by a factor which is unique for each plane in order to convert the measurements to the true distances in the level of the plane of the object radiographed. This factor is numerically equal to (H-h)/H and is recorded in Column F of Table 1. The area projection is changed by a factor equal to the square of the linear change and is recorded in Column F2. The changes in dimensions due to divergence of the beam are not to be confused with the shift produced by moving the target. To illustrate the practical use of this table, the methods of reconstruction and determination of planes and localization of objects are outlined herein. This work was done on a wax model (Fig. 1).

What the statistical slide rule will do

The statistical slide rule was first announced in July, 1934.1 It is an actual calculating instrument based on the engineers' log log duplex rule. Ten standard scales are identical on the two rules, while seven new scales have been devised for the statistical rule. This statistical slide rule has recently been revised, as a result of numerous suggestions offered in response to the original descriptive article and experimentation by the designer. A trial batch of the revised rules has been produced by the world's largest manufacturers of engineering slide rules who have named it the Enlow Statistical Slide Rule. Interested researchers here and there are trying it out. The slide rule, as an instrument to facilitate computation, is not new to many of the leading educational and psychological statisticians. In fact, they have made use of it in their laboratories and offices. Probably most of them have felt the inadequacy of the various engineering rules for statistical purposes. In a letter to the writer, one of the country's outstanding statistics teachers expressed discomfiture over the difficulty of going from the slide rule to tables and vice versa. The main object in the writer's mind in designing the statistical slide rule was to provide a device which would enable one to make all the ordinary statistical computations without having to refer to printed mathematical tables. That this has actually been achieved, the statistician may readily observe by noting the accompanying reproduction of both faces of the slide rule. There are seventeen scales, which are the equivalent of eighteen, since one serves a double

Mathematical Tables, Volume VI, Bessel Functions, Part I, Functions of Orders Zero and Unity.

Mathematical Tables, Volume VI, Bessel Functions, Part I, Functions of Orders Zero and Unity.