Mathematical Tables

Gaussian or Addition and Subtraction Logarithms.—In certain problems in astronomy and other subjects it is sometimes necessary to calculate the logarithm of (a+b) and (a—b) from log a and log b where a knowledge of the actual values of a and b is not required. Gaussian tables are intended to be used for this purpose. Leonelli in a very rare book, Supplement Logarithmique (Bordeaux, 1802-03), was the originator of a table which would simplify the calculations. A specimen table is given where, with log x as argument 0.00000(0.000000.00104, log (1-}-I) and log (I +x) are given to 14 places. Gauss, who took up the idea, con structed the first complete table of addition logarithms. It first appeared in Zach's Monatliche Correspondenz (vol. xxvi., 1812) and gives 5 places only in the tabular results. Three columns are headed A, B and C. The argument log x in column A is o(o.00r) 2.00(o-01)3.4(0.05.o; columns B and C give log (I+ and log (I -1-x) respectively to 5 places. By this table log(a+b) can be obtained by direct or inverse use of the table. Gauss's table has been reprinted in several collections of tables. There are several good tables to 6 and 7 places, but there are variations of the arrangement. Gray, Tables and Formulae (London, 1870) gives 6-place values of log(' -Fx) for log x=0.0000(0.00002 and log (I —x) for log x=3.00(o.00i)i.00(0.00001.899 with proportional parts; Cohn, Tafeln der Additions- und Subtraktions-logarithmen (Leipzig, 1909) gives a convenient table; log (r is given to 6 places for log x=0.0000(0.00001.500 (0.0003.00(0.005.0(0.06 with differences and log (I/I for log x=0.3000(0.00001.500 (0.0003.00(0.005.o. Similar tables are given by Bremiker, Gun delfinger and Jones. Seven-place tables are given by Matthiessen, Wittstein and Zech. The first of these is "nearly useless" (Gauss); the others tabulate in convenient form and for the greater part of the range the interval is 0.000r. Wittstein makes one table suffice for both addition and subtraction, while Zech has two. "Wittstein's table answers the purpose Gauss had in view the best of all" (Glaisher). By Zech's tables log (I+ and log — are tabulated with log x as argument. In 1922 in the Bulletin Astronomique, Deuxieme Serie (Paris, 1922), pp. 5-32, Andoyer has published "Tables Fondamentales pour les loga rithmes d'addition et de soustraction." These are to be regarded as basic tables for the future compilation of a table of Gaussian logarithms to n places when n<16. The chief table gives A and S to 16 decimal places corresponding to the argument Do.00(0.0i) 9.o0 where A = log(' S= —log(' [If D= log x, given with method.

Factor Tables.

The earliest extensive factor table is that of Chernac, Cribrum Arithmeticum (Deventer, 181I) ; it exhibits the factors of all numbers not divisible by 2, 3 or 5 up to 1,020,000 with the prime numbers indicated as they appear. Burckhardt published Table des Diviseurs (Paris, 1814-16-17), giving the lowest factor of all numbers indivisible by 2, 3 or 5 up to 3 mil lions (3,036,00o). The accuracy of this table is high. Dase fol lowed with a similar table for the range 6-9 millions, Factoren Tafeln fur alle Zahlen der siebenten Million (Hamburg, 1862), and two similar volumes. The Tables for the fourth, fifth and sixth millions were supplied by Glaisher and completed in 1883. Felkel (Vienna, 1776) probably aimed at io million, but the highest number is 408,00o. In Lehmer's table (Carnegie Institu tion, Washington, 1909) the least factor of numbers up to io mil lions indivisible by 2, 3 and 5 is given.

Product Tables.

These are of two distinct types: (i.) the simplest exhibits the products (a) to 9X 99,999, (b) to 99X999, (c) to 999X999. (ii.) The other consists of a table of quarter squares, the use of which depends on the formula (a =ab , viz., the product of two numbers is one-quarter

of the difference of the squares of their sum and difference. Bret schneider in Produktentafel (Hamburg-Gotha, 1841) gives the first 9 multiples of all integers to 99,999, while Crelle gives a similar table Erleichterungs-Tafel (Berlin, 1836) showing the first 9 multiples of numbers to ten million. The earliest extensive table is that of Herwart ab Hohenburg, Tabulae Arithmeticae (Munich, 161o), and this is of the type (C). Crelle, Rechentafeln (Berlin, 1820) gives a table for the same range; this has passed through many editions in English, French and German. The range 99X 999 has been chosen by Zimmermann, in Rechen Tafeln (Berlin, 1899) and Peters (Berlin, 1909).

Quarter-squares.

A number of tables of quarter-squares have been published, of which the most extensive is that of Blater, Tables of Quarter-Squares up to 200,000 (London, 1887). This is an English edition of the work originally published in Vienna. Other tables include Laundy, Table of Quarter-Squares (London, 1856) up to 100,000; Voisin, Tables de Multiplications (Paris, 1817) up to 20,000. Centnerschwer, Neu erfundene Multi plications- and Quadrat-Tafeln (Berlin, 1825) also takes the upper limit 20,00o. Several tables give quarter-squares for ranges below io,000.

Tables of Squares, Cubes and Higher Powers.

A fairly large number of extended tables of squares of integers have been published but few tables of cubes. The earliest combined tables were those of Babington, Pyrotechnic (London, 1635), which ex hibits the squares of integers to 25,00o and cubes to io,000, and Guldinus, De centro gravitatis. Appendix (Vienna, 1635), which exhibits squares and cubes of all integers to io,000. The most extensive table of this type is that of Kulik, Tafeln der Quadrat und Kubik-Zahlen (Leipzig, 1848), giving squares and cubes of all integers to io0,000. As far as squares are concerned, the tables of quarter-squares of Blater and Laundy mentioned above may be used as tables of squares to 200,000 and roo,000 respectively. Squares and cubes are given in the various editions of Barlow's Tables (1814), to io,000, in Buchner's Tabula 0700, to 12,000; Hutton, Tables of the Products and Powers of Numbers (London, 1781), gives squares to 25,400 and cubes to io,000 ; Jahn, Tafeln (Leipzig, 1839) gives squares to 27,00o and cubes to 24,000. Tables of squares (without cubes) include the following:— Gossart (Paris, 1865) to io,000, Maginus (Venice, 1592) to 10,100, and the most extensive table of squares is Ludolf's Tetra gonometria Tabularia (1689), of which there were several editions giving squares to ioo,000.

There are few tables of higher powers and those are, in general, of very limited extent. The 1814 edition of Barlow's Tables con tains the first ten powers of numbers i to ioo, fourth and fifth powers of numbers ioo to 1,000; Gain, Recueil de Tables (Namur, 1881) gives fourth, fifth, sixth, seventh and eighth powers of numbers to ioo; Hutton, Tables (1781) gives the first ten powers of numbers to ioo; Moore's Arithmetic (London, 166o) the fourth powers of numbers to 30o and the fifth and sixth powers to 200. Square and Cube Roots:—Barlow's Tables (1814) exhibit square and cube roots to 7 decimal places of all integers up to 10,000. Hhlsse's edition of Vega's Sammlung (Leipzig, 1840) gives square and cube roots to 12 and 7 places respectively of all integers to io,000. A most useful table of square roots has re cently been published by Milne-Thomson, Standard Table of Square Roots (London, Bell & Sons, 1929). This gives the square roots of x and lox for x= ioo (o.1) moo to 8 significant figures, i.e. 6 decimal places with first differences for the Vx and Viox columns. This table enables a computer to find the square root of any number with the minimum of interpolation.