James Whitbread Lee Glaisher

“ The invention of logarithms in 1614, followed immediately by the calculation of logarithmic tables, revolutionized all the methods of calculation; and the original work performed by Briggs and Vlacq in calculating logarithms 260 years ago has in effect formed a portion of every arithmetical operation that has since been carried out by means of logarithms. And not only has an incredible amount of labour been saved [1] but a vast number of calculations and researches have been rendered practicable which otherwise would have been quite beyond human reach. ”

“ The arrangement of a table on the page and all typographical details — such as the shape of the figures, their spacing, the thickness and placing of the rules, the colour and quality of the paper, &c. — are of the highest importance, as the computer has to spend hours with his eyes fixed upon the book; and the efforts of eye and brain required in finding the right numbers amidst a mass of figures on a page and in taking them out accurately, when the computer is tired as well as when he is fresh, are far more trying than the mechanical action of simple reading. ”

“ In some tables the labour required in the construction is the same as if all the tabular results had been calculated separately; but in the majority of instances a table can be formed by expeditious methods which are inapplicable to the calculation of an individual result. This is the case with tables of a continuous quantity, which may frequently be constructed by differences. The most striking instance perhaps is afforded by a factor table or a table of primes ”

“ In tables of logarithms of numbers to base 10 the mantissa only is in general tabulated, as the characteristic of the logarithm of a number can always be written down at sight, the rule being that, if the number is greater than unity, the characteristic is less by unity than the number of digits in the integral portion of it, and that if the number is less than unity the characteristic is negative, and is greater by unity than the number of ciphers between the decimal point and the first significant figure. ”

“ It is clear that the most suitable time for effecting such a change was when the natural canon was replaced by the logarithmic canon, and Briggs took advantage of this opportunity. He left the degree unaltered, but divided it centesimally instead of sexagesimally, thus ensurng the advantages of decimal division (a saving of work in interpolations, multiplications, &c.) with the minimum of change. The French mathematicians at the end of the 18th century divided the right angle centesimally, completely changing the whole system, with no appreciable advantages over Briggs’s system. ”

“ The best general method of calculating logarithms consists, in its simplest form, in resolving the number whose logarithm is required into factors of the form 1 - 14n, where n is one of the nine digits, and making use of subsidiary tables of logarithms of factors of this form. ”

“ The two systems of logarithms for which extensive tables have been calculated are the Napierian, or hyperbolic, or natural system, of which the base is e, and the Briggian, or decimal, or common system, of which the base is 10; and we see that the logarithms in the latter system may be deduced from those in the former by multiplication by the constant multiplier 1/log10, which is called the modulus of the common system of logarithms. ”

“ For example, a table of logarithms is a table of single entry, the numbers being the arguments and the logarithms the tabular results; an ordinary multiplication table is a table of double entry, giving as tabular result for and as arguments. Value of tables.The intrinsic value of a table may be estimated by the actual amount of time saved by consulting it; for example, a table of square roots to ten decimals is more valuable than a table of squares, as the extraction of the root would occupy more time than the multiplication of the number by itself. ”

“ If seven-figure logarithms do not give sufficiently accurate results, it is usual to have recourse to ten-figure tables; with one exception, there exist no tables giving 8 or 9 figures. The exception is John Newton's Trigonometria Britannica (London, 1658) , which gives logarithms of numbers to 100,000 to 8 places, and also log sines and tangents for every centesimal minute (i.e., the nine-thousandth part of a right angle) , and also log sines and tangents for the first three degrees of the quadrant to 5 places, the interval being the one-thousandth part of a degree. ”

“ Thus a table of log a/x, where x is the argument and a a constant, is called a table of logistic or proportional logarithms; and since log a/x = log a - log x it is clear that the tabular results differ from those given in an ordinary table of logarithms only by the subtraction of a constant and a change of sign. The first table of this kind appeared in Kepler’s Chilias logarithmorum (1624) already referred to. The object of a table of log a/x is to facilitate the working out of proportions in which the third term is a constant quantity a. ”

“ The integral part of a logarithm is called the index or characteristic, and the fractional part the mantissa. When the base is 10, the logarithms of all numbers in which the digits are the same, no matter where the decimal point may be, have the same mantissa ”

“ By far the greatest computer of pure trigonometrical tables is George Joachim Rheticus, whose work has never been superseded. His celebrated ten-decimal canon, the Opus Palatinum, was published by Valentine Otho at Neustadt in 1596, and in 1613 his fifteen-decimal table of sines by Pitiscus at Frankfort under the title Thesaurus Mathematicus. The Opus Palatinum contains a complete ten-decimal trigonometrical canon for every ten seconds of the quadrant, semi-quadrantally arranged, with differences for all the tabular results throughout. ”

“ The value of a table does not depend upon the difficulty of calculating it; for, once made, it is made for ever, and as far as the user is concerned the amount of labour devoted to its original construction is immaterial. ”

“ Tables relating to ordinary arithmetical operations are first described, and afterwards an account is given of the most useful and least technical of the more strictly mathematical tables, such as factorials, gamma functions, integrals, Bessel's functions, &c. It is difficult to classify the tables described in a perfectly satisfactory manner without prolixity, as many collections contain valuable sets belonging to a variety of classes. ”

“ As a rule seven-figure logarithms of numbers are not published separately, most tables of logarithms containing both the logarithms of numbers and a trigonometrical canon. Babbage’s and Sang’s logarithms are exceptional and give logarithms of numbers only. Babbage, Table of the Logarithms of the Natural Numbers from 1 to 108,000 (London, stereotyped in 1827; there are several tirages of later dates) , is the best for ordinary use. ”

“ In 1794 Vega reprinted at Leipsic Vlacq’s two works in a single folio volume, Thesaurus Logarithmorum Completus. The arrangement of the table of logarithms of numbers is more compendious than in Vlacq, being similar to that of an ordinary seven-figure table, but it is not so convenient, as mistakes in taking out the differences are more liable to occur. The trigonometrical canon gives log sines, cosines, tangents, and cotangents, from 0° to 2° at intervals of one second, to 10 places, without differences, and for the rest of the quadrant at intervals of ten seconds. ”

“ In 1742 a seven-figure table was published in quarto form by Gardiner, which is celebrated on account of its accuracy and of the elegance of the printing. A French edition, which closely resembles the original, was published at Avignon in 1770.
In 1783 appeared at Paris the first edition of Callet’s tables, which correspond to those of Hutton in England. These tables, which form perhaps the most complete and practically useful collection of logarithms for the general computer that has been published, have passed through many editions, and are still in use. ”

“ A list of the most convenient factors and their logarithms is given in a separate table. The principle of multiplying by a factor which is subsequently cancelled by subtracting its logarithm is one that is frequently employed in the calculation of logarithms, but the peculiarity of the present work is that it forms part of the process of using the table. ”

“ The table was calculated by interpolation from the Trigonometria Artificialis to 10 places and then reduced to 7, so that the last figure should always be correct. Partly on account of the absence of a mark to denote the change of figure in the column and partly on account of the size of the table and a somewhat inconvenient arrangement, the work seems never to have come into very general use. Computers have always preferred Bagay’s Nouvelles Tables Astronomiques et Hydrographiques (Paris, 1829) , which also contains a complete logarithmic canon to every second. ”

“ This gives seven figures of the reciprocal, and is arranged like a table of seven-figure logarithms, differences being added at the side of the page. The reciprocal of a number of five figures is therefore taken out at once, and two more figures may be interpolated for as in logarithms. Picarte, La Division réduite à une Addition (Paris, 1861) , gives to ten significant figures the reciprocals of the numbers from 10,000 to 100,000, and also the first nine multiples of these reciprocals. Small tables of reciprocals are not common. ”

“ They are generally to be found in collections of logarithmic tables, but rarely exceed a page in extent, and are very easy to construct, Schrön and Bruhns both give the first hundred multiples of the modulus and its reciprocal to 10 places, and Bremiker (in his edition of Vega and in his six -figure tables) and Dupuis to 7 places. Degen, Tabularum Enneas (Copenhagen, 1824) , gives the first hundred multiples of the modulus to 30 places.
Antilogarithms.Antilogarithms,—In the ordinary tables of logarithms the natural numbers are integers, while the logarithms are incommensurable. ”

“ Suppose that the ratio of 10, or any other particular number, to 1 is compounded of a very great number of equal ratios, as for example 1,000,000, then it can be shown that the ratio of 2 to 1 is very nearly equal to a ratio compounded of 301,030 of these small ratios, or ratiunculæ, that the ratio of 3 to 1 is very nearly equal to a ratio compounded of 477,121 of them, and so on. The small ratio, or ratiuncula, is in fact that of the millionth root of 10 to unity, and if we denote it by the ratio of a to 1, then the ratio of 2 to 1 will be nearly the same as that of a301,030 to 1 and so on ”

“ The function log x is not a uniform function, that is to say, if x denotes a complex variable of the form a + ib, and if complex quantities are represented in the usual manner by points in a plane, then it does not follow that if x describes a closed curve log x also describes a closed curve ”

“ The preceding methods are only appropriate for the calculation of isolated logarithms. If a complete table had to be reconstructed, or calculated to more places, it would undoubtedly be most convenient to employ the method of differences. A full account of this method as applied to the calculation of the Tables du Cadastre is given by M. Lefort in vol. ”

“ The system of Byrgius is greatly inferior to that of Napier, and it is to the latter alone that the world is indebted for the knowledge of logarithms. The claims of Byrgius are discussed in Kästner’s Geschichte der Mathematik, vol. ”

“ The first nine or ten powers are given in Vega, Tabulæ (1797) , and in Hülsse’s edition of the same (1840) , in Köhler, Handbuch (1848) , and in other collections. Faà de Bruno, Calcul des Erreurs (Paris, 1869) , and Müller, Vierstellige Logarithmen (1844) , give squares for use in connexion with the method of least squares. Small tables occur frequently in books intended for engineers and practical men. Drack (Messenger of Math., vol. vii., 1878, p. 87) has given to 33 places the cube roots (and the cube roots of the squares) of primes up to 127. ”

“ In 1811 Chernac published at Deventer his Cribrum Arithmeticum, which gives all the prime divisors of every number not divisible by 2, 3, or 5 up to 1,020,000. In 1814-17 Burckhardt published at Paris his Tables des Diviseurs, giving the least divisor of every number not divisible by 2, 3, or 5 up to 3,036,000. The second million was issued in 1814, the third in 1816, and the first in 1817. The corresponding tables for the seventh (in 1862) , eighth (1863) , and ninth (1865) millions were calculated by Dase and issued at Hamburg. ”

“ Small tables of powers of 2, 3, 5, 7 occur in various collections. In Vega’s Tabulæ (1797, and the subsequent editions, including Hülsse’s) the powers of 2, 3, 5 as far as the 45th, 36th, and 27th respectively are given ”