James Whitbread Lee Glaisher

“ But to form a table of prime numbers the process is theoretically simple and rapid, for we have only to range all the numbers in a line and strike out every second number beginning from 2, every third beginning from 3, and so on, those that remain being primes. Even when the tabular results are constructed separately, the method of differences or other methods connecting together different tabular results may afford valuable verifications. By having recourse to tables not only does the computer save time and labour, but he also obtains the certainty of accuracy. ”

“ It follows from this property of the function that we cannot have for log x a series which shall be convergent for all values of x, as is the case with sin x and cos x, for such a series could only represent a uniform function, and in fact the equation
log (1 + x) = x − 1/2x2 + 1/3x3 − 1/4x4 + ... is true only when the analytical modulus of x is less than unity. The exponential function, which may still be defined as the inverse of the logarithmic function, is, on the other hand, a uniform function of x, and its fundamental properties may be stated in the same form as for real values of x. ”

“ 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 ”

“ 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 xy as tabular result for x and y as arguments. 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. ”

“ 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. ”

“ Moreover, the trouble required by the computer to learn the use of a table need scarcely be considered; the important matter is the time and labour saved by it after he has learned its use. Tables are, as a rule, intended for professional and not amateur use; and it is of little moment whether the user who is unfamiliar with a table has to spend ten seconds or a minute in obtaining an isolated result, provided it can be used rapidly and without risk of error by a skilled computer. ”

“ The more one considers the condition of science at the time, and the state of the country in which the discovery took place, the more wonderful does the invention of logarithms appear. When algebra had advanced to the point where exponents were introduced, nothing would be more natural than that their utility as a means of performing multiplications and divisions should be remarked; but it is one of the surprises in the history of science that logarithms were invented as an arithmetical improvement years before their connexion with exponents was known. ”

“ 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. ”

“ It is evident that Wittich’s prosthaphaeresis could not be a good method of practically effecting multiplications unless the quantities to be multiplied were sines, on account of the labour of the interpolations. It satisfies the condition, however, equally with logarithms, of enabling multiplication to be performed by the aid of a table of single entry; and, analytically considered, it is not so different in principle from the logarithmic method. In fact, if we put xy = φ (X + Y) , X being a function of x only and Y a function of y only, we can show that we must have X = Aeqx, y = Beqy ”

“ 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. ”

“ Napier’s priority in the publication of the logarithms is unquestioned and only one other contemporary mathematician seems to have conceived the idea on which they depend. There is no anticipation or hint to be found in previous writers, [8] and it is very remarkable that a discovery or invention which was to exert so important and far-reaching an influence on astronomy and every science involving calculation was the work of a single mind. ”

“ Hutton’s suggestions are all the more to be regretted as they occur as a history which is the result of a good deal of investigation and which for years was referred to as an authority by many writers. His prejudice against Napier naturally produced retaliation, and Mark Napier in defending his ancestor has fallen into the opposite extreme of attempting to reduce Briggs to the level of a mere computer. In connexion with this controversy it should be noticed that the “Admonitio” on the last page of the Descriptio, containing the reference to the new logarithms, does not occur in all the copies. ”

“ 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. ”

“ In multiplications or divisions of any length it is generally convenient to begin by forming a table of the first nine multiples of the multiplicand or divisor, and Napier’s bones at best merely provide such a table, and in an incomplete form, for the additions of the two figures in the same parallelogram have to be performed each time the rods are used. ”

“ The work is a small quarto containing fifty-seven pages of explanatory matter and a table of ninety pages (see Napier, John) . The nature of logarithms is explained by reference to the motion of points in a straight line, and the principle upon which they are based is that of the correspondence of a geometrical and an arithmetical series of numbers. The table gives the logarithms of sines for every minute of seven figures; it is arranged semi-quadrantally, so that the differentiae, which are the differences of the two logarithms in the same line, are the logarithms of the tangents. ”

“ 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. ”

“ If is the smallest power of a for which the congruence is satisfied, then a is said to belong to the exponent x for modulus p, and x may be called the chief exponent (Haupt-exponent by Allan Cunningham) of the base a for the modulus p; so that (1) this exponent is the number of figures in the circulating period of the fraction 1/p in the scale of radix a, and (2) when , a is a primitive root of p. In Mess. of Math., 1904, 33, p. 145, Allan Cunningham has given a complete list of Haupt-exponent tables with lists of errata in them ”

“ 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 ”

“ The nine quarto volumes (Tables des diviseurs, Paris, 1814–1817; Factor Tables, London, 1879–1883; Factoren-Tafeln, Hamburg, 1862–1865) thus form one uniform table, giving the least divisor of every number not divisible by 2, 3, or 5, from unity to nine millions. The arrangement of the results on the page, which is due to Burckhardt, is admirable for its clearness and condensation, the least factors for 9000 numbers being given on each page. The tabular portion of each million occupies 112 pages. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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 platinum, 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 mathematics. ”

“ 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. ”

“ In some tables the line is broken where the change occurs; but the dislocation of the figures and the corresponding irregularity in the lines are very awkward. Babbage printed the fourth figure in small type after a change; and Bremiker placed a bar over it. The best method seems to be that of prefixing an asterisk to the fourth figure of each logarithm after the change, as is done in Schrön’s and many other modern tables. This is beautifully clear and the asterisk at once catches the eye. Shortrede and Sang replace 0 after a change by a noketa (resembling a diamond in a pack of cards) . ”

“ 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. ”

“ The Rabdologia attracted more general attention than the logarithms, and as has been mentioned, there were several editions on the Continent. Nothing shows more clearly the rude state of arithmetical knowledge at the beginning of the 17th century than the universal satisfaction with which Napier’s invention was welcomed by all classes and regarded as a real aid to calculation. Napier also describes in the Rabdologia two other larger rods to facilitate the extraction of square and cube roots. ”