James Whitbread Lee Glaisher

“ 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 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 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. ”

“ When Napier published the Canonis Descriptio England had taken no part in the advance of science, and there is no British author of the time except Napier whose name can be placed in the same rank as those of Copernicus, Tycho Brahe, Kepler, Galileo, or Stevinus. In England, Robert Recorde had indeed published his mathematical treatises, but they were of trifiing importance and without influence on the history of science. Scotland had produced nothing, and was perhaps the last country in Europe from which a great mathematical discovery would have been expected. ”

“ 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 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. ”

“ 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. ”

“ 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. ”

“ 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) . ”

“ 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. ”

“ In the following descriptions of tables an attempt is made to give an account of all those that a computer of the present day is likely to use in carrying out arithmetical calculations. 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. ”

“ Multiplication Tables.—A multiplication table is usually of double entry, the two arguments being the two factors; when so arranged it is frequently called a Pythagorean table. The largest and most useful work is A. L. Crelle’s Rechentafeln (Bremiker’s edition, 1857, stereotyped; many subsequent editions with German, French, and English title-pages) , which gives in one volume all the products up to 1000 ⨯ 1000, so arranged that all the multiples of any one number appear on the same page. The original edition was published in 1820 and consisted of two thick octavo volumes. ”

“ 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 general use. Computers have always preferred V. Bagay’s Nouvelles Tables astronomiques et hydrographiques (Paris, 1829) , which also contains a complete logarithmic canon to every second. ”

“ 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. ”

“ Mathematical Tables (W. & R. Chambers, Edinburgh) , containing logarithms of numbers to 100,000, and a canon to every minute of log sines, tangents, and secants and of natural sines to 7 places, besides proportional logarithms and other small tables, is cheap and suitable for schools, though not to be compared as regards matter or typography to the best tables described above.
Of six-figure tables C. Bremiker’s Logarithmorum VI. decimalium nova tabula Berolinensis (Berlin 1852) is probably one of the best. ”

“ Natural sines are also given for every second from 0° to 1° and from 89° to 90°, to 15 places, with first and second differences. The rescue of the manuscript of this work by Pitiscus forms a striking episode in the history of mathematical tables. The alterations and emendations in the earlier part of the corrected edition of the Opus palatinum were made by Pitiscus, who had his suspicions that Rheticus had himself calculated a ten second table of sines to 15 decimal places ”

“ In the Rabdologia the rods are called “virgulae,” but in the passage quoted above from the manuscript on arithmetic they are referred to as “bones” (ossa) .
Besides the logarithms and the calculating rods or bones, Napier’s name is attached to certain rules and formulae in spherical trigonometry. “Napier’s rules of circular parts,” which include the complete system of formulae for the solution of right-angled triangles, may be enunciated as follows. ”

“ These examples show that Napier was in possession of all the conventions and attributes that enable the decimal point to complete so symmetrically our system of notation, viz. (1) he saw that a point or separatrix was quite enough to separate integers from decimals, and that no signs to indicate primes, seconds, &c., were required; (2) he used ciphers after the decimal point and preceding the first significant figure; and (3) he had no objection to a decimal standing by itself without any integer. Napier thus had complete command over decimal fractions and the use of the decimal point. ”

“ 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. ”

“ He uses “radicatum” for power (for root, power, exponent, his words are radix, radicatum, index) .
Apart from the interest attaching to these manuscripts as the work of Napier, they possess an independent value as affording evidence of the exact state of his algebraical knowledge at the time when logarithms were invented. There is nothing to show whether the transcripts were sent to Briggs as intended and returned by him, or whether they were not sent to him. Among the Merchiston papers is a thin quarto volume in Robert Napier’s writing containing a digest of the principles of alchemy ”

“ In Glaisher, Factor Table for the Sixth Million (1883) , § iii., the values of li x are given from x = 0 to 9,000,000 at intervals of 50,000 to the nearest integer. J. P. Gram in the publications of the Copenhagen Academy, 1884, 2, No. 6 (pp. 268-272) , has given to 20 places the values of Ei x from x = 10 to x = 20 at intervals of a unit (thus carrying Bretschneider’s table to this extent) and to 8, 9, or 10 places, the values of the same function from x = 5 to x = 20 at intervals of 0·2 (thus extending Glaisher’s table in the Phil. Trans.) . ”

“ 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. C. F. Degen, Tabularum Enneas (Copenhagen, 1824) , gives the first hundred multiples of the modulus to 30 places.
Antilogarithms.—In the ordinary tables of logarithms the natural numbers are integers, while the logarithms are incommensurable. ”

“ Legendre shows that Maclaurin’s theorem with respect to confocal ellipsoids is true for any position of the external point when the ellipsoids are solids of revolution. Of this memoir Isaac Todhunter writes: “We may affirm that no single memoir in the history of our subject can rival this in interest and importance. During forty years the resources of analysis, even in the hands of d’Alembert, Lagrange and Laplace, had not carried the theory of the attraction of ellipsoids beyond the point which the geometry of Maclaurin had reached. ”

“ This work contains the first announcement of logarithms to the world, the first table of logarithms and the first use of the name logarithm, which was invented by Napier.
In 1617 Napier published his Rabdologia, [5] a duodecimo of one hundred and fifty-four pages; there is prefixed to it as preface a dedicatory epistle to the high chancellor of Scotland. The method which Napier terms “Rabdologia” consists in the use of certain numerating rods for the performance of multiplications and divisions. These rods, which were commonly called “Napier’s bones,” will be described further on. ”

“ It will be noticed that the rods only give the multiples of the number which is to be multiplied, or of the divisor when they are used for division, and it is evident that they would be of little use to any one who knew the multiplication table as far as 9×9. ”

“ A copy of Vlacq’s Arithmetica logarithmica (1628 or 1631) , with the errors in numbers, logarithms, and differences corrected, is still the best table for a calculator who has to perform work requiring ten-figure logarithms of numbers, but the book is not easy to procure, and Vega’s Thesaurus has the advantage of having log sines, &c., in the same volume. ”

“ The factor table has been already noticed. It is to be observed that the fourth edition is quite a distinct work from the third, which contained much fewer tables. J. B. Dale, Five-figure Tables of Mathematical Functions (London, 1903) , is a book of 92 pages containing a number of small five-figure tables of functions which are not elsewhere to be found in one volume. Among the functions tabulated are elliptic functions of the first and second kind, the gamma function, Legendre’s coefficients, Bessel’s functions, sine, cosine, and exponential integrals, &c. ”

“ Four-place tables of squares are frequently given in five- and four-figure collections of tables. Small tables often occur in books intended for engineers and practical men. S. M. Drach (Messenger of Math., 1878, 7, p. 87) has given to 33 places the cube roots (and the cube roots of the squares) of primes up to 127. Small tables of powers of 2, 3, 5, 7 occur in various collections. In Vega’s Tabulae (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 ”