1911 Encyclopædia Britannica/Table, Mathematical

by James Whitbread Lee Glaisher

TABLE, MATHEMATICAL. In any table the results tabulated are termed the “tabular results” or “respondents,” and the corresponding numbers by which the table is entered are termed the “arguments.” A table is said to be of single or double entry according as there are one or two arguments. 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. 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. 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, if it is required to determine whether a given number is prime or not, the only universally available method (in the absence of tables) is to divide it by every prime less than its square root or until one is found that divides it without remainder. 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.

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 Henry Briggs and Adrian Vlacq in calculating logarithms in the early part of the 17th century 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 beyond human reach. The mathematical process that underlies the tabular method of obtaining a result may be indirect and complicated; for example, the logarithmic method would be quite unsuitable for the multiplication of two numbers if the logarithms had to be calculated specially for the purpose and were not already tabulated for use. 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. 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.

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. Nearly all modern tables are stereotyped, and in giving their titles the accompanying date is either that of the original stereotyping or of the tirage in question. In tables that have passed through many editions the date given is that of the edition described. A much fuller account of general tables published previously to 1872, by the present writer, is contained in the British Association Report for 1873, pp. 1–175.

Tables of Divisors (Factor Tables) and Tables of Primes.—The existing factor tables extend to 10,000,000. In 1811 L. Chernac published at Deventer his Cribrum arithrneticum, which gives all the prime divisors of every number not divisible by 2, 3, or 5 up to 1,020,000. In 1814–1817 J. C. 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, eighth, and ninth millions were calculated by Z. Dase and issued at Hamburg in 1862, 1863, and 1865. Dase died suddenly in 1861 during the progress of the work, and it was completed by H. Rosenberg. Dase‘s calculation was performed at the instigation of Gauss, and he began at 6,000,000 because the Berlin Academy was in possession of a manuscript presented by Crelle extending Burckhardt‘s tables from 3,000,000 to 6,000,000. This manuscript was found on examination to be so inaccurate that the publication was not desirable, and accordingly the three intervening millions were calculated and published by James Glaisher, the Factor Table for the Fourth Million appearing at London in 1879, and those for the fifth and sixth millions in 1880 and 1883 respectively (all three millions stereotyped). The tenth million, though calculated by Dase and Rosenberg, has not been published. 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. The first three millions were issued separately, and also bound in one volume, but the other six millions are all separate. Burckhardt began the publication of his tables with the second million instead of the first, as Chernac‘s factor table for the first million was already in existence. Burckhardt‘s first million does not supersede Chernac‘s, as the latter gives all the prime divisors of numbers not divisible by 2, 3, or 5 up to 1,020,000. It occupies 1020 pages, and Burckhardt found it very accurate; he detected only thirty-eight errors, of which nine were due to the author, the remaining twenty-nine having been caused by the slipping of type in the printing. The errata thus discovered are given in Burckhardt‘s first million. Other errata are contained in Allan Cunningham‘s paper referred to below.

Burckhardt gives but a very brief account of the method by which he constructed his table; and the introduction to Dase‘s millions merely consists of Gauss‘s letter suggesting their construction. The Introduction to the Fourth Million (pp. 52) contains a full account of the method of construction and a history of factor tables, with a bibliography of writings on the subject. The Introduction (pp. 103) to the Sixth Million contains an enumeration of primes and a great number of tables relating to the distribution of primes in the whole nine millions, portions of which had been published in the Cambridge Philosophical Proceedings and elsewhere. A complete list of errors in the nine millions was published by J. P. Gram (Acta mathematica, 1893, 17, p. 310). These errors, 141 in number, and which affect principally the second, third, eighth, and ninth millions, should be carefully corrected in all the tables. In 1909 the Carnegie Institution of Washington published a factor table by Prof. D. N. Lehmer which gives the least factor of all numbers not divisible by 2, 3, 5, or 7, up to ten millions. This table, which covers a range of 21,000 numbers on a single page, was reproduced by photography from a type-written copy of the author‘s original manuscript. The introduction contains a list of errata in the nine millions previously published, completely confirming Gram‘s list.

The factor tables which have just been described greatly exceed both in extent and accuracy any others of the same kind, the largest of which only reaches 408,000. This is the limit of Anton Felkel‘s Tafel aller einfachen Factoren (Vienna, 1776), a remarkable and extremely rare book,^[2] nearly all the copies having been destroyed. Georg Vega (Tabulae, 1797) gave a table showing all the divisors of numbers not divisible by 2, 3, or 5 up to 102,000, followed by a list of primes from 102,000 to 400,313. In the earlier editions of this work there are several errors in the list, but these are no doubt corrected in J. A. Hülsse‘s edition (1840). J. Salomon (Vienna, 1827) gives the least divisor of all numbers not divisible by 2, 3, or 5, up to 102,011, and B. Goldberg (Primzahlen und Factoren-Tafeln, Leipzig, 1862) gives all factors of numbers not divisible by 2, 3, or 5 up to 251,650. H. G. Köhler (Logarithmiseh-trigonornetrisches Handbuch, 1848 and subsequent editions) gives all factors of numbers not prime or divisible by 2, 3, 5, or 11 up to 21,525. Peter Barlow (Tables, 1814) and F. Schaller (Prirnzahlen-Tafel, Weimar, 1855) give all factors of all numbers up to 10,000. Barlow‘s work also contains a list of primes up to 100,103. Both the factor table and the list of primes are omitted in the stereotyped (1840) reprint. Full lists of errata in Chernac (1811), Barlow (1814), Hülsse‘s Vega (1840), Köhler (1848), Schaller (1855), and Goldberg (1862) are contained in a paper by Allan Cunningham (Mess. of Math., 1904, 34, p. 24; 1905, 35, p. 24). V. A. Le Besgue (Tables difverses pour la decomposition des noinbres, Paris, 1864) gives in a table of twenty pages, the least factor of numbers not divisible by 2, 3, or 5 up to 115,500. In Rees‘s Cyclopaedia (1819), article “Prime Numbers,” there is a list of primes to 217,219 arranged in decades. The Fourth Million (1879) contains a list of primes up to 30,341. The fourth edition of the Logarithrmic Tables (London, and Ithaca, N.Y., 1893) of G. W. jones of Cornell University contains a table of all the factors of numbers not divisible by 2 or 5 up to 20,000. In the case of primes the ten-place logarithm is given. This table does not occur in the third edition (Ithaca, N.Y., 1891). On the first page of the Second Million Burckhardt gives the first nine multiples of the primes to 1423; and a smaller table of the same kind, extending only to 313, occurs in Lambert‘s Supplementa (1798). Several papers contain lists of high primes (i.e. beyond the range of the factor tables). Among these may be mentioned two, by Allan Cunningham and H. J. Woodall jointly, in the Mess. of Math., 1902, 31, p. 165; 1905, 34, p. 72. See also the papers on factorizations of high numbers referred to under Tables relating to the Theory of Numbers. The Vienna Academy possesses the manuscript of an immense factor table extending to 100,000,000, constructed many years ago by J. P. Kulik (1793–1863) (see Ency. math. Wiss., 1900–1904, i. 952, and Lehmer’s Factor Table, p. ix.).

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 second (stereotyped) edition is a convenient folio volume of 450 pages.^[3] In 1908 an entirely new edition, edited by O. Seeliger, was published in which the multiples of 10, 20, …, 990 (omitted in previous editions) are included. This adds 50 pages to the volume, but removes what has been a great drawback to the use of the tables. Other improvements are that the tables are divided off horizontally and vertically by lines and spaces, and that, for calculations in which the last two figures are rejected, a mark has been placed to show when the last figure retained should be increased. Two other tables of the same extent (1000⨯1000), but more condensed in arrangement, are H. C. Schmidt’s Zahlenbuch (Aschersleben, 1896), and A. Henselin’s Recherttafel (Berlin, 1897). An anonymous table, published at Oldenburg in 1860, gives products up to 500⨯509, and M. Cordier, Le Multiplicateur de trois cents carrés (Paris, 1872), gives a multiplication table to 300⨯300 (intended for commercial use). In both these works the product is printed in full. The four following tables are for the multiplication of a number by a single digit. (1) A. L. Crelle, Erleichterungstafel für jeden, der zu rechnen hat (Berlin, 1836), a work extending to 1000 pages, gives the product of a number of seven figures by a single digit, by means of a double operation of entry. Each page is divided into two tables: for example, to multiply 9382477 by 7 we turn to page 825, and enter the right-hand table at line 77, column 7, where we find 77339; we then enter the left-hand table on the same page at line 93, column 7, and find 656, so that the product required is 65677339. (2) C. A. Bretschneider, Prodztkientafel (Hamburg and Gotha, 1841), is somewhat similar to Crelle’s table, but smaller, the number of figures in the multiplicand being five instead of seven. (3) In S. L. Laundy, A Table of Products (London, 1865), the product of any five-figure number by a single digit is given by a double arrangement. The extent of the table is the same as that of Bretschneider’s, as also is the principle, but the arrangement is different, Laundy’s table occupying only 10 pages and Bretschneider’s 99 pages. (4) G. Diakow’s Multiplikations-Tabelle (St Petersburg, 1897) is of the same extent as Bretschneider’s table but occupies 1000 pages. Among tables extending to 100⨯1000 (i.e. giving the products of two figures by three) may be mentioned C. A. Müller’s Multiplications-Tabellen (Karlsruhe, 1891). The tables of L. Zimmermann (Recherttafeln, Liebenwerda, 1896) and J. Riem (Rechentabellen für Multiplication, Basel, 1897) extend to 100⨯10,000. In a folio volume of 500 pages Peters (Rechentafeln für Multiplikation und Division mit ein- bis vierstelligen Zahlen, Berlin, 1909) gives products of four figures by two. The entry is by the last three figures of the multiplicand, and there are 2000 products on each page. Among earlier tables. the interest of which is mainly historical, mention may be made of C. Hutton’s Table of Products and Powers of Numbers (London, 1781), which contains a table up to 100⨯1000, and J. P. Gruson's Grosses Einmaleins von Eins bis Hunderttausend (Berlin, 1799)—a table of products up to 9⨯10,000. The author's intention was to extend it to 100,000, but only the first part was published. In this book there is no condensation or double arrangement; the pages are very large, each containing 125 lines.

Quarter-Squares.—Multiplication may be performed by means of a table of single entry in the manner indicated by the formula—

ab = 1/4(a + b)²−1/4(a−b)².

Thus with a table of quarter-squares we can multiply together any two numbers by subtracting the quarter-square of their difference from the quarter-square of their sum. The largest table of quarter-squares is J. Blater’s Table of Quarter-Squares of all whole numbers from 1 to 200,000 (London, 1888),^[4] which gives quarter-squares of every number up to 200,000 and thus yields directly the product of any two five-figure numbers. This fine table is well printed and arranged. Previous to its publication the largest table was S. L. Laundy’s Table of Quarter-Squares of all numbers up to 100,000 (London, 1856), which is of only half the extent, and therefore is only directly available when the sum of the two numbers to be multiplied does not exceed 100,000.

Smaller works are J. J. Centnerschwer, Neuerfundene Multiplications- und Quadrat-Tafeln (Berlin, 1825), which extends to 20,000, and J. M. Merpaut, Tables arithmonomiques (Vannes, 1832), which extends to 40,000. In Merpaut’s work the quarter-square is termed the “arithmone." L. J. Ludolf, who published in 1690 a table of squares to 100,000 (see next paragraph), explains in his introduction how his table may be used to effect multiplications by means of the above formula; but the earliest book on quarter squares is A. Voisin, Tables des multiplications, ou logarithm es des nombres entiers depuis 1 jusqu'à 20,000 (Paris, 1817). By a logarithm Voisin means a quarter-square, i.e. he calls a a root and 1/4a² its logarithm. On the subject of quarter-squares, &c., see Phil. Mag. [v.] 6, p. 331.

Squares, Cubes, &c., and Square Roots and Cube Roots.—The most convenient table for general use is P. Barlow’s Tables (Useful Knowledge Society, London, from the stereotyped plates of 1840), which gives squares, cubes, square roots, cube roots, and reciprocals to 10,000. These tables also occur in the original edition of 1814. The largest table of squares and cubes is J. P. Kulik, Tafeln der Quadrat- und Kubik-Zahlen (Leipzig, 1848), which gives both as far as 100,000. Blater’s table of quarter-squares already mentioned gives squares of numbers up to 100,000 by dividing the number by 2; and up to 200,000 by multiplying the tabular result by 4. Two early tables give squares as far as 100,000, viz. Maginus, Tabula tetragonica (Venice, 1592), and Ludolf, Tetragonometria tabularia (Amsterdam, 1690); G. A. Jahn, Tafel der Quadrat- und Kubikwurzeln (Leipzig, 1839), gives squares to 27,000, cubes to 24,00c, and square and cube roots to 25,500, at first to fourteen decimals and above 1010 to five. E. Gélin (Recueil de tables numériques, Huy, 1894) gives square roots (to 15 places) and cube roots (to 10 places) of numbers up to 100. C. Hutton, Tables of Products and Powers of Numbers (London, 1781), gives squares up to 25,400, cubes to 10,000, and the first ten powers of the first hundred numbers. P. Barlow, Mathematical Tables (original edition, 1814), gives the first ten powers of the first hundred numbers. The first nine or ten powers are given in Vega, Tabulae (1797), and in Hülsse’s edition of the same (1840), in Köhler, Handbuch (1848), and in other collections. C. F. Faà de Bruno, Calcul des erreurs (Paris, 1869), and J. H. T. Müller, Vierstellige Logarithmen (1844), give squares for use in connexion with the method of least squares. 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; they also occur in Köhler’s Handbuch (1848). The first 25 powers of 2, 3, 5, 7 are given in Salomon, Logarithmische Tafeln (1827). W. Shanks, Rectification of the Circle (1853), gives every 12th power of 2 up to 2⁷²¹. A very valuable paper (“Power-tables, Errata") published by Allan Cunningham in the Messenger of Math., 1906, 35, p. 13, contains the results of a careful examination of 27 tables containing powers higher than the cube, with lists of errata found in each. Before using any power table this list should be consulted, not only in order to correct the errata, but for the sake of references and general information in regard to such tables. In an appendix (p. 23) Cunningham gives errata in the tables of squares and cubes of Barlow (1814), Jahn (1839), and Kulik (1848).

Triangular Numbers.—E. de Joncourt, De natura et praeclaro usu simplicissimae speciei numerorum trigonalium (The Hague, 1762), contains a table of triangular numbers up to 20,000: viz. 1/2n(n + 1) is given for all numbers from n=1 to 20,000. The table occupies 224 pages.

Reciprocals.—P. Barlow’s Tables (1814 and 1840) give reciprocals up to 10,000 to 9 or 10 places; and a table of ten times this extent is given by W. H. Oakes, Table of the Reciprocals of Numbers from 1 to 100,000 (London, 1865). This table 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. R. 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. J. C. Houzeau gives the reciprocals of numbers up to 100 to 20 places and their first nine multiples to 12 places in the Bulletin of the Brussels Academy, 1875, 40, p. 107. E. Gélin (Recueil de tables numériques, Huy, 1894) gives reciprocals of numbers to 1000 to 10 places.

Tables for the Expression of Vulgar Fractions as Decimals.—Tables of this kind have been given by Wucherer, Goodwyn and Gauss. W. F. Wucherer, Beyträge zum allgemeinern Gebrauch der Decimalbrziche (Carlsruhe, 1796), gives the decimal fractions (to 5 places) for all vulgar fractions whose numerator and denominator are each less than 50 and prime to one another, arranged according to denominators. The most extensive and elaborate tables that have been published are contained in Henry Goodwyn’s First Centenary of Tables of all Decimal Quotients (London, 1816), A Tabular Series of Decimal Quotients (1823), and A Table of the Circles arising from the Division of a Unit or any other Whole Number by all the Integers from 1 to 1024 (1823). The Tabular Series (1823), which occupies 153 pages, gives to 8 places the decimal corresponding to every vulgar fraction less than 99/991, whose numerator and denominator do not surpass 1000. The arguments are not arranged according to their numerators or denominators, but according to their magnitude, so that the tabular results exhibit a steady increase from ·001 (=1/1000) to ·09989909 (=99/991,). The author intended the table to include all fractions whose numerator and denominator were each less than 1000, but no more was ever published. The Table of Circles (1823) gives all the periods of the circulating decimals that can arise from the division of any integer by another integer less than 1024. Thus for 13 we find ·076923 and ·153846, which are the only periods in which a fraction whose denominator is 13 can circulate. The table occupies 107 pages, some of the periods being of course very long (e.g., for 1021 the period contains 1020 figures). The First Centenary (1816) gives the complete periods of the reciprocals of the numbers from 1 to 100. Goodwyn’s tables are very scarce, but as they are nearly unique of their kind they deserve special notice. A second edition of the First Centenary was issued in 1818 with the addition of some of the Tabular Series, the numerator not exceeding 50 and the denominator not exceeding 100. A posthumous table of C. F. Gauss’s, entitled “Tafel zur Verwandlung gemeiner Brüche mit Nennern aus dem ersten Tausend in Decimalbrüche,” occurs in vol. ii. pp. 412-434 of his Gesammelte Werke (Göttingen, 1863), and resembles Goodwyn's Table of Circles. On this subject see a paper “On Circulating Decimals, with special reference to Henry Goodwyn’s Table of Circles and Tabular Series of Decimal Quotients,” in Camb. Phil. Proc., 1878, 3, p. 185, where is also given a table of the numbers of digits in the periods of fractions corresponding to denominators prime to 10 from 1 to 1024 obtained by counting from Goodwyn’s table. See also under Circulating Decimals (below).

Sexagesimal and Sexcentenary Tables.—Originally all calculations were sexagesimal; and the relics of the system still exist in the division of the degree into 60 minutes and the minute into 60 seconds. To facilitate interpolation, therefore, in trigonometrical and other tables the following large sexagesimal tables were constructed. John Bernoulli, A Sexcentenary Table (London, 1779), gives at once the fourth term of any proportion of which the first term is 600″ and each of the other two is less than 600″; the table is of double entry, and may be described as giving the value of xy/600 correct to tenths of a second, x and y each containing a number of seconds less than 600. Michael Taylor, A Sexagesimal Table (London, 1780), exhibits at sight the fourth term of any proportion where the first term is 60 minutes, the second any number of minutes less than 60, and the third any number of minutes and seconds under 60 minutes; there is also another table in which the third term is any absolute number under 1000. Not much use seems to have been made of these tables, both of which were published by the Commissioners of Longitude. Small tables for the conversion of sexagesimals into centesimals and vice versa are given in a few collections, such as Hülsse’s edition of Vega. H. Schubert’s Fünfstellige Tafeln und Gegentafeln (Leipzig, 1897) contains a sexagesimal table giving xy/60 for x= 1 to 59 and y = 1 to 150.

Trigonometrical Tables (Natural).—Peter Apian published in 1533 a table of sines with the radius divided decimally. The first complete canon giving all the six ratios of the sides of a right-angled triangle is due to Rheticus (1551), who also introduced the semiquadrantal arrangement. Rheticus’s canon was calculated for every ten minutes to 7 places, and Vieta extended it to every minute (1579). In 1554 Reinhold published a table of tangents to every minute. The first complete canon published in England was by Thomas Blundeville (1594), although a table of sines had appeared four years earlier. Regiomontanus called his table of tangents (or rather cotangents) tabula foecunda on account of its great use; and till the introduction of the word “tangent” by Thomas Finck (Geometriae rotundi libri XIV., Basel, 1583) a table of tangents was called a tabula foecunda or canon foecundus. Besides “tangent,” Finck also introduced the word “secant,” the table of secants having previously been called tabula benefica by Maurolycus (1558) and tabula foecundissima by Vieta.

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. The Opus palatinum contains a complete ten-decimal trigonometrical canon for every ten seconds of the quadrant, semiquadrantally arranged, with differences for all the tabular results throughout. Sines, cosines, and secants are given on the left-hand pages in columns headed respectively “Perpendculum,” “Basis,” “Hypotenusa,” and on the right-hand pages appear tangents, cosecants, and cotangents in columns headed respectively “Perpendiculum,” “Hypotenusa,” “Basis.” At his death Rheticus left the canon nearly complete, and the trigonometry was finished and the whole edited by Valentine Otho; it was named in honour of the elector palatine Frederick IV., who bore the expense of publication. The Thesaurus of 1613 gives natural sines for every ten seconds throughout the quadrant, to 15 places, semiquadrantally arranged, with first, second, and third differences. 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; but it could not be found. Eventually the lost canon was discovered amongst the papers of Rheticus which had passed from Otho to James Christmann on the death of the former. Amongst these Pitiscus found (1) the ten-second table of sines to 15 places, with first, second, and third differences (printed in the Thesaurus); (2) sines for every second of the first and last degrees of the quadrant, also to 15 places, with first and second differences; (3) the commencement of a canon of tangents and secants, to the same number of decimal places, for every ten seconds, with first and second differences; (4) a complete minute canon of sines, tangents, and secants, also to 15 decimal places. This list, taken in connexion with the Opus platinum, gives an idea of the enormous labours undertaken by Rheticus; his tables not only remain to this day the ultimate authorities but formed the data from which Vlacq calculated his logarithmic canon. Pitiscus says that for twelve years Rheticus constantly had computers at work.

A history of trigonometrical tables by Charles Hutton was prefixed to all the early editions of his Tables of Logarithms, and forms Tract xix. of his Mathematical Tracts, vol. i. p. 278, 1812. A good deal of bibliographical information about the Opus platinum and earlier trigonometrical tables is given in A. De Morgan's article “Tables” in the English Cyclopaedia. The invention of logarithms the year after the publication of Rheticus’s volume by Pitiscus changed all the methods of calculation; and it is worthy of note that John Napier’s original table of 1614 was a logarithmic canon of sines and not a table of the logarithms of numbers. The logarithmic canon at once superseded the natural canon; and since Pitiscus’s time no really extensive table of pure trigonometrical functions has appeared, In recent years the employment of calculating machines has revived the use of tables of natural trigonometrical functions, it being found convenient for some purposes to employ such a machine in connexion with a natural canon instead of using a logarithmic canon. A. Junge’s Tafel der wirklichen Länge der Sinus und Cosinus (Leipzig, 1864) was published with this object. It gives natural sines and cosines for every ten seconds of the quadrant to 6 places. F. M. Clouth, Tables pour le calcul des coordonnées goniométriques (Mainz, n.d.), gives natural sines and cosines (to 6 places) and their first nine multiples (to 4 places) for every centesimal minute of the quadrant. Tables of natural functions occur in many collections, the natural and logarithmic values being sometimes given on opposite pages, sometimes side by side on the same page.

The following works contain tables of trigonometrical functions other than sines, cosines, and tangents. J. Pasquich, Tabulae logarithmico-trigonometricae (Leipzig, 1817), contains a table of sin²x, cos²x, tan²x, cot²x from x = 1° to 45° at intervals of 1′ to 5 places. J. Andrew, Astronomical and Nautical Tables (London, 1805), contains a table of “squares of natural semichords” i.e. of sin²1/2x from x = 0° to 120° at intervals of 10″ to 7 places. This table was greatly extended by Major-General Hannyngton in his Haversines, Natural and Logarithmic, used in computing Lunar Distances for the Nautical Almanac (London, 1876). The name “haversine,” frequently used in works upon navigation, is an abbreviation of “half versed sine”; viz., the haversine of x is equal to 1/2(1 − cos x), that is, to sin²1/2x. The table gives logarithmic haversines for every 15″ from 0° to 180°, and natural haversines for every 10″ from 0° to 180°, to 7 places, except near the beginning, where the logarithms are given to only 5 or 6 places. It occupies 327 folio pages, and was suggested by Andrew’s work, a copy of which by chance fell into Hannyngton’s hands Hannyngton recomputed the whole of it by a partly mechanical method, a combination of two arithmometers being employed. A table of haversines is useful for the solution of spherical triangles when two sides and the included angle are given, and in other problems in spherical trigonometry. Andrew’s original table seems to have attracted very little notice. Hannyngton’s was printed, on the recommendation of the superintendent of the Nautical Almanac office, at the public cost. Before the calculation of Hannyngton’s table R. Farley’s Natural Versed Sines (London, 1856) was used in the Nautical Almanac office in computing lunar distances. This fine table contains natural versed sines from 0° to 125° at intervals of 10″ to 7 places, with proportional parts, and log versed sines from 0° to 135° at intervals of 15″ to 7 places. The arguments are also given in time. The manuscript was used in the office for twenty-five years before it was printed. Traverse tables, which occur in most collections of navigation tables, contain multiples of sines and cosines.

Common or Briggian Logarithms of Numbers and Trigonometrical Ratios.—For an account of the invention and history of logarithms, see Logarithm. The following are the fundamental works which contain the results of the original calculations of logarithms of numbers and trigonometrical ratios:—Briggs, Arithmetica logarithmica (London, 1624), logarithms of numbers from 1 to 20,000 and from 90,000 to 100,000 to 14 places, with interscript differences; Vlacq, Arithmetica logarithmica (Gouda, 1628, also an English edition, London, 1631, the tables being the same), ten-figure logarithms of numbers from 1 to 100,000, with differences, also log sines, tangents, and secants for every minute of the quadrant to 10 places, with interscript differences; Vlacq, Trigonometria artificialis (Gouda, 1633), log sines and tangents to every ten seconds of the quadrant to 10 places, with differences, and ten-figure logarithms of numbers up to 20,000, with differences; Briggs, Trigonometria Britannica (London, 1633), natural sines to 15 places, tangents and secants to 10 places, log sines to 14 places, and tangents to 10 places, at intervals of a hundredth of a degree from 0° to 45°, with interscript differences for all the functions. In 1794 Vega reprinted at Leipzig Vacq’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. The trigonometrical canon is not wholly reprinted from the Trigonometria artiticialis, as the logarithms for every second of the first two degrees, which do not occur in Vlacq, were calculated for the work by Lieutenant Dorfmund. Vega devoted great attention to the detection of errors in Vacq’s logarithms of numbers, and has given several important errata lists. F. Lefort (Annales de l’Observatoire de Paris, vol. iv.) has given a full errata list in Vlacq’s and Vega’s logarithms of numbers, obtained by comparison with the great French manuscript Tables du cadastre (see Logarithm; comp. also Monthly Notices R.A.S., 32, pp. 255, 288; 33, p. 330; 34, p. 447). Vega seems not to have bestowed on the trigonometrical canon anything like the care that he devoted to the logarithms of numbers, as Gauss^[5] estimates the total number of last-figure errors at from 31,983 to 47,746, most of them only amounting to a unit, but some to as much as 3 or 4.

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 latter work also has been made more accessible by a photographic reproduction by the Italian government (Riproduzione fotozincografica dell’ Istituto Geografico Militare, Florence, 1896). In 1897 Max Edler von Leber published tables for facilitating interpolations in Vega’s Thesaurus (Tabularum ad fariliorem et breiiiorem in Georgii Vegae “Thesauri logarithmorum” magnis canonibus interpolation is computation em utilium Trias, Vienna, 1897). The object of these tables is to take account of second differences. Prefixed to the tables is a long list of errors in the Thesaurus, occupying twelve pages. From an examination of the tabular results in the trigonometrical canon corresponding to 1060 angles von Leber estimates that out of the 90,720 tabular results 40,396 are in error by ± 1, 2793 by ±2, and 191 by ±3. Thus his estimated value of the total number of last-figure errors is 43,326, which is in accordance with Gauss’s estimate. A table of ten-figure logarithms of numbers up to 100,009, the result of a new calculation, was published in the Report of the U.S. Coast and Geodetic Survey for 1895–6 (appendix 12, pp. 395–722) by W. W. Duffield, superintendent of the survey. The table was compared with Vega’s Thesaurus before publication.

S. Pineto’s Tables de logarithm es vulgaires a dix décimales, construites d’aprés un nouveau mode (St Petersburg, 1871), though a tract of only 80 pages, may be usefully employed when Vlacq and Vega are unprocurable. Pineto’s work consists of three tables: the first, or auxiliary table, contains a series of factors by which the numbers whose logarithms are required are to be multiplied to bring them within the range of table 2; it also gives the logarithms of the reciprocals of these factors to 12 places Table 1 merely gives logarithms to 1000 to 10 places. Table 2 gives logarithms from 1,000,000 to 1,011,000, with proportional parts to hundredths. The mode of using these tables is as follows. If the logarithm cannot be taken out directly from table 2, a factor M is found from the auxiliary table by which the number must be multiplied to bring it within the range of table 2 Then the logarithm can be taken out, and, to neutralize the effect of the multiplication, so far as the result is concerned, log 1/M must be added; this quantity is therefore given in an adjoining column to M in the auxiliary table. A similar procedure gives the number answering to any logarithm, another factor (approximately the reciprocal of M) being given, so that in both cases multiplication is used. The laborious part of the work is the multiplication by M; but this is somewhat compensated for by the ease with which, by means of the proportional parts, the logarithm is taken out. The factors are 300 in number, and are chosen so as to minimize the labour, only 25 of the 300 consisting of three figures all different and not involving 0 or 1. The principle of multiplying by a factor which is subsequently cancelled by subtracting its logarithm is used also in a tract, containing only ten pages, published by A. Namur and P. Mansion at Brussels in 1877 under the title Tables de logarithmes à 12 décimales jusqu’à 434 milliards. Here a table is given of logarithms of numbers near to 434,294, and other numbers are brought within the range of the table by multiplication by one or two factors. The logarithms of the numbers near to 434,294 are selected for tabulation because their differences commence with the figures 100 … and the presence of the zeros in the difference renders the interpolation easy.

The tables of S. Gundelhnger and A. Nell (Tafeln zur Berechnung neunstelliger Logarithmen, Darmstadt, 1891) afford an easy means of obtaining nine-figure logarithms, though of course they are far less convenient than a nine-figure table itself; The method in effect consists in the use of Gaussian logarithms, viz., if N = n + p, log N = log n+ log (1 + p/n)=log n+ B where B is log (1 + p/n) to argument A = log p − log n. The tables give log n from n = 1000 to n = 10,000, and values of B for argument A.^[6]

Until 1891, when the eight-decimal tables, referred to further on, were published by the French government, the computer who could not obtain sufficiently accurate results from seven—figure logarithms was obliged to have recourse to ten-figure tables, for, with only one exception, there existed no tables giving eight or nine figures. This 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. This table is also remarkable for giving the logarithms of the differences instead of the actual differences. The arrangement of the page now universal in seven-figure tables—with the fifth figures running horizontally along the top line of the page—is due to John Newton.

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. C. Babbage, Table of the Logarithms of the Natural Numbers from 1 to 108,000 (London, stereotyped in 1827; there are many tirages of later dates), is the best for ordinary use. Great pains were taken to get the maximum of clearness. The change of figure in the middle of the block of numbers is marked by a change of type in the fourth figure, which (with the sole exception of the asterisk) is probably the best method that has been used. Copies of the book were printed on paper of different colours—yellow, brown, green, &c.—as it was considered that black on a white ground was a fatiguing combination for the eye. The tables were also issued with title-pages and introductions in other languages. In 1871 E. Sang published A New Table of Seven-place Logarithms of all Numbers from 20,000 to 200,000 (London). In an ordinary table extending from 10,000 to 100,000 the differences near the beginning are so numerous that the proportional parts are either very crowded or some of them omitted; by making the table extend from 20,000 to 200,000 instead of from 10,000 to 100,000 the differences are halved in magnitude, while there are only one fourth as many in a page. There is also greater accuracy. A further peculiarity of this table is that multiples of the differences, instead of proportional parts, are given at the side of the page. Typographically the table is exceptional, as there are no rules, the numbers being separated from the logarithms by reversed commas—a doubtful advantage. This work was to a great extent the result of an original calculation; see Trans. Roy. Soc. Edin., 1871, 26. Sang proposed to publish a nine-figure table from 1 to 1,000,000, but the requisite support was not obtained. Various papers of Sang’s relating to his logarithmic calculations will be found in the Proc. Roy. Soc. Edin. subsequent to 1872. Reference should here be made to Abraham Sharp’s table of logarithms of numbers from 1 to 100 and of primes from 100 to 1100 to 61 places, also of numbers from 999,990 to 1,000,010 to 63 places. These first appeared in Geometry Improv’d . . . by A. S. Philomath (London, 1717). They have been republished in Sherwin’s, Callet’s, and the earlier editions of Hutton’s tables. H. M. Parkhurst, Astronomical Tables (New York, 1871), gives logarithms of numbers from 1 to 109 to 102 places.^[7]

In many seven-figure tables of logarithms of numbers the values of S and T are given at the top of the page, with V, the variation of each, for the purpose of deducing log sines and tangents. S and T denote log (sin x/x) and log (tan x/x) respectively, the argument being the number of seconds denoted by certain numbers (sometimes only the first, sometimes every tenth) in the number column on each page. Thus, in Callet’s tables, on the page on which the first number is 67200, S＝log (sin 6720″/6720) and T＝log (tan 6720″/6720), while the V’s are the variations of each for 10″. To find, for example, log sin 1°52′12″·7, or log sin 6732″·7, we have S＝4·6854980 and log 6732·7＝3·8281893, whence, by addition, we obtain 8·5136873; but V for 10″ is −2·29, whence the variation for 12″·7 is −3, and the log sine required is 8·5136870. Tables of S and T are frequently called, after their inventor, “Delambre’s tables.”

Some seven-figure tables extend to 100,000, and others to 108,000, the last 8000 logarithms, to 8 places, being given to ensure greater accuracy, as near the beginning of the numbers the differences are large and the interpolations more laborious and less exact than in the rest of the table. The eight-figure logarithms, however, at the end of a seven-figure table are liable to occasion error; for the computer who is accustomed to three leading figures, common to the block of figures, may fail to notice that in this part of the table there are four, and so a figure (the fourth) is sometimes omitted in taking out the logarithm. In the ordinary method of arranging a seven-figure table the change in the fourth figure, when it occurs in the course of the line, is a source of frequent error unless it is very clearly indicated. In the earlier tables the change was not marked at all, and the computer had to decide for himself, each time he took out a logarithm, whether the third figure had to be increased. 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). This is very clear in the case of the 0’s, but leaves unmarked the cases in which the fourth figure is 1 or 2. A method which finds favour in some recent tables is to underline all the figures after the increase, or to place a line over them.

Babbage printed a subscript point under the last figure of each logarithm that had been increased. Schrön used a bar subscript, which, being more obtrusive, seems less satisfactory. In some tables the increase of the last figure is only marked when the figure is increased to a 5, and then a Roman five (v) is used in place of the Arabic figure.

Hereditary errors in logarithmic tables are considered in two papers “On the Progress to Accuracy of Logarithmic Tables” and “On Logarithmic Tables,” in Monthly Notices R.A.S., 33, pp. 330, 440. See also vol. 34, p. 447; and a paper by Gernerth, Ztsch. f. d. österr. Gymm., Heft vi. p. 407.

Passing now to the logarithmic trigonometrical canon, the first great advance after the publication of the Trigonometria artificialis in 1633 was made in Michael Taylor’s Tables of Logarithms (London, 1792), which give log sines and tangents to every second of the quadrant to 7 places. This table contains about 450 pages with an average number of 7750 figures to the page, so that there are altogether nearly three millions and a half of figures. The change in the leading figures, when it occurs in a column, is not marked at all; and the table must be used with very great caution. In fact it is advisable to go through the whole of it, and fill in with ink the first 0 after the change, as well as make some mark that will catch the eye at the head of every column containing a change. 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 change in the column is very clearly marked by a large black nucleus, surrounded by a circle, printed instead of 0. Bagay’s work having become rare and costly was reprinted with the errors corrected. The reprint, however, bears the original title-page and date 1829, and there appears to be no means of distinguishing it from the original work except by turning to one of the errata in the original edition and examining whether the correction has been made.

The only other canon to every second that has been published is contained in R. Shortrede’s Logarithmic Tables (Edinburgh). This work was originally issued in 1844 in one volume, but being dissatisfied with it Shortrede issued a new edition in 1849 in two volumes. The first volume contains logarithms of numbers, antilogarithms, &c., and the second the trigonometrical canon to every second. The volumes are sold separately, and may be regarded as independent works; they are not even described on their title pages as vol. i. and vol. ii. The trigonometrical canon is very complete in every respect, the arguments being given in time as well as in arc, full proportional parts being added, &c. The change of figure in the column is denoted by a nokta, printed instead of 0 where the change occurs. The page is crowded and the print not very clear, so that Bagay is to be preferred for regular use.

Previous to 1891 the only important tables in which the quadrant is divided centesimally were J. P. Hobert and L. Ideler, Nouvelles tables trigonométriques (Berlin, 1799), and C. Borda and J. B. J. Delambre, Tables trigonométriques décimales (Paris, 1801). The former give, among other tables, natural and log sines, cosines, tangents, and cotangents, to 7 places, the arguments proceeding to 3° at intervals of 10″ and thence to 50° at intervals of 1′ (centesimal), and also natural sines and tangents for the first hundred ten-thousandths of a right angle to 10 places. The latter gives long sines, cosines, tangents, cotangents, secants, and cosecants from 0° to 3° at intervals of 10″ (with full proportional parts for every second), and thence to 50° at intervals of 1′ (centesimal) to 7 places. There is also a table of log sines, cosines, tangents, and cotangents from 0′ to 10′ at intervals of 10″ and from 0° to 50° at intervals of 10′ (centesimal) to 11 places. Hobert and Ideler give a natural as well as a logarithmic canon; but Borda and Delambre give only the latter. Borda and Delambre give seven-figure logarithms of numbers to 10,000, the line being broken when a change of figure takes place in it.

The tables of Borda and Delambre having become difficult to procure, and seven-figure tables being no longer sufficient for the accuracy required in astronomy and geodesy, the French government in 1891 issued an eight-figure table containing (besides logarithms of numbers to 120,000) log sines and tangents for every ten seconds (centesimal) of the quadrant, the latter being extracted from the Tables du cadastre of Prony (see Logarithm). The title of this fine and handsomely printed work is Service géographique de l’armée: Tables des logarithmes á huit décimales . . . publiées par ordre du ministre de la guerre (Paris, Imprimerie Nationale, 1891). These tables are now in common use where eight figures are required.

In Brigg’s Trigonometria Britannica of 1633 the degree is divided centesimally, and but for the appearance in the same year of Vlacq’s Trigonometria artificialis, in which the degree is divided sexagesimally, this reform might have been effected. It is clear that the most suitable time for making 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 ensuring 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. In fact the centesimal degree is as arbitrary a unit as the nonagesimal and it is only the non-centesimal subdivision of the degree that gives rise to inconvenience. Briggs’s example was followed by Roe, Oughtred, and other 17th-century writers; but the centesmal division of the degree seemed to have entirely passed out of use, till it was revived by C. Bremiker in his Logarithmisch-trigonometrische Tafeln mit fünf Decimalstellen (Berlin, 1872, 10th ed. revised by A. Kallius, 1906). This little book of 158 pages gives a five-figure canon to every hundredth of a degree with proportional parts, besides logarithms of numbers, addition and subtraction logarithms, &c.

The eight-figure table of 1891 has now made the use of a centesimal table compulsory, if this number of figures is required. The Astronomische Gesellschaft are, however, publishing an eight figure table on the sexagesimal system, under the charge of Dr. J. Bauschinger, the director of the k. Recheninstitut at Berlin. The arrangement is to be in groups of three as in Bremiker’s tables.

Collections of Tables.—For a computer who requires in one volume logarithms of numbers and a ten-second logarithmic canon, perhaps the two best books are L. Schrön, Seven-Figure Logarithms (London, 1865, stereotyped, an English edition of the German work published at Brunswick), and C. Bruhns, A New Manual of Logarithms to Seven Places of Decimals (Leipzig, 1870). Both these works (of which there have been numerous editions) give logarithms of numbers and a complete ten-second canon to 7 places; Bruhns also gives log sines, cosines, tangents, and cotangents to every second up to 6° with proportional parts. Schrön contains an interpolation table, of 75 pages, giving the first 100 multiples of all numbers from 40 to 420. The logarithms of numbers extend to 108,000 in Schron and to 100,000 in Bruhns. Almost equally convenient is Bremiker’s edition of Vega’s Logarithmic Tables (Berlin, stereotyped; the English edition was translated from the fortieth edition of Bremiker’s by W. L. F. Fischer). This book gives a canon to every ten seconds, and for the first five degrees to every second, with logarithms of numbers to 100,000. Schrön, Bruhns, and Bremiker all give the proportional parts for all the differences in the logarithms of numbers. In Babbage’s, Callet’s, and many other tables only every other table of proportional parts is given near the beginning for want of space. Schrön, Bruhns, and most modern tables published in Germany have title-pages and introductions in different languages. J. Dupuis, Tables de logarilhmes a sept décimales (stereotyped, third tirage, 1868, Paris), is also very convenient, containing a ten-second canon, besides logarithms of numbers to 100,000, hyperbolic logarithms of numbers to 1000, to 7 places, &c. In this work negative characteristics are printed throughout in the tables of circular functions, the minus sign being placed above the figure; for the mathematical calculator these are preferable to the ordinary characteristics that are increased by 10. The edges of the pages containing the circular functions are red, the rest being grey. Dupuis also edited Callet’s logarithms in 1862, with which this work must not be confounded. J. Salomon, Logarithimische Tafeln (Vienna, 1827), contains a ten-second canon (the intervals being one second for the first two degrees), logarithms of numbers to 108,000, squares, cubes, square roots, and cube roots to 1000, a factor table to 102,011, ten-place Briggian and hyperbolic logarithms of numbers to 1000 and of primes to 10,333, and many other useful tables. The work, which is scarce, is a well-printed small quarto volume.

Of collections of general tables among the most useful and accessible are Hutton, Callet, Vega, and Köhler. C. Hutton’s well-known Mathematical Tables (London) was first issued in 1785, but considerable additions were made in the fifth edition (1811). The tables contain seven-figure logarithms to 108,000, and to 1200 to 20 places, some antilogarithms to 20 places, hyperbolic logarithms from 1 to 10 at intervals of ·01 and to 1200 at intervals of unity to 7 places, logistic logarithms, log sines and tangents to every second of the first two degrees, and natural and log sines, tangents, secants, and versed sines for every minute of the quadrant to 7 places. The natural functions occupy the left-hand pages and the logarithmic the right-hand. The first six editions, published in Hutton’s lifetime (d. 1823), contain Abraham Sharp’s 61-figure logarithms of numbers. Olinthus Gregory, who brought out the 1830 and succeeding editions, omitted these tables and Hutton’s introduction, which contains a history of logarithms, the methods of constructing them, &c. F. Callet’s Tables portatives de logarithmes (stereotyped, Paris) seems to have been first issued in 1783, and has since passed through a great many editions. In that of 1853 the contents are seven-figure logarithms to 108,000, Briggian and hyperbolic logarithms to 48 places of numbers to 100 and of primes to 1097, log sines and tangents for minutes (centesimal) throughout the quadrant to 7 places, natural and log sines to 15 places for every ten minutes (centesimal) of the quadrant, log sines and tangents for every second of the first five degrees (sexagesimal) and for every ten seconds of the quadrant (sexagesimal) to 7 places, besides logistic logarithms, the first hundred multiples of the modulus to 24 places and the first ten to 70 places, and other tables. This is one of the most complete and practically useful collections of logarithms that have been published, and it is peculiar in giving a centesimally divided canon. The size of the page in the editions published in the 19th century is larger than that of the earlier editions, the type having been reset. G. Vega’s Tabulae logarithmo-trigonometricae was first published in 1797 in two volumes. The first contains seven-figure logarithms to 101,000, log sines, &c., for every tenth of a second to 1′, for every second to 1° 30′, for every 10″ to 6° 3′, and thence at intervals of a minute, also natural sines and tangents to every minute, all to 7 places. The second volume gives simple divisors of all numbers up to 102,000, a list of primes from 102,000 to 400,313, hyperbolic logarithms of numbers to 1000 and of primes to 10,000, to 8 places, e^x and log_10^ex to x= 10 at intervals of ·01 to 7 figures and 7 places respectively, the first nine powers of the numbers from 1 to 100, squares and cubes to 1000, logistic logarithms, binomial theorem coefficients, &c. Vega also published Manuale logarithmica-trigonometric am (Leipzig, 1800), the tables in which are identical with a portion of those contained in the first volume of the Tabitlae. The Tabulae went through many editions, a stereotyped issue being brought out by J. A. Hülsse (Sammlung mathematischer Tafeln, Leipzig) in one volume in 1840. The contents are nearly the same as those of the original work, the chief difference being that a large table of Gaussian logarithms is added. Vega differs from Hutton and Callet in giving so many useful non-logarithmic tables, and his collection is in many respects complementary to theirs. J. C. Schulze, Neue und erweiterte Sammlung logarithmischer, trigonometrischer, and anderer Tafeln (2 vols. Berlin, 1778), is a valuable collection, and contains seven-figure logarithms to 101,000, log sines and tangents to 2° at intervals of a second, and natural sines, tangents, and secants to 7 places, log sines and tangents and Napierian log sines and tangents to 8 places, all for every ten seconds to 4° and thence for every minute to 45°, besides squares, cubes, square roots, and cube roots to 1000, binomial theorem coefficients, powers of e, and other small tables. Wolfram’s hyperbolic logarithms of numbers below 10,000 to 48 places first appeared in this work. J. H. Lambert’s Supplementa tabularum logarithmicarum et trigonometricarum (Lisbon, 1798) contains a number of useful and curious non-logarithmic tables and bears a general resemblance to the second volume of Vega, but there are also other small tables of a more strictly mathematical character. A very useful collection of non-logarithmic tables is contained in Peter Barlow’s New Mathematical Tables (London, 1814). It gives squares, cubes, square roots, and cube roots (to 7 places), reciprocals to 9 or 10 places, and resolutions into their prime factors of all numbers from 1 to 10,000, the first ten powers of numbers to 100, fourth and fifth powers of numbers from 100 to 1000, prime numbers from 1 to 100,102, eight-place hyperbolic logarithms to 10,000, tables for the solution of the irreducible case in cubic equations, &c. In the stereotyped reprint of 1840 only the squares, cubes, square roots, cube roots, and reciprocals are retained. The first volume of Shortrede’s tables, in addition to the trigonometrical canon to every second, contains antilogarithms and Gaussian logarithms. F. R. Hassler, Tabulae logarithmicae et trigonometricae (New York, 1830, stereotyped), gives seven-figure logarithms to 100,000, log sines and tangents for every second to 1°, and log sines, cosines, tangents, and cotangents from 1° to 3° at intervals of 10” and thence to 45° at intervals of 30″, Every effort has been made to reduce the size of the tables without loss of distinctness, the page being only about 3 by 5 inches. Copies of the work were published with the introduction and title-page in different languages. A. D. Stanley, Tables of Logarithms (New Haven, U.S., 1860), gives seven-figure logarithms to 100,000, and log sines, cosines, tangents, cotangents, secants, and cosecants at intervals of ten seconds to 15° and thence at intervals of a minute to 45° to 7 places, besides natural sines and cosines, antilogarithms, and other tables. This collection owed its origin to the fact that Hassler’s tables were found to be inconvenient owing to the smallness of the type. G. Luvini, Tables of Logarithms (London, 1866, stereotyped, printed at Turin), gives seven-figure logarithms to 20,040, Briggian and hyperbolic logarithms of primes to 1200 to 20 places, log sines and tangents for each second to 9′, at intervals of 10″ to 2°, of 30″ to 9°, of 1′ to 45° to 7 places, besides square and cube roots up to 625. The book, which is intended for schools, engineers, &c., has a peculiar arrangement of the logarithms and proportional parts on the pages. 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. It gives logarithms of numbers to 100,000, with proportional parts, and log sines and tangents for every second to 5°, and beyond 5° for every ten seconds, with proportional parts. J. Hantschl, Logarithmisch-trigonometrisches Handbuch (Vienna, 1827), gives five figure logarithms to 10,000, log sines and tangents for every ten seconds to 6 places, natural sines, tangents, secants, and versed sines for every minute to 7 places, logarithms of primes to 15,391, hyperbolic logarithms of numbers to 11,273 to 8 places, least divisors of numbers to 18,277, binomial theorem coefficients, &c. R. Farley’s Six-Figure Logarithms (London, stereotyped, 1840), gives six-figure logarithms to 10,000 and log sines and tangents for every minute to 6 places.

Coming now to five-figure tables a very convenient little book is Tables of Logarithms (Useful Knowledge Society, London, from the stereotyped plates of 1839), which was prepared by De Morgan, though it has no name on the title-page. It contains five-figure logarithms to 10,000, log sines and tangents to every minute to 5 places, besides a few smaller tables. J. de Lalande’s Tables de logarithmes is a five-figure table with nearly the same contents as De Morgan’s, first published in 1805. It has since passed through many editions, and, after being extended from 5 to 7 places, passed through several more. J. Galbraith and S. Haughton, Manual of Mathematical Tables (London, 1860), give five-figure logarithms to 10,000 and log sines and tangents for every minute, also a small table of Gaussian logarithms. J. Houël, Tables de logarithmes à cinq décimales (Paris, 1871; new edition 1907), is a very convenient collection of five-figure tables; besides logarithms of numbers and circular functions, there are Gaussian logarithms, least divisors of numbers to 10,841, antilogarithms, &c. The work (118 pp.) is printed on thin paper. A. Gernerth, Fünfstellige gemeine Logarithmen (Vienna, 1866), gives logarithms to 10,800 and a ten-second canon. There are sixty lines on the page, so that the double page contains log sines, cosines, tangents, and cotangents extending over a minute. C. Bremiker, Logarithmisch-trigonometrische Tafeln mit fünf Decimalstellen (10th edition by A. Kallius, Berlin, 1906), which has been already referred to, gives logarithms to 10,009 and a logarithmic canon to every hundredth of a degree (sexagesimal), in a handy volume; the lines are divided into groups of three, an arrangement about the convenience of which there is a difference of opinion. H. Gravelius, Fünfstellige logarithmisch-trigonometrische Tafeln für die Decimalteilung des Quadranten (Berlin, 1886), is a well-printed five-figure table giving logarithms to 10,009, a logarithmic canon to every centesimal minute (i.e. ten-thousandth part of a right angle), and an extensive table (40 pp.) for the conversion of centesimally expressed arcs into sexagesimally expressed arcs and vice versa. Among the other tables is a four-place table of squares from 0 to 10 at intervals of −001 with proportional parts. E. Becker, Logarithmisch-trigonometrisches Handbuch auf fünf Decimalen (2nd stereo. ed., Leipzig, 1897), gives logarithms to 10,009 and a logarithmic canon for every tenth of a minute to 6° and thence to 45° for every minute. There are also Gaussian logarithms. V. E. Gamborg, Logaritmetabel (Copenhagen, 1897), is a well-printed collection of tables, which contains a five-figure logarithmic canon to every minute, five-figure logarithms of numbers to 10,000, and five-figure antilogarithms, viz., five-figure numbers answering to four-figure mantissa from ·0000 to ·9999 at intervals of ·0001. H. Schubert, Fünfstellige Tafeln und Gegentafeln (Leipzig, 1896), is peculiar in giving, besides logarithms of numbers and a logarithmic and natural canon, the three converse tables of numbers answering to logarithms, and angles answering to logarithmic and natural trigonometrical functions. The five-figure tables of F. G. Gauss (Berlin, 1870) have passed through very many editions, and mention should also be made of those of T. Wittstein (Hanover, 1859) and F. W. Rex (Stuttgart, 1884). S. W. Holman, Computation Rules and Logarithms (New York, 1896), contains a well-printed and convenient set of tables including five-figure logarithms of numbers to 10,000 and a five-figure logarithmic canon to every minute, the actual characteristics (with the negative sign above the number) being printed, as in the tables of Dupuis, 1868, referred to above. There is also a four-place trigonometrical canon and four-place antilogarithms, reciprocals, square and cube roots, &c. G. W. Jones, Logarithmic Tables (4th ed., London, and Ithaca, N.Y., 1893), contains a five-place natural trigonometrical canon and a six-place logarithmic canon to every minute, six-place Gaussian and hyperbolic logarithms, besides a variety of four-place tables, including squares, cubes, quarter-squares, reciprocals, &c. 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. J. Houël’s Recueil de formules et de tables numériques (Paris, 1868) contains 19 tables, occupying 62 pages, most of them giving results to 4 places; they relate to very varied subjects-antilogarithms, Gaussian logarithms, logarithms of 1+x/1−x elliptic integrals, squares for use in the method of least squares, &c. C. Bremiker, Tafel vierstelliger Logarithmen (Berlin, 1874), gives four figure logarithms, of numbers to 2009, log sines, cosines, tangents, and cotangents to 8° for every hundredth of a degree, and thence to 45° for every tenth of a degree, to 4 places. There are also Gaussian logarithms, squares from 0·000 to 13,500, antilogarithms, &c. The book contains 60 pages. It is not worth while to give a list of four figure tables or other tables of small extent, which are very numerous, but mention may be made of J. M. Peirce, Mathematical Tables chiefly to Four Figures (Boston, U.S., 1879), 42 pp., containing also hyperbolic functions; W. Hall, Four-figure Tables and Constants (Cambridge, 1905), 60 pp., chiefly for nautical computation; A. du P. Denning, Five-figure Mathematical Tables for School and Laboratory Purposes (12 pp. of tables, large octavo); A. R. Hinks, Cambridge Four-figure Mathematical Tables (12 pp.). C. M. Willich, Popular Tables (London, 1853), is a useful book for an amateur; it gives Briggian and hyperbolic logarithms to 1200 to 7 places, squares, &c., to 343, &c.

Hyperbolic or Napierian or Natural Logarithms.—The logarithms invented by Napier and explained by him in the Descriptio (1614) were not the same as those now called natural or hyperbolic (viz., to base e), and very frequently also Napierian, logarithms. Napierian logarithms, strictly so called, have entirely passed out of use and are of purely historic interest; it is therefore sufficient to refer to the article Logarithm), where a full account is given. Apart from the inventor’s own publications, the only strictly Napierian tables of importance are contained in Ursinus’s Trigonometria (Cologne, 1624–1625) and Schulze’s Sammlung (Berlin, 1778), the former being the largest that has been constructed. Logarithms to the base e, where e denotes 2·71828 . . ., were first published by J. Speidell, New Logarithmes (1619).

The most copious table of hyperbolic logarithms is Z. Dase, Tafel der natürlichen Logarithmen (Vienna, 1850), which extends from 1 to 1000 at intervals of unity and from 1000 to 10,500 at intervals of −1 to 7 places, with differences and proportional parts, arranged as in an ordinary seven-figure table. By adding log 10 to the results the range is from 10,000 to 105,000 at intervals of unity. The table formed part of the Annals of the Vienna Observatory for 1851, but separate copies were printed. The most elaborate table of hyperbolic logarithms is due to Wolfram, who calculated to 48 places the logarithms of all numbers up to 2200, and of all primes (also of a great many composite numbers) between this limit and 10,009. Wolfram’s results first appeared in Schulze’s Sammlung (1778). Six logarithms which Wolfram had been prevented from computing by a serious illness were supplied in the Berliner Jahrbuch, 1783, p. 191. The complete table was reproduced in Vega’s Thesaurus 1794), where several errors were corrected. Tables of hyperbolic logarithms are contained in the following collections:—Callet, all numbers to 100 and primes to 1097 to 48 places; Borda and Delambre (1801), all numbers to 1200 to 11 places; Salomon (1827), all numbers to 1000 and primes to 10,333 to 10 places; Vega, Tabulae (including Hülsse’s edition, 1840), and Köhler (1848), all numbers to 1000 and primes to 10,000 to 8 places; Barlow (1814), all numbers to 10,000; Hutton, Mathematical Tables, and Willich (1853), all numbers to 1200 to 7 places; Dupuis (1868), all numbers to 1000 to 7 places. Hutton also gives hyperbolic logarithms from 1 to 10 at intervals of ·01 to 7 places. Rees’s Cyclopaedia (1819), art, "Hyperbolic Logarithms," contains a table of hyperbolic logarithms of all numbers to 10,000 to 8 places.

Logarithms to base e are generally termed Napierian by English writers, and natural by foreign writers. There seems no objection to the former name, though the logarithms actually invented by Napier depended on the base e⁻¹, but it should be mentioned in text-books that so-called Napierian logarithms are not identical with those originally devised and calculated by Napier.

Tables to convert Briggian into Hyperbolic Logarithms, and vice versa.—Such tables merely consist of the first hundred (sometimes only the first ten) multiples of the modulus ·43429 44819 . . . and its reciprocal 2·30258 50929 . . . to 5, 6, 8, 10, or more places. 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. In an antilogarithmic canon the logarithms are exact quantities, such as ·00001, ·00002, &c., and the corresponding numbers are incommensurable. The largest and earliest work of this kind is J. Dodson’s Antilogarithmic Canon (London, 1742), which gives numbers to 11 places corresponding to logarithms from 0 to 1 at intervals of ·00001, arranged like a seven-figure logarithmic table, with interscript differences and proportional parts at the bottom of the page. This work was the only large antilogarithmic canon for more than a century, till in 1844 Shortrede published the first edition of his tables; in 1849 he published the second edition, and in the same year Filipowski’s tables appeared. Both these works contain seven-figure antilogarithms: Shortrede gives numbers to logarithms from 0 to 1 at intervals of ·00001, with differences and multiples at the top of the page, and H. E. Filipowski, A Table of Antilogarithms (London, 1849), contains a table of the same extent, the proportional parts being given to hundredths.

Small tables of antilogarithms to 20 places occur in several collections of tables, as Gardiner (1742), Callet, and Hutton. Four- and five-place tables are not uncommon in recent works, as e.g. in Houël (1871), Gamborg (1897), Schubert (1896), Holman (1896).

Addition and Subtraction, or Gaussian Logarithms.—The object of such tables is to give log (a ± b) by only one entry when log a and log b are given. Let

A＝log x, B＝log (1 + x⁻¹), C＝log (1 + x).

Leaving out the specimen table in Z. Leonelli’s Théorie des logarithmes additionnels et déductifs (Bordeaux, 1803), in which the first suggestion was made,^[8] the principal tables are the following: Gauss, in Zach’s Monatliche Correspondenz (1812), gives B and C for argument A from 0 to 2 at intervals of ·001, thence to 3·40 at intervals of ·01, and to 5 at intervals of ·1, all to 5 places. This table is reprinted in Gauss’s Werke, vol. iii. p. 244. E. A. Matthiessen, Tafel zur bequemern Berechnung (Altona, 1818), gives B and C to 7 places for argument A from 0 to 2 at intervals of ·0001, thence to 3 at intervals of ·001, to 4 at intervals of ·01, and to 5 at intervals of ·1; the table is not conveniently arranged. Peter Gray, Tables and Formulae (London, 18491 and “Addendum,” 1870), gives C for argument A from −3 to −1 at intervals of −001 and from −1 to 2 at intervals of ·0001, to 6 places, with proportional parts to hundredths, and log (1−x) for argument A from −3 to −1 at intervals of ·001 and from 1 to ·18999 at mtervals of ·0001, to 6 places, with proportional parts. J. Zech, Tafeln der Additions- und Subtractions-Logarithmen (Leipzig, 1849), gives B for argument A from 0 to 2 at intervals of −·0001, thence to 4 at intervals of ·001 and to 6 at intervals of -01; also C for argument A from o to -0003 at intervals of ·0000001, thence to ·05 at intervals of ·000001 and to ·303 at intervals of ·00001, all to 7 places, with proportional parts. These tables are reprinted from Hülsse’s edition of Vega (1849); the 1840 edition of Hülsse’s Vega contained a reprint of Gauss’s original table. T. Wittstein, Logarithmes de Gauss à sept décimales (Hanover, 1866), gives B for argument A from 3 to 4 at intervals of ·1, from 4 to 6 at intervals of ·01, from 6 to 8 at intervals of ·001, from 8 to 10 at intervals of ·0001, also from 0 to 4 at the same intervals. In this handsome work the arrangement is similar to that in a seven-figure logarithmic table. Gauss’s original five-place table was reprinted in Pasquich, Tabulae (Leipzig, 1817); Köhler, Jerome de la Lande’s Tafeln (Leipzig, 1832), and Handbuch (Leipzig, 1848); and Galbraith and Haughton, Manual (London, 1860). Houël, Tables de logarithmes (1871), also gives a small five-place table of Gaussian logarithms, the addition and subtraction logarithms being separated as in Zech. Modified Gaussian logarithms are given by J. H. T. Müller, Vierstellige Logarithmen (Gotha, 1844), viz., a four-place table of B and −log (1 − x⁻¹) from A = 0 to ·03 at intervals of ·001, thence to ·23 at intervals of ·001, to 2 at intervals of ·01, and to 4 at intervals of ·1; and by Shortrede, Logarithmic Tables (vol. i., 1849), viz., a five-place table of B and log (1 + x) from A=5 to 3 at intervals of ·1, from A=3 to 2·7 at intervals of ·01, to 5 at intervals of ·001, to 3 at intervals of ·01, and to 5 at intervals of ·1. Filipowski’s Antilogarithms (1849) contains Gaussian logarithms arranged in a new way. The principal table gives log (x + 1) as tabular result for log x as argument from 8 to 14 at intervals of ·001 to 5 places. Weidenbach, Tafel um den Logarithmen … (Copenhagen, 1829), gives log x + 1x − 1; for argument A from ·382 to 2·002 at intervals of ·001, to 3·6 at intervals of ·01, and to 5·5 at intervals of ·1 to 5 places. J. Houël’s Recueil de formules et de tables numériques (2nd ed., Paris, 1868) contains tables of log₁₀(x + 1), log₁₀11 − x and og₁₀1 + x1 − x from log x = −5 to −3 at intervals of ·1, from log x = −3 to −1 at intervals of ·01, from log x = −1 to 0 at intervals of ·001. F. W. Rex (Fünfstellige Logarithmen-Tafeln, Stuttgart, 1884) gives also a five-figure table of log 1 + x1 − x and E. Hammer in his Sechsstellige Tafel der Werthe für jeden Wert des Arguments log x (Leipzig, 1902) gives a six figure table of this function from log x = 7 to 1·99000, and thence to 1·999700 to 5 places. S. Gundelfinger’s Sechsstellige Gaussische und siebenstellige gemeine Logarithmen (Leipzig, 1902) contains a table of log₁₀(1 + x) to 6 places from log x = −2 to 2 at intervals of ·001. G. W. Jones’s Logarithmic Tables (4th ed., London, and Ithaca, N.Y., 1893) contain 17 pages of Gaussian six-figure tables; the principal of which give log (1 + x) to argument log x from log x = −2·80 to 0 at intervals of ·001, and thence to ·1999 at intervals of ·0001, and log (1 − x⁻¹) to argument log x from log x = ·4 to ·5 at intervals of ·0001, and thence to 2·8 at intervals of ·001. Gaussian logarithms to 5 or 4 places occur in many collections of five-figure or four-figure tables.

Quadratic Logarithms.—In a pamphlet Saggio di tavole dei logaritmi quadratic (Udine, 1885) Conte A. di Prampero has described a method of obtaining fractional powers (positive or negative) of any number by means of tables contained in the work. If

a^bx = N, then x = log log N − log log alog b

and if the logarithms are taken to be Briggian and a = 101/1024 and b = 2, then x = log₁₀ log₁₀N/log 2 + 10.

This quantity the author defines as the quadratic logarithm of N and denotes by L_qN. It follows from this definition that L_qN^r = L_qN + log_10^r/log_10². Thus the quadratic logarithms of N and N^s where s is any power (positive or negative) of 2 have the same mantissa.

A subsidiary table contains the values of the constant log_10^r/log_10² for 204 fractional values of r. The main table contains the values of 1000 mantissae corresponding to arguments N, N½, Ni, … (which all have the same mantissae). Among the arguments are the quantities 10·0, 10·1, 10·2, … 99·9 (the interval being ·1) and 10·00, 10·01, … 10·99 (the interval being ·01). As an example, to obtain the value of 122/3 we take from the first table

the constant −0·584962, which belongs to ⅔, and entering the main table with 12 we take out the quadratic logarithm 10·109937 which, by applying the constant, gives 9·524975 the quadratic logarithm of the quantity required.

An appendix (Tavola degli esponenti) gives the Briggian logarithms of the first 57 numbers to the first 50 numbers as base, viz. log_xN for N = 2, 3, …, 57 and x = 2, 3, …, 50. The results are generally given to 6 places.

Logistic and Proportional Logarithms.—In most collections of tables of logarithms a five-place table of logistic logarithms for every second to 1° is given. Logistic tables give log 3600 − log x at intervals of a second, x being expressed in degrees, minutes, and seconds. In Schulze (1778) and Vega (1797) the table extends to x = 3600″ and in Callet and Hutton to x = 5280″. Proportional logarithms for every second to 3° (i.e. log 10,800 − log x) form part of nearly all collections of tables relating to navigation, generally to 4 places, sometimes to 5. Bagay, Tables (1829), gives a five-place table, but such are not often to be found in collections of mathematical tables. The same remark applies to tables of proportional logarithms for every minute to 24^h, which give to 4 or 5 places the values of log 1440 − log x. The object of a proportional or logistic table, or a table of log a − log x, is to facilitate the calculation of proportions in which the third term is a.

Interpolation Tables.-—All tables of proportional parts may be regarded as interpolation tables. C. Bremiker, Tafel der Proportionalteile (Berlin, 1843), gives proportional parts to hundredths of all numbers from 70 to 699. Schrön, Logarithms, contains an interpolation table giving the first hundred multiples of all numbers from 40 to 410. Sexagesimal tables, already described, are interpolation tables where the denominator is 60 or 600. Tables of the values of binomial theorem coefficients, which are required when second and higher orders of differences are used, are described below. W. S. B. Woolhouse, On Interpolation, Summation, and the Adjustment of Numerical Tables (London, 1865), contains nine pages of interpolation tables. The book consists of papers extracted from vols. 11 and 12 of the Assurance Magazine.

Dual Logarithms.—This term was used by Oliver Byrne in his Dual Arithmetic, Young Dual Arithmetician, Tables of Dual Logarithms, &c. (London, 1863-67). A dual number of the ascending branch is a continued product of powers of 1·1, 1·01, 1·001, &c., taken in order, the owers only being expressed; thus ↓ 6,9,7,8 denotes (1·1)⁶(1·01)⁹(1·001)⁷(1·0001)⁸, the numbers following the ↓, being called dual digits. A dual number which has all but the last digit zeros is called a dual logarithm; the author uses dual logarithms in which there are seven ciphers between the ↓ and the logarithm. Thus since 1·00601502 is equal to ↓ 0,0,0,0,0,0,0,599702 the whole number 599702 is the dual logarithm of the natural number 1·00601502.

A dual number of the descending branch is a continued product of powers of ·9, ·99, &c.: for instance, (·9)³(·99)² is denoted by ’3 ’2 ↑. The Tables, which occupy 112 pages, give dual numbers and logarithms, both of the ascending and descending branches, and the corresponding natural numbers. The author claimed that his tables were superior to those of common logarithms.

Constants.—In nearly all tables of logarithms there is a page devoted to certain frequently used constants and their logarithms, such as π, π⁻¹, π², √π. A specially good collection is printed in W. Templeton’s Millwright’s and Engineer’s Pocket Companion (corrected by S. Maynard, London, 1871), which gives 58 constants involving π and their logarithms, generally to 30 places, and 13 others that may be properly called mathematical. A good list of constants involving π is given in Salomon (1827). A paper by G. Paucker in Grunert’s Archiv (vol. i. p. 9) has a number of constants involving π given to a great many places, and Gauss’s memoir on the lemniscate function (Werke, vol. iii.) has e^−π, e^−1/4π, e^−9/4π, &c., calculated to about 50 places. The quantity π has been worked out to 707 places (Shanks, Proc. Roy. Soc., 21, p. 319).

J. C. Adams has calculated Euler’s constant to 263 places (Proc. Roy. Soc., 27, p. 88) and the modulus ·43429 … to 272 places (Id. 42, p. 22). The latter value is quoted in extenso under Logarithm. J. Burgess on p. 23 of his paper of 1888, referred to under Tables of e^x, has given a number of constants involving π and ρ (the constant ·476936 … occurring in the Theory of Errors), and their Briggian logarithms, to 23 places.

Tables for the Solution of Cubic Equations.—Lambert, Suplementa (1798), gives ±(x − x³) from x = ·001 to 1·155 at intervals of ·001 to 7 places, and Barlow (1814) gives x³ − x from x = 1 to 1·1549 at intervals of ·0001 to 8 places. Very extensive tables for the solution of cubic equations are contained in a memoir “Beiträge zur Auflösung höherer Gleichungen” by J. P. Kulik in the Abh. der k. Böhm. Ges. der Wiss. (Prague, 1860), 11, pp. 1–123. The principal tables (pp. 58–123) give to 7 (or 6) places-the values of ± (x − x³) from x = 0 to x = 3·2800 at intervals of ·001. There are also tables of the even and uneven determinants of cubic equations, &c. Other tables for the solution of equations are by A. S. Guldberg in the Forhand. of the Videns-Selskab of Christiania for 1871 and 1872 (equations of the 3rd and 5th order), by S. Gundellinger, Tafeln zur Berechnung der reellen Wurzeln sämtlicher trinomischen Gleichungen (Leipzig, 1897), which depend on the use of Gaussian logarithms, and by R. Mehme, Schlömilch’s Zeitschrift, 1898, 43, p. 80 (quadratic equations). Binomial Theorem Coefficients.—Tables of the values of

x(x−1)/1.2, x(x−1)(x−2)/1.2.3,…x(x−1) … (x−5)/1.2 … 6

from x=·01 to x=1 at intervals of ·01 to 7 places (which are useful in interpolation by second and higher orders of differences), occur in Schulze (1778), Barlow (1814), Vega (1797 and succeeding editions), Hantschl (1827), and Köhler (1848). W. Rouse, Doctrine of Chances (London, n.d.), gives on a folding sheet (a+b)ⁿ for n = 1, 2,. . . 20.

H. Gyldén (Recueil des Tables, Stockholm, 1880) gives binomial coefficients to n=40 and their logarithms to 7 places. Lambert, Supplementa (1798), has the coefficients of the first 16 terms in (1+x)^1/2 and (1 − x)^1/2, their values being given accurately as decimals.

Vega (1797) has a page of tables giving 1/2.4, 1.3/2.4.6,. . .1/2.3,. . . and

similar quantities to 10 places, with their logarithms to 7 places, and a page of this kind occurs in other collections. Köhler (1848) gives the values of 40 such quantities.

Figurate Numbers.—Denoting n(n + 1) … (n +i − 1)/i! by [n]_i, Lambert, Supplementa, 1798, gives [n]_i.; from n = 1 to n = 30 and from i = 1 to i = 12; and G. W. Hill (Amer. Jour. Math., 1884, 6, p. 130) gives log₁₀[n]_i for n = 1/2, 3/2, 5/2, 7/2, 9/2, and from i = 1 to i = 30.

Trigonometrical Quadratic Surds.—The surd values of the sines of every third degree of the quadrant are given in some tables of logarithms; e.g., in Hutton’s (p. xxxix., ed. 1855), we find sin 3° =1/8{√(5+√5) + √15/2 + √(15 + 3√5) − √3/2 − √1/2 and the numerical values of the surds √(5 + √5), √(15/2), &c., are given to 10 places. These values were extended to 20 places by Peter Gray, Mess. of Math., 1877, 6, p. 105.

Circulating Decimals.—Goodwyn’s tables have been described already. Several others have been published giving the numbers of digits in the periods of the reciprocals of primes: Burckhardt, Tables des diviseurs du premier million (Paris, 1814-1817), gave one for all primes up to 2543 and for 22 primes exceedin that limit. E. Desmarest, Théorie des nombres (Paris, 1852), included all primes up to 10,000. C. G. Reuschle, Mathematische Abhandlung, enthaltend neue zahlentheoretische Tabellen (1856), contains a similar table to 15,000. This W. Shanks extended to 60,000; the portion from 1 to 30,000 is printed in the Proc. Roy. Soc., 22, p. 200, and the remainder is preserved in the archives of the society (Id., 23, p. 260 and 24, p. 392). The number of digits in the decimal period of 1p, is the same as the exponent to which 10 belongs for modulus p, so that, whenever the period has p − 1 digits, 10 is a primitive root of p. Tables of primes having a given number, n, of digits in their periods, i.e. tables of the resolutions of 10ⁿ−1 into factors and, as far as known, into prime factors, have been given by W. Looff (in Grunert’s Archiv, 16, p. 54; reprinted in Nouv. annales, 14, p. 115) and by Shanks (Proc. Roy. Soc., 22, p. 381). The former extends to n = 60 and the latter to n = 100, but there are gaps in both. Reuschle’s tract also contains resolutions of 10ⁿ − 1.

There is a similar table by C. E. Bickmore in Mess. of Math., 1896, 25, p. 43. A full account of all tables connecting n and p where 10ⁿ = 1, mod p, 10ⁿ being the least power for which this congruence holds good, is given by Allan Cunningham (Id., 1904, 33, p. 145). The paper by the same author, "Period-lengths of Circu1ates" (Id. 1900, 29, p. 145) relates to circulators in the scale of radix a. See also tables of the resolutions of aⁿ - 1 into factors under Tables relating to the Theory of Numbers (below). Some further references on circulating decimals are given in Proc. Camb. Phil. Soc., 1878, 3, p. 185.

Pythagorean Triangles.—Right-angled triangles in which the sides and hypotenuse are all rational integers are frequently termed Pythagorean triangles, as, for example, the triangles 3, 4, 5, and 5, 12, 13. Schulze, Sammlung (1778), contains a table of such triangles subject to the condition tan 1/2ω > 1/20(ω being one of the acute angles). About 100 triangles are given, but some occur twice. Large tables of right-angled rational triangles were given by C. A. Bretschneider, in Grunert’s Archiv, 1841, 1, p. 96, and by Sang, Trans. Roy. Soc. Edin., 1864, 33, p. 727. In these tables the triangles are arranged according to hypotenuses and extend to 1201, 1200, 49, and 1105, 1073, 264 respectively. W. A. Whitworth, in a paper read before the Lit. and Phil. Society of Liverpool in 1875, carried his list as far as 2465, 2337, 784. See also H. Rath, "Die rationalen Dreiecke," in Grunert’s Archiv, 1874, 56, p. 188. Sang’s paper also contains a table of triangles having an angle of 120° and their sides integers.

Powers of π.—G. Paucker, in Grunert’s Archiv, p. 10, gives π⁻¹ and π^1/2 to 140 places, and π⁻², π^−1/2, π^1/3 , π^2/3 to about 50 places; J. Burgess (Trans. Roy. Soc. Edin., 1898, 39, II., No. 9, p. 23) gives 1/2π^−1/2, 21/2π−l, and some other constants involving-π as well as their Briggian logarithms to 23 places, and in Maynard’s list of constants (see Constants, above) π² is given to 31 places. The first twelve powers of π and π−1 to 22 or more places were given by Glaisher, in Proc. Lond. Math. Soc., 8, p. 140, and the first hundred multiples of π and π⁻¹ to 12 places by J. P. Kulik, Tafel der Quadrat- und Kubik-Zahlen (Leipzig, 1848).

The Series 1⁻ⁿ + 2⁻ⁿ + 3⁻ⁿ + &c.—Let S_n s_n, σ_n denote respectively the sums of the series 1⁻ⁿ + 2⁻ⁿ + 3⁻ⁿ + &c., 1⁻ⁿ −2⁻ⁿ + 3⁻ⁿ − &c., 1⁻ⁿ + 3⁻ⁿ + 5⁻ⁿ + &c. Legendre (Traité des functions elliptiques, vol. 2, p. 432) computed S_n., to 16 places from n = 1 to 35, and Glaisher (Proc. Lond. Math. Soc., 4, p. 48) deduced s_n and σ_n for the same arguments and to the same number of places. The latter also gave S_n, s_n, σ_n for n = 2, 4, 6, … 12 to 22 or more places (Proc. Lond. Math. Soc., 8, p. 140), and the values of Σ_n, where Σ_n = 2⁻ⁿ + 3⁻ⁿ + 5⁻ⁿ + &c. (prime numbers only involved), for n = 2, 4, 6, … 36 to 15 places (Compte rendu de l’Assoc. Française, 1878, p. 172).

C. W. Merrifield (Proc. Roy. Soc., 1881, 33, p. 4) gave the values of log_e, S_n and Σ_n., for n= 1, 2, 3, …, 35 to 15 places, and Glaisher (Quar. Jour. Math., 1891, 25, p. 347) gave the values of the same quantities for n= 2, 4, 6,. . ., 80 to 24 places (last figure uncertain). Merrifield’s table was reprinted by J. P. Gram on p. 269 of the paper of 1884, referred to under Sine-integral, &c., who also added the values of log₁₀ S_n for the same arguments to 15 places. An error in Σ₃ in Merrifield’s table is pointed out in Quar. Jour. Math., 25, p. 373. This quantity is correctly given in Gram’s reprint. T. J. Stieljes has greatly extended Legendre’s table of S_n. His table (Acta math., 1887, 10, p. 299) gives S_n for all values of n up to n = 70 to 32 places. Except for six errors of a unit in the last figure he found Legendre’s table to be correct. Legendre’s table was reprinted in De Morganfs Diff. and Int. Calc. (1842), p. 554. Various small tables of other series, involving inverse powers of prime numbers, such as 3⁻ⁿ - 5⁻ⁿ + 7⁻ⁿ + 11ⁿ − 13ⁿ + …, are given in vols. 25 and 26 of the Quar. Jour. Math.

Tables of e^x and e^−x, or Hyperbolic Antilogarithms.—The largest tables are the following: C. Gudermann, Theorie der potential- oder cyklisch-hyperbolischen Functionen (Berlin, 1833), which consists of papers reprinted from vols. 8 and 9 of Crelle’s Journal, and gives log₁₀ sinh x, login cosh x, and log₁₀ tanh x from x = 2 to 5 at intervals of ·001 to 9 places and from x = 5 to 12 at intervals of ·01 to 10 places. Since sinh x=1/2(e^x − e^−x) and cosh x = 1/2(e^x + e^−x), the values of and e^−x are deducible at once by addition and subtraction. F. W. Newman, in Camb. Phil. Trans., 13, p. 145, gives values of e^−x from x = 0 to 15·349 at intervals of ·001 to 12 places, from x = 15·350 to 17·298 at intervals of ·002, and from x = 17·300 to 27·635 at intervals of ·005, to 14 places. Glaisher, in Camb. Phil. Trans., 13, p. 243, gives four tables of e^x, e^−x, log₁₀e^x, log₁₀e^−x, their ranges being from x = ·001 to ·1 at intervals of ·001, from ·01 to 2 at intervals of ·01, from ·1 to 10 at intervals of ·1, from 1 to 500 at intervals of unity. Vega, Tabulae (1797 and later ed.), has log₁₀e^x to 7 places and e^x to 7 figures from x = ·01 to 10 at intervals of ·01. Köhler’s Handbuch contains a small table of e^x. In Schulze’s Sammlung (1778) e^x is given for x = 1, 2, 3,… 24 to 28 or 29 figures and for x = 25, 30, and 60 to 32 or 33 figures; this table is reprinted in Glaisher’s paper (loc. cit.). In Salomon’s Tafeln (1827) the values of eⁿ, e^.n, e^·0n, e^·00n, … e^·000000n, where n has the values 1, 2,…9, are given to 12 places. Bretschneider, in Grunert’s Archiv, 3, p. 33, gave e^x and e^−x and also sin x and cos x for x = 1, 2,…10 to 20 places, and J. P. Gram (in his paper of 1884, referred to under Sine-integral, &c.), gives e^x for x = 10, 11,&hellp;20 to 24 places, and from x = 7 to x = 20 at intervals of 0·2 to 10, 13, 14, or 15 places. J. Burgess (Trans. Roy. Soc. Edin., 1888, 39, II. No. 9) has given (p. 26) the values of e^−x and √2/πe^−x for x = 1/2 and for x = 1, 2,&hellip, 10 to 30 places. In the same paper he also gives the values of √2/πe^−x2 from x = 0 to x = 1·250 to 9 places, and from x= 1·25 to x=1·50 at intervals of ·01, and thence at various intervals to x = 6 to 15 places, and the values of log₁₀ √2/πe^−x2 from x = 1 to x = 3 at intervals of ·001 to 16 places.

Factorials.—The values of log₁₀ (n!), where n! denotes 1 . 2 . 3... n, from n= 1 to 1200 to 18 places, are given by C. F. Degen, Tabularum Enneas (Copenhagen, 1824), and reprinted, to 6 places, at the end of De Morgan’s article "Probabilities" in the Encyclopaedia Metropolitana. Shortrede, Tables (1849, vol. i.), gives log (n!) to n= 1000 to 5 places, and for the arguments ending in 0 to 8 places. Degen also gives the complements of the logarithms. The first 20 figures of the values of n×n! and the values of −log (n×n!) to 10 places are given by Glaisher as far as n= 71 in the Phil. Trans. for 1870 (p. 370), and the values of 1/n! to 28 significant figures as far as n=50 in Camb. Phil. Trans., 13, p. 246.

Bernoullian Numbers.—The first fifteen Bernoullian numbers were given by Euler, Inst. Calc. Diff., part ii. ch. v. Sixteen more were calculated by Rothe, and the first thirty-one were published by M. Ohm in Crelle’s Journal, 20, p. 11. J. C. Adams calculated the next thirty-one, and a table of the first sixty-two was published by him in the Brit. Ass. Report for 1877 and in Crelle’s Journal, 85, p. 269. In the Brit. Ass. Report the numbers are given not only as vulgar fractions, but also expressed in integers and circulating decimals. The first nine figures of the values of the first 250 Bernoullian numbers, and their Briggian logarithms to 10 places, have been published by Glaisher, Camb. Phil. Trans., 12, p. 384.

Tables of log tan (1/4π+1/2φ).—C. Gudermann, Theorie der potenzialoder cyklisch-hyperbolischen Functionen (Berlin, 1833), gives (in 100 pages) log tan (1/4π+1/2φ) for every centesimal minute of the quadrant to 7 places. Another table contains the values of this function. also at intervals of a minute, from 88° to··00° (centesimal) to 11 places. A. M. Legendre, Traité des functions elliptiques (vol. ii. p. 256), gives the same function for every half degree (sexagesimal) of the quadrant to 12 places.

The Gamma Function.—Legendre’s great table appeared in vol. ii. of his Exercices de calcul integral (1816), p. 85, and in vol. ii. of his Traité des functions elliptiques (1826), p. 489. Log_ω Γ(x) is given from x＝1 to 2 at intervals of ·001 to 12 places, with differences to the third order. This table is reprinted in full in O. Schlömilch, Analytische Studien (1848), p. 183; an abridgment in which the arguments differ by ·01 is given by De Morgan, Dif. and Int. Calc., p. 587. The last figures of the values omitted are also supplied, so that the full table can be reproduced. A seven-place abridgment (without differences) is published in J. Bertrand, Calcul intégral (1870), p. 285, and a six-figure abridgment in B. Williamson, Integral Calculus (1884), p. 169. In vol. i. of his Exercices (1811), Legendre had previously published a seven-place table of log_ω Γ(x), without differences.

Tables connected with Elliptic Functions.—Legendre published elaborate tables of the elliptic integrals in vol. ii. of his Traité des functions elliptiques (1826). Denoting the modular angle by Θ, the amplitude by φ, the incomplete integral of the first and second kind by Fφ) and F₁φ), and the complete integrals by K and E, the tables are:—(1) log₁₀E and log₁₀K from Θ＝0° to 90° at intervals of 0°·1 to 12 or 14 places, with differences to the third order; (2) E(φ) and F(φ), the modular angle being 45°, from φ＝0° to 90° at intervals of 0°·5 to 12 places, with differences to the fifth order; (3) E₁(45°) and F(45°) from Θ＝0° to 90° at intervals of 1°, with differences to the sixth order, also E and K for the same arguments, all to 12 places; (4) E₁(φ) and F(φ) for every degree of both the amplitude and the argument to 9 or 10 places. The first three tables had been published previously in vol. iii. of the Exercices de calcul intégral (1816).

Tables involving q.-P. F. Verhulst, Traité des fonctions elliptiques (Brussels, 1841), contains a table of log_ω(log₁₀)q⁻¹ for argument Θ at intervals of 0°·1 to 12 or 14 places. C. G. Jacobi, in Crelle’s Journal, 26, p. 93, gives log₁₀ q from Θ＝0° to 90 at intervals of 0°·1 to 5 places. E. D. F. Meissel’s Sammlung mathematischer Tafeln, i. (Iserlohn, 1860), consists of a table of log₁₀q at intervals of 1′ from Θ＝0° to 90° to 8 places. Glaisher, in Month. Nat. R.A.S., 1877, 37, p. 372, gives log₁₀ q to 10 places and q to 9 places for every degree. In J. Bertrand’s Calcul Intégral (1870), a table of log₁₀ q from Θ＝0° to 90° at intervals of 5′ to 5 places is accompanied by tables of log₁₀ √(2K/π) and log₁₀ log₁₀ q⁻¹ and by abridgments of Legendre’s tables of the elliptic integrals. O. Schlömilch, Vorlesungen der höheren Analysis (Brunswick, 1879), p. 448, gives a small table of log₁₀ q for every degree to 5 places.

Legendrian Coefficients (Zonal Harmonies):-The values of Pⁿ(x) for n = 1, 2, 3, … 7 from x = 0 to 1 at intervals of ·01 are given by Glaisher, in Brit. Ass. Rep., 1879, pp. 54-57. The functions tabulated are P¹(x) = x, P²(x) = 1/2(3x² − 1), P³(x) = 1/2(5x² − 3x), P⁴(x) = ⅛(35x⁴ − 30x² + 3), P⁵(x) = ⅛(63x⁵ − 70x³ + 15x), P⁷(x) = 1/16(231x⁶ − 315x⁴ + 105x² − 5), P⁷(x) = 1/14(429x⁷ − 693x⁵ + 315x³ − 35x).

The values of Pⁿ(cos Θ) for n = 1, 2,…7 for Θ = 0°, 1°, 2°,…90° to 4 places are given by J. Perry in the Proc. Phys. Soc., 1892, 11, p. 221, and in the Phil. Mag., 1891, ser. 6, 32, p. 512. The functions Pⁿ occur in connexion with the theory of interpolation, the attraction of spheroids, and other physical theories.

Bessel’s Functions.—F. W. Bessel’s original table appeared at the end of his memoir, “Untersuchung des planetarischen Teils der Störungen, welche aus der Bewegung der Sonne entstehen” (in Abh. d. Berl. Akad. 1824; reprinted in vol. i. of his Abhandlungen, p. 84). It gives J₀(x) and J₁(x) from x = 0 to 3·2 at intervals of ·01. More extensive tables were calculated by P. A. Hansen in “Ermittelung der absolute Störungen in Ellipsen von beliebiger Excentricitat und Neigung” (in Schriften der Sternwarte Seeberg, part i., Gotha, 1843). They include an extension of Bessel’s original table to x＝20, besides smaller tables of J_n(x) for certain values of n as far as n = 28, all to 7 places. Hansen’s table was reproduced by O. Schlömilch, in Zeitschr. für Math., 2, p. 158, and by E. Lommel, Studien uber die Bessel’schen Functionen (Leipzig, 1868), p. 127. Hansen’s notation is slightly different from Bessel’s; the change amounts to halving each argument. Schlömilch gives the table in Hansen’s form; Lommel expresses it in Bessel’s.

Lord Rayleigh’s Theory of Sound (1894), 1, p. 321, gives J₀(x) and J₁(x) from x＝0 to x＝13·4 at intervals of 0·1 to 4 places, taken from Lommel. A large table of the same functions was given by E. D. F. Meissel in the Abh. d. Berlin Akad. for 1888 (published also separately). It contains the values of J₀(x) and J₁(x) from x＝0 to x＝15·50 at intervals of ·01. A. Lodge has calculated the values of the function I_n(x) where

I_n(x) = i⁻ⁿ J_n(ix) = xⁿ/2ⁿn!$\scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}$ 1 + x²/2(2n + 2) + x⁴/2·4.(2n + 2)(2n + 4) + . . . $\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}$

His tables give I_n(x) for n = 0, 1, 2, …, 11 from x = 0 to x = 6 at intervals of O·2 to 11 or 12 places (Brit. Ass. Rep., 1889, p. 29), I₁(x) and I₀(x) from x = 0 to x = 5·100 at intervals of ·001 to 9 places (Id., 1893, p. 229, and 1896, p. 99), and of J₀(x√i) from x = 0 to x = 6 at intervals of 0·2 (Id., 1893, p. 228) to 9 places. In all the tables the last figure is uncertain. Subsidiary tables for the calculation of Bessel’s functions are given by L. N. G. Filon and A. Lodge in Brit. Ass. Rep., 1907, p. 94. The work is being continued, the object being to obtain the values of J_n(x) for n = 0, 1/2, 1, 11/2, …, 61/2. A table by E. Jahnke has been announced, which, besides tables of other mathematical functions, is to contain values of Bessel’s functions of order ⅓ and roots of functions derived from Bessel’s functions.

Sine, Cosine, Exponential, and Logarithm Integrals.—The functions so named are the integrals $\int _{0}^{x}{\frac {sinx}{x}}\,dx$, $\int _{\infty }^{x}{\frac {cosx}{x}}\,dx$, $\int _{-\infty }^{x}{\frac {e^{x}}{x}}\,dx$, $\int _{0}^{x}{\frac {\,dx}{\log x'}}$ which are denoted by the functional signs Si x, Ci x, Ei x, li x respectively, so that Ei x =li e^x. J. von Soldner, Théorie et tables d’une nouvelle fonction transcendante (Munich, 1809), gave the values of li x from x = 0 to 1 at intervals of ·1 to 7 places, and thence at various intervals to 1220 to 5 or more places. This table is reprinted in De Morgan’s Diff. and Int. Calc., p. 662. Bretschneider, in Grunert’s Archiv, 3, p. 33, calculated Ei (≐ x), Si x, Ci x for x = 1, 2, … 10 to 20 places, and subsequently (in Schlömilch’s Zeitschrift, 6) worked out the values of the same functions from x＝0 to 1 at intervals of ·01 and from 1 to 7·5 at intervals of ·1 to 10 places. Two tracts by L. Stenberg, Tabulae logarithm integralis (Malmö, part i. 1861 and part ii. 1867), give the values of li 10^x from x = −15 to 3·5 at intervals of ·01 to 18 places. Glaisher, in Phil. Trans., 1870, p. 367, gives Ei (≐ x), Si x, Ci x from x = 0 to 1 at intervals of ·01 to 18 places, from x = 1 to 5 at intervals of ·1 and thence to 15 at intervals of unity, and for x = 20 to 11 places, besides seven-place tables of Si x and Ci x and tables of their maximum and minimum values. See also Bellavitis, “Tavole numeriche logaritmo-integral” (a paper in. Memoirs of the Venetian Institute, 1874). F. W. Bessel calculated the values of li 1000, li 10,000, li 100,000, li 200,000, … li 600,000, and li 1,000,000 (see Abhandlungen, 2, p. 339). 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.).

Values of $\int _{0}^{x}e^{-x^{2}}\,dx$ and $e^{x^{2}}\int _{0}^{x}e^{-x^{2}}\,dx$—These functions are employed in researches connected with refractions, theory of errors conduction of heat, &c. Let $\int _{0}^{x}e^{-x^{2}}\,dx$ and $\int _{x}^{\infty }e^{-x^{2}}\,dx$ be denoted by erf x and erfc x respectively, standing for “error function” and “error function complement,” so that erf x + erfc x = 1/2√π (Phil. Mag., Dec. 1871; it has since been found convenient to transpose as above the definitions there given of erf and erfc). The tables of the functions, and of the functions multiplied by e^x2, are as follows. C. Kramp, Analyse des Réfractions (Strasbourg, 1798), has erfc x from x = 0 to 3 at intervals of ·01 to 8 or more places, also log₁₀ (erfc x) and log₁₀ (e^x2 erfc x) for the same values to 7 places. F. W. Bessel, Fundamenta astronomiae (Königsberg, 1818), has log₁₀ (e^x2erfc x) from x = 0 to 1 at intervals of ·01 to 7 places, likewise for argument ₁₀ x, the arguments increasing from 0 to 1 at intervals of ·01. A. M. Legendre, Traité des functions elliptiques (1826), 2, p. 520, contains Γ(1/2, e − x²), that is, 2 erfc x from x = 0 to ·5 at intervals of ·01 to 10 places. J. F. Encke, Berliner ast. Jahrbuch for 1834, gives 2/√π erf x from x = 0 to 2 at intervals of ·01 to 7 places and 2/√π erf (ρx) from x = 0 to 3·4 at intervals of ·01 and thence to 5 at intervals of ·1 to 5 places, ρ being ·4769360. Glaisher, in Phil. Mag., December 1871, gives erfc x from x = 3 to 4·5 at intervals of ·01 to 11, 13, or 14 places. Encke’s tables and two of Kramp’s were reprinted in the Encyclopaedia Metropolitana, art. “Probabilities.” These tables have also been reprinted in many foreign works on probabilities, errors of observations, &c. In vol. 2 (1880) of his Lehrbuch zur Bahnbestimmung der Kometen und Planeten T. R. v. Oppolzer gives (p. 587) a table of erf x from x = 0 to 4·52 at intervals of ·01 to 10 places, and (p. 603) a table of 2/√π erf x from x = 0 to 2 at intervals of ·01 to 5 places. Both tables were the result of original calculations. A very large table of log₁₀ (e^x2 erfc x) was calculated by R. Radau and published in the Annales de l’observatoire de Paris (Mémoires, 1888, 18, B. 1-25) It contains the values of log₁₀ (e^x2 erfc x) from x= −0·120 to 1·000 at intervals of ·001 to 7 places, with differences. A; Markolf in a separate publication, Table des valeurs de l’intégrale $\int _{x}^{\infty }e^{-t^{2}}\,dt$ (St Petersburg, 1888), gives erfc x from x = 0 to 3 at intervals of ·001 and from x = 3 to 4·80 at intervals of ·001, with first, second, and third differences to 11 places. He also gives a table of 2/√π erf x from x = 0 to x = 2·499 at intervals of ·001 and thence to 3·79 at intervals of ·01, J. Burgess, Trans. Roy. Soc. Edin., 1888, 39, II., No. 9, published very extensive tables of 2/√π erf x, which were entirely the result of a new calculation. His tables give the values of this function from x = 0 to 1·250 at intervals of ·001 to 9 places with first and second differences, from x = 1 to 3 at intervals of ·001 to 15 places with differences to the fourth order, and from x = 3 to 5 at intervals of ·1 to 15 places. He also gives erfc x from x = 0 to x = 5 at intervals of ·1 to 15 places. B. Kämpfe in Wundt’s Phil. Stud., 1893, p. 147, gives ${\frac {2}{\sqrt {\pi }}}\mathrm {erf} \ x$ from x = 0 to x = 1·509 at intervals of ·001, and from x = 1·50 to x = 2·88 at intervals of ·01 to 4 places. G. T. Fechner’s Elemente der Psychophysik (Leipzig, 1860) contains (pp. 108, 110) some small four-place tables connecting r/n (as argument) and hD where ${\frac {r}{n}}={\tfrac {1}{2}}+{\frac {1}{\sqrt {\pi }}}{\text{erf}}{\frac {hD}{2}}$. A more detailed account of tables of erf x, e^x2 erf x, &c., is given in Mess. of Math., 1908, 38, p. 117.

Values of $\int _{0}^{x}e^{x^{2}}dx$.—The values of this integral have been calculated by H. G. Dawson from x = 0 to x = 2 to 7 places (last figure uncertain). The table is published in the Proc. Lond. Math. Soc., 1898, 29, p. 521.

Tables of Integrals, not Numerical.—Meyer Hirsch, Integraltafeln (1810; Eng. trans., 1823), and Minding, Integraltafeln (Berlin, 1849), give values of indefinite integrals and formulae of reduction; both are useful and valuable works. De Haan, Nouvelles tables d'intégrales définies (Leyden, 1867), is a quarto volume of 727 pages containing evaluations of definite integrals, arranged in 485 tables. The first edition appeared in vol. 4 of the Transactions of the Amsterdam Academy of Sciences. This edition, though not so full and accurate as the second, gives references to the original memoirs in which the different integrals are considered. B. O. Peirce’s A Short Table of Integrals (Boston, U.S.A., 1899) contains integrals, formulae, expansions, &c., as well as some four-place numerical tables, including those of hyperbolic sines and cosines and their logarithms.

Tables relating to the Theory of Numbers.—These are of so technical a character and so numerous that a comprehensive account cannot be attempted here. The reader is referred to Cayley’s report in the Brit. Ass. Rep. for 1875, p. 305. where a full description with references is given. Three tables published before that date may, however, be briefly noticed on account of their importance and because they form separate volumes: (1) C. F. Degen, Canon Pellianus (Copenhagen, 1817), relates to the indeterminate equation $y^{2}-ax^{2}=1$ for values of a from 1 to 1000. It in fact gives the expression for ${\sqrt {a}}$ as a continued fraction; (2) C. G. J. Jacobi, Canon arithmeticus (Berlin, 1839), is a quarto work containing 240 pages of tables, where we find for each prime up to 1000 the numbers corresponding to given indices and the indices corresponding to given numbers, a certain primitive root (10 is taken whenever it is a primitive root) of the prime being selected as base; (3) C. G. Reuschle, Tafeln complexer Primzahlen, welche aus Wurzeln der Einheit gebildet sind (Berlin, 1875), includes an enormous mass of results relating to the higher complex theories.

Passing now to tables published since the date of Cayley’s report, the two most important works are (1) Col. Allan Cunningham’s Binary Canon, (London, 1900), a quarto volume similar in construction, arrangement, purpose, and extent to Jacobi’s Canon arithmeticus, but differing from it in using the base 2 throughout, i.e. in Jacobi’s Canon the base of each table is always a primitive root of the modulus, while in Cunningham’s it is always 2. The latter tables in fact give the residues R of 22 (where x= 0, 1, 2, …) for every prime p or power of a prime, $p^{x}$, up to 1000, and also the indices x of $2^{x}$, which yield the residues R to the same moduli. This work contains a list of errors found in the Canon arithmeticus. (2) The same author’s Quadratic Partitions (London, 1904). These tables give for every prime p up to 100,000 the values of a, b; c, d; A, B; and L, M where $p=a^{2}+b^{2}=c^{2}+2d^{2}=A^{2}+3B^{2}={\frac {1}{4}}(L^{2}+27M^{2})$. They also give e, f where $p=e^{2}-2f^{2}$ up to 25,000 and resolutions of p into the forms $x^{2}-5y^{2}$, ${\frac {1}{4}}(X^{2}-5Y^{2})$, $t^{2}+7u^{2}$, ${\frac {1}{4}}(v^{2}+11w^{2})$, $A'^{2}-3B'^{2}$, $x'^{2}+5y'^{2}$, $G^{2}+6H^{2}$, $G'^{2}-6H'^{2}$, $t'^{2}-7u'^{2}$, $\xi ^{2}+10\eta ^{2}$, $\xi '^{2}-10\eta '^{2}$, $v'^{2}-11w'^{2}$ up to 10,000; as well as the least solutions of $\tau ^{2}-D\nu ^{2}=\pm 1$ up to D = 100 and least solutions of other similar equations. A complete list of errata in the previous partition tables of Jacobi, Reuschle, Lloyd Tanner, and in this table is given by Allan Cunningham in Mess. of Math., 1904, 34, p. 132. The resolution of $a^{n}-1$ into its numerical factors is treated in detail by C. E. Bickmore in Mess. of Math., 1896, 25, p. 1, and 1897, 26, p. 1. On p. 43 of the former volume he gives a table of the known factors of $a^{n}-1$ for a = 2, 3, 5, 6, 7, 10, 11, 12 and from n = 1 to n = 50. Other papers on the same subject contained in the same periodical are by Allan Cunningham, 1900, 29, p. 145; 1904, 33, p. 95; and F. B. Escott, ibid., p. 49. These papers contain references to other writings. Tables of the resolutions of $10^{n}-1$ are referred to separately in this article under Circulating Decimals. If $a^{x}$ is the smallest power of a for which the congruence $a^{x}\equiv 1$ 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 $x-p-1$, 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; and in Quar. Jour. Math., 1906, 37, p. 122, he gives a table of Haupt-exponents of 2 for all primes up to 10,000. In Acta Math. (1893, 17, p. 315; 1897, 20, p. 153; 1899, 22, p. 200) G. Wertheim has given the least primitive root of primes up to 5000 . The following papers contain lists of high primes or factorizations of high numbers: Allan Cunningham, Mess. of Math., 1906, 35, p. 166 (Pellian factorizations); 1907, 36, p. 145 (Quartan factorizations); 1908, 37, p. 65 (Trinomial binary factorizations); 1909, 38, pp. 81, 145 (Diophantive factorization of quartans); 1910, 39, pp. 33, 97; 1911, 40, p. 1 (Sextan factorizations); 1902, 31, p. 165; 1905, 34, p. 72 (High primes). The last three are joint papers by Cunningham and H. J. Woodall. Tables relating to the distribution of primes are contained in the introduction to the Sixth Million (see under Factor Tables), in J. P. Gram’s paper on the number of primes inferior to a given limit in the Vidensk. Selsk. Skr., 1884, II. 6, Copenhagen, and in Mess. of Math., 1902, 31, p. 172. A table of $\chi (n)$, the sum of the complex numbers having n for norm, for primes and powers of primes up to n = 13,000 by Glaisher, was published in Quar. Jour. Math., 1885, 20, p. 152, and a seven-place table of $f(x)$ and $\log _{10}f(x)$, where $f(x)$ denotes ${\tfrac {1}{2}}\cdot {\tfrac {2}{3}}\cdot {\tfrac {4}{5}}\cdots {\frac {x-1}{x}}$; the denominators being the series of prime numbers up to 10,000, in Mess. of Math., 1899, 28, p. 1.

Bibliography.—Bibliographical and historical information relating to tables is collected in Brit. Ass. Rep. for 1873, p. 6. The principal works are:—J. C. Heilbronner, Historia Matheseos (Leipzig, 1742), the arithmetical portion being at the end; J. E. Scheibel, Einleitung zur mathematischen Bücherkenntniss (Breslau, 1771–84); A. G. Kästner, Geschichte der Mathematik (Gottingen, 1796–1800), vol. iii.; F. G. A. Murhard, Bibliotheca Mathematica (Leipzig, 1797–1804), vol. ii.; J. Rogg, Bibliotheca Mathematica (Tübingen, 1830), and continuation from 1830 to 1854 by L. A. Sohnke (Leipzig and London, 1854); J. de Lalande, Bibliographie astronomique (Paris, 1803), a separate index on p. 960. A great deal of information upon early tables is given by J. B. J. Delambre, Histoire de l’astronomie moderne (Paris, 1821), vol. i.; and in Nos. xix. and xx. of C. Hutton'’s Mathematical Tracts (1812). For lists of logarithmic tables of all kinds see De Haan, Verslagen en Mededeelingen of the Amsterdam Academy of Sciences (Abt. Natuurkunde) 1862, xiv. 15, and Verhandelingen of the same academy, 1875, xv. separately paged.

De Morgan’s article “Tables,” which appeared first in the Penny Cyclopaedia, and afterwards with additions in the English Cyclopaedia, gives not only a good deal of bibliographical information, but also an account of tables relating to life assurance and annuities, astronomical tables, commercial tables, &c.

Reference should also be made to R. Mehmke’s valuable article “Numerisches Rechnen” in vol. i. pt. ii. pp. 941–1079 of the Encyk. der math. Wiss. (Leipzig, 1900–4), which besides tables includes calculating machines, graphical methods, &c.  (J. W. L. G.) 

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1. Referring to factor tables, J. H. Lambert wrote (Supplementa tabularum, 1798, p. xv.): “Universalis finis talium tabular um est ut semel pro semper computetur quod saepius de novo computandum foret, et ut pro omni casu computetur quod in futurum pro quovis casu computatum desiderabitur.” This applies to all tables.
2. For information about it, see a paper on “Factor Tables,” in Camb. Phil. Proc. (1878), iii. 99-138, or the Introduction to the Fourth Million.
3. Only one other multiplication table of the same extent as Crelle’s had appeared previously, viz. Herwart von Hohenburg’s Tabulae arithmeticae προσθαφαιρἐσεως universales (Munich, 1610), a huge folio volume of more than a thousand pages. It appears from a correspondence between Kepler and von Hohenburg, which took place at the end of 1608, that the latter used his table when in manuscript for the performance of multiplications in eneral, and that the occurrence of the word prosthaphaeresis on the title is due to Kepler, who pointed out that by means of the table spherical triangles could be solved more easily than by Wittich’s prosthaphaeresis. The invention of logarithms four years later afforded another means of performing multiplications, and von Hohenburg’s work never became generally known. On the method of prosthaphaeresis, see Napier, John, and on von Hohenburg’s table, see a paper “On multiplication by a Table of Single Entry,” Phil. Mag., 1878, ser. v., 6, p. 331.
4. The actual place of publication (with a German title, &c.) is Vienna. The copies with an English title, &c., were issued by Trübner; and those with a French title, &c., by Gauthier-Villars. All bear the date 1888.
5. See his “Einige Bemerkungen zu Vega’s Thesaurus logarithmorum,” in Astronomische Nachrichten for 1851 (reprinted in his Werke, vol. iii. pp. 257-64); also Monthly Notices R.A.S., 33, p. 440.
6. A seven-figure table of the same kind is contained in S. Gundelfinger’s Sechsstellige Gaussische und siebenstellige gemeinc Logarithmen (Leipzig, 1902).
7. Legendre (Traité des fonctions elliptiques, vol. ii., 1826) gives a table of natural sines to 15 places, and of log sines to 14 places, for every 15″ of the quadrant, and also a table of logarithms of uneven numbers from 1163 to 1501, and of primes from 1501 to 10,000 to 19 places. The latter, which was extracted from the Tables du cadastre, is a continuation of a table in W. Gardiner’s Tables of Logarithms (London, 1742; reprinted at Avignon, 1770), which gives logarithms of all numbers to 1000, and of uneven numbers from 1000 to 1143. Legendre’s tables also appeared in his Exercices de calcul intégral, vol. iii. (1816).
8. Leonelli’s original work of 1803, which is extremely scarce, was reprinted by J. Houël at Paris in 1875.