# Encyclopædia Britannica, Ninth Edition/Mathematical Tables

TABLES, 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 ${\displaystyle xy}$ as tabular result for ${\displaystyle x}$ and ${\displaystyle y}$ as arguments. Value of tables.The intrinsic value of a table may be estimated by the actual amount of time saved by consulting it; for example, a table of square roots to ten decimals is more valuable than a table of squares, as the extraction of the root would occupy more time than the multiplication of the number by itself. 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 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 beginning from two, every third beginning from three, 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: in fact, even when the tabular results are so easy to calculate that no time or mental effort would be saved by the use of a table, the certainty of accuracy might make it advantageous to employ it.

The invention of logarithms in 1614, followed immediately by the calculation of logarithmic tables, revolutionized all the methods of calculation; and the original work performed by Briggs and Vlacq in calculating logarithms 260 years ago has in effect formed a portion of every arithmetical operation that has since been carried out by means of logarithms. And not only has an incredible amount of labour been saved[1] but a vast number of calculations and researches have been rendered practicable which otherwise would have been quite beyond human reach. The 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. Tables are, as a rule, intended for professional and not amateur use; and it is of little moment whether the user who is unfamiliar with a table has to spend ten seconds or a minute in obtaining an isolated result, provided it can be used rapidly and without risk of error by a skilled computer.

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 of merely bibliographical or historical interest are not regarded as coming within the scope of this article, although for special reasons such tables are briefly noticed in some cases. Tables relating to ordinary arithmetical operations are first described, and afterwards an account is given of the most useful and least technical of the more strictly mathematical tables, such as factorials, gamma functions, integrals, Bessel's functions, &c. It is difficult to classify the tables described in a perfectly satisfactory manner without prolixity, as many collections contain valuable sets belonging to a variety of classes. 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; and to this the reader is referred.

Divisors and primes.Tables of Divisors (Factor Tables) and Tables of Primes.—The existing factor tables extend to 9,000,000. In 1811 Chernac published at Deventer his Cribrum Arithmeticum, which gives all the prime divisors of every number not divisible by 2, 3, or 5 up to 1,020,000. In 1814-17 Burckhardt published at Paris his Tables des Diviseurs, giving the least divisor of every number not divisible by 2, 3, or 5 up to 3,036,000. The second million was issued in 1814, the third in 1816, and the first in 1817. The corresponding tables for the seventh (in 1862), eighth (1863), and ninth (1865) millions were calculated by Dase and issued at Hamburg. Dase died suddenly during the progress of the work, and it was completed by 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, not having been published by 1877, 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. It is in the possession of the Berlin Academy, having been presented in 1878. The nine quarto volumes (Tables des Diviseurs, Paris, 1814–17; Factor Tables, London, 1879–83; Factoren-Tafeln, Hamburg, 1862–65) 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 with the second million instead of the first, as Chernac’s factor table for the first million was already in existence. Burckhard's first million does not supersede Chernac’s, as the latter gives all the prime divisors of numbers not divisible by 2, 3, or 6 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 the first million. 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. The factor tables which have been described greatly exceed in both extent and accuracy any others of the same kind, the largest of which only reaches 408,000. This is the limit of Felkel’s Tafel aller einfachen Factoren (Vienna, 1776), a remarable and extremely rare book, [2] nearly all the copies having been destroyed. Vega (Tabulæ, 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 Hülsse’s edition (1840). These are the largest and most convenient tables after those of Chernac. Salomon (1827) gives a factor table to 102,011, Köhler (Handbuch, 1848) all divisors up to 21,524, and Houël (Tables de Logarithmes, 1871) least divisors up to 10,841. Barlow ('Tables', 1814) gives the complete resolution of every number up to 10,000 into its factors; for example, corresponding to 4932 we have given ${\displaystyle 2^{2}.3^{2}.137}$. This table is unique so far as we know. The work also contains a list of primes up to 100,103. Both these tables are omitted in the stereotyped reprint of 1840. In Rees's Cyclopædia (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. 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.

Multiplication.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 Crelle's Rechentafeln (stereotyped, Bremiker’s edition, 1864), 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. Only one other multiplication table of the same extent has appeared, viz., Herwart von Hohenburg’s Tabulæ Arithmeticæ Προσθαφαιρἐσεως Universales[3] (Munich, 1610), on which see Napier, vol. xvii. p. 183. The invention of logarithms four years later afforded another means of performing multiplications, and Von Hohenburg’s work never became generally known. The three following tables are for the multiplication of a number by a single digit. (1) Crelle, Erleichterungs-Tafel 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) Bretschneider, Produktentafel (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 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. Among earlier works may be noticed Gruson, 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 we believe 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.Quarter-squares.—Multiplication may he performed by means of a table of single entry in the manner indicated by the formula—

${\displaystyle ab={\frac {1}{4}}(a+b)-{\frac {1}{4}}(a-b)^{2}.}$

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 Laundy, Table of Quarter-Squares of all Numbers up to 100,000 (London, 1856). Smaller works are Centnerschwer, Neuerfundene Multiplications- und Quadrat-Tafeln (Berlin, 1825), which extends to 20,000, and Merpaut, Tables Arithmonomiques (Vannes, 1832), which extends to 40,000. In Merpaut’s work the quarter-square is termed the "arithmone." 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 Voisin, Tables des Multiplications, ou logarithmes des nombres entiers depuis 1 jusqu'à 20,000 (Paris, 1817). By a logarithm Voison means a quarter-square, i.e., he calls ${\displaystyle a}$ a root and ${\displaystyle {\frac {1}{4}}a^{2}}$ its logarithm. On the subject of quarter-squares, &c., see the paper (already referred to) in Phil. Mag., November 1878.

Squares, Cubes, &c.Squares, Cubes, &c.—The most convenient table for general use, as well as the most extensive, is 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. The largest table of squares and cubes is Kulik, Tafeln der Quadrat- und Kubik-Zahlen (Leipsic, 1848), which gives both as far as 100,000. Two early tables also give squares as far as 100,000, viz., Maginus, Tabula Tetragonica (Venice, 1592), and Ludolf, Tetragonometria Tabularia (Amsterdam, 1690). 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. Barlow, Mathematical Tables (original edition, London, 1814), gives the first ten powers of the first hundred numbers. The first nine or ten powers are given in Vega, Tabulæ (1797), and in Hülsse’s edition of the same (1840), in Köhler, Handbuch (1848), and in other collections. Faà de Bruno, Calcul des Erreurs (Paris, 1869), and Müller, Vierstellige Logarithmen (1844), give squares for use in connexion with the method of least squares. Small tables occur frequently in books intended for engineers and practical men. Drack (Messenger of Math., vol. vii., 1878, p. 87) has given to 33 places the cube roots (and the cube roots of the squares) of primes up to 127. Small tables of powers of 2, 3, 5, 7 occur in various collections. In Vega’s Tabulæ (1797, and the subsequent editions, including Hülsse’s) the powers of 2, 3, 5 as far as the 45th, 36th, and 27th respectively are given; they also occur in Köhler’s Handbuch (1848). The first 25 powers of 2, 3, 5, 7 are given in Salomon, Logarithmische Tafeln (1827). Shanks, Rectification of the Circle (1853), gives powers of 2 up to 2721.

Triangular Numbers.Triangular Numbers.—E. de Joncourt, De Natura et Præclaro Usu Simplicissimæ Speciei Numerorum Trigonalium (The Hague, 1762), contains a table of triangular numbers up to 20,000: viz., ${\displaystyle {\frac {1}{2}}n(n+1)}$ is given for all numbers from n ${\displaystyle =}$ 1 to 20,000. The table occupies 224 pages.

Reciprocals.Reciprocals.—Barlow's Tables give reciprocals up to 10,000 to 9 or 10 places; and they have been carried to ten times this extent by Oakes, Table of the Reciprocals of Numbers from 1 to 100,000 (London, 1865). This gives seven figures of the reciprocal, and is arranged like a table of seven-figure logarithms, differences being added at the side of the page. The reciprocal of a number of five figures is therefore taken out at once, and two more figures may be interpolated for as in logarithms. Picarte, La Division réduite à une Addition (Paris, 1861), gives to ten significant figures the reciprocals of the numbers from 10,000 to 100,000, and also the first nine multiples of these reciprocals. Small tables of reciprocals are not common.

Vulgar Fractions as Decimals.Tables for the Expression of Vulgar Fractions as Decimals.—Tables of this kind have been given by Wucherer, Goodwyn, and Gauss. Wucherer, Beyträge zum allgemeinern Gebrauch der Decimalbrüche (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), running to 153 pages, gives to 8 places the decimal corresponding to every vulgar fraction less than 99991 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 (${\displaystyle =}$11000) to ·09989909 (${\displaystyle =}$99991). 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 ${\displaystyle \cdot 07692{\dot {3}}}$ and ${\displaystyle \cdot 15384{\dot {6}}}$, which are the only periods in which the fraction ${\displaystyle {\frac {x}{13}}}$ 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 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., vol. iii. (1878), pp. 185–206, where is also given a table of the periods of fractions corresponding to denominators prime to 10 from 1 to 1024 obtained by counting from Goodwyn’s table. See also the section on "Circulating Decimals," p. 13 below.

Sexagesimal and Sexcentenary.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 more fully described as giving the value of ${\displaystyle {\frac {xy}{600}}}$ correct to tenths of a second, ${\displaystyle x}$ and ${\displaystyle y}$ 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 those 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.

Trigonometrical.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 semi-quadrantal 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 Blunedvile (1594), although a table of sines had appeared four years earlier. Regiomontanus called his table of tangents (or rather cotangents) tabula fœcunda on account of its great use; and till the introduction of the word "tangent" by Finck (Geometriæ Rotundi Libri XIV., Basel, 1583) a table of tangents was called a tabula fœcunda or canon fœcundus. Besides "tangent," Finck also introduced the word "secant," the table of secants having previously been called tabula benefica by Maurolycus (1558) and tabula fœcundissima by Vieta.

A history of trigonometrical tables by Hutton was prefixed to all the early editions of his Tables of Logarithms, and forms Tract xix. of his Mathematical Tracts, vol. i. pp. 278–306, 1812. A good deal of bibliographical information about the Opus Palatinum and earlier trigonometrical tables is given in De Morgan’s article "Tables" in the English Cyclopædia. 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 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 the arithmometer of Thomas de Colmar has revived the use of tables of natural trigonometrical functions, it being found convenient for some purposes to employ an arithmometer and a natural canon instead of a logarithmic canon. Junge’s Tafel der wirklichen Länge der Sinus und Cosinus (Leipsic, 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. Pasquich, Tabulæ Logarithmico-Trigonometricæ (Leipsic, 1817), contains a table of ${\displaystyle sin^{2}x,\ cos^{2}x,\ tan^{2}x,\ cot^{2}x}$ from x${\displaystyle =}$1° to 45° at intervals of 1' to 5 places. Andrew, Astronomical and Nautical Tables (London, 1805), contains a table of "squares of natural semichords," i.e. ${\displaystyle sin^{2}{\frac {1}{2}}x}$ from x${\displaystyle =}$0° to 120° at intervals of 10" to 7 places. This table has recently been 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," now frequently used in works upon navigation, is an abbreviation of "half versed sine"; viz., the haversine of x is equal to ${\displaystyle {\frac {1}{2}}(1-cosx)}$, that is, to ${\displaystyle sin^{2}{\frac {1}{2}}x}$. The table gives logarithmic haversines for every 15ʺ from 0° to 180°, and natural haversines for every 10ʺ from 0° to 180°, to 7 places, except near the beginning, where the logarithms are given to only 5 or 6 places. The work itself occupies 327 folio pages, and was suggested by Andrew’s, 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 many 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 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 10ʺ to 7 places, with proportional parts, and log versed sines from 0° to 135° at intervals of 15ʺ 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.Common or Briggian Logarithms of Numbers and Trigonometrical Ratios.— For an account of the invention and history of logarithms, see Logarithms (vol. xiv. p. 773) and Napier. 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 Leipsic Vlacq’s two works in a single folio volume, Thesaurus Logarithmorum Completus. The arrangement of the table of logarithms of numbers is more compendious than in Vlacq, being similar to that of an ordinary seven-figure table, but it is not so convenient, as mistakes in taking out the differences are more liable to occur. The trigonometrical canon gives log sines, cosines, tangents, and cotangents, from 0° to 2° at intervals of one second, to 10 places, without differences, and for the rest of the quadrant at intervals of ten seconds. The trigonometrical canon is not wholly reprinted from the Trigonometria Artificialis, 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 Vlacq’s logarithms of numbers, and has given several important errata lists. M. 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 Logarithms, p. 776; comp, also Monthly Notices of Roy. Ast. Soc., for May 1872, June 1872, March 1873, and 1874, suppl. number). Vega seems not to have bestowed on the trigonometrical canon anything like the care that he devoted to the logarithms of numbers, as Gauss[4] estimates the total 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. As these errors in the Trigonometria Artificialis still remain uncorrected, it cannot be said that a reliable ten-place logarithmic trigonometrical canon exists. The calculator who has occasion to perform work requiring ten-figure logarithms of numbers should use Vlacq’s Arithmetica Logarithmica of 1628, after carefully correcting the errors pointed out by Vega and Lefort. After Vlacq, Vega’s Thesaurus is the next best table; and Pineto’s Tables de Logarithmes Vulgaires à Dix Decimales, construites d'après 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, ${\displaystyle log{\frac {1}{M}}}$ must be added; this quantity is therefore given in an adjoining column to ${\displaystyle M}$ in the auxiliary table. A similar procedure gives the number answering to any logarithm, another factor (approximately the reciprocal of ${\displaystyle M}$) being given, so that in both cases multiplication is used. The laborious part of the work is the multiplication, by ${\displaystyle M}$; 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 MM. Namur and 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.

If seven-figure logarithms do not give sufficiently accurate results, it is usual to have recourse to ten-figure tables; with one exception, there exist no tables giving 8 or 9 figures. The exception is John Newton's Trigonometria Britannica (London, 1658), which gives logarithms of numbers to 100,000 to 8 places, and also log sines and tangents for every centesimal minute (i.e., the nine-thousandth part of a right angle), and also log sines and tangents for the first three degrees of the quadrant to 5 places, the interval being the one-thousandth part of a degree. This table is also unique in that it gives 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. Babbage, Table of the Logarithms of the Natural Numbers from 1 to 108,000 (London, stereotyped in 1827; there are several tirages of later dates), is the best for ordinary use. 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 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. The book is not very easy to procure now. In 1871 Mr 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. This work was to a great extent the result of an original calculation; see Edinburgh Transactions, vol. xxvi. (1871). Mr Sang proposed to publish a nine-figure table from 1 to 1,000,000, but the requisite support was not obtained. Various papers of Mr Sang’s relating to his logarithmic calculations will be found in the Edinburgh Proceedings subsequent to 1872. In this connexion reference should he 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. Pankhurst, Astronomical Tables (New York, 1871), gives logarithms of numbers from 1 to 109 to 102 places. [5]

In many seven-figure tables of logarithms of numbers the values of ${\displaystyle S}$ and ${\displaystyle T}$ are given at the top of the page, with ${\displaystyle V}$, the variation of each, for the purpose of deducing log sines and tangents. ${\displaystyle S}$ and ${\displaystyle T}$ denote ${\displaystyle \log {\frac {sinx}{x}}}$ and ${\displaystyle \log {\frac {\tan x}{x}}}$ respectively, the arguments 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, ${\displaystyle S=\log {\frac {\sin 6720''}{6720}}}$ and ${\displaystyle T=\log {\frac {\tan 6720''}{6720}}}$, while the ${\displaystyle V}$'s are the variation of each for 10ʺ. To find, for example, log 1° 52ʹ 12ʺ·7 or log sin 6732ʺ·7 ${\displaystyle =}$ 3.8281893, whence, by addition, we obtain 8·5136873; but ${\displaystyle V}$ for 10″ is - 2·29, whence the variation for 12ʺ·7 is - 3, and the log sine required is 8·5136870. Tables of ${\displaystyle S}$ and ${\displaystyle T}$ 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 irregularity in the lines are very awkward, Babbage printed the fourth figure in small type after a change. 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 nokta (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. 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, is not so 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 of Roy. Ast Soc. for 1873. See also the Monthly Notices for 1874, p. 248; and a paper by Gernerth, Ztsch. f. d. österr. Gymn., 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 by Michael Taylor, Tables of Logarithms (London, 1792), which gives 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 very general use. Computers have always preferred Bagay’s Nouvelles Tables Astronomiques et Hydrographiques (Paris, 1829), which also contains a complete logarithmic canon to every second. The change in the column is very clearly marked by a largo black nucleus, surrounded by a circle, printed instead of zero. Bagay’s work has now become very rare. The only other canon to every second that has been published is contained in 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 trigonometricial 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.

Of tables in which the quadrant is divided centesimally, the principal are Hobort and Ideler, Nouvelles Tables Trigonométriques (Berlin, 1799), and Borda and 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 10ʺ and thence to 50° at intervals of 1ʹ (centesimal), and also natural sines and tangents for the first hundred ten-thousandths of a right angle to 10 places. The latter gives log sines, cosines, tangents, cotangents, secants, and cosecants from 0° to 3° at intervals of 10ʺ (with full proportional parts for every second), and thence to 50° at intervals of 1ʹ (centesimal) to 7 places. There is also a table of log sines, cosines, tangents, and cotangents from 0ʹ to 10ʹ at intervals of 10ʺ and from 0° to 50° at intervals of 10ʹ (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.

In Briggs’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 effecting such a change was when the natural canon was replaced by the logarithmic canon, and Briggs took advantage of this opportunity. He left the degree unaltered, but divided it centesimally instead of sexagesimally, thus ensurng the advantages of decimal division (a saving of work in interpolations, multiplications, &c.) with the minimum of change. The French mathematicians at the end of the 18th century divided the right angle centesimally, completely changing the whole system, with no appreciable advantages over Briggs’s system. 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 centesimal division of the degree seems to have entirely passed out of use, till it was recently revived by Bremiker in his Logarithmsch-trigonometrische Tafeln mit fünf Decimalstellen (Berlin, 1872). 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.

Collections.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 Schrön, Seven-Figure Logarithms (London, 1865, stereotyped, an English edition of the German work published at Brunswick), and Bruhns, A New Manual of Logarithms to Seven Places of Decimals (Leipsic, 1870). Both 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 Schrön and to 100,000 in Bruhns. Almost equally convenient is Bromiker’s edition of Vega’s Logarithmic Tables (Berlin, stereotyped; the English edition was translated from the fortieth edition of Dr 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. All these works 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. Dupuis, Tables de Logarithmes à 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; these are preferable to the ordinary characteristics that are increased by 10. This is the only work we know in which negative characteristics are used. 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. Salomon, Logarithmische 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 the most useful and accessible are Hutton, Callet, Vega, and Köhler. 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. 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. Vega’s Tabulæ Logarithmo-trigonometricæ 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 1ʹ, for every second to 1° 30ʹ, for every 10ʺ to 6° 3ʹ, 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, ${\displaystyle e^{x}}$ and ${\displaystyle \log _{10}e^{x}}$ to ${\displaystyle x=10}$ 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, &v. Vega also published Manuale Logarithmico-trigonometricum (Leipsic, 1800), the tables in which are identical with a portion of those contained in the first volume of the Tabulæ. The Tabulæ went through many editions, a stereotyped issue being brought out by J. A. Hülsse (Sammlung mathematischer Tafeln, Leipsic) 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. Schulze, Neue und erweiterte Sammlung logarithmischer, trigonometrischer, und anderer Tafeln (Berlin, 1778, 2 vols.), 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. Lambert, Supplementa Tabularum Logarithmicarum et Trigonometricarum (Lisbon, 1798), contains a number of useful and curious non-logarithmic tables; it bears a general resemblance to the second volume of Vega, but contains numerous other small tables of a more strictly mathematical character. A very useful collection of non-logarithmic tables is printed in 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 100,103, 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. Hassler, Tabulæ Logarithmicæ et Trigonometricæ (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 10ʺ and thence to 45° at intervals of 30ʺ. 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. 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. 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 9ʹ, at intervals of 10ʺ to 2°, of 30ʺ to 9°, of 1ʹ 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. Chambers’s Mathematical Tables (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 Bremiker’s Logarithmorum VI. Decimalium Nova Tabula BeroIinensis (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 this point for every ten seconds, with proportional parts. 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. 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. Of five-figure tables the most convenient is Tables of Logarithms (Useful Knowledge Society, London, from the stereotyped plates of 1839), which were prepared by De Morgan, though they have no name on the title-page. They contain five-figure logarithms to 10,000, log sines and tangents to every minute to 5 places, besides a few smaller tables. 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 6 to 7 places, passed through several more. Galbraith and Haughton, Manual of Mathematical Tables (London, 1860), give five-figure logarithms to 10,000 and sines and tangents for every minute, also a small table of Gaussian logarithms. Houël, Tables de Logarithmes à Cinq Décimales (Paris, 1871), 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 contains 118 pages of tables. The same author’s Recucil 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 ${\displaystyle {\frac {(1+x)}{(1-x)}}}$, elliptic integrals, squares for use in the method of least squares, &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. 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.

Napierian logarithms.Hyperbolic or Napierian 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 Logarithms and Napier, where a full account is given. Apart from the inventor's own publications, the only Napierian tables of importance are contained in Ursinus’s Trigonometria (Cologne, 1624–25) 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 Speidell, New Logarithmes (1619).

Hyperbolic logarithms.The most copious table of hyperbolic logarithms is 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 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), when 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 up to 1200 to 11 places; Salomon (1827), all numbers to 1000 and primes to 10,333 to 10 places; Vega, Tabulæ (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 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 Cyclopædia (1819), art. “Hyperbolic Logarithms,” contains a table of hyperbolic logarithms of all numbers up to 10,000 to 8 places.

Conversion of Briggian and hyperbolic logarithms.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. Degen, Tabularum Enneas (Copenhagen, 1824), gives the first hundred multiples of the modulus to 30 places.

Antilogarithms.Antilogarithms,—In the ordinary tables of logarithms the natural numbers are integers, while the logarithms are incommensurable. 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 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 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 Filipowski, A Table of Antilogarithms (London, 1849), contains a table of the same extent, the proportional parts being given to hundredths.

Addition and Subtraction, or Gaussian, Logarithms.—The object Gaussian logarithms.of such tables is to give ${\displaystyle \log \ (a\pm b)}$ by only one entry when ${\displaystyle \log \ a}$ and ${\displaystyle \log b}$ are given (see Logarithms, vol. xiv. p, 777). Let

${\displaystyle A=\log \ x\,}$${\displaystyle B=\log {\Bigl (}1+{\frac {1}{x}}{\Bigr )},}$${\displaystyle C=\log \ (1+x).}$

Leaving out the specimen table in Leonelli’s Théorie des Logarithms Additionnels et Déductifs (Bordeaux, 1803), the principal tables are the following. Gauss, in Zach’s Montliche Correspondenz (1812), giving ${\displaystyle B}$ and ${\displaystyle C}$ for argument ${\displaystyle A}$ 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. Matthiessen, Tafel zur bequemern Berechnung (Altona, 1818), giving ${\displaystyle B}$ and ${\displaystyle C}$ to 7 places for argument ${\displaystyle A}$ 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 Formulæ (London, 1849, and “Addendum,” 1870), giving ${\displaystyle C}$ for argument ${\displaystyle A}$ from 0 to 2 at intervals of ·0001 to 6 places, with proportional parts to hundredths, and ${\displaystyle \log \ (1-x)}$ for argument ${\displaystyle A}$ from ${\displaystyle {\bar {3}}}$ to ${\displaystyle {\bar {1}}}$ at intervals of ·001 and from ${\displaystyle {\bar {1}}}$ to ${\displaystyle {\bar {1}}\!\cdot \!9}$at intervals of ·0001, to 6 places, with proportional parts. Zech, Tafeln der Additions- und Subtractions- Logarithmen (Leipsic, 1849), giving ${\displaystyle B}$ for argument ${\displaystyle A}$ from 0 to 2 at intervals of ·0001, thence to 4 at intervals of ·001 and to 6 at intervals of ·01; also ${\displaystyle C}$ for argument ${\displaystyle A}$ from 0 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. Wittstein, Logarithmes de Gauss à Sept Decimals (Hanover, 1866), giving ${\displaystyle B}$ for argument ${\displaystyle A}$ 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, Tabulæ, (Leipsic, 1817); Köhler, Jerome de la Lande's Tafeln (Leipsic, 1832), and Handbuch (Leipsic, 1848); and Galbraith and Haughton, Manual (London, 1860). Houël, Tables de Logarithms (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 ${\displaystyle B}$ and ${\displaystyle -\log {\Bigl (}1-{\frac {1}{x}}{\Bigr )}}$ from ${\displaystyle A=0\ \mathrm {to} \cdot 03}$at intervals of ·0001, 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 ${\displaystyle B}$ and ${\displaystyle \log \ (1+x)}$ from ${\displaystyle A=5\ \mathrm {to} \ 3}$ at intervals of ·1, from ${\displaystyle A={\bar {3}}\ \mathrm {to} \ {\bar {2}}\!\cdot \!7}$ at intervals of ·01, to 1·3 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 ${\displaystyle \log \ (x+1)}$ as tabular result for ${\displaystyle \log x}$ as argument from 8 to 14 at intervals of ·001 to 5 places. Weidenbach, Tafel um den Logarithmen (Copenhagen, 1829), gives ${\displaystyle \log {\frac {x+1}{x-1}}}$ for argument ${\displaystyle A}$ 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.

Logistic and Proportional Logarithms.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 ${\displaystyle \log 3600-\log x}$ at intervals of a second, ${\displaystyle x}$ being expressed in degrees, minutes, and seconds; Schulze (1778) and Vega (1797) have them to ${\displaystyle x=3600''}$ and Callet and Hutton to ${\displaystyle x=5280''}$. Proportional logarithms for every second to 3° (i.e., ${\displaystyle 10,800-\log x}$) 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 24h, which give to 4 or 5 places the values of ${\displaystyle \log 1440-\log x}$. The object of a proportional or logistic table, or a table of ${\displaystyle \log a-\log x,}$ is to facilitate the calculation of proportions in which the third term is ${\displaystyle a}$.

Interpolation Tables.Interpolation Tables.—All tables of proportional parts may be regarded as interpolation tables. Bremiker, Tafel der Proportionaltheile (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. Tables of the values of binomial theorem coefficients, which are required when second and higher orders of differences are used, are described below. 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. xi. and xii. of the Assurance Magazine.

Dual Logarithms.Dual Logarithms.—This term is used by Mr 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 powers 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 logarithms. 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.Constants.—In nearly all tables of logarithms there is a page devoted to certain frequently used constants and their logarithms, such as π, 1π, π2, √π. A specially good collection is printed in Templeton's Millwrights 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 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, e-14π, e-94π, &c., calculated to about 50 places. The quantity π has been worked out to 707 places (Shanks, Proc. Roy. Soc., vol. xxi. p. 319) and Euler’s constant to 263 places (Adams, Proc. Roy. Soc., vol. xxvii. p. 88). The value of the modulus ${\displaystyle M}$, calculated by Prof. Adams, is given in Logarithms, vol. xiv. p. 779. This value is correct to 263 places; but the calculation has since been carried to 272 places (see Adams, Proc. Roy. Soc., vol. xlii. p. 22, 1887).

Irreducible cubic equations.Tables for the Solution of the Irreducible Case in Cubic Equations.—Lambert, Supplementa (1798), gives ${\displaystyle \pm (x-x^{3})}$ from ${\displaystyle x=}$ ·001 to 1·155 at intervals of ·001 to 7 places, and Barlow (1814) gives ${\displaystyle (x^{3}-x)}$ from ${\displaystyle x=}$1 to 1·1549 at intervals of ·0001 to 8 places.

Binomial Theorem Coefficients.Binomial Theorem Coefficients.—The values of ${\displaystyle {\frac {x(x-1)}{1\cdot 2}},{\frac {x(x-1)(x-2)}{1\cdot 2\cdot 3}},\ldots {\frac {x(x-1)(x-2)\ldots (x-5)}{1\cdot 2\ldots 6}},}$

from ${\displaystyle x=}$·01 to ${\displaystyle x=1}$ at intervals of ·01 to 7 places, are serviceable for use in interpolation by second and higher orders of differences. The table quoted above occurs in Schulze (1778), Barlow (1814), Vega (1797 and succeeding editions), Hantschl (1827), and Köhler (1848). Rouse, Doctrine of Chances (London, no date), gives on a folding sheet ${\displaystyle (a+b)^{n}}$ for ${\displaystyle n=1,2,...20}$. Lambert, Supplementa (1798), has the coefficients of the first 16 terms in ${\displaystyle (1+x)^{\frac {1}{2}}}$ and ${\displaystyle (1-x)^{\frac {1}{2}}}$, their accurate values being given as decimals. Vega (1797) has a page of tables giving ${\displaystyle {\frac {1}{2\mathrm {.} 4}},{\frac {1\mathrm {.} 3}{2\mathrm {.} 4\mathrm {.} 6}},\ldots {\frac {1}{2\mathrm {.} 3}},\ldots }$ 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.Figurate Numbers.—Lambert, Supplementa, gives ${\displaystyle x,{\frac {(x+1)}{1\mathrm {.} 2}},\ldots {\frac {(x+1)\ldots (x+11)}{1\mathrm {.} 2\ldots 12}}}$ from ${\displaystyle x=1\ \mathrm {to} \ 30}$.

Trigonometrical Quadratic Surds.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 ${\displaystyle \sin 3^{o}={\tfrac {1}{3}}\{\surd (5+\surd 5)+\surd {\tfrac {15}{2}}+\surd {\tfrac {5}{2}}-\surd (15+3\surd 5)-\surd {\tfrac {3}{2}}-\surd {\tfrac {1}{2}}\}}$; and the numerical values of the surds √(5 + √5), √(152), &c., are given to 10 places. These values were extended to 20 places by Peter Gray, Messenger of Math., vol. vi. (1877), p. 105.

Circulating Decimals.Circulating Decimals.—Goodwyn’s tables have been described above, p. 8. 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, 1814–17), gave one for all primes up to 2,543 and for 22 primes exceeding that limit. Desmarest, Théorie des Nombres (Paris, 1852), included all primes up to 10,000. Reuschle, Mathematische Abhandlung, enthaltend neue zahlentheoretische Tabellen (1856), contains a similar table to 15,000. This Shanks extended to 60,000; the portion from 1 to 30,000 is printed in the Proc. Roy. Soc., vol. xxii. p. 200, and the remainder is preserved in the archives of the society (Id., xiii, p. 260 and xxiv. p. 392). The number of digits in the decimal period of ${\displaystyle {\tfrac {1}{p}}}$ is the same as the exponent to which 10 belongs for modulus ${\displaystyle p,}$ so that, whenever the period has ${\displaystyle p-1}$ digits, 10 is a primitive root of ${\displaystyle p}$. Tables of primes having a given number, ${\displaystyle n,}$of digits in their periods, i.e., tables of the resolutions of ${\displaystyle 10^{n}-1}$into factors and, as far as known, into prime factors, have been given by Loof (in Grunert's Archiv, vol. xvi. p. 54; reprinted in Nouv. Annales, vol. xiv. p. 115) and by Shanks (Proc. Roy. Soc., vol. xxii. p. 381). The former extends to ${\displaystyle n=60}$ and the latter to ${\displaystyle n=100}$, but there are gaps in both. Reuschle's tract also contains resolutions of ${\displaystyle 10^{n}-1}$. For further references on circulating decimals, see Proc. Camb. Phil. Soc., vol. iii. p. 185 (1878).

Pythagorean Triangles.—Right-angled triangles in which the Pythagorean triangles.sides and hypothenuse 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 ${\displaystyle \tan {\tfrac {1}{2}}\omega >{\tfrac {1}{25}}}$ (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 Bretschneider, in Grunert's Archiv vol. i. p. 96 (1841), and by Sang, Edinburgh Transactions vol. xxiii. p. 727 (1864). In these the triangles are arranged according to hypothenuses and extend to 1201, 1200, 49, and 1105, 1073, 264 respectively. 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 Rath, "Die rationalen Dreiecke," in Gunert's Archiv vol. Ivi. p. 188 (1874). Sang’s paper also contains a table of triangles having an angle equal to 120" and their sides integers.

Powers of π.Powers of π.—Paucker, in Grunert's Archiv, vol. i. p. 10, gives of π-1 and π12 to 140 places, and π-2, π-12, π13, π23 to about 50 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 printed by Glaisher, Proc. Lond. Math. Soc., vol. viii. p. 140, and the first hundred multiples of π and π-1 to 12 places by Kulik, Tafel der Quadrat-und Kubik-Zahlen (Leipsic, 1848).

Series 1-n + 2-n, &c. The Series 1-n + 2-n + 3-n + &c.—Let Sn, sn, σn 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 Fonctions Elliptiques, vol. ii. p. 432) has computed ${\displaystyle S_{n}}$ to 16 places from ${\displaystyle n=\,\mathrm {1\,to\,35} }$, and Glaisher (Proc. Lond. Math. Soc., vol. iv. p. 48) has deduced ${\displaystyle s_{n}}$ and ${\displaystyle \sigma _{n}}$ for the same arguments and to the same number of places. The latter has also given ${\displaystyle S_{n},s_{n},\sigma _{n}}$for n=2, 4, 6, ... 12 to 22 or more places (Proc. Lond. Math. Soc., vol. viii. p. 140), and the values of ${\displaystyle \Sigma _{n}}$, where ${\displaystyle \Sigma _{n}=2^{-n}+3^{-n}+5^{-n}+}$ &c. (prime numbers only involved), for n = 2 , 4, 6, . . . 36 to 15 places (Compte Rendu de l'Assoc. Française for 1878, p. 172).

Hyperbolic Antilogarithms.Tables of ${\displaystyle e^{x},}$ or Hyperbolic Antilogarithms.—The largest tables are the following. Gudermann, Theorie der potenzial- oder cyklisch-hyperbolischen Functionen (Berlin, 1833), which consists of papers reprinted from vols. viii. and ix. of Crelle's Journal, and gives ${\displaystyle \log _{10}\sinh }$ , ${\displaystyle \log _{10}\cosh }$, and ${\displaystyle \log _{10}\tanh }$ from $x=$ 2 to 5 at intervals of ·001 to 9 places and from ${\displaystyle x=}$ 5 to 12 at intervals of ·01 to 10 places. Since ${\displaystyle \sinh x={\tfrac {1}{2}}(e^{x}-e^{-x})}$ and ${\displaystyle \cosh x={\tfrac {1}{2}}(e^{x}+e^{-x})}$, the values of ${\displaystyle e^{x}}$ and ${\displaystyle e^{-x}}$ are deducible at once by addition and subtraction. Newman, in Camb. Phil. Trans., vol. xiii. pp. 145-241, gives values of ${\displaystyle e^{-x}}$ from ${\displaystyle x=}$ 0 to 15·349 at intervals of ·001 to 12 places, from ${\displaystyle x=}$ 15·350 to 17·298 at intervals of ·002, and from ${\displaystyle x=}$17·300 to 27·635 at intervals of ·005, to 14 places. Glaisher, in Camb. Phil. Trans., vol, xiii. pp. 243-272, gives four tables of ${\displaystyle e^{x},\,e^{-x},\,\log _{10}e^{x},\,\log _{10}e^{-x}}$, their ranges being from ${\displaystyle x=}$·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, Tabulæ (1797 and later edd.), has ${\displaystyle \log _{10}e^{x}}$ to 7 places and ${\displaystyle e^{x}}$ to 7 figures from ${\displaystyle x=}$·01 to 10 at intervals of ·01. Köhler’s Handbuch contains a small table of ${\displaystyle e^{x}}$. In Schulze’s Sammlung (1778) ${\displaystyle e^{x}}$is given for ${\displaystyle x=}$1, 2, 3,. . .24 to 28 or 29 figures and for ${\displaystyle x=}$25, 30, and 60 to 32 or 33 figures; this table is printed in Glaisher’s paper (loc. cit.). In Salomon’s Tafeln (1827) the values of ${\displaystyle e^{n},\,e^{-n},\,e^{.0n},\,e^{.00n},...e^{.000000n}}$ where n has the values 1, 2,. . ., are given to 12 places. Bretschneider, in Gunert's Archiv, iii. p. 33, worked out ${\displaystyle e^{x}}$and ${\displaystyle e^{-x}}$ and also ${\displaystyle \sin x}$ and ${\displaystyle \cos x}$ for ${\displaystyle x=}$=1, 2, . . . 10 to 20 places.

Factorials.Factorials.—The values of ${\displaystyle \log _{10}(n!)}$, where ${\displaystyle n!}$ denotes ${\displaystyle 1,2,3...n}$, from ${\displaystyle n=}$1 to 1200 to 18 places, are given by Degen, Tabularum Enneas (Copenhagen, 1824), and reprinted, to 6 places, at the end of De Morgan’s article "Probabilities” in the Encyclopædia Metropolitana. Shortrede, Tables (1849, vol. i.), gives ${\displaystyle \log(n!)}$ to ${\displaystyle n=}$ 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 ${\displaystyle n\times n!}$ and the values of ${\displaystyle \log _{10}{\frac {1}{n\times n!}}}$are computed by Glaisher as far as ${\displaystyle n=71}$ in the Phil. Trans, for 1870 (p. 370), and the values of ${\displaystyle {\frac {1}{n!}}}$ to 28 significant figures as far as ${\displaystyle n=50}$ in Camb. Phil. Trans., vol. xiii. p. 246.

Bernoullian Numbers.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 Ohm in Crelle's Journal, vol. xx. p. 11. Prof. J. C. Adams has calculated the next thirty-one, and a table of the first sixty-two was published by him in the Brit. Assoc. Report for 1877 and in Crelle's Journal, vol. lxxxv. p. 269. The first nine figures of the values of the first 250 Bernoullian numbers, and their Briggian logarithms to 10 places, have been printed by Glaisher, Camb. Phil. Trans., vol. xii. p. 384.

Tables of ${\displaystyle \log \tan {\bigl (}{\tfrac {1}{4}}\pi +{\tfrac {1}{2}}\phi {\bigr )}}$.Tables of ${\displaystyle \log \tan {\bigl (}{\tfrac {1}{4}}\pi +{\tfrac {1}{2}}\phi {\bigr )}}$.—Guderman, Theorie der potenzial- oder cyklisch-hyperbolischen Functionen (Berlin, 1833), gives {in 100 pages} ${\displaystyle \log \tan {\bigl (}{\tfrac {1}{4}}\pi +{\tfrac {1}{2}}\phi {\bigr )}}$ for every centesimal minute of the quadrant to 7 places. Another table contains the values of this function, also at intervals of a minute, for 88° to 100° (centesimal) to 11 places. Legendre, Traite des Fonctions Elliptiques (vol. ii. p. 256), gives the same function for every half degree (sexagesimal) of the quadrant to 12 places.

Gamma function.The Gamma Function—Legendre’s great table appeared in vol. ii. of his Excercices de Calcul Intégral (1816), p. 85, and in vol. ii. of his Traité des Fonctions Elliptiques, (1826), p. 489. ${\displaystyle \log _{10}\Gamma (x)}$ is given from ${\displaystyle x=}$1 to 2 at intervals of ·001 to 12 places, with differences to the third order. This table is reprinted in full in Schlömilch, Analytische Studien (1848), p. 183; an abridgment in which the arguments differ by ·01 occurs in De Morgan, Diff. 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 Bertrand, Calcul Intégral (1870), p. 285, and a six-figure abridgment in Williamson, Integral Calculus (1884), p. 169. In vol. i. of his Exercices (1811), Legendre had previously published a seven-place table of ${\displaystyle \log _{10}\Gamma (x)}$, without differences.

Elliptic functions. Tables connected with Elliptic Functions.—Legendre calculated elaborate tables of the elliptic integrals in vol. ii. of Traité des Fonctions Elliptiques (1826). Denoting the modular angle by θ, the amplitude by φ, and the incomplete integral of the second kind by ${\displaystyle E_{1}(\phi )}$ the tables are— (1) ${\displaystyle \log _{10}E}$ and ${\displaystyle \log _{10}K}$ from ${\displaystyle \theta =}$ 0° to 90° at intervals of 0°·1 to 12 or 14 places, with differences to the third order; (2) ${\displaystyle E_{1}(\phi )}$ and ${\displaystyle F(\phi )}$, the modular angle being 45° from ${\displaystyle \phi =}$0° to 90° at intervals of 0°·5 to 12 places, with differences to the fifth order; (3) ${\displaystyle E_{1}(45^{o})}$ and ${\displaystyle F(45^{o})}$ from ${\displaystyle \theta =0^{o}\mathrm {to} \,90^{o}}$ at intervals of 1°, with differences to the sixth order, also ${\displaystyle E}$ and ${\displaystyle K}$ for the same arguments, all to 12 places; (4) ${\displaystyle E_{1}(\phi )}$ and ${\displaystyle F(\phi )}$ 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.Tables involving q.—Verhulst, Traité des Fonctions Elliptiques (Brussels, 1841), contains a table of ${\displaystyle \log _{10}\log _{10}\left({\frac {1}{q}}\right)}$ for argument ${\displaystyle \theta }$ at intervals of 0°·1 to 12 or 14 places. Jacobi, in Crelle's Journal, vol. xxvi. p. 93, gives ${\displaystyle \log _{10}q}$ from ${\displaystyle \theta =0^{o}\mathrm {to} \,90^{o}}$at intervals of 0°·1 to 5 places. Meissel, Sammlung mathematisher Tafeln, i. (Iserlohn, 1860), consists of a table of ${\displaystyle \log _{10}q}$ at intervals of 1' from ${\displaystyle \theta =0^{o}\mathrm {to} \,90^{o}}$to 8 places. Glaisher, in Month. Not. Roy. Ast. Soc., vol. xxxvii. p. 372 (1877), gives ${\displaystyle \log _{10}q}$ to 10 places and ${\displaystyle q}$ to 9 places for every degree. In Bertrand, Calcul Intégral (1870), a table of ${\displaystyle \log _{10}q}$ from ${\displaystyle \theta =0^{o}\mathrm {to} \,90^{o}}$ at intervals of 5' to 5 places is accompanied by tablesof ${\displaystyle \log _{10}{\sqrt {\frac {2K}{\pi }}}}$ and ${\displaystyle \log _{10}\log _{10}{\frac {1}{q}}}$ and by abridgements of Legendre’s tables of the elliptic integrals. Schlömilch, Vorlesungen der höheren Analysis (Brunswick, 1879), p. 448, gives a small table of ${\displaystyle \log _{10}q}$ for every degree to 5 places.

Legendrian Coefficients.Legendrian Coefficients.— The values of ${\displaystyle P^{n}(x)}$ for n=1, 2, 3, . .. 7 from ${\displaystyle x=}$0 to 1 at intervals of ·01 are given by Glaisher, in Brit. Assoc. Rep. for 1879, pp. 54–57. The functions tabulated are ${\displaystyle P^{1}(x)=x}$, ${\displaystyle P^{2}(x)={\tfrac {1}{2}}(3x^{2}-1)}$,${\displaystyle P^{3}(x)={\tfrac {1}{2}}(5x^{3}-3x)}$,${\displaystyle P^{4}(x)={\tfrac {1}{8}}(35x^{4}-30x^{2}+3)}$,${\displaystyle P^{5}(x)={\tfrac {1}{8}}(63x^{5}-70x^{3}+15x)}$,${\displaystyle P^{6}(x)={\tfrac {1}{16}}(231x^{6}-315x^{4}+105x^{2}-5)}$,${\displaystyle P^{7}(x)={\tfrac {1}{16}}(429x^{7}-693x^{5}+315x^{3}-35x)}$. The functions occur in connexion with the theory of interpolation, the attraction of spheroids, and other physical theories.

Bessel’s Functions.Bessel’s Functions. — Bessel’s original table appeared at the end of his memoir "Untersuchung des planetarischen Theils der Störungen, welche aus der Bewegung der Sonne entstchen" (in Abh. d. Berl. Akad., 1824; reprinted in vol. i. of his Abhandlungen, p. 84). It gives ${\displaystyle J_{o}(x)}$) and ${\displaystyle J_{1}(x)}$) from ${\displaystyle x=}$0 to 3·2 at intervals of ·01. More extensive tables were calculated by Hansen in "Ermittelung der absoluten Störungen in Ellipsen von beliebiger Excentricität und Neigung" (in Schriften der Sternwarte Seeberg, part i., Gotha, 1843). They include an extension of Bessel’s original table to ${\displaystyle x=}$20, besides smaller tables of ${\displaystyle J_{n}(x)}$for certain values of ${\displaystyle n}$ as far as ${\displaystyle n=}$28, all to 7 places. Hansen’s table was reproduced by Schlömilch, in Zeitschr. für Math., vol ii. p. 158, and by Lommel, Studien über die Bessel'schen Functionen (Leipsic, 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.

Sine, &c., integrals.Sine, Cosine, Exponential, and Logarithm Integrals.—The functions so named are the integrals ${\displaystyle \int _{0}^{x}{\frac {\sin x}{x}}dx,}$ ${\displaystyle \int _{\infty }^{x}{\frac {\cos x}{x}}dx,}$ ${\displaystyle \int _{-\infty }^{x}{\frac {e^{x}}{x}}dx,}$ ${\displaystyle \int _{0}^{x}{\frac {dx}{\log x}},}$which are denoted by the functional signs Si ${\displaystyle x}$, Ci ${\displaystyle x}$, Ei ${\displaystyle x}$, li ${\displaystyle x}$ respectively. Soldner, Théorie et Tables d'une Nouvelle Fonction Transcendante (Munich, 1809), gave the values of li ${\displaystyle x}$ from ${\displaystyle x=}$0 to 1 at intervals of ·1 to 7 places, and thence at various intervals to 1220 to 5 or more places. This table is reprinted in De Morgan’s Diff. and Int. Calc., p. 662. Bretschneider, in Grunert’s Archiv, vol. iii. p. 33, calculated Ei${\displaystyle (\pm x)}$, Si ${\displaystyle x}$, Ci ${\displaystyle x}$ for ${\displaystyle x=}$1, 2, . . . 10 to 20 places, and subsequently (in Schlömilch’s Zeitschrift, vol. vi.) worked out the values of the same functions from ${\displaystyle 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, Tabulæ Logarithmi Integralis (Malmö, part i. 1861 and part ii. 1867), give the values of li 10x 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, Cix 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-Integrale" (a paper in Memoirs of the Venetian Institute, 1874). 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, vol. ii. 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.

Values of ${\displaystyle \int _{0}^{x}e^{-x^{2}}dx}$ and ${\displaystyle e^{x^{2}}\int _{0}^{x}e^{-x^{2}}dx}$.Values of ${\displaystyle \int _{0}^{x}e^{-x^{2}}dx}$ and ${\displaystyle e^{x^{2}}\int _{0}^{x}e^{-x^{2}}dx}$ — These functions are employed in researches connected with refractions, theory of errors, conduction of heat, &c. Let ${\displaystyle \int _{0}^{x}e^{-x^{2}}dx}$ and ${\displaystyle \int _{x}^{\infty }e^{-x^{2}}dx}$be denoted Erf ${\displaystyle x}$; and Erfc ${\displaystyle x}$; respectively, standing for "error function" and "error function complement," so that Erf ${\displaystyle x}$+ Erfc ${\displaystyle x={\frac {1}{2}}\surd \pi }$ (Phil. Mag., Dec. 1871; it has since been found convenient to transpose as above the definitions of Erf and Erfc). The tables of the functions, and of the functions multiplied by ${\displaystyle e^{x^{2}}}$ are as follows. Kramp, Analyse des Réfractions (Strasburg, 1798), has Erfc ${\displaystyle x}$; from ${\displaystyle x=}$0 to 3 at intervals of ·01 to 8 or more places, also ${\displaystyle \log _{10}(\mathrm {E} \mathrm {r} \mathrm {f} \mathrm {c} \ x)}$ and ${\displaystyle \log _{10}(e^{x^{2}}\mathrm {E} \mathrm {r} \mathrm {f} \mathrm {c} \ x)}$for the same values to 7 places. Bessel, Fundamenta Astronomiæ (Königsberg, 1818), has ${\displaystyle \log _{10}(e^{x^{2}}\mathrm {E} \mathrm {r} \mathrm {f} \mathrm {c} \ x)}$from ${\displaystyle x=}$0 to 1 at intervals of ·01 to 7 places, likewise for argument ${\displaystyle \log _{10}x}$, the arguments increasing from 0 to 1 at intervals of ·01. Legendre, Traité des Fonctions Elliptigues (1826), vol. ii. p. 520, contains ${\displaystyle \Gamma ({\frac {1}{2}},e^{-x^{2}})}$, that is, ${\displaystyle 2\,\mathrm {E} \mathrm {r} \mathrm {f} \mathrm {c} \ x}$from ${\displaystyle x=}$0 to ·5 at intervals of ·01 to 10 places. Encke, Berliner Ast. Jahrbuch for 1834, prints -${\displaystyle {\frac {2}{\surd \pi }}\mathrm {E} \mathrm {r} \mathrm {f} \ x}$ from ${\displaystyle x=}$0 to 2 at intervals of ·01 to 7 places and ${\displaystyle {\frac {2}{\surd \pi }}\mathrm {E} \mathrm {r} \mathrm {f} \ (\rho x)}$ from ${\displaystyle x=}$ 0 to 3·4 at intervals of ·01 and thence to 5 at intervals of ·01 to 5 places, ρ being ·4769360. Glaisher, in Phil. Mag., December 1871, has calculated Erfc ${\displaystyle x}$ from ${\displaystyle x=}$ 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 Encyclopædia Metropolitana, art. "Probabilities."

Non-numerical integrals.Tables of Integrals, not Numerical.—Meyer Hirsch, Integraltafeln (1810; Eng. trans., 1823), and Minding, Integraltafeln (Berlin, 1849), give values of indefinite integrals and formulæ of reduction; both are useful and valuable works. De Haan, Nouvelles Tables d'lntégrales Défines (Leyden, 1867), is a quarto volume of 727 pages containing evaluations of definite integrals, arranged in 485 tables. The first edition appeared in vol. iv. of the Transactions of the Amsterdam Academy of Sciences. This, though not so full and accurate as the second edition, gives references to the original memoirs in which the different integrals are considered.

Theory of numbers.Tables relating to the Theory of Numbers.—These are of so technical a character and so numerous that a full account cannot be attempted here. The reader is referred to Cayley’s paper in the Brit. Assoc. Rep. for 1875, where a full description with references is given. Three tables may, however, be briefly noticed on account of their importance and because they form separate volumes : (1) Degen, Canon Pellianus (Copenhagen, 1817), relates to the indeterminate equation ${\displaystyle y^{2}-ax^{2}=1}$ for values of ${\displaystyle a}$ from 1 to 1000. It in fact gives the expression for ${\displaystyle \surd a}$ as a continued fraction; (2) 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) 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. A table of ${\displaystyle \chi ^{(n)}}$, where ${\displaystyle \chi ^{(n)}}$denotes the sum of the complex numbers which have ${\displaystyle n}$ for their norm for primes up to ${\displaystyle n=}$ 13,000 (cf. Quart. Journ., vol. xx. p. 152), has been published since the date of Cayley’s report. Some tables that belong to the theory of numbers have been described above under "Factor Tables" (p. 7).

Bibliography.—Full bibliographical and historical information relating to tables is collected in Brit. Assoc. Rep. for 1873, p. 6. The principal works are:— Heilbronner, Historia Matheseos (Leipsic, 1742), the arithmetical portion being at the end; Scheibel, Einleitung zur mathematischen Bücherkenntniss (Breslau, 1771–84); Kästner, Geschichte der Mathematik (Göttingen, 1796–1800), vol. iii.; Murhard, Bibliotheca Mathematica (Leipsic, 1797–1804), vol. ii,; Rogg, Bibliotheca Mathematica (Tübingen, 1830), and continuation from 1830 to 1854 by Sohnke (Leipsic and London, 1854); Lalande, Bibliographie Astronomique (Paris, 1803), a separate index on p. 960. A great deal of accurate information upon early tables is given by Delambre, Histoire de l'Astronomie Moderne (Paris, 1821), vol. i.; and Nos. xix. and xx. of Hutton’s Mathematical Tracts (1812). For a complete list of logarithmic tables of all kinds from 1614 to 1862, see De Haan, "lets over Logarithmentafels," in Verslagen en Mededeelingen der Koning. Akad. van Wetenschappen (Amsterdam, 1862), pt. xiv. De Morgan’s article "Tables," which appeared first in the Penny Cyclopædia, and afterwards with additions in the English Cyclopædia, 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.}}

1. Referring to factor tables, Lambert wrote (Supplementa Tabularum, 1798, p. xv.): "Universalis finis talium tabularum est ut semel pro semper computetur quod sæpius 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., vol. iii. (1878) pp. 99–138, or the Introduction to the Fourth Million.
3. See a paper "On Multiplication by a Table of Single Entry," in Phil. Mag., November 1878, for a notice of this book.
4. See his "Einige Bemerkungen zu Vega's Thesaurus Logarithmorum," in Astronomische Nachrichten for 1851 (reprinted in his Werke, vol. iii. pp. 257–264); also Monthly Notices Roy. Ast. Soc. for May 1873.
5. Legendre (Traité des Fonctions Elliptiques, vol. ii. 1826) gives a table of natural sines to 15 paces, and of log sines to 14 places, for every 15ʺ 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 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).