Page:Encyclopædia Britannica, Ninth Edition, v. 23.djvu/24

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 $\tan {\tfrac {1}{2}}\omega >{\tfrac {1}{25}}$ (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 π^1⁄₂ to 140 places, and π^-2, π^-1/2, π^1/3, π^2/3 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 S_n, s_n, σ_n denote respectively the sums of the series 1^-n + 2^-n + 3^-n &c., 1^-n - 2^-n + 3^-n - &c., 1^-n + 3^-n + 5^-n + &c. Legendre (Traité des Fonctions Elliptiques, vol. ii. p. 432) has computed $S_{n}$ to 16 places from $n=\,\mathrm {1\,to\,35}$, and Glaisher (Proc. Lond. Math. Soc., vol. iv. p. 48) has deduced $s_{n}$ and $\sigma _{n}$ for the same arguments and to the same number of places. The latter has also given $S_{n},s_{n},\sigma _{n}$for n=2, 4, 6, ... 12 to 22 or more places (Proc. Lond. Math. Soc., vol. viii. p. 140), and the values of $\Sigma _{n}$, where $\Sigma _{n}=2^{-n}+3^{-n}+5^{-n}+$ &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 $e^{x},$ 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 $\log _{10}\sinh$ , $\log _{10}\cosh$, and $\log _{10}\tanh$ from <math>x=</math> 2 to 5 at intervals of ·001 to 9 places and from $x=$ 5 to 12 at intervals of ·01 to 10 places. Since $\sinh x={\tfrac {1}{2}}(e^{x}-e^{-x})$ and $\cosh x={\tfrac {1}{2}}(e^{x}+e^{-x})$, the values of $e^{x}$ and $e^{-x}$ are deducible at once by addition and subtraction. Newman, in Camb. Phil. Trans., vol. xiii. pp. 145-241, gives values of $e^{-x}$ from $x=$ 0 to 15·349 at intervals of ·001 to 12 places, from $x=$ 15·350 to 17·298 at intervals of ·002, and from $x=$17·300 to 27·635 at intervals of ·005, to 14 places. Glaisher, in Camb. Phil. Trans., vol, xiii. pp. 243-272, gives four tables of $e^{x},\,e^{-x},\,\log _{10}e^{x},\,\log _{10}e^{-x}$, their ranges being from $x=$·001 to ·1 at intervals of ·001, from ·01 to 2 at intervals of ·01, from ·1 to 10 at intervals of ·1, from 1 to 500 at intervals of unity. Vega, Tabulæ (1797 and later edd.), has $\log _{10}e^{x}$ to 7 places and $e^{x}$ to 7 figures from $x=$·01 to 10 at intervals of ·01. Köhler’s Handbuch contains a small table of $e^{x}$. In Schulze’s Sammlung (1778) $e^{x}$is given for $x=$1, 2, 3,. . .24 to 28 or 29 figures and for $x=$25, 30, and 60 to 32 or 33 figures; this table is printed in Glaisher’s paper (loc. cit.). In Salomon’s Tafeln (1827) the values of $e^{n},\,e^{-n},\,e^{.0n},\,e^{.00n},...e^{.000000n}$ where n has the values 1, 2,. . ., are given to 12 places. Bretschneider, in Gunert's Archiv, iii. p. 33, worked out $e^{x}$and $e^{-x}$ and also $\sin x$ and $\cos x$ for $x=$=1, 2, . . . 10 to 20 places.

Factorials.Factorials.—The values of $\log _{10}(n!)$, where $n!$ denotes $1,2,3...n$, from $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 $\log(n!)$ to $n=$ 1000 to 5 places, and for the arguments ending in 0 to 8 places. Degen also gives the complements of the logarithms. The first 20 figures of the values of $n\times n!$ and the values of $\log _{10}{\frac {1}{n\times n!}}$are computed by Glaisher as far as $n=71$ in the Phil. Trans, for 1870 (p. 370), and the values of ${\frac {1}{n!}}$ to 28 significant figures as far as $n=50$ 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 $\log \tan {\bigl (}{\tfrac {1}{4}}\pi +{\tfrac {1}{2}}\phi {\bigr )}$.Tables of $\log \tan {\bigl (}{\tfrac {1}{4}}\pi +{\tfrac {1}{2}}\phi {\bigr )}$.—Guderman, Theorie der potenzial- oder cyklisch-hyperbolischen Functionen (Berlin, 1833), gives {in 100 pages} $\log \tan {\bigl (}{\tfrac {1}{4}}\pi +{\tfrac {1}{2}}\phi {\bigr )}$ 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. $\log _{10}\Gamma (x)$ is given from $x=$1 to 2 at intervals of ·001 to 12 places, with differences to the third order. This table is reprinted in full in 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 $\log _{10}\Gamma (x)$, 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 $E_{1}(\phi )$ the tables are— (1) $\log _{10}E$ and $\log _{10}K$ from $\theta =$ 0° to 90° at intervals of 0°·1 to 12 or 14 places, with differences to the third order; (2) $E_{1}(\phi )$ and $F(\phi )$, the modular angle being 45° from $\phi =$0° to 90° at intervals of 0°·5 to 12 places, with differences to the fifth order; (3) $E_{1}(45^{o})$ and $F(45^{o})$ from $\theta =0^{o}\mathrm {to} \,90^{o}$ at intervals of 1°, with differences to the sixth order, also $E$ and $K$ for the same arguments, all to 12 places; (4) $E_{1}(\phi )$ and $F(\phi )$ 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 $\log _{10}\log _{10}\left({\frac {1}{q}}\right)$ for argument $\theta$ at intervals of 0°·1 to 12 or 14 places. Jacobi, in Crelle's Journal, vol. xxvi. p. 93, gives $\log _{10}q$ from $\theta =0^{o}\mathrm {to} \,90^{o}$at intervals of 0°·1 to 5 places. Meissel, Sammlung mathematisher Tafeln, i. (Iserlohn, 1860), consists of a table of $\log _{10}q$ at intervals of 1' from $\theta =0^{o}\mathrm {to} \,90^{o}$to 8 places. Glaisher, in Month. Not. Roy. Ast. Soc., vol. xxxvii. p. 372 (1877), gives $\log _{10}q$ to 10 places and $q$ to 9 places for every degree. In Bertrand, Calcul Intégral (1870), a table of $\log _{10}q$ from $\theta =0^{o}\mathrm {to} \,90^{o}$ at intervals of 5' to 5 places is accompanied by tablesof $\log _{10}{\sqrt {\frac {2K}{\pi }}}$ and $\log _{10}\log _{10}{\frac {1}{q}}$ 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 $\log _{10}q$ for every degree to 5 places.

Legendrian Coefficients.Legendrian Coefficients.— The values of $P^{n}(x)$ for n=1, 2, 3, . .. 7 from $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 $P^{1}(x)=x$, $P^{2}(x)={\tfrac {1}{2}}(3x^{2}-1)$,$P^{3}(x)={\tfrac {1}{2}}(5x^{3}-3x)$,$P^{4}(x)={\tfrac {1}{8}}(35x^{4}-30x^{2}+3)$,$P^{5}(x)={\tfrac {1}{8}}(63x^{5}-70x^{3}+15x)$,$P^{6}(x)={\tfrac {1}{16}}(231x^{6}-315x^{4}+105x^{2}-5)$,$P^{7}(x)={\tfrac {1}{16}}(429x^{7}-693x^{5}+315x^{3}-35x)$. 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 $J_{o}(x)$) and $J_{1}(x)$) from $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 $x=$20, besides smaller tables of $J_{n}(x)$for certain values of $n$ as far as $n=$28, all to 7 places. Hansen’s table was reproduced by 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 $\int _{0}^{x}{\frac {\sin x}{x}}dx,$ $\int _{\infty }^{x}{\frac {\cos x}{x}}dx,$ $\int _{-\infty }^{x}{\frac {e^{x}}{x}}dx,$ $\int _{0}^{x}{\frac {dx}{\log x}},$which are denoted by the functional signs Si $x$, Ci $x$, Ei $x$, li $x$ respectively. Soldner, Théorie et Tables d'une Nouvelle Fonction Transcendante (Munich, 1809), gave the values of li $x$ from $x=$0 to 1 at intervals of ·1 to 7 places, and thence at various intervals to 1220 to 5 or more places. This table is reprinted in De Morgan’s Diff. and Int. Calc., p. 662. Bretschneider, in Grunert’s Archiv, vol. iii. p. 33, calculated Ei$(\pm x)$, Si $x$, Ci $x$ for $x=$1, 2, . . . 10 to 20 places, and subsequently (in Schlömilch’s Zeitschrift, vol. vi.) worked out the values of the same functions from $x=$0 to 1 at intervals of ·01 and from 1 to 7·5 at intervals of ·1 to 10 places. Two tracts by L. Stenberg, Tabulæ Logarithmi Integralis (Malmö, part i. 1861