Page:Encyclopædia Britannica, Ninth Edition, v. 23.djvu/25

and part ii. 1867), give the values of li 10^x from x;= - 15 to 3·5 at intervals of ·01 to 18 places. Glaisher, in Phil. Trans., 1870, p. 367, gives Ei(±x), Si x, 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 $\int _{0}^{x}e^{-x^{2}}dx$ and $e^{x^{2}}\int _{0}^{x}e^{-x^{2}}dx$.Values of $\int _{0}^{x}e^{-x^{2}}dx$ and $e^{x^{2}}\int _{0}^{x}e^{-x^{2}}dx$ — These functions are employed in researches connected with refractions, theory of errors, conduction of heat, &c. Let $\int _{0}^{x}e^{-x^{2}}dx$ and $\int _{x}^{\infty }e^{-x^{2}}dx$be denoted Erf $x$; and Erfc $x$; respectively, standing for "error function" and "error function complement," so that Erf $x$+ Erfc $x={\frac {1}{2}}\surd \pi$ (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 $e^{x^{2}}$ are as follows. Kramp, Analyse des Réfractions (Strasburg, 1798), has Erfc $x$; from $x=$0 to 3 at intervals of ·01 to 8 or more places, also $\log _{10}(\mathrm {E} \mathrm {r} \mathrm {f} \mathrm {c} \ x)$ and $\log _{10}(e^{x^{2}}\mathrm {E} \mathrm {r} \mathrm {f} \mathrm {c} \ x)$for the same values to 7 places. Bessel, Fundamenta Astronomiæ (Königsberg, 1818), has $\log _{10}(e^{x^{2}}\mathrm {E} \mathrm {r} \mathrm {f} \mathrm {c} \ x)$from $x=$0 to 1 at intervals of ·01 to 7 places, likewise for argument $\log _{10}x$, the arguments increasing from 0 to 1 at intervals of ·01. Legendre, Traité des Fonctions Elliptigues (1826), vol. ii. p. 520, contains $\Gamma ({\frac {1}{2}},e^{-x^{2}})$, that is, $2\,\mathrm {E} \mathrm {r} \mathrm {f} \mathrm {c} \ x$from $x=$0 to ·5 at intervals of ·01 to 10 places. Encke, Berliner Ast. Jahrbuch for 1834, prints -${\frac {2}{\surd \pi }}\mathrm {E} \mathrm {r} \mathrm {f} \ x$ from $x=$0 to 2 at intervals of ·01 to 7 places and ${\frac {2}{\surd \pi }}\mathrm {E} \mathrm {r} \mathrm {f} \ (\rho x)$ from $x=$ 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 $x$ from $x=$ 3 to 4·5 at intervals of ·01 to 11, 13, or 14 places. Encke’s tables and two of Kramp’s were reprinted in the 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 $y^{2}-ax^{2}=1$ for values of $a$ from 1 to 1000. It in fact gives the expression for $\surd a$ 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 $\chi ^{(n)}$, where $\chi ^{(n)}$denotes the sum of the complex numbers which have $n$ for their norm for primes up to $n=$ 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.}} (j. w. l. g.)

TABOO (also written Tabu and Tapu) is the name given to a system of religious prohibitions which attained its fullest development in Polynesia (from Hawaii to New Zealand; see vol. xix. p. 426), but of which under different names traces may be discovered in most parts of the world.

Meaning.The word "taboo" is common to the different dialects of Polynesia, and is perhaps derived from ta, "to mark," and pu, an adverb of intensity. The compound word "taboo" (tapu) would thus originally mean "marked thoroughly." Its ordinary sense is "sacred." It does not, however, imply any moral quality, but only "a connexion with the gods or a separation from ordinary purposes and exclusive appropriation to persons or things considered sacred; sometimes it means devoted as by a vow." Chiefs who trace their lineage to the gods are called arii tabu, "chiefs sacred," and a temple is called a wahi tabu, "place sacred." The converse of taboo is noa (in Tonga gnofoóa), which means "general" or "common." Thus the rule which forbade women to eat with men, as well as, except on special occasions, to eat any fruits or animals offered in sacrifice to the gods, was called ai tabu, "eating sacred"; while the present relaxation of the rule is called ai noa, eating generally, or having food in common. Although it was employed for civil as well as religious purposes, the taboo was essentially a religious observance. In Hawaii it could be imposed only by priests; but elsewhere in Polynesia kings and chiefs, and even to a certain extent ordinary individuals, exercised the same power. The strictness with which the taboo was observed depended largely on the influence of the person who imposed it; if he was a great chief it would not be broken ; but a powerful man often set at nought the taboo of an inferior.

General and particular taboos.A taboo might be general or particular, permanent or temporary. A general taboo applied, e.g., to a whole class of animals; a particular taboo was confined to one or more individuals of the class. Idols, temples, the persons and names of kings and of members of the royal family, the persons of chiefs and priests, and the property (canoes, houses, clothes, &c.) of all these classes of persons were always taboo or sacred. By a somewhat arbitrary extension of this principle a chief could render taboo to (i.e., in favour of) himself anything which took his fancy by merely calling it by the name of a part of his person. Thus, if he said "That axe is my backbone," or "is my head," the axe was his; if he roared out "That canoe! my skull shall be the baler to bale it out," the canoe was his likewise. The names of chiefs and still more of kings were taboo, and could not be uttered. If the name of a king of Tahiti was a common word or even resembled a common word, that word dropped out of use and a new name was substituted for it. Thus in course of time most of the common words in the language underwent considerable modifications or were entirely changed.

Duration.Certain foods were permanently taboo to (i.e., in favour of or for the use of) gods and men, but were forbidden to women. Thus in Hawaii the flesh of hogs, fowls, turtle, and several kinds of fish, cocoa-nuts, and nearly everything offered in sacrifice were reserved for gods and men, and could not, except in special cases, be consumed by women. In the Marquesas Islands human flesh was tabooed from women. Sometimes certain fruits, animals, and fish were taboo for months together from both men and women. In the Marquesas houses were tabooed against water: nothing was washed in them; no drop of water might be spilled in them. If an island or a district was tabooed, no canoe or person might approach it while the taboo lasted; if a path was tabooed, no one might walk on it. Seasons generally kept taboo were the approach of a great religious ceremony, the time of preparation for war, and the sickness of chiefs. The time during which they lasted varied from