at intervals of 01, and to 5 at intervals of 1, all to 5 places. This table is reprinted in Gauss's Werke, vol. iii. p. 244. E. A. Matthiessen, Tafel zur bequemern Berechnung (Altona, 1818), gives B and C to 7 places for argument A from 0 to 2 at intervals of .0001, thence to 3 at intervals of 001, to 4 at intervals of 01, and to 5 at intervals of 1; the table is not conveniently arranged. Peter Gray, Tables and Formulae (London, 1849, and "Addendum,' 1870), gives C for argument A from -3 to 1 at intervals of .001 and from 1 to 2 at intervals of 0001, to 6 places, with proportional parts to hundredths, and log (1-x) for argument A from-3 to 1 at intervals of 001 and from I to 18999 at intervals of .0001, to 6 places, with proportional parts. J. Zech, Tafeln der Additionsund Subtractions-Logarithmen (Leipzig, 1849), gives B for argument A from 0 to 2 at intervals of 0001, thence to 4 at intervals of 001 and to 6 at intervals of 01; also C for argument A from 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. T. Wittstein, Logarithmes de Gauss à sept décimales (Hanover, 1866), gives B for argument A from 3 to 4 at intervals of 1, from 4 to 6 at intervals of 01, from 6 to 8 at intervals of 001, from 8 to 10 at intervals of 0001, also from 0 to 4 at the same intervals. In this handsome work the arrangement is similar to that in a seven-figure logarithmic table. Gauss's original five-place table was reprinted in Pasquich, Tabulae (Leipzig, 1817); Köhler, Jerome de la Lande's Tafeln (Leipzig, 1832), and Handbuch (Leipzig, 1848); and Galbraith and Haughton, Manual (London, 1860). Houel, Tables de logarithmes (1871), also gives a small five-place table of Gaussian logarithms, the addition and subtraction logarithms being separated as in Zech. Modified Gaussian logarithms are given by J. H. T. Müller, Vierstellige Logarithmen (Gotha, 1844), viz., a four-place table of B and -fog (1-x-1) from A=0 to 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 B and log (1+x) from A-5 to 3 at intervals of 1, from A=3 to 2-7 at intervals of 01, to 1.3 at intervals of 001, to 3 at intervals of 01, and to 5 at intervals of I. Filipowski's Antilogarithms (1849) contains Gaussian logarithms arranged in a new way. The principal table gives log (x+1) as tabular result for log x as argument from 8 to 14 at intervals of 001 to 5 places. Weidenbach, Tafel um den Logarithx+1 men... (Copenhagen, 1829), gives log for argument A from -382 to 2.002 at intervals of 001, to 3.6 at intervals of 01, and to 5.5 at intervals of 1 to 5 places. J. Houel's Recueil de formules et de tables numériques (2nd ed., Paris, 1868) contains tables of log10(x+1), logo and logic from log x=-5 to -3 at intervals of I, from log x=-3 to 1 at intervals of 01, from log x1 to o at intervals of 001. F. W. Rex (Fünfstellige Logarithmen Tafeln, Stuttgart, 1884) gives also a five-figure table

I+x

-x

the constant -0.584962, which belongs to }, and entering the main table with 12 we take out the quadratic logarithm 10.109937 which, by applying the constant, gives 9.524975 the quadratic logarithm of the quantity required. An appendix (Tavola degli esponenti) gives the Briggian logarithms of the first 57 numbers to the first 50 numbers as base, viz. log, N for N=2, 3, . . ., 57 and x=2, 3,..., 50. The results are generally given to 6 places.

Logistic and Proportional Logarithms.-in most collections of tables of logarithms a five-place table of logistic logarithms for every second to 1 is given. Logistic tables give log 3600-log x at intervals of a second, x being expressed in degrees, minutes, and seconds. In Schulze (1778) and Vega (1797) the table extends to x=3600" and in Callet and Hutton to x=5280". Proportional logarithms for every second to 3° (i.e. log 10,800-log x) form part of nearly all collections of tables relating to navigation, generally to 4 places, sometimes to 5 Bagay, Tables (1829), gives a five-place table, but such are not often to be found in collections of mathematical tables. The same remark applies to tables of proportional logarithms for every minute to 24h, which give to 4 or 5 places the values of log 1440-log x. The object of a proportional or logistic table, or a table of log a-log x, is to facilitate the calculation of proportions in which the third term is a. Interpolation Tables.-All tables of proportional parts may be regarded as interpolation tables. C. Bremiker, Tafel der Proportionalteile (Berlin, 1843), gives proportional parts to hundredths of all numbers from 70 to 699. Schrön, Logarithms, contains an interpolation table giving the first hundred multiples of all numbers from 40 to 410. Sexagesimal tables, already described, are interpolation tables where the denominator is 60 or 600. Tables of the values of binomial theorem coefficients, which are required when second and higher orders of differences are used, are described below. W. S. B. Woolhouse, On Interpolation, Summation, and the Adjustment of Numerical Tables (London, 1865), contains nine pages of interpolation tables. The book consists of papers extracted from vols. 11 and 12 of the Assurance Magazine. Dual Logarithms.-This term was used by Oliver Byrne in his Dual Arithmetic, Young Dual Arithmetician, Tables of Dual Logarithms, &c. (London, 1863-67). A dual number of the ascending branch is a continued product of powers of 1.1, 101, 1.001, &c., taken in order, the powers only being expressed; thus 6,9,7,8 denotes (1.1)(1-01) (1·001) (1-0001), the numbers following the being called dual digits. A dual number which has all but the last digit zeros is called a dual logarithm; the author uses dual logarithms in which there are seven ciphers between the ↓ and the logarithm. Thus since 1.00601502 is equal to 0,0,0,0,0,0,0,599702 the whole number 599702 is the dual logarithm of the natural number 100601502.

A dual number of the descending branch is a continued product of powers of 9, 99, &c.: for instance, (9)(99) is denoted by '3'2 1. arithms, both of the ascending and descending branches, and the The Tables, which occupy 112 pages, give dual numbers and logcorresponding natural numbers. The author claimed that his tables were superior to those of common logarithms.

devoted to certain frequently used constants and their logarithms, Constants. In nearly all tables of logarithms there is a page

Quadratic Logarithms.-In a pamphlet Saggio di tavole dei loga: ritmi quadratici (Udine, 1885) Conte A. di Prampero has described a method of obtaining fractional powers (positive or negative) of any number by means of tables contained in the work. If log log N-log log a log b

a=N, then x=

and if the logarithms are taken to be Briggian and a=1OT7 and b=2, then x = log10 log10 N/log 2+10.

This quantity the author defines as the quadratic logarithm of N and denotes by LN. It follows from this definition that LNLN+logiolog102. Thus the quadratic logarithms of N and N where s is any power (positive or negative) of 2 have

the same mantissa.

A subsidiary table contains the values of the constant logo/logi02 for 204 fractional values of r. The main table contains the values of 1000 mantissae corresponding to arguments N, N, Nt,... (which all have the same mantissae). Among the arguments are the quantities 10-0, 101, 10-2,... 99.9 (the interval being 1) and 10.00, 10.01, ... 10.99 (the interval being 01). As an example, to obtain the value of 123 we take from the first table XXVI 6⭑

Templeton's Millwright's and Engineer's Pocket Companion (corrected by S. Maynard, London, 1871), which gives 58 constants involving and their logarithms, generally to 30 places, and 13 others that may be properly called mathematical. A good list of constants involving is given in Salomon (1827). A paper by G. Paucker in Grunert's Archiv ( vol. i. p. 9) has a number of constants involving given to a great many places, and Gauss's memoir on the lemniscate function (Werke, vol. iii.) has e, e-i-, et", &c., calculated to about 50 places. The quantity has been worked out to 707 places (Shanks, Proc. Roy. Soc., 21, p. 319).

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

Tables for the Solution of Cubic Equations.-Lambert, Supplementa (1798), gives (x-x3) from x=001 to 1.155 as intervals of 001 to 7 places, and Barlow (1814) gives x-x from x=1 to 1-1549 at intervals of 0001 to 8 places. Very extensive tables for the solution of cubic equations are contained in a memoir" Beiträge zur Auflösung höherer Gleichungen " by J. P. Kulik in the Abh, der k. Böhm. Ges. der Wiss. (Prague, 1860), 11, pp. 1-123. The principal tables (pp. 58-123) give to 7 (or 6) places the values of (x-x3) from x=0 to x=3-2800 at intervals of 001. There are also tables of the even and uneven determinants of cubic equations, &c. Other tables for the solution of equations are by A. S. Guldberg in the Forhand of the Videns-Selskab of Christiania for 1871 and 1872 (equations of the 3rd and 5th order), by S. Gundelfinger, Tafeln zur Berechnung der reellen Wurzeln sämtlicher trinomischen Gleichungen (Leipzig, 1897), which depend on the use of Gaussian logarithms, and by R. Mehme, Schlömülch's Zeitschrift, 1898, 43, p. 80 (quadratic equations).

from x=-01 to x=1 at intervals of 01 to 7 places (which are useful in interpolation by second and higher orders of differences), occur in Schulze (1778), Barlow (1814). Vega (1797 and succeeding editions), Hantschl (1827), and Kohler (1848). W. Rouse, Doctrine of Chances (London, n.d.), gives on a folding sheet (a+b)" for n=1, 2,...20. H. Gyldén (Recueil des Tables, Stockholm, 1880) gives binomial coefficients to n=40 and their logarithms to 7 places. Lambert, Supplementa (1798), has the coefficients of the first 16 terms in (1+x) and (1-x), their values being given accurately as decimals. 1 1.3 Vega (1797) has a page of tables giving

and 2.4 2.4.6 2.3" similar quantities to 10 places, with their logarithms to 7 places, and a page of this kind occurs in other collections. Köhler (1848) gives the values of 40 such quantities. Figurate Numbers.-Denoting n(n+1)...(n+i-1)/i! by [l, Lambert, Supplementa, 1798, gives [n], from n = 1 to n = 30 and from i=1 to i=12; and G. W. Hill (Amer. Jour. Math., 1884, 6, p. 130) gives log[] for n=1, 1, 1, 1. 1, and from i=1 to -30.

Trigonometrical Quadratic Surds.-The surd values of the sines of every third degree of the quadrant are given in some tables of logarithms; e.g., in Hutton's (p. xxxix., ed. 1855), we find sin 3v (5+ √5) + √ 12 2 + √ {−√(15+3√5) −√√; and the numerical values of the surds √(5+√5), √(Y), &c., are given to 10 places. These values were extended to 20 places by Peter Gray, Mess. of Math., 1877, 6, p. 105. Circulating Decimals. Goodwyn's tables have been described already. Several others have been published giving the numbers of digits in the periods of the reciprocals of primes: Burckhardt, Tables des diviseurs du premier million (Paris, 1814-1817), gave one for all primes up to 2543 and for 22 primes exceeding that limit. E. Desmarest, Théorie des nombres (Paris, 1852), included all primes up to 10,000. C. G. Reuschle, Mathematische Abhandlung, enthaltend neue zahlentheoretische Tabellen (1856), contains a similar table to 15,000. This W. Shanks extended to 60,000; the portion from I to 30,000 is printed in the Proc. Roy. Soc., 22, p. 200, and the remainder is preserved in the archives of the society (Id., 23, p. 260 and 24, p. 392). The number of digits in the decimal period of 1p. is the same as the exponent to which 10 belongs for modulus p, so that, whenever the period has p-1 digits, 10 is a primitive root of p. Tables of primes having a given number, n, of digits in their periods, i.e. tables of the resolutions of 10"-1 into factors and, as far as known, into prime factors, have been given by W. Looff (in Grunert's Archiv, 16, p. 54; reprinted in Nouv. annales, 14, p. 115) and by Shanks (Proc. Roy. Soc., 22, p. 381). The former extends to n = 60 and the latter to n=100, but there are gaps in both. Reuschle's tract also contains resolutions of 10"-1.

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

Pythagorean Triangles.-Right-angled triangles in which the sides and 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 (w being one of the acute angles). About 100 triangles are given, but some occur twice. Large tables of right-angled rational triangles were given by C. A. Bretschneider, in Grunert's Archiv, 1841, 1, p. 96, and by Sang, Trans. Roy. Soc. Edin., 1864, 33, p. 727. In these tables the triangles are arranged according to hypothenuses and extend to 1201, 1200, 49, and 1105, 1073, 264 respectively. W. A. Whitworth, in a paper read before the Lit. and Phil. Society of Liverpool in 1875, carried his list as far as 2465, 2337, 784. See also H. Rath, "Die rationalen Dreiecke," in Grunert's Archiv, 1874, 56, p. 188. Sang's paper also contains a table of triangles having an angle of 120 and their sides integers.

Powers of .-G. Paucker, in Grunert's Archiv, p. 10, gives 1 and to 140 places, and a,, i, i to about 50 places; J. Burgess (Trans. Roy. Soc. Edin., 1898, 39. II., No. 9. p. 23) gives (), 2, and some other constants involving as well as their Briggian logarithms to 23 places, and in Maynard's list of constants (see Constants, above) 2 is given to 31 places. The first twelve powers of and to 22 or more places were given by Glaisher, in Proc. Lond. Math. Soc., 8, p. 140, and the first hundred multiples of and to 12 places by J. P. Kulik, Tafel der Quadrat- und Kubik-Zahlen (Leipzig, 1848). The Series 1+2+3+&c.—Let S, Sa, on denote respectively the sums of the series 1+2+3+ &c., 1-2"+3"-&c., !

1+3+5+ &c. Legendre (Traité des fonctions elliptiques, vol. 2, p. 432) computed S. to 16 places from 1 to 35, and Glaisher (Proc. Lond. Math. Soc., 4, p. 48) deduced s, and on for the same arguments and to the same number of places. The latter also gave S. Sn, on for n=2, 4, 6,...12 to 22 or more places (Proc. Lond. Math. Soc., 8, p. 140), and the values of 2, where En=2+3+ 5+&c. (prime numbers only involved), for n=2, 4, 6,... 36 to 15 places (Compte rendu de l'Assoc. Française, 1878, p. 172). C. W. Merrifield (Proc. Roy. Soc., 1881, 33, p. 4) gave the values of log. S. and Zn for n=1,2,3,.... 35 to 15 places, and Glaisher (Quar. Jour. Math., 1891, 25, p. 347) gave the values of the same quantities for n=2,4,6, 80 to 24 places (last figure uncertain). Merrifield's table was reprinted by J. P. Gram on p. 269 of the paper of 1884, referred to under Sine-integral, &c., who also added the values of logo S for the same arguments to 15 places. An error in E, in Merrifield's table is pointed out in Quar. Jour. Math., 25, p. 373. This quantity is correctly given in Gram's reprint. T. J. Stieljes has greatly extended Legendre's table of S. His table (Acta math., 1887, 10, p. 299) gives S for all values of n up to n = 70 to 32 places. Except for six errors of a unit in the last figure he found Legendre's table to be correct. Legendre's table was reprinted in De Morgan's Diff. and Int. Calc. (1842), p. 554. Various small tables of other series, involving inverse powers of prime numbers, such as 3-5+7+11"-13"+..., are given in vols. 25 and 26 of the Quar. Jour. Math.

Tables of e and e, or Hyperbolic Antilogarithms.-The largest tables are the following: C. Gudermann, Theorie der potenzial- oder cyklisch-hyperbolischen Functionen (Berlin, 1833), which consists of papers reprinted from vols. 8 and 9 of Crelle's Journal, and gives logo sinh x, logo cosh x, and logo tanh x from x=2 to 5 at intervals of 001 to 9 places and from x=5 to 12 at intervals of or to 10 places. Since sinh x=4(e-e) and cosh x=(e+), the values of e and are deducible at once by addition and subtraction. F. W. Newman, in Camb. Phil. Trans., 13, p. 145, gives values of e from x=0 to 15.349 at intervals of 001 to 12 places, from x=15.350 to 17-298 at intervals of ⚫002, and from x = 17.300 to 27-635 at intervals of 005, to 14 places. Glaisher, in Camb. Phil. Trans., 13, p. 243, gives four tables of e2, es, logo e, logo e, 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 I, from 1 to 500 at intervals of unity. Vega, Tabulae (1797 and later ed.), has logo e to 7 places and e* to 7 figures from x=-01 to 10 at intervals of or. Köhler's Handbuch contains a small table of e. In Schulze's Sammlung (1778) e is given for x=1, 2, 3,... 24 to 28 or 29 figures and for x=25, 30, and 60 to 32 or 33 figures; this table is reprinted in Glaisher's paper (loc. cit.). In Salomon's Tafeln (1827) the values of e, c", en, e·00,.. e000000, where n has the values 1, 2,...9, are given to 12 places. Bretschneider, in Grunert's Archiv, 3, p. 33. gave e and eand also sin x and cos x for x=1, 2,...10 to 20 places, and J. P. Gram (in his paper of 1884, referred to under Sineintegral, &c.), gives et for x=10, 11,...20 to 24 places, and from Burgess (Trans. Roy. Soc. Edin., 1888, 39, II. No. 9) has given (p. 26) x=7 to x=20 at intervals of 0-2 to 10, 13, 14, or 15 places. J. the values of and for x= and for x=1, 2,..., 10 to 30

16 places.

Factorials.-The values of log10 (n!), where n! denotes 1.2.3... n, from n=1 to 1200 to 18 places, are given by C. F. Degen, Tabularum Enncas (Copenhagen, 1824), and reprinted, to 6 places, at the end of De Morgan's article "Probabilities" in the Encyclopaedia Metropolitana. Shortrede, Tables (1849, vol. i.), gives log (n!) to n = 1000 to 5 places, and for the arguments ending in 0 to 8 places. Degen also gives the complements of the logarithms. The first 20 figures of the values of nXn! and the values of -log (nXn!) to 10 places are given by Glaisher as far as n = 71 in the Phil. Trans. for 1870 (p. 370), and the values of 1/n! to 28 significant figures as far as n=50 in Camb. Phil. Trans., 13, p. 246.

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

Tables of log tan (}+14).—C. Gudermann, Theorie der potenzialoder cyklisch-hyperbolischen Functionen (Berlin, 1833), gives (in 100 pages) log tan (+14) for every centesimal minute of the quadrant to 7 places. Another table contains the values of this function.

beauty. The view from the summit overlooking Table Bay is also one of much grandeur.

The south-east winds which sweep over Table Mountain frequently cause the phenomenon known as " The Table-cloth." The summit of the mountain is then covered by a whitish-grey cloud, which is being constantly forced down the northern face towards Cape Town, but never reaches the lower slopes. The clouds (not always caused by the south-easter) form very suddenly, and the weather on the mountain is exceedingly changeable. The rainfall on the summit is heavy, 72-14 inches a year being the average of twelve years' observations. This compares with an average of 54.63 inches at Bishop's Court, Newlands, at the foot of the mountain on the east and with 25.43 inches at Cape Town at the northern foot of the mountain. The relative luxuriance of the vegetation on the upper part of the mountain, compared with that of its lower slopes, is due not only to the rainfall, but to the large additional moisture condensed from clouds. The result of experiments conducted by Dr Marloth (Trans. S. Afrn. Phil. Soc. for 1903 and 1905) goes to show that during cloudy weather the summit of the mountain resembles an immense sponge, and that this condensation of moisture considerably influences the yield of the springs in the lower part of the mountain.

TABLE-TURNING. When the movement of modern spirit: ualism first reached Europe from America in the winter of 1852-3, the most popular method of consulting the "spirits" was for several persons to sit round a table, with their hands resting on it, and wait for the table to move. If the experiment was successful the table would rotate with considerable rapidity, and would occasionally rise in the air, or perform other movements. Whilst by many the movements were ascribed to the agency of spirits, two investigators-count de Gasparin and Professor Thury of Geneva-conducted a careful series of experiments by which they claimed to have demonstrated that the movements of the table were due to a physical force emanating from the bodies of the sitters, for which they proposed the name "ectenic force." Their conclusion rested on the supposed elimination of all known physical causes for the movements; but it is doubtful from the description of the experiments whether the precautions taken were sufficient to exclude unconscious muscular action or even deliberate fraud.

In England table-turning became a fashionable diversion and was practised all over the country in the year 1853. Dr John Elliotson and his followers attributed the phenomena to mesmerism. The general public were content to find the explanation of the movements in spirits, animal magnetism, odic force, galvanism, electricity, or even the rotation of the earth. James Braid, W. B. Carpenter and others pointed out, however, that the phenomena obviously depended upon the expectation of the sitters, and could be stopped altogether by appropriate suggestion. And Faraday devised some simple apparatus which conclusively demonstrated that the movements were due to unconscious muscular action. The apparatus consisted of two small boards, with glass rollers between them, the whole fastened together by indiarubber bands in such a manner that the upper board could slide under lateral pressure to a limited extent over the lower one. The occurrence of such lateral movement was at once indicated by means of an upright haystalk fastened to the apparatus. When by this means it was made clear to the experimenters that it was the fingers which moved the table, not the table the fingers, the phenomena generally ceased. The movements were in fact simply an illustration of automatism. But Faraday's demonstration did little to stop the popular craze.

By believers the table was made to serve as a means of communicating with the spirits; the alphabet would be slowly called over and the table would tilt at the appropriate letter, thus spelling out words and sentences. Some Evangelical clergymen discovered by this means that the spirits who caused the movements were of a diabolic nature, and some amazing accounts were published in 1853 and 1854 of the revelations obtained from the talking tables.

Table-turning is still in vogue amongst spiritualist circles. The device was employed with success by Professor Charles Richet and others in thought-transference experiments.

See A. E. de Gasparin, Des Tables tournantes, du Surnaturel, &c. Faraday's letter on Table-turning in The Times, 30th June 1853. (Paris, 1854): Thury, Des Tables tournantes (Geneva, 1855); Quarterly Review, Sept. 1853-article by Carpenter on Spiritualism, &c.; Mrs De Morgan, From Matter to Spirit (London, 1863); Ch. Richet, Proceedings S.P.R., vol. v. F. Podmore, Modern Spiritualism (London, 1902), ii. 7-21, gives an account of the movement in 1853, with references to contemporary pamphlets and newspaper articles. (F. P.)

TABLINUM (or tabulinum, from tabula, board, picture), in Roman architecture, the name given to an apartment generally situated on one side of the atrium and opposite to the entrance; it opened in the rear on to the peristyle, with either a large window or only an anteroom or curtain. The walls were richly decorated with fresco pictures, and busts of the family were arranged on pedestals on the two sides of the room.

TABOO (also written tapu and tabu), the Polynesian name given to prohibitions enforced by religious or magical sanctions. As a verb it means to “prohibit," as an adjective “prohibited, sacred, dangerous, unclean."

1. The word "taboo" or its dialectical forms are found throughout Polynesia; in Melanesia the term is tambu; in various parts of Malaysia and the East Indies pantang, bobosso, pamalli, &c.; in Madagascar fadi includes taboo; in North America the Dakota term wakan bears a similar meaning. Taboo is perhaps derived from ta, to mark, and pu, an adverb of intensity.

2. Fundamental Ideas.-In taboo proper are combined two notions which with the progress of civilization have become differentiated—(i) sacred and (ii.) impure, or unclean; it must be borne in mind that the impurity is sacred, and is not derived from contact with common things. It does not imply any moral quality; it has been defined as an indication of "a connexion with the gods, or a separation from ordinary pur poses and exclusive appropriation to persons or things considered sacred; sometimes it means devoted by a vow." This definition does not cover the whole connotation of taboo as it is employed at the present day, but it indicates clearly the non-moral character of the idea. The ordinary usage is perhaps best defined-the statement that taboo is "negative magic," i.e. abstinence from certain acts, in order that undesired magical results may not follow; in this sense a taboo is simply a ritual prohibition. Properly speaking taboo includes only (a) the sacred (or unclean) character of persons or things, (b) the kind of prohibition which results from this character, and (c) the sanctity (or uncleanness) which results from a violation of the prohibition. The converse of taboo in Polynesia is noa and allied forms, which mean" general "or" common "; by a curious coincidence noa is the term used in Central Australia to express the relation of persons of opposite sexes on whose intercourse there is no restriction.

3. Classification.-Various classes of taboo in the wider sense may be distinguished: (i) natural or direct, the result of mana (mysterious power) inherent in a person or thing; (ii.) communicated or indirect, equally the result of mana, but (a) acquired or (b) imposed by a priest, chief or other person; (iii.) intermediate, where both factors are present, as in the appropriation of a wife to her husband. These three classes are those of taboo proper. The term taboo is also applied to ritual prohibitions of a different nature; but its use in these senses is better avoided. It might be argued that the term should be extended to embrace cases in which the sanction of the prohibition is the creation of a god or spirit, i.e. to religious interdictions as distinguished from magical, but there is neither automatic action nor contagion in such a case, and a better term for it is Religious interdiction.

4. Objects. The objects of taboo are many: (i.) direct taboos aim at (a) the protection of important persons-chiefs, priests, &c. and things against harm; (b) the safeguarding