Chapter III: Front Matter (3)

is invariant for all equivalent representations, when written as a
polynomial in [lambda]. Moreover, it has the same value for S and S',
if these are two conjugate operations in G. Of the various invariants
that thus arise the most important is s11 + s22 + ... + s_(mm), which
is called the "characteristic" of S. If S is an operation of order p,
its characteristic is the sum of m pth roots of unity; and in
particular, if S is the identical operation its characteristic is m.
If r is the number of sets of conjugate operations in G, there is, for
each representation of G as an irreducible group, a set of r
characteristics: X1, X2, ... X_r, one corresponding to each conjugate
set; so that for the r irreducible representations just r such sets of
characteristics arise. These are distinct, in the sense that if
[Psi]1, [Psi]2, ..., [Psi]_r are the characteristics for a distinct
representation from the above, then X_i and [Psi]_i are not equal for
all values of the suffix i. It may be the case that the r
characteristics for a given representation are all real. If this is so
the representation is said to be self-inverse. In the contrary case
there is always another representation, called the "inverse"
representation, for which each characteristic is the conjugate
imaginary of the corresponding one in the original representation. The
characteristics are subject to certain remarkable relations. If h_p
denotes the number of operations in the pth conjugate set, while
X^_i{p}, and X^j{p} are the characteristics of the pth conjugate set
in [Gamma]_i and [Gamma]_j, then

__p=r
\ h_p X_p^i X^_p^j = 0 or n,
/__p=1

according to [Gamma]_i and [Gamma]_j are not or are inverse
representations, n being the order of G.

Again

__i=r
\ X_p^i X^_q^i = 0 or n/h_p
/__i=1

according as the pth and qth conjugate sets are not or are inverse;
the qth set being called the inverse of the pth if it consists of the
inverses of the operations constituting the pth.

Linear homogeneous groups.

Another form in which every group of finite order can be represented
is that known as a linear homogeneous group. If in the equations

x'_r = a_(r1)x1 + a_(r2)x2 + ... +a_(rm)x_m, (r = 1, 2, ..., m),

which define a linear homogeneous substitution, the coefficients are
integers, and if the equations are replaced by congruences to a finite
modulus n, the system of congruences will give a definite operation,
provided that the determinant of the coefficients is relatively prime
to n. The product of two such operations is another operation of the
same kind; and the total number of distinct operations is finite,
since there is only a limited number of choices for the coefficients.
The totality of these operations, therefore, constitutes a group of
finite order; and such a group is known as a _linear homogeneous_
group. If n is a prime the order of the group is

(n^m - 1)(n^m - n) ... (n^m - n^(m-1)).

The totality of the operations of the linear homogeneous group for
which the determinant of the coefficients is congruent to unity forms
a subgroup. Other subgroups arise by considering those operations
which leave a function of the variables unchanged (mod. n). All such
subgroups are known as linear homogeneous groups.

When the ratios only of the variables are considered, there arises a
_linear fractional_ group, with which the corresponding linear
homogeneous group is isomorphic. Thus, if p is a prime the totality of
the congruences

az + b
z' [equiv] ------, ad - bc [/=] 0, (mod. p)
cz + d

constitutes a group of order p(p^2 - 1). This class of groups for
various values of p is almost the only one which has been as yet
exhaustively analysed. For all values of p except 3 it contains a
simple self-conjugate subgroup of index 2.

A great extension of the theory of linear homogeneous groups has been
made in recent years by considering systems of congruences of the form

x'_r [equiv] a_(r1)x1 + a_(r2)x2 + ... + a_(rm)x_m,
(r = 1, 2, ..., m),

in which the coefficients a_(rs), are integral functions with real
integral coefficients of a root of an irreducible congruence to a
prime modulus. Such a system of congruences is obviously limited in
numbers and defines a group which contains as a subgroup the group
defined by the same congruences with ordinary integral coefficients.

Applications.

The chief application of the theory of groups of finite order is to
the theory of algebraic equations. The analogy of equations of the
second, third and fourth degrees would give rise to the expectation
that a root of an equation of any finite degree could be expressed in
terms of the coefficients by a finite number of the operations of
addition, subtraction, multiplication, division, and the extraction of
roots; in other words, that the equation could be solved by radicals.
This, however, as proved by Abel and Galois, is not the case: an
equation of a higher degree than the fourth in general defines an
algebraic irrationality which cannot be expressed by means of
radicals, and the cases in which such an equation can be solved by
radicals must be regarded as exceptional. The theory of groups gives
the means of determining whether an equation comes under this
exceptional case, and of solving the equation when it does. When it
does not, the theory provides the means of reducing the problem
presented by the equation to a normal form. From this point of view
the theory of equations of the fifth degree has been exhaustively
treated, and the problems presented by certain equations of the sixth
and seventh degrees have actually been reduced to normal form.

Galois (see EQUATION) showed that, corresponding to every irreducible
equation of the nth degree, there exists a transitive
substitution-group of degree n, such that every function of the roots,
the numerical value of which is unaltered by all the substitutions of
the group can be expressed rationally in terms of the coefficients,
while conversely every function of the roots which is expressible
rationally in terms of the coefficients is unaltered by the
substitutions of the group. This group is called the group of the
equation. In general, if the equation is given arbitrarily, the group
will be the symmetric group. The necessary and sufficient condition
that the equation may be soluble by radicals is that its group should
be a soluble group. When the coefficients in an equation are rational
integers, the determination of its group may be made by a finite
number of processes each of which involves only rational arithmetical
operations. These processes consist in forming resolvents of the
equation corresponding to each distinct type of subgroup of the
symmetric group whose degree is that of the equation. Each of the
resolvents so formed is then examined to find whether it has rational
roots. The group corresponding to any resolvent which has a rational
root contains the group of the equation; and the least of the groups
so found is the group of the equation. Thus, for an equation of the
fifth degree the various transitive subgroups of the symmetric group
of degree five have to be considered. These are (i.) the alternating
group; (ii.) a soluble group of order 20; (iii.) a group of order 10,
self-conjugate in the preceding; (iv.) a cyclical group of order 5,
self-conjugate in both the preceding. If x0, x1, x2, x3, x4 are the
roots of the equation, the corresponding resolvents may be taken to be
those which have for roots (i.) the square root of the discriminant;
(ii.) the function (x0x1 + x1x2 + x2x3 + x3x4 + x4x0)(x0x2 + x2x4 +
x4x1 + x1x3 + x3x0); (iii.) the function x0x1 + x1x2+ x2x3 + x3x4 +
x4x0; and (iv.) the function x0^2x1 + x1^2x2 + x2^2x3 + x3^2x4 +
x4^2x0. Since the groups for which (iii.) and (iv.) are invariant are
contained in that for which (ii.) is invariant, and since these are
the only soluble groups of the set, the equation will be soluble by
radicals only when the function (ii.) can be expressed rationally in
terms of the coefficients. If

(x0x1 + x1x2 + x2x3 + x3x4 + x4x0)(x0x2 + x2x4 + x4x1 + x1x3 + x3x0)

is known, then clearly x0x1 + x1x2 + x2x3 + x3x4 + x4x0 can be
determined by the solution of a quadratic equation. Moreover, the sum
and product (x0 + [epsilon]x1 + [epsilon]^2x2 + [epsilon]^3x3 +
[epsilon]^4x4)^5 and (x0 + [epsilon]^4x1+[epsilon]^3x2 + [epsilon]^2x3
+ [epsilon]x4)^5 can be expressed rationally in terms of x0x1 + x1x2 +
x2x3 + x3x4 + x4x0, [epsilon], and the symmetric functions; [epsilon]
being a fifth root of unity. Hence (x0 + [epsilon]x1 + [epsilon]^2x2 +
[epsilon]^3x3 + [epsilon]^4X4)^5 can be determined by the solution of
a quadratic equation. The roots of the original equation are then
finally determined by the extraction of a fifth root. The problem of
reducing an equation of the fifth degree, when not soluble by
radicals, to a normal form, forms the subject of Klein's _Vorlesungen
uber das Ikosaeder_. Another application of groups of finite order is
to the theory of linear differential equations whose integrals are
algebraic functions. It has been already seen, in the discussion of
discontinuous groups in general, that the groups of such equations
must be groups of finite order. To every group of finite order which
can be represented as an irreducible group of linear substitutions on
n variables will correspond a class of irreducible linear differential
equations of the nth order whose integrals are algebraic. The complete
determination of the class of linear differential equations of the
second order with all their integrals algebraic, whose group has the
greatest possible order, viz. 120, has been carried out by Klein.

AUTHORITIES.--Continuous groups: Lie and Engel, _Theorie der
Transformationsgruppen_ (Leipzig, vol. i., 1888; vol. ii., 1890; vol.
iii., 1893); Lie and Scheffers, _Vorlesungen uber gewohnliche
Differentialgleichungen mit bekannten infinitesimalen
Transformationen_ (Leipzig, 1891); _Idem, Vorlesungen uber
continuierliche Gruppen_ (Leipzig, 1893); _Idem, Geometrie der
Beruhrungstransformationen_ (Leipzig, 1896); Klein and Schilling,
_Hohere Geometrie_, vol. ii. (lithographed) (Gottingen, 1893, for both
continuous and discontinuous groups). Campbell, _Introductory Treatise
on Lie's Theory of Finite Continuous Transformation Groups_ (Oxford,
1903). Discontinuous groups: Klein and Fricke, _Vorlesungen uber die
Theorie der elliptischen Modulfunktionen_ (vol. i., Leipzig, 1890)
(for a full discussion of the modular group); _Idem, Vorlesungen uber
die Theorie der automorphen Funktionen_ (vol. i., Leipzig, 1897; vol.
ii. pt. i., 1901) (for the general theory of discontinuous groups);
Schoenflies, _Krystallsysteme und Krystallstruktur_ (Leipzig, 1891)
(for discontinuous groups of motions); Groups of finite order: Galois,
_[OE]uvres mathematiques_ (Paris, 1897, reprint); Jordan, _Traite des
substitutions et des equations algebriques_ (Paris, 1870); Netto,
_Substitutionentheorie und ihre Anwendung auf die Algebra_ (Leipzig,
1882; Eng. trans. by Cole, Ann Arbor, U.S.A., 1892); Klein,
_Vorlesungen uber das Ikosaeder_ (Leipzig, 1884; Eng. trans. by
Morrice, London, 1888); H. Vogt, _Lecons sur la resolution algebrique
des equations_ (Paris, 1895); Weber, _Lehrbuch der Algebra_
(Braunschweig, vol. i., 1895; vol. ii., 1896; a second edition
appeared in 1898); Burnside, _Theory of Groups of Finite Order_
(Cambridge, 1897); Bianchi, _Teoria dei gruppi di sostituzioni e delle
equazioni algebriche_ (Pisa, 1899); Dickson, _Linear Groups with an
Exposition of the Galois Field Theory_ (Leipzig, 1901); De Seguier,
_Elements de la theorie des groupes abstraits_ (Paris, 1904), A
summary with many references will be found in the _Encyklopadie der
mathematischen Wissenschaften_ (Leipzig, vol. i., 1898, 1899).
(W. Bu.)

FOOTNOTE:

[1] The word "group," which appears first in English in the sense of
an assemblage of figures in an artistic design, picture, &c., is
adapted from the Fr. _groupe_, which is to be referred to the
Teutonic word meaning "knot," "mass," "bunch," represented in English
by "crop" (q.v.). The technical mathematical sense is not older than
1870.

GROUSE, a word of uncertain origin,[1] now used generally by ornithologists to include all the "rough-footed" Gallinaceous birds, but in common speech applied almost exclusively, when used alone, to the _Tetrao scoticus_ of Linnaeus, the _Lagopus scoticus_ of modern systematists--more particularly called in English the red grouse, but till the end of the 18th century almost invariably spoken of as the Moor-fowl or Moor-game. The effect which this species is supposed to have had on the British legislature, and therefore on history, is well known, for it was the common belief that parliament always rose when the season for grouse-shooting began (August 12th); while according to the _Orkneyinga Saga_ (ed. Jonaeus, p. 356; ed. Anderson, p. 168) events of some importance in the annals of North Britain followed from its pursuit in Caithness in the year 1157.

The red grouse is found on moors from Monmouthshire and Derbyshire northward to the Orkneys, as well as in most of the Hebrides. It inhabits similar situations throughout Wales and Ireland, but it does not naturally occur beyond the limits of the British Islands,[2] and is the only species among birds peculiar to them. The word "species" may in this case be used advisedly (since the red grouse invariably "breeds true," it admits of an easy diagnosis, and it has a definite geographical range); but scarcely any zoologist can doubt of its common origin with the willow-grouse, _Lagopus albus_ (_L. subalpinus_ or _L. saliceti_ of some authors), that inhabits a subarctic zone from Norway across the continents of Europe and Asia, as well as North America from the Aleutian Islands to Newfoundland. The red grouse indeed is rarely or never found away from the heather on which chiefly it subsists; while the willow-grouse in many parts of the Old World seems to prefer the shrubby growth of berry-bearing plants (_Vaccinium_ and others) that, often thickly interspersed with willows and birches, clothes the higher levels or the lower mountain-slopes, and it flourishes in the New World where heather scarcely exists, and a "heath" in its strict sense is unknown. It is true that the willow-grouse always becomes white in winter, which the red grouse never does; but in summer there is a considerable resemblance between the two species, the cock willow-grouse having his head, neck and breast of nearly the same rich chestnut-brown as his British representative, and, though his back be lighter in colour, as is also the whole plumage of his mate, than is found in the red grouse, in other respects the two species are precisely alike. No distinction can be discovered in their voice, their eggs, their build, nor in their anatomical details, so far as these have been investigated and compared.[3] Moreover, the red grouse, restricted as is its range, varies in colour not inconsiderably according to locality.

Though the red grouse does not, after the manner of other members of the genus _Lagopus_, become white in winter, Scotland possesses a species of the genus which does. This is the ptarmigan, _L. mutus_ or _L. alpinus_, which differs far more in structure, station and habits from the red grouse than that does from the willow-grouse, and in Scotland is far less abundant, haunting only the highest and most barren mountains. It is said to have formerly inhabited both Wales and England, but there is no evidence of its appearance in Ireland. On the continent of Europe it is found most numerously in Norway, but at an elevation far above the growth of trees, and it occurs on the Pyrenees and on the Alps. It also inhabits northern Russia. In North America, Greenland and Iceland it is represented by a very nearly allied form--so much so indeed that it is only at certain seasons that the slight difference between them can be detected. This form is the _L. rupestris_ of authors, and it would appear to be found also in Siberia (_Ibis_, 1879, p. 148). Spitzbergen is inhabited by a large form which has received recognition as _L. hemileucurus_, and the northern end of the chain of the Rocky Mountains is tenanted by a very distinct species, the smallest and perhaps the most beautiful of the genus, _L. leucurus_, which has all the feathers of the tail white.

The bird, however, to which the name of grouse in all strictness belongs is probably the _Tetrao tetrix_ of Linnaeus--the blackcock and greyhen, as the sexes are respectively called. It is distributed over most of the heath-country of England, except in East Anglia, where attempts to introduce it have been only partially successful. It also occurs in North Wales and very generally throughout Scotland, though not in Orkney, Shetland or the Outer Hebrides, nor in Ireland. On the continent of Europe it has a very wide range, and it extends into Siberia. In Georgia its place is taken by a distinct species, on which a Polish naturalist (_Proc. Zool. Society_, 1875, p. 267) has conferred the name of _T. mlokosiewiczi_. Both these birds have much in common with their larger congener the capercally and its eastern representative.

The species of the genus _Bonasa_, of which the European _B. sylvestris_ is the type, does not inhabit the British Islands. It is perhaps the most delicate game-bird that comes to table. It is the _gelinotte_ of the French, the _Haselhuhn_ of Germans, and _Hjerpe_ of Scandinavians. Like its transatlantic congener _B. umbellus_, the ruffed grouse or birch-partridge (of which there are two other local forms, _B. umbelloides_ and _B. sabinii_), it is purely a forest-bird. The same may be said of the species of _Canace_, of which two forms are found in America, _C. canadensis_, the spruce-partridge, and _C. franklini_, and also of the Siberian _C. falcipennis_. Nearly allied to these birds is the group known as _Dendragapus_, containing three large and fine forms _D. obscurus_, _D. fuliginosus_, and _D. richardsoni_--all peculiar to North America. Then there are _Centrocercus urophasianus_, the sage-cock of the plains of Columbia and California, and _Pedioecetes_, the sharp-tailed grouse, with its two forms, _P. phasianellus_ and _P. columbianus_, while finally _Cupidonia_, the prairie-hen, also with two local forms, _C. cupido_ and _C. pallidicincta_, is a bird that in the United States of America possesses considerable economic value, enormous numbers being consumed there, and also exported to Europe.

The various sorts of grouse are nearly all figured in Elliot's
_Monograph of the Tetraoninae_, and an excellent account of the
American species is given in Baird, Brewer and Ridgway's _North
American Birds_ (iii. 414-465). See also SHOOTING. (A. N.)

FOOTNOTES:

[1] It seems first to occur (O. Salusbury Brereton, _Archaeologia_,
iii. 157) as "grows" in an ordinance for the regulation of the royal
household dated "apud Eltham, mens. Jan. 22 Hen. VIII.," i.e. 1531,
and considering the locality must refer to black game. It is found in
an Act of Parliament 1 Jac. I. cap. 27, S 2, i.e. 1603, and, as
reprinted in the _Statutes at Large_, stands as now commonly spelt,
but by many writers or printers the final e was omitted in the 17th
and 18th centuries. In 1611 Cotgrave had "Poule griesche. A
Moore-henne; the henne of the Grice [in ed. 1673 "Griece"] or
Mooregame" (_Dictionarie of the French and English Tongues, s.v.
Poule_). The most likely derivation seems to be from the old French
word _griesche_, _greoche_ or _griais_ (meaning speckled, and cognate
with _griseus_, grisly or grey), which was applied to some kind of
partridge, or according to Brunetto Latini (_Tres._ p. 211) to a
quail, "porce que ele fu premiers trovee en Grece." The Oxford
Dictionary repudiates the possibility of "grouse" being a spurious
singular of an alleged plural "grice," and, with regard to the
possibility of "grows" being a plural of "grow," refers to Giraldus
Cambrensis (c. 1210), _Topogr. Hib. opera_ (Rolls) v. 47: "gallinae
campestres, quas vulgariter _grutas_ vocant."

[2] It was successfully, though with much trouble, introduced by Mr
Oscar Dickson on a tract of land near Gottenburg in Sweden (_Svenska
Jagarforbundets Nya Tidskrift_, 1868, p. 64 _et alibi_).

[3] A very interesting subject for discussion would be whether
_Lagopus scoticus_ or _L. albus_ has varied most from the common
stock of both. Looking to the fact that the former is the only
species of the genus which does not assume white clothing in winter,
an evolutionist might at first deem the variation greatest in its
case; but then it must be borne in mind that the species of _Lagopus_
which turn white differ in that respect from all other groups of the
family _Tetraonidae_. Furthermore every species of _Lagopus_ (even
_L. leucurus_, the whitest of all) has its first set of _remiges_
coloured brown. These are dropped when the bird is about half-grown,
and in all the species but _L. scoticus_ white _remiges_ are then
produced. If therefore the successive phases assumed by any animal in
the course of its progress to maturity indicate the phases through
which the species has passed, there may have been a time when all the
species of _Lagopus_ wore a brown livery even when adult, and the
white dress donned in winter has been imposed upon the wearers by
causes that can be easily suggested. The white plumage of the birds
of this group protects them from danger during the snows of a
protracted winter. But the red grouse, instead of perpetuating
directly the more ancient properties of an original _Lagopus_ that
underwent no great seasonal change of plumage, may derive its
ancestry from the widely-ranging willow-grouse, which in an epoch
comparatively recent (in the geological sense) may have stocked
Britain, and left descendants that, under conditions in which the
assumption of a white garb would be almost fatal to the preservation
of the species, have reverted (though doubtless with some
modifications) to a comparative immutability essentially the same as
that of the primal _Lagopus_.

GROVE, SIR GEORGE (1820-1900), English writer on music, was born at Clapham on the 13th of August 1820. He was articled to a civil engineer, and worked for two years in a factory near Glasgow. In 1841 and 1845 he was employed in the West Indies, erecting lighthouses in Jamaica and Bermuda. In 1849 he became secretary to the Society of Arts, and in 1852 to the Crystal Palace. In this capacity his natural love of music and enthusiasm for the art found a splendid opening, and he threw all the weight of his influence into the task of promoting the best music of all schools in connexion with the weekly and daily concerts at Sydenham, which had a long and honourable career under the direction of Mr (afterwards Sir) August Manns. Without Sir George Grove that eminent conductor would hardly have succeeded in doing what he did to encourage young composers and to educate the British public in music. Grove's analyses of the Beethoven symphonies, and the other works presented at the concerts, set the pattern of what such things should be; and it was as a result of these, and of the fact that he was editor of _Macmillan's Magazine_ from 1868 to 1883, that the scheme of his famous _Dictionary of Music and Musicians_, published from 1878 to 1889 (new edition, edited by J. A. Fuller Maitland, 1904-1907), was conceived and executed. His own articles in that work on Beethoven, Mendelssohn and Schubert are monuments of a special kind of learning, and that the rest of the book is a little thrown out of balance owing to their great length is hardly to be regretted. Long before this he had contributed to the _Dictionary of the Bible_, and had promoted the foundation of the Palestine Exploration Fund. On a journey to Vienna, undertaken in the company of his lifelong friend, Sir Arthur Sullivan, the important discovery of a large number of compositions by Schubert was made, including the music to _Rosamunde_. When the Royal College of Music was founded in 1882 he was appointed its first director, receiving the honour of knighthood. He brought the new institution into line with the most useful European conservatoriums. On the completion of the new buildings in 1894 he resigned the directorship, but retained an active interest in the institution to the end of his life. He died at Sydenham on the 28th of May 1900.

His life, a most interesting one, was written by Mr Charles Graves.
(J. A. F. M.)

GROVE, SIR WILLIAM ROBERT (1811-1896), English judge and man of science, was born on the 11th of July 1811 at Swansea, South Wales. After being educated by private tutors, he went to Brasenose College, Oxford, where he took an ordinary degree in 1832. Three years later he was called to the bar at Lincoln's Inn. His health, however, did not allow him to devote himself strenuously to practice, and he occupied his leisure with scientific studies. About 1839 he constructed the platinum-zinc voltaic cell that bears his name, and with the aid of a number of these exhibited the electric arc light in the London Institution, Finsbury Circus. The result was that in 1840 the managers appointed him to the professorship of experimental philosophy, an office which he held for seven years. His researches dealt very largely with electro-chemistry and with the voltaic cell, of which he invented several varieties. One of these, the Grove gas-battery, which is of special interest both intrinsically and as the forerunner of the secondary batteries now in use for the "storage" of electricity, was based on his observation that a current is produced by a couple of platinum plates standing in acidulated water and immersed, the one in hydrogen, the other in oxygen. At one of his lectures at the Institution he anticipated the electric lighting of to-day by illuminating the theatre with incandescent electric lamps, the filaments being of platinum and the current supplied by a battery of his nitric acid cells. In 1846 he published his famous book on _The Correlation of Physical Forces_, the leading ideas of which he had already put forward in his lectures: its fundamental conception was that each of the forces of nature--light, heat, electricity, &c.--is definitely and equivalently convertible into any other, and that where experiment does not give the full equivalent, it is because the initial force has been dissipated, not lost, by conversion into other unrecognized forces. In the same year he received a Royal medal from the Royal Society for his Bakerian lecture on "Certain phenomena of voltaic ignition and the decomposition of water into its constituent gases." In 1866 he presided over the British Association at its Nottingham meeting and delivered an address on the continuity of natural phenomena. But while he was thus engaged in scientific research, his legal work was not neglected, and his practice increased so greatly that in 1853 he became a Q.C. One of the best-known cases in which he appeared as an advocate was that of William Palmer, the Rugeley poisoner, whom he defended. In 1871 he was made a judge of the Common Pleas in succession to Sir Robert Collier, and remained on the bench till 1887. He died in London on the 1st of August 1896.

A selection of his scientific papers is given in the sixth edition of
_The Correlation of Physical Forces_, published in 1874.

GROVE (O.E. _graf_, cf. O.E. _groefa_, brushwood, later "greave"; the word does not appear in any other Teutonic language, and the _New English Dictionary_ finds no Indo-European root to which it can be referred; Skeat considers it connected with "grave," to cut, and finds the original meaning to be a glade cut through a wood), a small group or cluster of trees, growing naturally and forming something smaller than a wood, or planted in particular shapes or for particular purposes, in a park, &c. Groves have been connected with religious worship from the earliest times, and in many parts of India every village has its sacred group of trees. For the connexion of religion with sacred groves see TREE-WORSHIP.

The word "grove" was used by the authors of the Authorized Version of
the Bible to translate two Hebrew words: (1) _'eshel_, as in Gen. xxi.
33, and 1 Sam. xxii. 6; this is rightly given in the Revised Version
as "tamarisk"; (2) _asherah_ in many places throughout the Old
Testament. Here the translators followed the Septuagint [Greek: alsos]
and the Vulgate _lucus_. The _'[)a]sherah_ was a wooden post erected
at the Canaanitish places of worship, and also by the altars of
Yahweh. It may have represented a tree.

GROZNYI, a fortress and town of Russia, North Caucasia, in the province of Terek, on the Zunzha river, 82 m. by rail N.E. of Vladikavkaz, on the railway to Petrovsk. There are naphtha wells close by. The fortifications were constructed in 1819. Pop. (1897) 15,599.

GRUB, the larva of an insect, a caterpillar, maggot. The word is formed from the verb "to grub," to dig, break up the surface of the ground, and clear of stumps, roots, weeds, &c. According to the _New English Dictionary_, "grub" may be referred to an ablaut variant of the Old Teutonic _grab_-, to dig, cf. "grave." Skeat (_Etym. Dict._ 1898) refers it rather to the root seen in "grope," "grab," &c., the original meaning "to search for." The earliest quotation of the slang use of the word in the sense of food in the _New English Dictionary_ is dated 1659 from _Ancient Poems, Ballads_, &c., Percy Society Publications. "Grub-street," as a collective term for needy hack-writers, dates from the 17th century and is due to the name of a street near Moorfields, London, now Milton Street, which was as Johnson says "much inhabited by writers of small histories, dictionaries and temporary poems."

GRUBER, JOHANN GOTTFRIED (1774-1851), German critic and literary historian, was born at Naumburg on the Saale, on the 29th of November 1774. He received his education at the town school of Naumburg and the university of Leipzig, after which he resided successively at Gottingen, Leipzig, Jena and Weimar, occupying himself partly in teaching and partly in various literary enterprises, and enjoying in Weimar the friendship of Herder, Wieland and Goethe. In 1811 he was appointed professor at the university of Wittenberg, and after the division of Saxony he was sent by the senate to Berlin to negotiate the union of the university of Wittenberg with that of Halle. After the union was effected he became in 1815 professor of philosophy at Halle. He was associated with Johann Samuel Ersch in the editorship of the great work _Allgemeine Encyklopadie der Wissenschaften und Kunste_; and after the death of Ersch he continued the first section from vol. xviii. to vol. liv. He also succeeded Ersch in the editorship of the _Allgemeine Literaturzeitung_. He died on the 7th of August 1851.

Gruber was the author of a large number of works, the principal of
which are _Charakteristik Herders_ (Leipzig, 1805), in conjunction
with Johann T. L. Danz (1769-1851), afterwards professor of theology
at Jena; _Geschichte des menschlichen Geschlechts_ (2 vols., Leipzig,
1806); _Worterbuch der altklassischen Mythologie_ (3 vols., Weimar,
1810-1815); _Wielands Leben_ (2 parts, Weimar, 1815-1816), and
_Klopstocks Leben_ (Weimar, 1832). He also edited Wieland's _Samtliche
Werke_ (Leipzig, 1818-1828).

GRUMBACH, WILHELM VON (1503-1567), German adventurer, chiefly known through his connexion with the so-called "Grumbach feuds" (_Grumbachsche Handel_), the last attempt of the German knights to destroy the power of the territorial princes. A member of an old Franconian family, he was born on the 1st of June 1503, and having passed some time at the court of Casimir, prince of Bayreuth (d. 1527), fought against the peasants during the rising in 1524 and 1525. About 1540 Grumbach became associated with Albert Alcibiades, the turbulent prince of Bayreuth, whom he served both in peace and war. After the conclusion of the peace of Passau in 1552, Grumbach assisted Albert in his career of plunder in Franconia and was thus able to take some revenge upon his enemy, Melchior von Zobel, bishop of Wurzburg. As a landholder Grumbach was a vassal of the bishops of Wurzburg, and had held office at the court of Conrad of Bibra, who was bishop from 1540 to 1544. When, however, Zobel was chosen to succeed Conrad the harmonious relations between lord and vassal were quickly disturbed. Unable to free himself and his associates from the suzerainty of the bishop by appealing to the imperial courts he decided to adopt more violent measures, and his friendship with Albert was very serviceable in this connexion. Albert's career, however, was checked by his defeat at Sievershausen in July 1553 and his subsequent flight into France, and the bishop took advantage of this state of affairs to seize Grumbach's lands. The knight obtained an order of restitution from the imperial court of justice (_Reichskammergericht_), but he was unable to carry this into effect; and in April 1558 some of his partisans seized and killed the bishop. Grumbach declared he was innocent of this crime, but his story was not believed, and he fled to France. Returning to Germany he pleaded his cause in person before the diet at Augsburg in 1559, but without success. Meanwhile he had found a new patron in John Frederick, duke of Saxony, whose father, John Frederick, had been obliged to surrender the electoral dignity to the Albertine branch of his family. Chafing under this deprivation the duke listened readily to Grumbach's plans for recovering the lost dignity, including a general rising of the German knights and the deposition of Frederick II., king of Denmark. Magical charms were employed against the duke's enemies, and communications from angels were invented which helped to stir up the zeal of the people. In 1563 Grumbach attacked Wurzburg, seized and plundered the city and compelled the chapter and the bishop to restore his lands. He was consequently placed under the imperial ban, but John Frederick refused to obey the order of the emperor Maximilian II. to withdraw his protection from him. Meanwhile Grumbach sought to compass the assassination of the Saxon elector, Augustus; proclamations were issued calling for assistance; and alliances both without and within Germany were concluded. In November 1566 John Frederick was placed under the ban, which had been renewed against Grumbach earlier in the year, and Augustus marched against Gotha. Assistance was not forthcoming, and a mutiny led to the capitulation of the town. Grumbach was delivered to his foes, and, after being tortured, was executed at Gotha on the 18th of April 1567.

See F. Ortloff, _Geschichte der Grumbachschen Handel_ (Jena,
1868-1870), and J. Voigt, _Wilhelm von Grumbach und seine Handel_
(Leipzig, 1846-1847).

GRUMENTUM, an ancient town in the centre of Lucania, 33 m. S. of Potentia by the direct road through Anxia, and 52 m. by the Via Herculia, at the point of divergence of a road eastward to Heraclea. It seems to have been a native Lucanian town, not a Greek settlement. In 215 B.C. the Carthaginian general Hanno was defeated under its walls, and in 207 B.C. Hannibal made it his headquarters. In the Social War it appears as a strong fortress, and seems to have been held by both sides at different times. It became a colony, perhaps in the time of Sulla, at latest under Augustus, and seems to have been of some importance. Its site, identified by Holste from the description of the martyrdom of St Laverius, is a ridge on the right bank of the Aciris (Agri) about 1960 ft. above sea-level, 1/2 m. below the modern Saponara, which lies much higher (2533 ft.). Its ruins (all of the Roman period) include those of a large amphitheatre (arena 205 by 197 ft.), the only one in Lucania, except that at Paestum. There are also remains of a theatre. Inscriptions record the repair of its town walls and the construction of _thermae_ (of which remains were found) in 57-51 B.C., the construction in 43 B.C., of a portico, remains of which may be seen along an ancient road, at right angles to the main road, which traversed Grumentum from S. to N.

See F. P. Caputi in _Notizie degli scavi_ (1877), 129, and G. Patroni,
ibid. (1897) 180. (T. As.)

GRUN. HANS BALDUNG (c. 1470-1545), commonly called Grun, a German painter of the age of Durer, was born at Gmund in Swabia, and spent the greater part of his life at Strassburg and Freiburg in Breisgau. The earliest pictures assigned to him are altarpieces with the monogram H. B. interlaced, and the date of 1496, in the monastery chapel of Lichtenthal near Baden. Another early work is a portrait of the emperor Maximilian, drawn in 1501 on a leaf of a sketch-book now in the print-room at Carlsruhe. The "Martyrdom of St Sebastian" and the "Epiphany" (Berlin Museum), fruits of his labour in 1507, were painted for the market-church of Halle in Saxony. In 1509 Grun purchased the freedom of the city of Strassburg, and resided there till 1513, when he moved to Freiburg in Breisgau. There he began a series of large compositions, which he finished in 1516, and placed on the high altar of the Freiburg cathedral. He purchased anew the freedom of Strassburg in 1517, resided in that city as his domicile, and died a member of its great town council 1545.

Though nothing is known of Grun's youth and education, it may be inferred from his style that he was no stranger to the school of which Durer was the chief. Gmund is but 50 m. distant on either side from Augsburg and Nuremberg. Grun prints were often mistaken for those of Durer; and Durer himself was well acquainted with Grun's woodcuts and copper-plates in which he traded during his trip to the Netherlands (1520). But Grun's prints, though Dureresque, are far below Durer, and his paintings are below his prints. Without absolute correctness as a draughtsman, his conception of human form is often very unpleasant, whilst a questionable taste is shown in ornament equally profuse and "baroque." Nothing is more remarkable in his pictures than the pug-like shape of the faces, unless we except the coarseness of the extremities. No trace is apparent of any feeling for atmosphere or light and shade. Though Grun has been commonly called the Correggio of the north, his compositions are a curious medley of glaring and heterogeneous colours, in which pure black is contrasted with pale yellow, dirty grey, impure red and glowing green. Flesh is a mere glaze under which the features are indicated by lines. His works are mainly interesting because of the wild and fantastic strength which some of them display. We may pass lightly over the "Epiphany" of 1507, the "Crucifixion" of 1512, or the "Stoning of Stephen" of 1522, in the Berlin Museum. There is some force in the "Dance of Death" of 1517, in the museum of Basel, or the "Madonna" of 1530, in the Liechtenstein Gallery at Vienna. Grun's best effort is the altarpiece of Freiburg, where the "Coronation of the Virgin," and the "Twelve Apostles," the "Annunciation, Visitation, Nativity and Flight into Egypt," and the "Crucifixion," with portraits of donors, are executed with some of that fanciful power which Martin Schon bequeathed to the Swabian school. As a portrait painter he is well known. He drew the likeness of Charles V., as well as that of Maximilian; and his bust of Margrave Philip in the Munich Gallery tells us that he was connected with the reigning family of Baden as early as 1514. At a later period he had sittings from Margrave Christopher of Baden, Ottilia his wife, and all their children, and the picture containing these portraits is still in the grand-ducal gallery at Carlsruhe. Like Durer and Cranach, Grun became a hearty supporter of the Reformation. He was present at the diet of Augsburg in 1518, and one of his woodcuts represents Luther under the protection of the Holy Ghost, which hovers over him in the shape of a dove.

GRUNBERG, a town of Germany, in Prussian Silesia, beautifully situated between two hills on an affluent of the Oder, and on the railway from Breslau to Stettin via Kustrin, 36 m. N.N.W. of Glogau. Pop. (1905) 20,987. It has a Roman Catholic and two Evangelical churches, a modern school and a technical (textiles) school. There are manufactures of cloth, paper, machinery, straw hats, leather and tobacco. The prosperity of the town depends chiefly on the vine culture in the neighbourhood, from which, besides the exportation of a large quantity of grapes, about 700,000 gallons of wine are manufactured annually.

GRUNDTVIG, NIKOLAI FREDERIK SEVERIN (1783-1872), Danish poet, statesman and divine, was born at the parsonage of Udby in Zealand on the 8th of September 1783. In 1791 he was sent to live at the house of a priest in Jutland, and studied at the free school of Aarhuus until he went up to the university of Copenhagen in 1800. At the close of his university life he made Icelandic his special study, until in 1805 he took the position of tutor in a house on the island of Langeland. The next three years were spent in the study of Shakespeare, Schiller and Fichte. His cousin, the philosopher Henrik Steffens, had returned to Copenhagen in 1802 full of the teaching of Schelling and his lectures and the early poetry of Ohlenschlager opened the eyes of Grundtvig to the new era in literature. His first work, _On the Songs in the Edda_, attracted no attention. Returning to Copenhagen in 1808 he achieved greater success with his _Northern Mythology_, and again in 1809-1811 with a long epic poem, the _Decline of the Heroic Life in the North_. The boldness of the theological views expressed in his first sermon in 1810 offended the ecclesiastical authorities, and he retired to a country parish as his father's assistant for a while. From 1812 to 1817 he published five or six works, of which the _Rhyme of Roskilde_ is the most remarkable. From 1816 to 1819 he was editor of a polemical journal entitled _Dannevirke_, and in 1818 to 1822 appeared his Danish paraphrases (6 vols.) of Saxo Grammaticus and Snorri. During these years he was preaching against rationalism to an enthusiastic congregation in Copenhagen, but he accepted in 1821 the country living of Praesto, only to return to the metropolis the year after. In 1825 he published a pamphlet, _The Church's Reply_, against H. N. Clausen, who was professor of theology in the university of Copenhagen. Grundtvig was publicly prosecuted and fined, and for seven years he was forbidden to preach, years which he spent in publishing a collection of his theological works, in paying two visits to England, and in studying Anglo-Saxon. In 1832 he obtained permission to preach again, and in 1839 he became priest of the workhouse church of Vartov hospital, Copenhagen, a post he continued to hold until his death. In 1837-1841 he published _Songs for the Danish Church_, a rich collection of sacred poetry; in 1838 he brought out a selection of early Scandinavian verse; in 1840 he edited the Anglo-Saxon poem of the _Phoenix_, with a Danish translation. He visited England a third time in 1843. From 1844 until after the first German war Grundtvig took a very prominent part in politics. In 1861 he received the titular rank of bishop, but without a see. He went on writing occasional poems till 1866, and preached in the Vartov every Sunday until a month before his death. His preaching attracted large congregations, and he soon had a following. His hymn-book effected a great change in Danish church services, substituting the hymns of the national poets for the slow measures of the orthodox Lutherans. The chief characteristic of his theology was the substitution of the authority of the "living word" for the apostolic commentaries, and he desired to see each congregation a practically independent community. His patriotism was almost a part of his religion, and he established popular schools where the national poetry and history should form an essential part of the instruction. His followers are known as Grundtvigians. He was married three times, the last time in his seventy-sixth year. He died on the 2nd of September 1872. Grundtvig holds a unique position in the literature of his country; he has been styled the Danish Carlyle. He was above all things a man of action, not an artist; and the formless vehemence of his writings, which have had a great influence over his own countrymen, is hardly agreeable or intelligible to a foreigner. The best of his poetical works were published in a selection (7 vols., 1880-1889) by his eldest son, Svend Hersleb Grundtvig (1824-1883), who was an authority on Scandinavian antiquities, and made an admirable collection of old Danish poetry (_Danmarks gamle Folkeviser_, 1853-1883, 5 vols.; completed in 1891 by A. Olrik).

His correspondence with Ingemann was edited by S. Grundtvig (1882);
his correspondence with Christian Molbech by L. Schroder (1888); see
also F. Winkel Horn, _Grundtvigs Liv og Gjerning_ (1883); and an
article by F. Nielsen in Bricka's _Dansk Biografisk Lexikon_.

GRUNDY, SYDNEY (1848- ), English dramatist, was born at Manchester on the 23rd of March 1848, son of Alderman Charles Sydney Grundy. He was educated at Owens College, Manchester, and was called to the bar in 1869, practising in Manchester until 1876. His farce, _A Little Change_, was produced at the Haymarket Theatre in 1872. He became well known as an adapter of plays, among his early successes in this direction being _The Snowball_ (Strand Theatre, 1879) from _Oscar, ou le mari qui trompe sa femme_ by MM. Scribe and Duvergne, and _In Honour Bound_ (1880) from Scribe's _Une Chaine_. In 1887 he made a popular success with _The Bells of Haslemere_, written with Mr H. Pettitt and produced at the Adelphi. In 1889-1890 he produced two ingenious original comedies, _A White Lie_ (Court Theatre) and _A Fool's Paradise_ (Gaiety Theatre), which had been played two years earlier at Greenwich as _The Mouse-Trap_. These were followed by _Sowing the Wind_ (Comedy, 1893), _An Old Jew_ (Garrick, 1894), and by an adaptation of Octave Feuillet's _Montjoye as A Bunch of Violets_ (Haymarket, 1894). In 1894 he produced _The New Woman_ and _The Slaves of the Ring_; in 1895, _The Greatest of These_, played by Mr and Mrs Kendal at the Garrick Theatre; _The Degenerates_ (Haymarket, 1899), and _A Debt of Honour_ (St James's 1900). Among Mr Grundy's most successful adaptations were the charming _Pair of Spectacles_ (Garrick, 1890) from _Les Petits Oiseaux_ of MM. Labiche and Delacour. Others were _A Village Priest_ (Haymarket, 1890) from _Le Secret de la terreuse_, a melodrama by MM. Busnach and Cauvin; _A Marriage of Convenience_ (Haymarket, 1897) from _Un Mariage de Louis XV_, by Alex. Dumas, pere, _The Silver Key_ (Her Majesty's, 1897) from his _Mlle de Belle-isle_, and _The Musqueteers_ (1899) from the same author's novel; _Frocks and Frills_ (Haymarket, 1902) from the _Doigts de fees_ of MM. Scribe and Legouve; _The Garden of Lies_ (St James's Theatre, 1904) from Mr Justus Miles Forman's novel; _Business is Business_ (His Majesty's Theatre, 1905), a rather free adaptation from Octave Mirbeau's _Les Affaires sont les affaires_; and _The Diplomatists_ (Royalty Theatre, 1905) from _La Poudre aux yeux_, by Labiche.

GRUNDY, MRS, the name of an imaginary English character, who typifies the disciplinary control of the conventional "proprieties" of society over conduct, the tyrannical pressure of the opinion of neighbours on the acts of others. The name appears in a play of Thomas Morton, _Speed the Plough_ (1798), in which one of the characters, Dame Ashfield, continually refers to what her neighbour Mrs Grundy will say as the criterion of respectability. Mrs Grundy is not a character in the play, but is a kind of "Mrs Harris" to Dame Ashfield.

GRUNER, GOTTLIEB SIGMUND (1717-1778), the author of the first connected attempt to describe in detail the snowy mountains of Switzerland. His father, Johann Rudolf Gruner (1680-1761), was pastor of Trachselwald, in the Bernese Emmenthal (1705), and later (1725) of Burgdorf, and a great collector of information relating to historical and scientific matters; his great _Thesaurus topographico-historicus totius ditionis Bernensis_ (4 vols. folio, 1729-1730) still remains in MS., but in 1732 he published a small work entitled _Deliciae urbis Bernae_, while he possessed an extensive cabinet of natural history objects. Naturally such tastes had a great influence on the mind of his son, who was born at Trachselwald, and educated by his father and at the Latin school at Burgdorf, not going to Berne much before 1736, when he published a dissertation on the use of fire by the heathen. In 1739 he qualified as a notary, in 1741 became the archivist of Hesse-Homburg, and in 1743 accompanied Prince Christian of Anhalt-Schaumburg to Silesia and the university of Halle. He returned to his native land before 1749, when he obtained a post at Thorberg, being transferred in 1764 to Landshut and Fraubrunnen. It was in 1760 that he published in 3 vols. at Berne his chief work, _Die Eisgebirge des Schweizerlandes_ (bad French translation by M. de Keralio, Paris, 1770). The first two volumes are filled by a detailed description of the snowy Swiss mountains, based not so much on personal experience as on older works, and a very large number of communications received by Gruner from numerous friends; the third volume deals with glaciers in general, and their various properties. Though in many respects imperfect, Gruner's book sums up all that was known on the subject in his day, and forms the starting-point for later writers. The illustrations are very curious and interesting. In 1778 he republished (nominally in London, really at Berne) much of the information contained in his larger work, but thrown into the form of letters, supposed to be written in 1776 from various spots, under the title of _Reisen durch die merkwurdigsten Gegenden Helvetiens_ (2 vols.). (W. A. B. C.)

GRUNEWALD, MATHIAS. The accounts which are given of this German painter, a native of Aschaffenburg, are curiously contradictory. Between 1518 and 1530, according to statements adopted by Waagen and Passavant, he was commissioned by Albert of Brandenburg, elector and archbishop of Mainz, to produce an altarpiece for the collegiate church of St Maurice and Mary Magdalen at Halle on the Saale; and he acquitted himself of this duty with such cleverness that the prelate in after years caused the picture to be rescued from the Reformers and brought back to Aschaffenburg. From one of the churches of that city it was taken to the Pinakothek of Munich in 1836. It represents St Maurice and Mary Magdalen between four saints, and displays a style so markedly characteristic, and so like that of Lucas Cranach, that Waagen was induced to call Grunewald Cranach's master. He also traced the same hand and technical execution in the great altarpieces of Annaberg and Heilbronn, and in various panels exhibited in the museums of Mainz, Darmstadt, Aschaffenburg, Vienna and Berlin. A later race of critics, declining to accept the statements of Waagen and Passavant, affirm that there is no documentary evidence to connect Grunewald with the pictures of Halle and Annaberg, and they quote Sandrart and Bernhard Jobin of Strassburg to show that Grunewald is the painter of pictures of a different class. They prove that he finished before 1516 the large altarpiece of Issenheim, at present in the museum of Colmar, and starting from these premises they connect the artist with Altdorfer and Durer to the exclusion of Cranach. That a native of the Palatinate should have been asked to execute pictures for a church in Saxony can scarcely be accounted strange, since we observe that Hans Baldung (Grun) was entrusted with a commission of this kind. But that a painter of Aschaffenburg should display the style of Cranach is strange and indeed incredible, unless vouched for by first-class evidence. In this case documents are altogether wanting, whilst on the other hand it is beyond the possibility of doubt, even according to Waagen, that the altarpiece of Issenheim is the creation of a man whose teaching was altogether different from that of the painter of the pictures of Halle and Annaberg. The altarpiece of Issenheim is a fine and powerful work, completed as local records show before 1516 by a Swabian, whose distinguishing mark is that he followed the traditions of Martin Schongauer, and came under the influence of Altdorfer and Durer. As a work of art the altarpiece is important, being a poliptych of eleven panels, a carved central shrine covered with a double set of wings, and two side pieces containing the Temptation of St Anthony, the hermits Anthony and Paul in converse, the Virgin adored by Angels, the Resurrection, the Annunciation, the Crucifixion, St Sebastian, St Anthony, and the Marys wailing over the dead body of Christ. The author of these compositions is also the painter of a series of monochromes described by Sandrart in the Dominican convent, and now in part in the Saalhof at Frankfort, and a Resurrection in the museum of Basel, registered in Amerbach's inventory as the work of Grunewald.

GRUTER (or GRUYTERE), JAN (1560-1627), a critic and scholar of Dutch parentage by his father's side and English by his mother's, was born at Antwerp on the 3rd of December 1560. To avoid religious persecution his parents while he was still young came to England; and for some years he prosecuted his studies at Cambridge, after which he went to Leiden, where he graduated M. A. In 1586 he was appointed professor of history at Wittenberg, but as he refused to subscribe the _formula concordiae_ he was unable to retain his office. From 1589 to 1592 he taught at Rostock, after which he went to Heidelberg, where in 1602 he was appointed librarian to the university. He died at Heidelberg on the 20th of September 1627.

Gruter's chief works were his _Inscriptiones antiquae totius orbis
Romani_ (2 vols., Heidelberg, 1603), and _Lampas, sive fax artium
liberalium_ (7 vols., Frankfort, 1602-1634).