Chapter I: Part 1

Transcriber's note: A few typographical errors have been corrected: they
are listed at the end of the text. Due to space constraints, italics
denoting underscores were not used in the tables.

THE ENCYCLOPÆDIA BRITANNICA

A DICTIONARY OF ARTS, SCIENCES, LITERATURE AND GENERAL INFORMATION

ELEVENTH EDITION

VOLUME VIII slice III

Destructor to Diameter

DESTRUCTOR (_continued from volume 8, slice 2, page 0108._)
... in main flues, &c. (g) The chimney draught must be assisted with
forced draught from fans or steam jet to a pressure of 1½ in. to 2 in.
under grates by water-gauge. (h) Where a destructor is required to
work without risk of nuisance to the neighbouring inhabitants, its
efficiency as a refuse destructor plant must be primarily kept in view
in designing the works, steam-raising being regarded as a secondary
consideration. Boilers should not be placed immediately over a furnace
so as to present a large cooling surface, whereby the temperature of
the gases is reduced before the organic matter has been thoroughly
burned. (i) Where steam-power and a high fuel efficiency are desired a
large percentage of CO_{2} should be sought in the furnaces with as
little excess of air as possible, and the flue gases should be
utilized in heating the air-supply to the grates, and the feed-water
to the boilers. (j) Ample boiler capacity and hot-water storage
feed-tanks should be included in the design where steam-power is
required.

[Sidenote: Cost.]

As to the initial cost of the erection of refuse destructors, few
trustworthy data can be given. The outlay necessarily depends, amongst
other things, upon the difficulty of preparing the site, upon the
nature of the foundations required, the height of the chimney-shaft,
the length of the inclined or approach roadway, and the varying prices
of labour and materials in different localities. As an example may be
mentioned the case of Bristol, where, in 1892, the total cost of
constructing a 16-cell Fryer destructor was £11,418, of which £2909
was expended on foundations, and £1689 on the chimney-shaft; the cost
of the destructor proper, buildings and approach road was therefore
£6820, or about £426 per cell. The cost per ton of burning refuse in
destructors depends mainly upon--(a) The price of labour in the
locality, and the number of "shifts" or changes of workmen per day;
(b) the type of furnace adopted; (c) the nature of the material to be
consumed; (d) the interest on and repayment of capital outlay. The
cost of burning ton for ton consumed, in high-temperature furnaces,
including labour and repairs, is not greater than in slow-combustion
destructors. The average cost of burning refuse at twenty-four
different towns throughout England, exclusive of interest on the cost
of the works, is 1s. 1½d. per ton burned; the minimum cost is 6d. per
ton at Bradford, and the maximum cost 2s. 10d. per ton at Battersea.
At Shoreditch the cost per ton for the year ending on the 25th of
March 1899, including labour, supervision, stores, repairs, &c. (but
exclusive of interest on cost of works), was 2s. 6.9d. The quantity of
refuse burned per cell per day of 24 hours varies from about 4 tons up
to 20 tons. The ordinary low-temperature destructor, with 25 sq. ft.
grate area, burns about 20 lb. of refuse per square foot of grate
area per hour, or between 5 and 6 tons per cell per 24 hours. The
Meldrum destructor furnaces at Rochdale burn as much as 66 lb. per
square foot of grate area per hour, and the Beaman and Deas destructor
at Llandudno 71.7 lb. per square foot per hour. The amount, however,
always depends materially on the care observed in stoking, the nature
of the material, the frequency of removal of clinker, and on the
question whether the whole of the refuse passed into the furnace is
thoroughly cremated.

[Sidenote: Residues:]

The amount of residue in the shape of clinker and fine ash varies from
22 to 37% of the bulk dealt with. From 25 to 30% is a very usual
amount. At Shoreditch, where the refuse consists of about 8% of straw,
paper, shavings, &c., the residue contains about 29% clinker, 2.7%
fine ash, .5% flue dust, and .6% old tins, making a total residue of
32.8%. As the residuum amounts to from one-fourth to one-third of the
total bulk of the refuse dealt with, it is a question of the utmost
importance that some profitable, or at least inexpensive, means should
be devised for its regular disposal. Among other purposes, it has been
used for bottoming for macadamized roads, for the manufacture of
concrete, for making paving slabs, for forming suburban footpaths or
cinder footwalks, and for the manufacture of mortar. The last is a
very general, and in many places profitable, mode of disposal. An
entirely new outlet has also arisen for the disposal of good
well-vitrified destructor clinker in connexion with the construction
of bacteria beds for sewage disposal, and in many districts its value
has, by this means, become greatly enhanced.

[Sidenote: Forced draught.]

Through defects in the design and management of many of the early
destructors complaints of nuisance frequently arose, and these have,
to some extent, brought destructor installations into disrepute.
Although some of the older furnaces were decided offenders in this
respect, that is by no means the case with the modern improved type of
high-temperature furnace; and often, were it not for the great
prominence in the landscape of a tall chimney-shaft, the existence of
a refuse destructor in a neighbourhood would not be generally known to
the inhabitants. A modern furnace, properly designed and worked, will
give rise to no nuisance, and may be safely erected in the midst of a
populous neighbourhood. To ensure the perfect cremation of the refuse
and of the gases given off, forced draught is essential. This is
supplied either as air draught delivered from a rapidly revolving fan,
or as steam blast, as in the Horsfall steam jet or the Meldrum blower.
With a forced blast less air is required to obtain complete combustion
than by chimney draught. The forced draught grate requires little more
than the quantity theoretically necessary, while with chimney draught
more than double the theoretical amount of air must be supplied. With
forced draught, too, a much higher temperature is attained, and if it
is properly worked, little or no cold air will enter the furnaces
during stoking operations. As far as possible a balance of pressure in
the cells during clinkering should be maintained just sufficient to
prevent an inrush of cold air through the flues. The forced draught
pressure should not exceed 2 in. water-gauge. The efficiency of the
combustion in the furnace is conveniently measured by the
"Econometer," which registers continuously and automatically the
proportion of CO_{2} passing away in the waste gases; the higher the
percentage of CO_{2} the more efficient the furnace, provided there is
no formation of CO, the presence of which would indicate incomplete
combustion. The theoretical maximum of CO_{2} for refuse burning is
about 20%; and, by maintaining an even clean fire, by admitting
secondary air over the fire, and by regulating the dampers or the
air-pressure in the ash-pit, an amount approximating to this
percentage may be attained in a well-designed furnace if properly
worked. If the proportion of free oxygen (i.e. excess of air) is
large, more air is passed through the furnace than is required for
complete combustion, and the heating of this excess is clearly a waste
of heat. The position of the econometer in testing should be as near
the furnace as possible, as there may be considerable air leakage
through the brickwork of the flues.

The air supply to modern furnaces is usually delivered hot, the inlet
air being first passed through an air-heater the temperature of which
is maintained by the waste gases in the main flue.

[Sidenote: Calorific value.]

The modern high-temperature destructor, to render the refuse and gases
perfectly innocuous and harmless, is worked at a temperature varying
from 1250° to 2000° F., and the maintenance of such temperatures has
very naturally suggested the possibility of utilizing this heat-energy
for the production of steam-power. Experience shows that a
considerable amount of energy may be derived from steam-raising
destructor stations, amply justifying a reasonable increase of
expenditure on plant and labour. The actual calorific value of the
refuse material necessarily varies, but, as a general average, with
suitably designed and properly managed plant, an evaporation of 1 lb.
of water per pound of refuse burned is a result which may be readily
attained, and affords a basis of calculation which engineers may
safely adopt in practice. Many destructor steam-raising plants,
however, give considerably higher results, evaporations approaching 2
lb. of water per pound of refuse being often met with under
favourable conditions.

From actual experience it may be accepted, therefore, that the
calorific value of unscreened house refuse varies from 1 to 2 lb. of
water evaporated per pound of refuse burned, the exact proportion
depending upon the quality and condition of the material dealt with.
Taking the evaporative power of coal at 10 lb. of water per pound of
coal, this gives for domestic house refuse a value of from {1/10} to
{1/5} that of coal; or, with coal at 20s. per ton, refuse has a
commercial value of from 2s. to 4s. per ton. In London the quantity of
house refuse amounts to about 1¼ million tons per annum, which is
equivalent to from 4 cwt. to 5 cwt. per head per annum. If it be
burned in furnaces giving an evaporation of 1 lb. of water per pound
of refuse, it would yield a total power annually of about 138 million
brake horse-power hours, and equivalent cost of coal at 20s. per ton
for this amount of power even when calculated upon the very low
estimate of 2 lb.[1] of coal per brake horse-power hour, works out at
over £123,000. On the same basis, the refuse of a medium-sized town,
with, say, a population of 70,000 yielding refuse at the rate of 5
cwt. per head per annum, would afford 112 indicated horse-power per
ton burned, and the total indicated horse-power hours per annum would
be

70,000 × 5 cwt.
--------------- × 112 = 1,960,000 I.H.P. hours annually.
20

If this were applied to the production of electric energy, the
electrical horse-power hours would be (with a dynamo efficiency of
90%)

1,960,000 × 90
-------------- = 1,764,000 E.H.P. hours per annum;
100

and the watt-hours per annum at the central station would be

1,764,000 × 746 = 1,315,944,000.

Allowing for a loss of 10% in distribution, this would give
1,184,349,600 watt-hours available in lamps, or with 8-candle-power
lamps taking 30 watts of current per lamp, we should have

1,184,349,600 watt-hours
------------------------ = 39,478,320 8-c.p. lamp-hours per annum;
30 watts

39,478,320
that is, ----------------- = 563 8-c.p. lamp hours per annum per
70,000 population head of population.

Taking the loss due to the storage which would be necessary at 20% on
three-quarters of the total or 15% upon the whole, there would be 478
8-c.p. lamp-hours per annum per head of the population: i.e. if the
power developed from the refuse were fully utilized, it would supply
electric light at the rate of one 8-c.p. lamp per head of the
population for about 1{1/3} hours for every night of the year.

[Sidenote: Difficulties.]

In actual practice, when the electric energy is for the purposes of
lighting only, difficulty has been experienced in fully utilizing the
thermal energy from a destructor plant owing to the want of adequate
means of storage either of the thermal or of the electric energy. A
destructor station usually yields a fairly definite amount of thermal
energy uniformly throughout the 24 hours, while the consumption of
electric-lighting current is extremely irregular, the maximum demand
being about four times the mean demand. The period during which the
demand exceeds the mean is comparatively short, and does not exceed
about 6 hours out of the 24, while for a portion of the time the
demand may not exceed {1/20}th of the maximum. This difficulty, at
first regarded as somewhat grave, is substantially minimized by the
provision of ample boiler capacity, or by the introduction of feed
thermal storage vessels in which hot feed-water may be stored during
the hours of light load (say 18 out of the 24), so that at the time of
maximum load the boiler may be filled directly from these vessels,
which work at the same pressure and temperature as the boiler.
Further, the difficulty above mentioned will disappear entirely at
stations where there is a fair day load which practically ceases at
about the hour when the illuminating load comes on, thus equalizing
the demand upon both destructor and electric plant throughout the 24
hours. This arises in cases where current is consumed during the day
for motors, fans, lifts, electric tramways, and other like purposes,
and, as the employment of electric energy for these services is
rapidly becoming general, no difficulty need be anticipated in the
successful working of combined destructor and electric plants where
these conditions prevail. The more uniform the electrical demand
becomes, the more fully may the power from a destructor station be
utilized.

In addition to combination with electric-lighting works, refuse
destructors are now very commonly installed in conjunction with
various other classes of power-using undertakings, including tramways,
water-works, sewage-pumping, artificial slab-making and
clinker-crushing works and others; and the increasingly large sums
which are being yearly expended in combined undertakings of this
character is perhaps the strongest evidence of the practical value of
such combinations where these several classes of work must be carried
on.

For further information on the subject, reference should be made to
William H. Maxwell, _Removal and Disposal of Town Refuse, with an
exhaustive treatment of Refuse Destructor Plants_ (London, 1899), with
a special _Supplement_ embodying later results (London, 1905).

See also the _Proceedings of the Incorporated Association of Municipal
and County Engineers_, vols. xiii. p. 216, xxii. p. 211, xxiv. p. 214
and xxv. p. 138; also the _Proceedings of the Institution of Civil
Engineers_, vols. cxxii. p. 443, cxxiv. p. 469, cxxxi. p. 413,
cxxxviii. p. 508, cxxix. p. 434, cxxx. pp. 213 and 347, cxxiii. pp.
369 and 498, cxxviii. p. 293 and cxxxv. p. 300. (W. H. MA.)

[1] With medium-sized steam plants, a consumption of 4 lb. of coal per brake horse-power per hour is a very usual performance.

DE TABLEY, JOHN BYRNE LEICESTER WARREN, 3rd BARON (1835-1895), English poet, eldest son of George Fleming Leicester (afterwards Warren), 2nd Baron De Tabley, was born on the 26th of April 1835. He was educated at Eton and Christ Church, Oxford, where he took his degree in 1856 with second classes in classics and in law and modern history. In the autumn of 1858 he went to Turkey as unpaid attaché to Lord Stratford de Redcliffe, and two years later was called to the bar. He became an officer in the Cheshire Yeomanry, and unsuccessfully contested Mid-Cheshire in 1868 as a Liberal. After his father's second marriage in 1871 he removed to London, where he became a close friend of Tennyson for several years. From 1877 till his succession to the title in 1887 he was lost to his friends, assuming the life of a recluse. It was not till 1892 that he returned to London life, and enjoyed a sort of renaissance of reputation and friendship. During the later years of his life Lord De Tabley made many new friends, besides reopening old associations, and he almost seemed to be gathering around him a small literary company when his health broke, and he died on the 22nd of November 1895 at Ryde, in his sixty-first year. He was buried at Little Peover in Cheshire. Although his reputation will live almost exclusively as that of a poet, De Tabley was a man of many studious tastes. He was at one time an authority on numismatics; he wrote two novels; published _A Guide to the Study of Book Plates_ (1880); and the fruit of his careful researches in botany was printed posthumously in his elaborate _Flora of Cheshire_ (1899). Poetry, however, was his first and last passion, and to that he devoted the best energies of his life. De Tabley's first impulse towards poetry came from his friend George Fortescue, with whom he shared a close companionship during his Oxford days, and whom he lost, as Tennyson lost Hallam, within a few years of their taking their degrees. Fortescue was killed by falling from the mast of Lord Drogheda's yacht in November 1859, and this gloomy event plunged De Tabley into deep depression. Between 1859 and 1862 De Tabley issued four little volumes of pseudonymous verse (by G. F. Preston), in the production of which he had been greatly stimulated by the sympathy of Fortescue. Once more he assumed a pseudonym--his _Praeterita_ (1863) bearing the name of William Lancaster. In the next year he published _Eclogues and Monodramas_, followed in 1865 by _Studies in Verse_. These volumes all displayed technical grace and much natural beauty; but it was not till the publication of _Philoctetes_ in 1866 that De Tabley met with any wide recognition. _Philoctetes_ bore the initials "M.A.," which, to the author's dismay, were interpreted as meaning Matthew Arnold. He at once disclosed his identity, and received the congratulations of his friends, among whom were Tennyson, Browning and Gladstone. In 1867 he published _Orestes_, in 1870 _Rehearsals_ and in 1873 _Searching the Net_. These last two bore his own name, John Leicester Warren. He was somewhat disappointed by their lukewarm reception, and when in 1876 _The Soldier of Fortune_, a drama on which he had bestowed much careful labour, proved a complete failure, he retired altogether from the literary arena. It was not until 1893 that he was persuaded to return, and the immediate success in that year of his _Poems, Dramatic and Lyrical_, encouraged him to publish a second series in 1895, the year of his death. The genuine interest with which these volumes were welcomed did much to lighten the last years of a somewhat sombre and solitary life. His posthumous poems were collected in 1902. The characteristics of De Tabley's poetry are pre-eminently magnificence of style, derived from close study of Milton, sonority, dignity, weight and colour. His passion for detail was both a strength and a weakness: it lent a loving fidelity to his description of natural objects, but it sometimes involved him in a loss of simple effect from over-elaboration of treatment. He was always a student of the classic poets, and drew much of his inspiration directly from them. He was a true and a whole-hearted artist, who, as a brother poet well said, "still climbed the clear cold altitudes of song." His ambition was always for the heights, a region naturally ice-bound at periods, but always a country of clear atmosphere and bright, vivid outlines.

See an excellent sketch by E. Gosse in his _Critical Kit-Kats_ (1896).
(A. WA.)

DETAILLE, JEAN BAPTISTE ÉDOUARD (1848- ), French painter, was born in Paris on the 5th of October 1848. After working as a pupil of Meissonier's, he first exhibited, in the Salon of 1867, a picture representing "A Corner of Meissonier's Studio." Military life was from the first a principal attraction to the young painter, and he gained his reputation by depicting the scenes of a soldier's life with every detail truthfully rendered. He exhibited "A Halt" (1868); "Soldiers at rest, during the Manoeuvres at the Camp of Saint Maur" (1869); "Engagement between Cossacks and the Imperial Guard, 1814" (1870). The war of 1870-71 furnished him with a series of subjects which gained him repeated successes. Among his more important pictures may be named "The Conquerors" (1872); "The Retreat" (1873); "The Charge of the 9th Regiment of Cuirassiers in the Village of Morsbronn, 6th August 1870" (1874); "The Marching Regiment, Paris, December 1874" (1875); "A Reconnaissance" (1876); "Hail to the Wounded!" (1877); "Bonaparte in Egypt" (1878); the "Inauguration of the New Opera House"--a water-colour; the "Defence of Champigny by Faron's Division" (1879). He also worked with Alphonse de Neuville on the panorama of Rezonville. In 1884 he exhibited at the Salon the "Evening at Rezonville," a panoramic study, and "The Dream" (1888), now in the Luxemburg. Detaille recorded other events in the military history of his country: the "Sortie of the Garrison of Huningue" (now in the Luxemburg), the "Vincendon Brigade," and "Bizerte," reminiscences of the expedition to Tunis. After a visit to Russia, Detaille exhibited "The Cossacks of the Ataman" and "The Hereditary Grand Duke at the Head of the Hussars of the Guard." Other important works are: "Victims to Duty," "The Prince of Wales and the Duke of Connaught" and "Pasteur's Funeral." In his picture of "Châlons, 9th October 1896," exhibited in the Salon, 1898, Detaille painted the emperor and empress of Russia at a review, with M. Félix Faure. Detaille became a member of the French Institute in 1898.

See Marius Vachon, _Detaille_ (Paris, 1898); Frédéric Masson,
_Édouard Detaille and his work_ (Paris and London, 1891); J. Claretie,
_Peintres et sculpteurs contemporains_ (Paris, 1876); G. Goetschy,
_Les Jeunes peintres militaires_ (Paris, 1878).

DETAINER (from _detain_, Lat. _detinere_), in law, the act of keeping a person against his will, or the wrongful keeping of a person's goods, or other real or personal property. A writ of detainer was a form for the beginning of a personal action against a person already lodged within the walls of a prison; it was superseded by the Judgment Act 1838.

DETERMINANT, in mathematics, a function which presents itself in the solution of a system of simple equations.

1. Considering the equations

ax + by + cz = d,
a'x + b'y + c'z = d',
a"x + b"y + c"z = d",

and proceeding to solve them by the so-called method of cross multiplication, we multiply the equations by factors selected in such a manner that upon adding the results the whole coefficient of y becomes = 0, and the whole coefficient of z becomes = 0; the factors in question are b'c" - b"c', b"c - bc", bc' - b'c (values which, as at once seen, have the desired property); we thus obtain an equation which contains on the left-hand side only a multiple of x, and on the right-hand side a constant term; the coefficient of x has the value

a(b'c" - b"c') + a'(b"c - bc") + a"(bc' - b'c),

and this function, represented in the form

|a, b, c |,
|a', b', c'|
|a", b", c"|

is said to be a determinant; or, the number of elements being 3², it is called a determinant of the third order. It is to be noticed that the resulting equation is

|a, b, c | x = |d, b, c |
|a', b', c'| |d', b', c'|
|a", b", c"| |d", b", c"|

where the expression on the right-hand side is the like function with d, d', d" in place of a, a', a" respectively, and is of course also a determinant. Moreover, the functions b'c" - b"c', b"c - bc", bc' - b'c used in the process are themselves the determinants of the second order

|b', c'|, |b", c"|, |b, c |.
|b", c"| |b, c | |b', c'|

We have herein the suggestion of the rule for the derivation of the determinants of the orders 1, 2, 3, 4, &c., each from the preceding one, viz. we have

|a| = a,

|a, b | = a|b'| - a'|b|.
|a', b'|

|a, b, c | = a|b', c'| + a'|b", c"| + a"|b, c |,
|a', b', c'| |b", c"| |b , c | |b', c'|
|a", b", c"|

|a, b , c , d | = a|b', c', d' | - a'|b" , c" , d" | +
|a', b' , c' , d' | |b", c", d" | |b"', c"', d"'|
|a", b" , c" , d" | |b"', c"', d"'| |b , c , d |
|a"', b"', c"', d"'|

+ a"|b"', c"', d"'| - a"'|b , c, d |,
|b , c , d | |b', c', d'|
|b' , c' , d' | |b", c", d"|

and so on, the terms being all + for a determinant of an odd order, but alternately + and - for a determinant of an even order.

2. It is easy, by induction, to arrive at the general results:--

A determinant of the order n is the sum of the 1.2.3...n products which can be formed with n elements out of n² elements arranged in the form of a square, no two of the n elements being in the same line or in the same column, and each such product having the coefficient ± unity.

The products in question may be obtained by permuting in every possible manner the columns (or the lines) of the determinant, and then taking for the factors the n elements in the dexter diagonal. And we thence derive the rule for the signs, viz. considering the primitive arrangement of the columns as positive, then an arrangement obtained therefrom by a single interchange (inversion, or derangement) of two columns is regarded as negative; and so in general an arrangement is positive or negative according as it is derived from the primitive arrangement by an even or an odd number of interchanges. [This implies the theorem that a given arrangement can be derived from the primitive arrangement only by an odd number, or else only by an even number of interchanges,--a theorem the verification of which may be easily obtained from the theorem (in fact a particular case of the general one), an arrangement can be derived from itself only by an even number of interchanges.] And this being so, each product has the sign belonging to the corresponding arrangement of the columns; in particular, a determinant contains with the sign + the product of the elements in its dexter diagonal. It is to be observed that the rule gives as many positive as negative arrangements, the number of each being = ½ 1.2...n.

The rule of signs may be expressed in a different form. Giving to the columns in the primitive arrangement the numbers 1, 2, 3 ... n, to obtain the sign belonging to any other arrangement we take, as often as a lower number succeeds a higher one, the sign -, and, compounding together all these minus signs, obtain the proper sign, + or - as the case may be.

Thus, for three columns, it appears by either rule that 123, 231, 312 are positive; 213, 321, 132 are negative; and the developed expression of the foregoing determinant of the third order is

= ab'c" - ab"c' + a'b"c - a'bc" + a"bc' - a"b'c.

3. It further appears that a determinant is a linear function[1] of the elements of each column thereof, and also a linear function of the elements of each line thereof; moreover, that the determinant retains the same value, only its sign being altered, when any two columns are interchanged, or when any two lines are interchanged; more generally, when the columns are permuted in any manner, or when the lines are permuted in any manner, the determinant retains its original value, with the sign + or - according as the new arrangement (considered as derived from the primitive arrangement) is positive or negative according to the foregoing rule of signs. It at once follows that, if two columns are identical, or if two lines are identical, the value of the determinant is = 0. It may be added, that if the lines are converted into columns, and the columns into lines, in such a way as to leave the dexter diagonal unaltered, the value of the determinant is unaltered; the determinant is in this case said to be _transposed_.

4. By what precedes it appears that there exists a function of the n² elements, linear as regards the terms of each column (or say, for shortness, linear as to each column), and such that only the sign is altered when any two columns are interchanged; these properties completely determine the function, except as to a common factor which may multiply all the terms. If, to get rid of this arbitrary common factor, we assume that the product of the elements in the dexter diagonal has the coefficient +1, we have a complete definition of the determinant, and it is interesting to show how from these properties, assumed for the definition of the determinant, it at once appears that the determinant is a function serving for the solution of a system of linear equations. Observe that the properties show at once that if any column is = 0 (that is, if the elements in the column are each = 0), then the determinant is = 0; and further, that if any two columns are identical, then the determinant is = 0.

5. Reverting to the system of linear equations written down at the beginning of this article, consider the determinant

|ax + by + cz - d , b , c |;
|a'x + b'y + c'z - d', b', c'|
|a"x + b"y + c"z - d", b", c"|

it appears that this is

= x|a , b , c | + y|b , b , c | + z|c , b , c | - |d , b , c |;
|a', b', c'| |b', b', c'| |c', b', c'| |d', b', c'|
|a", b", c"| |b", b", c"| |c", b", c"| |d", b", c"|

viz. the second and third terms each vanishing, it is

= x|a , b , c | - |d , b , c |.
|a', b', c'| |d', b', c'|
|a", b", c"| |d", b", c"|

But if the linear equations hold good, then the first column of the original determinant is = 0, and therefore the determinant itself is = 0; that is, the linear equations give

x|a , b , c | - |d , b , c | = 0;
|a', b', c'| |d', b', c'|
|a", b", c"| |d", b", c"|

which is the result obtained above.

We might in a similar way find the values of y and z, but there is a more symmetrical process. Join to the original equations the new equation

[alpha]x + [beta]y + [gamma]z = [delta];

a like process shows that, the equations being satisfied, we have

|[alpha], [beta], [gamma], [delta]| = 0;
| a , b , c , d |
| a' , b' , c' , d' |
| a" , b" , c" , d" |

or, as this may be written,

|[alpha], [beta], [gamma] | - [delta]| a , b , c | = 0:
| a , b , c , d | | a', b', c'|
| a' , b' , c' , d'| | a", b", c"|
| a" , b" , c" , d"| | |

which, considering [delta] as standing herein for its value [alpha]x + [beta]y + [gamma]z, is a consequence of the original equations only: we have thus an expression for [alpha]x + [beta]y + [gamma]z, an arbitrary linear function of the unknown quantities x, y, z; and by comparing the coefficients of [alpha], [beta], [gamma] on the two sides respectively, we have the values of x, y, z; in fact, these quantities, each multiplied by

|a , b , c |,
|a', b', c'|
|a", b", c"|

are in the first instance obtained in the forms

|1 |, | 1 |, | 1 |;
|a , b , c , d | |a , b , c , d | |a , b , c , d |
|a', b', c', d'| |a', b', c', d'| |a', b', c', d'|
|a", b", c", d"| |a", b", c", d"| |a", b", c", d"|

but these are

= |b , c , d |, - |c , d , a |, |d , a , b |,
|b', c', d'| |c', d', a'| |d', a', b'|
|b", c", d"| |c", d", a"| |d", a", b"|

or, what is the same thing,

= |b , c , d |, |c , a , d |, |a , b , d |
|b', c', d'| |c', a', d'| |a', b', d'|
|b", c", d"| |c", a", d"| |a", b", d"|

respectively.

6. _Multiplication of two Determinants of the same Order._--The theorem is obtained very easily from the last preceding definition of a determinant. It is most simply expressed thus--

([alpha], [alpha]', [alpha]"),
([beta],[beta]',[beta]"),
([gamma],[gamma]',[gamma]")
+---------------------------------------+
(a , b , c )| " " " | =
(a', b', c')| " " " |
(a", b", c")| " " " |

= |a , b , c |. |[alpha] , [beta] , [gamma] |,
|a', b', c'| |[alpha]', [beta]', [gamma]'|
|a", b", c"| |[alpha]", [beta]", [gamma]"|

where the expression on the left side stands for a determinant, the terms of the first line being (a, b, c)([alpha], [alpha]', [alpha]"), that is, a[alpha] + b[alpha]' + c[alpha]", (a, b, c)([beta], [beta]', [beta]"), that is, a[beta] + b[beta]' + c[beta]", (a, b, c)([gamma], [gamma]', [gamma]"), that is a[gamma] + b[gamma]' + c[gamma]"; and similarly the terms in the second and third lines are the life functions with (a', b', c') and (a", b", c") respectively.

There is an apparently arbitrary transposition of lines and columns; the result would hold good if on the left-hand side we had written ([alpha], [beta], [gamma]), ([alpha]', [beta]', [gamma]'), ([alpha]", [beta]", [gamma]"), or what is the same thing, if on the right-hand side we had transposed the second determinant; and either of these changes would, it might be thought, increase the elegance of the form, but, for a reason which need not be explained,[2] the form actually adopted is the preferable one.

To indicate the method of proof, observe that the determinant on the left-hand side, _qua_ linear function of its columns, may be broken up into a sum of (3³ =) 27 determinants, each of which is either of some such form as

= [alpha][beta][gamma]'|a , a , b |,
|a', a', b'|
|a", a", b"|

where the term [alpha][beta][gamma]' is not a term of the [alpha][beta][gamma]-determinant, and its coefficient (as a determinant with two identical columns) vanishes; or else it is of a form such as

= [alpha][beta]'[gamma]"|a , b , c |,
|a', b', c'|
|a", b", c"|

that is, every term which does not vanish contains as a factor the abc-determinant last written down; the sum of all other factors ± [alpha][beta]'[gamma]" is the [alpha][beta][gamma]-determinant of the formula; and the final result then is, that the determinant on the left-hand side is equal to the product on the right-hand side of the formula.

7. _Decomposition of a Determinant into complementary Determinants._--Consider, for simplicity, a determinant of the fifth order, 5 = 2 + 3, and let the top two lines be

a , b , c , d , e
a', b', c', d', e'

then, if we consider how these elements enter into the determinant, it is at once seen that they enter only through the determinants of the second order |a , b |, &c., which can be formed by selecting any two |a', b'| columns at pleasure. Moreover, representing the remaining three lines by

a" , b" , c" , d" , e"
a"', b"', c"', d"', e"'
a"", b"", c"", d"", e""

it is further seen that the factor which multiplies the determinant formed with any two columns of the first set is the determinant of the third order formed with the complementary three columns of the second set; and it thus appears that the determinant of the fifth order is a sum of all the products of the form

= |a , b | |c" , d" , e" |,
|a', b"| |c"', d"', e"'|
|c"", d"", e""|

the sign ± being in each case such that the sign of the term ± ab'c"d'"e"" obtained from the diagonal elements of the component determinants may be the actual sign of this term in the determinant of the fifth order; for the product written down the sign is obviously +.

Observe that for a determinant of the n-th order, taking the decomposition to be 1 + (n - 1), we fall back upon the equations given at the commencement, in order to show the genesis of a determinant.

8. Any determinant |a , b | formed out of the elements of the original |a', b'| determinant, by selecting the lines and columns at pleasure, is termed a _minor_ of the original determinant; and when the number of lines and columns, or order of the determinant, is n-1, then such determinant is called a _first minor_; the number of the first minors is = n², the first minors, in fact, corresponding to the several elements of the determinant--that is, the coefficient therein of any term whatever is the corresponding first minor. The first minors, each divided by the determinant itself, form a system of elements _inverse_ to the elements of the determinant.

A determinant is _symmetrical_ when every two elements symmetrically situated in regard to the dexter diagonal are equal to each other; if they are equal and opposite (that is, if the sum of the two elements be = 0), this relation not extending to the diagonal elements themselves, which remain arbitrary, then the determinant is _skew_; but if the relation does extend to the diagonal terms (that is, if these are each = 0), then the determinant is _skew symmetrical_; thus the determinants

|a, h, g|; | a , [nu], - [mu]|; | 0, [nu], - [mu]|
|h, b, f| |- [nu], b,[lambda]| |- [nu], 0,[lambda]|
|g, f, c| | [mu],-[lambda], c | | [mu],- [lambda], 0|

are respectively symmetrical, skew and skew symmetrical:

The theory admits of very extensive algebraic developments, and applications in algebraical geometry and other parts of mathematics. For further developments of the theory of determinants see ALGEBRAIC FORMS. (A. CA.)

9. _History._--These functions were originally known as "resultants,"
a name applied to them by Pierre Simon Laplace, but now replaced by
the title "determinants," a name first applied to certain forms of
them by Carl Friedrich Gauss. The germ of the theory of determinants
is to be found in the writings of Gottfried Wilhelm Leibnitz (1693),
who incidentally discovered certain properties when reducing the
eliminant of a system of linear equations. Gabriel Cramer, in a note
to his _Analyse des lignes courbes algébriques_ (1750), gave the rule
which establishes the sign of a product as _plus_ or _minus_ according
as the number of displacements from the typical form has been even or
odd. Determinants were also employed by Étienne Bezout in 1764, but
the first connected account of these functions was published in 1772
by Charles Auguste Vandermonde. Laplace developed a theorem of
Vandermonde for the expansion of a determinant, and in 1773 Joseph
Louis Lagrange, in his memoir on _Pyramids_, used determinants of the
third order, and proved that the square of a determinant was also a
determinant. Although he obtained results now identified with
determinants, Lagrange did not discuss these functions systematically.
In 1801 Gauss published his _Disquisitiones arithmeticae_, which,
although written in an obscure form, gave a new impetus to
investigations on this and kindred subjects. To Gauss is due the
establishment of the important theorem, that the product of two
determinants both of the second and third orders is a determinant. The
formulation of the general theory is due to Augustin Louis Cauchy,
whose work was the forerunner of the brilliant discoveries made in the
following decades by Hoëné-Wronski and J. Binet in France, Carl Gustav
Jacobi in Germany, and James Joseph Sylvester and Arthur Cayley in
England. Jacobi's researches were published in _Crelle's Journal_
(1826-1841). In these papers the subject was recast and enriched by
new and important theorems, through which the name of Jacobi is
indissolubly associated with this branch of science. The far-reaching
discoveries of Sylvester and Cayley rank as one of the most important
developments of pure mathematics. Numerous new fields were opened up,
and have been diligently explored by many mathematicians.
Skew-determinants were studied by Cayley; axisymmetric-determinants by
Jacobi, V. A. Lebesque, Sylvester and O. Hesse, and centro-symmetric
determinants by W. R. F. Scott and G. Zehfuss. Continuants have been
discussed by Sylvester; alternants by Cauchy, Jacobi, N. Trudi, H.
Nagelbach and G. Garbieri; circulants by E. Catalan, W. Spottiswoode
and J. W. L. Glaisher, and Wronskians by E. B. Christoffel and G.
Frobenius. Determinants composed of binomial coefficients have been
studied by V. von Zeipel; the expression of definite integrals as
determinants by A. Tissot and A. Enneper, and the expression of
continued fractions as determinants by Jacobi, V. Nachreiner, S.
Günther and E. Fürstenau. (See T. Muir, _Theory of Determinants_,
1906).

[1] The expression, a linear function, is here used in its narrowest sense, a linear function without constant term; what is meant is that the determinant is in regard to the elements a, a', a", ... of any column or line thereof, a function of the form Aa + A'a' + A"a" + ... without any term independent of a, a', a" ...

[2] The reason is the connexion with the corresponding theorem for the multiplication of two matrices.

DETERMINISM (Lat. _determinare_, to prescribe or limit), in ethics, the name given to the theory that all moral choice, so called, is the determined or necessary result of psychological and other conditions. It is opposed to the various doctrines of Free-Will, known as voluntarism, libertarianism, indeterminism, and is from the ethical standpoint more or less akin to necessitarianism and fatalism. There are various degrees of determinism. It may be held that every action is causally connected not only externally with the sum of the agent's environment, but also internally with his motives and impulses. In other words, if we could know exactly all these conditions, we should be able to forecast with mathematical certainty the course which the agent would pursue. In this theory the agent cannot be held responsible for his action in any sense. It is the extreme antithesis of Indeterminism or Indifferentism, the doctrine that a man is absolutely free to choose between alternative courses (the _liberum arbitrium indifferentiae_). Since, however, the evidence of ordinary consciousness almost always goes to prove that the individual, especially in relation to future acts, regards himself as being free within certain limitations to make his own choice of alternatives, many determinists go so far as to admit that there may be in any action which is neither reflex nor determined by external causes solely an element of freedom. This view is corroborated by the phenomenon of remorse, in which the agent feels that he ought to, and could, have chosen a different course of action. These two kinds of determinism are sometimes distinguished as "hard" and "soft" determinism. The controversy between determinism and libertarianism hinges largely on the significance of the word "motive"; indeed in no other philosophical controversy has so much difficulty been caused by purely verbal disputation and ambiguity of expression. How far, and in what sense, can action which is determined by motives be said to be free? For a long time the advocates of free-will, in their eagerness to preserve moral responsibility, went so far as to deny all motives as influencing moral action. Such a contention, however, clearly defeats its own object by reducing all action to chance. On the other hand, the scientific doctrine of evolution has gone far towards obliterating the distinction between external and internal compulsion, e.g. motives, character and the like. In so far as man can be shown to be the product of, and a link in, a long chain of causal development, so far does it become impossible to regard him as self-determined. Even in his motives and his impulses, in his mental attitude towards outward surroundings, in his appetites and aversions, inherited tendency and environment have been found to play a very large part; indeed many thinkers hold that the whole of a man's development, mental as well as physical, is determined by external conditions.

In the Bible the philosophical-religious problem is nowhere discussed, but Christian ethics as set forth in the New Testament assumes throughout the freedom of the human will. It has been argued by theologians that the doctrine of divine fore-knowledge, coupled with that of the divine origin of all things, necessarily implies that all human action was fore-ordained from the beginning of the world. Such an inference is, however, clearly at variance with the whole doctrine of sin, repentance and the atonement, as also with that of eternal reward and punishment, which postulates a real measure of human responsibility.

For the history of the free-will controversy see the articles, WILL, PREDESTINATION (for the theological problems), ETHICS.

DETINUE (O. Fr. _detenue_, from _detenir_, to hold back), in law, an action whereby one who has an absolute or a special property in goods seeks to recover from another who is in actual possession and refuses to redeliver them. If the plaintiff succeeds in an action of detinue, the judgment is that he recover the chattel or, if it cannot be had, its value, which is assessed by the judge and jury, and also certain damages for detaining the same. An order for the restitution of the specific goods may be enforced by a special writ of execution, called a writ of delivery. (See CONTRACT; TROVER.)

DETMOLD, a town of Germany, capital of the principality of Lippe-Detmold, beautifully situated on the east slope of the Teutoburger Wald, 25 m. S. of Minden, on the Herford-Altenbeken line of the Prussian state railways. Pop. (1905) 13,164. The residential château of the princes of Lippe-Detmold (1550), in the Renaissance style, is an imposing building, lying with its pretty gardens nearly in the centre of the town; whilst at the entrance to the large park on the south is the New Palace (1708-1718), enlarged in 1850, used as the dower-house. Detmold possesses a natural history museum, theatre, high school, library, the house in which the poet Ferdinand Freiligrath (1810-1876) was born, and that in which the dramatist Christian Dietrich Grabbe (1801-1836), also a native, died. The leading industries are linen-weaving, tanning, brewing, horse-dealing and the quarrying of marble and gypsum. About 3 m. to the south-west of the town is the Grotenburg, with Ernst von Bandel's colossal statue of Hermann or Arminius, the leader of the Cherusci. Detmold (Thiatmelli) was in 783 the scene of a conflict between the Saxons and the troops of Charlemagne.

DETROIT, the largest city of Michigan, U.S.A., and the county-seat of Wayne county, on the Detroit river opposite Windsor, Canada, about 4 m. W. from the outlet of Lake St Clair and 18 m. above Lake Erie. Pop. (1880) 116,340; (1890) 205,876; (1900) 285,704, of whom 96,503 were foreign-born and 4111 were negroes; (1910 census) 465,766. Of the foreign-born in 1900, 32,027 were Germans and 10,703 were German Poles, 25,403 were English Canadians and 3541 French Canadians, 6347 were English and 6412 were Irish. Detroit is served by the Michigan Central, the Lake Shore & Michigan Southern, the Wabash, the Grand Trunk, the Père Marquette, the Detroit & Toledo Shore Line, the Detroit, Toledo & Ironton and the Canadian Pacific railways. Two belt lines, one 2 m. to 3 m., and the other 6 m. from the centre of the city, connect the factory districts with the main railway lines. Trains are ferried across the river to Windsor, and steamboats make daily trips to Cleveland, Wyandotte, Mount Clemens, Port Huron, to less important places between, and to several Canadian ports. Detroit is also the S. terminus for several lines to more remote lake ports, and electric lines extend from here to Port Huron, Flint, Pontiac, Jackson, Toledo and Grand Rapids.

The city extended in 1907 over about 41 sq. m., an increase from 29 sq. m. in 1900 and 36 sq. m. in 1905. Its area in proportion to its population is much greater than that of most of the larger cities of the United States. Baltimore, for example, had in 1904 nearly 70% more inhabitants (estimated), while its area at that time was a little less and in 1907 was nearly one-quarter less than that of Detroit. The ground within the city limits as well as that for several miles farther back is quite level, but rises gradually from the river bank, which is only a few feet in height. The Detroit river, along which the city extends for about 10 m., is here ½ m. wide and 30 ft. to 40 ft. deep; its current is quite rapid; its water, a beautiful clear blue; at its mouth it has a width of about 10 m., and in the river there are a number of islands, which during the summer are popular resorts. The city has a 3 m. frontage on the river Rouge, an estuary of the Detroit, with a 16 ft. channel. Before the fire by which the city was destroyed in 1805, the streets were only 12 ft. wide and were unpaved and extremely dirty. But when the rebuilding began, several avenues from 100 ft. to 200 ft. wide were--through the influence of Augustus B. Woodward (c. 1775-1827), one of the territorial judges at the time and an admirer of the plan of the city of Washington--made to radiate from two central points. From a half circle called the Grand Circus there radiate avenues 120 ft. and 200 ft. wide. About ¼ m. toward the river from this was established another focal point called the Campus Martius, 600 ft. long and 400 ft. wide, at which commence radiating or cross streets 80 ft. and 100 ft. wide. Running north from the river through the Campus Martius and the Grand Circus is Woodward Avenue, 120 ft. wide, dividing the present city, as it did the old town, into nearly equal parts. Parallel with the river is Jefferson Avenue, also 120 ft. wide. The first of these avenues is the principal retail street along its lower portion, and is a residence avenue for 4 m. beyond this. Jefferson is the principal wholesale street at the lower end, and a fine residence avenue E. of this. Many of the other residence streets are 80 ft. wide. The setting of shade trees was early encouraged, and large elms and maples abound. The intersections of the diagonal streets left a number of small, triangular parks, which, as well as the larger ones, are well shaded. The streets are paved mostly with asphalt and brick, though cedar and stone have been much used, and kreodone block to some extent. In few, if any, other American cities of equal size are the streets and avenues kept so clean. The Grand Boulevard, 150 ft. to 200 ft. in width and 12 m. in length, has been constructed around the city except along the river front. A very large proportion of the inhabitants of Detroit own their homes: there are no large congested tenement-house districts; and many streets in various parts of the city are faced with rows of low and humble cottages often having a garden plot in front.

Of the public buildings the city hall (erected 1868-1871), overlooking the Campus Martius, is in Renaissance style, in three storeys; the flagstaff from the top of the tower reaches a height of 200 ft. On the four corners above the first section of the tower are four figures, each 14 ft. in height, to represent Justice, Industry, Art and Commerce, and on the same level with these is a clock weighing 7670 lb--one of the largest in the world. In front of the building stands the Soldiers' and Sailors' monument, 60 ft. high, designed by Randolph Rogers (1825-1892) and unveiled in 1872. At each of the four corners in each of three sections rising one above the other are bronze eagles and figures representing the United States Infantry, Marine, Cavalry and Artillery, also Victory, Union, Emancipation and History; the figure by which the monument is surmounted was designed to symbolize Michigan. A larger and more massive and stately building than the city hall is the county court house, facing Cadillac Square, with a lofty tower surmounted by a gilded dome. The Federal building is a massive granite structure, finely decorated in the interior. Among the churches of greatest architectural beauty are the First Congregational, with a fine Byzantine interior, St John's Episcopal, the Woodward Avenue Baptist and the First Presbyterian, all on Woodward Avenue, and St. Anne's and Sacred Heart of Mary, both Roman Catholic. The municipal museum of art, in Jefferson Avenue, contains some unusually interesting Egyptian and Japanese collections, the Scripps' collection of old masters, other valuable paintings, and a small library; free lectures on art are given here through the winter. The public library had 228,500 volumes in 1908, including one of the best collections of state and town histories in the country. A large private collection, owned by C. M. Burton and relating principally to the history of Detroit, is also open to the public. The city is not rich in outdoor works of art. The principal ones are the Merrill fountain and the soldiers' monument on the Campus Martius, and a statue of Mayor Pingree in West Grand Circus Park.