Chapter V: Part 5

To prove this, let us consider the catenary as the form of equilibrium of a chain suspended between two fixed points A and B. Suppose the chain hanging between A and B to be of very great length, then the tension at A or B will be very great. Let the chain be hauled in over a peg at A. At first the tension will diminish, but if the process be continued the tension will reach a minimum value and will afterwards increase to infinity as the chain between A and B approaches to the form of a straight line. Hence for every tension greater than the minimum tension there are two catenaries passing through A and B. Since the tension is measured by the height above the directrix these two catenaries have the same directrix. Every catenary lying between them has its directrix higher, and every catenary lying beyond them has its directrix lower than that of the two catenaries.

Now let us consider the surfaces of revolution formed by this system of catenaries revolving about the directrix of the two catenaries of equal tension. We know that the radius of curvature of a surface of revolution in the plane normal to the meridian plane is the portion of the normal intercepted by the axis of revolution.

The radius of curvature of a catenary is equal and opposite to the portion of the normal intercepted by the directrix of the catenary. Hence a catenoid whose directrix coincides with the axis of revolution has at every point its principal radii of curvature equal and opposite, so that the mean curvature of the surface is zero.

The catenaries which lie between the two whose direction coincides with the axis of revolution generate surfaces whose radius of curvature convex towards the axis in the meridian plane is less than the radius of concave curvature. The mean curvature of these surfaces is therefore convex towards the axis. The catenaries which lie beyond the two generate surfaces whose radius of curvature convex towards the axis in the meridian plane is greater than the radius of concave curvature. The mean curvature of these surfaces is, therefore, concave towards the axis.

Now if the pressure is equal on both sides of a liquid film, and if its mean curvature is zero, it will be in equilibrium. This is the case with the two catenoids. If the mean curvature is convex towards the axis the film will move from the axis. Hence if a film in the form of the catenoid which is nearest the axis is ever so slightly displaced from the axis it will move farther from the axis till it reaches the other catenoid.

If the mean curvature is concave towards the axis the film will tend to approach the axis. Hence if a film in the form of the catenoid which is nearest the axis be displaced towards the axis, it will tend to move farther towards the axis and will collapse. Hence the film in the form of the catenoid which is nearest the axis is in unstable equilibrium under the condition that it is exposed to equal pressures within and without. If, however, the circular ends of the catenoid are closed with solid disks, so that the volume of air contained between these disks and the film is determinate, the film will be in stable equilibrium however large a portion of the catenary it may consist of.

The criterion as to whether any given catenoid is stable or not may be obtained as follows:--

Let PABQ and ApqB (fig. 14) be two catenaries having the same directrix and intersecting in A and B. Draw Pp and Qq touching both catenaries, Pp and Qq will intersect at T, a point in the directrix; for since any catenary with its directrix is a similar figure to any other catenary with its directrix, if the directrix of the one coincides with that of the other the centre of similitude must lie on the common directrix. Also, since the curves at P and p are equally inclined to the directrix, P and p are corresponding points and the line Pp must pass through the centre of similitude. Similarly Qq must pass through the centre of similitude. Hence T, the point of intersection of Pp and Qq, must be the centre of similitude and must be on the common directrix. Hence the tangents at A and B to the upper catenary must intersect above the directrix, and the tangents at A and B to the lower catenary must intersect below the directrix. The condition of stability of a catenoid is therefore that the tangents at the extremities of its generating catenary must intersect before they reach the directrix.

_Stability of a Plane Surface._--We shall next consider the limiting conditions of stability of the horizontal surface which separates a heavier fluid above from a lighter fluid below. Thus, in an experiment of F. Duprez ("Sur un cas particulier de l'equilibre des liquides," _Nouveaux Mem. del' Acad. de Belgique, 1851 et 1853_), a vessel containing olive oil is placed with its mouth downwards in a vessel containing a mixture of alcohol and water, the mixture being denser than the oil. The surface of separation is in this case horizontal and stable, so that the equilibrium is established of itself. Alcohol is then added very gradually to the mixture till it becomes lighter than the oil. The equilibrium of the fluids would now be unstable if it were not for the tension of the surface which separates them, and which, when the orifice of the vessel is not too large, continues to preserve the stability of the equilibrium.

When the equilibrium at last becomes unstable, the destruction of equilibrium takes place by the lighter fluid ascending in one part of the orifice and the heavier descending in the other. Hence the displacement of the surface to which we must direct our attention is one which does not alter the volume of the liquid in the vessel, and which therefore is upward in one part of the surface and downward in another. The simplest case is that of a rectangular orifice in a horizontal plane, the sides being a and b.

Let the surface of separation be originally in the plane of the
orifice, and let the co-ordinates x and y be measured from one corner
parallel to the sides a and b respectively, and let z be measured
upwards. Then if [rho] be the density of the upper liquid, and [sigma]
that of the lower liquid, and P the original pressure at the surface
of separation, then when the surface receives an upward displacement
z, the pressure above it will be P - [rho]gz, and that below it will
be P - [sigma]gz, so that the surface will be acted on by an upward
pressure ([rho] - [sigma])gz. Now if the displacement z be everywhere
very small, the curvature in the planes parallel to xz and yz will be
d^2z/dx^2 and d^2z/dy^2 respectively, and if T is the surface-tension
the whole upward force will be

/ d^2z d^2z \
T ( ---- + ---- ) + ([rho] - [sigma])gz.
\ dx^2 dy^2 /

If this quantity is of the same sign as z, the displacement will be
increased, and the equilibrium will be unstable. If it is of the
opposite sign from z, the equilibrium will be stable. The limiting
condition may be found by putting it equal to zero. One form of the
solution of the equation, and that which is applicable to the case of
a rectangular orifice, is

z = C sin px sin qy.

Substituting in the equation we find the condition

/ + ^(ve) stable.
(p^2 + q^2)T - ([rho] - [sigma])g = < 0 neutral.
\ - ^(ve) unstable.

That the surface may coincide with the edge of the orifice, which is
a rectangle, whose sides are a and b, we must have

pa = m[pi], qb = n[pi],

when m and n are integral numbers. Also, if m and n are both unity,
the displacement will be entirely positive, and the volume of the
liquid will not be constant. That the volume may be constant, either n
or m must be an even number. We have, therefore, to consider the
conditions under which

/ m^2 n^2\
[pi]^2 ( --- + --- )T - ([rho] - [sigma])g
\ a^2 b^2/

cannot be made negative. Under these conditions the equilibrium is
stable for all small displacements of the surface. The smallest
admissible value of

m^2 n^2 4 1
--- + --- is --- + ---,
a^2 b^2 a^2 b^2

where a is the longer side of the rectangle. Hence the condition of
stability is that

/ 4 1 \
[pi]^2 ( --- + --- )T - ([rho] - [sigma])g
\a^2 + b^2/

is a positive quantity. When the breadth b is less than
__________________
/ [pi]^2 T
/ ------------------
\/ ([rho] - [sigma])g

the length a may be unlimited.

When the orifice is circular of radius a, the limiting value of a is
_______
/ T
/ ------- z, where z is the least root of the equation
\/ g [rho]

2 z^2 z^4 z^6
--- J1(z) = 1 - --- + ------- + ----------- + &c., = 0.
z 2.4 2.4^2.6 2.4^2.6^2.8

The least root of this equation is

z = 3.83171.

If h is the height to which the liquid will rise in a capillary tube
of unit radius, then the diameter of the largest orifice is
____ ___
2a = 3.83171 \/(2h) = 5.4188 \/(h).

Duprez found from his experiments
___
2a = 5.485 \/(h).

[The above theory may be well illustrated by a lecture experiment. A thin-walled glass tube of internal diameter equal to 14-1/2 mm. is ground true at the lower end. The upper end is contracted and is fitted with a rubber tube under the control of a pinch-cock. Water is sucked up from a vessel of moderate size, the rubber is nipped, and by a quick motion the tube and vessel are separated, preferably by a downward movement of the latter. The inverted tube, with its suspended water, being held in a clamp, a beaker containing a few drops of ether is brought up from below until the free surface of the water is in contact with ether vapour. The lowering of tension, which follows the condensation of the vapour, is then strikingly shown by the sudden precipitation of the water.]

_Effect of Surface-tension on the Velocity of Waves._--When a series of waves is propagated on the surface of a liquid, the surface-tension has the effect of increasing the pressure at the crests of the waves and diminishing it in the troughs. If the wave-length is [lambda], the equation of the surface is

y = b sin 2[pi](x/[lambda]).

The pressure due to the surface tension T is

d^2y 4[pi]^2
p = - T ---- = ---------- Ty.
dx^2 [lambda]^2

This pressure must be added to the pressure due to gravity g [rho] y. Hence the waves will be propagated as if the intensity of gravity had been

4[pi]^2 T
f = g + ---------- -----
[lambda]^2 [rho]

instead of g. Now it is shown in hydrodynamics that the velocity of propagation of waves in deep water is that acquired by a heavy body falling through half the radius of the circle whose circumference is the wave-length, or

f[lambda] g[lambda] 2[pi]T
v^2 = --------- = --------- + -------------. (1)
2[pi] 2[pi] [rho][lambda]

This velocity is a minimum when
______
/ T
[lambda] = 2 [pi] / ------,
\/ g[rho]

and the minimum value is
_______
4 / Tg
v = / 4 -----.
\/ [rho]

For waves whose length from crest to crest is greater than [lambda], the principal force concerned in the motion is that of gravitation. For waves whose length is less than [lambda] the principal force concerned is that of surface-tension. Lord Kelvin proposed to distinguish the latter kind of waves by the name of ripples.

When a small body is partly immersed in a liquid originally at rest, and moves horizontally with constant velocity V, waves are propagated through the liquid with various velocities according to their respective wave-lengths. In front of the body the relative velocity of the fluid and the body varies from V where the fluid is at rest, to zero at the cutwater on the front surface of the body. The waves produced by the body will travel forwards faster than the body till they reach a distance from it at which the relative velocity of the body and the fluid is equal to the velocity of propagation corresponding to the wave-length. The waves then travel along with the body at a constant distance in front of it. Hence at a certain distance in front of the body there is a series of waves which are stationary with respect to the body. Of these, the waves of minimum velocity form a stationary wave nearest to the front of the body. Between the body and this first wave the surface is comparatively smooth. Then comes the stationary wave of minimum velocity, which is the most marked of the series. In front of this is a double series of stationary waves, the gravitation waves forming a series increasing in wave-length with their distance in front of the body, and the surface-tension waves or ripples diminishing in wave-length with their distance from the body, and both sets of waves rapidly diminishing in amplitude with their distance from the body.

If the current-function of the water referred to the body considered as origin is [psi], then the equation of the form of the crest of a wave of velocity w, the crest of which travels along with the body, is

d[psi] = w ds

where ds is an element of the length of the crest. To integrate this equation for a solid of given form is probably difficult, but it is easy to see that at some distance on either side of the body, where the liquid is sensibly at rest, the crest of the wave will approximate to an asymptote inclined to the path of the body at an angle whose sine is w/V, where w is the velocity of the wave and V is that of the body.

The crests of the different kinds of waves will therefore appear to diverge as they get farther from the body, and the waves themselves will be less and less perceptible. But those whose wave-length is near to that of the wave of minimum velocity will diverge less than any of the others, so that the most marked feature at a distance from the body will be the two long lines of ripples of minimum velocity. If the angle between these is 2[theta], the velocity of the body is w sec[theta], where w for water is about 23 centimetres per second.

[Lord Kelvin's formula (1) may be applied to find the surface-tension of a clean or contaminated liquid from observations upon the length of waves of known periodic time, travelling over the surface. If v = [lambda]/[tau] we have

[rho][lambda]^3 2[pi]h g[lambda]^2[rho]
T = --------------- - coth-------- - ---------------- (2)
2[pi][tau]^2 [lambda] 4[pi]^2

h denoting the depth of the liquid. In observations upon ripples the factor involving h may usually be omitted, and thus in the case of water ([rho] = 1)

[lambda]^3 g[lambda]^2
T = ------------ - ------------ (3)
2[pi][tau]^2 4[pi]^2

simply. The method has the advantage of independence of what may occur at places where the liquid is in contact with solid bodies.

The waves may be generated by electrically maintained tuning-forks from which dippers touch the surface; but special arrangements are needed for rendering them visible. The obstacles are (1) the smallness of the waves, and (2) the changes which occur at speeds too rapid for the eye to follow. The second obstacle is surmounted by the aid of the stroboscopic method of observation, the light being intermittent in the period of vibration, so that practically only one phase is seen. In order to render visible the small waves employed, and which we may regard as deviations of a plane surface from its true figure, the method by which Foucault tested reflectors is suitable. The following results have been obtained

Clean 74.0
Greasy to the point where camphor motions nearly cease 53.0
Saturated with olive oil 41.0
Saturated with sodium oleate 25.0

(_Phil. Mag._ November 1890) for the tensions of various water-surfaces at 18 deg. C., reckoned in C.G.S. measure.

The tension for clean water thus found is considerably lower than that (81) adopted by Quincke, but it seems to be entitled to confidence, and at any rate the deficiency is not due to contamination of the surface.

A calculation analogous to that of Lord Kelvin may be applied to find the frequency of small transverse vibrations of a cylinder of liquid under the action of the capillary force. Taking the case where the motion is strictly in two dimensions, we may write as the polar equation of the surface at time t

r = a + a_n cos n[theta] cos pt, (4)

where p is given by

T
p^2 = (n^3 - n)--------. (5)
[rho]a^3

If n = 1, the section remains circular, there is no force of restitution, and p = 0. The principal vibration, in which the section becomes elliptical, corresponds to n = 2.

Vibrations of this kind are observed whenever liquid issues from an elliptical or other non-circular hole, or even when it is poured from the lip of an ordinary jug; and they are superposed upon the general progressive motion. Since the phase of vibration depends upon the time elapsed, it is always the same at the same point in space, and thus the motion is _steady_ in the hydrodynamical sense, and the boundary of the jet is a fixed surface. In so far as the vibrations may be regarded as isochronous, the distance between consecutive corresponding points of the recurrent figure, or, as it may be called, the _wave-length_ of the figure, is directly proportional to the velocity of the jet, i.e. to the square root of the head. But as the head increases, so do the _lateral_ velocities which go to form the transverse vibrations. A departure from the law of isochronism may then be expected to develop itself.

The transverse vibrations of non-circular jets allow us to solve a problem which at first sight would appear to be of great difficulty. According to Marangoni the diminished surface-tension of soapy water is due to the formation of a film. The formation cannot be instantaneous, and if we could measure the tension of a surface not more than 1/100 of a second old, we might expect to find it undisturbed, or nearly so, from that proper to pure water. In order to carry out the experiment the jet is caused to issue from an elliptical orifice in a thin plate, about 2 mm. by 1 mm., under a head of 15 cm. A comparison under similar circumstances shows that there is hardly any difference in the wave-lengths of the patterns obtained with pure and with soapy water, from which we conclude that at this initial stage, the surface-tensions are the same. As early as 1869 Dupre had arrived at a similar conclusion from experiments upon the vertical rise of fine jets.

A formula, similar to (5), may be given for the frequencies of vibration of a spherical mass of liquid under capillary force. If, as before, the frequency be p/2[Pi], and a the radius of the sphere, we have

T
p^2 = n(n - 1)(n + 2)--------, (6)
[rho]a^3

n denoting the order of the spherical harmonic by which the deviation from a spherical figure is expressed. To find the radius of the sphere of water which vibrates seconds, put p = 2[Pi], T = 81, [rho] = 1, n = 2. Thus a = 2.54 cms., or one inch very nearly.]

TABLES OF SURFACE-TENSION

In the following tables the units of length, mass and time are the centimetre, the gramme and the second, and the unit of force is that which if it acted on one gramme for one second would communicate to it a velocity of one centimetre per second:--

_Table of Surface-Tension at 20 deg. C. (Quincke)._

+----------------------------------+---------+----------------------------+---------------------------------------+
| | | Tension of surface | Angle of contact with |
| Liquid. | Specific| separating the liquid from | glass in presence of |
| | Gravity.+-------+---------+----------+-------------+------------+------------+
| | | Air. | Water. | Mercury. | Air. | Water. | Mercury. |
+----------------------------------+---------+-------+---------+----------+-------------+------------+------------|
| Water | 1 | 81 | . . | 418 | 25 deg. 32' | . . | 25 deg. 6' |
| Mercury | 13.5432 | 540 | 418 | . . | 51 deg. 8' | 26 deg. 8' | . . |
| Bisulfuride of Carbon | 1.2687 | 32.1 | 41.75 | 372.5 | 32 deg. 16' | 15 deg. 8' | . . |
| Chloroform | 1.4878 | 30.6 | 29.5 | 399 | . . | . . | . . |
| Alcohol | 0.7906 | 25.5 | . . | 399 | 25 deg. 12' | . . | . . |
| Olive Oil | 0.9136 | 36.9 | 20.56 | 335 | 21 deg. 50' | 17 deg. | 47 deg. 2' |
| Turpentine | 0.8867 | 29.7 | 11.55 | 250.5 | 37 deg. 44' | 37 deg. 44'| 47 deg. 2' |
| Petroleum | 0.7977 | 31.7 | 27.8 | 284 | 36 deg. 20' | 42 deg. 46'| . . |
| Hydrochloric Acid | 1.1 | 70.1 | . . | 377 | . . | 42 deg. 46'| . . |
| Solution of Hyposulphite of Soda | 1.1248 | 77.5 | . . | 442.5 | 23 deg. 20' | . . | 10 deg. 42'|
+----------------------------------+---------+-------+---------+----------+-------------+------------+------------+

Olive Oil and Alcohol, 12.2.

Olive oil and aqueous alcohol (sp. g. .9231, tension of free surface
25.5), 6.8, angle 87 deg. 48'.

Quincke has determined the surface-tension of a great many substances near their point of fusion or solidification. His method was that of observing the form of a large drop standing on a plane surface. If K is the height of the flat surface of the drop, and k that of the point where its tangent plane is vertical, then

T = 1/2(K - k)^2 g[rho]

_Surface-Tensions of Liquids at their Point of Solidification. From
Quincke._

+--------------------+-----------------+----------+
| Substance. | Temperature of | Surface- |
| | Solidification. | Tension. |
+--------------------+-----------------+----------+
| Platinum | 2000 deg. C. | 1658 |
| Gold | 1200 deg. | 983 |
| Zinc | 360 deg. | 860 |
| Tin | 230 deg. | 587 |
| Mercury | -40 deg. | 577 |
| Lead | 330 deg. | 448 |
| Silver | 1000 deg. | 419 |
| Bismuth | 265 deg. | 382 |
| Potassium | 58 deg. | 364 |
| Sodium | 90 deg. | 253 |
| Antimony | 432 deg. | 244 |
| Borax | 1000 deg. | 212 |
| Carbonate of Soda | 1000 deg. | 206 |
| Chloride of Sodium | . . | 114 |
| Water | 0 deg. | 86.2 |
| Selenium | 217 deg. | 70.4 |
| Sulphur | 111 deg. | 41.3 |
| Phosphorus | 43 deg. | 41.1 |
| Wax | 68 deg. | 33.4 |
+--------------------+-----------------+----------+

Quincke finds that for several series of substances the surface-tension is nearly proportional to the density, so that if we call (K - k)^2 = 2T/g[rho] the specific cohesion, we may state the general results of his experiments as follows:--

The bromides and iodides have a specific cohesion about half that of mercury. The nitrates, chlorides, sugars and fats, as also the metals lead, bismuth and antimony, have a specific cohesion nearly equal to that of mercury. Water, the carbonates and sulphates, and probably phosphates, and the metals platinum, gold, silver, cadmium, tin and copper have a specific cohesion double that of mercury. Zinc, iron and palladium, three times that of mercury, and sodium, six times that of mercury.

RELATION OF SURFACE-TENSION TO TEMPERATURE

It appears from the experiments of Brunner and of Wolf on the ascent of water in tubes that at the temperature t deg. centigrade

T = 75.20 (1 - 0.00187t) (Brunner);
= 76.08 (1 - 0.002t + 0.00000415t^2), for a tube .02346 cm. diameter
(Wolf);
= 77.34 (1 - 0.00181t), for a tube .03098 cm. diameter (Wolf).

Lord Kelvin has applied the principles of Thermodynamics to determine the thermal effects of increasing or diminishing the area of the free surface of a liquid, and has shown that in order to keep the temperature constant while the area of the surface increases by unity, an amount of heat must be supplied to the liquid which is dynamically equivalent to the product of the absolute temperature into the decrement of the surface-tension per degree of temperature. We may call this the _latent heat of surface-extension_.

It appears from the experiments of C. Brunner and C.J.E. Wolf that at ordinary temperatures the latent heat of extension of the surface of water is dynamically equivalent to about half the mechanical work done in producing the surface-extension.

REFERENCES.--Further information on some of the matters discussed
above will be found in Lord Rayleigh's _Collected Scientific Papers_
(1901). In its full extension the subject of capillarity is very wide.
Reference may be made to A.W. Reinold and Sir A.W. Rucker (_Phil.
Trans._ 1886, p. 627); Sir W. Ramsay and J. Shields (_Zeitschr.
physik. Chem._ 1893, 12, p. 433); and on the theoretical side, see
papers by Josiah Willard Gibbs; R. Eotvos (_Wied. Ann._, 1886, 27, p.
452); J.D. Van der Waals, G. Bakker and other writers of the Dutch
school. (J. C. M.; R.)

FOOTNOTES:

[1] In this revision of James Clerk Maxwell's classical article in
the ninth edition of the _Encyclopaedia Britannica_, additions are
marked by square brackets.

[2] See Enrico Betti, _Teoria della Capillarita: Nuovo Cimento_
(1867); a memoir by M. Stahl, "Ueber einige Punckte in der Theorie
der Capillarerscheinungen," _Pogg. Ann._ cxxxix. p. 239 (1870); and
J.D. Van der Waal's _Over de Continuiteit van den Gasen
Vloeistoftoestand_. A good account of the subject from a mathematical
point of view will be found in James Challis's "Report on the Theory
of Capillary Attraction," _Brit. Ass. Report_, iv. p. 235 (1834).

[3] _Nouvelle theorie de l'action capillaire_ (1831).

[4] _Determinatio superficiei minimae rotatione curvae data duo
puncta jungentis circa datum axem ortae_ (Gottingen, 1831).

[5] _Lecons de calcul des variations_ (Paris, 1861).

[6] "Sur la surface de revolution dont la courbure moyenne est
constante," _Liouville's Journal_, vi.

[7] "Theorie geometrique des rayons et centres de courbure," _Bullet,
de l'Acad. de Belgique_, 1857.

[8] _Tractatus de Theoria Mathematica Phaenomenorum in Liquidis
actioni gravitatis detractis observatorum_ (Bonn, 1857).

[9] _Journal de l'Institut_, No. 1260.

[10] _Statique experimental et theorique des liquides_, 1873.

CAPISTRANO, GIOVANNI DI (1386-1456), Italian friar, theologian and inquisitor, was born in the little village of Capistrano in the Abruzzi, of a family which had come to Italy with the Angevins. He lived at first a wholly secular life, married, and became a successful magistrate; he took part in the continual struggles of the small Italian states in such a way as to compromise himself. During his captivity he was practically ruined and lost his young wife. He then in despair entered the Franciscan order and at once gave himself up to the most rigorous asceticism, violently defending the ideal of strict observance. He was charged with various missions by the popes Eugenius IV. and Nicholas V., in which he acquitted himself with implacable violence. As legate or inquisitor he persecuted the last Fraticelli of Ferrara, the Jesuati of Venice, the Jews of Sicily, Moldavia and Poland, and, above all, the Hussites of Germany, Hungary and Bohemia; his aim in the last case was to make conferences impossible between the representatives of Rome and the Bohemians, for every attempt at conciliation seemed to him to be conniving at heresy. Finally, after the taking of Constantinople, he succeeded in gathering troops together for a crusade against the Turks (1455), which at least helped to raise the siege of Belgrade, which was being blockaded by Mahommed II. He died shortly afterwards (October 23, 1456), and was canonized in 1690. Capistrano, in spite of this restless life, found time to work both in the lifetime of his master St Bernardino of Siena and after, at the reform of the order of the minor Franciscans, and to uphold both in his writings and his speeches the most advanced theories upon the papal supremacy as opposed to that of the councils.

See E. Jacob, _Johannes von Capistrano_, vol. i.: "Das Leben und
Wirken Capistrans;" vol. ii.: "Die handschriftlichen Aufzeichnungen
von Reden und Tractaten Capistrans," (1st series, Breslau, 1903-1905).
(P. A.)

CAPITAL (Lat. _caput_, head), in architecture, the crowning member of the column, which projects on each side as it rises, in order to support the abacus and unite the square form of the latter with the circular shaft. The bulk of the capital may either be convex, as in the Doric capital; concave, as in the bell of the Corinthian capital; or bracketed out, as in the Ionic capital. These are the three principal types on which all capitals are based. The capitals of Greek, Doric, Ionic and Corinthian orders are given in the article ORDER.

From the prominent position it occupies in all monumental buildings, it has always been the favourite feature selected for ornamentation, and consequently it has become the clearest indicator of any style.

The two earliest capitals of importance are those which are based on the lotus (fig. 1) and papyrus (fig. 2) plants respectively, and these, with the palm tree capital, were the chief types employed by the Egyptians down to the 3rd century B.C., when, under the Ptolemaic dynasties, various river plants were employed decoratively and the lotus capital goes through various modifications (fig 3) Some kind of volute capital is shown in the Assyrian bas-reliefs, but no Assyrian capital has ever been found, those exhibited as such in the British Museum are bases.

The Persian capital belongs to the third class above mentioned, the brackets are carved with the lion (fig. 4) or the griffin projecting right and left to support and lessen the bearing of the architrave, and on their backs carry other brackets at right angles to support the cross timbers. The profuse decoration underneath the bracket capital in the palace of Xerxes and elsewhere, serves no structural function, but gives some variety to the extenuated shaft.

The earliest Greek capital is that shown in the Temple-fresco at Cnossus in Crete (1600 B.C.); it was of the first type--convex, and was probably moulded in stucco: the second is represented by the richly carved example of the columns (fig 5) flanking the tomb of Agamemnon in Mycenae (c. 1100 B.C.), also convex, carved with the chevron device, and with an apophyge on which the buds of some flowers are sculptured. The Doric capital of the temple of Apollo at Syracuse (c. 700 B.C.) follows, in which the echinus moulding has become a more definite form: this in the Parthenon reaches its culmination, where the convexity is at the top and bottom with a delicate uniting curve The sloping side of the echinus becomes flatter in the later examples, and in the Colosseum at Rome forms a quarter round.

In the Ionic capital of the Archaic temple of Diana at Ephesus (560 B.C.) the width of the abacus is twice that of its depth, consequently the earliest Ionic capital known was virtually a bracket capital. A century later, in the temple on the Ilissus, published in Stuart and Revett, the abacus has become square. One of the most beautiful Corinthian capitals is that from the Tholos of Epidaurus (400 B.C.) (fig. 6); it illustrates the transition between the earlier Greek capital of Bassae and the Roman version of the temple of Mars Ultor (fig. 7).

The foliage of the Greek Corinthian capital was based on the Acanthus spinosus, that of the Roman on the Acanthus mollis; the capital of the temple of Vesta and other examples at Pompeii are carved with foliage of a different type.

Byzantine capitals are of endless variety; the Roman composite capital would seem to have been the favourite type they followed at first: subsequently, the block of stone was left rough as it came from the quarry, and the sculptor, set to carve it, evolved new types of design to his own fancy, so that one rarely meets with many repetitions of the same design. One of the most remarkable is the capital in which the leaves are carved as if blown by the wind; the finest example being in Sta Sophia, Thessalonica; those in St Mark's, Venice (fig. 8) specially attracted Ruskin's fancy. Others are found in St Apollinare-in-classe, Ravenna. The Thistle and Pine capital is found in St Mark's, Venice; St Luke's, Delphi; the mosques of Kairawan and of Ibn Tulun, Cairo, in the two latter cases being taken from Byzantine churches. The illustration of the capital in S. Vitale, Ravenna (figs. 9 and 10) shows above it the dosseret required to carry the arch, the springing of which was much wider than the abacus of the capital.

The Romanesque and Gothic capitals throughout Europe present the same variety as in the Byzantine and for the same reason, that the artist evolved his conception of the design from the block he was carving, but in these styles it goes further on account of the clustering of columns and piers.

The earliest type of capital in Lombardy and Germany is that which is known as the cushion-cap, in which the lower portion of the cube block has been cut away to meet the circular shaft (fig. 11). These early types were generally painted at first with various geometrical designs, afterwards carved.

In Byzantine capitals, the eagle, the lion and the lamb are occasionally carved, but treated conventionally.

In the Romanesque and Gothic styles, in addition to birds and beasts, figures are frequently introduced into capitals, those in the Lombard work being rudely carved and verging on the grotesque; later, the sculpture reaches a higher standard; in the cloisters of Monreale (fig. 12) the birds being wonderfully true to nature. In England and France (figs. 13 and 14), the figures introduced into the capitals are sometimes full of character. These capitals, however, are not equal to those of the Early English school, in which the foliage is conventionally treated as if it had been copied from metal work, and is of infinite variety, being found in small village churches as well as in cathedrals.

Reference has only been made to the leading examples of the Roman capitals; in the Renaissance period (fig. 15) the feature became of the greatest importance and its variety almost as great as in the Byzantine and Gothic styles. The pilaster, which was employed so extensively in the Revival, called for new combinations in the designs for its capitals. Most of the ornament can be traced to Roman sources, and although less vigorous, shows much more delicacy and refinement in its carving. (R. P. S.)

CAPITAL (i.e. capital stock or fund), in economics, generally, the accumulated wealth either of a man or a community, that is available for earning interest and producing fresh wealth. In social discussion it is sometimes treated as antithetical to labour, but it is in reality the accumulated savings of labour and of the profits accruing from the savings of labour. It is that portion of the annual produce reserved from consumption to supply future wants, to extend the sphere of production, to improve industrial instruments and processes, to carry out works of public utility, and, in short, to secure and enlarge the various means of progress necessary to an increasing community. It is the increment of wealth or means of subsistence analogous to the increment of population and of the wants of civilized man. Hence J.S. Mill and other economists, when seeking a graphic expression of the service of capital, have called it "abstinence." The labourer serves by giving physical and mental effort in order to supply his means of consumption. The capitalist, or labourer-capitalist, serves by abstaining from consumption, by denying himself the present enjoyment of more or less of his means of consumption, in the prospect of a future profit. This quality, apparent enough in the beginnings of capital, applies equally to all its forms and stages; because whether a capitalist stocks his warehouse with goods and produce, improves land, lends on mortgage or other security, builds a factory, opens a mine, or orders the construction of machines or ships, there is the element of self-deprival for the present, with the risk of ultimate loss of what is his own, and what, instead of saving and embodying income productive form, he might choose to consume. On this ground rests the justification of the claims of capital to its industrial rewards, whether in the form of rent, interest or profits of trade and investment.

To any advance in the arts of industry or the comforts of life, a rate of production exceeding the rate of consumption, with consequent accumulation of resources, or in other words, the formation of capital, is indispensable. The primitive cultivators of the soil, whether those of ancient times or the pioneers who formed settlements in the forests of the New World, soon discovered that their labour would be rendered more effective by implements and auxiliary powers of various kinds, and that until the produce from existing means of cultivation exceeded what was necessary for their subsistence, there could be neither labour on their part to produce such implements and auxiliaries, nor means to purchase them. Every branch of industry has thus had a demand for capital within its own circles from the earliest times. The flint arrow-heads, the stone and bronze utensils of fossiliferous origin, and the rude implements of agriculture, war and navigation, of which we read in Homer, were the forerunners of that rich and wonderful display of tools, machines, engines, furnaces and countless ingenious and costly appliances, which represent so large a portion of the capital of civilized countries, and without the pre-existing capital could not have been developed. Nor in the cultivation of land, or the production simply of food, is the need of implements, and of other auxiliary power, whether animal or mechanical, the only need immediately experienced. The demands on the surplus of produce over consumption are various and incessant. Near the space of reclaimed ground, from which the cultivator derives but a bare livelihood, are some marshy acres that, if drained and enclosed, would add considerably in two or three years to the produce; the forest and other natural obstructions might also be driven farther back with the result, in a few more years, of profit; fences are necessary to allow of pasture and field crops, roads have to be made and farm buildings to be erected; as the work proceeds more artificial investments follow, and by these successive outlays of past savings in improvements, renewed and enhanced from generation to generation, the land, of little value in its natural state either to the owner and cultivator or the community, is at length brought into a highly productive condition. The history of capital in the soil is substantially the history of capital in all other spheres. No progress can be made in any sphere, small or large, without reserved funds possessed by few or more persons, in small or large amounts, and the progress in all cases is adventured under self-deprival in the meanwhile of acquired value, and more or less risk as to the final result.

Capital is necessarily to be distinguished from money, with which in ordinary nomenclature it is almost identical. Wealth may be in other things than money; oxen, wives, tools, have at different stages of civilization represented the recognized form of capital; and modern usage only treats capital as meaning the command of money because money is the ordinary form of it nowadays. The capital of a country can scarce be said to be less than the whole sum of its investments in a productive form, and possessing a recognized productive value.

Adam Smith's distinction of "fixed" and "circulating" capital in the _Wealth of Nations_ (book ii. c. i.) cannot fail to be always useful in exhibiting the various forms and conditions under which capital is employed. Yet the principal phenomena of capital are found to be the same, whether the form of investment be more or less permanent or circulable. The machinery in which capital is "fixed," and which yields a profit without apparently changing hands, is in reality passing away day by day, until it is worn out, and has to be replaced. So also of drainage and other land improvements. When the natural forests have been consumed and the landowners begin to plant trees on the bare places, the plantations while growing are a source of health, shelter and embellishment--they are not without a material profit throughout their various stages to maturity--and when, at the lapse of twenty or more years, they are ready to be cut down, and the timber is sold for useful purposes, there is a harvest of the original capital expended as essentially as in the case of the more rapid yearly crops of wheat or oats. The chief distinction would appear to rest in the element of time elapsing between the outlay of capital and its return. Capital may be employed in short loans or bills of exchange at two or three months, in paying wages of labour for which there may be return in a day or not in less than a year or more, or in operations involving within themselves every form of capital expenditure, and requiring a few years or ninety-nine years for the promised fructification on which they proceed. But the common characteristic of capital is that of a fund yielding a return and reproducing itself whether the time to this end be long or short. The division of expenditure or labour (all expenditure having a destination to labour of one kind or another) into "productive" and "unproductive" by the same authority (book ii. c. 3) is also apposite both for purposes of political economy and practical guidance, though economists have found it difficult to define where "productive expenditure" ends and "unproductive expenditure" begins. Adam Smith includes in his enumeration of the "fixed capital" of a country "the acquired and useful abilities of all the inhabitants"; and in this sense expenditure on education, arts and sciences might be deemed expenditure of the most productive value, and yet be wanting in strict commercial account of the profit and loss. It must be admitted that there is a personal expenditure among all ranks of society, which, though not in any sense a capital expenditure, may become capital and receive a productive application, always to be preferred to the grossly unproductive form, in the interest both of the possessors and of the community.

The subject in its details is full of controversies, and a discussion
of it at any length would embrace the whole field of economics. The
subject will be found fully dealt with in every important economic
work, but the following may be specially consulted:--J.S. Mill,
_Principles of Political Economy_; J.E. Cairns, _Some Leading
Principles of Political Economy_; F.A. Walker, _Political Economy_; A.
Marshall, _Principles of Economics_; E. Bohm v. Bawerk, _Capital and
Interest_; K. Marx, _Capital_; J.B. Clark, _Capital and its Earnings_;
see also the economic works of W.H. Mallock (_Critical Examination of
Socialism_, 1908, &c.) for an insistence on the importance of
"ability," or brain-work, as against much of modern socialist
theorizing against "capitalism."

CAPITAL PUNISHMENT. By this term is now meant the infliction of the penalty of death for crime under the sentence of some properly constituted authority, as distinguished from killing the offender as a matter of self-defence or private vengeance, or under the order of some self-constituted or irregular tribunal unknown to the law, such as that of the Vigilantes of California, or of lynch law (q.v.). In the early stages of society a man-slayer was killed by the "avenger of blood" on behalf of the family of the man killed, and not as representing the authority of the state (Pollock and Maitland, _Hist. Eng. Law_, ii. 447.) This mode of dealing with homicide survives in the vendetta of Corsica and of the Mainotes in Greece, and in certain of the southern states of North America. The obligation or inclination to take vengeance depends on the fact of homicide, and not on the circumstances in which it was committed, i.e. it is a part of the _lex talionis_. The mischief of this system was alleviated under the Levitical law by the creation of cities of refuge, and in Greece and Italy, both in Pagan and Christian times, by the recognition of the right of sanctuary in temples and churches. A second mode of dealing with homicide was that known to early Teutonic and early Celtic law, where the relatives of the deceased, instead of the life of the slayer, received the wer of the deceased, i.e. a payment in proportion to the rank of the slain, and the king received the blood-wite for the loss of his man. But even under this system certain crimes were in Anglo-Saxon law bot-less, i.e. no compensation could be paid, and the offender must suffer the penalty of death. In the laws of Khammurabi, king of Babylon (2285-2242 B.C.), the death penalty is imposed for many offences. The modes for executing it specially named are burning, drowning and impalement (_Oldest Code of Laws_, by C.H.W. Johns, 1903). Under the Roman law, "capital" punishment also included punishments which deprived the offender of the status of Roman citizen (_capitis deminutio, capitis amissio_), e.g. condemnation to servitude in the mines or to deportation to an island (_Dig._ 48. 19).

British and foreign laws and methods.

_United Kingdom._--The modes of capital punishment in England under the Saxon and Danish kings were various: hanging, beheading, burning, drowning, stoning, and precipitation from rocks. The principle on which this variety depended was that where an offence was such as to entitle the king to outlaw the offender, he forfeited all, life and limb, lands and goods, and that the king might take his life and choose the mode of death. William the Conqueror would not allow judgment of death to be executed by hanging and substituted mutilation; but his successors varied somewhat in their policy as to capital punishment, and by the 13th century the penalty of death became by usage (without legislation) the usual punishment for high and petty treason and for all felonies (except mayhem and petty larceny, i.e. theft of property worth less than 1s.); see Stephen, _Hist. Cr. Law_, vol. i. 458; Pollock and Maitland, _Hist. Eng. Law_, vol. ii. 459. It therefore included all the more serious forms of crime against person or property, such as murder, manslaughter, arson, highway robbery, burglary (or hamesucken) and larceny; and when statutory felonies were created they were also punishable by death unless the statute otherwise provided. The death penalty was also extended to heretics under the writ _de heretico comburendo_, which was lawfully issuable under statute from 1382 (5 Ric. II. stat. 5) until 1677 (29 Chas. II. c. 9). For this purpose the legislature had adopted the civil law of the Roman Empire, which was not a part of the English common law (Stephen, _Hist. Cr. Law_, vol. ii. 438-469).

The methods of execution by crucifixion (as under the Roman law), or breaking on the wheel (as under the Roman Dutch law and the Holy Roman Empire), were never recognized by the common law, and would fall within the term "cruel and unusual punishments" in the English Bill of Rights, and in the United States would seem to be unconstitutional (see _Wilkinson v. Utah_, 1889, 136 U.S. 436, 446).

The severity of barbarian and feudal laws was mitigated, so far as common-law offences were concerned, by the influence of the Church as the inheritor of Christian traditions and Roman jurisprudence. The Roman law under the empire did not allow the execution of citizens except under the _Lex Porcia_. But the right of the emperors to legislate _per rescriptum principis_ enabled them to disregard the ordinary law when so disposed. The 83rd novel of Justinian provided that criminal causes against clerics should be tried by the judges, and that the convicted cleric should be degraded by his bishop before his condemnation by the secular power, and other novels gave the bishops considerable influence, if not authority, over the lay judiciary. In western Europe the right given by imperial legislation in the Eastern Empire was utilized by the Papacy to claim privilege of clergy, i.e. that clerks must be remitted to the bishop for canonical punishment, and not subjected to civil condemnation at all. The history of benefit of clergy is given in Pollock and Maitland, _Hist. English Law_, vol. i. pp. 424-440, and Stephen, _Hist. Cr. Law_, vol. iii. 459, 463. By degrees the privilege was extended not only to persons who could prove ordination or show a genuine tonsure, but all persons who had sufficient learning to be able to read the neck-verse (Ps. li. v. 1). Before the Reformation the ecclesiastical courts had ceased to take any effective action with respect to clerks accused of offences against the king's laws; and by the time of Henry VII. burning on the hand under the order of the king's judges was substituted for the old process of compurgation in use in the spiritual courts.

The effect of the claim of benefit of clergy is said to have been to increase the number of convictions, though it mitigated the punishment; and it became, in fact, a means of showing mercy to certain classes of individuals convicted of crime as a kind of privilege to the educated, i.e. to all clerks whether secular or religious (25 Edw. III. stat. 3); and it was allowed only in case of a first conviction, except in the case of clerks who could produce their letters of orders or a certificate of ordination. To prevent a second claim it was the practice to brand murderers with the letter M, and other felons with the Tyburn T, and Ben Jonson was in 1598 so marked for manslaughter.

The reign of Henry VIII. was marked by extreme severity in the execution of criminals--as during this time 72,000 persons are said to have been hanged. After the formation of English settlements in America the severity of the law was mitigated by the practice of reprieving persons sentenced to death on condition of their consenting to be transported to the American colonies, and to enter into bond service there. The practice seems to have been borrowed from Spain, and to have been begun in 1597 (39 Eliz. c. 4). It was applied by Cromwell after his campaign in Ireland, and was in full force immediately after the Restoration, and is recognized in the Habeas Corpus Act 1677, and was used for the Cameronians during Claverhouse's campaign in south-west Scotland. In the 18th century the courts were empowered to sentence felons to transportation (see DEPORTATION) instead of to execution, and this state of the law continued until 1857 (6 _Law Quarterly Review_, p. 388). This power to sentence to transportation at first applied only to felonies with benefit of clergy; but in 1705, on the abolition of the necessity of proving capacity to read, all criminals alike became entitled to the benefit previously reserved to clerks. Benefit of clergy was finally abolished in 1827 as to all persons not having privilege of peerage, and in 1841 as to peers and peeresses. Its beneficial effect had now been exhausted, since no clergyable offences remained capital crimes.

At the end of the 18th century the criminal law of all Europe was ferocious and indiscriminate in its administration of capital punishment for almost all forms of grave crime; and yet owing to poverty, social conditions, and the inefficiency of the police, such forms of crime were far more numerous than they now are. The policy and righteousness of the English law were questioned as early as 1766 by Goldsmith through the mouth of the vicar of Wakefield: "Nor can I avoid even questioning the validity of that right which social combinations have assumed of capitally punishing offences of a slight nature. In cases of murder their right is obvious, as it is the duty of us all from the law of self-defence to cut off that man who has shown a disregard for the life of another. Against such all nature rises in arms; but it is not so against him who steals my property." He adds later: "When by indiscriminate penal laws the nation beholds the same punishment affixed to dissimilar degrees of guilt, the people are led to lose all sense of distinction in the crime, and this distinction is the bulwark of all morality."