Chapter VII: Part 7

DHULEEP SINGH (1837-1893), maharaja of Lahore, was born in February 1837, and was proclaimed maharaja on the 18th of September 1843, under the regency of his mother the rani Jindan, a woman of great capacity and strong will, but extremely inimical to the British. He was acknowledged by Ranjit Singh and recognized by the British government. After six years of peace the Sikhs invaded British territory in 1845, but were defeated in four battles, and terms were imposed upon them at Lahore, the capital of the Punjab. Dhuleep Singh retained his territory, but it was administered to a great extent by the British government in his name. This arrangement increased the regent's dislike of the British, and a fresh outbreak occurred in 1848-49. In spite of the valour of the Sikhs, they were utterly routed at Gujarat, and in March 1849 Dhuleep Singh was deposed, a pension of £40,000 a year being granted to him and his dependants. He became a Christian and elected to live in England. On coming of age he made an arrangement with the British government by which his income was reduced to £25,000 in consideration of advances for the purchase of an estate, and he finally settled at Elvedon in Suffolk. While passing through Alexandria in 1864 he met Miss Bamba Müller, the daughter of a German merchant who had married an Abyssinian. The maharaja had been interested in mission work by Sir John Login, and he met Miss Müller at one of the missionary schools where she was teaching. She became his wife on the 7th of June 1864, and six children were the issue of the marriage. In the year after her death in 1890 the maharaja married at Paris, as his second wife, an English lady, Miss Ada Douglas Wetherill, who survived him. The maharaja was passionately fond of sport, and his shooting parties were celebrated, while he himself became a _persona grata_ in English society. The result, however, was financial difficulty, and in 1882 he appealed to the government for assistance, making various claims based upon the alleged possession of private estates in the Punjab, and upon the surrender of the Koh-i-nor diamond to the British Crown. His demand was rejected, whereupon he started for India, after drawing up a proclamation to his former subjects. But as it was deemed inadvisable to allow him to visit the Punjab, he remained for some time as a guest at the residency at Aden, and was allowed to receive some of his relatives to witness his abjuration of Christianity, which actually took place within the residency itself. As the climate began to affect his health, the maharaja at length left Aden and returned to Europe. He stayed for some time in Russia, hoping that his claim against England would be taken up by the Russians; but when that expectation proved futile he proceeded to Paris, where he lived for the rest of his life on the pension allowed him by the Indian government. His death from an attack of apoplexy took place at Paris on the 22nd of October 1893. The maharaja's eldest son, Prince Victor Albert Jay Dhuleep Singh (b. 1866), was educated at Trinity and Downing Colleges, Cambridge. In 1888 he obtained a commission in the 1st Royal Dragoon Guards. In 1898 he married Lady Anne Coventry, youngest daughter of the earl of Coventry. (G.F.B.)

DHULIA, a town of British India, administrative headquarters of West Khandesh district in Bombay, on the right bank of the Panjhra river. Pop. (1901) 24,726. Considerable trade is done in cotton and oil-seeds, and weaving of cotton. A railway connects Dhulia with Chalisgaon, on the main line of the Great Indian Peninsula railway.

DIABASE, in petrology, a rock which is a weathered form of dolerite. It was long widely accepted that the pre-Tertiary rocks of this group differed from their Tertiary and Recent representatives in certain essential respects, but this is now admitted to be untenable, and the differences are known to be merely the result of the longer exposure to decomposition, pressure and shearing, which the older rocks have experienced. Their olivine tends to become serpentinized; their augite changes to chlorite and uralite; their felspars are clouded by formation of zeolites, calcite, sericite and epidote. The rocks acquire a green colour (from the development of chlorite, uralite and epidote); hence the older name of "greenstones," which is now little used. Many of them become somewhat schistose from pressure ("greenstone-schists," meta-diabase, &c.). Although the original definition of the group can no longer be justified, the name is so well established in current usage that it can hardly be discarded. The terms diabase and dolerite are employed really to designate distinct facies of the same set of rocks.

The minerals of diabase are the same as those of dolerite, viz.
olivine, augite, and plagioclase felspar, with subordinate quantities
of hornblende, biotite, iron oxides and apatite.

There are olivine-diabases and diabases without olivine;
quartz-diabases, analcite-diabases (or teschenites) and hornblende
diabases (or proterobases). Hypersthene (or bronzite) is
characteristic of another group. Many of them are ophitic, especially
those which contain olivine, but others are intersertal, like the
intersertal dolerites. The last include most quartz-diabases,
hypersthene-diabases and the rocks which have been described as
tholeites. Porphyritic structure appears in the diabase-porphyrites,
some of which are highly vesicular and contain remains of an abundant
fine-grained or partly glassy ground-mass (_diabas-mandelstein_,
amygdaloidal diabase). The somewhat ill-defined spilites are regarded
by many as modifications of diabase-porphyrite. In the intersertal and
porphyrite diabases, fresh or devitrified glassy base is not
infrequent. It is especially conspicuous in some tholeites
(hyalo-tholeites) and in weisselbergites. These rocks consist of
augite and plagioclase, with little or no olivine, on a brown,
vitreous, interstitial matrix. Devitrified forms of tachylyte
(sordawilite, &c.) occur at the rapidly chilled margins of dolerite
sills and dikes, and fine-grained spotted rocks with large spherulites
of grey or greenish felspar, and branching growths of brownish-green
augite (variolites).

To nearly every variety in composition and structure presented by the
diabases, a counterpart can be found among the Tertiary dolerites. In
the older rocks, however, certain minerals are more common than in the
newer. Hornblende, mostly of pale green colours and somewhat fibrous
habit, is very frequent in diabase; it is in most cases secondary
after pyroxene, and is then known as uralite; often it forms
pseudomorphs which retain the shape of the original augite. Where
diabases have been crushed or sheared, hornblende readily develops at
the expense of pyroxene, sometimes replacing it completely. In the
later stages of alteration the amphibole becomes compact and well
crystallized; the rocks consist of green hornblende and plagioclase
felspar, and are then generally known as epidiorites or amphibolites.
At the same time a schistose structure is produced. But transition
forms are very common, having more or less of the augite remaining,
surrounded by newly formed hornblende which at first is rather fibrous
and tends to spread outwards through the surrounding felspar. Chlorite
also is abundant both in sheared and unsheared diabases, and with it
calcite may make its appearance, or the lime set free from the augite
may combine with the titanium of the iron oxide and with silica to
form incrustations or borders of sphene around the original crystals
of ilmenite. Epidote is another secondary lime-bearing mineral which
results from the decomposition of the soda lime felspars and the
pyroxenes. Many diabases, especially those of the teschenite
sub-group, are filled with zeolites.

Diabases are exceedingly abundant among the older rocks of all parts
of the globe. Popular names for them are "whinstone," "greenstone,"
"toadstone" and "trap." They form excellent road-mending stones and
are much quarried for this purpose, being tough, durable and resistant
to wear, so long as they are not extremely decomposed. Many of them
are to be preferred to the fresher dolerites as being less brittle.
The quality of the Cornish greenstones appears to have been distinctly
improved by a smaller amount of recrystallization where they have been
heated by contact with intrusive masses of granite. (J. S. F.)

DIABETES (from Gr. [Greek: dia], through, and [Greek: bainein], to pass), a constitutional disease characterized by a habitually excessive discharge of urine. Two forms of this complaint are described, viz. Diabetes Mellitus, or Glycosuria, where the urine is not only increased in quantity, but persistently contains a greater or less amount of sugar, and Diabetes Insipidus, or Polyuria, where the urine is simply increased in quantity, and contains no abnormal ingredient. This latter, however, must be distinguished from the polyuria due to chronic granular kidney, lardaceous disease of the kidney, and also occurring in certain cases of hysteria.

_Diabetes mellitus_ is the disease to which the term is most commonly applied, and is by far the more serious and important ailment. It is one of the diseases due to altered metabolism (see METABOLIC DISEASES). It is markedly hereditary, much more prevalent in towns and especially modern city life than in more primitive rustic communities, and most common among the Jews. The excessive use of sugar as a food is usually considered one cause of the disease, and obesity is supposed to favour its occurrence, but many observers consider that the obesity so often met with among diabetics is due to the same cause as the disease itself. No age is exempt, but it occurs most commonly in the fifth decade of life. It attacks males twice as frequently as females, and fair more frequently than dark people.

The symptoms are usually gradual in their onset, and the patient may suffer for a length of time before he thinks it necessary to apply for medical aid. The first symptoms which attract attention are failure of strength, and emaciation, along with great thirst and an increased amount and frequent passage of urine. From the normal quantity of from 2 to 3 pints in the 24 hours it may be increased to 10, 20 or 30 pints, or even more. It is usually of pale colour, and of thicker consistence than normal urine, possesses a decidedly sweet taste, and is of high specific gravity (1030 to 1050). It frequently gives rise to considerable irritation of the urinary passages.

By simple evaporation crystals of sugar may be obtained from diabetic urine, which also yields the characteristic chemical tests of sugar, while the amount of this substance can be accurately estimated by certain analytical processes. The quantity of sugar passed may vary from a few ounces to two or more pounds per diem, and it is found to be markedly increased after saccharine or starchy food has been taken. Sugar may also be found in the blood, saliva, tears, and in almost all the excretions of persons suffering from this disease. One of the most distressing symptoms is intense thirst, which the patient is constantly seeking to allay, the quantity of liquid consumed being in general enormous, and there is usually, but not invariably, a voracious appetite. The mouth is always parched, and a faint, sweetish odour may be evolved from the breath. The effect of the disease upon the general health is very marked, and the patient becomes more and more emaciated. He suffers from increasing muscular weakness, the temperature of his body is lowered, and the skin is dry and harsh. There is often a peculiar flush on the face, not limited to the malar eminences, but extending up to the roots of the hair. The teeth are loosened or decay, there is a tendency to bleeding from the gums, while dyspeptic symptoms, constipation and loss of sexual power are common accompaniments. There is in general great mental depression or irritability.

Diabetes as a rule advances comparatively slowly except in the case of young persons, in whom its progress is apt to be rapid. The complications of the disease are many and serious. It may cause impaired vision by weakening the muscles of accommodation, or by lessening the sensitiveness of the retina to light. Also cataract is very common. Skin affections of all kinds may occur and prove very intractable. Boils, carbuncles, cellulitis and gangrene are all apt to occur as life advances, though gangrene is much more frequent in men than in women. Diabetics are especially liable to phthisis and pneumonia, and gangrene of the lungs may set in if the patient survives the crisis in the latter disease. Digestive troubles of all kinds, kidney diseases and heart failure due to fatty heart are all of common occurrence. Also patients seem curiously susceptible to the poison of enteric fever, though the attack usually runs a mild course. The sugar temporarily disappears during the fever. But the most serious complication of all is known as diabetic coma, which is very commonly the final cause of death. The onset is often insidious, but may be indicated by loss of appetite, a rapid fall in the quantity of both urine and sugar, and by either constipation or diarrhoea. More rarely there is most acute abdominal pain. At first the condition is rather that of collapse than true coma, though later the patient is absolutely comatose. The patient suffers from a peculiar kind of dyspnoea, and the breath and skin have a sweet ethereal odour. The condition may last from twenty-four hours to three days, but is almost invariably the precursor of death.

Diabetes is a very fatal form of disease, recovery being exceedingly rare. Over 50% die of coma, another 25% of phthisis or pneumonia, and the remainder of Bright's disease, cerebral haemorrhage, gangrene, &c. The most favourable cases are those in which the patient is advanced in years, those in which it is associated with obesity or gout, and where the social conditions are favourable. A few cures have been recorded in which the disease supervened after some acute illness. The unfavourable cases are those in which there is a family history of the disease and in which the patient is young. Nevertheless much may be done by appropriate treatment to mitigate the severity of the symptoms and to prolong life.

There are two distinct lines of treatment, that of diet and that of drugs, but each must be modified and determined entirely by the idiosyncrasy of the patient, which varies in this condition between very wide limits. That of diet is of primary importance inasmuch as it has been proved beyond question that certain kinds of food have a powerful influence in aggravating the disease, more particularly those consisting largely of saccharine and starchy matter; and it may be stated generally that the various methods of treatment proposed aim at the elimination as far as possible of these constituents from the diet. Hence it is recommended that such articles as bread, potatoes and all farinaceous foods, turnips, carrots, parsnips and most fruits should be avoided; while animal food and soups, green vegetables, cream, cheese, eggs, butter, and tea and coffee without sugar, may be taken with advantage. As a substitute for ordinary bread, which most persons find it difficult to do without for any length of time, bran bread, gluten bread and almond biscuits. A patient must never pass suddenly from an ordinary to a carbohydrate-free diet. Any such sudden transition is extremely liable to bring on diabetic coma, and the change must be made quite gradually, one form of carbohydrate after another being taken out of the diet, whilst the effect on the quantity of sugar passed is being carefully noted meanwhile. The treatment may be begun by excluding potatoes, sugar and fruit, and only after several days is the bread to be replaced by some diabetic substitute. When the sugar excretion has been reduced to its lowest point, and maintained there for some time, a certain amount of carbohydrate may be cautiously allowed, the consequent effect on the glycosuria being estimated. The best diet can only be worked out experimentally for each individual patient. But in every case, if drowsiness or any symptom suggesting coma supervene, all restrictions must be withdrawn, and carbohydrate freely allowed. The question of alcohol is one which must be largely determined by the previous history of the patient, but a small quantity will help to make up the deficiencies of a diet poor in carbohydrate. Scotch and Irish whisky, and Hollands gin, are usually free from sugar, and some of the light Bordeaux wines contain very little. Fat is beneficial, and can be given as cream, fat of meat and cod-liver oil. Green vegetables are harmless, but the white stalks of cabbages and lettuces and also celery and endive yield sugar. Laevulose can be assimilated up to 1½ ozs. daily without increasing the glycosuria, and hence apples, cooked or raw, are allowable, as the sugar they contain is in this form. The question of milk is somewhat disputed; but it is usual to exclude it from the rigid diet, allowing a certain quantity when the diet is being extended. Thirst is relieved by anything that relieves the polyuria. But hypodermic injections of pilocarpine stimulate the flow of saliva, and thus relieve the dryness of the mouth. Constipation appears to increase the thirst, and must always be carefully guarded against. The best remedies are the aperient mineral waters.

Numerous medicinal substances have been employed in diabetes, but few of them are worthy of mention as possessed of any efficacy. Opium is often found of great service, its administration being followed by marked amelioration in all the symptoms. Morphia and codeia have a similar action. In the severest cases, however, these drugs appear to be of little or no use, and they certainly increase the constipation. Heroin hydrochloride has been tried in their place, but this seems to have more power over slight than over severe cases. Salicylate of sodium and aspirin are both very beneficial, causing a diminution in the sugar excretion without counterbalancing bad effects.

In _diabetes insipidus_ there is constant thirst and an excessive flow of urine, which, however, is not found to contain any abnormal constituent. Its effects upon the system are often similar to those of diabetes mellitus, except that they are much less marked, the disease being in general very slow in its progress. In some cases the health appears to suffer very slightly. It is rarely a direct cause of death, but from its debilitating effects may predispose to serious and fatal complications. It is best treated by tonics and generous diet. Valerian has been found beneficial, the powdered root being given in 5-grain doses.

DIABOLO, a game played with a sort of top in the shape of two cones joined at their apices, which is spun, thrown, and caught by means of a cord strung to two sticks. The idea of the game appears originally to have come from China, where a top (_Kouengen_), made of two hollow pierced cylinders of metal or wood, joined by a rod--and often of immense size,--was made by rotation to hum with a loud noise, and was used by pedlars to attract customers. From China it was introduced by missionaries to Europe; and a form of the game, known as "the devil on two sticks," appears to have been known in England towards the end of the 18th century, and Lord Macartney is credited with improvements in it. But its principal vogue was in France in 1812, where the top was called "le diable." Amusing old prints exist (see _Fry's Magazine_, March and December 1907), depicting examples of the popular craze in France at the time. The _diable_ of those days resembled a globular wooden dumb-bell with a short waist, and the sonorous hum when spinning--the _bruit du diable_--was a pronounced feature. At intervals during the century occasional attempts to revive the game of spinning a top of this sort on a string were made, but it was not till 1906 that the sensation of 1812 began to be repeated. A French engineer, Gustave Phillipart, discovering some old implements of the game, had experimented for some time with new forms of top with a view to bringing it again into popularity; and having devised the double-cone shape, and added a miniature bicycle tire of rubber round the rims of the two ends of the double-cone, with other improvements, he named it "diabolo." The use of celluloid in preference to metal or wood as its material appears to have been due to a suggestion of Mr C. B. Fry, who was consulted by the inventor on the subject. The game of spinning, throwing and catching the diabolo was rapidly elaborated in various directions, both as an exercise of skill in doing tricks, and in "diabolo tennis" and other ways as an athletic pastime. From Paris, Ostend and the chief French seaside resorts, where it became popular in 1906, its vogue spread in 1907 so that in France and England it became the fashionable "rage" among both children and adults.

The mechanics of the diabolo were worked out by Professor C. V. Boys in the _Proc. Phys. Soc._ (London), Nov. 1907.

DIACONICON, in the Greek Church, the name given to a chamber on the south side of the central apse, where the sacred utensils, vessels, &c., of the church were kept. In the reign of Justin II. (565-574), owing to a change in the liturgy, the diaconicon and protheses were located in apses at the east end of the aisles. Before that time there was only one apse. In the churches in central Syria of slightly earlier date, the diaconicon is rectangular, the side apses at Kalat-Seman having been added at a later date.

DIADOCHI (Gr. [Greek: diadechesthai], to receive from another), i.e. "Successors," the name given to the Macedonian generals who fought for the empire of Alexander after his death in 323 B.C. The name includes Antigonus and his son Demetrius Poliorcetes, Antipater and his son Cassander, Seleucus, Ptolemy, Eumenes and Lysimachus. The kingdoms into which the Macedonian empire was divided under these rulers are known as Hellenistic. The chief were Asia Minor and Syria under the SELEUCID DYNASTY (q.v.), Egypt under the PTOLEMIES (q.v.), Macedonia under the successors of Antigonus Gonatas, PERGAMUM (q.v.) under the Attalid dynasty. Gradually these kingdoms were merged in the Roman empire. (See MACEDONIAN EMPIRE.)

DIAGONAL (Gr. [Greek: dia], through, [Greek: gônia], a corner), in geometry, a line joining the intersections of two pairs of sides of a rectilinear figure.

DIAGORAS, of Melos, surnamed the Atheist, poet and sophist, flourished in the second half of the 5th century B.C. Religious in his youth and a writer of hymns and dithyrambs, he became an atheist because a great wrong done to him was left unpunished by the gods. In consequence of his blasphemous speeches, and especially his criticism of the Mysteries, he was condemned to death at Athens, and a price set upon his head (Aristoph. _Clouds_, 830; _Birds_, 1073 and Schol.). He fled to Corinth, where he is said to have died. His work on the Mysteries was called [Greek Phrygioi logoi] or [Greek: Apopyrgizontes], in which he probably attacked the Phrygian divinities.

DIAGRAM (Gr. [Greek: diagramma], from [Greek: diagraphein], to mark out by lines), a figure drawn in such a manner that the geometrical relations between the parts of the figure illustrate relations between other objects. They may be classed according to the manner in which they are intended to be used, and also according to the kind of analogy which we recognize between the diagram and the thing represented. The diagrams in mathematical treatises are intended to help the reader to follow the mathematical reasoning. The construction of the figure is defined in words so that even if no figure were drawn the reader could draw one for himself. The diagram is a good one if those features which form the subject of the proposition are clearly represented.

Diagrams are also employed in an entirely different way--namely, for purposes of measurement. The plans and designs drawn by architects and engineers are used to determine the value of certain real magnitudes by measuring certain distances on the diagram. For such purposes it is essential that the drawing be as accurate as possible. We therefore class diagrams as diagrams of illustration, which merely suggest certain relations to the mind of the spectator, and diagrams drawn to scale, from which measurements are intended to be made. There are some diagrams or schemes, however, in which the form of the parts is of no importance, provided their connexions are properly shown. Of this kind are the diagrams of electrical connexions, and those belonging to that department of geometry which treats of the degrees of cyclosis, periphraxy, linkedness and knottedness.

_Diagrams purely Graphic and mixed Symbolic and Graphic._--Diagrams may also be classed either as purely graphical diagrams, in which no symbols are employed except letters or other marks to distinguish particular points of the diagrams, and mixed diagrams, in which certain magnitudes are represented, not by the magnitudes of parts of the diagram, but by symbols, such as numbers written on the diagram. Thus in a map the height of places above the level of the sea is often indicated by marking the number of feet above the sea at the corresponding places on the map. There is another method in which a line called a contour line is drawn through all the places in the map whose height above the sea is a certain number of feet, and the number of feet is written at some point or points of this line. By the use of a series of contour lines, the height of a great number of places can be indicated on a map by means of a small number of written symbols. Still this method is not a purely graphical method, but a partly symbolical method of expressing the third dimension of objects on a diagram in two dimensions.

In order to express completely by a purely graphical method the relations of magnitudes involving more than two variables, we must use more than one diagram. Thus in the arts of construction we use plans and elevations and sections through different planes, to specify the form of objects having three dimensions. In such systems of diagrams we have to indicate that a point in one diagram corresponds to a point in another diagram. This is generally done by marking the corresponding points in the different diagrams with the same letter. If the diagrams are drawn on the same piece of paper we may indicate corresponding points by drawing a line from one to the other, taking care that this line of correspondence is so drawn that it cannot be mistaken for a real line in either diagram. (See GEOMETRY: _Descriptive_.)

In the stereoscope the two diagrams, by the combined use of which the form of bodies in three dimensions is recognized, are projections of the bodies taken from two points so near each other that, by viewing the two diagrams simultaneously, one with each eye, we identify the corresponding points intuitively. The method in which we simultaneously contemplate two figures, and recognize a correspondence between certain points in the one figure and certain points in the other, is one of the most powerful and fertile methods hitherto known in science. Thus in pure geometry the theories of similar, reciprocal and inverse figures have led to many extensions of the science. It is sometimes spoken of as the method or principle of Duality. GEOMETRY: _Projective_.)

DIAGRAMS IN MECHANICS.

The study of the motion of a material system is much assisted by the
use of a series of diagrams representing the configuration,
displacement and acceleration of the parts of the system.

_Diagram of Configuration._--In considering a material system it is
often convenient to suppose that we have a record of its position at
any given instant in the form of a diagram of configuration. The
position of any particle of the system is defined by drawing a
straight line or vector from the origin, or point of reference, to the
given particle. The position of the particle with respect to the
origin is determined by the magnitude and direction of this vector. If
in the diagram we draw from the origin (which need not be the same
point of space as the origin for the material system) a vector equal
and parallel to the vector which determines the position of the
particle, the end of this vector will indicate the position of the
particle in the diagram of configuration. If this is done for all the
particles we shall have a system of points in the diagram of
configuration, each of which corresponds to a particle of the material
system, and the relative positions of any pair of these points will be
the same as the relative positions of the material particles which
correspond to them.

We have hitherto spoken of two origins or points from which the
vectors are supposed to be drawn--one for the material system, the
other for the diagram. These points, however, and the vectors drawn
from them, may now be omitted, so that we have on the one hand the
material system and on the other a set of points, each point
corresponding to a particle of the system, and the whole representing
the configuration of the system at a given instant.

This is called a diagram of configuration.

_Diagram of Displacement._--Let us next consider two diagrams of
configuration of the same system, corresponding to two different
instants. We call the first the initial configuration and the second
the final configuration, and the passage from the one configuration to
the other we call the displacement of the system. We do not at present
consider the length of time during which the displacement was
effected, nor the intermediate stages through which it passed, but
only the final result--a change of configuration. To study this change
we construct a diagram of displacement.

Let A, B, C be the points in the initial diagram of configuration, and
A', B', C' be the corresponding points in the final diagram of
configuration. From o, the origin of the diagram of displacement, draw
a vector oa equal and parallel to AA', ob equal and parallel to BB',
oc to CC', and so on. The points a, b, c, &c., will be such that the
vector ab indicates the displacement of B relative to A, and so on.
The diagram containing the points a, b, c, &c., is therefore called
the diagram of displacement.

In constructing the diagram of displacement we have hitherto assumed
that we know the absolute displacements of the points of the system.
For we are required to draw a line equal and parallel to AA', which we
cannot do unless we know the absolute final position of A, with
respect to its initial position. In this diagram of displacement there
is therefore, besides the points a, b, c, &c., an _origin_, o, which
represents a point absolutely fixed in space. This is necessary
because the two configurations do not exist at the same time; and
therefore to express their relative position we require to know a
point which remains the same at the beginning and end of the time.

But we may construct the diagram in another way which does not assume
a knowledge of absolute displacement or of a point fixed in space.
Assuming any point and calling it a, draw ak parallel and equal to BA
in the initial configuration, and from k draw kb parallel and equal to
A'B' in the final configuration. It is easy to see that the position
of the point b relative to a will be the same by this construction as
by the former construction, only we must observe that in this second
construction we use only vectors such as AB, A'B', which represent the
relative position of points both of which exist simultaneously,
instead of vectors such as AA', BB', which express the position of a
point at one instant relative to its position at a former instant, and
which therefore cannot be determined by observation, because the two
ends of the vector do not exist simultaneously.

It appears therefore that the diagram of displacements, when drawn by
the first construction, includes an origin o, which indicates that we
have assumed a knowledge of absolute displacements. But no such point
occurs in the second construction, because we use such vectors only as
we can actually observe. Hence the diagram of displacements _without
an origin_ represents neither more nor less than all we can ever know
about the displacement of the material system.

_Diagram of Velocity._--If the relative velocities of the points of
the system are constant, then the diagram of displacement
corresponding to an interval of a unit of time between the initial and
the final configuration is called a diagram of relative velocity. If
the relative velocities are not constant, we suppose another system in
which the velocities are equal to the velocities of the given system
at the given instant and continue constant for a unit of time. The
diagram of displacements for this imaginary system is the required
diagram of relative velocities of the actual system at the given
instant. It is easy to see that the diagram gives the velocity of any
one point relative to any other, but cannot give the absolute velocity
of any of them.

_Diagram of Acceleration._--By the same process by which we formed the
diagram of displacements from the two diagrams of initial and final
configuration, we may form a diagram of changes of relative velocity
from the two diagrams of initial and final velocities. This diagram
may be called that of total accelerations in a finite interval of
time. And by the same process by which we deduced the diagram of
velocities from that of displacements we may deduce the diagram of
rates of acceleration from that of total acceleration.

We have mentioned this system of diagrams in elementary kinematics
because they are found to be of use especially when we have to deal
with material systems containing a great number of parts, as in the
kinetic theory of gases. The diagram of configuration then appears as
a region of space swarming with points representing molecules, and the
only way in which we can investigate it is by considering the number
of such points in unit of volume in different parts of that region,
and calling this the _density_ of the gas.

In like manner the diagram of velocities appears as a region
containing points equal in number but distributed in a different
manner, and the number of points in any given portion of the region
expresses the number of molecules whose velocities lie within given
limits. We may speak of this as the velocity-density.

_Diagrams of Stress._--Graphical methods are peculiarly applicable to
statical questions, because the state of the system is constant, so
that we do not need to construct a series of diagrams corresponding to
the successive states of the system. The most useful of these
applications, collectively termed Graphic Statics, relates to the
equilibrium of plane framed structures familiarly represented in
bridges and roof-trusses. Two diagrams are used, one called the
diagram of the frame and the other called the diagram of stress. The
structure itself consists of a number of separable pieces or links
jointed together at their extremities. In practice these joints have
friction, or may be made purposely stiff, so that the force acting at
the extremity of a piece may not pass exactly through the axis of the
joint; but as it is unsafe to make the stability of the structure
depend in any degree upon the stiffness of joints, we assume in our
calculations that all the joints are perfectly smooth, and therefore
that the force acting on the end of any link passes through the axis
of the joint.

The axes of the joints of the structure are represented by points in
the diagram of the frame. The link which connects two joints in the
actual structure may be of any shape, but in the diagram of the frame
it is represented by a straight line joining the points representing
the two joints. If no force acts on the link except the two forces
acting through the centres of the joints, these two forces must be
equal and opposite, and their direction must coincide with the
straight line joining the centres of the joints. If the force acting
on either extremity of the link is directed towards the other
extremity, the stress on the link is called pressure and the link is
called a "strut." If it is directed away from the other extremity, the
stress on the link is called tension and the link is called a "tie."
In this case, therefore, the only stress acting in a link is a
pressure or a tension in the direction of the straight line which
represents it in the diagram of the frame, and all that we have to do
is to find the magnitude of this stress. In the actual structure
gravity acts on every part of the link, but in the diagram we
substitute for the actual weight of the different parts of the link
two weights which have the same resultant acting at the extremities of
the link.

We may now treat the diagram of the frame as composed of links without
weight, but loaded at each joint with a weight made up of portions of
the weights of all the links which meet in that joint. If any link has
more than two joints we may substitute for it in the diagram an
imaginary stiff frame, consisting of links, each of which has only two
joints. The diagram of the frame is now reduced to a system of points,
certain pairs of which are joined by straight lines, and each point is
in general acted on by a weight or other force acting between it and
some point external to the system. To complete the diagram we may
represent these external forces as links, that is to say, straight
lines joining the points of the frame to points external to the frame.
Thus each weight may be represented by a link joining the point of
application of the weight with the centre of the earth.

But we can always construct an imaginary frame having its joints in
the lines of action of these external forces, and this frame, together
with the real frame and the links representing external forces, which
join points in the one frame to points in the other frame, make up
together a complete self-strained system in equilibrium, consisting of
points connected by links acting by pressure or tension. We may in
this way reduce any real structure to the case of a system of points
with attractive or repulsive forces acting between certain pairs of
these points, and keeping them in equilibrium. The direction of each
of these forces is sufficiently indicated by that of the line joining
the points, so that we have only to determine its magnitude. We might
do this by calculation, and then write down on each link the pressure
or the tension which acts in it.

We should in this way obtain a mixed diagram in which the stresses are
represented graphically as regards direction and position, but
symbolically as regards magnitude. But we know that a force may be
represented in a purely graphical manner by a straight line in the
direction of the force containing as many units of length as there are
units of force in the force. The end of this line is marked with an
arrow head to show in which direction the force acts. According to
this method each force is drawn in its proper position in the diagram
of configuration of the frame. Such a diagram might be useful as a
record of the result of calculation of the magnitude of the forces,
but it would be of no use in enabling us to test the correctness of
the calculation.

But we have a graphical method of testing the equilibrium of any set
of forces acting at a point. We draw in series a set of lines parallel
and proportional to these forces. If these lines form a closed polygon
the forces are in equilibrium. (See MECHANICS.) We might in this way
form a series of polygons of forces, one for each joint of the frame.
But in so doing we give up the principle of drawing the line
representing a force from the point of application of the force, for
all the sides of the polygon cannot pass through the same point, as
the forces do. We also represent every stress twice over, for it
appears as a side of both the polygons corresponding to the two joints
between which it acts. But if we can arrange the polygons in such a
way that the sides of any two polygons which represent the same stress
coincide with each other, we may form a diagram in which every stress
is represented in direction and magnitude, though not in position, by
a single line which is the common boundary of the two polygons which
represent the joints at the extremities of the corresponding piece of
the frame.

We have thus obtained a pure diagram of stress in which no attempt is
made to represent the configuration of the material system, and in
which every force is not only represented in direction and magnitude
by a straight line, but the equilibrium of the forces at any joint is
manifest by inspection, for we have only to examine whether the
corresponding polygon is closed or not.

The relations between the diagram of the frame and the diagram of
stress are as follows:--To every link in the frame corresponds a
straight line in the diagram of stress which represents in magnitude
and direction the stress acting in that link; and to every joint of
the frame corresponds a closed polygon in the diagram, and the forces
acting at that joint are represented by the sides of the polygon taken
in a certain cyclical order, the cyclical order of the sides of the
two adjacent polygons being such that their common side is traced in
opposite directions in going round the two polygons.

The direction in which any side of a polygon is traced is the
direction of the force acting on that joint of the frame which
corresponds to the polygon, and due to that link of the frame which
corresponds to the side. This determines whether the stress of the
link is a pressure or a tension. If we know whether the stress of any
one link is a pressure or a tension, this determines the cyclical
order of the sides of the two polygons corresponding to the ends of
the links, and therefore the cyclical order of all the polygons, and
the nature of the stress in every link of the frame.

_Reciprocal Diagrams._--When to every point of concourse of the lines
in the diagram of stress corresponds a closed polygon in the skeleton
of the frame, the two diagrams are said to be reciprocal.

The first extensions of the method of diagrams of forces to other
cases than that of the funicular polygon were given by Rankine in his
_Applied Mechanics_ (1857). The method was independently applied to a
large number of cases by W. P. Taylor, a practical draughtsman in the
office of J. B. Cochrane, and by Professor Clerk Maxwell in his
lectures in King's College, London. In the _Phil. Mag._ for 1864 the
latter pointed out the reciprocal properties of the two diagrams, and
in a paper on "Reciprocal Figures, Frames and Diagrams of Forces,"
_Trans. R.S. Edin._ vol. xxvi., 1870, he showed the relation of the
method to Airy's function of stress and to other mathematical methods.
Professor Fleeming Jenkin has given a number of applications of the
method to practice (_Trans. R.S. Edin._ vol. xxv.).

L. Cremona (_Le Figure reciproche nella statica grafica_, 1872)
deduced the construction of reciprocal figures from the theory of the
two components of a wrench as developed by Möbius. Karl Culmann, in
his _Graphische Statik_ (1st ed. 1864-1866, 2nd ed. 1875), made great
use of diagrams of forces, some of which, however, are not
reciprocal. Maurice Levy in his _Statique graphique_ (1874) has
treated the whole subject in an elementary but copious manner, and R.
H. Bow, in his _The Economics of Construction in Relation to Framed
Structures_ (1873), materially simplified the process of drawing a
diagram of stress reciprocal to a given frame acted on by a system of
equilibrating external forces.

Instead of lettering the joints of the frame, as is usually done, or
the links of the frame, as was the custom of Clerk Maxwell, Bow places
a letter in each of the polygonal areas enclosed by the links of the
frame, and also in each of the divisions of surrounding space as
separated by the lines of action of the external forces. When one link
of the frame crosses another, the point of apparent intersection of
the links is treated as if it were a real joint, and the stresses of
each of the intersecting links are represented twice in the diagram of
stress, as the opposite sides of the parallelogram which corresponds
to the point of intersection.

This method is followed in the lettering of the diagram of
configuration (fig. 1), and the diagram of stress (fig. 2) of the
linkwork which Professor Sylvester has called a quadruplane.

In fig. 1 the real joints are distinguished from the places where one
link appears to cross another by the little circles O, P, Q, R, S, T,
V. The four links RSTV form a "contraparallelogram" in which RS = TV
and RV = ST. The triangles ROS, RPV, TQS are similar to each other. A
fourth triangle (TNV), not drawn in the figure, would complete the
quadruplane. The four points O, P, N, Q form a parallelogram whose
angle POQ is constant and equal to [pi] - SOR. The product of the
distances OP and OQ is constant. The linkwork may be fixed at O. If
any figure is traced by P, Q will trace the inverse figure, but turned
round O through the constant angle POQ. In the diagram forces Pp, Qq
are balanced by the force Co at the fixed point. The forces Pp and Qq
are necessarily inversely as OP and OQ, and make equal angles with
those lines.

Every closed area formed by the links or the external forces in the
diagram of configuration is marked by a letter which corresponds to a
point of concourse of lines in the diagram of stress. The stress in
the link which is the common boundary of two areas is represented in
the diagram of stress by the line joining the points corresponding to
those areas. When a link is divided into two or more parts by lines
crossing it, the stress in each part is represented by a different
line for each part, but as the stress is the same throughout the link
these lines are all equal and parallel. Thus in the figure the stress
in RV is represented by the four equal and parallel lines HI, FG, DE
and AB. If two areas have no part of their boundary in common the
letters corresponding to them in the diagram of stress are not joined
by a straight line. If, however, a straight line were drawn between
them, it would represent in direction and magnitude the resultant of
all the stresses in the links which are cut by any line, straight or
curved, joining the two areas. For instance the areas F and C in fig.
1 have no common boundary, and the points F and C in fig. 2 are not
joined by a straight line. But every path from the area F to the area
C in fig. 1 passes through a series of other areas, and each passage
from one area into a contiguous area corresponds to a line drawn in
the diagram of stress. Hence the whole path from F to C in fig. 1
corresponds to a path formed of lines in fig. 2 and extending from F
to C, and the resultant of all the stresses in the links cut by the
path is represented by FC in fig. 2.

Many examples of stress diagrams are given in the article on BRIDGES
(q.v.).

_Automatic Description of Diagrams._

There are many other kinds of diagrams in which the two co-ordinates
of a point in a plane are employed to indicate the simultaneous values
of two related quantities. If a sheet of paper is made to move, say
horizontally, with a constant known velocity, while a tracing point is
made to move in a vertical straight line, the height varying as the
value of any given physical quantity, the point will trace out a curve
on the paper from which the value of that quantity at any given time
may be determined. This principle is applied to the automatic
registration of phenomena of all kinds, from those of meteorology and
terrestrial magnetism to the velocity of cannon-shot, the vibrations
of sounding bodies, the motions of animals, voluntary and involuntary,
and the currents in electric telegraphs.

In Watt's indicator for steam engines the paper does not move with a
constant velocity, but its displacement is proportional to that of the
piston of the engine, while that of the tracing point is proportional
to the pressure of the steam. Hence the co-ordinates of a point of the
curve traced on the diagram represent the volume and the pressure of
the steam in the cylinder. The indicator-diagram not only supplies a
record of the pressure of the steam at each stage of the stroke of the
engine, but indicates the work done by the steam in each stroke by the
area enclosed by the curve traced on the diagram. (J. C. M.)

DIAL and DIALLING. Dialling, sometimes called gnomonics, is a branch of applied mathematics which treats of the construction of sun-dials, that is, of those instruments, either fixed or portable, which determine the divisions of the day (Lat. _dies_) by the motion of the shadow of some object on which the sun's rays fall. It must have been one of the earliest applications of a knowledge of the apparent motion of the sun; though for a long time men would probably be satisfied with the division into morning and afternoon as marked by sun-rise, sun-set and the greatest elevation.

_History._--The earliest mention of a sun-dial is found in Isaiah xxxviii. 8: "Behold, I will bring again the shadow of the degrees which is gone down in the _sun-dial_ of Ahaz ten degrees backward." The date of this would be about 700 years before the Christian era, but we know nothing of the character or construction of the instrument. The earliest of all sun-dials of which we have any certain knowledge was the hemicycle, or hemisphere, of the Chaldaean astronomer Berossus, who probably lived about 300 B.C. It consisted of a hollow hemisphere placed with its rim perfectly horizontal, and having a bead, or globule, fixed in any way at the centre. So long as the sun remained above the horizon the shadow of the bead would fall on the inside of the hemisphere, and the path of the shadow during the day would be approximately a circular arc. This arc, divided into twelve equal parts, determined twelve equal intervals of time for that day. Now, supposing this were done at the time of the solstices and equinoxes, and on as many intermediate days as might be considered sufficient, and then curve lines drawn through the corresponding points of division of the different arcs, the shadow of the bead falling on one of these curve lines would mark a division of time for that day, and thus we should have a sun-dial which would divide each period of daylight into twelve equal parts. These equal parts were called _temporary hours_; and, since the duration of daylight varies from day to day, the temporary hours of one day would differ from those of another; but this inequality would probably be disregarded at that time, and especially in countries where the variation between the longest summer day and the shortest winter day is much less than in our climates.

The dial of Berossus remained in use for centuries. The Arabians, as appears from the work of Albategnius, still followed the same construction about the year A.D. 900. Four of these dials have in modern times been found in Italy. One, discovered at Tivoli in 1746, is supposed to have belonged to Cicero, who, in one of his letters, says that he had sent a dial of this kind to his villa near Tusculum. The second and third were found in 1751--one at Castel-Nuovo and the other at Rignano; and a fourth was found in 1762 at Pompeii. G. H. Martini in his _Abhandlungen von den Sonnenuhren der Alten_ (Leipzig, 1777), says that this dial was made for the latitude of Memphis; it may therefore be the work of Egyptians, perhaps constructed in the school of Alexandria.

Herodotus recorded that the Greeks derived from the Babylonians the use of the gnomon, but the great progress made by the Greeks in geometry enabled them in later times to construct dials of great complexity, some of which remain to us, and are proof not only of extensive knowledge but also of great ingenuity.