The general equations occurring in the application of mathematics to natural phenomena are equally true whatever units we employ for the measurement of the different quantities which enter into them, provided we employ the same units throughout the equation. Hence such equations must be homogeneous with respect to any system of variables which is referred to the same unit, and all quantities essentially numerical, such as exponents and exponentials, logarithms, angles, and circular and elliptic functions, must be of zero dimensions.

There are two methods of interpreting the equations relating to geometry and other concrete sciences.

We may regard the symbols which occur in the equation as of themselves denoting lines, masses, times, &c.; or we may consider each symbol as denoting only the numerical value of the corresponding quantity, the concrete unit to which it is referred being tacitly understood.

measurements of a system in which the centimetre, gramme, and second are the units. This is sometimes referred to as the C. G. S. system, and a copious list of examples of the measurement of physical quantities on this system, of its comparison with other systems, and of the dimensions of quantities occurring in all branches of physics, has been prepared by Dr Everett, and published by the Physical Society of London and by Taylor and Francis, under the title Illustrations of the C. G. S. System of Units. The three fundamental units may be selected each independently of the others, in an entirely arbitrary manner. It is possible, however, by taking advantage of the permanence of the properties of natural substances, so to define the units that one or more of them may be reproduced without reference to any material standard at present existing.

Thus, if the density of a standard substance in a standard state, such as water when at its maximum density under the pressure of its own vapour, is defined as the unit of density, then the unit of mass may be derived from the unit of length, or vice versa. In this system, therefore, the dimensions of mass in terms of length are L3, or of length in terms of mass, M.

Thus if we write [L] for the unit of length, that is to say, the actual concrete centimetre or foot, and if x denotes the numerical value of a certain line, then the complete expression for the line is x [L]; and if y, z, &c., are the numerical values of other lines, then the complete expression for the quantity whose numerical value is xay is

and this quantity is said to be of a+B+y dimensions with respect to [L], the unit of length.

There must be as many different units as there are different kinds of quantities to be measured, but in all dynamical sciences it is possible to define these units in terms of the three fundamental units of length, time, and mass. We therefore suppose these three fundamental units

to be given, and deduce all the others from these by the simplest attainable definitions.

The equations at which we arrive must be such that a person of any nation, by substituting for the different symbols the numerical values of the quantities as measured by his own national units, would obtain a true result.

This can only be the case if the equation is homogeneous with respect to each of the fundamental units. To ascertain if it is so we must count the dimensions of every term, and for this purpose we must know the dimensions of any derived units which enter into the equation. The theory of the dimensions of physical quantities were first stated by Fourier, Théorie de Chaleur, sec. 160.

By knowing the dimensions of any quantity we are able at once to deduce its numerical value as expressed in terms of one system of units from its numerical value as given in terms of another system.

Thus. magnetic measurements have been made according to the British system, in which the foot, the grain, and the second of mean time are the fundamental units. Other magnetic measurements have been made according to systems derived from the French metric system, using the metre, centimetre, or millimetre as unit of length, the kilogramme, gramme, or milligramme as unit of mass, and the second as unit of time. In recent times an effort has

of the system are determinate. We may therefore define the unit of density in terms of the density of the liquid water under these conditions, and the unit of pressure in terms of the pressure in the vessel. We may deduce the third unit from the law of gravitation, and define the unit of time in terms of the time of revolution of a satellite about a sphere having the unit density at a distance equal to the radius. This time must be calculated from the results of experiments on attraction. Having thus obtained a density, a pressure, and a time, the magnitudes of which are the same under all circumstances, we can derive from them standards of length and mass. For the dimensions of the unit of density [D] are [ML]), and those of the unit of pressure [F] are [ML 'T'], so that the dimensions of [L] are [PDT), and those of [M] are [P* D ̄T'].

This method of defining the three fundamental units is suggested, not as being at all comparable in point of accuracy with the usual methods, but as being an example of a method independent of the preservation of any material standards, whether artificial, as those lept by Government, or natural, as the earth, and its time of revolution. (J. C. M.)

DINAJPUR, a district of British India, within the Rájsháhí Kuch-Behar division or commissionership, under the lieutenant-governor of Bengal, is situated between 24° 43′ 40′′ and 26° 22′ 50′′ N. lat., and between 88° 4' 0" and 89° 21′ 5′′ E. long. The district, which occupies an area of 4126 square miles, is a triangular tract of country with the acute angle towards the north, lying between the districts of Jalpaiguri and Rangpur on the E., and Purniah on the W.; on the S. it is bounded by the districts of Bográ, Rájsháhí, and Maldah. The country is generally flat, but towards the south becomes undulating, some of the elevations being about 100 feet in height. The district is traversed in every direction by a network of channels and

water courses.

Along the banks of the Kulik river, the undulating ridges and long lines of mango-trees give the landscape an aspect of beauty which is not found elsewhere VII -31

Macedonian control. Little as the man's life, so far as we know of it, cngages our respect or esteem, his position must at least be broadly distinguished from that of such a man as Eschines, an Athenian citizen who, while his city could still be saved, abetted its enemies-or from that of such a hireling as Demades. In the Harpalus affair, Demosthenes was, beyond all reasonable doubt, innocent, and so, probably, were others of the accused. Yet Hyperides, the most fiery of the patriots, was on the same side as Dinarchus.

Under the regency-for such it really was-of his old master, Demetrius Phalereus, Dinarchus had much political influence. The years 317-306 B.C. were the most prosperous of his life. On the fall of Demetrius Phalereus, Dinarchus withdrew into exile at Chalcis in Euboea. About 292 B.C. he ventured to return to Attica, and took up his abode with a former associate, Proxenus, in the country, against whom he afterwards brought an action, on the ground that Proxenus had robbed him of some money and plate which he had brought with him. He died at Athens, at the age of about seventy, i.e., about 291 B.O. Dionysius held that, out of 85 extant speeches bearing the name of Dinarchus, 58 were genuine,-28 in public causes, 30 in private causes, In addition to the three speeches above mentioned, we have scanty fragments of 88 more which passed, with at least some authors, under his name. The number need not surprise us, wher. we remember that Suidas speaks of 160 speeches of Dinarchus, and (following Cæcilius probably) allows 60 as genuine. No orator of the Attic decade had so little of an individual style, and to no other, consequently, was alien work so largely ascribed by the Alexandrian critics. Dinarchus imitated by turns the style of Lysias, of Hyperides, of Demostheneз3. As Dionysius says of him, oudev ovTE KOVOV our tolov čoxev, he had no general stamp of his own, no distinctive trait. He was neither an inventor, like Lysias, Isocrates, and Isæus, nor a perfecter like Eschines, Hyperides, and Demosthenes. He is called by Hermogenes kpitivos Anpooléns,-a metaphor taken either from barley compared with wheat, or, better perhaps, from beer compared with wine,-a Demosthenes whose strength is rougher, and who has neither the flavour nor the sparkle.

Our best MSS. are the Codex Crippsianus and the Codex Oxoniensis (containing also Antiphon, Andocides, Isæus, Lycurgus.) The three extant orations, with the fragment ascribed to Demades, ed. F. Blass, Lips. 1871. The fragments in Baiter and Sauppe's

Oratores Attici, vol. ii.

DINGWALL, a royal burgh of Scotland, the county town of Ross-shire, 15 miles north-west of Inverness, at the junction of the Sutherland and Dingwall and Skye railways. It occupies a low situation at the upper end of Cromarty Firth, where the valley of Strathpeffer unites with the alluvial lands at the mouth of the Conan. Though a neatly built and thriving place, it has nothing special to show except the curious old town-house, a few remains of the ancient mansion-house of the powerful family of Ross, and an obelisk 57 feet in height, erected to the memory of George, first earl of Cromarty. Dingwall, like so many towns on the same coast, is of Norse origin, and its name in Scandinavian signifies the Court Hill. In Gaelic it is known as Inbhir-pheoran, or the mouth of the Peffer. Its charter, granted by Alexander II., was renewed by James IV. It unites with Tain, Dornoch, Wick, Kirkwall, and Cromarty in returning one member to Parliament. Population in 1871, 2125.

DINKELSBÜHL, a town of Bavaria, in the department of Mittelfranken, or Middle Franconia, on the Wörnitz, about 40 miles by rail from Donauwörth, where the river joins the Danube. It is an important centre both of civil and ecclesiastical administration, and has a Roman catholic and a Protestant church, a Latin and

industrial school, and several benevolent institutions. The inhabitants carry on the manufacture of gloves, stockings, and other articles, and deal largely in cattle. Fortified by Henry I., Dinkelsbühl received in 1305 the same municipal rights as Ulm, and obtained in 1351 the position of a free imperial city, which it retained till 1803. Its municipal code, the Dinkelsbühler Recht, printed in 1536, and republished in a revised form in 1738, contained a very extensive collection of laws on matters both of public and private interest. Population in 1875, 5238.

DINOCRATES (called by Pliny Dinochares), a Greek architect, who lived in the reign of Alexander the Great. He applied to that king's courtiers for an introduction to the Macedonian king, but was put off from time to time with vain promises. Impatient at the delay, he is said to have laid aside his usual dress, besmeared his body with oil in the manner of an athlete, thrown a lion's skin over his shoulders, and, with his head adorned with a wreath of palm branches, and a club in his hand, made his way through a dense crowd which surrounded the royal tribunal to the place where the king was dispensing justice. Amazed at the strange sight, Alexander asked him who he was. He replied that he had come into the royal presence to make known a scheme which would be worthy of the consideration of the greatest monarch in the world. Out of Mount Athos, a mountain rising like a pyramid to a height of 6780 feet topped with a cone of white limestone, he proposed to construct the gigantic figure of a man, holding a large city in his right hand, while in his left he held a gigantic tank large enough to contain all the water from the brooks in the peninsula. The story goes that the king was not displeased with the idea, but, as he thought it chimerical, it came to nothing. Alexander, however, was so delighted with the man, and with his bold and daring conceptions, that he carried Dinocrates with him when he went on his campaigns against Darius. He was employed by the king to design and lay out the city of Alexandria. This city was founded in 332 B.C., but the untimely death of Dinocrates prevented it from assuming the propor tions intended by its designer. The Ephesians, whose temple of Diana had just been burnt down, employed him in its reconstruction. But perhaps the most original of all his conceptions was his design for a temple to Arsinoë, wife of Ptolemy II., king of Egypt. The roof of the building was to have been composed of a mass of loadstones, strong enough to hold floating in the air, and suspended within it, an iron statue of the queen.

DINORNIS (Selvós, terrible, and opvis, bird), a genus of gigantic Struthious birds, believed to be extinct, which in post-Pliocene times must have formed a principal feature in the fauna of New Zealand. Their remains are found in greatest abundance in the provinces of Otago and Canterbury, often strewn in great profusion over the surface of the ground, but more usually met with buried in alluvial deposits, and in swamps; and they indicate that many of the species attained a huge size-thus the tibia of Dinornis giganteus measures about a yard in length, and the bird itself must have stood 10 or 11 feet high. Another species, Dinornis elephantopus, although less in height, possessed, according to Professor Owen, the most massive skeleton in the entire order of birds, its toe bones almost rivalling those of the elephant. Wing bones are believed to have been entirely wanting in those species which now constitute the genus Dinornis, as also the fourth toe, which is present along with rudimentary wing bones in the species which have been placed in the new genus Palapteryx. Among living birds Dinornis agrees most closely with the Apteryx, the diminutive living representative in New Zealand of this gigantic race of bipeds, while somewhat resembling the emeu and cassowary in the formation of

the feathers. Judging from their general structure, and from the habits of their nearest living allies, these great wingless birds may be supposed to have inhabited the plains and hillsides rather than the forests of New Zealand, and to have been omnivorous, feeding indiscriminately on seeds and roots, lizards and insects. Crop-stones are often found in little heaps beside their skeletons, and as these are generally such stones as occur in the neighbourhood, it has been inferred that the Dinornis was comparatively stationary in its habits. New Zealand has been so thoroughly explored in recent years as to render it highly improbable that the moa, as the Dinornis is called by the Maoris, will yet be found alive, but there seems sufficient reason for believing that its final extinction may have taken place since the arrival of the Maori race in New Zealand. The Maoris have only been settled there for about five centuries, yet they have traditions regarding moa hunting, its bones are found in ancient cooking ovens, and many specimens have been obtained in which portions of the skin with feathers attached are still preserved. An egg has also been recently found containing the bones of the chick, and another measuring 10 inches long and 7 inches broad was taken from a grave, where it rested in the hands of a human skeleton. There is evidence of the coexistence in New Zealand of about 20 species of moas during post-Pliocene times, and this, as A. R. Wallace remarks, points to the conclusion that New Zealand was at one time a much more extensive land than it now is; while the fact, that recently remains of the Dinornis have been found in a postPliocene deposit in Queensland strengthens the supposition that when the moa flourished Australia and New Zealand formed portions of one continent.

DINOTHERIUM, an extinct mammal, fossil remains of which occur in the Miocene beds of France, Germany, Greece, and Northern India. These until lately consisted exclusively of teeth and the bones of the head. An entire skull, obtained from the Epplesheim beds of Hesse Darmstadt in 1836, measured 4 feet in length and 3 feet in breadth, and thus indicated an animal exceeding the elephant in size. Its upper jaw was destitute of incisor and canine teeth, but possessed 5 molars on each side, with a corresponding number in the jaw beneath. Its most remarkable feature, however, consisted in the front part of the lower jaw being bent downwards and bearing two tusklike incisors also directed downwards and backwards. Judging from these remains Professor Owen placed the Dinotherium among the proboscidean mammals; De Blainville, on the other hand, regarded it as an aquatic animal, destitute of legs, and somewhat resembling the manatee, its reversed tusks having probably been used to moor the creature to the bank of the streams it frequented, or to assist it in leaving the water. The recent discovery, however, of limb bones, decidedly proboscidean in type, and supposed to belong to the Dinotherium, supports the view that these creatures were more akin to the elephant and mastodon than to the manatee.

DIOCESE, from the Greek Stoiknous—primarily meaning administration, then the territorial circumscription in which administration was exercised-was first used to denote the Greek provinces of the Roman empire, or more properly the portion of a province ruled by a proprætor. Thus Cicero had, besides Cilicia, three "dioceses" in Asia. Bingham (lib. ix. c. 1) says that the division of the empire into clerical dioceses was in the time of Constantine, whereas the division into provinces was much anterior. He goes on to show that the primitive church followed exactly the example of the empire in her territorial arrange ments. As in every metropolis of each province there was magistrate with authority over the magistrates of each city, so in every metropolis there was a bishop, whose

authority extended over the entire province, who was thenee called "metropolitan," or "primate," as being the first or principal bishop of the province. And everywhere the episcopal sees were under the authority of the bishop of the civil metropolis, except in Africa, where the primate was usually the senior bishop of the province. The term "diocese," however, was sometimes used in the more comprehensive, and the term province in the less comprehensive sense, as appears from the Notitia dignitatum Imperii, drawn up, as it would seem, in the time of the emperors Arcadius and Honorius (see Bingham, loc. cit.) The territorial division, however, as given in the Notitia, was purely civil. But Bingham tells us that, though we have no equally ancient account of the ecclesiastical division of the empire, yet if we compare the fragmentary bits of information which may be picked out of the acts of and subscriptions to the carlier councils with later notices, it will be seen that the ecclesiastical very exactly followed the civil distribution.

It may be mentioned that, before the 4th century, the term " parish"-apoixía-was often used indiscriminately with the word "diocese," a circumstance which has caused ecclesiastical antiquarians to expend much erudition in showing that, despite the confusion of terms, the thing intended corresponded to our idea of a diocese, and not to our idea of a parish.

The uncertainty with regard to the number and circumscription of the English ecclesiastical dioceses under the Romans is great, and the information attainable fragmentary. At the council of Arles, held in the year 314, the bishop of York, the bishop of London, and the bishop "de colonia Lindi," probably Lincoln, are recorded to have been present. But the changes in the number and territorial circumscription following the Saxon invasion-and not yet finally completed-were so great that volumes of minute antiquarian investigation would be needed to trace in so far as it may be still possible to trace-the progress of nomenclature and delimitation of the various dioceses of Britain from the first establishment of them to the present day.

The division of dioceses found to be too large to be conveniently administered by one bishop was practised from very early times, as may be seen by the decrees of a council held in Portugal about the middle of the 6th century. Another reason for dividing a diocese, and establishing a new see, has been recognized by the church as duly existing "if the sovereign should think fit to endow some principal village or town with the rank and privileges of a city" (Bingham, lib. xvii. c. 5). But there are canons for the punishment of such as might induce the sovereign so to erect any town into a city, solely with the view of becoming bishop thereof. Nor could any diocese be divided without the consent of the primate.

In the countries more immediately subjected to the Roman pontiff the multiplication of dioceses has been excessive, the number of them in the apostolic dominions being no less than 68, while the Roman Church reckons in the whole of Europe (exclusive of the English, but inclusive of the Irish sees) 578 sees.

DIOCLETIAN. VALERIUS DIOCLETIANUS (245-313), Roman emperor, was born of obscure parents near Salons, in Dalmatia, and reigned from 284 to 305 A.D: entered the army and served with high distinction, held important commands under the emperors Probus and Aurelian, and accompanied Carus to the Persian war. After the death of Numerianus he was chosen emperor by the troops at Chalcedon, and slew with his own hands Arrius Aper, the prefect of the prætorians. His advent to the throne marks the commencement of the era of Diocletian, August 29, 284. Having been installed at

Nicomedia, he received general acknowledgment after the murder of Carinus. He appointed Maximian. Augustus in 286. and Constantius Chlorus and Galerius, Caesars in 292. Each of the four rules was placed at a separate capital Treves. Sirmium. Milan. Nicomedin. This amounted to an entirely new organization of the empire. on a plan commensurate with the work of government which it now had to effect. At the age of fifty-nine. exhausted with labour, be abdicated his sovereignty on May 1, 305, and retired to Salona, the place of his birth, where he died eight years afterwards. His reign was memorable for the persecution of the Christians.

DIODATI, GIOVANNI (1576-1649), a Swiss theologian of the Reformed Church, was born at Geneva on the 6th June 1576 of a noble family originally belonging to Lucca, which had been expatriated for the profession of Protestantism. In his youth he distinguished himself as a biblical scholar, and at the age of twenty-one he was nominated Professor of Hebrew at Geneva on the recommendation of Beza. In 1608 he became a pastor, or parish minister, at Geneva, and in the following year he succeeded Beza as professor of theology. As a preacher he was eloquent, bold, and fearless, with his full share of the intolerance that prevailed among his party at Geneva. He held a high place among the reformers of Geneva, by whom he was sent on a mission to France in 1614. He had previously visited Italy, and made the acquaintance of Sarpi and Fulgenzio, whom he endeavoured unsuccessfully to engage in a reformation movement. In 1618-19 he attended the Synod of Dort, and took a prominent part in its deliberations, being one of the six divines appointed to draw up the account of its proceedings. He was a thorough Calvinist, and entirely sympathized with the condemnation of the Arminians. In 1645 he resigned his professorship, and he died at Geneva on the 3d October 1649. Diodati is chiefly celebrated as the author of the translation of the Bible into Italian which appeared in 1603. Another edition with notes was issued in 1607. As a translator he possessed the primary qualification of a competent knowledge of the original, but his work was rather a paraphrase than a translation, and his notes were those of a theologian rather than of a critic. He also undertook a translation of the Bible into French. which appeared with notes in 1644. Among his other works were his Annotationes in Biblia (1607), of which an English translation was published in London in 1648, and various polemical treatises, such as De fictitio Pontificiorum Purgatorio, 1619; De justa Secessione Reformatorum ab Ecclesia Romana, 1628; De Antichristo, &c. He also published French translations of Sarpi's History of the Council of Trent, and of Edwin Sandys's Account of the State of Religion in the West.

DIODORUS, named SICULUS, a Greek historian, born at Agyrium in Sicily. Of his life we know nothing except what he himself has narrated, that, in prosecution of his historical researches, he undertook frequent and dangerous journeys, and studied Latin at Rome. His history occupied thirty years in writing, and was at last completed in forty books. From internal evidence it is certain that it was written after the death of Julius Caesar; but the passages which show him to have survived the alteration of the calendar by Augustus are generally regarded as spurious. His history, to which, from its comprehensive plan, he has given the title of Bibliotheca, is divided into three parts. The first treats of the mythic history of the non-Hellenic, and afterwards of the Hellenic tribes; the second section ends with Alexander's death; and the third continues the history as far as the beginning of Cæsar's Gallic war. Of this extensive work there are still extant only the first five books, treating of the mythic history of the Egyptians,

Assyrians, Ethiopians, and Greeks; and also fi nu the 11th to the 20th book inclusive, beginning with the soul Persian war, and ending with the history of the successors of Alexander. previously to the partition of the Macedonian empire. The rest exists only in fragments which have been collected by Photius. The faults of Diodorus arise principally from the gigantic nature of the undertaking, the cumbrous nature of the materials. and the awkward form of annals into which he has thrown his narrative. | He has been at little pains to sift his materials, and hence frequent repetitions and contradictions may be found in the body of the work. As a critic, he seems to have been altogether ignorant of the ethical advantages of history, and shrinks from administering praise or blame to the persons whose history he writes. In the chronology of the strictly historical period he is occasionally inaccurate; and the poetical myths which take the place of the early history are related with all the gravity of historical detail. His narrative is without colouring, and monotonous; and his simple and clear diction, which stands intermediate between pure Attic and the colloquial Greek of his time, enables us to detect in the narrative the nndigested fragments of the materials which he employed. The particulars, however, which he has handed down are valuable, as enabling us in several points to rectify the errors of Livy.

The best editions of Diodorus are Wesseling's, 2 vols., Amstel. 1745; that printed at Deux-Ponts, 11 vols., 1795-1801; Eichstadt' (to book xiv.) 2 vols., Halle, 1802-4; and Dindorf's, 5 vols., Leips. 1828-81.

DIOGENES, of Apollonia in Crete, a celebrated natural philosopher who flourished at Athens about 460 B.C. He was a pupil of Anaximenes and a contemporary of Anaxagoras, The fragments of his writings have been collected together by Panzerbieter. He believed air to be the source of all being, and all other substances to be derived from it by condensation and rarefaction. His chief advance upon the doctrines of his master is that he asserted air, the primal force, to be intelligence-"the air which stirred within him not only prompted but instructed. The air as the origin of all things is necessarily an eternal, imperishable substance, but as soul it is also necessarily endowed with consciousness." Mr Lewes and Mr Grote assign to him a higher place in the evolution of philosophy than either Hegel or Schwegler.

DIOGENES (about 412-223 B.C.), the famous Cynic philosopher, was the son of Icesias, a money-changer of Sinope in Pontus. Having been detected in adulterating coin, his father and he were compelled to leave their native city. According to another account, however, Icesias died in prison, and Diogenes fled to Athens with a single attendant. On his arrival in that city he dismissed hi attendant with the piquant question, "If Manes could live without Diogenes, why not Diogenes without him?" and on the same principle he denuded himself of all superfluous dress, furniture, and even ideas. A wooden bowl, which, with his cloak and wallet, formed his only movables, is said to have been immediately discarded when he saw a boy drinking water from the hollow of his hand. The fame of Antisthenes soon attracted him to Cynosarges, and the pertinacity with which, for the sake of wisdom, he not only endured the scoffs but volunteered to submit to the blows of the great teacher, soon procured him a favourable reception from the whole Cynical school. The favourite pupil, however, soon outstripped his master in the extravagancies of his life, and the pungent keenness of his sarcasms. That he took up his abode in a cask belonging to the temple of Cybele is a circumstance liable to suspicion, from being more frequently alluded to by the satirists than by the biographers of Diogenes. That he