BYZANTINE ART (continued 8)

Up to 1707, when Calcutta was first declared a presidency, it had been dependent upon the older English settlement at Madras. From 1707 to 1773 the presidencies were maintained on a footing of equality; but in the latter year the act of parliament was passed, which provided that the presidency of Bengal should exercise a control over the other possessions of the Company; that the chief of that presidency should be styled governor-general; and that a supreme court of judicature should be established at Calcutta. In the previous year, 1772, Warren Hastings had taken under the immediate management of the Company's servants the general administration of Bengal, which had hitherto been left in the hands of the old Mahommedan officials, and had removed the treasury from Murshidabad to Calcutta. The latter town thus became the capital of Bengal and the seat of the supreme government in India. In 1834 the governor-general of Bengal was created governor-general of India, and was permitted to appoint a deputy-governor to manage the affairs of Lower Bengal during his occasional absence. It was not until 1854 that a separate head was appointed for Bengal, who, under the style of lieutenant-governor, exercises the same powers in civil matters as those vested in the governors in council of Madras or Bombay, although subject to closer supervision by the supreme government. Calcutta is thus at present the seat both of the supreme and the local government, each with an independent set of offices. (See Bengal.)

See A.K. Ray, A Short History of Calcutta (Indian Census, 1901); H.B. Hyde, Parochial Annals of Bengal (1901); K. Blechynden, Calcutta, Past and Present (1905); H.E. Busteed, Echoes from Old Calcutta (1897); G.W. Forrest, Cities of India (1903); C.R. Wilson, Early Annals of the English in Bengal (1895); and Old Fort William in Bengal (1906); Imperial Gazetteer of India (Oxford, 1908), s.v. "Calcutta."

CALDANI, LEOPOLDO MARCO ANTONIO (1725-1813), Italian anatomist and physician, was born at Bologna in 1725. After studying under G.B. Morgagni at Padua, he began to teach practical medicine at Bologna, but in consequence of the intrigues of which he was the object he returned to Padua, where in 1771 he succeeded Morgagni in the chair of anatomy. He continued to lecture until 1805 and died at Padua in 1813. His works include Institutiones pathologicae (1772), Institutiones physiologicae (1773) and Icones anatomicae (1801-1813).

His brother, Petronio Maria Caldani (1735-1808), was professor of mathematics at Bologna, and was described by J. le R. D'Alembert as the "first geometer and algebraist of Italy."

CALDECOTT, RANDOLPH (1846-1886), English artist and illustrator, was born at Chester on the 22nd of March 1846. From 1861 to 1872 he was a bank clerk, first at Whitchurch in Shropshire, afterwards at Manchester; but devoted all his spare time to the cultivation of a remarkable artistic faculty. In 1872 he migrated to London, became a student at the Slade School and finally adopted the artist's profession. He gained immediately a wide reputation as a prolific and original illustrator, gifted with a genial, humorous faculty, and he succeeded also, though in less degree, as a painter and sculptor. His health gave way in 1876, and after prolonged suffering he died in Florida on the 12th of February 1886. His chief book illustrations are as follows:—Old Christmas (1876) and Bracebridge Hall (1877), both by Washington Irving; North Italian Folk (1877), by Mrs Comyns Carr; The Harz Mountains (1883); Breton Folk (1879), by Henry Blackburn; picture-books (John Gilpin, The House that Jack Built, and other children's favourites) from 1878 onwards; Some Aesop's Fables with Modern Instances, &c. (1883). He held a roving commission for the Graphic, and was an occasional contributor to Punch. He was a member of the Royal Institute of Painters in Water-colours.

See Henry Blackburn, Randolph Caldecott, Personal Memoir of his Early Life (London, 1886).

CALDER, SIR ROBERT, Bart. (1745-1818), British admiral, was born at Elgin, in Scotland, on the 2nd of July 1745 (o.s.). He belonged to a very ancient family of Morayshire, and was the second son of Sir Thomas Calder of Muirton. He was educated at the grammar school of Elgin, and at the age of fourteen entered the British navy as midshipman. In 1766 he was serving as lieutenant of the "Essex," under Captain the Hon. George Faulkner, in the West Indies. Promotion came slowly, and it was not till 1782 that he attained the rank of post-captain. He acquitted himself honourably in the various services to which he was called, but for a long time had no opportunity of distinguishing himself. In 1796 he was named captain of the fleet by Sir John Jervis, and took part in the great battle off Cape St Vincent (February 14, 1797). He was selected as bearer of the despatches announcing the victory, and on that occasion was knighted by George III. He also received the thanks of parliament, and in the following year was created a baronet. In 1799 he became rear-admiral; and in 1801 he was despatched with a small squadron in pursuit of a French force, under Admiral Gantheaume, conveying supplies to the French in Egypt. In this pursuit he was not successful, and returning home at the peace he struck his flag. When the war again broke out he was recalled to service, was promoted vice-admiral in 1804, and was employed in the following year in the blockade of the ports of Ferrol and Corunna, in which (amongst other ports) ships were preparing for the invasion of England by Napoleon I. He held his position with a force greatly inferior to that of the enemy, and refused to be enticed out to sea. On its becoming known that the first movement directed by Napoleon was the raising of the blockade of Ferrol, Rear-Admiral Stirling was ordered to join Sir R. Calder and cruise with him to intercept the fleets of France and Spain on their passage to Brest. The approach of the enemy was concealed by a fog; but on the 22nd of July 1805 their fleet came in sight. It still outnumbered the British force; but Sir Robert entered into action. After a combat of four hours, during which he captured two Spanish ships, he gave orders to discontinue the action. He offered battle again on the two following days, but the challenge was not accepted. The French admiral Villeneuve, however, did not pursue his voyage, but took refuge in Ferrol. In the judgment of Napoleon, his scheme of invasion was baffled by this day's action; but much indignation was felt in England at the failure of Calder to win a complete victory. In consequence of the strong feeling against him at home he demanded a court-martial. This was held on the 23rd of December, and resulted in a severe reprimand of the vice-admiral for not having done his utmost to renew the engagement, at the same time acquitting him of both cowardice and disaffection. False expectations had been raised in England by the mutilation of his despatches, and of this he indignantly complained in his defence. The tide of feeling, however, turned again; and in 1815, by way of public testimony to his services, and of acquittal of the charge made against him, he was appointed commander of Portsmouth. He died at Holt, near Bishop's Waltham, in Hampshire, on the 31st of August 1818.

See Naval Chronicle, xvii.; James, Naval History, iii. 356-379 (1860).

CALDER, an ancient district of Midlothian, Scotland. It has been divided into the parishes of Mid-Calder (pop. in 1901 3132) and West-Calder (pop. 8092), East-Calder belonging to the parish of Kirknewton (pop. 3221). The whole locality owes much of its commercial importance and prosperity to the enormous development of the mineral oil industry. Coal-mining is also extensively pursued, sandstone and limestone are worked, and paper-mills flourish. Mid-Calder, a town on the Almond (pop. 703), has an ancient church, and John Spottiswood (1510-1585), the Scottish reformer, was for many years minister. His sons—John, archbishop of St Andrews, and James (1567-1645), bishop of Clogher—were both born at Mid-Calder. West-Calder is situated on Breich Water, an affluent of the Almond, 15½ m. S.W. of Edinburgh by the Caledonian railway, and is the chief centre of the district. Pop. (1901) 2652. At Addiewell, about 1½ m. S.W., the manufacture of ammonia, naphtha, paraffin oil and candles is carried on, the village practically dating from 1866, and having in 1901 a population of 1591. The Highland and Agricultural Society have an experimental farm at Pumpherston (pop. 1462). The district contains several tumuli, old ruined castles and a Roman camp in fair preservation.

CALDERÓN, RODRIGO (d. 1621), Count of Oliva and Marques de las Siete Iglesias, Spanish favourite and adventurer, was born at Antwerp. His father, Francisco Calderón, a member of a family ennobled by Charles V., was a captain in the army who became afterwards comendador mayor of Aragon, presumably by the help of his son. The mother was a Fleming, said by Calderón to have been a lady by birth and called by him Maria Sandelin. She is said by others to have been first the mistress and then the wife of Francisco Calderón. Rodrigo is said to have been born out of wedlock. In 1598 he entered the service of the duke of Lerma as secretary. The accession of Philip III. in that year made Lerma, who had unbounded influence over the king, master of Spain. Calderón, who was active and unscrupulous, made himself the trusted agent of Lerma. In the general scramble for wealth among the worthless intriguers who governed in the name of Philip III., Calderón was conspicuous for greed, audacity and insolence. He was created count of Oliva, a knight of Santiago, commendador of Ocaña in the order, secretary to the king (secretario de cámara), was loaded with plunder, and made an advantageous marriage with Ines de Vargas. As an insolent upstart he was peculiarly odious to the enemies of Lerma. Two religious persons, Juan de Santa Mariá, a Franciscan, and Mariana de San José, prioress of La Encarnacion, worked on the queen Margarita, by whose influence Calderón was removed from the secretaryship in 1611. He, however, retained the favour of Lerma, an indolent man to whom Calderón's activity was indispensable. In 1612 he was sent on a special mission to Flanders, and on his return was made marques de las Siete Iglesias in 1614. When the queen Margarita died in that year in childbirth, Calderón was accused of having used witchcraft against her. Soon after it became generally known that he had ordered the murder of one Francisco de Juaras. When Lerma was driven from court in 1618 by the intrigues of his own son, the duke of Uceda, and the king's confessor, the Dominican Aliaga, Calderón was seized upon as an expiatory victim to satisfy public clamour. He was arrested, despoiled, and on the 7th of January 1620 was savagely tortured to make him confess to the several charges of murder and witchcraft brought against him. Calderón confessed to the murder of Juaras, saying that the man was a pander, and adding that he gave the particular reason by word of mouth since it was more fit to be spoken than written. He steadfastly denied all the other charges of murder and the witchcraft. Some hope of pardon seems to have remained in his mind till he heard the bells tolling for Philip III. in March 1621. "He is dead, and I too am dead" was his resigned comment. One of the first measures of the new reign was to order his execution. Calderón met his fate firmly and with a show of piety on the 21st of October 1621, and this bearing, together with his broken and prematurely aged appearance, turned public sentiment in his favour. The magnificent devotion of his wife helped materially to placate the hatred he had aroused. Lord Lytton made Rodrigo Calderón the hero of his story Calderon the Courtier.

See Modests de la Fuente, Historia General España (Madrid, 1850-1867), vol. xv. pp. 452 et seq.; Quevedo, Obras (Madrid, 1794), vol. x.—Grandes Anales de Quince Dias. A curious contemporary French pamphlet on him, Histoire admirable et declin pitoyable advenue en la personne d'unfawory de la Cour d'Espagne, is reprinted by M.E. Fournier in Variétés historiques (Paris, 1855), vol. i.

(D. H.)

CALDERÓN DE LA BARCA, PEDRO (1600-1681), Spanish dramatist and poet, was born at Madrid on the 17th of January 1600. His mother, who was of Flemish descent, died in 1610; his father, who was secretary to the treasury, died in 1615. Calderón was educated at the Jesuit College in Madrid with a view to taking orders and accepting a family living; abandoning this project, he studied law at Salamanca, and competed with success at the literary fêtes held in honour of St Isaidore at Madrid (1620-1632). According to his biographer, Vera Tassis, Calderón served with the Spanish army in Italy and Flanders between 1625 and 1635; but this statement is contradicted by numerous legal documents which prove that Calderón resided at Madrid during these years. Early in 1629 his brother Diego was stabbed by an actor who took sanctuary in the convent of the Trinitarian nuns; Calderón and his friends broke into the cloister and attempted to seize the offender. This violation was denounced by the fashionable preacher, Hortensio Félix Paravicino (q.v.), in a sermon preached before Philip IV.; Calderón retorted by introducing into El Príncipe constante a mocking reference (afterwards cancelled) to Paravicino's gongoristic verbiage, and was committed to prison. He was soon released, grew rapidly in reputation as a playwright, and, on the death of Lope de Vega in 1635, was recognized as the foremost Spanish dramatist of the age. A volume of his plays, edited by his brother José in 1636, contains such celebrated and diverse productions as La Vida es sueño, El Purgatorío de San Patricia, La Devoción de la cruz, La Dama duende and Peor está que estaba. In 1636-1637 he was made a knight of the order of Santiago by Philip IV., who had already commissioned from him a series of spectacular plays for the royal theatre in the Buen Retiro. Calderón was almost as popular with the general public as Lope de Vega had been in his zenith; he was, moreover, in high favour at court, but this royal patronage did not help to develop the finer elements of his genius. On the 28th of May 1640 he joined a company of mounted cuirassiers recently raised by Olivares, took part in the Catalonian campaign, and distinguished himself by his gallantry at Tarragona; his health failing, he retired from the army in November 1642, and three years later was awarded a special military pension in recognition of his services in the field. The history of his life during the next few years is obscure. He appears to have been profoundly affected by the death of his mistress—the mother of his son Pedro José—about the year 1648-1649; his long connexion with the theatre had led him into temptations, but it had not diminished his instinctive spirit of devotion, and he now sought consolation in religion. He became a tertiary of the order of St Francis in 1650, and finally reverted to his original intention of joining the priesthood. He was ordained in 1651, was presented to a living in the parish of San Salvador at Madrid, and, according to his statement made a year or two later, determined to give up writing for the stage. He did not adhere to this resolution after his preferment to a prebend at Toledo in 1653, though he confined himself as much as possible to the composition of autos sacramentales—allegorical pieces in which the mystery of the Eucharist was illustrated dramatically, and which were performed with great pomp on the feast of Corpus Christi and during the weeks immediately ensuing. In 1662 two of Calderón's autos—Las órdenes militares and Místicay real Babilonia—were the subjects of an inquiry by the Inquisition; the former was censured, the manuscript copies were confiscated, and the condemnation was not rescinded till 1671. Calderón was appointed honorary chaplain to Philip IV, in 1663, and the royal favour was continued to him in the next reign. In his eighty-first year he wrote his last secular play, Hado y Divisa de Leonido y Marfisa, in honour of Charles II.'s marriage to Marie-Louise de Bourbon. Notwithstanding his position at court and his universal popularity throughout Spain, his closing years seem to have been passed in poverty. He died on the 25th of May 1681.

Like most Spanish dramatists, Calderón wrote too much and too speedily, and he was too often content to recast the productions of his predecessors. His Saber del mal y del bien is an adaptation of Lope de Vega's play, Las Mudanzas de la fortuna y sucesos de Don Beltran de Aragón; his Selva confusa is also adapted from a play of Lope's which bears the same title; his Encanto sin encanto derives from Tirso de Molina's Amar par señas, and, to take an extreme instance, the second act of his Cabellos de Absalón is transferred almost bodily from the third act of Tirso's Venganza de Tamar. It would be easy to add other examples of Calderón's lax methods, but it is simple justice to point out that he committed no offence against the prevailing code of literary morality. Many of his contemporaries plagiarized with equal audacity, but with far less success. Sometimes, as in El Alcalde de Zalamea, the bold procedure is completely justified by the result; in this case by his individual treatment he transforms one of Lope de Vega's rapid improvisations into a finished masterpiece. It was not given to him to initiate a great dramatic movement; he came at the end of a literary revolution, was compelled to accept the conventions which Lope de Vega had imposed on the Spanish stage, and he accepted them all the more readily since they were peculiarly suitable to the display of his splendid and varied gifts. Not a master of observation nor an expert in invention, he showed an unexampled skill in contriving ingenious variants on existing themes; he had a keen dramatic sense, an unrivalled dexterity in manipulating the mechanical resources of the stage, and in addition to these minor indispensable talents he was endowed with a lofty philosophic imagination and a wealth of poetic diction. Naturally, he had the defects of his great qualities; his ingenuity is apt to degenerate into futile embellishment; his employment of theatrical devices is the subject of his own good-humoured satire in No hay burlas con el amor; his philosophic intellect is more interested in theological mysteries than in human passions; and the delicate beauty of his style is tinged with a wilful preciosity. Excelling Lope de Vega at many points, Calderón falls below his great predecessor in the delineation of character. Yet in almost every department of dramatic art Calderón has obtained a series of triumphs. In the symbolic drama he is best represented by El Principe constante, by El Mágico prodigioso (familiar to English readers in Shelley's free translation), and by La Vida es sueño, perhaps the most profound and original of his works. His tragedies are more remarkable for their acting qualities than for their convincing truth, and the fact that in La Niña de Gomez Arias he interpolates an entire act borrowed from Velez de Guevara's play of the same title seems to indicate that this kind of composition awakened no great interest in him; but in El Médico de sa honra and El Mayor monstruo los celos the theme of jealousy is handled with sombre power, while El Alcalde de Zalamea is one of the greatest tragedies in Spanish literature. Calderón is seen to much less advantage in the spectacular plays—dramas de tramoya—which he wrote at the command of Philip IV.; the dramatist is subordinated to the stage-carpenter, but the graceful fancy of the poet preserves even such a mediocre piece as Los Tres Mayores prodigies (which won him his knighthood) from complete oblivion. A greater opportunity is afforded in the more animated comedias palaciegas, or melodramatic pieces destined to be played before courtly audiences in the royal palace: La Banda y la flor and El Galán fantasma are charming illustrations of Calderón's genial conception and refined artistry. His historical plays (La Gran Cenobia, Las armas de la hermosura, &c.) are the weakest of all his formal dramatic productions; El Golfo de la sirenas and La Púrpura de la rosa are typical zarzuelas, to be judged by the standard of operatic libretti, and the entremeses are lacking in the lively humour which should characterize these dramatic interludes. On the other hand, Calderón's faculty of ingenious stagecraft is seen at its best in his "cloak-and-sword" plays (comedias de capa y espada) which are invaluable pictures of contemporary society. They are conventional, no doubt, in the sense that all representations of a specially artificial society must be conventional; but they are true to life, and are still as interesting as when they first appeared. In this kind No siempre lo peor es cierto, La Dama duende, Una casa con dos puertas mala es de guardar and Guárdate del agua mansa are almost unsurpassed. But it is as a writer of autos sacramentales that Calderón defies rivalry: his intense devotion, his subtle intelligence, his sublime lyrism all combine to produce such marvels of allegorical poetry as La Cena del rey Baltasar, La Viña del Senor and La Serpiente de metal. The autos lingered on in Spain till 1765, but they may be said to have died with Calderón, for his successors merely imitated him with a tedious fidelity. Almost alone among Spanish poets, Calderón had the good fortune to be printed in a fairly correct and readable edition (1682-1691), thanks to the enlightened zeal of his admirer, Juan de Vera Tassis y Villaroel, and owing to this happy accident he came to be regarded generally as the first of Spanish dramatists. The publication of the plays of Lope de Vega and of Tirso de Molina has affected the critical estimate of Calderón's work; he is seen to be inferior to Lope de Vega in creative power, and inferior to Tirso de Molina in variety of conception. But, setting aside the extravagances of his admirers, he is admittedly an exquisite poet, an expert in the dramatic form, and a typical representative of the devout, chivalrous, patriotic and artificial society in which he moved.

Bibliography.—H. Breymann, Calderon-Studien (München and Berlin, 1905), i. Teil, contains a fairly exhaustive list of editions, translations and arrangements; Autos sacramentales (Madrid, 1759-1760, 6 vols.), edited by Juan Fernandez de Apontes; Comedias (Madrid, 1848-1850, 4 vols.), edited by Juan Eugenio Hartzenbuch; Max Krenkel, Klassische Buhnendichtungen der Spanier, containing La Vida es sueño, El mágico prodigioso and El Alcalde de Zalamca (Leipzig, 1881-1887, 3 vols.); Teatro selecto (Madrid, 1884, 4 vols.), edited by M. Menéndez y Pelayo; El Mágico prodigioso (Heilbronn, 1877), edited by Alfred Morel-Fatio; Select Plays of Calderón (London, 1888), edited by Norman MacColl; F.W.V. Schmidt, Die Schauspiele Calderon's (Elberfeld, 1857); E. Günthner, Calderon und seine Werke (Freiburg i. B., 1888, 2 vols.); Felipe Picatoste y Rodriguez, Biografia de Don Pedro Calderón de la Barca in Homenage á Calderón (Madrid, 1881); Antonio Sánchez Moguel, Memoria acerca de "El Mágico prodigioso" (Madrid, 1881); M. Menéndez y Pelayo, Calderón y su teatro (Madrid, 1881); Ernest Martinenche, La Comedia espagnole en France de Hardy á Racine (Paris, 1900).

(J. F.-K.)

CALDERWOOD, DAVID (1575-1650), Scottish divine and historian, was born in 1575. He was educated at Edinburgh, where he took the degree of M.A. in 1593. About 1604 he became minister of Crailing, near Jedburgh, where he became conspicuous for his resolute opposition to the introduction of Episcopacy. In 1617, while James was in Scotland, a Remonstrance, which had been drawn up by the Presbyterian clergy, was placed in Calderwood's hands. He was summoned to St Andrews and examined before the king, but neither threats nor promises could make him deliver up the roll of signatures to the Remonstrance. He was deprived of his charge, committed to prison at St Andrews and afterwards removed to Edinburgh. The privy council ordered him to be banished from the kingdom for refusing to acknowledge the sentence of the High Commission. He lingered in Scotland, publishing a few tracts, till the 27th of August 1619, when he sailed for Holland. During his residence in Holland he published his Altare Damascenum. Calderwood appears to have returned to Scotland in 1624, and he was soon afterwards appointed minister of Pencaitland, in the county of Haddington. He continued to take an active part in the affairs of the church, and introduced in 1649 the practice, now confirmed by long usage, of dissenting from the decision of the Assembly, and requiring the protest to be entered in the record. His last years were devoted to the preparation of a History of the Church of Scotland. In 1648 the General Assembly urged him to complete the work he had designed, and voted him a yearly pension of £800. He left behind him a historical work of great extent and of great value as a storehouse of authentic materials for history. An abridgment, which appears to have been prepared by himself, was published after his death. An excellent edition of the complete work was published by the Wodrow Society, 8 vols., 1842-1849. The manuscript, which belonged to General Calderwood Durham, was presented to the British Museum. Calderwood died at Jedburgh on the 29th of October 1650.

CALDERWOOD, HENRY (1830-1897), Scottish philosopher and divine, was born at Peebles on the 10th of May 1830. He was educated at the Royal High school, and later at the university of Edinburgh. He studied for the ministry of the United Presbyterian Church, and in 1856 was ordained pastor of the Greyfriars church, Glasgow. He also examined in mental philosophy for the university of Glasgow from 1861 to 1864, and from 1866 conducted the moral philosophy classes at that university, until in 1868 he became professor of moral philosophy at Edinburgh. He was made LL.D. of Glasgow in 1865. He died on the 19th of November 1897. His first and most famous work was The Philosophy of the Infinite (1854), in which he attacked the statement of Sir William Hamilton that we can have no knowledge of the Infinite. Calderwood maintained that such knowledge, though imperfect, is real and ever-increasing; that Faith implies Knowledge. His moral philosophy is in direct antagonism to Hegelian doctrine, and endeavours to substantiate the doctrine of divine sanction. Beside the data of experience, the mind has pure activity of its own whereby it apprehends the fundamental realities of life and combat. He wrote in addition A Handbook of Moral Philosophy, On the Relations of Mind and Brain, Science and Religion, The Evolution of Man's Place in Nature. Among his religious works the best-known is his Parables of Our Lord, and just before his death he finished a Life of David Hume in the "Famous Scots" series. His interests were not confined to religious and intellectual matters; as the first chairman of the Edinburgh school board, he worked hard to bring the Education Act into working order. He published a well-known treatise on education. In the cause of philanthropy and temperance he was indefatigable. In politics he was at first a Liberal, but became a Liberal Unionist at the time of the Home Rule Bill.

A biography of Calderwood was published in 1900 by his son W.C. Calderwood and the Rev. David Woodside, with a special chapter on his philosophy by Professor A.S. Pringle-Pattison.

CALEB (Heb. kēleb, "dog"), in the Bible, one of the spies sent by Moses from Kadesh in South Palestine to spy out the land of Canaan. For his courage and confidence he alone was rewarded by the promise that he and his seed should obtain a possession in it (Num. xiii. seq.). The later tradition includes Joshua, the hero of the conquest of the land. Subsequently Caleb settled in Kirjath-Arba (Hebron), but the account of the occupation is variously recorded. Thus (a) Caleb by himself drove out the Anakites, giants of Hebron, and promised to give his daughter Achsah to the hero who could take Kirjath-Sepher (Debir). This was accomplished by Othniel, the brother of Caleb (Josh. xv. 14-19). Both are "sons" of Kenaz, and Kenaz is an Edomite clan (Gen. xxxvi. 11, 15, 42). Elsewhere (b) Caleb the Kenizzite reminds Joshua of the promise at Kadesh; he asks that he may have the "mountain whereof Yahweh spake," and hopes to drive out the giants from its midst. Joshua blesses him and thus Hebron becomes the inheritance of Caleb (Josh. xiv. 6-15). Further (c) the capture of Hebron and Debir is ascribed to Judah who gives them to Caleb (Judg. i. 10 seq. 20); and finally (d) these cities are taken by Joshua himself in the course of a great and successful campaign against South Canaan (Josh. x. 36-39). Primarily the clan Caleb was settled in the south of Judah but formed an independent unit (i Sam. xxv., xxx. 14). Its seat was at Carmel, and Abigail, the wife of the Calebite Nabal, was taken by David after her husband's death. Not until later are the small divisions of the south united under the name Judah, and this result is reflected in the genealogies where the brothers Caleb and Jerahmeel are called "sons of Hezron" (the name typifies nomadic life) and become descendants of Judah.

Similarly in Num. xiii. 6, xxxiv. 19 (post-exilic), Caleb becomes the representative of the tribe of Judah, and also in c (above) Caleb's enterprise was later regarded as the work of the tribe with which it became incorporated, b and d are explained in accordance with the aim of the book to ascribe to the initiation or the achievements of one man the conquest of the whole of Canaan (see Joshua). The mount or hill-country in b appears to be that which the Israelites unsuccessfully attempted to take (Num. xiv. 41-45), but according to another old fragment Hormah was the scene of a victory (Num. xxi. 1-3), and it seems probable that Caleb, at least, was supposed to have pushed his way northward to Hebron. (See Jerahmeel, Kenites, Simeon.)

The genealogical lists place the earliest seats of Caleb in the south of Judah (1 Chron. ii. 42 sqq.; Hebron, Maon, &c.). Another list numbers the more northerly towns of Kirjath-jearim, Bethlehem, &c., and adds the "families of the scribes," and the Kenites (ii. 50 seq.). This second move is characteristically expressed by the statements that Caleb's first wife was Azūbah ("abandoned," desert region)—Jerīōth ("tent curtains") appears to have been another—and that after the death of Hezron he united with Ephrath (p. 24 Bethlehem). On the details in 1 Chron. ii., iv., see further, J. Wellhausen, De Gent. et Famil. Judaeorum (1869); S. Cook, Critical Notes on O.T. History, Index, s.v.; E. Meyer, Israeliten, pp. 400 sqq.; and the commentaries on Chronicles (q.v.).

(S. A. C.)

CALEDON (1) a town of the Cape Province, 81 m. by rail E.S.E. of Cape Town. Pop. (1904) 3508. The town is 15 m. N. of the sea at Walker Bay and is built on a spur of the Zwartberg, 800 ft. high. The streets are lined with blue gums and oaks. From the early day of Dutch settlement at the Cape Caledon has been noted for the curative value of its mineral springs, which yield 150,000 gallons daily. There are seven springs, six with a natural temperature of 120° F., the seventh being cold. The district is rich in flowering heaths and everlasting flowers. The name Caledon was given to the town and district in honour of the 2nd earl of Caledon, governor of the Cape 1807-1811. (2) A river of South Africa, tributary to the Orange (q.v.), also named after Lord Caledon.

CALEDONIA, the Roman name of North Britain, still used especially in poetry for Scotland. It occurs first in the poet Lucan (A.D. 64), and then often in Roman literature. There were (1) a district Caledonia, of which the southern border must have been on or near the isthmus between the Clyde and the Forth, (2) a Caledonian Forest (possibly in Perthshire), and (3) a tribe of Caledones or Calidones, named by the geographer Ptolemy as living within boundaries which are now unascertainable. The Romans first invaded Caledonia under Agricola (about A.D. 83). They then fortified the Forth and Clyde Isthmus with a line of forts, two of which, those at Camelon and Barhill, have been identified and excavated, penetrated into Perthshire, and fought the decisive battle of the war (according to Tacitus) on the slopes of Mons Graupius.^[1] The site—quite as hotly contested among antiquaries as between Roman and Caledonian—may have been near the Roman encampment of Inchtuthill (in the policies of Delvine, 10 m. N. of Perth near the union of Tay and Isla), which is the most northerly of the ascertained Roman encampments in Scotland and seems to belong to the age of Agricola. Tacitus represents the result as a victory. The home government, whether averse to expensive conquests of barren hills, or afraid of a victorious general, abruptly recalled Agricola, and his northern conquests—all beyond the Tweed, if not all beyond Cheviot—were abandoned. The next advance followed more than fifty years later. About A.D. 140 the district up to the Firth of Forth was definitely annexed, and a rampart with forts along it, the Wall of Antoninus Pius, was drawn from sea to sea (see Britain: Roman; and Graham's Dyke). At the same time the Roman forts at Ardoch, north of Dunblane, Carpow near Abernethy, and perhaps one or two more, were occupied. But the conquest was stubbornly disputed, and after several risings, the land north of Cheviot seems to have been lost about A.D. 180-185. About A.D. 208 the emperor Septimius Severus carried out an extensive punitive expedition against the northern tribes, but while it is doubtful how far he penetrated, it is certain that after his death the Roman writ never again ran north of Cheviot. Rome is said, indeed, to have recovered the whole land up to the Wall of Pius in A.D. 368 and to have established there a province, Valentia. A province with that name was certainly organized somewhere. But its site and extent is quite uncertain and its duration was exceedingly brief. Throughout, Scotland remained substantially untouched by Roman influences, and its Celtic art, though perhaps influenced by Irish, remained free from Mediterranean infusion. Even in the south of Scotland, where Rome ruled for half a century (A.D. 142-180), the occupation was military and produced no civilizing effects. Of the actual condition of the land during the period of Roman rule in Britain, we have yet to learn the details by excavation. The curious carvings and ramparts, at Burghead on the coast of Elgin, and the underground stone houses locally called "wheems," in which Roman fragments have been found, may represent the native forms of dwelling, &c., and some of the "Late Celtic" metal-work may belong to this age. But of the political divisions, the boundaries and capitals of the tribes, and the like, we know nothing. Ptolemy gives a list of tribe and place-names. But hardly one can be identified with any approach to certainty, except in the extreme south. Nor has any certainty been reached about the ethnological problems of the population, the Aryan or non-Aryan character of the Picts and the like. That the Caledonians, like the later Scots, sometimes sought their fortunes in the south, is proved by a curious tablet of about A.D. 220, found at Colchester, dedicated to an unknown equivalent of Mars, Medocius, by one "Lossio Veda, nepos [ = kin of] Vepogeni, Caledo." The name Caledonia is said to survive in the second syllable of Dunkeld and in the mountain name Schiehallion (Sith-chaillinn).

Authorities.—Tacitus, Agricola; Hist. Augusta, Vita Severi; Dio lxxvi.; F. Haverfield, The Antonine Wall Report (Glasgow, 1899), pp. 154-168; J. Rhys, Celtic Britain (ed. 3). On Burghead, see H.W. Young, Proc. of Scottish Antiq. xxv., xxvii.; J. Macdonald, Trans. Glasgow Arch. Society. The Roman remains of Scotland are described in Rob. Stuart's Caled. Romana (Edinburgh, 1852), the volumes of the Scottish Antiq. Society, the Corpus Inscriptionum Latinarum, vol. vii., and elsewhere.

(F. J. H.)

CALEDONIAN CANAL. The chain of fresh-water lakes—Lochs Ness, Oich and Lochy—which stretch along the line of the Great Glen of Scotland in a S.W. direction from Inverness early suggested the idea of connecting the east and west coasts of Scotland by a canal which would save ships about 400 m. of coasting voyage round the north of Great Britain through the stormy Pentland Firth. In 1773 James Watt was employed by the government to make a survey for such a canal, which again was the subject of an official report by Thomas Telford in 1801. In 1803 an act of parliament was passed authorizing the construction of the canal, which was begun forthwith under Telford's direction, and traffic was started in 1822. From the northern entrance on Beauly Firth to the southern, near Fort William, the total length is about 60 m., that of the artificial portion being about 22 m. The number of locks is 28, and their standard dimensions are:—length 160 ft, breadth 38 ft., water-depth 15 ft. Their lift is in general about 8 ft., but some of them are for regulating purposes only. A flight of 8 at Corpach, with a total lift of 64 ft., is known as "Neptune's Staircase." The navigation is vested in and managed by the commissioners of the Caledonian Canal, of whom the speaker of the House of Commons is ex officio chairman. Usually the income is between £7000 and £8000 annually, and exceeds the expenditure by a few hundred pounds; but the commissioners are not entitled to make a profit, and the credit balances, though sometimes allowed to accumulate, must be expended on renewals and improvements of the canal. They have not, however, always proved sufficient for their purposes, and parliament is occasionally called upon to make special grants. In the commissioners is also vested the Crinan Canal, which extends from Ardrishaig on Loch Gilp to Crinan on Loch Crinan. This canal was made by a company incorporated by act of parliament in 1793, and was opened for traffic in 1801. At various times it received grants of public money, and ultimately in respect of these it passed into the hands of the government. In 1848 it was vested by parliament in the commissioners of the Caledonian Canal (who had in fact administered it for many years previously); the act contained a proviso that the company might take back the undertaking on repayment of the debt within 20 years, but the power was not exercised. The length of the canal is 9 m., and it saves vessels sailing from the Clyde a distance of about 85 m. as compared with the alternative route round the Mull of Kintyre. Its highest reach is 64 ft. above sea level, and its locks, 15 in number, are 96 ft. long, by 24 ft. wide, the depth of water being such as to admit vessels up to a draught of 9½ ft. The revenue is over £6000 a year, and there is usually a small credit balance which, as with the Caledonian Canal, must be applied to the purposes of the undertaking.

CALENBERG, or Kalenberg, the name of a district, including the town of Hanover, which was formerly part of the duchy of Brunswick. It received its name from a castle near Schulenburg, and is traversed by the rivers Weser and Leine, its area being about 1050 sq. m. The district was given to various cadets of the ruling house of Brunswick, one of these being Ernest Augustus, afterwards elector of Hanover, and the ancestor of the Hanoverian kings of Great Britain and Ireland.

CALENDAR, so called from the Roman Calends or Kalends, a method of distributing time into certain periods adapted to the purposes of civil life, as hours, days, weeks, months, years, &c.

Of all the periods marked out by the motions of the celestial bodies, the most conspicuous, and the most intimately connected with the affairs of mankind, are the solar day, which is distinguished by the diurnal revolution of the earth and the alternation of light and darkness, and the solar year, which completes the circle of the seasons. But in the early ages of the world, when mankind were chiefly engaged in rural occupations, the phases of the moon must have been objects of great attention and interest,—hence the month, and the practice adopted by many nations of reckoning time by the motions of the moon, as well as the still more general practice of combining lunar with solar periods. The solar day, the solar year, and the lunar month, or lunation, may therefore be called the natural divisions of time. All others, as the hour, the week, and the civil month, though of the most ancient and general use, are only arbitrary and conventional.

Day.—The subdivision of the day (q.v.) into twenty-four parts, or hours, has prevailed since the remotest ages, though different nations have not agreed either with respect to the epoch of its commencement or the manner of distributing the hours. Europeans in general, like the ancient Egyptians, place the commencement of the civil day at midnight, and reckon twelve morning hours from midnight to midday, and twelve evening hours from midday to midnight. Astronomers, after the example of Ptolemy, regard the day as commencing with the sun's culmination, or noon, and find it most convenient for the purposes of computation to reckon through the whole twenty-four hours. Hipparchus reckoned the twenty-four hours from midnight to midnight. Some nations, as the ancient Chaldeans and the modern Greeks, have chosen sunrise for the commencement of the day; others, again, as the Italians and Bohemians, suppose it to commence at sunset. In all these cases the beginning of the day varies with the seasons at all places not under the equator. In the early ages of Rome, and even down to the middle of the 5th century after the foundation of the city, no other divisions of the day were known than sunrise, sunset, and midday, which was marked by the arrival of the sun between the Rostra and a place called Graecostasis, where ambassadors from Greece and other countries used to stand. The Greeks divided the natural day and night into twelve equal parts each, and the hours thus formed were denominated temporary hours, from their varying in length according to the seasons of the year. The hours of the day and night were of course only equal at the time of the equinoxes. The whole period of day and night they called νυχθήμερον.

Week.—The week is a period of seven days, having no reference whatever to the celestial motions,—a circumstance to which it owes its unalterable uniformity. Although it did not enter into the calendar of the Greeks, and was not introduced at Rome till after the reign of Theodosius, it has been employed from time immemorial in almost all eastern countries; and as it forms neither an aliquot part of the year nor of the lunar month, those who reject the Mosaic recital will be at a loss, as Delambre remarks, to assign it to an origin having much semblance of probability. It might have been suggested by the phases of the moon, or by the number of the planets known in ancient times, an origin which is rendered more probable from the names universally given to the different days of which it is composed. In the Egyptian astronomy, the order of the planets, beginning with the most remote, is Saturn, Jupiter, Mars, the Sun, Venus, Mercury, the Moon. Now, the day being divided into twenty-four hours, each hour was consecrated to a particular planet, namely, one to Saturn, the following to Jupiter, the third to Mars, and so on according to the above order; and the day received the name of the planet which presided over its first hour. If, then, the first hour of a day was consecrated to Saturn, that planet would also have the 8th, the 15th, and the 22nd hour; the 23rd would fall to Jupiter, the 24th to Mars, and the 25th, or the first hour of the second day, would belong to the Sun. In like manner the first hour of the 3rd day would fall to the Moon, the first of the 4th day to Mars, of the 5th to Mercury, of the 6th to Jupiter, and of the 7th to Venus. The cycle being completed, the first hour of the 8th day would return to Saturn, and all the others succeed in the same order. According to Dio Cassius, the Egyptian week commenced with Saturday. On their flight from Egypt, the Jews, from hatred to their ancient oppressors, made Saturday the last day of the week.

The English names of the days are derived from the Saxon. The ancient Saxons had borrowed the week from some Eastern nation, and substituted the names of their own divinities for those of the gods of Greece. In legislative and justiciary acts the Latin names are still retained.

Latin. English. Saxon.

Dies Solis. Sunday. Sun's day.

Dies Lunae. Monday. Moon's day.

Dies Martis. Tuesday. Tiw's day.

Dies Mercurii. Wednesday. Woden's day.

Dies Jovis. Thursday. Thor's day.

Dies Veneris. Friday. Frigg's day.

Dies Saturni. Saturday. Seterne's day.

Month.—Long before the exact length of the year was determined, it must have been perceived that the synodic revolution of the moon is accomplished in about 29½ days. Twelve lunations, therefore, form a period of 354 days, which differs only by about 11¼ days from the solar year. From this circumstance has arisen the practice, perhaps universal, of dividing the year into twelve months. But in the course of a few years the accumulated difference between the solar year and twelve lunar months would become considerable, and have the effect of transporting the commencement of the year to a different season. The difficulties that arose in attempting to avoid this inconvenience induced some nations to abandon the moon altogether, and regulate their year by the course of the sun. The month, however, being a convenient period of time, has retained its place in the calendars of all nations; but, instead of denoting a synodic revolution of the moon, it is usually employed to denote an arbitrary number of days approaching to the twelfth part of a solar year.

Among the ancient Egyptians the month consisted of thirty days invariably; and in order to complete the year, five days were added at the end, called supplementary days. They made use of no intercalation, and by losing a fourth of a day every year, the commencement of the year went back one day in every period of four years, and consequently made a revolution of the seasons in 1461 years. Hence 1461 Egyptian years are equal to 1460 Julian years of 365¼ days each. This year is called vague, by reason of its commencing sometimes at one season of the year, and sometimes at another.

The Greeks divided the month into three decades, or periods of ten days,—a practice which was imitated by the French in their unsuccessful attempt to introduce a new calendar at the period of the Revolution. This division offers two advantages: the first is, that the period is an exact measure of the month of thirty days; and the second is, that the number of the day of the decade is connected with and suggests the number of the day of the month. For example, the 5th of the decade must necessarily be the 5th, the 15th, or the 25th of the month; so that when the day of the decade is known, that of the month can scarcely be mistaken. In reckoning by weeks, it is necessary to keep in mind the day of the week on which each month begins.

The Romans employed a division of the month and a method of reckoning the days which appear not a little extraordinary, and must, in practice, have been exceedingly inconvenient. As frequent allusion is made by classical writers to this embarrassing method of computation, which is carefully retained in the ecclesiastical calendar, we here give a table showing the correspondence of the Roman months with those of modern Europe.

Instead of distinguishing the days by the ordinal numbers first, second, third, &c., the Romans counted backwards from three fixed epochs, namely, the Calends, the Nones and the Ides. The Calends (or Kalends) were invariably the first day of the month, and were so denominated because it had been an ancient custom of the pontiffs to call the people together on that day, to apprize them of the festivals, or days that were to be kept sacred during the month. The Ides (from an obsolete verb iduare, to divide) were at the middle of the month, either the 13th or the 15th day; and the Nones were the ninth day before the Ides, counting inclusively. From these three terms the days received their denomination in the following manner:—Those which were comprised between the Calends and the Nones were called the days before the Nones; those between the Nones and the Ides were called the days before the Ides; and, lastly, all the days after the Ides to the end of the month were called the days before the Calends of the succeeding month. In the months of March, May, July and October, the Ides fell on the 15th day, and the Nones consequently on the 7th; so that each of these months had six days named from the Nones. In all the other months the Ides were on the 13th and the Nones on the 5th; consequently there were only four days named from the Nones. Every month had eight days named from the Ides. The number of days receiving their denomination from the Calends depended on the number of days in the month and the day on which the Ides fell. For example, if the month contained 31 days and the Ides fell on the 13th, as was the case in January, August and December, there would remain 18 days after the Ides, which, added to the first of the following month, made 19 days of Calends. In January, therefore, the 14th day of the month was called the nineteenth before the Calends of February (counting inclusively), the 15th was the 18th before the Calends and so on to the 30th, which was called the third before the Calend (tertio Calendas), the last being the second of the Calends, or the day before the Calends (pridie Calendas).

Days of the Month. March. May. July. October. January. August. December. April. June. September. November. February.

1 Calendae. Calendae. Calendae. Calendae.

2 6 4 4 4

3 5 3 3 3

4 4 Prid. Nonas. Prid. Nonas. Prid. Nonas.

5 3 Nonae. Nonae. Nonae.

6 Prid. Nonas. 8 8 8

7 Nonae. 7 7 7

8 8 6 6 6

9 7 5 5 5

10 6 4 4 4

11 5 3 3 3

12 4 Prid. Idus. Prid. Idus. Prid. Idus.

13 3 Idus. Idus. Idus.

14 Prid. Idus. 19 18 16

15 Idus. 18 17 15

16 17 17 16 14

17 16 16 15 13

18 15 15 14 12

19 14 14 13 11

20 13 13 12 10

21 12 12 11 9

22 11 11 10 8

23 10 10 9 7

24 9 9 8 6

25 8 8 7 5

26 7 7 6 4

27 6 6 5 3

28 5 5 4 Prid. Calen.

29 4 4 3 Mart.

30 3 3 Prid. Calen.

31 Prid. Calen. Prid. Calen.

Year.—The year is either astronomical or civil. The solar astronomical year is the period of time in which the earth performs a revolution in its orbit about the sun, or passes from any point of the ecliptic to the same point again; and consists of 365 days 5 hours 48 min. and 46 sec. of mean solar time. The civil year is that which is employed in chronology, and varies among different nations, both in respect of the season at which it commences and of its subdivisions. When regard is had to the sun's motion alone, the regulation of the year, and the distribution of the days into months, may be effected without much trouble; but the difficulty is greatly increased when it is sought to reconcile solar and lunar periods, or to make the subdivisions of the year depend on the moon, and at the same time to preserve the correspondence between the whole year and the seasons.

Of the Solar Year.—In the arrangement of the civil year, two objects are sought to be accomplished,—first, the equable distribution of the days among twelve months; and secondly, the preservation of the beginning of the year at the same distance from the solstices or equinoxes. Now, as the year consists of 365 days and a fraction, and 365 is a number not divisible by 12, it is impossible that the months can all be of the same length and at the same time include all the days of the year. By reason also of the fractional excess of the length of the year above 365 days, it likewise happens that the years cannot all contain the same number of days if the epoch of their commencement remains fixed; for the day and the civil year must necessarily be considered as beginning at the same instant; and therefore the extra hours cannot be included in the year till they have accumulated to a whole day. As soon as this has taken place, an additional day must be given to the year.

The civil calendar of all European countries has been borrowed from that of the Romans. Romulus is said to have divided the year into ten months only, including in all 304 days, and it is not very well known how the remaining days were disposed of. The ancient Roman year commenced with March, as is indicated by the names September, October, November, December, which the last four months still retain. July and August, likewise, were anciently denominated Quintilis and Sextilis, their present appellations having been bestowed in compliment to Julius Caesar and Augustus. In the reign of Numa two months were added to the year, January at the beginning and February at the end; and this arrangement continued till the year 452 B.C., when the Decemvirs changed the order of the months, and placed February after January. The months now consisted of twenty-nine and thirty days alternately, to correspond with the synodic revolution of the moon, so that the year contained 354 days; but a day was added to make the number odd, which was considered more fortunate, and the year therefore consisted of 355 days. This differed from the solar year by ten whole days and a fraction; but, to restore the coincidence, Numa ordered an additional or intercalary month to be inserted every second year between the 23rd and 24th of February, consisting of twenty-two and twenty-three days alternately, so that four years contained 1465 days, and the mean length of the year was consequently 366¼ days. The additional month was called Mercedinus or Mercedonius, from merces, wages, probably because the wages of workmen and domestics were usually paid at this season of the year. According to the above arrangement, the year was too long by one day, which rendered another correction necessary. As the error amounted to twenty-four days in as many years, it was ordered that every third period of eight years, instead of containing four intercalary months, amounting in all to ninety days, should contain only three of those months, consisting of twenty-two days each. The mean length of the year was thus reduced to 365¼ days; but it is not certain at what time the octennial periods, borrowed from the Greeks, were introduced into the Roman calendar, or whether they were at any time strictly followed. It does not even appear that the length of the intercalary month was regulated by any certain principle, for a discretionary power was left with the pontiffs, to whom the care of the calendar was committed, to intercalate more or fewer days according as the year was found to differ more or less from the celestial motions. This power was quickly abused to serve political objects, and the calendar consequently thrown into confusion. By giving a greater or less number of days to the intercalary month, the pontiffs were enabled to prolong the term of a magistracy or hasten the annual elections; and so little care had been taken to regulate the year, that, at the time of Julius Caesar, the civil equinox differed from the astronomical by three months, so that the winter months were carried back into autumn and the autumnal into summer.

In order to put an end to the disorders arising from the negligence or ignorance of the pontiffs, Caesar abolished the use of the lunar year and the intercalary month, and regulated the civil year entirely by the sun. With the advice and assistance of Sosigenes, he fixed the mean length of the year at 365¼ days, and decreed that every fourth year should have 366 days, the other years having each 365. In order to restore the vernal equinox to the 25th of March, the place it occupied in the time of Numa, he ordered two extraordinary months to be inserted between November and December in the current year, the first to consist of thirty-three, and the second of thirty-four days. The intercalary month of twenty-three days fell into the year of course, so that the ancient year of 355 days received an augmentation of ninety days; and the year on that occasion contained in all 445 days. This was called the last year of confusion. The first Julian year commenced with the 1st of January of the 46th before the birth of Christ, and the 708th from the foundation of the city.

In the distribution of the days through the several months, Caesar adopted a simpler and more commodious arrangement than that which has since prevailed. He had ordered that the first, third, fifth, seventh, ninth and eleventh months, that is January, March, May, July, September and November, should have each thirty-one days, and the other months thirty, excepting February, which in common years should have only twenty-nine, but every fourth year thirty days. This order was interrupted to gratify the vanity of Augustus, by giving the month bearing his name as many days as July, which was named after the first Caesar. A day was accordingly taken from February and given to August; and in order that three months of thirty-one days might not come together, September and November were reduced to thirty days, and thirty-one given to October and December. For so frivolous a reason was the regulation of Caesar abandoned, and a capricious arrangement introduced, which it requires some attention to remember.

The additional day which occured every fourth year was given to February, as being the shortest month, and was inserted in the calendar between the 24th and 25th day. February having then twenty-nine days, the 25th was the 6th of the calends of March, sexto calendas; the preceding, which was the additional or intercalary day, was called bis-sexto calendas,—hence the term bissextile, which is still employed to distinguish the year of 366 days. The English denomination of leap-year would have been more appropriate if that year had differed from common years in defect, and contained only 364 days. In the modern calendar the intercalary day is still added to February, not, however, between the 24th and 25th, but as the 29th.

The regulations of Caesar were not at first sufficiently understood; and the pontiffs, by intercalating every third year instead of every fourth, at the end of thirty-six years had intercalated twelve times, instead of nine. This mistake having been discovered, Augustus ordered that all the years from the thirty-seventh of the era to the forty-eighth inclusive should be common years, by which means the intercalations were reduced to the proper number of twelve in forty-eight years. No account is taken of this blunder in chronology; and it is tacitly supposed that the calendar has been correctly followed from its commencement.

Although the Julian method of intercalation is perhaps the most convenient that could be adopted, yet, as it supposes the year too long by 11 minutes 14 seconds, it could not without correction very long answer the purpose for which it was devised, namely, that of preserving always the same interval of time between the commencement of the year and the equinox. Sosigenes could scarcely fail to know that this year was too long; for it had been shown long before, by the observations of Hipparchus, that the excess of 365¼ days above a true solar year would amount to a day in 300 years. The real error is indeed more than double of this, and amounts to a day in 128 years; but in the time of Caesar the length of the year was an astronomical element not very well determined. In the course of a few centuries, however, the equinox sensibly retrograded towards the beginning of the year. When the Julian calendar was introduced, the equinox fell on the 25th of March. At the time of the council of Nice, which was held in 325, it fell on the 21st; and when the reformation of the calendar was made in 1582, it had retrograded to the 11th. In order to restore the equinox to its former place, Pope Gregory XIII. directed ten days to be suppressed in the calendar; and as the error of the Julian intercalation was now found to amount to three days in 400 years, he ordered the intercalations to be omitted on all the centenary years excepting those which are multiples of 400. According to the Gregorian rule of intercalation, therefore, every year of which the number is divisible by four without a remainder is a leap year, excepting the centurial years, which are only leap years when divisible by four after omitting the two ciphers. Thus 1600 was a leap year, but 1700, 1800 and 1900 are common years; 2000 will be a leap year, and so on.

As the Gregorian method of intercalation has been adopted in all Christian countries, Russia excepted, it becomes interesting to examine with what degrees of accuracy it reconciles the civil with the solar year. According to the best determinations of modern astronomy (Le Verrier's Solar Tables, Paris, 1858, p. 102), the mean geocentric motion of the sun in longitude, from the mean equinox during a Julian year of 365.25 days, the same being brought up to the present date, is 360° + 27″.685. Thus the mean length of the solar year is found to be 360°/(360° + 27".685) × 365.25 = 365.2422 days, or 365 days 5 hours 48 min. 46 sec. Now the Gregorian rule gives 97 intercalations in 400 years; 400 years therefore contain 365 × 400 + 97, that is, 146,097 days; and consequently one year contains 365.2425 days, or 365 days 5 hours 49 min. 12 sec. This exceeds the true solar year by 26 seconds, which amount to a day in 3323 years. It is perhaps unnecessary to make any formal provision against an error which can only happen after so long a period of time; but as 3323 differs little from 4000, it has been proposed to correct the Gregorian rule by making the year 4000 and all its multiples common years. With this correction the rule of intercalation is as follows:—

Every year the number of which is divisible by 4 is a leap year, excepting the last year of each century, which is a leap year only when the number of the century is divisible by 4; but 4000, and its multiples, 8000, 12,000, 16,000, &c. are common years. Thus the uniformity of the intercalation, by continuing to depend on the number four, is preserved, and by adopting the last correction the commencement of the year would not vary more than a day from its present place in two hundred centuries.

In order to discover whether the coincidence of the civil and solar year could not be restored in shorter periods by a different method of intercalation, we may proceed as follows:—The fraction 0.2422, which expresses the excess of the solar year above a whole number of days, being converted into a continued fraction, becomes

1

4 + 1

7 + 1

1 + 1

3 + 1

4 + 1

1 + , &c.

which gives the series of approximating fractions,

1 4,7 29,8 33,31 128,132 545,163 673, &c.

The first of these, 1/4, gives the Julian intercalation of one day in four years, and is considerably too great. It supposes the year to contain 365 days 6 hours.

The second, 7/29, gives seven intercalary days in twenty-nine years, and errs in defect, as it supposes a year of 365 days 5 hours 47 min. 35 sec.

The third, 8/33, gives eight intercalations in thirty-three years or seven successive intercalations at the end of four years respectively, and the eighth at the end of five years. This supposes the year to contain 365 days 5 hours 49 min. 5.45 sec.

The fourth fraction, 31/128 = (24 + 7) / (99 + 29) = (3 × 8 + 7) / (3 × 33 + 29) combines three periods of thirty-three years with one of twenty-nine, and would consequently be very convenient in application. It supposes the year to consist of 365 days 5 hours 48 min. 45 sec., and is practically exact.

The fraction 8/33 offers a convenient and very accurate method of intercalation. It implies a year differing in excess from the true year only by 19.45 sec., while the Gregorian year is too long by 26 sec. It produces a much nearer coincidence between the civil and solar years than the Gregorian method; and, by reason of its shortness of period, confines the evagations of the mean equinox from the true within much narrower limits. It has been stated by Scaliger, Weidler, Montucla, and others, that the modern Persians actually follow this method, and intercalate eight days in thirty-three years. The statement has, however, been contested on good authority; and it seems proved (see Delambre, Astronomie Moderne, tom. i. p.81) that the Persian intercalation combines the two periods 7/29 and 8/33. If they follow the combination (7 + 3 × 8) / (29 + 3 × 33) = 31/128 their determination of the length of the tropical year has been extremely exact. The discovery of the period of thirty-three years is ascribed to Omar Khayyam, one of the eight astronomers appointed by Jelāl ud-Din Malik Shah, sultan of Khorasan, to reform or construct a calendar, about the year 1079 of our era.

If the commencement of the year, instead of being retained at the same place in the seasons by a uniform method of intercalation, were made to depend on astronomical phenomena, the intercalations would succeed each other in an irregular manner, sometimes after four years and sometimes after five; and it would occasionally, though rarely indeed, happen, that it would be impossible to determine the day on which the year ought to begin. In the calendar, for example, which was attempted to be introduced in France in 1793, the beginning of the year was fixed at midnight preceding the day in which the true autumnal equinox falls. But supposing the instant of the sun's entering into the sign Libra to be very near midnight, the small errors of the solar tables might render it doubtful to which day the equinox really belonged; and it would be in vain to have recourse to observation to obviate the difficulty. It is therefore infinitely more commodious to determine the commencement of the year by a fixed rule of intercalation; and of the various methods which might be employed, no one perhaps is on the whole more easy of application, or better adapted for the purpose of computation, than the Gregorian now in use. But a system of 31 intercalations in 128 years would be by far the most perfect as regards mathematical accuracy. Its adoption upon our present Gregorian calendar would only require the suppression of the usual bissextile once in every 128 years, and there would be no necessity for any further correction, as the error is so insignificant that it would not amount to a day in 100,000 years.

Of the Lunar Year and Luni-solar Periods.—The lunar year, consisting of twelve lunar months, contains only 354 days; its commencement consequently anticipates that of the solar year by eleven days, and passes through the whole circle of the seasons in about thirty-four lunar years. It is therefore so obviously ill-adapted to the computation of time, that, excepting the modern Jews and Mahommedans, almost all nations who have regulated their months by the moon have employed some method of intercalation by means of which the beginning of the year is retained at nearly the same fixed place in the seasons.

In the early ages of Greece the year was regulated entirely by the moon. Solon divided the year into twelve months, consisting alternately of twenty-nine and thirty days, the former of which were called deficient months, and the latter full months. The lunar year, therefore, contained 354 days, falling short of the exact time of twelve lunations by about 8.8 hours. The first expedient adopted to reconcile the lunar and solar years seems to have been the addition of a month of thirty days to every second year. Two lunar years would thus contain 25 months, or 738 days, while two solar years, of 365¼ days each, contain 730½ days. The difference of 7½ days was still too great to escape observation; it was accordingly proposed by Cleostratus of Tenedos, who flourished shortly after the time of Thales, to omit the biennary intercalation every eighth year. In fact, the 7½ days by which two lunar years exceeded two solar years, amounted to thirty days, or a full month, in eight years. By inserting, therefore, three additional months instead of four in every period of eight years, the coincidence between the solar and lunar year would have been exactly restored if the latter had contained only 354 days, inasmuch as the period contains 354 × 8 + 3 × 30 = 2922 days, corresponding with eight solar years of 365¼ days each. But the true time of 99 lunations is 2923.528 days, which exceeds the above period by 1.528 days, or thirty-six hours and a few minutes. At the end of two periods, or sixteen years, the excess is three days, and at the end of 160 years, thirty days. It was therefore proposed to employ a period of 160 years, in which one of the intercalary months should be omitted; but as this period was too long to be of any practical use, it was never generally adopted. The common practice was to make occasional corrections as they became necessary, in order to preserve the relation between the octennial period and the state of the heavens; but these corrections being left to the care of incompetent persons, the calendar soon fell into great disorder, and no certain rule was followed till a new division of the year was proposed by Meton and Euctemon, which was immediately adopted in all the states and dependencies of Greece.

The mean motion of the moon in longitude, from the mean equinox, during a Julian year of 365.25 days (according to Hansen's Tables de la Lune, London, 1857, pages 15, 16) is, at the present date, 13 × 360° + 477644″.409; that of the sun being 360° + 27″.685. Thus the corresponding relative mean geocentric motion of the moon from the sun is 12 × 360° + 477616″.724; and the duration of the mean synodic revolution of the moon, or lunar month, is therefore 360° / (12 × 360° + 477616″.724) × 365.25 = 29.530588 days, or 29 days, 12 hours, 44 min. 2.8 sec.

The Metonic Cycle, which may be regarded as the chef-d'œuvre of ancient astronomy, is a period of nineteen solar years, after which the new moons again happen on the same days of the year. In nineteen solar years there are 235 lunations, a number which, on being divided by nineteen, gives twelve lunations for each year, with seven of a remainder, to be distributed among the years of the period. The period of Meton, therefore, consisted of twelve years containing twelve months each, and seven years containing thirteen months each; and these last formed the third, fifth, eighth, eleventh, thirteenth, sixteenth, and nineteenth years of the cycle. As it had now been discovered that the exact length of the lunation is a little more than twenty-nine and a half days, it became necessary to abandon the alternate succession of full and deficient months; and, in order to preserve a more accurate correspondence between the civil month and the lunation, Meton divided the cycle into 125 full months of thirty days, and 110 deficient months of twenty-nine days each. The number of days in the period was therefore 6940. In order to distribute the deficient months through the period in the most equable manner, the whole period may be regarded as consisting of 235 full months of thirty days, or of 7050 days, from which 110 days are to be deducted. This gives one day to be suppressed in sixty-four; so that if we suppose the months to contain each thirty days, and then omit every sixty-fourth day in reckoning from the beginning of the period, those months in which the omission takes place will, of course, be the deficient months.

The number of days in the period being known, it is easy to ascertain its accuracy both in respect of the solar and lunar motions. The exact length of nineteen solar years is 19 × 365.2422 = 6939.6018 days, or 6939 days 14 hours 26.592 minutes; hence the period, which is exactly 6940 days, exceeds nineteen revolutions of the sun by nine and a half hours nearly. On the other hand, the exact time of a synodic revolution of the moon is 29.530588 days; 235 lunations, therefore, contain 235 × 29.530588 = 6939.68818 days, or 6939 days 16 hours 31 minutes, so that the period exceeds 235 lunations by only seven and a half hours.

After the Metonic cycle had been in use about a century, a correction was proposed by Calippus. At the end of four cycles, or seventy-six years, the accumulation of the seven and a half hours of difference between the cycle and 235 lunations amounts to thirty hours, or one whole day and six hours. Calippus, therefore, proposed to quadruple the period of Meton, and deduct one day at the end of that time by changing one of the full months into a deficient month. The period of Calippus, therefore, consisted of three Metonic cycles of 6940 days each, and a period of 6939 days; and its error in respect of the moon, consequently, amounted only to six hours, or to one day in 304 years. This period exceeds seventy-six true solar years by fourteen hours and a quarter nearly, but coincides exactly with seventy-six Julian years; and in the time of Calippus the length of the solar year was almost universally supposed to be exactly 365¼ days. The Calippic period is frequently referred to as a date by Ptolemy.

Ecclesiastical Calendar.—The ecclesiastical calendar, which is adopted in all the Catholic, and most of the Protestant countries of Europe, is luni-solar, being regulated partly by the solar, and partly by the lunar year,—a circumstance which gives rise to the distinction between the movable and immovable feasts. So early as the 2nd century of our era, great disputes had arisen among the Christians respecting the proper time of celebrating Easter, which governs all the other movable feasts. The Jews celebrated their passover on the 14th day of the first month, that is to say, the lunar month of which the fourteenth day either falls on, or next follows, the day of the vernal equinox. Most Christian sects agreed that Easter should be celebrated on a Sunday. Others followed the example of the Jews, and adhered to the 14th of the moon; but these, as usually happened to the minority, were accounted heretics, and received the appellation of Quartodecimans. In order to terminate dissensions, which produced both scandal and schism in the church, the council of Nicaea, which was held in the year 325, ordained that the celebration of Easter should thenceforth always take place on the Sunday which immediately follows the full moon that happens upon, or next after, the day of the vernal equinox. Should the 14th of the moon, which is regarded as the day of full moon, happen on a Sunday, the celebration Of Easter was deferred to the Sunday following, in order to avoid concurrence with the Jews and the above-mentioned heretics. The observance of this rule renders it necessary to reconcile three periods which have no common measure, namely, the week, the lunar month, and the solar year; and as this can only be done approximately, and within certain limits, the determination of Easter is an affair of considerable nicety and complication. It is to be regretted that the reverend fathers who formed the council of Nicaea did not abandon the moon altogether, and appoint the first or second Sunday of April for the celebration of the Easter festival. The ecclesiastical calendar would in that case have possessed all the simplicity and uniformity of the civil calendar, which only requires the adjustment of the civil to the solar year; but they were probably not sufficiently versed in astronomy to be aware of the practical difficulties which their regulation had to encounter.

Dominical Letter.—The first problem which the construction of the calendar presents is to connect the week with the year, or to find the day of the week corresponding to a given day of any year of the era. As the number of days in the week and the number in the year are prime to one another, two successive years cannot begin with the same day; for if a common year begins, for example, with Sunday, the following year will begin with Monday, and if a leap year begins with Sunday, the year following will begin with Tuesday. For the sake of greater generality, the days of the week are denoted by the first seven letters of the alphabet, A, B, C, D, E, F, G, which are placed in the calendar beside the days of the year, so that A stands opposite the first day of January, B opposite the second, and so on to G, which stands opposite the seventh; after which A returns to the eighth, and so on through the 365 days of the year. Now if one of the days of the week, Sunday for example, is represented by E, Monday will be represented by F, Tuesday by G, Wednesday by A, and so on; and every Sunday through the year will have the same character E, every Monday F, and so with regard to the rest. The letter which denotes Sunday is called the Dominical Letter, or the Sunday Letter; and when the dominical letter of the year is known, the letters which respectively correspond to the other days of the week become known at the same time.

Solar Cycle.—In the Julian calendar the dominical letters are readily found by means of a short cycle, in which they recut in the same order without interruption. The number of years in the intercalary period being four, and the days of the week being seven, their product is 4 × 7 = 28; twenty-eight years is therefore a period which includes all the possible combinations of the days of the week with the commencement of the year. This period is called the Solar Cycle, or the Cycle of the Sun, and restores the first day of the year to the same day of the week. At the end of the cycle the dominical letters return again in the same order on the same days of the month; hence a table of dominical letters, constructed for twenty-eight years, will serve to show the dominical letter of any given year from the commencement of the era to the Reformation. The cycle, though probably not invented before the time of the council of Nicaea, is regarded as having commenced nine years before the era, so that the year one was the tenth of the solar cycle. To find the year of the cycle, we have therefore the following rule:—Add nine to the date, divide the sum by twenty-eight; the quotient is the number of cycles elapsed, and the remainder is the year of the cycle. Should there be no remainder, the proposed year is the twenty-eighth or last of the cycle. This rule is conveniently expressed by the formula ((x + 9) / 28)_r, in which x denotes the date, and the symbol r denotes that the remainder, which arises from the division of x + 9 by 28, is the number required. Thus, for 1840, we have (1840 + 9) / 28 = 66-1/28; therefore ((1840 + 9) / 28)_r = 1, and the year 1840 is the first of the solar cycle. In order to make use of the solar cycle in finding the dominical letter, it is necessary to know that the first year of the Christian era began with Saturday. The dominical letter of that year, which was the tenth of the cycle, was consequently B. The following year, or the 11th of the cycle, the letter was A; then G. The fourth year was bissextile, and the dominical letters were F, E; the following year D, and so on. In this manner it is easy to find the dominical letter belonging to each of the twenty-eight years of the cycle. But at the end of a century the order is interrupted in the Gregorian calendar by the secular suppression of the leap year; hence the cycle can only be employed during a century. In the reformed calendar the intercalary period is four hundred years, which number being multiplied by seven, gives two thousand eight hundred years as the interval in which the coincidence is restored between the days of the year and the days of the week. This long period, however, may be reduced to four hundred years; for since the dominical letter goes back five places every four years, its variation in four hundred years, in the Julian calendar, was five hundred places, which is equivalent to only three places (for five hundred divided by seven leaves three); but the Gregorian calendar suppresses exactly three intercalations in four hundred years, so that after four hundred years the dominical letters must again return in the same order. Hence the following table of dominical letters for four hundred years will serve to show the dominical letter of any year in the Gregorian calendar for ever. It contains four columns of letters, each column serving for a century. In order to find the column from which the letter in any given case is to be taken, strike off the last two figures of the date, divide the preceding figures by four, and the remainder will indicate the column. The symbol X, employed in the formula at the top of the column, denotes the number of centuries, that is, the figures remaining after the last two have been struck off. For example, required the dominical letter of the year 1839? In this case X = 18, therefore (X/4)_r = 2; and in the second column of letters, opposite 39, in the table we find F, which is the letter of the proposed year.

It deserves to be remarked, that as the dominical letter of the first year of the era was B, the first column of the following table will give the dominical letter of every year from the commencement of the era to the Reformation. For this purpose divide the date by 28, and the letter opposite the remainder, in the first column of figures, is the dominical letter of the year. For example, supposing the date to be 1148. On dividing by 28, the remainder is 0, or 28; and opposite 28, in the first column of letters, we find D, C, the dominical letters of the year 1148.

Lunar Cycle and Golden Number.—In connecting the lunar month with the solar year, the framers of the ecclesiastical calendar adopted the period of Meton, or lunar cycle, which they supposed to be exact. A different arrangement has, however, been followed with respect to the distribution of the months. The lunations are supposed to consist of twenty-nine and thirty days alternately, or the lunar year of 354 days; and in order to make up nineteen solar years, six embolismic or intercalary months, of thirty days each, are introduced in the course of the cycle, and one of twenty-nine days is added at the end. This gives 19 × 354 + 6 × 30 + 29 = 6935 days, to be distributed among 235 lunar months. But every leap year one day must be added to the lunar month in which the 29th of February is included. Now if leap year happens on the first, second or third year of the period, there will be five leap years in the period, but only four when the first leap year falls on the fourth. In the former case the number of days in the period becomes 6940 and in the latter 6939. The mean length of the cycle is therefore 6939¾ days, agreeing exactly with nineteen Julian years.

Table I.—Dominical Letters.

Years of the Century. X 4_r = 1 X 4_r = 2 X 4_r = 3 X 4_r = 0

0 C E G B, A

1 29 57 85 B D F G

2 30 58 86 A C E F

3 31 59 87 G B D E

4 32 60 88 F, E A, G C, B D, C

5 33 61 89 D F A B

6 34 62 90 C E G A

7 35 63 91 B D F G

8 36 64 92 A, G C, B E, D F, E

9 37 65 93 F A C D

10 38 66 94 E G B C

11 39 67 95 D F A B

12 40 68 96 C, B E, D G, F A, G

13 41 69 97 A C E F

14 42 70 98 G B D E

15 43 71 99 F A C D

16 44 72 E, D G, F B, A C, B

17 45 73 C E G A

18 46 74 B D F G

19 47 75 A C E F

20 48 76 G, F B, A D, C E, D

21 49 77 E G B C

22 50 78 D F A B

23 51 79 C E G A

24 52 80 B, A D, C F, E G, F

25 53 81 G B D E

26 54 82 F A C D

27 55 83 E G B C

28 56 84 D, C F, E A, G B, A

Table II.—The Day of the Week.

Month. Dominical Letter.

Jan. Oct. A B C D E F G

Feb. Mar. Nov. D E F G A B C

April July G A B C D E F

May B C D E F G A

June E F G A B C D

August C D E F G A B

Sept. Dec. F G A B C D E

1 8 15 22 29 Sun. Sat Frid. Thur. Wed. Tues Mon.

2 9 16 23 30 Mon. Sun. Sat. Frid. Thur. Wed. Tues.

3 10 17 24 31 Tues. Mon. Sun. Sat. Frid. Thur. Wed.

4 11 18 25 Wed. Tues. Mon. Sun. Sat. Frid. Thur.

5 12 19 26 Thur. Wed. Tues. Mon. Sun. Sat. Frid.

6 13 20 27 Frid. Thur. Wed. Tues. Mon. Sun. Sat.

7 14 21 28 Sat. Frid. Thur. Wed. Tues. Mon. Sun.

By means of the lunar cycle the new moons of the calendar were indicated before the Reformation. As the cycle restores these phenomena to the same days of the civil month, they will fall on the same days in any two years which occupy the same place in the cycle; consequently a table of the moon's phases for 19 years will serve for any year whatever when we know its number in the cycle. This number is called the Golden Number, either because it was so termed by the Greeks, or because it was usual to mark it with red letters in the calendar. The Golden Numbers were introduced into the calendar about the year 530, but disposed as they would have been if they had been inserted at the time of the council of Nicaea. The cycle is supposed to commence with the year in which the new moon falls on the 1st of January, which took place the year preceding the commencement of our era. Hence, to find the Golden Number N, for any year x, we have N = ((x + 1) / 19)_r, which gives the following rule: Add 1 to the date, divide the sum by 19; the quotient is the number of cycles elapsed, and the remainder is the Golden Number. When the remainder is 0, the proposed year is of course the last or 19th of the cycle. It ought to be remarked that the new moons, determined in this manner, may differ from the astronomical new moons sometimes as much as two days. The reason is that the sum of the solar and lunar inequalities, which are compensated in the whole period, may amount in certain cases to 10°, and thereby cause the new moon to arrive on the second day before or after its mean time.

Dionysian Period.—The cycle of the sun brings back the days of the month to the same day of the week; the lunar cycle restores the new moons to the same day of the month; therefore 28 × 19 = 532 years, includes all the variations in respect of the new moons and the dominical letters, and is consequently a period after which the new moons again occur on the same day of the month and the same day of the week. This is called the Dionysian or Great Paschal Period, from its having been employed by Dionysius Exiguus, familiarly styled "Denys the Little," in determining Easter Sunday. It was, however, first proposed by Victorius of Aquitain, who had been appointed by Pope Hilary to revise and correct the church calendar. Hence it is also called the Victorian Period. It continued in use till the Gregorian reformation.

Cycle of Indiction.—Besides the solar and lunar cycles, there is a third of 15 years, called the cycle of indiction, frequently employed in the computations of chronologists. This period is not astronomical, like the two former, but has reference to certain judicial acts which took place at stated epochs under the Greek emperors. Its commencement is referred to the 1st of January of the year 313 of the common era. By extending it backwards, it will be found that the first of the era was the fourth of the cycle of indiction. The number of any year in this cycle will therefore be given by the formula ((x + 3) / 15)_r, that is to say, add 3 to the date, divide the sum by 15, and the remainder is the year of the indiction. When the remainder is 0, the proposed year is the fifteenth of the cycle.

Julian Period.—The Julian period, proposed by the celebrated Joseph Scaliger as an universal measure of chronology, is formed by taking the continued product of the three cycles of the sun, of the moon, and of the indiction, and is consequently 28 × 19 × 15 = 7980 years. In the course of this long period no two years can be expressed by the same numbers in all the three cycles. Hence, when the number of any proposed year in each of the cycles is known, its number in the Julian period can be determined by the resolution of a very simple problem of the indeterminate analysis. It is unnecessary, however, in the present case to exhibit the general solution of the problem, because when the number in the period corresponding to any one year in the era has been ascertained, it is easy to establish the correspondence for all other years, without having again recourse to the direct solution of the problem. We shall therefore find the number of the Julian period corresponding to the first of our era.

We have already seen that the year 1 of the era had 10 for its number in the solar cycle, 2 in the lunar cycle, and 4 in the cycle of indiction; the question is therefore to find a number such, that when it is divided by the three numbers 28, 19, and 15 respectively the three remainders shall be 10, 2, and 4.

Let x, y, and z be the three quotients of the divisions; the number sought will then be expressed by 28 x + 10, by 19 y + 2, or by 15 z + 4. Hence the two equations

To solve the equations 28 x + 10 = 19 y + 2, or y = x + 9 x + 8 19, let m = 9 x + 8 19, we have then x = 2 m + m - 8 9.

Let m - 8 9= m′; then m = 9 m′ + 8; hence

Again, since 28 x + 10 = 15 z + 4, we have

15 z = 28 x + 6, or z = 2 x - 2 x - 6 15.

Let 2 x - 6 15= n; then 2 x = 15 n + 6, and x = 7 n + 3 + n 2.

Let n 2= n′; then n = 2 n′; consequently

Equating the above two values of x, we have

15 n′ + 3 = 19 m′ + 16; whence n′ = m′ + 4 m′ + 13 15.

Let 4 m′ + 13 15= p; we have then

4 m′ = 15 p - 13, and m′ = 4 p - p + 13 4.

Let p + 13 4= p′; then p = 4 p′ - 13;

Now in this equation p′ may be any number whatever, provided 15 p′ exceed 52. The smallest value of p′ (which is the one here wanted) is therefore 4; for 15 × 4 = 60. Assuming therefore p′ = 4, we have m′ = 60 - 52 = 8; and consequently, since x = 19 m′ + 16, x = 19 × 8 + 16 = 168. The number required is consequently 28 × 168 + 10 = 4714.