Algebra is a kind of generalized arithmetic, in which numbers or quantities and operations, often also the results of operations, are represented by symbols. Algebra is an invaluable instrument in intricate calculations of all kinds, and enables operations to be performed and results obtained that by arithmetic would be impossible, and its scope is still being extended.
The beginnings of algebraic method are to be found in Diophantus, a Greek of the fourth century of our era, but it was the Arabians that introduced algebra to Europe and from them it received its name. The first Arabian treatise on algebra was published in the reign of the great Kaliph Al Mamun (813-833) by Mohammed Ben Musa. In 1202 Leonardo Fibonacci of Pisa, who had travelled and studied in the East, published a work treating of algebra as then understood in the Arabian school. From this time to the discovery of printing considerable attention was given to algebra, and the work of Ben Musa and another Arabian treatise, called the Rule of Algebra, were translated into Italian.
The first printed work treating on algebra (also on arithmetic, etc) appeared at Venice in 1494, the author being a monk called Luca Pacioli da Bergo. Rapid progress now began to be made, and among the names of those to whom advances are to be attributed are Tarfcaglia and Cardan. About the middle of the sixteenth century the German Stifel introduced the plus, minus and square root symbols, and Recorde the equals sign. Recorde wrote the first English work on algebra. Francois Vieta, a French mathematician (1540-1603), first adopted the method which has led to so great an extension of modern algebra, by being the first who used general symbols for known quantities as well as for unknown. It was he also who first made the application of algebra to geometry.
Albert Girard extended the theory of equations by the supposition of imaginary quantities. The Englishman Harriot, early in the seventeenth century, discovered negative roots, and established the equality between the number of roots and the units in the degree of the equation. He also invented the less than and greater than signs, and Oughthred that of the x multiplication symbol. Descartes, though not the first to apply algebra to geometry, has, by the extent and importance of his applications, commonly acquired the credit of being so. The same discoveries have also been attributed to him as to Harriot, and their respective claims have caused much controversy. He obtained by means of algebra the definition and description of curves. Since his time algebra has been applied so widely in geometry and higher mathematics that we need only mention the names of Fermat, Wallis, Newton, Leibnitz, De Moivre, MacLaurin, Taylor, Euler, D'Alembert, Lagrange, Laplace, Fourier, Poisson, Gauss, Horner, De Morgan, Sylvester, Cayley. Boole, Jevons, and others have applied the algebraic method not only to formal logic but to political economy. Research Algebra More information about Algebra
The Byzantine Empire, the Eastern Roman Empire, so called from its capital Byzantium or Constantinople was founded in 395 AD when Theodosius at his death divided the Roman Empire between his sons Arcadius and Honorius. In this empire the Greek language and civilization were prevalent; but the rulers claimed still to be Roman emperors, and under their sway the laws and official forms of Rome were maintained. It lasted for about a thousand years after the downfall of the Western Empire.
The Eastern Empire, then comprising Asia Minor, Syria, Egypt, Greece, Thrace, Moesia, Macedonia, and Crete, fell to Theodosius's elder son Arcadius, through whose weakness and that of several of his immediate successors it suffered severely from the encroachments of Huns, Goths, Bulgarians, and Persians. In 527 the celebrated Justinian succeeded,, whose reign is famous for the codification of Roman law, and the victories of his generals Belisarius and Narses over the Vandals in Africa, and the Goths in Italy, which was henceforth governed for the Eastern Empire by an exarch residing at Ravenna. But his energy could not revive the decaying strength of the empire, and Justin II his successor, a weak and avaricious prince, lost his reason by the reverses encountered in his conflicts with plundering Lombards, Avars, and Persians.
Tiberius, a captain of the guard, succeeded in 578, and in 582 Mauricius; both were men of ability. In 602 Phocas, proclaimed emperor by the army, succeeded, and produced by his incapacity tlie greatest disorder in the empire. Heraclius, son of the governor of Africa, who headed a conspiracy, conquered Constantinople, and caused Phocas to be executed in 610. He was an excellent general, and finally succeeded in repressing the Avars and recovering the provinces lost to the Persians, whose power indeed he overthrew. But a far more dangerous enemy to the Byzantine empire now appeared in the Muslim power, founded amongst the Arabians by Mohammed and the caliphs, which gradually extended its conquests over Phoenicia, the countries on the Euphrates, Judea, Syria, and Egypt from 635-641.
The empire was in sore straits when Leo the Isaurian (Leo III), general of the army of the East, mounted the throne in 716, and a new period of comparative prosperity began. Some writers date the beginning of the Byzantine Empire proper, and the end of the Eastern Roman Empire, from this era.
Numerous reforms, civil and military, were now introduced, and the worship of images was prohibited. Leo repelled the Arabians or Saracens from Constantinople, but allowed the Lombards to seize the Italian provinces, while the Arabians plundered the Eastern ones. Constantine V in 741 recovered part of Syria and Armenia from the Arabians; and the struggle was carried on not unsuccessfully by his son Leo IV. Under his grandson, Constantine VI, Irene, the ambitious mother of the latter, raised a large faction by the restoration of image worship, and, in conjunction with her paramour Stauratius, deposed her son, and had his eyes put out in 797.
A revolt of the patricians placed one of their order, Nicephorus, on the throne, who fell in the war against the Bulgarians in 811. Stauratius, Michael, Leo V and Michael II in 820 ascended the throne in rapid succession. During the reign of the latter the Arabians conquered Sicily, Lower Italy, Crete, and other countries. The long dispute as to image-worship was brought to a close in 842, when the practice was finally sanctioned at the council of Nicaea, under Michael III.
He was put to death by Basil the Macedonian, who came to the throne as Basil I in 867, and whose reign formed a period of great glory in the history of the Byzantine Empire. He founded a dynasty (the Macedonian) which lasted until 1056. Among the greatest of his successors were Nicephorus II (Phocas), and John Zimisces in 969, who carried on successful wars against the Muslims, Bulgarians, and Russians.
Basil II succeeded this prince in 976. He vanquished the Bulgarians and the Arabians. His brother, Constantine IX was succeeded by Romanus III in 1028, who married Zoe, daughter of Constantine. This dissolute but able princess caused her husband to be executed, and successively raised to the throne Michael IV, Michael V, and Constantine X. Russians and Muslims meanwhile devastated the empire. Her sister Theodora succeeded her on the throne in 1054.
After the short reign of Michael VI from 1054 until 1057 Isaac Comnenus, the first of the Comnenian dynasty, ascended the throne, but soon after became a monk. The three chief emperors of this dynasty were Alexius, John, and Manuel Comnenus. During the reign of Alexius I from 1081 to 1118 the Crusades commenced. His son, John II, and grandson, Manuel I, fought with success against the Turks, whose progress also was considerably checked by the Crusades. The Latins, the name given to the French, Venetian, etc, crusaders, now forced their way to Constantinople in 1204, conquered the city, and retained it, together with most of the European territories of the empire.
Baldwin, count of Flanders, was made emperor; Boniface, marquis of Montferrat, obtained Thessalonica as a kingdom, and the Venetians acquired a large extent of territory. Theodore Lascaris seized on the Asiatic provinces, in 1206 made Nice (Nicaea) the capital of the empire, and was at first more powerful than Baldwin. Neither Baldwin nor his successors, Henry, Peter, and Robert of Courtenay, were able to secure the tottering throne. John, emperor of Nice, conquered all the remaining Byzantine territory except Constantinople, and at last, in 1261, Michael Palaeologus, king of Nice, conquered Constantinople, and thus overthrew the Latin dynasty.
Thus again the vast but exhausted Byzantine Empire was united under Michael Palaeologus, founder of the last Byzantine dynasty. Internal disturbances and wars with the Turks disturbed the reigns of his descendants Andronicus II and Andronicus III. For a time the Cantacuzenes shared the crown with John Palasologus, son of Andronicus III; but in 1355 John again became sole emperor. In his reign the Turks first obtained a firm footing in Europe, and conquered Gallipoli in 1357. In 1361 Sultan Amurath took Adrianople. Bajazet conquered almost all the European provinces except Constantinople, and was pressing it hard when Timur's invasion of the Turkish provinces saved Constantinople for this time in 1402. Manuel then recovered his throne, and regained some of the lost provinces from the contending sons of Bajazet. To him succeeded his son John, Palaeologus II whom Amurath II stripped of all his territories except Constantinople, and laid under tribute in 1444.
To the Emperor John succeeded his brother Constantine Palaeologus. With the assistance of his general Giustiniani, a Genoese, he withstood the superior forces of the enemy with fruitless courage, and fell in the defence of Constantinople, by the conquest of which on May the 29th, 1453 Mohammed II put an end to the Greek or Byzantine Empire. The Byzantine Empire, which thus lasted for over a thousand years, stemmed the tide of the advance of Islam and instead spread Christianity and maintained a regular system of government, law, and policy in the midst of surrounding conflicting systems. Research Byzantine Empire
Gesta Romanorum ('Deeds of the Romans'), was the usual title of a collection of short tales, legends, etc, written in Latin, which was very popular during the middle ages. The book was probably written about the close of the 13th century by a certain monk Elinandus, an Englishman or a German. The separate tales making up the Gesta are of very various contents, and belong to different times and countries, the sources from which they are derived being partly classical, partly oriental, and partly western. Whatever may have been the intention of the original compiler, they very soon were adapted to the moralizing tendencies of the time, and moral reflections and allegorical interpretations were added to them, it is said, by a Petrus Bercorius or Pierre Bercaire of Poitou, a Benedictine prior. After the Reformation the book fell into oblivion. Research Gesta Romanorum
In Christian theology, Pelagianism is a rationalistic and naturalistic heretical doctrine concerning grace and morals, which emphasises human free will as the decisive element in human perfectibility and minimises or denies the need for divine grace and redemption. The doctrine was formulated by the Romano-British monk Pelagius. Research Pelagianism
Tonsure is the religious practice of having the head shaved before entering the priesthood or becoming a monk. Until 1973 in the Roman Catholic Church, the crown was shaved (leaving a surrounding fringe to resemble Jesus' crown of thorns); in the Eastern Orthodox Church the hair is merely shorn close. For Buddhist monks, the entire head is shaved except for a topknot. Research Tonsure
Albertus Magnus (Albert the Great) was Count of Bollstadt and a distinguished German scholar of the thirteenth century. He was born in 1193 and died in 1280. He studied at Padua, became a monk of the Dominican order, teaching in the schools of Hildeslioini, Ratisbon, and Cologne, where Thomas Aquinas became his pupil. In 1245 he went to Paris and publicly expounded the doctrines of Aristotle, notwithstanding the prohibition of the church. He became rector of the school of Cologne in 1249; in 1254 he was made provincial of his order in Germany; and in 1260 he received from Pope Alexander IV the appointment of Bishop of Ratisbon. In 1263 he retired to his convent at Cologne, where he composed many works, especially commentaries on Aristotle. Owing to his profound knowledge he did not escape the imputation of using magical arts and trafficking with the 'Evil One'. Research Albertus Magnus
Alessandro Gavazzi was a popular Italian preacher and religious reformer. He was born in 1909 at and died in 1889. At the age of fifteen he became a monk of the Barnabite order, at twenty he was professor of rhetoric in the College of Naples, and soon after made his mark as a pulpit orator. In 1846 he was chaplain-general of the Roman patriotic league. Subsequently he threw off his papal allegiance and joined the agitation which ended in the short-lived republic. The French occupation of Rome drove him into exile, when he travelled through Britain and America lecturing against the Church of Rome, his power as an orator evoking much enthusiasm. He was with Garibaldi in 1860, and made subsequent visits to Britain gathering funds for the Free Italian Church, in the interests of which he lectured, preached, and travelled on deputation work until his death. Research Alessandro Gavazzi
Alexander Barclay was a British poet. He was born about 1475 probably in Scotland and died in 1552. For some years he was a priest and chaplain of St Mary Ottery, in Devonshire, afterwards he was a Benedictinemonk of Ely, subsequently a Franciscan, and latterly the holder of one or two livings. His principal work was a satire, entitled The Shyp of Folys of this Worlde, part translation and part imitation of Brandt's Narrenschiff (Ship of Fools), and printed by Pynson in 1509. He also wrote a Myrrour of Good Manors, and some Egloges (Eclogues), both printed by Pynson, as well as translations, etc. Research Alexander Barclay
Alfonso IV (Alfonso The Monk) was king of Leon and Asturias (Spain). He died in 933. Known as Alfonso The Monk because he abdicated to become a monk, he tried to reclaim his throne but chaos ensued. Research Alfonso IV
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