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SERAPHIM, ORDER OF THE. The oldest Swedish order, also called the Blue Ribbon. Its foundation is ascribed to Magnus Ladulås in 1260, and it was renewed by Frederick I. in 1748. The decoration, worn on a blue ribbon, consists of an eight-pointed cross with seraphs' heads and patriarchal crosses, bearing the letters JHS with three Swedish crowns.
SERA'PIS, or SARAPIS (Lat., from Gk. Σέραπις, Σάραπις). An Egyptian deity worshiped especially at Memphis and at Alexandria. The name is a compound of Osiris and Apis and in its earliest Greek form occurs as Osirapis, of which Serapis (Sarapis) is a corruption. The god, in fact, was the sacred bull Apis (q.v.), who, after his death, became one with Osiris and, under the name of Osiris-Apis (Egyptian Oser-Hapi), was worshiped as a god of the dead. The Sera
peum, or temple of Serapis, at Memphis, enjoyed
the reputation of special holiness and was visited by pilgrims from all parts of Egypt. The Greeks identified Serapis with their Hades, the King of the Underworld, and Ptolemy I. built the famous Serapeum of Alexandria upon the site of an older temple of the Egyptian god. This temple seems to have contained two statues of the god; one said to have come from Sinope, the other, representing the god as Hades with Cerberus, brought from Seleucia. The Alexandrian Serapis was therefore a fusion of the Greek and Egyptian divinities. Under the Romans when the worship of Serapis spread beyond its original territory, he, rather than Osiris, was regarded as the consort of Isis. Con sult: Wiedemann, Religion of the Ancient Egyptians, translated (New York, 1897); Mahaffy, The Empire of the Ptolemies (New York, 1898); id., A History of Egypt Under the Ptolemaic Dynasty (ib., 1899); Milne, A History of Egypt Under Roman Rule (ib., 1898). See SERAPEUM, SERBIAN LANGUAGE AND LITERATURE. See SERVIAN LANGUAGE AND LITERATURE. SERBO-CROATIAN (or SERBO-HORVATIAN) LANGUAGE. The speech of about 8,000,000 people inhabiting the Kingdom of Servia, the Principality of Montenegro, the provinces of Bosnia and Herzegovina, Old Servia (Novibazar Kossovo), Croatia, and Slavonia, the southern part of Hungary proper, Istria, and Dalmatia.
With the Bulgarian and Slovenian it forms the so-called southern group of the family of Slavic languages (q.v.). Among the phonetic peculiarities of Serbo-Croatian are the frequent occurrence of the broad a for the e or o in the other Slavic languages, as Serbo-Croatian otats, 'father,' Russian oshets; the vocalic r, as SerboCroatian srtse, 'heart,' Russian serdtse; the change of 7 into u, when in the middle of a word, as Serbo-Croatian vuk, 'wolf,' Russian volk, and into o when final, as Serbo-Croatian pisao, 'I wrote,' Russian pisal. In morphology, the loss of the dual is almost complete, and the locative of nouns, as well as the supine and present passive participle in verbs, has also disappeared. The accent is entirely free, the Croatian, or Horvatian, generally agreeing with the Russian accentuation, the Servian proper usually following almost rigid laws. The existence of long and short vowels along with a musical pitch accent makes Serbo-Croatian one of the most expressive among the Slavic languages. The characters used vary with the religion prevailing; in the Croatian Catholic
lands, the Roman alphabet is used, while the bulk of the Servians, belonging to the Greek Orthodox Church, use the ancient Kirillitsa (q.v.), modified by Karajitch (q.v.) in the early part of the nineteenth century. Consult: Vymazal, Serbische Grammatik (Brünn, 1882); Budmani, Grammatica della lingua serbo-croata (illirica) (Vienna, 1867); Partchitch, Grammaire de la langue serbo-croate, trans. by Feuvrier (Paris, 1877); Karajitch, Serbisch-deutschlateinisches Wörterbuch (3d ed., Belgrade, 1898); id., Deutsch-serbisches Wörterbuch (Vienna, 1872); Popovitch, Wörterbuch der serbischen und deutschen Sprache (2d ed., Pansova, 188695).
SERENA, sâ-ra'nyȧ, LA. A town of Chile. See LA SERENA.
SERENADE (OF. serenade, Fr. sérénade,
from It. serenata, serenade, from serenare, to
make serene, from sereno, from Lat. serenus, calm, serene). Originally music performed on a calm night; hence a song given under the window of a lady by her lover. The modern serenade (or serenata) is a cyclical composition for full orchestra. It differs from the symphony in the greater number of its movements (5, 6, 7, or more) and in their freer construction.
SERES, sĕr'ěs. A town in the Vilayet of Saloniki, European Turkey, 43 miles northeast of Saloniki (Map: Balkan Peninsula, D 4). It is protected by high walls, and contains a citadel, many handsome villas, and several mosques and churches. It is the centre of the Turkish woolen industry, and exports skins, cotton, wool, and tobacco. Population, 30,000.
SERETH, sĕr'ět. An important affluent of the Danube. It rises as the Great Sereth in the Austrian Crownland of Bukowina, flows southward through almost the whole length of Moldavia, and joins the Danube five miles above Galatz (Map: Balkan Peninsula, F 1). Its principal tributaries are the Little Sereth on the right, and the Suczava, Moldava, and Bistritz on the left. Total length, 291 miles.
SERF (OF., Fr. serf, from Lat. servus, servant, slave; connected with Av. har, to protect). In common usage, an unfree feudal dependent, who occupies a place in the social scale above the slave. The serf was usually a peasant bound to the land which he cultivated and for which he owed service and obedience to the lord in whom the ownership of the land was vested. The serf was frequently the product of the feudal system, and under a feudal organization of society the institution of serfdom, or villeinage, is seen in its most developed form. This article will treat chiefly of serfdom or villeinage as it existed in Western Europe.
The origin and development of villeinage in Western Europe has been a subject of violent dispute among historians. With the decay of the Roman power in the fourth and fifth centuries anarchy became prevalent, and there were many who were compelled to seek the protection of their more powerful neighbors. In return they performed such services as a freeman may perform. This institution was known as the patrocinium, and at first the relation terminated with the death of either party. Some of those who sought protection were also owners of small parcels of land, and such land was frequently
handed over to the more powerful to be received back by the former proprietor as precarium; that is to say, the latter had the usufruct, his protector the ownership. Among the early Germans also there probably existed some such relation between men. In the middle of the nineteenth century it was generally held that the organization of society described in the Germania of Tacitus was that of the free village community, by which is meant that the villages were inhabited by freemen, who held land in common, and who annually distributed the land anew. Various writers, especially Fustel de Coulanges and Seebohm, have attacked this theory, and hold that the manorial system was prevalent in Germany (see MANOR), by which is implied that the peasants held their land from a lord, and in return for the use of the property owed service of some kind or other to the owner. In the Frankish kingdom the German and Roman elements met. Again historians are unable to agree whether the chief elements in the feudalism which developed among the Franks were German or Roman or even Celtic. It suffices, however, to state that by the tenth century there were few free peasants or artisans left in what is now France. Probably the institutions of patrocinium and precarium had been joined together, and after some further development we have serfdom as it existed in France with comparatively slight changes until abolished by the Revolution of 1789. (See FEUDALISM.) In regard to his general condition the French serf may be taken as typical.
The relationship which in France bound the serf to the lord had at first been merely a contract between the two persons in question. The general tendency, however, was toward the establishment of the principle of inheritance, and by the end of the eleventh century son inherited from father in nearly all cases. Still the laws and customs which regulated the relationship between the serf and his lord varied greatly at different periods, and in the different provinces of France, as well as in the rest of Europe. Moreover, the dividing line between the serf and the slave on the one hand and the serf and the freeman on the other is not always very clear. In general, a serf was distinguished from the slave in that he had a definite piece of land for his own use, and was protected to some extent even against his lord by fixed customs. He was distinguished from the free peasant proprietor in that he could not leave his lord without the latter's consent, and was subject to some exactions from which the freeman was exempt. The chief burdens of the serf were: (1) The census, or rent, which, "though estimated in money, was usually paid in the form of a large percentage of the crop, what remained over being nominally the property of the serf." (2) The capitagium, or census capitis, which was an annual poll-tax. (3) The taille, or arbitrary tax, which permitted the owner to demand money of the serf whenever he chose. Besides these three taxes the serf had to work on the lord's domain several days in each week. This was the corvée. Also, since the lord's consent was necessary for the serf to marry, permission had usually to be purchased by a fee, known as the formariage. Finally, when the serf died, his heir had to pay a fixed sum known as the mortmain, since according to the legal theory the property really be
longed to the lord and not to the serf, and the latter's heir paid to retain the land.
The question arises, How could the serf become free? In answering this question, it must be noted that at first the serf had little desire to become a freeman. His condition was not much improved thereby, for in the absence of any central authority to which the weak could successfully appeal, the strong could exact from him what they pleased; while, on the other hand, the lord had sufficient interest in his serf to protect him from others. Later, however, at least from the time of Philip Augustus (1180-1223), conditions improved and the weak no longer needed the protection of the nobles in all cases. The lord could bring back his runaway serf, though in some places the theory prevailed that the serf might surrender all his property, both real and movable, to his lord, renounce his bond, and depart. Also some town charters had a clause which declared that an unfree person who came to the town and remained there unclaimed for a year and a day was free. These two methods of emancipation did not meet the demands of improving times, and more regular means developed by which the serf might obtain manumission. The most common came, in time, to be the payment of a fixed sum to the lord, and when the noble was in pressing need of money, as during the Crusades, he sometimes compelled his serfs to buy their freedom.
In recent years an active controversy has been waged concerning villeinage in England. The battle has been fought between the great German and French scholars; between Löbell, Waitz, and Roth on the one hand, and Raynouard, Guérard, and Fustel de Coulanges on the other. In England the scholars were chiefly Germanists. Kemble, Karl Maurer, Freeman, Stubbs, and Gneist held, to all intents and purposes, the same views as Waitz and his school. In general, they believed that the Roman and Celtic civilizations played no rôle in the development of England; that the Anglo-Saxon brought with him his institutions from Germany, such as the free village community or mark. In time, however, "with the growth of population, of inequalities, of social competition, the relations of dependency are seen constantly gaining on the field of freedom," the ceorl becomes a serf, manors arise, and by the time of the Norman Conquest the transformation has been completed. In 1883 Seebohm in his English Village Community declared that there never was a mark system in England, and that "the Saxon invasion did not Roman destroy what it found in the island. villas and their laborers passed from one lord to the other that is all. The ceorls of Saxon times are the direct descendants of Roman slaves and coloni, some of them personally free, but all in agrarian subjection. Indeed, social development is a movement from serfdom to freedom, and the village community of its early stages is connected not with freedom, but with serfdom." Since the appearance of Seebohm's book numerous works have appeared on both sides, and the The condition question is far from settled. of the English serf did not differ essentially from the condition of the French serf. But the English bondsman received valuable privileges much earlier than the French villein. As early as the reign of Edward IV. the serf had the right to plead in the royal courts, a privilege
which the French serf never obtained. Moreover, in England the last known act of enfranchisement took place in the reign of Elizabeth.
In Germany serfdom was generally not of a very harsh kind, though it varied considerably in different parts of the country. In some portions of Prussia, however, peasants were, until 1773, in a state of absolute slavery. Serfdom was abolished in Prussia by the decree of October 9, 1807, which was issued through the influence of Stein and his associates. This declared that from Martinmas, 1810, all persons should be free in the States of Prussia. Subsequent enactments removed the social and property distinctions, which had separated the classes, and gave to every citizen the power to possess in fee simple all kinds of property. This legislation was generally imitated in the other German States. The remains of the German system of serfdom lingered until 1836 in Saxony, and until
1848 in Austria.
In Russia, where the feudal system never prevailed, and the early condition of the peasant was not a servile one, the reduction of the peasantry to a state of serfdom and their attach
ment to the soil were gradually effected, and did not prevail to a very great extent till the close of the sixteenth century. Peter the Great strength ened the attachment of the serf to the soil for fiscal reasons, and under Catharine II. the system reached its highest development, the serf being reduced to so low a level that he differed little, if at all, from a slave. Serfs were regarded by law as a part of the proprietor's working capital, and as such were bought and sold, sometimes with the land, and sometimes without it. The serf had no legal means of self-defense. Alexander I. introduced various improvements in the condition of the peasantry, particularly those belonging to the Crown, and in his reign serfdom was abolished in Courland and Livonia in order
to weaken the power of the German nobles of those districts. The entire abolition of villeinage was effected by Alexander II. (q.v.) by a very sweeping measure. The manifesto of March 3 (February 19), 1861, gave personal freedom to more than twenty millions of serfs.
SERGEANT (OF. sergeant, Fr. sergent, Prov. servent, sirvent, servant, from Lat. serviens, pres. part. of servire, to serve; connected with servus, slave). An important non-commissioned rank in the army; the next rank above that of corporal. Modern conditions demand more intelligence and military training than ever before, and have consequently greatly increased the duties of the grade. In extended movements, the sergeant is frequently compelled to act on his own initiative. In both the United States and the British armies, sergeants are distinguished by three chevrons; in the former they are of the color appropriate to the arm of the service and are worn on both sleeves of the coat. British sergeants wear three gold stripes or chevrons on the left arm only, and wear a silk sash, similar to that worn by the commissioned officer, except that it is worn over the right shoulder. See CHEVRONS; NON-COMMISSIONED OFFICER. SERGEANT-AT-ARMS. In the English the Lord Chancellor with a mace, and executes Court of Chancery, an officer who attends upon various writs of process directed to him, apprehending, for example, persons pronounced in contempt of the court. A similar officer is attached whom the House orders to be arrested. Serto each House of Parliament and arrests those geants-at-arms are also attached to the United States Senate and House of Representatives. They receive a salary of $4500 a year. They are authorized to preserve order in both Houses, and also have charge of the payment of members.
SERGEANT-AT-LAW. See SERJEANT-AT
SERGEANT-FISH (so called from its lateral stripes, which resemble a sergeant's chevrons). A large strong voracious fish (Rachycentron canadus), of the southeastern coast of the United States, related to the mackerels, but superficially resembling a remora. Its habit of lingering about large fishes has led to its being named 'shark's waiting-boy;' and it is also called cobia and crab-eater. It reaches a length of 5 feet, and is olive brown, with obscurely striped sides.
BIBLIOGRAPHY. Waitz, Deutsche Verfassungsgeschichte, vol. i. (3d ed., Berlin, 1880); Brunner, Deutsche Rechtsgeschichte (Leipzig, 1887-92); Fustel de Coulanges, Histoire des institutions politiques de l'ancienne France (Paris, 1890); id., Questions historiques (ib., 1893); Kemble, Saxons in England (new ed., London, 1876); Nasse, Ueber die mittelalterliche Feldgemeinschaft (Bonn., 1869); Seebohm, The English Vil. lage Community (4th ed., London, 1890); Meitzen, Siedelung und Agrarwesen (Berlin, 1895); Maitland, Domesday Book and Beyond (Cambridge, 1897); Vinogradoff, Villainage in England (Oxford, 1892); Hallam, View of the State of Europe During the Middle Ages (11th ed., London, 1855); Knapp, Die Bauernbefreiung und der Ursprung der Landarbeiter in den älteren Teilen Preussens (Leipzig, 1887); id., Die Landarbeiter in Knechtschaft und Freiheit (ib., 1891); Engelmann, Die Leibeigenschaft in Russland (ib., 1884); Leroy-Beaulieu, L'empire des Tsars et les Russes (Eng. trans., New York, 1893); Wallace, Russia (2d ed., ib., 1881); Page, End of Villainage in England (ib., 1900); Sée. Les classes rurales et le régime domanial en France en moyen âge (ib., 1901).
SERGEANTY, GRAND (OF. sergentie, serA jantie, from sergeant, sergeant, servant). species of tenure by which many of the nobility of England held their lands of the King under the feudal system. After the Conquest the land was parceled out among the followers of the Conqueror according to their rank. At that time two species of tenure were introduced: tenure by knight service, consisting of an obligation to perform military service in time of war; and tenure by sergeanty, grand and petit, which involved, in addition to military service, some further service to the King in time of peace. A tenant by grand sergeanty was bound to render some personal service to the King, as to be his standard-bearer, cupbearer, or chamberlain, and to attend Court during certain seasons. Such tenure was also said to be per baroniam; the tenants became known as barons, and were higher in rank than the others. Although originally lands so held could not be divided or alienated, this was quietly done from time to time, and the burdens of the tenure gradually became extinct, and were finally abolished with the military tenures. However, the hereditary privileges and honors, as to be standard-bearer, etc., are still claimed by the great
nobility on great occasions, as coronations. Petty sejeanty was an inferior service, as to render an arrow, or a pair of spurs, etc., to the King annually, and was, therefore, more in the nature of a socage tenure. See TENURE.
SERGEL, ser gel, JOHAN TOBIAS (1740-1814). A Swedish sculptor, born at Stockholm. First a pupil of L'Archevêcque, he studied afterwards in Paris, and after 1767 in Rome, where during a sojourn of twelve years he acquired great reputation. Upon his return to Stockholm, whither he had been summoned by Gustavus III., he was appointed Court sculptor, professor, and in 1810 director of the Academy. The fifteen works of his preserved in the National Museum at Stockholm include a "Faun;" "Cupid and Psyche;" his masterpiece, "Diomedes Stealing the Palladium;" "The Muse of History Recording the Deeds of Gustavus Adolphus," a group of heroic size; and a colossal "Bust of Gustavus III." Besides these the "Monument of Gustavus III." (1808), at the foot of the Slottsbacke (Palace Hill), the "Resurrection," an altar-piece, and the "Monument to Descartes," both in the Adolf-Fredriks Kyrka, should be mentioned. For his biography, consult Nyblom (Upsala, 1877).
SERGI, sérʼjê, GIUSEPPE (1841-). An Italian
anthropologist, born in Messina, Sicily. He was educated at the University of Messina, where afterwards he became an instructor. Later he taught in Milan. In 1880 he was appointed to the chair of anthropology in the University of Bologna; in 1884 he accepted a similar professorship in the Royal University of Rome, and at the same time became director of the Anthropological Institute. He has devoted particular attention to the psychic traits as well as to the physical characters of the peoples of the East: Mediterranean region. His publications treat of archæology, criminal anthropology, and education. His best known works are Elementi di psicologia (1879), Psychologie physiologique (1887), Principi di psicologia (1894), Specie e varietà umane (1900), and The Mediterranean Race (1901), in Italian, English, and German editions.
SERGINSK, ser-gensk', UPPER AND LOWER. Two industrial settlements in the Government of Perm, East Russia, 43 miles west-southwest of Ekaterinburg. They were founded by Demidoff (q.v.) in 1742 and still belong to a private company. Most of the inhabitants are engaged in the extensive iron works and the iron mines in the vicinity. The population of Upper Serginsk is 14,000, and of Lower Serginsk 8,000. The annual production of both towns amounts to over 15,000 tons of pig iron and 26,000 tons of steel.
SER'ĠIUS. The name of four popes. SERGIUS I., SAINT, Pope 687-701. He was born at Palermo of a Syrian family and was ordained priest in 683. On the death of Pope Conon there was a contested election, and both factions finally united on Sergius. He refused to confirm the acts of the Trullan Council (see QUINISEXT), and the Emperor Justinian II. sent officers to Rome to seize him; but the soldiery of the exarchate rallied to his defence, and the Imperial emissary's life was only saved by the Pope's intervention. He consecrated Saint Willibrord, the Apostle of Frisia, and succeeded in terminating the schism in Northern Italy which grew out of the pretensions of the Patriarch of Aquileia.-SERGIUS II., Pope 844-47. He was of a Roman family and became archipresbyter under Gregory IV., whom he succeeded. Lothair I., displeased that he had been consecrated without waiting for Imperial sanction, sent his son Louis with an army to Rome. The Pope and the Roman nobles refused to swear fidelity to Lothair as King of Italy, but recog nized him as Emperor, and Louis was solemnly crowned as King of the Lombards. In 846 Rome was attacked and devastated by Saracen hordes, who were finally driven off by Duke Guido of Spoleto, summoned by the Pope.-SERGIUS III., Pope 904-11. He was a Roman by birth, consecrated Bishop of Care against his will by Formosus in 892 or 893, and elected Pope, on the death of Theodore II. in 897, by the Tuscan faction, but not recognized by the Emperor Lambert, who set up John IX. He returned to Rome in 904, overthrew the Antipope Christopher, and gained possession of the See. His pontificate was troubled, and his own character is said by some ancient writers to have been stained by the prevailing immorality.-SERGIUS IV., Pope 1009-12. He was made Bishop of Albano in 1004. On his election to the Papacy he changed his own name of Peter, being unwilling out of reverence to call himself Peter II. His power was limited in secular matters by the domination in Rome of the patrician John Crescentius and his family.
SERGIPE, sĕr-zhēpe. A maritime State of Brazil, bounded on the north by Alagoas, on the west and south by Bahia, and on the east by the Atlantic (Map: Brazil, K 6). Area, 15,090 square miles. It is the smallest State of the Republic. The coast region is flat and sandy; the interior is a sparsely watered plateau. The climate is hot and dry. The southwestern part affords good grazing land and is the seat of extensive stock-raising. In the eastern portion are cultivated sugar, cacao, tobacco, cotton, and manioc. The chief exports are sugar and rubber, and the centre of the commerce is the capital, Aracajú (q.v.). Population, in 1890, 310,926.
SERI, să'rê. A wild and warlike tribe formerly holding a considerable territory on the west coast of Sonora, Mexico, together with the adjacent island of Tiburon, in the Gulf of California, but now restricted to the island. They are in the lowest culture condition, live in mere brushwood shelters, and shift constantly from place to place. Their ordinary implements are of stone or shell, their weapons being bows, clubs, and stones. The arrows are sometimes poisoned. They wear kilts of pelican skin and paint their faces with elaborate designs. They twist ropes from hair and vegetable fibre, make baskets and rude pottery, and use rafts or balsas woven from reeds. They know the use of the fire drill. Physically they are tall, well made, and of great agility. They seem to be untamably hostile to all aliens, and have no alliance or friendship with any other tribe. On the strength of a short vocabulary obtained by Bartlett in 1852 they were at first classed with the Yuman stock (q.v.), but later study of more adequate material shows that they form a distinct stock, which probably also included the now extinct Tepoca. They were formerly a large tribe, but have been nearly exterminated by the Mexicans. In 1852 they were still estimated at 500, but in 1894 had been
reduced to less than 300.
Seri Indians (Washington, 1899).
Consult McGee, The feet, in the third 4 feet, and so on indefinitely,
SERICITE (from Lat. sericum, silk, from Gk. onpixos, serikos, silky, seric, from p, Ser, Chinaman). A fine scaly variety of muscovite, characterized by a silky lustre. It is found chiefly near Wiesbaden, Germany. See MUSCOVITE.
SERICITE GNEISS, or SERICITE SCHIST. A metamorphic rock, composed essentially of the hydro-micaceous mineral sericite (q.v.) with quartz or quartz and feldspar. In some cases at least sericite gneiss has been produced by the mashing of granite and rhyolite (q.v.) under the action of mountain-building forces.
tion by means of the formula &=
An infinite series in which Sn as n increases indefinitely, has a finite limit is called a convergent series, otherwise a divergent series. A series in which the sum is finite, but takes alternate values as n increases, as in 1-1+1 -1+..., is called an oscillating series.
The ability to determine what series are convergent and to determine the limit of sn evidently conditions the utility of any series for the purpose of pure and applied mathematics. Thus the trigonometric functions sinx =x- +
SERIES (Lat. series, row, succession, from serere, to bind; connected with Gk. elper, eirein, Skt. sā, to bind). In mathematics, a succession of terms formed according to some common law; e.g.: (1) in the series 1, 3, 5, 7,... each term is formed from the preceding by adding 2; (2) in 3, 9, 27, 81,...by multiplying by 3. A series in which each term after the first is formed by adding a constant to the preceding term is called series ex=1+x+ an arithmetic series or progression; e.g. series (1) above. A series in which each term after the first is found by multiplying the preceding term by a constant is called a geometric series or progression; e.g. series (2) above. Any term of an arithmetic series is given by the formula t = a + (n-1)d, in which a is the first term, d the common difference, and n the number of terms.
The sum of ʼn terms is given by the
geometric series the corresponding formulas are
Although the above are the chief series treated in elementary algebra, there is an unlimited number of kinds. E.g. a type to which considerable interest is attached is the arithmetico-geometric series, in which the coefficients are in arithmetic series and the variable in geometric series. E.g. 1 + 2x + 3x2 + ... ( n − 1) x12 + næ11. If the number of terms in a series is unlimited, it is called an infinite series. The general or nth term in such a series and the sum of n terms, n being indefinitely great, may or may not be determinate.
Infinite series in which the values of t and 8 (no) are indeterminate are of little value, but those in which a limit for s can be found are important. Thus in an infinite geomet
x3 x5 3! 51
and the logarith
+ are avail
1 2 3
able for those values only of the variables which render the series convergent.
A knowledge of elementary series is very old, the Pythagoreans (B.C. 550) having treated them Euclid quite comprehensively. (See NUMBER.) (c.300 B.C.) used geometric series, and infinite frequently in the works of Archimedes (c.280 convergent series of the geometric type appear B.C.). Among the Hindus, Aryabhatta, Brahmagupta, and Bhaskara treated arithmetic series, and Bhaskara discussed geometric series. Arabs did little to advance the subject and the Europeans up to the sixteenth century had made no further progress. Saint-Vincent (1584-1667) and Mercator (c.1620-87) developed the series for log(1+x), and Gregory (1668) those for tan -c, sing, cost, secx, cscx. The terms convergent and divergent appear in the writings of Gregory.
The theory of infinite series may be said to begin with Newton and Leibnitz, and to have been further advanced by Euler. In 1812 Gauss published his celebrated memoir on the hypergeometric series (name due to Pfaff), which has since occupied the attention of Jacobi, Kummer, Schwarz, Cayley, Goursat, and numerous others. Cauchy (1821) may be considered the founder of the theory of convergence and divergence of series. advanced the theory of power series by his expansion of a complex function in such a form. Abel was the next important contributor, and he corrected certain of Cauchy's conclusions. General criteria began with Kummer (1835), and have been studied by Eisenstein (1847), Weierstrass in his various contributions to the theory of functions, Dini (1867), Du Bois-Reymond (1873), and many others. Pringsheim's (from 1889) memoirs present the most complete general theory.