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Rh investiture than Henry IV. had been. Several attempts at settlement failed. In February 1111 legates of Paschal II. met Henry V. at Sutri and declared that the pope was ready to surrender all the temporalities that had been bestowed on the clergy since the days of Charlemagne in return for freedom of election and the abolition of lay investiture. Henry, having agreed to the proposal, entered Rome to receive his crown. The bishops and clergy who were present at the coronation protested against this surrender, and a tumult arising, the ceremony had to be abandoned. The king then seized pope and curia and left the city. After two months of close confinement Paschal consented to an unqualified renunciation on his part of the right of investiture. In the following year, however, a Lateran council repudiated this compact as due to violence, and a synod held at Vienne with papal approval declared lay investiture to be heresy and placed Henry under the ban. The struggle was complicated throughout its course by political and other considerations; there were repeated rebellions of German nobles, constant strife between rival imperial and papal factions in the Lombard cities and at Rome, and creation of several anti-popes, of whom Guibert of Ravenna (Clement III.) and Gregory VIII. were the most important. Final settlement of the struggle was retarded, moreover, by the question of the succession to the lands of the great Countess Matilda, who had bequeathed all her property to the Holy See, Henry claiming the estates as suzerain of the fiefs and as heir of the allodial lands. The efforts of Gelasius II. to settle the strife by a general council were rendered fruitless by his death (1119).

At length in 1122 the struggle was brought to an end by the concordat of Worms, the provisions of which were incorporated in the eighth and ninth canons of the general Lateran council of 1123. The settlement was a compromise. The emperor, on the one hand, preserved feudal suzerainty over ecclesiastical benefices; but, on the other, he ceased to confer ring and crozier, and thereby not only lost the right of refusing the elect on the grounds of unworthiness, but also was deprived of an efficacious means of maintaining vacancies in ecclesiastical offices. Few efforts were made to undo the compromise. King Lothair the Saxon demanded of Innocent II. the renewal of lay investiture as reward for driving the antipope Anacletus from Rome, but the opposition of St Bernard and the German prelates was so potent that the king dropped his demand, and Innocent in 1133 confirmed the concordat. In fact, the imperial control over the election of bishops in Germany came later to be much curtailed in practice, partly by the tacitly changed relations between the empire and its feudatories, partly by explicit concessions wrung at various times from individual emperors, such as Otto IV. in 1209 and Frederick II. in 1213; but the principles of the concordat of Worms continued theoretically to regulate the tenure of bishoprics and abbacies until the dissolution of the empire on 1806.

In France the course of the struggle was somewhat different. As in the empire, the king and the nobles, each within his own sphere of influence, claimed the right of investing with ring and crozier and of exacting homage and oaths of fealty. The struggle, however, was less bitter chiefly because France was not a united country, and it was eventually terminated without formal treaty. The king voluntarily abandoned lay investiture and the claim to homage during the pontificate of Paschal II., but continued to interfere with elections, to appropriate the revenues of vacant benefices, and to exact an oath of fealty before admitting the elect to the enjoyment of his temporalities. Most of the great feudal lords followed the king’s example, but their concessions varied considerably, and in the south of France some of the bishops were still doing homage for their sees until the closing years of the 13th century; but long before then the right of investing with ring and crozier had disappeared from every part of France.

England was the scene of an investiture contest in which the chief actors were Henry I. and Anselm. The archbishop, in obedience to the decrees of Gregory VII. and Urban II., not only refused to perform homage to the king (1100), but also refused to consecrate newly-chosen bishops who had received investiture from Henry. The dispute was bitter, but was carried on without any of the violence which characterized the conflict between papacy and empire; and it ended in a compromise which closely foreshadowed the provisions of the concordat of Worms and received the confirmation of Paschal II. in 1106. Freedom of election, somewhat similar in form to that which still exists, was formally conceded under Stephen, and confirmed by John in Magna Carta.

Many documents relating to the investiture struggle have been edited by E. Dümmler in Monumenta Germaniae historica, Libelli de ''lite imperatorum et pontificum saeculis xi. et xii.'' (3 vols., 1891–1897), See Ducange, Glossarium, s.v. “Investitura.”

On investiture in the empire consult C. Mirbt, Die Publizistik im Zeitalter Gregors VII. (Leipzig, 1894); E. Bernheim, Das Wormser Konkordat (Breslau, 1906); R. Boerger, Die Belehnungen der deutschen geistlichen Fürsten (Leipzig, 1901); K. E. Benz, Die Stellung der Bischöfe von Meissen, Merseburg und Naumburg im ''Investiturstreite unter Heinrich IV. und Heinrich V.'' (Dresden, 1899); W. Martens, Gregor VII., sein Leben und Wirken (2 vols., Leipzig, 1894); P. Fisher, The Medieval Empire, c. 10 (London, 1898). For France, see P. Imbart de la Tour, Les Élections épiscopales dans l’église de France du XI&#8202;e au XII&#8202;e siècle (Paris, 1891); A. Luchaire, Histoire des institutions monarchiques de la France sous les premiers Capétiens 987–1180 (2nd ed., Paris, 1891); P. Viollet, Histoire des institutions politiques et administratives de la France (Paris, 1898); Ibach, Der Kampf zwischen Papsttum und Königtum ''von Gregor VII. bis Calixto II.'' (Frankfort, 1884). For England, see J. F. Böhmer, Kirche und Staat in England und in der Normandie in ''XI. und XII. Jahrhundert (Leipzig, 1899); E. A. Freeman, The'' ''Reign of William II. Rufus and the Accession of Henry I. (London, 1882); H. W. C. Davis, England under the Normans and Angevins'' (London, 1905).

INVOICE (originally a plural, Invoyes or Invoys, of Invoy, a variant of “envoy,” from the French envoyer, to send), a statement giving full particulars of goods sent or shipped by a trader to a customer, with the quantity, quality and prices, and the charges upon them. Consular invoices, i.e. invoices signed at the port of shipment by a consul of the country to which the goods are being consigned, are generally demanded by those countries which impose ad valorem duties.

 INVOLUTION (Lat. involvere, to roll up), a rolling up or complication. In arithmetic, involution is the operation of raising a quantity to any power; it is the converse of evolution, which is the operation of extracting any root of a quantity (see ; ). In geometry, an involution is a one-to-one correspondence between two ranges of points or between two pencils (see : Projective). The “involute” of a curve may be regarded as the locus of the extremity of a string when it is unwrapped from the curve (see ).

 IO, in Greek mythology, daughter of Inachus, the river-god of Argos and its first king. As associated with the oldest worship of Hera she is called the daughter of Peiren, who made the first image of that goddess out of a pear-tree at Tiryns; and under the name of Callithyia Io was regarded as the first priestess of Hera. Zeus fell in love with her, and, to protect her from the wrath of Hera, changed her into a white heifer (Apollodorus ii. 1; Hyginus, Fab. 145; Ovid, Metam. i. 568–733); according to Aeschylus (Supplices, 299) the metamorphosis was the work of Hera herself. Hera, having persuaded Zeus to give her the heifer, set Argus Panoptes to watch her. Zeus thereupon sent Hermes, who lulled Argus to sleep and cut off his head with the sword with which Perseus afterwards slew the Gorgon. In another account Argus is killed by a stone thrown by Hermes. But the wrath of Hera still pursued Io. Maddened by a gadfly sent by the goddess she wandered all over the earth, swam the strait known on this account as the Bosporus (Ox-ford), and crossed the Ionian sea (traditionally called after her) until at last she reached Egypt, where she was restored to her original form and became the mother of Epaphus. Accounts of her wanderings (differing considerably in detail) are given in the Supplices and Prometheus Vinctus of Aeschylus. Various interpretations are given of the latter part of her story, which dates from the 7th century, when intercourse was frequent between Greece and Egypt, and when much influence was 