intersection of two circles or of a line and a circle is unaltered by inversion. The method obviously affords a ready means for converting theorems involving lines and circles into other propositions involving the same, but differently placed, figures; in mathematical physics it is of special value in solving geometrically electrostatical and optical problems.
INVERURIE, a royal, municipal and police burgh of Aberdeenshire, Scotland, situated at the confluence of the rivers Don and Ury, 16% m. N.W. of Aberdeen by rail, on the Great North of Scotland railway. Pop. (1901) 3624. Paper-making, milling, and the making of mineral waters are the chief manufactures, but the town is an important centre of the cattle trade with London, markets being held at frequent intervals. It also contains the workshops of the Great North of Scotland railway. Inverurie belongs to the Elgin district group of parliamentary burghs. At Harlaw, about 3 m. to the N.W., was fought in 1411 the great battle between Donald, lord of the Isles, and the royal forces under the earl of Mar. Not far from the scene of this conflict stands Balquhain Castle, a seat of the Leslies, now a mere shell, which was occupied by Queen Mary in September 1562 before the fight at Corrichie between her forces, led by the earl of Moray, and those of the earl of Huntly. The granite block from which she is said to have viewed the combat is still called the Queen's Chair or the Maiden Stone. Near Bennachie (1619 ft.) are stone circles and monoliths supposed to be of Druidical origin. There is a branch line from Inverurie to Old Meldrum, 5% m. to the by rail, a market town with a charter dating from 1672, where brewing and distilling are carried on.
INVESTITURE (Late Lat. inveslilura), the formal installation into an office or estate, which constituted in the middle ages one of the acts that betokened the feudal relation between suzerain and vassal. The suzerain, after receiving the vassal's homage and oath of fealty, invested him with his land or office by presenting some symbol, such as a clod, a banner, a branch, or some other object according to the custom of the fief. Otto of Freising says: “ It is customary when a kingdom is delivered over to any one that a sword be given to represent it, and when a province is transferred a standard is given.” As feudal customs grew more stereotyped, the sword and sceptre, emblematic respectively of service and military command and of judicial prerogatives, became the usual embiems of investiture of laymen. The word investiture (from veslire, to put in possession) is later than the 9th century; the thing itself was an outcome of feudal society.
It is in connexion with the Church that investiture has its greatest historical interest. The Church quite naturally shared in feudal land-holding; in addition to the tithes she possessed immense estates which had been given her by the faithful from early times, and for the defence of which she resorted to secular means. The bishops and abbots, by confiding their domains to laymen on condition of assistance with the sword in case of need, became temporal lords and suzerains with vassals to fight for them, with courts of justice, and in short with all the rights and privileges exercised by lay lords. On the other hand there were bishop-dukes, bishop-counts, &c., themselves vassals of other lords, and especially of the king, from whom they received the investiture of their temporalities. Many of the faithful founded abbeys and churches on condition that the right of patronage, that is the choice of beneficiaries, should be reserved to them and their heirs. Thus in various ways ecclesiastical benefices were gradually transformed into fiefs, and lay suzerains claimed the same rights over ecclesiastics as over other vassals from whom they received homage, and whom they invested with lands. This ecclesiastical investiture by lay princes dates at least from the time of Charlemagne. It did not seem fitting at first to confer ecclesiastical investiture by such military and worldly emblems as the sword and sceptre, nor to exact an oath of fealty. The emperor Henry I. invested bishops with a glove; Otto II. presented the pastoral staff; Conrad II., according to Wipo, went farther and required from the archbishop of Milan an oath of fealty. By the time of Henry III. investiture with ring and crozier had become the general practice: it probably had been customary in some places since Otto II.
Investiture of ecclesiastics by laymen had certain serious effects which were bound to bring on a conflict between the temporal and spiritual authorities. In the first place the lay authorities often rendered elections uncanonical by interfering in behalf of some favourite, thereby impairing the freedom of the electors. Again, benefices were kept vacant for long periods in order to ensure to the lord as long as possible the exercise of his regalian rights. And, nnally, control by temporal princes of investiture, and indirectly of election, greatly increased simony. Otto II. is charged with having practised simony in this connexion, and under Conrad II. the abuse grew prevalent. At a synod at Reims in 1049, the bishops of Nevers and Coutances affirmed that they had bought their bishoprics, and the bishop of Nantes stated that his father had been a bishop and that on his decease he himself had purchased the see. At a synod at Toulouse in 1056, Berengar of N arbonne accused the bishop of having purchased his see for 100,000 solidi, and of having plundered his church and sold relics and crucifixes to Spanish Jews in order to secure another 100,000 solidi with which to buy for his brother the bishopric of Urgel. Innumerable similar cases appear in acts of synods and in chronicles during the 11th century. Ecclesiastical investiture was further complicated by the considerable practice of concubinage. There was always the tendency for clerics in such cases to invest their sons with the temporalities of the Church; and the synod convened by Benedict VIII. at Pavia in 1018 (or 1022 according to some authorities) was mainly concerned with the issue of decrees against clerics who lived with wives or concubines and bestowed Church goods on their children. In time the Church came to perceive how closely lay investiture was bound up with simony. The sixth decree of the Lateran synod of 1059 forbade any cleric to accept Church office from a layman. In the following year this decree was reaffirmed by synods held at Vienne and Toulouse under the presidency of a legate of Nicholas II. The main investiture struggle with the empire did not take place, however, until Hildebrand became Pope Gregory VII. To Gregory it was intolerable that a layman, whether emperor, king or baron, should invest a churchman with the emblems of spiritual office; ecclesiastical investiture should come only from ecclesiastics. To the emperor Henry IV. it was highly undesirable that the advantages and revenues accruing from lay investiture should be surrendered; it was reasonable that ecclesiastics should receive investiture of temporalities from their temporal protectors and suzerains.
Although the full text of the decrees of the famous Lenten synod of 1075 has not been preserved, it is known that Gregory on that occasion denounced the marriage of the clergy, excommunicated five of Henry IV.'s councillors on the ground that they had gained church offices through simony, and forbade the emperor and all laymen to grant investiture of bishopric or inferior dignity. The pope immediately summoned Henry to appear at Rome in order to justify his private misconduct, and Henry replied by causing the partisan synod of Worms (1076) to pronounce Gregory's deposition. The pope excommunicated the emperor and stirred up civil war against him in Saxony with such success that he brought about Henry's bitter humiliation at Canossa in the following year. The papal prohibition of lay investiture was renewed at synods in 1078 and 1080, and although Gregory's death in exile (1085) prevented him from realizing his aim in the matter, his policy was steadfastly maintained by his successors. Victor III. condemned lay investiture at the synod of Benevento in 1087, and Urban II. at that of Melfi in 1089. At the celebrated council of Clermont (1095), at which the first crusade was preached, Urban strengthened the former prohibitions by declaring that no one might accept any spiritual office from a layman, or take an oath of fealty to any layman. Urban's immediate successor, Paschal II., stirred up the rebellion of the emperor's son, but soon found Henry V. even more persistent in the claim of