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 (1903). For examples of measurement of areas by geometrical construction, see G. C. Turner, Graphics applied to Arithmetic, Mensuration and Statics (1907). Discussions of the approximate calculation of definite integrals will be found in works on the infinitesimal calculus; see e.g. E. Goursat, A Course in Mathematical Analysis (1905; trans. by E. R. Hedrick). For the methods involving finite differences, see references under ; and. On calculation of moments of graphs, see W. P. Elderton, Frequency-Curves and Correlation (1906); as to the formulae of § 82, see also Biometrika, v. 450. For mechanical methods of calculating areas and moments see.

 MENTAWI, a chain of islands in the Dutch East Indies, off the west coast of Sumatra, between 1° and 3° 30′ S. There are twenty-one islands in all, of which the majority lie close to or between the four largest—Siberut, Sikaban or Sipora, North Pageh and South Pageh. The two last (also called Pagi or Poggy) are sometimes termed the Nassau Islands. The total land area is 1224 sq. m. The islands are included in the administration of Padang, Sumatra. They are apparently volcanic. Coral reefs lie off the coasts and render them difficult of access. The natives in language and customs present affinities with some Polynesians, and have been held to be a survival of the eastward immigration of people of Caucasian stock which took place before those which established the “pre-Malay” peoples (such as the Dyaks and Battas) in the Malay Archipelago. The islands produce some coco-nuts, sago, trepang and timber.  MENTEITH, or, a district of south Perthshire, Scotland, roughly comprising the territory between the Teith and the Forth. Formerly it was a stewartry and gave the title to an earldom. The title was first held by Gilchrist, a Celtic chief ennobled by Malcolm IV., and passed successively to Walter Comyn (d. 1258), to a branch of the Stewarts, and finally to the Grahams, becoming extinct in 1694. The lake of Menteith, situated of Loch Vennachar measures  long by 1 m. broad, and contains three islands. On Inchmahome (Gaelic, “the Isle of Rest”) are the ruins of an Augustinian priory founded in 1238 by Walter Comyn. It is Early English, with an ornate western doorway. The island was the residence of Queen Mary, when a child of five, for a few months before her departure to France in 1548. On Inch Talla stands the ruined tower of the earls of Menteith, dating from 1428. The village of Port of Monteith (pop. of parish, 1088), on the north shore of the lake, is north by west of the station of the same name on the North British Railway Company’s Forth & Clyde line.  MENTONE (Fr. Menton), a town in the department of the Alpes Maritimes in south-east France, situated on the shore of the Mediterranean, about 15 m. by rail E. of Nice. Pop. (1901), 9944. It is built in the form of an amphitheatre on a rocky promontory, which divides its semicircular bay into two portions. The main town is composed of two parts. Below, along the seashore, is the town of hotels and foreigners, while above, and inaccessible to wheeled vehicles, is that of the native Mentonese, with steep, narrow and dark streets, clinging to the mountain side around the strong castle which was once its protection against pirates. In the old town is the church of St Michel, rebuilt in great part since an earthquake in 1887, while below, in the principal street, the Corniche road, is the monument set up in 1896 to commemorate the union (in 1860) of Mentone with France. East of the main town is the suburb of Garavan, sheltered by cliffs, and filled with hotels. A mile and a half farther on is the Pont St Louis, which marks the frontier between France and Italy, while beyond it Sir Thomas Hanbury’s villa at La Murtola is soon reached, with its marvellous gardens of 250 acres. West of the main town more hotels and villas are scattered along the coast towards Cap Martin. This is a pine-covered promontory which shelters the Bay of Mentone on the west, and is crowned by a great hotel, not far from which is the villa of the ex-empress Eugénie. Facing south-east, and sheltered on the north and west by mountains, the Bay of Mentone has a delicious climate and is frequented by invalids. The mean for the year is 61° F., while that for the winter is 72° in the sun, and 55° in the shade. Frost occurs on the average only once in ten years. Besides the charms of its climate Mentone offers those of an almost tropical vegetation. Lemon-trees, olive-trees and pines rise in successive stages on surrounding slopes. The district produces 40,000,000 lemons yearly, and this is its principal natural wealth. In the east bay is the harbour, constructed in 1890. It has a depth of about 26 ft., and is sheltered by a jetty about 400 yds. in length. The harbour is frequented by pleasure yachts and a few coasting vessels.

 MENTOR, in Greek legend, the son of Alcimus and the faithful friend of Odysseus. During the absence of the latter, Mentor was entrusted with the care of his household and the guardianship of his son Telemachus. The word “mentor” is now used in the sense of a wise and trustworthy adviser, a meaning probably connected with the etymology of the name, from the root mon-, seen, in Lat. monere, to advise, monitor, adviser.

 MENTOR OF RHODES, brother of (q.v.), a Greek condottiere who appears first in the service of the rebellious satrap Artabazus of Phrygia in 363. When Artabazus had rebelled a second time and was in 353 forced to flee with Memnon into Macedonia, Mentor entered the service of the Egyptian king Nectanebus, and was sent by him with a body of Greek mercenaries to support the rebellious king Tennes (Tabnit) of Sidon against Artaxerxes III. But Tennes and Mentor betrayed the besieged town to the Persians (344 ). Tennes was killed after his treason, but Mentor gained the favour of the king. It was due largely to him that Egypt was conquered in 343 (Diod. xvi. 45 sqq.). He now closely allied himself with the eunuch (q.v.), the all-powerful vizier of Artaxerxes III. He was appointed general in Asia Minor, and with the help of Artabazus and Memnon, whose pardon and recall he obtained from the king, subdued the rebels and local dynasts. The most famous among them was Hermias of Atarneus, the protector of Aristotle, who had become master of some towns of Aeolis and Troas. By treachery he made him prisoner and occupied his towns (342 ); Hermias was executed by order of the king (Diod. xvi. 52; Polyaen. vi. 48; pseudo-Arist. Oecon. ii. 27; Strabo xiii. 610; Didymus’ commentary on Demosthenes Phil. 4, p. 6; cf. Diog. Laërt. vi. 9). Shortly afterwards Mentor died, and was succeeded by his brother Memnon. His son Thymondas commanded in the naval war against Alexander and at Issus (Arrian ii. 2, 1; 13, 2).

 MENZEL, ADOLPH FRIEDRICH ERDMANN VON (1815–1905), German artist, was born at Breslau on the 8th of December 1815. His father was at the head of a school for girls, and intended to educate his son as a professor; but he would not thwart his taste for art. Left an orphan in 1832, Menzel had to maintain his family. In 1833 Sachse of Berlin published his first work, an album of pen-and-ink drawings reproduced on stone, to illustrate Goethe’s little poem, “Künstlers Erdenwallen.” He executed lithographs in the same manner to illustrate Denkwürdigkeiten aus der brandenburgisch-preussischen Geschichte, pp. 834–836; “The Five Senses” and “The Prayer,” as well as diplomas for various corporations and societies. From 1839 to 1842 he produced 400 drawings, reviving at the same time the technique of engraving on wood, to illustrate the Geschichte Friedrichs des Grossen (“History of Frederick the Great”) by Franz Kugler. He subsequently brought out Friedrichs des Grossen Armee in ihrer Uniformirung (“The Uniforms of the Army under Frederick the Great”), Soldaten Friedrichs des Grossen (“The Soldiers of Frederick the Great”); and finally, by