helped to find a sum of money that had been lost. Jerome (the authority for the date of his death) calls him Pythagoricus et magus. The abstruse nature of his studies, the mystical character of his writings, and the general indifference of the Romans to such subjects, caused his works to be soon forgotten. Amongst his scientific, theological and grammatical works mention may be made of De diis, containing an examination of various cults and ceremonials; treatises on divination and the interpretation of dreams; on the sphere, the winds and animals. His Commentarii grammatici in at least 29 books was an ill-arranged collection of linguistic, grammatical and antiquarian notes. In these he expressed the opinion that the meaning of words was natural, not fixed by man. He paid especial attention to orthography, and sought to differentiate the meanings of cases of like ending by distinctive marks (the apex to indicate a long vowel is attributed to him). In etymology he endeavoured to find a Roman explanation of words where possible (according to him frater was = fere alter). Quintilian (Instit. orat. xi, 3. 143) speaks of a rhetorical treatise De gestu by him.
See Cicero, Ad Fam. iv. 13; scholiast on Lucan i. 639; several references in Aulus Gellius; Teuffel, Hist. of Roman Literature, 170; M. Hertz, De N. F. studiis atque operibus (1845); Quaestiones Nigidianae (1890), and edition of the fragments (1889) by A. Swoboda.
FIGURATE NUMBERS, in mathematics. If we take the sum
of n terms of the series 1 + 1 + 1 + . . ., i.e. n, as the nth term of
a new series, we obtain the series 1 + 2 + 3 + . . ., the sum
of n terms of which is 12n · n + 1. Taking this sum as the nth
term, we obtain the series 1 + 3 + 6 + 10 + . . ., which has
for the sum of n terms n (n + 1) (n + 2) / 3![1] This sum is taken as
the nth term of the next series, and proceeding in this way we
obtain series having the following nth terms:—
 |
The numbers obtained by giving n any value in these expressions are of the first, second, third, . . . or r th order of figurate numbers.
Pascal treated these numbers in his Traité du triangle arithmetique (1665), using them to develop a theory of combinations and to solve problems in probability. His table is here shown in its simplest form. It is to be noticed that each number is the sum of the numbers immediately above and to the left of it; and that the numbers along a line, termed a base, which cuts off an equal number of units along the top row and column are the coefficients in the binomial expansion of (1 + x)r−1, where r represents the number of units cut off.
FIJI (Viti), a British colony consisting of an archipelago in the Pacific Ocean, the most important in Polynesia, between 15° and 20° S., and on and about the meridian of 180°. The islands number about 250, of which some 80 are inhabited. The total land area is 7435 sq. m. (thus roughly equalling that of Wales), and the population is about 121,000. The principal island is Viti Levu, 98 m. in length (E. to W.) and 67 in extreme breadth, with an area of 4112 sq. m. Forty miles N.E. lies Vanua Levu, measuring 117 m. by 30, with an area of 2432 sq. m. Close off the south-eastern shore of Vanua Levu is Taviuni, 26 m. in length by 10 in breadth; Kandavu or Kadavu, 36 m. long and very narrow, is 41 m. S. of Viti Levu, and the three other main islands, lying east of Viti Levu in the Koro Sea, are Koro, Ngau or Gau, and Ovalau. South-east from Vanua Levu a loop of islets extends nearly to 20° S., enclosing the Koro Sea. North-west of Viti Levu lies another chain, the Yasawa or western group; and, finally, the colony includes the island of Rotumah (q.v.), 300 m. N.W. by N. of Vanua Levu.
The formation of the larger islands is volcanic, their surface rugged, their vegetation luxuriant, and their appearance very beautiful; their hills rise often above 3000, and, in the case of a few summits, above 4000 ft., and they contrast strongly with the low coral formation of the smaller members of the group. There is not much level country, except in the coral islets, and certain rich tracts along the coasts of the two large islands, especially near the mouths of the rivers. The large islands have a considerable extent of undulating country, dry and open on their lee sides. Streams and rivers are abundant, the latter very large in proportion to the size of the islands, affording a waterway to the rich districts along their banks. These and the extensive mud flats and deltas at their mouths are often flooded, by which their fertility is increased, though at a heavy cost to the cultivator. The Rewa, debouching through a wide delta at the south-east of Viti Levu, is navigable for small vessels for 40 m. There are also in this island the Navua and Sigatoka (flowing S.), the Nandi (W.), and the Ba (N.W.). The Dreketi, flowing W., is the chief stream of Vanua Levu. It breaches the mountains in a fine valley; for this island consists practically of one long range, whereas the main valleys and ranges separating them in Viti Levu radiate for the most part from a common centre. With few exceptions the islands are surrounded by barriers of coral, broken by openings opposite the mouths of streams. Viti Levu is the most important island not only from its size, but from its fertility, variety of surface, and population, which is over one-third of that of the whole group. The town of Suva lies on an excellent harbour at the south-east of the island, and has been the capital of the colony since 1882, containing the government buildings and other offices. Vanua Levu is less fertile than Viti Levu; it has good anchorages along its entire southern coast. Of the other islands, Taviuni, remarkable for a lake (presumably a crater-lake) at the top of its lofty central ridge, is fertile, but exceptionally devoid of harbours; whereas the well-timbered island of Kandavu has an excellent one. On the eastern shore of Ovalau, an island which contains in a small area a remarkable series of gorge-like valleys between commanding hills, is the town of Levuka, the capital until 1882. It stands partly upon the narrow shore, and partly climbs the rocky slope behind. The chief islands on the west of the chain enclosing the Koro Sea are Koro, Ngau, Moala and Totoya, all productive, affording good anchorage, elevated and picturesque. The eastern islands of the chain are smaller and more numerous, Vanua Batevu (one of the Exploring Group) being a centre of trade. Among others, Mago is remarkable for a subterranean outlet of the waters of the fertile valley in its midst.
The land is of recent geological formation, the principal ranges being composed of igneous rock, and showing traces of much volcanic disturbance. There are boiling springs in Vanua
- ↑ The notation n! denotes the product 1 · 2 · 3 · . . . n, and is termed “factorial n.”
