FIGURATE NUMBERS

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 1/2 n(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:--1, n, n(n + 1)/2!, n(n + 1) (n + 2)/3!, ... n(n + 1) ... (n + r - 2)/(r - 1)!. the numbers obtained by giving n any value in these expressions are of the first, second, third, ... or rth order of figurate numbers. 1 /| 1 /| 1 /| 1 /| 1 /| 1 /| 1 /| 1 / | / | / | / | / | / | / | 1 /| 2 /| 3 /| 4 /| 5 /| 6 /| 7 | / | / | / | / | / | / | 1 /| 3 /| 6 /| 10/| 15/| 21 | / | / | / | / | / | 1 /| 4 /| 10/| 20/| 35 | / | / | / | / | 1 /| 5 /| 15/| 35 | / | / | / | 1 /| 6 /| 21 | / | / | 1 /| 7 | / | 1 | pascal treated these numbers in his _traite 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. footnote: [1] the notation n! denotes the product 1 . 2 . 3.... n, and is termed "factorial n." fiji (_viti_), a british colony consisting of an archipelago in the pacific ocean, the most important in polynesia, between 15 deg. and 20 deg. s., and on and about the meridian of 180 deg. 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 deg.