[2] The first part appeared in 1838.
BINOMIAL (from the Lat. bi-, bis, twice, and nomen, a name or term), in mathematics, a word first introduced by Robert Recorde (1557) to denote a quantity composed of the sum or difference to two terms; as a + b, a − b. The terms trinomial, quadrinomial, multinomial, &c., are applied to expressions composed similarly of three, four or many quantities.
The binomial theorem is a celebrated theorem, originally due to Sir Isaac Newton, by which any power of a binomial can be expressed as a series. In its modern form the theorem, which is true for all values of n, is written as
| (x + a)n = xn + naxn−1 + | n·(n − 1) | a2xn−2 | n·(n − 1)·(n − 2) | a3xn−3 ... + an. |
| 1·2 | 1·2·3 |
The reader is referred to the article [Algebra] for the proof and applications of this theorem; here we shall only treat of the history of its discovery.
The original form of the theorem was first given in a letter, dated the 13th of June 1676, from Sir Isaac Newton to Henry Oldenburg for communication to Wilhelm G. Leibnitz, although Newton had discovered it some years previously. Newton there states that
| (p + pq)m/n = pm/n + | m | aq + | m − n | bq + | m − 2n | cq ... &c., |
| n | 2n | 3n |
where p + pq is the quantity whose (m/n)th power or root is required, p the first term of that quantity, and q the quotient of the rest divided by p, m/n the power, which may be a positive or negative integer or a fraction, and a, b, c, &c., the several terms in order, e.g.
| a = pm/n, b = | m | aq, c = | m − n | bq, and so on. |
| n | 2n |