BINOMIAL (from the

See also:

• LAT

Lat. bi-, bis, twice, and nomen, a name or

See also:

• TERM

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)"=xn+nax"—I+n 1.2 I.ax" -+n.n 1.2.3 -2asxn—a...+a" . 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)" = pn + naq+ Znnbq+m3nan cg . . . &c., where p+pq is the quantity whosentnpower or root is required, p the first term of that quantity, and q the quotient of the rest divided by p, me 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 = p'", b= aq, c = 2n nbq, and so on . In a second letter, dated the 24th of October 1676, to Olden-burg, Newton gave the train of reasoning by which he devised the theorem . " In the beginning of my mathematical studies, when I was perusing the works of the celebrated Dr Wallis, and considering the series by the interpolation of which he exhibits the area of the circle and hyperbola (for instance, in this series of curves whose common base or axis is x, and the ordinates respectively (1-xx), (1-xx)i, (1-xx)i, (1-xx)I, &c), I perceived that if the areas of the alternate curves, which are x,x-3x3,x-x3x3+ix5,x- &c., could be interpolated, we should obtain the areas of the intermediate ones, the first of which (I -xx)I is the area of the circle .

Now in order to [do] this, it appeared that in all the series the first term was x; that the second terms 1x3, ;x3, &c., were in arithmetical progression; and consequently that the first two terms of all the series ixs sxs to be interpolated would be x- 3 , x- 3 , x- $3xs , &c . " Now for the interpolation of the rest, I considered that the de-nominators 1, 3, 5, &c., were in arithmetical progression; and that therefore only the numerical coefficients of the numerators were to be investigated . But these in the alternate areas, which are given, were the sarfle with the figures of which the several

powers of II consist, viz., of I I°, I I1, 112, II', &c., that is, the first 1; the second, 1, 1; the third, 1, 2, 1, ; the fourth 1, 3, 3, I ; and so on . I enquired therefore how, in these series, the rest of the terms may be derived from the first two being given; and I found that by putting m for the second figure or term, the rest should be produced by the con- tinued multiplication of the terms of this seriesm I o x'--'L---1 xm3-2 ... , &c . . . . This rule I therefore applied to the series to be interpolated . And since, in the series for the circle, the second term was z3 , I put m = ... And hence I found the required area of the circular segment to be x-233-955-;7 , &c .... And in the same manner might be produced the interpolated areas of other curves; as also the area of the hyperbola and the other alternates in this series (I+xx)1, (1+xx)I, (1+xx)I, &c . . . . Having proceeded so far, I considered that the terms (1-xx), (I -xx)I, (s -xx)t, (1-xxA &c., that is 1, I -x2, I - 2X2+X4, I -3x2+3x4-xs, &c., might be interpolated in the same manner as the areas generated by them, and for this, nothing more was required than to omit the denominators 1, 3, 5, 7, &c.; in the terms expressing the areas; that is, the coefficients of the terms of the quantity to be interpolated (1—xx)l or (I —xx)3/2, or generally (1—xx)"' will be produced by the continued multiplication of this series mXm2 11 X3 22 Xm4 3 ...

&a" The binomial theorem was thus discovered as a development of

John Wallis's investigations in the method of interpolation . Newton gave no proof, and it was in the Ars Conjectandi (r 713) that James Bernoulli's proof for positive integral values of the exponent was first published, although Bernoulli must have discovered it many years previously . A rigorous demonstration was wanting for many years, Leonhard Euler's proof for negative and fractional values being faulty, and was finally given by Niels Heinrik Abel . The multi- (or poly-) nomial theorem has for its object the expansion of any power of a multinomial and was discussed in 1697 by Abraham Demoivre (see COMBINATORIAL ANALYSIS) . gondence of Scientific Men of the 17th Century (1841); M . Cantor, eschichte der Mathematik (1894-1901) . BINTURONG (Arctictis binturong), the single species of the viverrine genus Arctictis, ranging from Nepal through the Malay Peninsula to Sumatra and Java . This animal, also called the bear-cat, is allied to the palm-civets, or paradoxures, but differs from the rest of the family (Viverridae) by its tufted ears and long, bushy, prehensile tail, which is thick at the root and almost equals in length the head and body together (from 28 to 33 inches) . The fur is long and coarse, of a dull black hue with a grey wash on the head and fore-limbs . In habits the binturong is nocturnal and arboreal, inhabiting forests, and living on small vertebrates, worms, insects and fruits . It is said to be naturally fierce, but when taken young is easily tamed and becomes gentle and playful .