In a second letter, dated the 24th of October 1676, to Oldenburg, 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)0/2, (1 − xx)1/2, (1 − xx)2/2, (1 − xx)3/2, &c), I perceived that if the areas of the alternate curves, which are x, x − 1⁄3x3, x − 2⁄3x3 + 1⁄5x5, x − 3⁄3x3 + 3⁄5x5 − 1⁄7x7, &c., could be interpolated, we should obtain the areas of the intermediate ones, the first of which (1 − xx)1/2 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 0⁄3x³, 1⁄3x³, 2⁄3x³, &c., were in arithmetical progression; and consequently that the first two terms of all the series to be interpolated would be x − ½x³/3, x − 3⁄2x³/3, x − 5⁄2x³/3, &c.
“Now for the interpolation of the rest, I considered that the denominators 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 same with the figures of which the several powers of 11 consist, viz., of 11º, 11¹, 11², 11³, that is, the first 1; the second, 1, 1; the third, 1, 2, 1,; the fourth 1, 3, 3, 1; 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 continued multiplication of the terms of this series (m − 0)/1 × (m − 1)/2 × (m − 2)/3 ..., &c. ... This rule I therefore applied to the series to be interpolated. And since, in the series for the circle, the second term was (½x³)/3, I put m = ½.... And hence I found the required area of the circular segment to be x − (½x3)/3 − (1⁄8x5)/5 − (1⁄16x7)/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 (1 + xx)0/2, (1 + xx)1/2, (1 + xx)2/2, &c. ... Having proceeded so far, I considered that the terms (1 − xx)0/2, (1 − xx)2/2, (1 − xx)4/2, (1 − xx)6/2, &c., that is 1, 1 − x2, 1 − 2x2 + x4, 1 − 3x2 + 3x4 − x6, &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)1/2 or (1 − xx)3/2, or generally (1 − xx)m will be produced by the continued multiplication of this series m × (m − 1)/2 × (m − 2)/3 × (m − 3)/4 ... &c.”
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 (1713) 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]).
References.—For the history of the binomial theorem, see John Collins, Commercium Epistolicum (1712); S.P. Rigaud, The Correspondence of Scientific Men of the 17th Century (1841); M. Cantor, Geschichte 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.
BINYON, LAURENCE (1869- ), English poet, born at Lancaster on the 10th of August 1869, was educated at St Paul’s school, London, and Trinity College, Oxford, where he won the Newdigate prize in 1890 for his Persephone. He entered the department of printed books at the British Museum in 1893, and was transferred to the department of prints and drawings in 1895, the Catalogue of English Drawings in the British Museum (1898, &c.) being by him. As a poet he is represented by Lyric Poems (1894), Poems (Oxford, 1895), London Visions (2 vols., 1895-1898), The Praise of Life (1896), Porphyrion and other Poems (1898), Odes (1900), The Death of Adam (1903), Penthesilea (1903), Dream come true (1905), Paris and Oenone (1906), a one-act tragedy, and Attila, a poetical drama (1907); as an art critic by monographs on the 17th-century Dutch etchers, on John Crome and John Sell Cotman, contributed to the Portfolio, &c. In 1906 he published the first volume of a series of reproductions from William Blake, with a critical introduction.